elliptic functions and modular forms - lmu münchenwehler/ellipticfunctionsscript.pdf · release...

Elliptic Functionsand Modular Forms

DRAFT, Release 0.8

March 22, 2018

2

I have prepared these notes for the students of my lecture. The lecture took placeduring the winter term 2017/18 at the mathematical department of LMU (Ludwig-Maximilians-Universitat) at Munich.

Compared to the oral lecture in class these written notes contain some additionalmaterial.

Please report any errors or typos to [email protected]

Release notes:

• Release 0.8: Chapter 5 revised, Chapter 6 added.

• Release 0.7: Introduction and preliminary version of Chapter 5 added.

• Release 0.6: Chapter 4 added. Update list of references.

• Release 0.5: Chapter 3 added. Update list of references. Minor revisions passim.

• Release 0.4: Chapter 2 added.

• Release 0.3: Update list of references.

• Release 0.2: Completion of Chapter 1. Update list of references.

• Release 0.1: Document created, Chapter 1.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 The elliptic functions of a lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Doubly periodic meromorphic functions . . . . . . . . . . . . . . . . . . . . . . . 31.2 The ℘-function of Weierstrass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Abel’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Complex tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Compact Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Isomorphism classes of complex tori and the modular group . . . . . . . 322.3 The modular curve of the modular group . . . . . . . . . . . . . . . . . . . . . . . 37

3 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.1 Plane cubics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2 The group structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.3 Reduction mod p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4 Modular forms of the modular group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.1 Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2 The algebra of modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.3 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5 Application to Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.1 The τ-function of Ramanujan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.2 Complex multiplication and imaginary quadratic number fields . . . . 1295.3 Applications of the modular polynomial . . . . . . . . . . . . . . . . . . . . . . . . 139

6 Outlook to Congruence Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.1 Modular forms of congruence subgroups . . . . . . . . . . . . . . . . . . . . . . . 1536.2 The four squares theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1586.3 Modularity of elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

v

vi Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Introduction

Modular Forms and the theory of Elliptic Functions are a classical domain frommathematics. At least, since the proof of Fermat’s last conjecture the domain attractswidespread attention. The domain of Modular Forms integrates the three mathe-matical disciplines Complex Analysis, Algebraic Geometry, and Algebraic NumberTheory.

1. Modular Forms

• Modular group• Proper group operation• Fundamental domain• Congruence subgroup• Modular curve• Modular form• Fourier explansion• Eisenstein series and ∆ and j• Mk(Γ ) = H0(P1,ΩΓ ,k)• Weight formula• Dimension formula• Hecke operator• Uniformization of elliptic curves over C• Congruence subgroups• Modularity of elliptic curves over Q

2. Complex Analysis

• Laurent series• Residue theorem• Elliptic functions• Weierstrass ℘-function• Abel’s theorem• Lemma of Schwarz

1

2 Introduction

• Compact Riemann surface• Complex torus• Line bundle• Divisor• Riemann-Roch Theorem• Serre duality• Very ample line bundle

3. Algebraic Geometry

• Elliptic curve• Embedding elliptic curves into P2

• Weierstrass equation• Discriminant• Group structure• Class group• Reduction mod prime• ζ -function• L-series

4. Algebraic Number Theory

• Fractional ideal• Ideal class group• Imaginary quadratic number field• Ramanujan τ-function• Class number of quadratic imaginary number fields• Four squares theorem• Legendre symbol• Law of quadratic reciprocity

Chapter 1The elliptic functions of a lattice

The present chapter deals with complex analysis, i.e. we study holomorphic andmeromorphic functions on domains in the complex plane.

1.1 Doubly periodic meromorphic functions

A meromorphic function on a domain G⊂ C is a holomorphic function

f : G\S→ C

with a discrete subset S⊂ G of poles of f - but no essential singularities.

Definition 1.1 (Period of a meromorphic function). Consider a meromorphicfunction f on a domain G. A number ω ∈ C is a period of f if

• for all z ∈ G also z±ω ∈ G

• and for all z ∈ G\Pf (z+ω) = f (z).

It looks like Definition 1.1 excludes the poles from the definition of periodicity.A pole z0 of f is an isolated singularity of f . Therefore in a neighbourhood of z0 thefunction f expands into a convergent Laurent series around z0

f (z) =∞

∑n≥n0

an · (z− z0)n

with coefficients

an =1

2πi

∫|z−z0|=ε

f (z)(z− z0)n+1 dz with suitable ε > 0.

3

4 1 The elliptic functions of a lattice

If f has the period ω then the Laurent expansions of f around z0 and z0 +ω havethe same coefficients because the integrands with z and z+ω give the same value.Hence periodicity of f includes also the poles.

Example 1.2 (Periodic meromorphic functions).

1. The functions sin,cos : C→ C are holomorphic with period ω = 2π .

2. The tangent-function is a meromorphic function on C with the singularities π/2+ kπ, k ∈ Zand period ω = π .

3. The function C→ C∗,z 7→ e2πiz, is holomorphic with period ω = 1.

If a meromorphic function f has a period ω 6= 0 one can ask for the set of allperiods. Apparently elements of the form kω, k ∈ Z are also periods of f . Howmany independent periods of f exist?

Definition 1.3 (Discrete subgroup). A subset Γ ⊂ C is a discrete subgroup, if itsatisfies the following conditions

1. Γ is a subgroup of the additive group (C,+).

2. Γ is discrete, i.e. a discrete topological space when equipped with the subspacetopology of C: A neighbourhood U of 0 in C exists such that

U ∩Γ = 0.

Proposition 1.4 (Period group). Consider a non-constant meromorphic function fon a domain G⊂ C. The set Γf of all periods of f is a discrete subgroup of C. It isnamed the period group of f .

Proof. i) Apparently Γf is a subgroup.

ii) In case Γf were not discrete, a sequence (ωn)n∈N of periods ωn ∈ Γf \ 0would exist with

limn→∞

ωn = 0.

Then for any arbitrary but fixed z0 ∈ G\P and for all n ∈ N

f (z0) = f (z0 +ωn) = f (ωn).

The identity theorem implies that f is constant, a contradiction, q.e.d.

1.1 Doubly periodic meromorphic functions 5

Proposition 1.5 (Discrete subgroups of (C,+)). Each discrete subgroup Γ ⊂ (C,+)belongs to one of the following types:

1. Rank = 0:Γ = 0

2. Rank = 1:Γ = Zω = m ·ω : m ∈ Z

with a suitable element ω ∈ C∗.

3. Rank = 2:

Γ = Zω1 +Zω2 = m1 ·ω1 +m2 ·ω2 : m1,m2 ∈ Z

with ω1,ω2 ∈ C linearly independent over the base field R. The subgroup Γ isnamed a lattice.

For a proof see [1, Chap. 7, Sect. 2.1].

Note. If we specify a lattice in the form Λ = Zω1 +Zω2 we will assume that thebasis (ω1,ω2) is positively oriented, i.e. det (ω1 ω2)> 0.

Definition 1.6 (Elliptic function). Consider a non-constant meromorphic function fon a domain G⊂ C with period group Γf .

• If rank Γf ≥ 1 then f is named periodic.

• If rank Γf = 2 then f is named doubly periodic. Its period group

Γf = Zω1 +Zω2

is also named the period lattice of f . The subset

P = λ1ω1 +λ2ω2 : 0≤ λ1,λ2 < 1

is the fundamental period parallelogram of f . A doubly periodic meromorphicfunction defined on C is an elliptic function with respect to a lattice Λ if its periodgroup Γf contains Λ , i.e. all lattice points ω ∈Λ are periods of f .

Proposition 1.7 (Constant elliptic functions). Each holomorphic elliptic function

f : C→ C

is constant.


Proof. An entire function f is holomorphic and attends the maximum of its moduluson the closure of its period parallelogram which is compact set. Being a boundedentire function, f is constant according to Liouville’s theorem, q.e.d.

Proposition 1.8 (Residue theorem for elliptic functions). An elliptic function fwith period parallelogram P satisfies

∑ζ∈P

resζ ( f ) = 0.

Proof. The boundary ∂P comprises only finitely many poles of f . Therefore we canchoose a number a ∈ C such the boundary of the translated period parallelogram Pa := a+Pcontains no poles of f . The residue theorem implies

12πi

∫∂Pa

f (z) dz = ∑ζ∈Pa

resζ ( f ) = ∑ζ∈P

resζ ( f ).

The integrand on the left-hand side is doubly periodic, therefore the integral van-ishes, q.e.d.

Corollary 1.9 (Poles of elliptic functions). There are no elliptic functions f withone pole of order = 1 modulo the period lattice Γf , and no other pole modulo Γf .

Proof. A pole of order 1 at a∈C has resa( f ) 6= 0, which contradicts Proposition 1.8,q.e.d.

Corollary 1.10 (Counting poles and zeros of elliptic functions). Any non-constantelliptic function f assumes modulo its period lattice Γf each value a∈C∪∞ withthe same multiplicity. In particular, f has modulo Γf the same number of poles andzeros.

Proof. The case a 6=∞ reduces to the second claim by considering the function f −a.It has the same poles as f . Therefore it suffices to show that f has the same numberof zeros and poles, taken with multiplicity. We apply Proposition 1.8 to the mero-morphic function f ′/ f and obtain

∑ζ∈P

resζ

(f ′

f

)= 0.

Each residue evaluates to

resζ

(f ′

f

)=

k f has a zero of order k at ζ

−k f has a pole of order k at ζ

q.e.d.

1.2 The ℘-function of Weierstrass 7

Corollary 1.11 (Zeros of an elliptic function). There are no elliptic functions fwith only one zero of order = 1 modulo the period lattice Γf and no other zeromodulo Γf .

Proof. The proof follows from Corollary 1.9 and 1.10, q.e.d.

1.2 The ℘-function of Weierstrass

In the present section we consider an arbitrary but fixed lattice

Λ := Z ω1 +Z ω2.

We use the notation Λ ′ := Λ \ 0. Moreover P denotes the fundamental periodparallelogram of Λ .

Lemma 1.12 (Lattice constants). The infinite series

GΛ ,k := ∑ω∈Λ ′

1ωk, with k ∈ N, k > 2,

is absolutely convergent. Its values for k > 2 are named the lattice constants of Λ .

Proof. We choose the exhaustion of Λ by the sequence (Λn)n∈N of subsets

Λn := µ ·ω1 +ν ·ω2 ∈Λ : µ,ν ∈ Z; |µ|, |ν | ≤ n and (|µ|= n or |ν |= n),

i.e.Λ =

⋃n∈N

Λn.

Then card Λn = 8n. A suitable constant c exists with |ω| ≥ c · n for all n ∈ N andfor all ω ∈Λn. Therefore

∑ω∈Λn

1|ω|k≤

8 ·nck ·nk ≤

(8ck

)·

1nk−1.

For k > 2∞

∑n=1

1nk−1 < ∞.

Therefore the claim follows from

∑ω∈Λ ′

1ωk =

∞

∑n=1

(∑

ω∈Λn

1ωk

), q.e.d.


We will study elliptic functions with period lattice Λ . According to Corollary 1.11a candidate f for the most simple example would have only one pole mod Λ , thepole having order = 2 and residue = 0. The Laurent expansion of f around z0 = 0would start

f (z) =1z2 +a1 · z+O(2).

Being doubly periodic, f would have poles exactly at the points ω ∈Λ . The follow-ing Theorem 1.13 shows that such a function actually exists.

Theorem 1.13 (Weierstrass ℘-function). The series

℘(z) :=1z2 + ∑

ω∈Λ ′

(1

(z−ω)2−1

ω2

)

is absolutely and compactly convergent for all z ∈ C\Λ . It defines an even ellipticfunction ℘ with respect to Λ , named the Weierstrass ℘-function of Λ . The pole setof ℘ is Λ , each pole has order = 2.

If we consider only the first summand with fixed z then

1(z−ω)2 ∼

1ω2.

Therefore the series with these summands alone does not converge. We will showthat the difference

1(z−ω)2−

1ω2

is proportional to1

ω3.

Therefore the series ℘(z) converges.

Note: TeX reserves the separate symbol “backslash wp” to denote the Weier-strass function ℘.

Proof. i) Meromorphic with pole set Λ : For each compact set K ⊂ C only finitelymany summands of ℘have a pole in K. We show that the remaining series convergesabsolutely and uniform on K: Choose an arbitrary but fixed R > 0. For

|z| ≤ R and 2R≤ |ω|

holds|ω|2−|z| ≥ 0.


The triangle inequality

|z−ω|= |ω− z| ≥ ||ω|− |z||=

∣∣∣∣∣ |ω|2−|z|+

|ω|2

∣∣∣∣∣≥ |ω|2

implies∣∣∣∣∣ 1(z−ω)2−

1ω2

∣∣∣∣∣=∣∣∣∣∣ω2− (z−ω)2

ω2 · (z−ω)2)

∣∣∣∣∣≤ |− z2 +2zω|

|ω|2 ·

(|ω|2

)2 ≤ 4 ·|z|2 +2 · |z| · |ω|

|ω|4≤

≤ 4 ·|z|2

|ω|4+8 ·

|z||ω|3≤

4 ·R2

2 ·R · |ω|3+8 ·

R|ω|3

= 10 ·R|ω|3

.

According to Lemma 1.12 the series

∑ω∈Λ ′

1ω3

converges absolutely. For |z| ≤ R we decompose the series with respect to the sum-mation over ω ∈Λ[

1z2 + ∑

ω∈Λ ′,|ω|<R

(1

(z−ω)2−1

ω2

)]+

[∑

ω∈Λ ,|ω|≥2R

(1

(z−ω)2−1

ω2

)]

The first summand is meromorphic with poles exactly at the points z ∈ Λ ∩∆(R).The second summand converges absolutely and compactly on ∆(R) \Λ due toLemma 1.12, and defines a holomorphic function on ∆(R).

Because R can be choosen arbitrary, the series ℘ is compact convergent on C\Λ

and defines a meromorphic function on C with pole set Λ .

ii) Even function with periods from Λ :

• ℘ is even: After rearranging the summation, which is admissible due to the ab-solute convergence,

℘(−z)=1z2+ ∑

ω∈Λ ′

(1

(−z−ω)2−1

ω2

)=

1z2+ ∑

ω∈Λ ′

(1

(z− (−ω))2−1

(−ω)2

)=℘(z).

• ℘ has periods from Λ : The derivative is

℘′(z) =

−2z3 + ∑

ω∈Λ ′

−2(z−ω)3 =−2 ∑

ω∈Λ

1(z−ω)3


Apparently for all ω ∈Λ

℘′(z+ω) =℘

′(z)

after rearranging the summation. As a consequence

℘(z+ω j) =℘(z)+ c j

with two suitable constants c j ∈ C, j = 1,2.

For j = 1,2 and the specific arguments z :=−ω j/2:

On one hand℘((−ω j/2)+ω j) =℘(ω j/2)+ c j,

on the other hand due to periodicity

℘(−(ω j/2)+ω j) =℘(ω j/2),

which impliesc j = 0, q.e.d.

Lemma 1.14 (Half-lattice points). Denote by

Z :=

ω1

2,

ω2

2,

ω1 +ω2

2

the set of half-lattice points from the fundamental period parallelogram PΛ .

1. Consider an even elliptic function f with period lattice Λ . Any u ∈ Z has evenorder

ordu( f ).

2. Modulo Λ , the odd elliptic function ℘ ′ has a pole of order = 3 at 0 ∈ C and noother poles, and exactly three zeros mod Λ , namely the points u ∈ Z.

Proof. 1. Set u := ω/2 ∈ Z,ω ∈ Λ . We compute the Laurent expansion of faround u: On one hand, because ω is a period

f (u+ z) = f (u+ z−ω) = f (−u+ z).

On the other hand, because f is even

f (−u+ z) = f (u− z).

As a consequencef (u+ z) = f (u− z).

Therefore the Laurent expansion around u


f (w) =∞

∑n=n0

cn(w−u)n

satisfies cn = 0 for odd n. In particular ordu( f ) is even.

2. Set u := ω/2∈ Z,ω ∈Λ . According to Theorem 1.13 the function℘has no poleat u. The derivative ℘ ′ has the same pole set Λ as ℘. Because ℘ is even, thefunction ℘ ′ is an odd elliptic function

℘′(u) =−℘

′(−u) =−℘′(−u+2u) =−℘

′(u).

Therefore℘′(u) = 0.

Because ℘ ′ has a single pole at 0 ∈ C of order = 3 mod Λ and no other polemod Λ . Corollary 1.10 implies that Z is the zero set of ℘ ′ mod Λ , and each zerohas order = 1, q.e.d.

Note that the determination of the two zeros modulo Λ of ℘ is much more deli-cate [9].

Corollary 1.15 (Period lattice of ℘ and ℘ ′). The Weierstrass function ℘ of thelattice Λ and its derivative ℘ ′ have the period lattice Λ , i.e.

Λ = Λ℘ = Λ℘ ′ ,

the only periods of ℘ and of ℘ ′ are the points from Λ .

Proof. We knowΛ ⊂Λ℘⊂Λ℘ ′

Here the first inclusion has been shown in Theorem 1.13, while the second inclusionis obvious. We show

Λ℘ ′ ⊂Λ :

Let ω ∈Λ℘ ′ be a period of ℘ ′. Hence

℘′(ω/2) =℘

′((ω/2)−ω) =℘′(−ω/2) =−℘

′(ω/2)

which implies ℘ ′(ω/2) = 0. Lemma 1.14 implies ω/2 ∈Λ/2 or ω ∈Λ , q.e.d.

See the PARI-file Weierstrass_p_function_08 as motivation for the fol-lowing Proposition 1.16.


Proposition 1.16 (Laurent expansion of ℘). The Weierstrass function ℘ of Λ hasthe Laurent expansion around zero

℘(z) =1z2 +

∞

∑k=1

a2k · z2k

with coefficientsa2k = (2k+1) ·GΛ ,2k+2.

Proof. We determine the Taylor series of

1(z−ω)2−

1ω2

around 0 ∈ C: Taking the derivative

1(z−ω)2 =

ddz

(−1

z−ω

)

and expanding into a geometric series

−1z−ω

=1

ω− z=

1/ω

1− (z/ω)= (1/ω) ·

∞

∑ν=0

(1/ω)ν · zν

shows

1(z−ω)2 = (1/ω) ·

∞

∑ν=1

ν · (1/ω)ν · zν−1 =∞

∑ν=0

(ν +1) ·1

ων+2 · zν

and1

(z−ω)2−1

ω2 =∞

∑ν=1

(ν +1)1

ων+2 · zν .

Because℘ is even and the series converges absolutely, see Theorem 1.13, we obtainafter rearrangement

℘(z)=1z2+

∞

∑ν=1

(ν+1)

(∑

ω∈Λ ′

1ων+2

)zν =

1z2+

∞

∑k=1

(2k+1)

(∑

ω∈Λ ′

1ω2k+2

)z2k, q.e.d.

Theorem 1.17 (Differential equation of ℘). The Weierstrass function ℘ of the lat-tice Λ satisfies the differential equation

℘′2 = 4 ·℘3−g2 ·℘−g3

with the constants


g2 := 60 ·GΛ ,4, g3 := 140 ·GΛ ,6

derived from the lattice constants of Λ .

Proof. To simplify the notation we omit the index Λ from the lattice constants GΛ ,k.Due to Proposition 1.16 the Laurent expansions start

℘(z) =1z2 + ∑

k≥1(2k+1) ·G2k+2 · z2k =

1z2 +3 ·G4 · z2 +5 ·G6 · z4 +O(6)

℘′(z) =−

2z3 +6 ·G4 · z+20 ·G6 · z3 +O(5)

℘′(z)2 =

4z6−

24 ·G4

z2 −80 ·G6 +O(2)

℘(z)2 =1z4 +6 ·G4 +10 ·G6 · z2 +O(4)

℘(z)3 =1z6 +

6 ·G4

z2 +10 ·G6 +3 ·G4

z2 +5 ·G6 +O(2) =1z6 +

9 ·G4

z2 +15 ·G6 +O(2)

Therefore℘′2−4 ·℘3 +g2 ·℘+g3 = O(2).

The left-hand side is an elliptic function. The right-hand side shows that the functionis even holomorphic and vanishes at 0 ∈ C. According to Proposition 1.7 the left-hand side is constant, i.e. zero, q.e.d.

Theorem 1.18 (Field of elliptic functions). The field M (Λ) of elliptic functionswith respect to the lattice Λ is a field extension of C of transcendence degree = 1,more precisely

M (Λ) = C(℘)[℘ ′]

with ℘ ′ algebraic over the field C(℘) with quadratic minimal polynomial

f (T ) = T 2−4 ·℘3 +g2 ·℘+g3 ∈ C(℘)[T ]

with the lattice constants

g2 := 60 ·GΛ ,4, g3 := 140 ·GΛ ,6.

Note the equality C(℘)[℘ ′] = C(℘)(℘ ′) because℘ ′ is algebraic over C(℘).

Proof. i) Due to Theorem 1.17 we have

f (℘ ′) = 0 ∈ C(℘).


The derivative ℘ ′ is odd while ℘ is even. Therefore ℘ ′ /∈ C(℘), and the minimalpolynomial must have degree ≥ 2.

ii) Consider an elliptic function f ∈M (Λ), f 6= 0. We show f ∈ C(℘, ℘ ′).

• We first consider the case that f has pole set Λ .

The unique pole of f mod Λ has order k ≥ 2 due to Corollary 1.9. Induction onthe pole order k. If k = 2 then for suitable c ∈ C the function

f − c ·℘

has at most a single pole mod Λ of order < 2. Therefore the function is holomor-phic, and due to Corollar 1.9 even constant.

For the induction step assume k > 2. On one hand, for even k = 2m a suitableconstant c ∈C exists such that f −c ·℘ m has only poles of order < k. Therefore

f − c ·℘ m ∈ C(℘,℘ ′)

by induction assumption. On the other hand, for odd k = 2m+1 a suitable con-stant c ∈ C exists such that

f − c ·℘ m−1 ·℘ ′

has only poles of order < k. By induction assumption

f − c ·℘ m−1℘′ ∈ C(℘,℘ ′).

In both casesf ∈ C(℘,℘ ′).

• Eventually, we consider the case of an arbitrary pole set of f .

Then f has only finitely many poles zν ∈P\Λ ,ν = 1, ...,n. For suitable constants k1, ...,kn ∈ Nthe function

f (z) ·n

∏ν=1

(℘(z)−℘(zν))kν

has poles at most in Λ . According to the first part

f (z) ·n

∏ν=1

(℘(z)−℘(zν))kν ∈ C(℘,℘ ′).

Thereforef ∈ C(℘,℘ ′), q.e.d.

1.3 Abel’s Theorem 15

1.3 Abel’s Theorem

We continue with fixing an arbitrary lattice

Λ := Z ω1 +Z ω2,

keeping the notations Λ ′ :=Λ \0 and P for the fundamental period parallelogramof Λ .

Proposition 1.19 (Zero and pole divisor). Consider an elliptic function and denoteby A = (ai)i=1,...,n its family of zeros in P and by B = (bi)i=1,...,n its family of polesin P. Then n≥ 2 and

∑i=1,...,n

ai− ∑i=1,...,n

bi ∈Λ .

Note that each point p of A and B appears as often as its multiplicity as zero or poleof f indicates. By Corollary 1.9 holds n≥ 2, and by Corollary 1.10 both families Aand B have the same cardinality.

Proof. We consider a translation Pa := P+a of the fundamental parallelogram suchthat f has neither zeros nor poles on the boundary ∂Pa. With a counter-clockwiseorientation of ∂Pa the residue theorem gives

12πi

∫∂Pa

z ·f ′(z)f (z)

dz = ∑p∈Pa

resp

(z ·

f ′(z)f (z)

).

• Right-hand side: Assume ordp( f ) = k ∈ Z∗. Thenp is a zero of f of order k if k > 0p is a pole of f of order k if k < 0

In a neighbourhood of pf ′(z)f (z)

=k

z− p+ f1(z)

with f1 holomorph. As a consequence

z ·f ′(z)f (z)

= (p+(z− p)) ·f ′(z)f (z)

=p · k

z− p+ f2(z)

with f2 holomorph. Therefore

resp

(z ·

f ′(z)f (z)

)= p · k

and


∑p∈Pa

resp(z ·f ′(z)f (z)

) =n

∑i=1

ai−n

∑i=1

bi.

• Left-hand side:∫−→AB+−→CD

z ·f ′(z)f (z)

=∫−→AB

(z− (z+ω2)) ·f ′(z)f (z)

=−ω2

∫−→AB

f ′(z)f (z)

=−ω2 [log f (z)]BA

Here log denotes a branch of the logarithm in a simply-connected neighbourhoodof the line AB. ∫

−→AB+−→CD

z ·f ′(z)f (z)

dz = ω2 · (log f (A)− log(B)).

Because f (A) = f (B) an integer m2 ∈ Z exists with

log f (A)− log f (B) = 2πi ·m2.

A similar argument for the lines BC and DA provides a second constant m1 ∈ Zsuch that ∫

Pa

z ·f ′(z)f (z)

dz = m1 ·ω1 +m2 ·ω2 ∈Λ , q.e.d.

Theorem 1.20 (Abel’s theorem). Consider n≥ 2 and two finite families of complexpoints

A := (ai)i=1,...,n, B := (bi)i=1,...,n ⊂ P

with A disjoint to B. Then an elliptic function f ∈M (Λ) exist with its zeros A andits poles B mod Λ if and only if

n

∑i=1

ai−n

∑i=1

bi ∈Λ .

Proof. Proposition 1.19 proves that the zero set and the pole set of f satisfy the con-dition stated above. In order to prove the opposite direction of the theorem assumetwo sets A and B with the property stated above. Without restriction we may assume

n

∑j=1

(a j−b j) = 0,

otherwise replace a1 by a suitable point which is congruent mod Λ . The proof forthe existence of a suitable elliptic function f is by induction, starting with n = 2. If

a1−b1 = b2−a2

1.3 Abel’s Theorem 17

then consider

z0 :=a1 +a2 +b1 +b2

4.

Geometrically the point z0 is the center of the parallelogram with vertices thepoints a1,a2,b1,b2. We may assume z0 = 0, otherwise we translate all coordinatesby −z0. As a consequence

a1 =−a2 and b1 =−b2.

The even functiong(z) :=℘(z)−℘(a1)

has a

• zero of order = 1 at a1 and at a2 if a1 6≡ a2 mod Λ , and of order = 2 otherwise,

• and a pole of order = 2 at z = 0 .

An analogous result holds for the function

h(z) :=℘(z)−℘(b1).

Hence the quotient f :=gh

solves the problem.

Induction step n 7→ n+1: If mod Λ

(a1−b1)+ ...+(an−bn)+(an+1−bn+1)≡ 0

then choose an ∈ C with

an ≡ (a1−b1)+ ...+(an−1−bn−1)+an.

Then

(a1−b1)+ ...+(an−1−bn−1)+(an− an)≡ 0.

By induction assumption an elliptic function f1 ∈M (Λ) exists with zeros (a1, ...,an)and poles (b1, ...,bn−1, an). A second elliptic function f2 ∈ M (Λ) exists withzeros (an,an+1) and poles (bn,bn+1). Hence the function

f := f1 · f2

solves the problem, q.e.d.

Abel’s theorem generalizes to all compact Riemann surfaces, it is not restrictedto tori, see [10, §20].


Corollary 1.21 (Elliptic functions with pole of order = 3). An elliptic function f ∈M (Λ)with a pole mod Λ at 0 ∈ Λ of order = 3 and no other poles mod Λ , has exactlythree zeros a1,a2,a3 mod Λ . They satisfy

a1 +a2 +a3 = 0 mod Λ .

Chapter 2Complex tori

2.1 Compact Riemann surfaces

This section recalls some necessary prerequisites from the theory of Riemann sur-faces. A reference is the textbook [10].

Fig. 2.1 Riemann surface X

19

20 2 Complex tori

Definition 2.1 (Riemann surface). A Riemann surface X a connected complexmanifold of complex dimension 1.

We assume all manifolds to be Hausdorff spaces. By a theorem of Rado everyRiemann surface has a countable basis of its topology.

Example 2.2 (Riemann surfaces).

1. Any domain G⊂ C is a Riemann surface.

2. The complex projective space P1 = P1(C) (Riemann sphere) is a compact Rie-mann surface.

3. According to Riemann’s mapping theorem any simply-connected Riemann sur-face is biholomorphically equivalent to

• either the unit disc ∆ := z ∈ C : |z|< 1,• or to C,• or to P1.

The upper half-plane, also called Poincare upper half-plane,

H := τ ∈ C : Im τ > 0

of complex numbers with positive imaginary part is a Riemann surface, biholo-morphically equivalent to ∆ .

4. The complex torus C/Λ , with Λ ⊂ (C,+) a lattice, is a compact Riemann sur-face. Note that the torus is also an Abelian group

(C,+)/(Λ ,+)

with respect to addition.

Remark 2.3 (Meromorphic function).

1. Consider a Riemann surface X and an open subset U ⊂ X . A meromorphic func-tion f defined on U is determined by a discrete set S ⊂ U and a holomorphicfunction

f : U \S→ C,

such that for all p ∈ Slimz→p| f (z)|= ∞.

The set S is the pole set of f : With respect to a local coordinate φ around apoint p ∈ S the function f φ−1 has a pole at φ(p). The field of all meromorphicfunctions defined on U is denoted M (U).

2.1 Compact Riemann surfaces 21

The attachmentU 7→M (U), U ⊂ X open,

defines on X the sheaf MX of meromorphic functions. It is sheaf of rings. Themeromorphic functions, which are non-zero on any connected component of theirdomain of definition, define the subsheaf M ∗

X . The latter is a sheaf of multiplica-tive Abelian groups. Analogously one defines the sheaves OX and O∗X of respec-tively holomorphic functions and holomorphic functions without zeros.

2. Consider the canonical embedding

C → P1,z 7→ (z : 1),

and the point ∞ := (1 : 0) ∈ P1.

Any meromorphic function f defined on a Riemann surface X with pole set Scan be considered a holomorphic map

f : X → P1 = C∪∞

by defining for all poles z ∈ S

f (z) := ∞ ∈ P1.

Vice versa, any holomorphic map f : X → P1 with f (X) 6= ∞ defines a mero-morphic function f on X .

In the following we will not distinguish between a meromorphic function f de-fined on X and the corresponding holomorphic map f : X → P1.

3. Consider a lattice Λ ⊂ C and the corresponding torus T := C/Λ . Holomorphicmaps

f : T → P1

correspond bijectively to elliptic functions f ∈M (Λ):

f ([z]) = f (z),z ∈ C.

If there is no risk of confusion we will denote both functions with the sameletter f . In particular, the Weierstrass function ℘ of the lattice Λ will be consid-ered also a meromorphic function on T .

Proposition 2.4 (Non-constant holomorphic maps). Any non-constant holomor-phic map f : X → Y between two compact Riemann surfaces is surjective.

It attains each value y ∈ Y with the same multiplicity. The value is named deg f ,the degree of f : For all y ∈ Y the fibre Xy := f−1(y) has cardinality

22 2 Complex tori

card Xy = deg f .

Proof. A non-constant holomorphic map between two Riemann surfaces is an openmap. Locally f is the map z 7→ zk for a suitable k ∈ N∗. If X is compact then f isalso closed. The open and closed subset f (X)⊂Y equals Y because Y is connected.

The set of S ⊂ X of ramification points of f , i.e. with k ≥ 2 in a neighbourhoodof the point, is a discrete subset, hence a finite set. Denote by f (S) ⊂ Y the set ofbranch points. Then

f |X \ f−1( f (S)) : X \ f−1( f (S))→ Y \ f (S)

is a covering projection. Because Y \ f (S) is connected, each fibre of

f (X \ f−1( f (S))

has the same finite cardinality, and a posteriori also each fibre of f, q.e.d.

Note: The degree of a non-constant map f according to the topological definitionfrom Proposition 2.4 equals the degree of the field extension

[M (X) : f ∗M (Y )]

which is an algebraic concept.

Definition 2.5 (Divisor). Consider a compact Riemann surface X .

1. A divisor on X is a formal sum

D := ∑x∈X

nx(x)

with integers nx ∈ Z, and nx = 0 for all but finitely many x ∈ X . The degree of Dis

deg D := ∑x∈X

nx.

For each x ∈ X set D(x) := nx. The divisors of meromorphic functions f ∈M (X)∗

( f ) := ∑x∈X

nx(x), nx := ordx( f ),

are named principal divisors. Note that f is non-zero, hence has only finitelymany zeros and poles. Here


ordx( f ) =

n if f has zero at x of order n−n if f has pole at x of order n

Principal divisors satisfy deg ( f ) = 0 according to Proposition 2.4.

2. Consider two divisorsDi = ∑

x∈Xnx.i(x), i = 1,2.

• Their sum is defined as

D1 +D2 := ∑x∈X

nx(x), nx := nx.1 +nx,2.

• One definesD1 ≥ D2 ⇐⇒ nx,1 ≥ nx,2 f or all x ∈ X .

If D1 ≥ D2 then one names D1 a multiple of D2.

• Two divisors D1,D2 ∈ Div(X) are equivalent if their difference D2−D1 is aprincipal divisor.

3. The Abelian group of all divisors on X , the free Abelian group on the set X , isdenoted Div(X). The divisors of degree = 0 form the subgroup

Div0(X)⊂ Div(X).

The quotient of Div(X) with respect to the subgroup of pricipal divisors isthe group of divisor classes Cl(X). The notation Cl0(X) denotes the subgroupof Cl(X) generated by the classes from Div0(X).

Example 2.6 (Abel’s Theorem: Version with divisors). The families A and B fromAbel’s Theorem 1.20 define on the torus T two divisors

DA := ∑a∈A

na(a) and DB := ∑b∈B

nb(b)

with respectively na and nb the multiplicities of the points a in the family A and bin B. The theorem proves the existence of a meromorphic function f ∈M (T )∗ withdivisor

( f ) = D := DA−DB ∈ Div(T )

if and only if∑a∈A

na ·a = ∑b∈B

nb ·b ∈ T

with respect to addition and multiples from the group structure on T - not withrespect to the group structure due to Div(T ).

24 2 Complex tori

Remark 2.7 (Exact sequence of divisor class groups). On a compact Riemann sur-face X the following sequence of group morphisms is exact

1→ O∗(X)→M ∗(X)→ Div0(X)→Cl0(X)→ 0

HereO∗(X) := H0(X ,O∗X ) = C∗.

The mapM ∗(X)→ Div0(X), f 7→ ( f ),

is well-defined due to Proposition 2.4. For a torus T Abel’s theorem shows Cl0(T ) 6= 0.Later Theorem 3.17 will compute this group.

Proposition 2.8 (Divisors and line bundles). Consider a compact Riemann surface X.

1. Consider a divisorD = ∑

x∈Xnx(x) ∈ Div(X).

For each open set U ⊂ X define the Abelian group

OD(U) := f ∈M ∗(U) : ( f )≥−D|U∪0

i.e. for all x ∈U

f has in x

a pole of order at most nx i f nx ≥ 0a zero of order at least −nx i f nx ≤ 0

Then the attachmentU 7→ OD(U), U ⊂ X open,

defines a subsheaf OD ⊂MX , named the sheaf of meromorphic multiples of−D.

2. The sheaf OD is isomorphic to the sheaf LD of sections of the holomorphic linebundle LD on X, which is defined by the following cocycle

g := (gi j)i, j∈I ∈ Z1(U,O∗X ) :

Choose an open covering U = (Ui)i∈I of X by open subsets Ui ⊂ X with mero-morphic functions ψi ∈M ∗(Ui) such that

(ψi) = D|Ui, i ∈ I,

and set

gi j :=ψi

ψ j∈ O∗X (Ui∩U j).


3. Two divisors D1, D2 ∈ Div(X) are equivalent iff

OD1 'OD2 .

The mapF : (Cl(X),+)→ (Pic(X),⊗), [D] 7→ [LD],

is a group isomorphism from the additive group of divisor classes to the multi-plicative Picard group of isomorphism classes of holomorphic line bundles.

Proof. 1. Apparently,OD ⊂M

is a subsheaf of the sheaf of meromorphic functions.

2. For i, j ∈ I the quotient

gi j :=ψi

ψ j∈ O∗X (Ui∩U j)

is holomorphic without zeros because ψi and ψ j define the same divisor on Ui∩U j.Apparently the family g := (gi j)i, j∈I satisfies the cocycle relation

gi j ·g jk = gik f or i, j,k ∈ I.

Consider an open subset U ⊂ X and a meromorphic function f ∈ OD(U). Thenfor all i ∈ I:

( f |(U ∩Ui) ·ψi|(U ∩Ui))≥ 0,

i.e. the function

fi := f |(U ∩Ui) ·ψi|(U ∩Ui) ∈ OX (U ∩Ui)

is holomorphic. Apparently on the intersections U ∩ (Ui∩U j):

gi j · f j =ψi

ψ j· ( f ·ψ j) = ψi · f = fi,

i.e. the family ( fi)i∈I is a section of the line bundle LD on the open set U ⊂ X .

3. In order to obtain the inverse

G : Pic(X)→Cl(X)

of the morphism F we use the following deep fact: Any line bundle on a compactRiemann surface X has a non-constant meromorphic section [10, Satz 29.16].For a line bundle L on X choose a non-constant meromorphic section ψ of Land define

G(L ) := [(ψ)] ∈Cl(X).

26 2 Complex tori

One checks: The definition of G is independent from the choice of ψ , and G = F−1,q.e.d.

The sheaf LD of sections of a line bundle LD is a locally free sheaf of rank = 1.These sheaves are also named invertible sheaves. We shall often identify line bun-dles with their sheaf of sections, i.e. we shall not distinguish between LD and LD.

Corollary 2.9 (Divisors of negative degree). Consider a compact Riemann surfaceand a divisor D ∈ Div(E) with deg D < 0. Then

H0(X ,OD) = 0.

Proof. A non-zero section f ∈H0(X ,OD) is a meromorphic function f with divisor ( f )≥−D,in particular

deg ( f )≥−deg D > 0,

a contradiction to Proposition 2.4, q.e.d.

A deep result from the theory of compact Riemann surfaces is the following the-orem on the finiteness of the cohomology of coherent sheaves, in particular the co-homology of invertible sheaves like OD, Ω 1, Θ , but in general not the cohomologyof M .

Remark 2.10 (Finiteness of cohomology groups). Consider a compact Riemann sur-face X and a coherent sheaf F on X . Then for all i ∈ N the complex vectorspace H i(X ,F ) has finite dimension.

Definition 2.11 (Genus of a compact Riemann surface). The (geometric) genusof a compact Riemann surface X is defined as

g(X) := dim H0(X ,Ω 1X ),

the dimension of the complex vector space of holomorphic differential forms.

Example 2.12 (Genus of projective space). The projective space P1 is a compactRiemann surface of genus g = 0.

Proof. To compute H0(P1,Ω 1P1) we use the standard covering (U0,U1) of P1

Ui := (z0 : z1) ∈ P1 : zi 6= 0, i = 0,1,

and the corresponding charts


w0 = φ0(z0 : z1) :=z1

z0and w1 = φ1(z0 : z1) :=

z0

z1.

A differential form ω ∈H0(P1,Ω 1P1) is a pair ( fi ·dwi)i=0,1 with holomorphic func-

tions fi, which on

C∗ 'U0∩U1 = P1 \(0 : 1), (1 : 0),

satisfyf0(w0) ·dw0 = f1(w1) ·dw1.

Because w1 = 1/w0 we have

dw1 =−dw0

w20.

Therefore

−w20 · f (w0) ·dw0 = f1(1/w0) ·dw0 or −w2

0 · f (w0) = f1(1/w0).

The Taylor expansions of f0 and f1 shows

f0 = f1 = 0.

Therefore ω = 0, q.e.d.

We recall the main result to compute the dimension of cohomology groups oncompact Riemann surfaces.

Theorem 2.13 (Theorem of Riemann-Roch). Consider a compact Riemann sur-face X of genus g and a line bundle on X with sheaf of sections L . Then:

dimH0(X ,L )−dimH1(X ,L ) = 1−g+ c1(L ).

Here c1(L ) ∈ Z denotes the first Chern number of L . If L = OD for thedivisor D ∈ Div(X), then

c1(L ) = deg D.

For the proof of Theorem 2.13 see [10, Satz 16.9].

A second basic result is Serre’s theorem on duality:

Theorem 2.14 (Serre duality). Consider a compact Riemann surface X. Then thesheaf ωX := Ω 1

X is the dualizing sheaf of X, i.e. for any line bundle L on X andits dual line bundle L∗ with invertible sheaves of sections L and L −1 a canonicalpairing of vector spaces exists

H0(X ,L −1⊗ωX )×H1(X ,L )→ C.

In particular

28 2 Complex tori

dim H0(X ,L −1⊗ωX ) = dim H1(X ,L ).

For a proof of Theorem 2.14 see [10, Satz 17.9].

Due to Theorem 2.14 the genus of a compact Riemann surface X satisfies theequality

g(X) := dim H0(X ,Ω 1X ) = dim H1(X ,OX ).

Moreover from the viewpoint of algebraic topology

2 ·g(X) = dim H1(X ,C).

Combining Theorem 2.13 and 2.14 the theorem of Riemann-Roch takes on thefollowing form:

Corollary 2.15 (Riemann-Roch with Serre duality). Consider a compact Rie-mann surface X of genus g. Then for any line bundle on X with sheaf L of sections:

dim H0(X ,L )−dim H0(X ,L −1⊗Ω1X ) = 1−g+ c1(L ).

Corollary 2.16 (Degree of differential forms). Consider a compact Riemann sur-face X of genus g ∈ N. Then

c1(Ω1X ) = 2g−2.

Proof. According to Corollary 2.15

dim H0(X ,Ω 1X )−dim H0(X ,OX ) = 1−g+ c1(L ).

Due to the maximum principle all holomorphic functions on X are constant, hence

dim H0(X ,OX ) = 1.

According to the definition

g := dim H0(X ,Ω 1X ).

As a consequence

1−g+ c1(L ) = dim H0(X ,Ω 1X )−dim H0(X ,OX ) = g−1

i.ec1(L ) = 2g−2, q.e.d.

A divisor K ∈ Div(X) with OK 'Ω 1X is named canonical divisor of X .


Theorem 2.17 (Theorem of Riemann-Hurwitz). Consider a non-constant holo-morphic map

f : X → Y

between two compact Riemann surfaces and denote by S⊂ X the set of its ramifica-tion points. Then

2g(X)−2 = (deg f )(2g(Y )−2)+ ∑x∈S

rx f .

Recall from the proof of Proposition 2.4 the local representation of f around anarbitrary point x ∈ X

z 7→ zn with n≥ 2.

The numberrx f := n−1 ∈ N

is the ramification order of f at x. The point x ∈ X is a ramification point of fif rx( f )≥ 1.

Recall the holomorphic Euler characteristic

χ(OX ) := dim H0(X ,OX )−dim H1(X ,OX )

and the topological Euler characteristic

χ(X) :=2

∑i=0

dim H i(X ,C).

They relate as χ(X) = 2 ·χ(OX ). Then

2g(x)−2 =−(χ(X)) =−2 ·χ(OX ).

Theorem 2.17 shows that a non-constant holomorphic map f : X → Y cannot in-crease the genus. It is either an isomorphism or it decreases the genus, i.e. g(Y ) <g(X).

The following theorem 2.18 describes how to obtain holomorphic maps

f : X → Pn

into the projective space from suitable line bundles L on X . Actually, any holomor-phic map f : X → Pn arises in a similar vein.

Theorem 2.18 (Maps into a projective space). Consider a compact Riemann sur-face X and a line bundle L on X with sheaf of sections L .

30 2 Complex tori

1. If L has sufficiently many sections, then L defines an embedding

φL : X → Pn(C)

of X as a closed submanifold of Pn(C), more precisely:

Choose a basis (si)i=0,...,n of H0(X ,L ) and consider the following conditions:

• The sheaf L has no base points, i.e. for all p ∈ X a section s ∈ H0(X ,L )with s(p) 6= 0 exists: Then define

φL : X → Pn(C), p 7→ (s0(p) : ... : sn(p)).

• The sheaf L separates points, i.e. for any two distinct points p 6= q ∈ X asection s ∈ H0(X ,L ) exists with s(p) = 0 but s(q) 6= 0: Then φL is injective.

• The sheaf L separates tangent vectors, i.e. for all points p ∈ X the map

s ∈ H0(X ,L ) : s(p) = 0→mpLp/m2pLp, s 7→ [sp],

is surjective: Then φL is an immersion.

If L satisfies all three conditions, then φL : X→ Pn(C) is defined and embeds Xinto Pn(C) as a closed submanifold.

2. Assume L = OD for the divisor D ∈ Div(X). Then:

• L has no base points iff for all points p ∈ X

dim H0(X ,OD−(p)) = dim H0(X ,OD)−1.

• L separates points and tangent lines iff for all points p,q∈X, not necessarilydistinct,

dim H0(X ,OD−(p)−(q)) = dim H0(X ,OD)−2.

The definition of the map φL uses explicitly a basis of H0(X ,L ). This makesthe definition easy to understand but dependent on the choice of the basis. To obtaina map which is independent from any basis, one uses the more refined definitionwith the dual space:

X → P(H0(X ,L )∗),x 7→ [λx],

withλx : H0(X ,L )→L (x) := Lx/mxLx ' C,σ 7→ [σx].

Here Lx denotes the stalk of the sheaf L at the point x ∈ X . The class of λx doesnot depend on the choice of a trivialization of L around x.

For a proof of Theorem 2.18 in the context of Algebraic Geometry see [15, Prop. 7.3].


Remark 2.19 (Very ample line bundle). A line bundle L on a compact Riemannsurface X is named very ample if

φL : X → P1(H0(X ,L )∗)

is defined and is a closed embedding.

Lemma 2.20 (Genus of a torus). Any complex torus T = C/Λ is a compact Rie-mann surface of genus g = 1.

Proof. The holomorphic differential form dz ∈ H0(C,Ω 1C) on C has no zeros. It is

invariant with respect to translation by arbitrary but fixed constants a ∈ C:

d(z+a) = dz.

Therefore dz induces a holomorphic differential form

ω ∈ H0(T,Ω 1T )

on T with deg (ω) = 0. According to Corollary 2.16 the genus g of T satisfies

2g−2 = deg (ω) = 0, q.e.d.

Note: In the proof of Lemma 2.20 the reference to Corollary 2.16 can be avoided.Instead one argues that the line bundle Ω 1

T is trivial because it has a holomorphicsection without zeros. From Ω 1

T ' OT follows

H0(T,Ω 1T )' H0(T,OT ) = C.

We now apply the theorem of Riemann-Roch to the question on the existence ofmeromorphic functions on a torus.

Example 2.21 (Meromorphic functions on tori). Consider a complex torus X . In or-der to prove the existence or non-existence of meromorphic functions f ∈M (X)with a single pole at a given point x ∈ X we consider the divisors

D(n) := n(x) ∈ Div(X), n ∈ N.

They satisfydeg D(n) = n.

According to Corollary 2.15 with genus g = 1 and according to Lemma 2.20

dim H0(X ,OD(n))−dim H0(X ,OD(−n)) = n.

32 2 Complex tori

For all n≥ 1H0(X ,OD(−n)) = 0

according to Corollary 2.9. Hence for all n≥ 1

dim H0(X ,OD(n)) = n,

and in additiondim H0(X ,OD(0)) = dim H0(X ,OX ) = 1

because H0(X ,OX )' C.

From these dimension formulas we now rediscover the result from Corollary 1.9and the result about the existence of non-constant elliptic functions:

i) According to Proposition 2.8 the 1-dimensional vector space H0(X ,OD(1)) isthe vector space of all meromorphic functions with at most one pole at x of order = 1and no other pole. The vector space comprises the 1-dimensional subspace of con-stant functions. Therefore

H0(X ,OD(1)) = C,

in particular there do not exist meromorphic functions with only one pole at x oforder = 1 and no other pole.

ii) Similarly, the 2-dimensional vector space H0(X ,OD(2)) of meromorphic func-tions with at most one pole at x of order = 2 and no other pole comprises

H0(X ,OD(1)) = H0(X ,OD(0))

as a proper subspace. Hence at least one meromorphic function

f ∈ H0(X ,M (X))

exists with a pole at x of order = 2 and no other pole.

2.2 Isomorphism classes of complex tori and the modular group

The basis of a latticeΛ = Zω1 +Zω2

is not uniquely determined. Two positively oriented bases generate the same latticeif and only if they are equivalent modulo SL(2,Z), i.e. iff they belong to the sameorbit of the following SL(2,Z)-operation:

Lemma 2.22 (Positively oriented bases of a lattice). Denote by

M :=(ω1,ω2) basis o f R2 with det

(ω1 ω2| |

)> 0

2.2 Isomorphism classes of complex tori and the modular group 33

the set of positively oriented bases of R2. The set R of all lattices Λ ⊂C is the orbitset

SL(2,Z)\M

of the left SL(2,Z)-operation

SL(2,Z)×M→M,

(γ,

(ω1ω2

))7→(

ω ′1ω ′2

):= γ ·

(ω1ω2

).

Proof. The invertible matrices γ ∈M(2×2,Z) have determinant ±1. Because bothbases are positively oriented we have det γ = 1, q.e.d.

Two tori belonging to different lattices may be biholomorphically equivalent. Thefollowing Definition 2.23 introduces a name for the relation between lattices withisomorphic tori.

Definition 2.23 (Similar lattices). Two lattices Λ , Λ ′ ⊂C are similar if a complexnumber µ ∈ C∗ exists with

Λ′ = µ ·Λ .

Proposition 2.24 (Lattices of biholomorphically equivalent tori). Two tori

T = C/Λ and T ′ = C/Λ′

are biholomorphically equivalent iff their lattices Λ and Λ ′ are similar.

Proof. i) Assume the existence of a biholomorphic map f : T → T ′ satisfying with-out restriction f (0) = 0. Because C is simply connected and the canonical map

p′ : C→ C/Λ′

is a covering projection, the map f extends to a commutative diagram

C C

C/Λ C/Λ ′

F

p p′

f

with a unique holomorphic map F such that F(Λ)⊂Λ ′ and F(0) = 0. For arbitrarybut fixed ω ∈Λ consider the holomorphic map

Fω : C→ C, z 7→ F(z+ω)−F(z).

34 2 Complex tori

By assumption Fω(C) ⊂ Λ ′. Because the lattice Λ ′ is a discrete space and the do-main of definition C is connected, the map Fω is constant. Hence the derivativesatisfies

F ′ω(z) =dFdz

(z+ω)−dFdz

(z) = 0

for all z ∈ C. As a consequence, the holomorphic map F ′ is elliptic with respectto Λ . Proposition 1.7 implies that F ′ is constant, i.e. a number µ ∈ C exists with

F(z) = µ · z

for all z ∈ C. Analogously the unique lift F∗ of f−1 with F∗(0) = 0 satisfies

F∗(z) = µ∗ · z, z ∈ C,

for a suitable µ∗ ∈ C. Then

C→ C,z 7→ (µ∗ ·µ) · z,

is the unique lift of id = f−1 f which maps 0 7→ 0. Hence µ∗ ·µ = 1, in particular

µ ·Λ = Λ′.

ii) Assume Λ ′ = µ ·Λ with µ ∈ C∗. Then the biholomorphic map

F : C→ C, z 7→ µ · z,

induces a biholomorphic map f : T → T ′, q.e.d.

Theorem 2.25 (Biholomorphic equivalence classes of complex tori).

1. Any complex torus is biholomorphic equivalent to a torus T = C/Λ with a nor-malized lattice

Λ = Z ·1+Z · τ ⊂ C

with τ ∈H in the upper half-plane.

2. The period τ of a normalized lattice is determined up to a transformationof SL(2,Z), i.e. two tori

C/Λτ and C/Λτ ′

with normalized lattices

Λτ = Z ·1+Z · τ and Λτ ′ = Z ·1+Z · τ ′

are biholomorphic equivalent iff

τ′ =

a τ +bc τ +d

2.2 Isomorphism classes of complex tori and the modular group 35

with integers a,b,c,d ∈ Z satisfying ad−bc = 1.

Proof. 1. Consider an arbitrary torus T = C/Λ with a lattice satisfying

Λ = Zω1 +Zω2

and det (ω1 ω2)> 0. Then τ :=ω2

ω1satisfies Im τ > 0 and the lattice Λ is similar

to the normalized latticeΛτ := Z ·1+Z · τ

due to1

ω1·Λ = Λτ .

According to Proposition 2.24 the two tori T and T ′ :=C/Λτ are biholomorphicequivalent.

2. i) Assume

τ′ =

a τ +bc τ +d

with γ :=(

a bc d

)∈ SL(2,Z).

Set (ω2ω1

):= γ ·

(τ

1

).

ThenZ ·ω1 +Z ·ω2 = Z ·1+Z · τ = Λτ .

As a consequence1

ω1·Λτ = Λτ ′

because

τ′ =

ω2

ω1.

The lattices Λ and Λ ′ are similar. Hence the tori C/Λτ and C/Λτ ′ are biholomor-phic equivalent according to Proposition 2.24.

ii) For the opposite direction assume that two tori with normalized lattices

C/Λτ and C/Λτ ′ ,τ,τ′ ∈H,

are biholomorphic equivalent. According to Proposition 2.24 a number µ ∈ C∗exists with

Λτ ′ = µ ·Λτ .

As a consequenceΛτ ′ = Z ·µ +Z ·µτ.

Therefore a matrix

36 2 Complex tori

γ =

(a bc d

)∈ SL(2,Z)

exists with (τ ′

1

)= γ ·

(µτ

µ

),

i. e.

τ′ =

a ·µτ +b ·µc ·µτ +d

=a · τ +bc · τ +d

, q.e.d.

Theorem 2.25 shows the consequences when two points of the upper half-planeare related by an element of the group SL(2,Z). We formalize this relation by theconcept of a group operation of SL(2,Z).

Definition 2.26 (Operation of the modular group).

1. The groupSL(2,Z) := γ ∈M(2×2,Z) : det γ = 1

is named the modular group.

2. The SL(2,Z)-left operation on the upper half-plane is the map

Φ : SL(2,Z)×H→H,(γ,τ) 7→ γ(τ) :=aτ +bcτ +d

, γ :=(

a bc d

).

3. The isotropy group of a point τ ∈H is the subgroup

SL(2,Z)τ := γ ∈ SL(2,Z) : γ(τ) = τ.

The point τ ∈H is an elliptic point of Φ if its isotropy group

Γτ := γ ∈ Γ : γ(τ) 6= τ

is not trivial, i.e. ifΓτ 6= ±id.

The period of an arbitrary, not necessary elliptic τ ∈H is the number

hτ := (1/2) ·ord Γτ .

4. The orbit of a point τ ∈H is the set

γ(τ) ∈H : γ ∈ SL(2,Z).

5. To points τ1,τ2 ∈H are equivalent with respect to Φ iff they belong to the sameorbit. The orbit set of Φ is the set of all equivalence classes

2.3 The modular curve of the modular group 37

Y := SL(2,Z)\H.

Note that Φ is well-defined because γ(τ) ∈H for τ ∈H: One checks

Im γ(τ) =Im τ

|cτ +d|2.

The operation Φ is continuous. The element−id ∈ SL(2,Z) operates trivially on H,i.e. −id ∈ SL(2,Z)τ for all τ ∈H. Therefore some authors apply the name modulargroup to the quotient

PSL(2,Z) := SL(2,Z)/±id.

We will not follow this convention. Later we will study also the operation of certainsubgroups of SL(2,Z) which do not contain the element −id.

2.3 The modular curve of the modular group

In the present section we denote by

Φ : SL(2,Z)×H→H,(γ,τ) 7→ γ(τ) :=aτ +bcτ +d

, γ :=(

a bc d

)the continuous SL(2,Z)-left operation introduced in Definition 2.26. We will oftenuse the shorthand notation for the modular group

Γ := SL(2,Z).

Our aim is to provide the orbit space Y := Γ \H with an analytic structure and tocompactify the resulting Riemann surface to obtain a compact Riemann surface X .The latter will be named the modular curve of the operation Φ . We proceed alongthe following steps:

• Constructing a Hausdorf topology on Y , see Corollary 2.30.

• Constructing an analytic structure on Y , see Proposition 2.33.

• Compactifying Y to a compact Riemann surface X , see Theorem 2.34.

To provide the orbit space of a group operation with a suitable topological ordifferentiable structure is always a subtle task. In the present context the prob-lem is intensified by the fact that the modular group does not operate freely, seeTheorem 2.31: There are elliptic points. The difficulty consist in dealing with theelliptic points with respect to the Hausdorff property and with respect to complexcharts on the quotient.

38 2 Complex tori

The following Lemma 2.27 covers the main step for proving the Hausdorff prop-erty for the quotient topoplogy on Y .

Lemma 2.27 (Separating orbits). For any two points τ1,τ2 ∈ H exist neighbour-hoods

Ui ⊂H of τi, i = 1,2,

such that for all γ ∈ Γ :

γ(U1)∩U2 6= /0 =⇒ γ(τ1) = τ2.

In particular: Any point τ ∈H has a neighbourhood U ⊂H such that for all γ ∈ Γ :

γ(U)∩U 6= /0 =⇒ γ ∈ Γτ .

The set U has no elliptic points except possibly τ .

Lemma 2.27 can be paraphrased as follows: If an element γ ∈ Γ maps a point ofthe distinguished neighbourhood U1 of τ1 to the distinguished neighbourhhood U2of τ2, then γ maps τ1 to τ2. The add-on states: If γ ∈ Γ does not fix τ , then γ movesall points from U away from U .

Proof. 1. Finiteness: We choose two relatively compact neighbourhoods

Vi ⊂⊂H of τi, i = 1,2,

and show: Only finitely many matrices γ ∈ Γ exists with

γ(V1)∩V2 6= /0.

The idea is to extend the operation of the modular group defined over Z to anoperation of SL(2,R) defined over R, hereby embedding each isotropy groupof Γ into a compact isotropy group of SL(2,R).

Working out this idea we show that H is a homogeneous space for the SL(2,R)-left operation

φR : SL(2,R)×H→H,(γ,τ) 7→ γ(τ) :=aτ +bcτ +d

, γ :=(

a bc d

).

The point i has the isotropy group

SL(2,R)i = SO(2,R)

because one checks

γ(i) = i ⇐⇒ a = d and b =−c.


The operation φR is transitive because any point τ ∈ H belongs to the orbit of i:If

τ = x+ iy

then γτ(i) = τ for the matrix

γτ :=1√

y·(

y x0 1

)∈ SL(2,R).

Note that γτ depends continuously on τ .

As a consequence for any two points e1,e2 ∈H and any γ ∈ SL(2,R):

γ(e1) = e2 ⇐⇒ γ ∈ γe2 ·SO(2,R) · γ−1e1

.

e1 e2

i i

γ

γ−1e1

γe2

If a matrix γ ∈ Γ satisfies γ(V1)∩V2 6= /0 then also

γ(V1)∩V2 6= /0

and

G := γ ∈ SL(2,R) : γ(V1∩V2 6= /0=⋃

e1∈V1,e2∈V2

γe2 ·SO(2,R) · γ−1e1

.

The latter set is the continuous image of the compact set

V2×SO(2,R)×V1

and therefore compact. Its intersection G∩Γ with the discrete group Γ is finite.

2. Shrinking V1 and V2: According to the first part only finitely many γ ∈ Γ existwith

γ(V1)∩V2 6= /0.

In particular, the set

F := γ ∈ Γ : γ(V1)∩V2 6= /0 and γ(τ1) 6= τ2

is finite. For each γ ∈ F we have γ(τ1) 6= τ2. Because H is a Hausdorff space wecan choose two disjoint neighbourhoods

U1,γ ⊂H of γ(τ1) and U2,γ ⊂H of τ2.

40 2 Complex tori

We define two neighbourhoods of respectively τ1 and τ2 in H

U1 :=V1∩

(⋂γ∈F

γ−1(U1,γ)

), U2 :=V2∩

(⋂γ∈F

U2,γ

).

Then for all γ ∈ Fγ(U1)∩U2 = /0.

Eventually, consider an arbitrary element

γ ∈ Γ with γ(U1)∩U2 6= /0.

We show γ(τ1) = τ2 by an indirect proof: Otherwise γ ∈ F and by construction

U1 ⊂ γ−1(U1,γ) and U2 ⊂U2,γ

andU1,γ ∩U2,γ = /0.

Then(γ(U1)∩U2)⊂ (U1,γ ∩U2,γ) = /0,

a contradiction to the choice of γ .

3. τ1 = τ2: The add-on referring to the isotropy group Γτ is the specific case τ1 = τ2and U :=U1∩U2, q.e.d.

Definition 2.28 (Proper group operation). Consider a continuous operation

F : G×X → X , (g,x) 7→ g(x),

of a locally compact group G on a compact space X . The operation is a proper groupoperation if

σ : G×X → X → X×X , (g,x) 7→ (g(x),x),

is a proper map.

Corollary 2.29 (Proper operation of the modular group). The operation

φ : Γ ×H→H

is a proper group operation.

Proof. Consider a compact subset K = K1 ×K2 ⊂ X × X . We have to show thecompactness of

σ−1(K)⊂ G×X .


It suffices to show that σ−1(K) is sequentially compact. We claim: Any sequence

(γν ,τν)ν∈N

in σ−1(K) has a convergent subsequence. Due to compactness of K we may assumethat the image sequence

σ((γν ,τν))ν∈N = (γν(τν),τν)ν∈N

is convergent. Setx1 := lim

ν→∞τν , x2 := lim

ν→∞γν(τν).

Choose open neighbourhoods Ui⊂X of xi, i= 1,2 with the properties from Lemma 2.27.For all but finitely many ν ∈ N we have

τν ∈U1 and γν(τν) ∈U2.

The Lemma implies for all but finitely many ν ∈ N

γν(x1) = x2.

The setγ ∈ Γ : γ(x1) = x2

has the same cardinality like the isotropy group Γx1 . Hence it is a finite set. Asa consequence: A suitable subsequence (γτν k)k∈N is constant, hence a posterioriconvergent, q.e.d.

Corollary 2.30 (Hausdorff topology of the orbit space). The quotient topologyon the orbit space Y := Γ \H is a connected Hausdorff space and the canonicalprojection

p : H→ Y

is an open map.

Proof. i) Open map: We provide Y with the quotient topology with respect to p,i.e. a subset U ⊂ Y is open iff p−1(U) ⊂ H is open. Connectedness of H impliesconnectedness of Y . Eventually, p is open: For an open set U ⊂H the inverse image

p−1(p(U)) =⋃

γ∈Γ

γ(U)

is open as union of open sets. By definition of the quotient topology p(U) ⊂ Y isopen.

ii) Hausdorff topology: We use the following general result: Consider a topolog-ical space X and an equivalence relation R⊂ X×X . If the canonical projection

42 2 Complex tori

p : X → X/R

is an open map with respect to the quotient topology, then X/R is Hausdorff iff

R⊂ X×X

is closed. According to Corollary 2.29 the map

σ : Γ ×H→H×H

is proper. It satisfiesσ(Γ ×H) = R⊂H×H

with R the equivalence relation on H induced from the operation of Γ , and thequotient space

Y =H/R.

Therefore R⊂H×H is closed, and Y is Hausdorff, q.e.d.

The modular group contains the two distinguished elements

• Reflection at the y-axis and dividing by squared absolute value:

S :=(

0 −11 0

)∈ Γ , S(τ) =−1/τ =

− τ

|τ|2.

• Translation by one:

T :=(

1 10 1

)∈ Γ , T (τ) = τ +1.

Theorem 2.31 (Fundamental domain and elliptic points).

1. The modular group Γ is generated by the two elements S and T . Their relationsare generated by the two relations

S4 = id, (ST )6 = id.

2. The orbit space Y = Γ \H maps bijectively to the set

D := τ ∈H : |τ|> 1 and −1/2≤Re τ < 1/2∪τ ∈H : |τ|= 1 and −1/2≤Re τ ≤ 0,

the fundamental domain of Φ .

3. The operation Φ has exactly two elliptic points z0 ∈ D . Their isotropy groupsare cyclic:

• z0 = i withΓi =< S >⊂ Γ , ord Γi = 4,


• z0 = ρ := e2πi/3 with

Γρ =< ST >⊂ Γ ,ord Γρ = 6.

44 2 Complex tori

Fig. 2.2 Fundamental domain of the modular group [15]

Figure 2.2 shows the fundamental domain D of the modular group. In order toexclude pairs of congruent points within D one has to exclude part of the boundaryof D . This convention is followed by [15] but not by all references. Most referencesname fundamental domain the closure D .

Proof. Note that the three parts of the subsequent proof do not correspond one toone to the three parts of the theorem.

i) The domain D : Denote by

G :=< S,T >⊂ Γ

the subgroup generated by the two distinguished elements. We show: Any givenpoint τ ∈H is equivalent to a point τ ′ ∈D mod G.

We choose an element γ0 ∈ G with Im γ0(τ) maximal. Such choice is possiblebecause

Im γ(τ) =Im τ

|c · τ +d|2for γ =

(a bc d

)∈ G.


The translation T ∈Γ does not change the imaginary part of its argument. Thereforewe may assume

−1/2≤ Re γ0(τ)< 1/2.

In addition, |γ0(τ)| ≥ 1 because otherwise

Im (S γ0)(τ) = Im

(− γ0(τ)

|γ0(τ)|2

)> Im γ0(τ).

• If |γ0(τ)|> 1 then τ ′ := γ0(τ).

• If |γ0(τ)|= 1 but 0 < Re γ0(τ)≤ 1/2 then

τ′ := (S γ0)(τ).

In both cases τ ′ ∈D .

ii) Isotropy groups and fundamental domain: Consider two points τ,τ ′ ∈D with

τ′ = γ(τ),γ =

(a bc d

)∈ Γ .

We show τ = τ ′, and the latter equation implies γ ∈ Γτ .

Because

Im τ′ =

Im τ

|c · τ +d|2.

we may assume without restriction Im τ ′ ≥ Im τ , i.e.

|c · τ +d| ≤ 1.

Because Im τ ≥√

3/2 only the cases c = 0,±1 are possible:

• Case 1, c = 0: Then d =±1 and a = d. Without restriction a = d = 1, hence

τ′ = τ +b, b ∈ Z.

Because Re τ ′ = (Re τ)+b we obtain b = 1; i.e. τ = τ ′ and

γ =±(

1 00 1

)=± id ∈ Γτ .

• Case 2, c = 1:

Case 2a, d = 0: If d = 0 then b =−1 and

τ′ =

a · τ−1τ

= a− (1/τ), a ∈ Z.

46 2 Complex tori

Due to τ ∈D the equation

|c · τ +d|= |τ| ≤ 1

implies |τ|= 1. Hence

1/τ = τ and τ + τ = 2 ·Re τ = a ∈ Z.

Therefore: Either a = 0 and τ = i which implies

τ′ =−(1/τ) = i = τ

and

γ =±(

0 −11 0

)=±S ∈ Γi.

Or a =−1 and τ = ρ which implies

τ′ =−1− (1/τ) = ρ = τ

and

γ =±(−1 −11 0

)=±(ST )2 ∈ Γρ .

Case 2b, d =±1: If d =±1 then

|c · τ +d|= |τ±1| ≤ 1.

Because of τ,τ ′ ∈D the case d =−1 with |τ−1| ≤ 1 is excluded. Therefore d = 1leaving

|τ +1|= |τ− (−1)| ≤ 1.

As a consequence, τ = ρ and

ad−bc = a−b = 1 and τ′ =

a ·ρ +bρ +1

= a+ρ,

hence a = 0, τ ′ = ρ = τ and

γ =

(0 −11 1

)= ST ∈ Γρ .

• Case 3, c =−1: This case reduces to case 2, c = 1, by considering −γ0.

iii) G = Γ : Consider an arbitrary element γ ∈ Γ and the point

τ0 := 2 · i ∈D .

Due to part i) an element γ0 ∈ G exists with (γ0 γ)(τ0) ∈D . Due to part ii)


τ0 = (γ0 γ)(τ0) ∈D ,

i.e.γ0 γ ∈ Γτ0 = ±id.

Because S2 =−id we obtainγ0 γ =±S2,

which implies γ ∈ G, q.e.d.

It are the elliptic points in particular which complicate the introduction of a com-plex structure on the orbit space Y . To overcome this difficulty we make a detailedstudy of the operation in the neighbourhood of an elliptic point. Fortunately, herethe operation restricts to the operation of the isotropy group and the latter turns outas a group of rotation. The statement of the following Lemma 2.32 is trivial fornon-elliptic points. For the elliptic points it relies on the Lemma of Schwarz.

Lemma 2.32 (Isotropy groups operate as rotations). For any point τ ∈H exist

• a neighbourhood Uτ ⊂H of τ such that for all γ ∈ Γ

γ(Uτ)∩Uτ 6= /0 =⇒ γ ∈ Γτ ,

• and a fractional linear transformation λτ ∈ GL(2,C) satisfying

λτ : Uτ

'−→ ∆τ

with a disc ∆τ = z ∈ ∆ : |z|< rτ for a suitable radius rτ > 0

such that: For all γ ∈ Γτ the conjugate map

γτ := λτ γ λ−1τ : ∆τ → ∆τ

from the commutative diagram

Uτ γτ(Uτ)

∆τ ∆τ

γ

λτ

γτ

λτ

is a rotation around 0 ∈ ∆τ . The angle of rotation is an integer multiple of 2π/hτ .

Proof. Denote by

λτ : P1→ P1, λτ(z) :=z− τ

z− τ,

the fractional linear automorphism of P1 with λ (H) = ∆ and

48 2 Complex tori

λτ(τ) = 0, λτ(τ) = ∞.

We choose a radius rτ > 0 and a neighbourhood Uτ of τ with the properties fromLemma 2.27 such that the restriction

λτ |Uτ : Uτ

'−→ ∆τ

is biholomorphic. The conjugate

γτ := λτ γ λ−1τ : ∆τ → ∆

has the fixed point 0 ∈ ∆ . Moreover |γτ′(0)|= |γ ′(τ)|= 1 because in particular

|S ′(i)|= |(ST )′(ρ)|= 1.

Therefore the Lemma of Schwarz implies that γτ is a rotation-dilation around 0∈∆τ .Because |γτ

′(0)|= 1 we obtain that γτ reduces to a rotation. The angle of rotation isa multiple of 2π/hτ , because the isotropy group Γτ is cyclic with order 2hτ .

Note that the whole statement is trivial for all non-elliptic points τ ∈ H becausehere Γτ = ±id, q.e.d.

Proposition 2.33 (Analytic structure of the orbit space). On the orbit space

Y = SL(2,Z)\H

exists an analytic structure such that Y becomes a Riemann surface with holomor-phic canonical projection

p : H→ Y.

Considered from a general point of view the complex manifold Y is the coarsemoduli space of complex tori.

Proof. i) Homeomorphisms: For each τ ∈ H choose a neighbourhood Uτ ⊂ H of τ

with the properties from Lemma 2.32. Using the notation from the Lemma define

powτ : ∆τ → C,z 7→ zhτ ,

and set Wτ := powτ(∆τ). Eventually, define a chart of Y around y = p(τ) ∈ Y

φτ : p(Uτ)→Wτ

as the commutative completion of the following diagram


Uτ p(Uτ)

∆τ Wτ

p|Uτ

λτ

powτ

φτ

The map φτ is

• well-defined and bijective according to Lemma 2.32,• continuous, because p is open,• open, because λτ and powτ are open.

As a consequence, φτ is a homeomorphism.

ii) Holomorphic transition functions: We show that the family

A := (φτ : p(Uτ)→Wτ)τ∈H

is a complex atlas of Y .

Assume two chartsφτi : p(Uτi)→Wτi

around points yi = p(τi) ∈ Y, i = 1,2, and assume

p(Uτ1)∩ p(Uτ2) 6= /0.

To simplify the subscripts we set

Ui :=Uτi ,φi := φτi ,λi := λτi ,hi := hτi ,Wτi :=Wi, i = 1,2.

Consider a pointx ∈ p(U1)∩ p(U2).

We show that the mapφ2,1 := φ2 φ

−11

is holomorphic in a neighbourhood of φ1(x) ∈W1 ⊂ C: By definition of φi twopoints τi ∈Ui, i = 1,2, exist with

λi(τi)hi = φi(x), i = 1,2.

They satisfy τ2 = γ(τ1) for a suitable γ ∈ Γ . For q ∈C in a neighbourhood of φ1(x)

φ2,1(q) =((λ2 γ λ

−11 )(q1/h1)

)h2

50 2 Complex tori

Therefore we have to prove: Taking the h1-root does not prevent the holomorphy ofthe map.

We prove the non-trivial case h1 > 1, i.e. the case of an elliptic point τ1 ∈H:


• Case φ1(x) = 0 ∈W1, i.e. τ1 = τ1:

The point τ1 = τ1 is elliptic. Therefore also τ2 = γ(τ1) ∈U2 is elliptic.Because τ2 is the only elliptic point in U2 we obtain τ2 = τ2, and

τ2 = γ(τ1) and h := h2 = h1.

By construction of λi, i = 1,2, the composition

λ2 γ λ−11

is a fractional linear transformation of P1 mapping 0 to 0 and ∞ to ∞. Thereforea matrix

A =

(a bc d

)∈ GL(2,C)

exists withλ2 γ λ

−11 = A

andA · (0 1)> = (0 1)>, A · (1 0)> = (1 0)>.

As a consequence

A =

(a 00 d

)and

φ2,1(q) =((λ2 γ λ

−11 )(q1/h)

)h=(

A(q1/h))h

= ((a ·q1/h)/d)h = (a/d)h ·q.

Therefore φ2,1 is holomorphic in a neighbourhood of 0 = φ1(x).

• Case φ2(x) = 0 ∈W2, i.e. τ2 = τ2:

The previous case shows the holomorphy of φ1,2 in a suitable neighbourhood.Because the map is bijective, its inverse φ2,1 is holomorphic.

• Case φ1(x) ∈W1 arbitrary, i.e. τ1 ∈U1 arbitrary:

We reduce this case to the other two cases. Choose τ3 ∈D with p(τ3) = x andconsider the chart

φτ3 : Uτ3 →Wτ3

with the shorthandU3 :=Uτ3 ,φ3 := φτ3 ,W3 :=Wτ3 .

In a suitable neighbourhood W of y with

W ⊂ (p(U1)∩ p(U2)∩ p(U3))

52 2 Complex tori

Fig. 2.3 Transition functions of Y


we haveφ1,2 = φ1,3 φ3,2.

The right-hand side is holomorphic due to part i) and ii). Therefore φ1,2 isholomorphic, q.e.d.

The important steps in the proof of Proposition 2.33 and its prerequisites:

• The modular group operates properly.

• Isotropy groups operate as rotations (Lemma of Schwarz).

• Existence of fractional linear maps λ : H '−→ ∆ .

Theorem 2.34 (The modular curve of the modular group).

1. The operation Φ of the modular group Γ extends to a continuous Γ -left operation

Φ∗ : Γ ×H∗→H∗

on the extended upper half-plane

H∗ :=H∪Q∪∞.

2. On the orbit spaceX := Γ \H∗

exists the structure of a compact Riemann surface such that the following dia-gram commutes with canonical projections p and p∗

H H∗

Y X

p p∗

3. The modular curve X is biholomorphic equivalent to P1.

The orbit set Y compactifies because the operation of the modular group extendsto the point ∞. The orbit of ∞ encloses all points of Q. Therefore one has to considerthe extended operation on the extended half-plane H∗. Apparently, Q is not discreteas a subspace of the Euclidean space. This fact hinders the quotient topology frombeing Hausdorff. Therefore one introduces a coarser topology on the orbit of ∞. Thenew topology derives from the usual topology around the point ∞.

54 2 Complex tori

Proof. i) Extending φ to H∗: The complex plane embedds canonically into the pro-jective space P1 via

C → P1, z 7→ (z : 1).

The complementP1 \C

is the singleton with the point ∞ := (1 : 0). The Γ -left operation

φ : Γ ×H→H

extends via the canonical Γ -left operation by matrix multiplication

Γ ×C2→ C2,(γ,z) 7→ γ · z

to the Γ -left operation on P1

Γ ×P1→ P1, (γ,(z1 : z2)) 7→ (a · z1 +b · z2 : c · z1 +d · z2).

In particular(γ,∞) = (γ,(1 : 0)) 7→ (a : c) = (a/c : 1).

The last expression makes precise the intuitive expression

a ·∞+bc ·∞+d

=a+(b/∞)

c+(d/∞)=

ac.

For any rational number q = a/c ∈ Q with coprime a,c ∈ Z exist integers b,d ∈ Zwith

da−bc = 1.

As a consequence, the matrix

γ :=(

a bc d

)∈ Γ

maps(γ,∞) 7→ γ(∞) = (a/c : 1) = q ∈Q.

If we compactify H by adding the point ∞ we have to add also the whole orbit of ∞,i.e. the set

Q∪∞.

This additional orbit is named the cusp of the extended operation on the extendedupper half-plane

Φ∗ : Γ ×H∗→H∗,(γ,τ) 7→ γ(τ) :=

a · τ +bc · τ +d

,

with


γ :=(

a bc d

).

ii) Topology on H∗: Take as neighbourhood basis of ∞ in the new topology on H∗the sets

UN := τ ∈H : Im τ > N∪∞, N ∈ N,

and as neighbourhood basis of a/b ∈Q, gcd(a,c) = 1, the sets

γ(UN), N ∈ N, γ =

(a ∗c ∗

)∈ Γ .

These neighbourhood bases together with the open sets of H generate a uniquetopology on H∗. Apparently H⊂H∗ is an open subspace.

Note. Because fractional linear transformations from Γ take circles to circles orlines the neighbourhoods of γ(UN) are circles tangent to the real axis at the distin-guished point a/b ∈ Q. The topology in the neighbourhood of ∞ is not the usualtopology with the complement of compact sets (with respect to the Euclidean topol-ogy) as neighbourhoods of ∞.

iii) Hausdorff topology on Γ \H∗: We provide X :=Γ \H∗ with the quotient topol-ogy with respect to the canonical projection

p∗ : H∗→ X .

Then p∗ is an open map: The proof is the same as the corresponding proof forCorollary 2.30. The proof uses only the fact that the quotient arises from a groupoperation.

In order to show that X is Hausdorff we consider two points

xi = p∗(τi) ∈ X , τi ∈H∗, i = 1,2, and x1 6= x2,

and distinguish the following cases:

• τ1 ∈H, τ2 = ∞: For each

γ =

(a bc d

)∈ Γ

the formula

Im γ(τ) =Im τ

|c · τ +d|2=

Im τ

(c ·Re τ +d)2 + c2 · Im2 τ≤

≤ Im τ ·

1d2 c = 0

1c2 · Im2 τ

c 6= 0≤

Im τ c = 01

Im τc 6= 0

≤ maxIm τ,1/Im τ

shows

56 2 Complex tori

Im γ(τ)≤ maxIm τ,1/(Im τ).

Therefore a relatively compact neighbourhood U1 ⊂⊂H of τ1 and a neighbour-hood U2 ⊂H∗ of τ2 exist such that for all γ ∈ Γ

γ(U1)∩U2 = /0.

As a consequencep∗(U1)∩ p∗(U2) = /0.

• τ1,τ2 ∈H: This case has been considered already in the proof of Corollary 2.30.

iv) Compactness of X : Denote by D ⊂ H the closure of D with respect to H.Note that

D ∪∞ ⊂H∗

is compact with respect to H∗: Consider an open covering (Ui)i∈I of D ∪ ∞.If ∞ ∈Ui0 then for a number N ∈ R:

τ ∈D ∪∞ : Im τ > N ⊂Ui0 .

The remaining set

τ ∈D ∪∞ : Im τ ≤ N = τ ∈D : Im τ ≤ N ⊂H

is compact in H, and therefore covered by a finite subcovering of (Ui)i∈I .

As a consequence X is compact as continuous image of the compact set D∪∞.One has

Y = X \p∗(∞).

Therefore Y ⊂ X is an open subset.

v) Charts of the modular curve: In order to define a chart

φ∞ : p∗(U∞)→W∞

around the point x = p∗(∞) ∈ X we consider the neighbourhood of ∞ in H∗

U∞ := τ ∈H : Im τ > 2∪∞.

We define

pow∞ : U∞→ ∆ , τ 7→

e2πi·τ if τ 6= ∞

0 if τ = ∞

and setW∞ := pow∞(U∞)⊂ C.

For two points τi ∈U∞, i = 1,2,


τ2 = p∗(τ1) ⇐⇒ τ2 = γ(τ1) for a suitable γ ∈ Γ

⇐⇒ τ2 = τ1 +m for a suitable m ∈ Z

⇐⇒ pow∞(τ2) = pow∞(τ1).

We define the chart φ∞ as the commutative completion of the diagram

U∞ p∗(U∞)

W∞

p∗|U∞

φ∞pow∞

One shows analogously to the proof of Proposition 2.33 that the map φ∞ is a home-omorphism.

Eventually we have to show that the charts of Y and the chart φ∞ are compatible,It suffices to prove the compatibility of the chart

φ∞ : p∗(U∞)→W∞

around p∗(∞) with the chart

φτ : p(Uτ) = p∗(Uτ)→Wτ

around an arbitrary point p(τ), τ ∈H. The fact that

pow∞ : U∞→W∞

is locally biholomorphic, proves that the transition function of the two charts isholomorphic.

vi) Modular curve X ' P1: The image of the fundamental domain

Y = p(D)

is homeomorphic to the plane C: Identify points on the left and right boundary of Dand equivalent points of D on the unit circle.

Therefore the compactification of Y is homeomorphic to the 1-point compactifi-cation of C, i.e. X is homeomorphic to S2. By the Riemann mapping theorem theonly complex structure on S2 is the complex projective space P1(C), q.e.d.

For the fact X ' P1(C) a second proof will be given in Corollary 4.16. Thissecond proof uses the modular invariant from the theory of modular forms, whichprovides an explicit identification.

Chapter 3Elliptic curves

The first two sections of the present chapter deal with Algebraic Geometry. We leavethe analytical domain of complex manifolds and investigate non-singular projective-algebraic varieties. The focus switches from holomorphic and meromorphic func-tions to polynomials and rational functions. A torus, being a Riemann surface, cor-responds to a projective-algebraic curve, named an elliptic curve.

Complex Analysis with several variables and Algebraic Geometry are theorieswith many concepts in common and distinguished concepts in parallel. We compilethe following dictionary.

Proposition 3.1 (Dictionary Complex Analysis - Algebraic Geometry).

Complex Analysis Algebraic GeometryComplex Manifold Smooth varietyEuclidean topology Zariski topology

Compact complex manifold Smooth projective-algebraic varietyCompact Riemann surface Smooth projectiv-algebraic curve

Stein manifold Smooth affine varietyStructure sheaf OX

Holomorphic function Regular functionMeromorphic function Rational function

Local ring OX ,xLine bundleCohomology

Riemann-Roch TheoremSerre Duality

Note that not any compact manifold, and not even any compact Kaehler manifoldhas a counter part in Algebraic Geometry.

59

60 3 Elliptic curves

In Algebraic Geometry a variety is assumed to be irreducible. Irreducibility cor-responds to connectedness in the case of manifolds. If one skips the requirementof smoothness on the right-hand side one obtains the irreducible reduced complexspace as the analogue of a variety.

Remark 3.2 (Algebraic and analytic geometry).

1. Any smooth projective-algebraic variety X ⊂ Pn(C) can be equipped with thestructure of a connected compact complex manifold. In particular, a projective-algebraic curve C ∈ Pn(C) carries also the structure of a compact Riemann sur-face. The idea is to cover X by affine subsets (Ui)i∈I and to consider the regulartransition functions of their charts with respect to the Zariski topology as holo-morphic functions with respect to the Euclidean topology of Ui, i ∈ I. Muchdeeper is the result in the opposite direction: By a theorem of Chow any con-nected submanifold of X ⊂ Pn(C) carries the structure of a projective-algebraicvariety, i.e. X can be defined by homogeneous polynomials.

It is a deep result that many invariants X from the algebraic context translateto invariants in the analytical context. E.g., the genus of a projective-algebraiccurve is the same as the genus of its corresponding compact Riemann surface.Relations of this type are subsumed under the keyword GAGA. The latter is ashorthand for Geometrie Algebrique et Geometrie Analytique, the title of Serre’sseminal paper [31].

2. In both contexts, in the algebraic and in the analytic context the same form ofthe Riemann-Roch theorem (Theorem 2.13) and Serre duality (Theorem 2.14)are valid. For the phrasing of these theorems and their corollaries we will alwayspoint to the statements from the analytic category. But in general the proof doesnot translate from the algebraic category to the analytic category. This is due tothe fact that a general compact complex manifold does not embed into a pro-jective space. The case is different for compact Riemann surfaces: All compactRiemann surfaces X embed into P3(C). But to prove this result one first needsthe Riemann-Roch theorem on X .

3. We will use the analytic theory to deal with modular forms. They turn out to bemeromorphic differential forms on the modular curves. The letter are compactRiemann surfaces. The algebraic theory is the context for elliptic curves definedover Q and their reductions to curves over the finite fields Fp. Elliptic curves arethe bridge to Number Theory.

The base field for elliptic curves is no longer restricted to C. Instead we willmake a refined study if the curve is defined over the prime field Q or a numberfield Q⊂ k ⊂ C. If the curve is defined by a polynomial with coefficients from Q

3.1 Plane cubics 61

one can clear the denominators and obtain polynomials with even integer coeffi-cients. They can be reduced modulo arbitrary primes. As a consequence, the origi-nal curve defined in characteristics 0 is accompanied by a family of curves in primecharacteristics. They arise from reducing the coefficients of the defining polynomi-als.

Apparently a rational projective-algebraic curve encodes more information thanthe set of its complex points. This fact suggests that the basic objects of algebraic ge-ometry are not the point-sets but instead the defining polynomials, or more preciselythe corresponding ideal of polynomials.

The analogy between Riemann manifolds and projective-algebraic curves is farreaching. But nevertheless, both live in different categories: As a topological spacea Riemann manifold carries locally the Euclidean topology of C' R2. On the con-trary, a projective-algebraic variety carries the Zariski topology: Its closed subsetsare the zero sets of polynomials, i.e. the the algebraic subsets. Both topologies aredifferent in a qualitative aspect: E.g., in dimension = 1 proper closed subsets in theZariski topology are finite sets. And in general, the Zariski topology is not Hausdoff,in particular the concept of compacteness is not defined. Nevertheless the Zariskihas available analogoue concepts: Any projective-algebraic variety X is separared,i.e. the diagonal morphism ∆ : X → X ×X is a closed immersion. In addition, anymorphisms f : X → Y between projective-algebraic varieties is closed, i.e. it mapsZariski-closed subsets of X to Zariski-closed subsets of Y .

We use the notation k for a perfect field, i.e. a field with all finite field exten-sions separable. The algebraic closure of k is denoted k. Typical perfect fields in ourcontext are:

• Fields of characteristic = 0 like Q or C, more general number fields Q⊂ K ⊂C.

• Finite fields like Fp or Fpn .

A typical imperfect field is the field Fp(T ) of rational functions with coefficientsfrom Fp.

As a textbook for the present chapter we recommend [35]. A more general refer-ence to algebraic geometry is the classic [15].

3.1 Plane cubics

Many properties of elliptic curves derive from the Riemann-Roch Theorem. Its nu-merical results rely on the fact that elliptic curves have genus g = 1. As a first con-sequence any elliptic curve is a plane curve. It is the zero set of one polynomial


of degree = 3. Such polynomials are named Weierstrass polynomials. They are themain data structure to represent an elliptic curve.

Elliptic curves are distinguished from other projective-algebraic curves by theexistence of a group structure on its set of points. We introduce this group structurein two different ways. The first uses the Weierstrass polynomial, the second appliesdirectly the Riemann-Roch Theorem. Theorem 3.17 shows that both methods leadto the same result.

Definition 3.3 (Variety with structure sheaf). Consider a field k.

1. A projective-algebraic variety X is the zero set

X =V (a)⊂ Pn(k)

of a prime ideal a⊂ k[X0, ...,Xn] generated by homogeneous polynomials, i.e.

X := (x = (x0 : ... : xn) ∈ Pn(k) : f (x) = 0 for all homogeneous f ∈ a.

If the ideal a can be generated by polynomials with coefficients from k then X isdefined over k, denoted X/k.

2. The Zariski topology on X is the subspace topology induced from the Zariskitopology of Pn(k). Closed sets of Pn(k) are the projective-algebraic subvarieties.

3. The sheaf OX of regular functions on X is the functor

U 7→ OX (U) := f : U → k| f is regular , U ⊂ X affine,

with the restriction maps. A function f : U → k is regular if locally it has theform

f =gh

with polynomials g,h ∈ k[X1, ...,Xn],h without zeros.

4. The local ring OX ,x at a point x ∈ X is the ring of germs of regular functionsdefined in a neighbourhood of x ∈ X .

5. The projective-algebraic variety X ⊂Pn(k) is non-singular or smooth at a point x ∈ Xif the local ring OX ,x is a regular local ring, i.e. if

dim OX ,x = dimk(x) (mx/mx2) (Ring dimension equals tangent dimension).

Here k(x) := OX ,x/mx denotes the residue field at the point x.

6. A projective-algebraic curve C/k is a 1-dimensional, non-singular projective-algebraic variety defined over k. If

C ⊂ P2(k)

3.1 Plane cubics 63

then C/k is a plane projective-algebraic curve.

Definition 3.3 for X/k involves two fields:

• First, the field of definition k is not necessarily algebraically closed. It is the fieldwhere the coefficients of a set of generators of a may be taken from.

• Secondly, the algebraically closed field k is used to visualize the point set of theideal a⊂ k[X0, ...,Xn]. Also the condition on smoothness refers to k.

But the set X ⊂ Pn(k) is not the only point set attached to a. Definition 3.4 willdefine point sets X(K) attached to any intermediate field K between k and k.

We have included the requirement of non-singularity into our definition of aprojective-algebraic curve because all curves will be assumed smooth if not statedotherwise. Note that any plane projective algebraic curve is defined by a single ho-mogeneous polynomial. Its degree is the degree of C/k.

Definition 3.4 (K-valued points). Consider a projective-algebraic variety X/k with

X ⊂ Pn(k).

Then for any intermediate field k⊂ K ⊂ k the set of K-valued points of X is definedas

X(K) := X ∩Pn(K) = (x0 : ... : xn) ∈ Pn(k) : x ∈ X and xi ∈ K f or all i = 0, ...,n.

Definition 3.5 (Elliptic curve). An elliptic curve defined over k is a pair (E,O) with

• a projective-algebraic curve E/k of genus g = 1

• and a k-valued point O ∈ E

In Definition 3.5 “O” denotes the upper-case letter O, not the digit 0.

The genus of a projective-algebraic curve C is defined as

g = dimk H0(C,Ω 1C),

the dimension of the k-vector space of regular differential forms, compare Definition 2.11.


Note that there exist smooth projective-algebraic curves C/Q of genus g = 1without rational points, i.e. with C(Q) = /0. An example is the Selmer curve in P2(C)defined by the inhomogeneous equation

3X3 +4Y 3 +5Z3 = 0.

The fact g = 1 for an elliptic curve E allows a simple application of the Riemann-Roch theorem to compute the vector spaces of multiples of a divisor on E.

Lemma 3.6 (Divisors on an elliptic curve). Consider an elliptic curve E and adivisor D ∈ Div(E). Then

dim H0(E,OD) =

0 if deg D < 00 if deg D = 0, D not principal1 if deg D = 0, D principaldeg D if deg D > 0

Proof. An elliptic curve E has genus g = 1. Corollary 2.16 implies for the canonicaldivisor

deg K = deg Ω1E = 2g−2 = 0.

Hence Corollary 2.15 implies for any divisor D ∈ Div(E)

dim H0(E,OD)−dim H1(E,OD) = dim H0(E,OD)−dim H0(E,O−D+K) =

= 1−g+deg D = deg D

If deg D > 0 then deg (−D)< 0. As a consequence

dim H0(E,OD) =

0 if deg D < 0deg D if deg D > 0

The statement for deg D = 0 follows from the fact that a non-trivial line bundle hasno holomophic sections, q.e.d.

Proposition 3.7 (Rational functions and function field). Consider a projective al-gebraic variety X/k.

1. Any non-empty open subset U ⊂ X is dense in X with respect to the Zariskitopology.

2. Each ring OX (U),U ⊂ X affine, not empty, is an integral domain. All these inte-gral domains have the same quotient field. It is named the field k(X) of rationalfunctions of X or the function field of X.

3.1 Plane cubics 65

According to Proposition 3.7 for any rational function f ∈ k(X) exist a non-empty affine subset U ⊂ X and two regular functions g,h ∈ OX (U), h 6= 0, suchthat

f =gh.

The rational functions with all coefficients from k form the subfield k(X)⊂ k(X).

The following Lemma 3.8 will be used in the proof of Theorem 3.9. It indi-cates why the existence of a k-valued point is requested for an elliptic curve definedover k.

Lemma 3.8 (Divisors defined over a subfield). Consider a curve X/k and a divisor

D = ∑x∈X

nx(x) ∈ Div(X)

which is fixed under the canonical action of the Galois group Gk/k, i.e.

D = Dσ := ∑x∈X

nx(xσ ) f or all σ ∈ Gk/k.

Then the finite dimensional k-vector space

OD(X) = f ∈ k∗(X) : ( f )≥−D∪0

has a basis of rational functions from k(X).

For the proof see [35, II, Prop. 5.8].

Elliptic curves are by definition subvarieties of a projective space Pn(k). Theorem 3.9proves that they are even plane curves, i.e. that they embed as hypersurfacesinto P2(k).

Theorem 3.9 (Elliptic curves are plane cubics). For an elliptic curve(E,O) de-fined over k exist two rational functions x,y ∈ k(E) with poles only at O such thatthe map

φ : E→ P2(k), p 7→

(1 : x(p) : y(p)) p 6= O(0 : 0 : 1) p = O

induces an isomorphism onto the plane cubic C ⊂ P2(k), which is defined by ahomogeneous polynomial Fhom ∈ k[X0,X1,X2] of the form

Fhom(X0,X1,X2) = X0X22 +a1X0X1X2 +a3X2

0 X2− (X31 +a2X0X2

1 +a4X20 X1 +a6X3

0 )

with five suitable constants ai ∈ k, i = 1,2,3,4,6. In particular, C is defined over k.

Proof. i) We construct a very ample line bundle L =OD on E and define φ := φL .Therefore consider the divisor


D := 3(O) ∈ Div(E).

We claim: The line bundleL := OD ∈ Pic(E)

satisfies the ampleness conditions from Theorem 2.18.

The proof follows from the dimension formula in Lemma 3.6 because for allpoints p,q ∈ X

dim H0(E,OD−(p))= dim H0(E,OD)−1 and dim H0(E,OD−(p)−(q))= dim H0(E,OD)−2.

For n≥ 0

H0(E,On(O)) = f ∈ k∗(E) : f has pole at O of order at most = n∪0

and the dimension formula implies the proper inclusion

H0(E,O3(O))⊃ H0(E,O2(O))⊃ H0(E,O(O)) = H0(E,OE).

Hence a basis (1,x,y) of H0(E,L ) exists with

• y ∈ k(E) has a pole of order = 3 at O and no other pole,

• x ∈ k(E) has a pole of order = 2 at O and no other pole.

By assumption O ∈ E(k). Therefore the divisor D is defined over k. According toLemma 3.8 we may choose even x,y ∈ k(E). With respect to the basis (1,x,y) theline bundle L defines the closed embedding

φ := φL : E→ P2(k), p 7→ (1 : x(p) : y(p))

onto a curve C/k in P2(k). The family

(1, x, y, x2, xy, x3, y2)

of seven rational functions from the 6-dimensional vector space H0(E,O6(O)) islinearly dependent. Therefore the functions satisfy a linear relation with coefficientsfrom k. Among them the two functions x3 and y2 are the only ones with a pole atO ∈ E of order = 6. Hence in the linear relation their coefficients are not zero. Afterdivision we may assume that their coefficients are =±1 and

y2 +a1xy+a3y = x3 +a2x2 +a4x+a6

with coefficients ai ∈ k. This relation defines the inhomogeneous polynomial (Weier-strass polynomial)

F(X ,Y ) := Y 2 +a1XY +a3Y − (X3 +a2X2 +a4X +a6) ∈ k[X ,Y ]

and its homogenization, the polynomial Fhom ∈ k[X0,X1,X2] with

3.1 Plane cubics 67

Fhom(X0,X1,X2) := X0X21 +a1X0X1X2+a3X2X2

0 −(X31 +a2X2

1 X0+a4X1X20 +a6X3

0 ).

The inhomogeneous variables (X ,Y ) relate to the homogeneous variables (X0,X1,X2)according to

X =X1

X0and Y =

X2

X0.

The curve φ(E) is a subvariety of the 1-dimensional cubic hypersurface

C :=V (Fhom)⊂ p ∈ P2(k) : Fhom(p) = 0.

The cubic C intersects the line at infinity

P1(k)' (x0 : x1 : x2) ∈ P2(k) : x0 = 0

in the single point (0 : 0 : 1). Its affine part is the variety of the Weierstrass polyno-mial F(X ,Y ) ∈ k[X ][Y ]

Caff = (x0 : x1 : x2) ∈ P2(k) : Fhom((x0 : x1 : x2)) = 0 and x0 6= 0=

= (x,y) ∈ A2(k) : F(x,y) = 0=V (F).

The Weierstrass polynomial F ∈ k[X ,Y ], considered as polynomial in Y with coef-ficients from the ring k[X ], is irreducible: It does not split as

F(X ,Y ) = (Y +b1)(Y +b2)

with polynomials b1,b2 ∈ k[X ]. Therefore Caff and also C are irreducible. Hence thenon-constant regular map

φ : E→C

is surjective, cf. [15, Chap. II, Prop. 6.8] as an analogue of Proposition 2.4. As aconsequence φ(E) =C and E 'C over k.

ii) In order to show that φ is a morphism also in a neighbourhood of O choose anaffine neighbourhood U ⊂ E of O such that

x =fx

gxand y =

fy

gy

with regular functionsfx,gx, fy,gy ∈ OE(U)

satisfying

fx(O), fy(O) 6= 0, gx has a zero of order 2 at O, gy has a zero of order 3 at O.

Multiplying for p ∈ U \ O the homogenous components of φ(p) by gy(p) 6= 0implies


φ(p) = (1 : x(p) : y(p)) = (gy(p) : gy(p)/gx(p) : 1).

The singularity of φ at O is removable by defining

φ(O) := (0 : 0 : 1), q.e.d.

In the following we will identify the elliptic curve (E,O) defined over k with theplane cubic from Theorem 3.9. We say that (E,O), more precisely its affine part, isdefined over k by the cubic Weierstrass equation F(X ,Y ) = 0.

Definition 3.10 (Discriminant of a Weierstrass polynomial). Consider a Weier-strass polynomial

F(X ,Y ) = Y 2 +a1XY +a3Y − (X3 +a2X2 +a4X +a6) ∈ k[X ,Y ]

Its discriminant ∆F ∈ k is defined as

∆F :=−b22b8−8b3

4−27b26 +9b2b4b6

with

b2 = a21+4a2, b4 = 2a4+a1a3, b6 = a2

3+4a6, b8 = a21a6+4a2a6−a1a3a4+a2a2

3−a24.

Proposition 3.11 (Weierstrass equation in short form). Assume char k 6= 2,3 andconsider a plane elliptic curve (E/k,(0 : 0 : 1)) defined by a homogeneous cubicequation.

1. After applying a projective-linear coordinate transformation from PGL(3,k) theelliptic curve E is defined by a homogeneous polynomial of the form

Fhom(X0,X1,X2) = X0X2− (X31 +AX2

0 X1 +BX30 ) ∈ k[X0,X1,X2], A, B ∈ k.

Its affine part Eaff is defined by the Weierstrass polynomial in short form

F(X ,Y ) = Y 2− (X3 +AX +B) ∈ k[X ,Y ].

Its discriminant is∆F =−16 · (4A3 +27B2) ∈ k.

2. Any two Weierstrass polynomials F(X ,Y ) and F ′(X ′,Y ′) in short form of (E/k,(0 : 0 : 1))relate to each other by a linear coordinate change(

X ′

Y ′

)7→(

XY

)=

(u2 ·X ′+ r

u3 ·Y ′+ s ·u2 ·X ′+ t

)with constants

3.1 Plane cubics 69

u,r,s, t ∈ k,u 6= 0

The corresponding transformation of the discriminant is

∆F = u12 ·∆F ′ .

3. In particular, any two Weierstrass polynomials F(X ,Y ) and F ′(X ′,Y ′), whichdefine (E/k,(0 : 0 : 1)) in short form, relate to each other by a linear coordinatetransformation (

XY

)7→(

X ′

Y ′

):=(

u2 00 u3

)·(

XY

).

If

F(X ,Y ) = Y 2− (X3 +AX +B) and F ′(X ′,Y ′) = Y ′2− (X ′3 +A′X ′+B′)

thenA′ = u4 ·A, B′ = u6 ·B, ∆F ′ := u12 ·∆F .

and the j-invariant

jF := 1728 ·(4A)3

∆F= 1728 ·

(4A′)3

∆F ′=: jF ′

is independent from the choice of a Weierstrass polynomial in short form.

For the proof of Proposition 3.11 see [35]: For part 1) cf. Chap. III, §1, for part 2)cf. Chap. III, Prop. 3.1(b)

Proposition 3.11 shows that a coordinate transformation allows to eliminate all12th-power from the discriminant of a Weierstrass polynomial in short form. Ex-ample 3.22 will show why it is necessary to represent elliptic curves by a minimalWeierstrass polynomial in short form.

Definition 3.12 (Minimal Weierstrass polynomial). For an elliptic curve

E/Q= (E/Q,(0 : 0 : 1))

defined over Q a Weierstrass polynomial F ∈ Z[X ,Y ] in short form is minimal if thediscriminant ∆F ∈ Z is minimal, i.e. for all Weierstrass polynomials F ′ of E in shortform and for all primes p ∈ Z

ordp(∆F)≤ ordp(∆F ′).

Every elliptic curve defined over Q has a minimal Weierstrass polynomial F . Ingeneral, F itself is not uniquely determined, but its discriminant ∆F is unique. It isnamed ∆min(E).


Because minimality of the discriminant is a property referring to all primes of Zthat property is named global minimality in a more general context. For a numberfield k with class number = 1, in particular for k = Q, any elliptic curve definedover k has a global minimal model [35, Chap. VIII, Cor. 8.3].

See PARI-file Elliptic_curve_weierstrass_equation_12.

The discriminant ∆F ∈ k of a Weierstrass polynomial F(X ,Y ) ∈ k[X ,Y ] indicateswhether the projective-algebraic variety

X =V (Fhom)⊂ P2(k)

is non-singular or not. If ∆F 6= 0 then (C/k,(0 : 0 : 1)) is non-singular, i.e. an ellipticcurve.

For an elliptic curve (E/Q,(0 : 0 : 1)) the prime factors of ∆min(E) are theprimes p ∈ N with singular reduction mod p of E/Q, see Section 3.3.

Proposition 3.13 (Singularities of cubics). Assume k 6= 2,3 and consider a planecubic curve

C :=V (Fhom) =⊂ P2(k)

defined by homogenization of the Weierstrass polynomial in short form

F(X ,Y ) = Y 2− (X3 +AX +B) ∈ k[X ,Y ]

with A, B ∈ k and discriminant

∆F =−16 · (4A3 +27B2).

The curve C has at most one singular point. The curve

• is non-singular iff ∆F 6= 0

• has a node iff ∆F = 0 and A 6= 0

• has a cusp iff ∆F = A = 0.

For a proof of Proposition 3.13 see [35, Chap. III, Prop. 1.4]. See PARI-fileElliptic_curve_plot_06 for a graphical illustration of Proposition 3.13.The illustrations show the

• elliptic curves Y 2 = X3−3X +3,Y 2 = X3 +X ,Y 2 = X3−X

• and the singular cubics Y 2 = X3(cusp),Y 2 = X3 +X2(node).

3.2 The group structure 71

3.2 The group structure

In the present section we provide an elliptic curves (E,O) with the structure of anAbelian group with the point O as neutral element This properties distinguishescurves of genus = 1 from curves of any other genus.

Due to Theorem 3.9 and Proposition 3.11 we may assume that a given ellipticcurve defined over the field k with char k 6= 2,3 is a plane cubic in P2(k) defined bythe homogenization of a Weierstrass polynomial in short form

F(X ,Y ) = Y 2− (X3 +AX +B) ∈ k[X ,Y ].

We introduce the following notation: For two points P 6= Q ∈ E we denoteby L(P,Q) the line in Pn(k) passing through P and Q. The symbol L(P,P) denotesthe tangent to E at the point P.

Definition 3.14 (Composition Law). Consider an elliptic curve (E,O) definedover k. Then a law of composition

⊕ : E×E→ E,(P,Q) 7→ P⊕Q,

is defined by the following construction:

• Step 1: For arbitrary points P, Q ∈ E denote by R ∈ E the third point of intersec-tion from L(P,Q)∩E.

• Step 2: SetP⊕Q ∈ E

as the third point of intersection from L(R,O)∩E.


Fig. 3.1 Group law on an elliptic curve

Theorem 3.15 (Group structure: Geometric version). Assume chark 6= 2,3 andconsider an elliptic curve (E,O) defined over k. The composition

⊕ : E×E→ E


from Definition 3.14 is well-defined and has the following properties, in particularit defines an Abelian group on E:

i) For all P,Q ∈ E:

(P⊕Q)⊕R = O

ii) Neutral element: For all P ∈ E:

P⊕O = P

iii) Abelian: For all P,Q ∈ E:

P⊕Q = Q⊕P

iv) Inverse: For all P ∈ E exists a point P ∈ E satisfying

P⊕ (P) = O

If P = (x0 : x1 : x2) ∈ E then

P = (x0 : x1 :−x2).

v) Associativity: For all P,Q,S ∈ E:

(P⊕Q)⊕S = P⊕ (Q⊕S).

vi) For any intermediate fieldk ⊂ K ⊂ k

the group law ⊕ restricts to a group structure on E(K).

Proof. We use a simple version of Bezout’s theorem ([15, Chap. I, Cor. 7.8]), namely:Any line in P2(k) intersects the cubic E in three points. Apparently the statement isequivalent to the fact that any cubic polynomial has three zeros over an algebraicallyclosed field. Therefore a line passing through two points P,Q ∈ E intersects E in athird point R. In general R is not the sum P⊕Q but the construction of R is anintermediate step in the construction of the point P⊕Q. To obtain three points of in-tersection requires to consider points with coordinates from an algebraically closedfield.

Some remarks to fix the notation:

• Homogeneous coordinates (X0 : X1 : X2) on P2(k)

• Plane cubic E =V (Fhom)⊂ P2(k) with

Fhom(X0,X1,X2) = X0X2− (X31 +AX1X2

0 +BX30 ) ∈ k[X0,X1,X2]

homogenization of a Weierstrass polynomial F(X ,Y ) ∈ k[X ,Y ] in short form.


• Inhomogeneous coordinates around (1 : 0 : 0) ∈ P2(k) on the affine space

A2(k) = P2(k)\X0 = 0 :

X :=X1

X0and Y :=

X2

X0.

• Inhomogeneous coordinates around O = (0 : 0 : 1) =: ∞ ∈ P2(k) on the affinespace

P2(k)\X2 = 0 ' A2(k) :

V :=X0

X2and U :=

X1

X2.

Then E := E ∩X2 6= 0=V (F) with

F(U,V ) =V − (U3 +AUV 2 +BV 3)

• Tangent to E at ∞ is L∞ =V (G) with

G(U,V ) =∂ F∂U

(0,0) ·U +∂ F∂V

(0,0) ·V,

i.e.G(U,V ) =V.

HenceL∞∩ E = (u,0) : u3 = 0= (0,0)

with multiplicity = 3. The tangent to E at the point ∞ has no further point ofintersection with E.

i) We compute(P⊕Q)⊕R.

The third point of intersection from

L(P⊕Q,R)∩E

is R1 = O. Then L(R1,O) = L(O,O) is the tangent to E at ∞. The preliminary remarksimply

L(O,O)∩E = O.

Hence by construction(P⊕Q)⊕R = O.

ii) Denote by R the third point of intersection from

L(P,O)∩E.


The third point of intersection from L(R,O)∩E is P, because the two lines LP,O andLR,O are the same. Hence by construction

P⊕O = P.

iii) The proof is obvious by construction.

iv) Due to part ii) holds O = O. Assume

P = (1 : x1 : x2) 6= O.

Then L(P,Q) intersect E in the third point R = O. The line L(O,O) is tangent at E at O,hence it does not intersect E in a point different from O. Therefore P⊕Q = O.

v) To verify the law of associativity is a tedious task. One proof is to calculateusing coordinates on E as done in [35, Chap. III, Algorithm 2.3]. A second proofemploys a calculation with intersection numbers asin [12, Chap. III, Sect. 6, Prop. 4]. A third proof follows from Theorem 3.17.

vi) For two points P,Q ∈ E(K) the line L(P,Q) is defined over K. Hence R, its pointof intersection with E has coordinates from K. Analogously the line L(R,O) isdefined over K and intersects E in the point P⊕Q with coordinates from K. Inaddition, part iv) shows that P ∈ E(K) implies P ∈ E(K), q.e.d.

Definition 3.14 uses a geometric construction to introduce a group law on an el-liptic curve. The construction relies on the fact, that any line intersects the cubiccurve E in three points, counted with multiplicity. We now give a second construc-tion of the group structure. It links the points of E to the divisors of degree 0. Thesecond construction relies on the Riemann-Roch theorem and the fact, that an el-liptic curve has genus g = 1. The second construction does not make use of anygeometric argument.

Lemma 3.16 prepares the proof of Theorem 3.17. The lemma shows that apair (P,Q) of distinct points from E never defines the divisor

D := (P)− (Q)

of a rational function, though deg D = 0. Note that the lemma gives also a secondproof of Corollary 1.9.

Lemma 3.16 (No single pole of order one). Consider an elliptic curve (E,0) de-fined over a field k. If two points P,Q ∈ E satisfy

(P)− (Q) = ( f )

for a rational function f ∈ k(E)∗, then P = Q.


Proof. Consider the divisor D := (Q) ∈Div(E) with degree deg D = 1. Lemma 3.6implies

dim H0(E,OD) = 1.

The inclusionk ⊂ H0(E,OD)

impliesk = H0(E,OD).

Hence f is a constant function and ( f ) = 0 ∈ Div(E), which implies P = Q, q.e.d.

Lemma 3.16 shows: The obstructions for a divisor D ∈ Div0(E) to be principalare divisors of type (P)− (Q) or even of type (P)− (O). A divisor D ∈ Div0(E) isthe divisor of a rational function up to a divisor of type

(P)− (O).

Because the point P ∈ E is uniquely determined by D, the obstruction is expressedby P alone.

Theorem 3.17 (Group structure: Version with divisor class group). Consider anelliptic curve (E,O) defined over a field k.

i) For any divisor D ∈ Div0(E) a unique point P ∈ E exists with

D∼ (P)− (O).

Denote the corresponding map by

σ : Div0(E)→ E, D 7→ P.

ii) The map σ is surjective.

iii) For two divisors D1,D2 ∈ Div0(E)

σ(D1) = σ(D2) ⇐⇒ D1 ∼ D2.

Therefore σ induces a bijective map - also denoted by σ -

Cl0(E) '−→ E.

The inverse of σ is the map

ρ : E '−→Cl0(E), P 7→ [(P)− (O)].

iv) The maps


σ : Cl0(E) '−→ E and ρ : E '−→Cl0(E)

are isomorphisms between the additive group of classes of divisors of degree = 0on E and the group (E,⊕) from Definition 3.14.

Proof. i) The divisor D+(O) has degree = 1. Lemma 3.6 implies

dim H0(E,OD+(O)) = 1.

We choose a non-zero rational function f ∈ k(E)∗, f 6= 0, with divisor satisfying

( f )≥−D− (O).

Becausedeg ( f ) = 0 but deg(−D− (O)) =−1,

a point P ∈ E exists with ( f ) =−D− (O)+(P), i.e.

D∼ (P)− (O).

If also D∼ (Q)− (O) for a point Q ∈ E, then

0 = D−D∼ (P)− (Q).

Hence for a suitable rational function g ∈ k∗(E)

(P)− (Q) = (g).

Lemma 3.16 implies P = Q.

ii) Consider an arbitrary point P ∈ E. Set

D := (P)− (O) ∈ Div0(E).

Then σ(D) is the unique point Q ∈ E with

D∼ (Q)− (O).

Therefore σ(D) = P, which proves the surjectivity of σ .

iii) Set Pj := σ(D j), j = 1,2, i.e. D j ∼ Pj− (O). Then

D1−D2 ∼ (P1)− (P2).

On one hand, if σ(D1) = σ(D2) then P1 = P2. Hence D1−D2 ∼ 0 i.e.

D1 ∼ D2.

On the other hand, if D1 ∼ D2 then


(P1)− (P2)∼ 0

which implies P1 = P2 according to Lemma 3.16.

iv) Consider two arbitrary but fixed points P,Q ∈ E. We claim:

ρ(P⊕Q) = ρ(P)+ρ(Q) ∈Cl(E).

The idea is now to translate classical geometry into algebrai geometry. Starting at thegeometric construction from Definition 3.14 we represent the distinguished pointsof E as divisors of suitable rational functions on E. The translation proceeds alongthe following steps:

• Convert the information about the points into information about lines: Considerthe line L passing through P and Q and its third point of intersection R ∈ L∩Eand the line L′ passing through R and 0 and its third point of intersection

P⊕Q ∈ L′∩E.

• Convert the lines into linear homogeneous polynomials: Denote by

Ghom(X0,X1,X2) ∈ k[X0,X1,X2]

the linear homogeneous polynomial with L =V (Ghom) and by

G′hom(X0,X1,X2) ∈ k[X0,X1,X2]

the linear homogeneous polynomial with V (G′hom) = L′. Eventually the line atinfinite distance

L∞ := (x0 : x1 : x2) ∈ P2(k) : x0 = 0

is L∞ = V (X0). These three lines incorporate the gemometrically constructedgroup structure of E.

• Construct suitable rational functions from these linear polynomials: Consideron E the rational functions

Ghom

X0,

G′hom

X0∈ k(E)∗.

• Determine the divisors of these rational functions: The rational functionGhom

X0has zeros of order = 1 at the points P, Q, R of its affine part and at infinitedistance a pole of order = 3 at O . Hence its divisor is(

Ghom

X0

)= (P)+(Q)+(R)−3(O).


Similar the rational functionG′hom

X0has the divisor

(G′hom

X0

)= (R)+(O)+(P⊕Q)−3(O).

Hence

0∼

(G′hom

Ghom

)=

(G′hom

X0

)−

(Ghom

X0

)=

= (P⊕Q)− (O)−(P)− (O)−(Q)− (O) ∈ Div0(E).

As a consequence

0 = ρ(P⊕Q)−ρ(P)−ρ(Q) ∈Cl0(E)

i.e.ρ(P⊕Q) = ρ(P)+ρ(Q) ∈Cl0(E), q.e.d.

Corollary 3.18 (Abel’s Theorem for elliptic curves). Consider an elliptic curve (E,O)defined over k. A divisor

D = ∑P∈E

nP(P) ∈ Div(E)

is a principal divisor iff ⊕P∈E

[nP]P = 0 ∈ E.

Here the last equation refers to the group structure ⊕ on E: [k]P denotes taking k-times the sum of a point P ∈ E if k ≥ 0 and taking (−k)-times the sum of P ∈ Eif k ≤ 0.

Proof. Because deg D = 0 we have

D = D− ∑P∈E

nP(O) = ∑p∈E

np((P)− (O)) ∈ Div0(E)

Due to Theorem 3.17, part v)

D principal ⇐⇒ D∼ 0∈Div0(E) ⇐⇒ σ(D)=O∈E ⇐⇒ σ(∑p∈E

np((P)−(O)))=O∈E

⇐⇒⊕p∈E

[nP](σ((P)− (O)) = O ⇐⇒⊕p∈E

[nP]P = 0 ∈ (E,⊕), q.e.d.


Remark 3.19 (The theorems of Mordell and Faltings).

1. Consider an elliptic curve (E,O) defined over Q. Mordell proved: The Abeliangroup E(Q) of Q-valued points of E is finitely generated, see [35, Chap. VIII,Theor. 4.1]. As a consequence, E(Q) decomposes as the product of an Abeliantorsion group and a free Abelian group of finite rank

E(Q) = Etors×Zr.

Many open questions are related to the rank r.

2. Elliptic curves have genus g = 1. Curves of genus g > 1 do no longer carrya group structure. Mordell conjectured: Curves C/Q with genus g > 1 haveonly finitely many points with rational coordinates, i.e. card C(Q) is finite. TheMordell-conjecture has been proved by Faltings, see [37].

Any elliptic curve E/k ⊂ P2(C) is not only a projective algebraic curve. Whenequipped with the Euclidean topology it defines also a compact Riemann manifold.To indicate the different point of view we denote the latter by Ean.

Theorem 3.20 (Isomorphism between tori and elliptic curves). Consider a torus T := C/Λ

with latticeΛ = Zω1 +Zω2.

Setg2 := 60 ·GΛ ,4, g3 := 140 ·GΛ ,6,

(Lemma 1.12), and denote by ℘=℘Λ the Weierstrass function of Λ . The map

φ : T → P2an(C), p 7→

(1 :℘(p) :℘ ′(p)) if p 6= 0(0 : 0 : 1) if p = 0

maps the torus T biholomorphically onto the elliptic curve Ean ⊂ P2an(C) defined by

the cubic polynomial

Fhom(X0,X1,X2) = X22 X0− (4X3

1 −g2 ·X1X20 −g3 ·X3

0 ) ∈ C[X0,X1,X2],

the homogenization of

F(X ,Y ) = Y 2− (4X3−g2X−g3) ∈ C[X ,Y ].

Moreover, φ is an isomorphism of groups between (T,+) and (Ean,⊕).

Note. The coordinate transformation

X ′ := X , Y ′ := 2 ·Y


transforms the inhomogeneous polynomial F(X ,Y ) to the Weierstrass polynomial

F ′(X ′,Y ′) = Y ′2− (X ′3− (1/4) ·g2 ·X ′− (1/4) ·g3).

Proof. The proof of Theorem 3.20 makes use of the properties of ℘ and ℘ ′ fromSection 1.2.

i) Biholomorphic map: The proof relies on the Riemann-Roch theorem. Its state-ment is literally the same for both complex analysis and algebraic geometry, cf.Proposition 3.1. Therefore we can copy the corresponding part of the proof of The-orem 3.9 with the substitutions from Proposition 3.1. Hereby, the two rational func-tions

x, ,y ∈ k(E)∗

can be made explicit by the two meromorphic functions

℘, ℘′ ∈M (T ).

ii) Isomorphism of groups: For the sake of clarity we distinguish the group composition ⊕/on the elliptic curve Ean from the group composition +/− on the torus T . As a short-hand we denote the image of a point p ∈ T by the corresponding upper-case letter

P := φ(p) ∈ Ean.

We prove: For all p, q ∈ TP⊕Q = φ(p+q).

Assume first p 6= q and denote by L(P,Q) ⊂ P1(C) the line passing through P and Q.The line L(P,Q) has the equation

c0X0 + c1X1 + c2X2 = 0

with constants c0,c1,c2 ∈ C. We attach to the pair (p,q) the meromophic functionon T

h(P,Q) : T → P1(C), z 7→ c0 + c1 ·℘(z)+ c2 ·℘ ′(z).

The zeros of h(P,Q) correspond to the points of intersection from LP,Q∩Ean:

z ∈ T : h(P,Q)(z) = 0 '−→ L(P,Q)∩Ean,z 7→ φ(z).

With respect to L(P,Q) we distinguish the following cases :

• If c2 6= 0, i.e. the line L(P,Q) does not pass through O = (0 : 0 : 1), then the mero-morphic function h(P,Q) has a pole of order 3 at z = 0 ∈ T . On one hand, thefunction has three zeros p,q,r, see Proposition 1.19, which satisfy

p+q+ r = 0 i.e. p+q =−r

according to Theorem 1.20.


On the other hand, according to Theorem 3.15 the three intersection points P,Q,Rfrom L∩Ean satisfy

(P⊕Q)⊕R = 0 i.e. P⊕Q =R.

Using that the function ℘ is even and ℘ ′ is odd we obtain

P⊕Q =R =(1 :℘(r) :℘′(r)) = (1 :℘(r) :−℘

′(r)) =

= (1 :℘(−r) :℘′(−r)) = φ(−r) = φ(p+q).

• If c2 = 0 but c1 6= 0, then the third intersection point of L(P,Q)∩Ean is the point

O = (0 : 0 : 1) = φ(0)

at infinite distance. Therefore

P⊕Q⊕O = O i.e. P⊕Q = O.

The functionh(P,Q) = c0 + c1 ·℘

has two zeros pi 6= 0, i = 1,2. They satisfy

p2 =−p1 i.e. p1 + p2 = 0

because ℘ is an even function. We obtain

φ(p1 + p2) = φ(0) = O = P⊕Q.

• If c1 = c2 = 0 then L(P,Q) is the line at infinite distance. This case is excluded: p 6= qimplies

P := φ(p) 6= Q := φ(q),

but the line at infinite distance contains only one single point from Ean, namelythe point O.

So far we have proved the claim for the case of two points p 6= q. From this followsthe claim also for the case p = q by a limit argument because all involved maps arecontinuous, q.e.d.

Conversely, we will show in Theorem 4.18 that any elliptic curve can be obtainedfrom a torus with a normalized period lattice via a biholomorphic map Φ accordingto Theorem 3.20.

3.3 Reduction mod p 83

3.3 Reduction mod p

Reduction mod p is a powerful tool to investigate the information encoded in anelliptic curve defined over Q. The method initiates the arithmetic theory of ellipticcurves.

Consider an elliptic curve (E,O) defined over Q by a homogeneous polynomial

Fhom ∈Q[X0,X1,X2]

being the homogenization of a Weierstrass polynomial

F(X ,Y ) ∈Q[X ,Y ].

As noted subsequently to Definition 3.12 we may assume that F is even a minimalWeierstrass polynomial in short form

F(X ,Y ) ∈ Z[X ,Y ].

ThenFhom ∈ Z[X0,X1,X2].

Because the coefficients of Fhom are integers, for any prime p reducing mod p thesecoefficients provides a homogeneous polynomial

Fhom,p ∈ Fp[X0,X1,X2].

It defines the projective variety

Ep := (x0 : x1 : x2) ∈ P2(Fp) : Fhom,p(x0,x1,x2) = 0 ⊂ P2(Fp).

Definition 3.21 (Stable, semistable, unstable). Consider an elliptic curve (E,O)over Q. At a prime p ∈ Z the elliptic curve is

1. stable (good reduction) if (Ep,O) is non-singular, i.e. an elliptic curve over Fp.

2. semistable (bad reduction of multiplicative type) if (Ep,O) is singular with anode. If both tangents at the singularity have slope in Fp, then the reduction is asplit reduction, otherwise a non-split reduction.

3. unstable (bad reduction of additive type) if (Ep,O) is singular with a cusp.

Note. According to Proposition 3.13 the distinction between stable and not-stableat p depends on the reduction of the minimal discriminant.


The following example 3.22 shows why it is necessary to study elliptic curves byconsidering minimal Weierstrass equations.

Example 3.22 (Minimal Weierstrass polynomial and reduction mod p). Consider theelliptic curve (E,O) defined over Q by the inhomogeneous Weierstrass polynomial

F(X ,Y ) = Y 2− (X3 +15.625)

with discriminant∆F =−24 ·33 ·512 ∈ Z.

Note 15.625 = 56.

i) The Weierstrass polynomial F ∈ Z[X ,Y ] is in short form but it is not minimal.Reducing its coefficients mod p = 5 gives the Weierstrass polynomial

Fp(X ,Y ) = Y 2−X3 ∈ Fp[X ,Y ]

with discriminant∆Fp = 0 ∈ Fp.

Hence the Weierstrass polynomial Fp defines a singular cubic with a cusp. Note thatthe discriminant ∆F contains the 12th-power factor u = 5.

ii) The coordinate transformation with the matrix(1/u2 0

0 1/u3

)transforms F(X ,Y ) to the minimal Weierstrass polynomial

F ′(X ′,Y ′) = Y ′2− (X ′3 +1)

with discriminant∆F ′ =−24 ·33 ∈ Z.

Reducing the coefficients of F ′ mod p = 5 creates the Weierstrass polynomial

F ′5(X′,Y ′) = Y 2− (X ′3 +1)

with discriminant∆F ′5

= 3 ∈ F5.

The reduction of E mod p = 5 defines an elliptic curve E5 over F5.

Remark 3.23 (Hasse estimate).


1. One of the most interesting properties of an elliptic curve E/Q is the function,which attaches to each prime p of good reduction the order of the groups Ep(Fpn),n ∈ N.These groups collect those points from Ep ⊂ P2(F p) with all coordinates in thefinite fields Fpn . We recall the following result of Hasse for the case n = 1:∣∣1+ p− card Ep(Fp)

∣∣≤ 2 ·√p.

For a proof see [18, Theor. 10.5].

See the PARI-file Elliptic_curve_Hasse_estimate_05 for a numeri-cal example.

Hasses result had been conjectured by Artin. The value

1+ p

for comparison can be motivated as follows: The field Fp has p elements. AWeierstrass equation

Y 2 = X3 +AX +B

is quadratic in Y . Hence for each x∈Fp there exist at most two solutions (x,y) ∈ A2(Fp).Taking into account the point at infinity (0 : 0 : 1) ∈ Ep(Fp) gives the estimate∣∣card Ep(Fp)

∣∣≤ (1+2p).

In the average, i.e. by choosing the Weierstrass polynomial at random, there isa 50% chance to find a solution (x,y) at all. Therefore 1+ p is a plausible averagefor card Ep(Fp).

2. One defines for any prime p the difference

ap(E) := 1+ p− card Ep(Fp)

between the average number and the actual number of Fp-valued points of E.The sequence

(ap(E))p prime

will turn out to be a fundamental characteristic of the elliptic curve E, seeRemark 3.25. These numbers determine also the cardinalities card E(Fpn) for n≥ 2.

The local information about an elliptic curve E/Q, which is stable at a prime p, isencoded by the ζ -function Z(Ep,T ) of the reduction Ep. Therefore we first considerelliptic curves defined over the finite field Fp.

Definition 3.24 (Zeta-function). Consider an elliptic curve E/Fp defined over theprime field Fp. The ζ -function of E is the generating formal power series of thesequence (card E(Fpn))n∈N∗


Z(E,T ) :=∞

∑n=1

card E(Fpn) ·T n

n∈Q[[T ]].

Remark 3.25 (Properties of the Zeta-function). The ζ -function of an elliptic curve E/Fp, p prime,has the following properties:

1. Rationality: The ζ -function is a rational function, more precisely:

Z(E,T ) =L(E,T )

(1−T ) · (1− p ·T )∈ Z(T )

with the quadratic polynomial

L(E,T ) = 1−ap(E) ·T + p ·T 2 ∈ Z[T ].

2. Functional equation: The ζ -function satisfies the functional equation

Z(E,1/(pT )) = Z(E,T ).

3. Riemann hypothesis: Both complex zeros α, β ∈ C of L(E,T ) have modulus

|α|= |β |=√p.

The proof of these properties is due to Weil, [35, Chap. V, Theor. 2.2]. He alsoproved an analogue for projective-algebraic curves of arbitrary genus. Morevoer,Weil made a conjecture for the general case of higher-dimensional smooth projective-algebraic varieties. Deligne proved the Weil conjecture, cf. [15, Appendix C].

The L-series L(E) of an elliptic curve E/Q collects for each prime p ∈ Z thelocal information about the reduction Ep. For stable primes of E this information isthe quadratic polynomial L(Ep,T ) from the ζ -function of Ep. For non-stable primesof E information is added which characterizes the type of singularity.

Definition 3.26 (L-series of an elliptic curve). Consider an elliptic curve E/Q de-fined over Q. For each prime p set

Lp(T ) :=

L(Ep,T ) E stable at p1−T E semistable at p with split multiplicative reduction1+T E semistable at p with non-split multiplicative reduction1 E unstable at p

The L-series of E is the function


L(E) : s ∈ C : Re s > 3/2→ C, L(E,s) := ∏p prime

1Lp(1/ps)

.

Note: The absolute convergence of the L-series follows from Hasses estimation inRemark 3.23 and the theory of Euler products, see [18, Chap. X, Cor. 10.6]. HenceL(E) is a holomorphic function. Current research deals with the task of extendingthe domain of definition for E/K with K a number field, see [35, App. C, § 16], [13].

Chapter 4Modular forms of the modular group

Chapter 1 investigated elliptic functions. Elliptic functions are those meromorphicfunctions on C which are invariant under the left operation on C of the discreteAbelian group

Λ = Zω1 +Zω2.

We proved that the field M (Λ) of elliptic functions is determined by the Weierstrass ℘-functionof Λ , namely

M (Λ) = C(℘Λ )[℘′

Λ ].

Similar to the groups Λ we now consider the modular group

Γ := SL(2,Z)

which is also discrete but no longer Abelian. We proved in Chapter 2 that Γ actsproper on the upper half-plane H. Moreover this actions extends to H∗. We showedthat the orbit space

X := Γ \H∗

is a compact Riemann surface, more precisely

X ' P1(C).

An obvious question asks for the meromorphic functions on the upper half-plane,invariant under the left operation of the modular group. The complete answer isgiven by the modular invariant j.

The modular invariant is the most prominent example of the larger class of mero-morphic functions on H which satisfy a certain symmetry with respect to the mod-ular group and extend meromorphically to ∞. The homolomorphic functions fromthis class are named modular forms.

In this chapter we denote the modular group by Γ := SL(2,Z).

89

90 4 Modular forms of the modular group

4.1 Eisenstein series

Modular forms are holomorphic functions on H which transform in a simple mannerunder the operation of Γ . These functions have a discrete non-Abelian group ofsymmetries. In particular, they are invariant under the translation

T :=(

1 10 1

)∈ Γ .

Holomorphic functions with period = 1 develop into a convergent Fourier series

f (τ) =∞

∑n=−∞

an ·qn, q = e2πi·τ ,

similar to the trigonometric function

cos(2πz) =12(e2πi·z + e−2πi·z).

The holomorphic mapH→ ∆

∗,τ 7→ e2πiτ ,

surjects to the punctured unit disk

∆∗ := z ∈ C : 0 < |z|< 1.

A meromorphic function on H with period = 1 induces a corresponding meromor-phic function f on ∆ ∗ by the definition

f (q) := f (τ), q = e2πi·τ .

Definition 4.1 (Fourier expansion around ∞). Consider a meromorphic functionf on H invariant with respect to the operation of T and the induced meromorphicfunction f on ∆ ∗ with

f (e2πiτ) := f (τ).

1. The function f is named meromorphic at ∞

• if f has an isolated singularity at 0 and• if this singularity is at most a pole.

2. If f is meromorphic at ∞ then the induced function f has a Laurent expansionaround 0

f (q) =∞

∑n=n0

an ·qn, q = e2πiτ ,

with a suitable n0 ∈ Z. The elements (an)n≥n0 are the Fourier coefficients of f .3. A function f , meromorphic at ∞, is named holomorphic at ∞ if n0 ∈ N, i.e.

if 0 ∈ ∆ is a removable singularity of f . In this case one defines

4.1 Eisenstein series 91

f (∞) := f (0).

Note: For a meromorphic function f on H, invariant with respect to T :

f holomorphic at ∞ ⇐⇒ limIm τ→∞

f (τ)<∞ ⇐⇒ ∃ R> 0 : | f (τ)| bounded for Im τ ≥R.

Modular forms are not necessarily invariant under the operation of the modulargroup, but they multiply with an automorphy factor which is a holomorphic functionwithout zeros.

Definition 4.2 (Automorphic and modular forms). Consider an integer k ∈ Z anda meromorphic function f on H.

1. The function f is weakly modular of weight k if for all γ ∈ Γ and for all τ ∈H

1(cτ +d)k · f (γ(τ)) = f (τ), γ =

(a bc d

).

2. The function f is an automorphic form of weight k if f

• is weakly modular of weight k• and is meromorphic at ∞.

3. The function f is a modular form of weight k if f

• is weakly modular of weight k,• is holomorphic on H,• and is holomorphic at ∞.

4. A modular form f of weight k satisfying f (∞) = 0 is a cusp form of weight k.

5. An automorphic function is an automorphic form of weight = 0, i.e. satisfying f γ = f .Similarly, a modular function is a modular form of weight = 0.

On denotes byAk(Γ )⊃Mk(Γ )⊃ Sk(Γ )

the vector spaces of automorphic forms of weight k, the subspace of modular forms,and the subspace of cusp forms.

Note that Definition 4.2 uses the weight convention from [7, Chap. 1.1]. It differsfrom Serre’s convention in [33, Chap. VII, §2] by the factor 2. In the literature alsothe notation to discriminate between modular functions and modular forms is notuniform.

According to Definition 4.2, a modular form of weight k > 0, when considereda function f on H, is not invariant under the operation of the modular group. But


for even k > 0 one easily finds a related object which is actually invariant: Theseinvariant objects are holomorphic differential forms on H.

Proposition 4.3 (Modular forms are invariant differential forms).

1. The Γ -left operation on H induces the Γ -left operation on H0(H,Ω 1H)

ΦΩ : Γ ×H0(H,Ω 1H)→ H0(H,Ω 1

H), (γ,ω) 7→ γ∗(ω),

withγ∗( f ·dτ) := ( f γ) ·dγ.

A differential formω = f ·dτ ∈ H0(H,Ω 1

H)

is invariant under ΦΩ if and only if f is weakly modular of weight = 2 andholomorphic.

2. More general: For any even k ∈ N denote by

Ωk/2H := (Ω 1

H)⊗k/2

the (k/2)-th tensor power. A differential form

ω = f ·dτ⊗k/2 ∈ H0(H,Ω

⊗k/2H )

is invariant underΦ

Ω⊗k/2 := (ΦΩ )⊗k/2,

the (k/2)-th tensor power of ΦΩ , if and only if f is weakly modular of weight kand holomorphic.

Proof. 1) For

γ =

(a bc d

)∈ Γ

we compute

(dγ)(τ) = d

(aτ +bcτ +d

)=

a(cτ +d)− (aτ +b)c(cτ +d)2 ·dτ =

1(cτ +d)2 ·dτ

Therefore

ΦΩ (γ, f ·dτ) = ( f (γ(τ)) ·d(γ(τ)) = f (τ) · (cτ +d)2 ·1

(cτ +d)2 ·dτ

= f (τ) ·dτ.

2) For even k > 2 the proof is analogous, q.e.d.


Remark 4.4 (Factor of automorphy). The following definition is slightly more gen-eral than needed in the present context. We consider the action of

GL(2,R)+ := A ∈ GL(2,R) : det A > 0

on the upper half-plane

ΦGL+ : GL(2,R)+×H→H,(A =

(a bc d

),τ) 7→ A(τ) :=

aτ +bcτ +d

,

which extends the operation Φ of the modular group from Definition 2.26.

1. The maph : GL(2,R)+×H→H, h(A,τ) := cτ +d,

is named the automorphy factor of ΦGL+ .

2. We denote by M (H) the complex vector space of meromorphic functions on H.For a matrix A ∈ GL(2,R)+ and an integer k ∈ Z we define the linear operator

[A]k : M (H)→M (H), f 7→ f [A]k,

withf [A]k(τ) := (det A)k/2 ·h(A,τ)−k · f (A(τ)).

Employing the new notation, the condition of weak modularity of weight k fromDefinition 4.2 reads:

f [γ]k = f , γ ∈ Γ .

3. The factors of automorphy satisfy the composition rules

•h(B ·A,τ) = h(B,A(τ)) ·h(A,τ)

•[B ·A]k = [B]k [A]k.

Note. The latter operators apply to a function written on their left-hand side.

Proof. i) For A,B ∈ GL(2,R)+ and τ ∈H:

A ·(

τ

1

)=

(A(τ)

1

)·h(A,τ).

Here we distinguish between the matrix product of A with a vector on the left-hand side and the fractional linear transformation A(τ) on the right-hand side.

Applying this identity for the product B ·A

(B ·A) ·(

τ

1

)=

((B ·A)(τ)

1

)·h(B ·A,τ).


Left-hand side:

B ·(

A ·(

τ

1

))= B ·

((A(τ)

1

)·h(A,τ)

)=

((B(A(τ))

1

)·h(B,A(τ)) ·h(A,τ).

Equating with the right-hand side gives for the lower row

h(B ·A,τ) = h(B,A(τ)) ·h(A,τ),

hence for the upper row(B ·A)(τ) = B(A(τ)).

ii) Starting with the right-hand side we compute

( f [B]k)[A]k(τ) = ( f [B]k)(A(τ)) ·h(A,τ)−k√

det A =

= f (B(A(τ))) ·h(B,A(τ))−k ·h(A,τ)−k ·√

det B ·√

det A =

= f ((B ·A)(τ)) ·h(B ·A,τ)−k ·√

det (B ·A) = f [B ·A]k(τ),

using the result from part i) to obtain the penultimate equality, q.e.d.

As a consequence of Theorem 2.31 and Remark 4.4 it is sufficient to require inDefinition 4.2 the transformation law only for the two generators S,T of Γ .

Modular forms f are distinguished by their two properties

1. On one hand, they have the discrete group SL(2,Z) as group of symmetries.

2. On the other hand, they expand into a Fourier series around the point ∞.

The interplay of these two properties is coined “The magic of modular forms” [39].Therefrom a series of remarkable identities between the Fourier coefficients of mod-ular forms arises. In many cases these identities imply interesting statements aboutarithmetic functions. We will see examples in Chapter 5 and Chapter 6.

Remark 4.5 (Modular functions of odd weight). A modular function of odd weight k ∈ Zsatisfies for all τ ∈H and for the particular element

γ =

(−1 0

0 −1

)∈ Γ

the equationf (γ(τ)) = f (τ) = (−1)k · f (τ).

Therefore f = 0 is the only modular function of odd weight.


We will now construct modular forms Gk of even weight k ≥ 4 and a distin-guished holomorphic function G2, which behaves in a different way under symme-tries. To achieve the construction we need the convergence of certain infinite series.Lemma 4.6 provides the necessary prerequisites.

Lemma 4.6 (Some convergent series). Assume k ∈ N. For τ ∈ C set q := e2πi·τ .

1. For k ≥ 2 and Im τ 6= 0

∑n∈Z

1(τ +n)k =

(−2πi)k

(k−1)!·

∞

∑r=1

rk−1 ·qr Im τ > 0

(−1)k ·∞

∑r=1

rk−1 ·q−r Im τ < 0

2. For Im τ > 0 the double series

∞

∑m=1

(∞

∑d=1

d ·qmd

)

is absolutely convergent.

3. For even k ≥ 2, Im τ > 0,

∑m∈Z∗

(∑n∈Z

1(mτ +n)k

)=

2 (2πi)k

(k−1)!·

∞

∑n=1

σk−1(n) qn.

Here for any k ≥ 2 the arithmetic function

σk−1 : N→ N

has the valuesσk−1(n) := ∑

d|ndk−1, n ∈ N,

the sum of the (k−1)-powers of the non-negative integer divisors of n.

Proof. 1. i) We first consider the case Im τ > 0. We expand the cotangent functioninto a partial fraction

π · cot(πτ) =1τ+

∞

∑m=1

(1

τ +m+

1τ−m

)

and also into a Fourier series

π · cot(πτ) = πcos(πτ)

sin(πτ)=−πi−

2πi1−q

=−πi−2πi∞

∑m=1

qm.


Here the second equality results from the Euler formulas

cos πτ =12(eiπτ + e−iπτ) and sin πτ =

12i(eiπτ − e−iπτ)

which imply

cot(πτ) = i ·eiπτ + e−iπτ

eiπτ − e−iπτ= i ·

q+1q−1

= (−i) ·1+q1−q

The third equality uses

1+q1−q

= (1+q)∞

∑m=0

qm = 1+2∞

∑m=1

qm

The geometric series is absolutely convergent because |q|< 1 due to τ ∈H.

Equating both expansions gives

1τ+

∞

∑m=1

(1

τ +m+

1τ−m

)=−πi−2πi

∞

∑m=1

qm,

and successive (k−1)-times differentiation with respect to τ shows

∑m∈Z

1(τ +m)k =

(−2πi)k

(k−1)!·

∞

∑m=1

mk−1 qm.

ii) For Im τ < 0 we first use

cot(πτ)= i ·eiπτ + e−iπτ

eiπτ − e−iπτ= i ·

1+(1/q)1− (1/q)

= i(1+(1/q)) ·∞

∑m=1

q−m = i+2i ·∞

∑m=1

q−m

Equating the partial fraction expansion and the Fourier expansion gives

1τ+

∞

∑m=1

(1

τ +m+

1τ−m

)= πi+2πi

∞

∑m=1

q−m,

and successive (k−1)-times differentiation with respect to τ shows

∑m∈Z

1(τ +m)k = (−1)1+(k−1) (−2πi)k

(k−1)!·

∞

∑m=1

mk−1 q−m.

2. For any complex number x ∈ C with |x| < 1 the geometric series is absolutelyconvergent satisfying

∞

∑d=0

xd =1

1− x


Taking the derivative and multiplying by x shows

∞

∑d=1

d · xd =x

(1− x)2.

For m≥ 1 set x = qm with |q|= e−2π·Im τ < 1 gives

∞

∑m=1

(∞

∑d=1

d ·qmd

)=

∞

∑m=1

(∞

∑d=1

d · (qm)d

)=

∞

∑m=1

qm

(1−qm)2.

The last series is absolutely convergent because for fixed q

limm→∞|1−qm|2 = 1.

3. We split the summation ∑m∈Z∗

into ∑m<0

and ∑m>0

. Part 1) justifies the second equal-

ity sign:

∑m∈Z∗

(∑n∈Z

1(mτ +n)k

)=

∑m<0

(∑n∈Z

1(mτ +n)k

)+ ∑

m>0

(∑n∈Z

1(mτ +n)k

)=

(2πi)k

(k−1)!·

(∑

m<0

(∞

∑r=1

rk−1 ·q−rm

)+ ∑

m>0

(∞

∑r=1

rk−1 ·qrm

))=

= 2 ·(2πi)k

(k−1)!·

∞

∑m=1

(∞

∑r=1

rk−1 ·qrm

).

For k ≥ 4 any rearrangement is permitted. For k = 2 the last series is absolutelyconvergent according to part 2). We rearrange the series by changing the indicesof summation from (m,r) to (n = rm,d|n):

∞

∑m=1

(∞

∑r=1

rk−1 ·qrm

)=

∞

∑n=1

(∑d|n

dk−1 ·qn

)=

∞

∑n=1

σk−1(n) ·qn, q.e.d.

Theorem 4.7 (Eisenstein series, k ≥ 4). Consider an even integer k > 2. Then:

1. The Eisenstein series

Gk(τ) := ∑(c,d)∈Z2

′ 1(cτ +d)k

is absolutely and compactly convergent for τ ∈H.

The series defines a modular form Gk of weight k with


Gk(∞) = 2ζ (k).

Here

ζ (s) :=∞

∑n=1

(1/ns), Re s > 1,

denotes the Riemann ζ -function.

2. The Eisenstein series Gk ∈Mk(Γ ) has the Fourier expansion

Gk(τ) = 2ζ (k)+2(−2πi)k

(k−1)!

∞

∑n=1

σk−1(n) ·qn, q = e2πiτ .

The normalized Eisenstein series of weight k is the function

Ek :=Gk

2ζ (k).

The identity between the Riemann ζ -function and the Bernoullis numbers

2 ·ζ (k) =−(2πi)k

k!·Bk

shows

Ek(τ) = 1−2kBk·

∞

∑n=1

σk−1(n) ·qn ∈Mk(Γ ).

In particular, all Fourier coefficients of Ek are rational.

The first values (k,ζ (k),Bk,−2kBk

) are

(4,(π4/2 ·32 ·5),−(1/30),240)

(6,(π6/33 ·5 ·7),(1/42),−504)

(8,(π8/2 ·33 ·52 ·7),−(1/30),480)

(10,(π10/35 ·5 ·7 ·11),(5/66),−264)

(12,(691 π12/36 ·53 ·72 ·11 ·13),−(691/2730),(65520/691))

Proof. i) Modularity: For τ ∈H consider the normalized lattice

Λτ := Z+Zτ

and for k ≥ 2 its lattice constants GΛτ ,k . Apparently

Gk(τ) := GΛτ ,k


The series GΛ ,k is absolute convergent due to Lemma 1.12, hence allows arbitraryrearrangement. With respect to the translation

T : H→H, τ 7→ T (τ) := τ +1,

we obtainGk(τ) = Gk(τ +1).

With respect to the reflection

S : H→H, τ 7→ S(τ) :=−1τ

we have

1τk ·Gk(−(1/τ)) =

1τk · ∑

(c,d)∈Z2

′ 1(−c(1/τ)+d)k =

1τk · ∑

(c,d)∈Z2

′ τk

(−c+dτ)k = Gk(τ).

Because S and T generate the modular group we have for all γ ∈Γ and for all τ ∈H

Gk(τ) = h(γ,τ)−k ·Gk(γ(τ)).

Note that h(γ,τ)−k depends holomorphically on τ .

ii) Holomorphy: The lattice constant

GΛρ,k , ρ = e2πi/3,

dominates the Eisenstein series and shows its compact convergence on D : We obtainfor τ ∈D and c, d ∈ Z using |τ| ≥ 1 and |Re τ| ≤ 1/2

|c · τ +d|2 = c2|τ|2 +2cd ·Re τ +d2 ≥

≥

c2− cd +d2 cd ≥ 0

c2 + cd +d2 cd < 0=

|c ·ρ +d|2 cd ≥ 0

|c ·ρ−d|2 cd < 0

Therefore

∑(c,d)∈Z2

′ 1|(cτ +d)|k

≤ ∑(c,d)∈Z2

′ 1|(cρ +d)|k

+ ∑(c,d)∈Z2

′ 1|(cρ−d)|k

=

= 2 · ∑(c,d)∈Z2

′ 1|(cρ +d)|k

< ∞

Hence the restriction Gk|D is compact convergent. Each summand of Gk is holo-morphic in H. Therefore Gk|D is holomorphic. For any γ ∈ Γ and τ ∈D we have


Gk(γ(τ)) = h(γ,τ) ·Gk(τ)

with h(γ,τ) 6= 0 and holomorphic with respect to τ . As a consequence, the restric-tion Gk|γ(D) is absolutely and compactly convergent, hence holomorphic. The rep-resentation

H=⋃

γ∈Γ

γ(D)

shows that Gk : H→ C is absolutely and compactly convergent, hence also holo-morphic.

iii) Fourier expansion: We split the summation over

(c,d) ∈ Z2,(c,d) 6= (0,0),

into the simple sum for (c,d) with c = 0 and d ∈Z∗ and the double sum with c ∈ Z∗and d ∈ Z.

Gk(z) = ∑(c,d)∈Z2

′ 1(cτ +d)k = ∑

d∈Z∗

1dk + ∑

c∈Z∗,d∈Z

1(cτ +d)k =

= 2ζ (k)+2 (2πi)k

(k−1)!·

∞

∑n=1

σk−1(n) qn, q = e2πi·τ ,

due to Lemma 4.6, part 3). The Fourier expansion proves the holomorphy of Gk atthe point ∞, q.e.d.

Remark 4.8 (Eisenstein series and lattice constants). For even k > 2 the Eisensteinseries Gk relate to the lattice constants GΛ ,k of an arbitrary lattice with positivelyoriented basis

Λ = Zω1 +Zω2

as follows: Set τ := ω2/ω1 ∈H. Then

GΛ ,k =1

ω k1

Gk(τ).

Similar to the series∞

∑n=1

(1/n)

in one dimension, in two dimensions the series

∑(m,n)∈Z2

′ 1(mτ +n)2


is not convergent with respecto to any order of summation. Anyhow, in both cases aspecial order of summation exists which provides a finite value. In the 2-dimensionalcase we obtain the Eisenstein series G2.

Remark 4.9 (Eisenstein series, k = 2).

1. The Eisenstein series

G2(τ) :=∞

∑d=1

1d2 + ∑

c∈Z∗

(∑d∈Z

1(cτ +d)2

)

is holomorphic on H.

2. The Eisenstein series has the Fourier expansion

G2(τ) = 2ζ (2)−8π2 ·

∞

∑n=1

σ1(n) ·qn, q = e2πi·τ ,

and the normalized Eisenstein series of weight k = 2 is

E2(τ) :=G2(τ)

2ζ (2)= 1−24 ·

∞

∑n=1

σ1(n) ·qn.

3. The Eisenstein series G2 is invariant with respect to the translation T ∈ Γ . Withrespect to the reflection S ∈ Γ it transforms as

G2(S(τ)) = τ2 ·G2(τ)−2πiτ.

In particular, G2 is not weakly modular.

Proof. ad 1) and 2): Lemma 4.6, part 3) applied to the second summand shows

∑c∈Z

(∑

d∈Z∗

1(cτ +d)2

)= ∑

d∈Z∗

1d2 + ∑

c∈Z∗

(∑d∈Z

1(cτ +d)2

)=

= 2ζ (2)−8π2 ·

∞

∑n=1

σ1(n) ·qn,q = e2πi·τ .

The Fourier representation shows: G2 is holomorphic on H∪∞ and invariant withrespect to translation.

ad 3) Cf. [20, Kap. III, § 6, Abschn. 1]. The lack of modularity is due to theconditional convergence G2 of G2.

Definition 4.10 (Discriminant form).


The discriminant form is the modular form of weight k = 12 defined as

∆ := g32−27g2

3 : H→ C

derived from the Eisenstein series

g2 := 60 ·G4, g3 := 140 ·G6.

Proposition 4.11 (Fourier coefficients of the dicriminant form). All Fourier co-efficients of the normalized discriminant modular form

∆

(2π)12

are integers:∆(τ)

(2π)12 =∞

∑n=0

τn ·qn ∈ Zq, q = e2πi·τ ,

with τ0 = 0 and τ1 = 1. In particular ∆ ∈ S12(Γ ) is a cusp form of weight 12.

Proof. We refer to the Fourier series expansion from Theorem 4.7 for the normal-ized Eisenstein series

E4 = 32 ·5 ·G4

π4 = 1+240 ·∞

∑n=1

σ3(n) ·qn

E6 =33 ·5 ·7

2·

G6

π6 = 1−504 ·∞

∑n=1

σ5(n) ·qn.

In addition,

g2 = 60 ·G4 =22

3π

4 ·E4 and g3 = 140 ·G6 =23

33 π6 ·E6.

Hence

∆ = g32−33g2

3 = π12 ·

26

33(E34 −E2

6 ) = (2π)12 ·E3

4 −E26

26 ·33 = (2π)12 ·E3

4 −E26

1728

and∆(τ)

(2π)12 =E3

4 −E26

26 ·33 =E3

4 −E26

1728.

We set

Xk(q) :=∞

∑n=1

σk(n) ·qn

and obtain


∆

(2π)12 =(1+240 ·X3)

3− (1−504 ·X5)2

26 ·33

Denotbe by Q = Q(q) the numerator series Q of the latter fraction. The explicitform of Q(q) proves τ0 = 0. Moreover

Q = 3 ·240 ·X3 +3 ·2402 ·X23 +2403 ·X3

3 +1008 ·X5−5042 ·X25

In lowest order with X3(q) = q+O(2) and X5(q) = q+O(2) we get

Q(q) = 3 ·240 ·q+1008 ·q+O(2) = 1728 ·q+O(2)

which proves τ1 = 1.

The remaining part of the lemma reduces to the claim that also all other coeffi-cients of the series Q are divisible by

26 ·33.

We compute

Q = 3 ·240 ·X3 +3 ·2402 ·X23 +2403 ·X3

3 +1008 ·X5−5042 ·X25

Q≡ 122(5 ·X3 +7 ·X5) mod 123

To complete the proof we have to show that all coefficients of the power series

5 ·X3 +7 ·X5

are divisible by 12. We show even more: For all d ∈ Z

12 divides 5 ·d3 +7 ·d5.

According to the Chinese remainder theorem the claim reduces to the two congru-ences

5 ·d3 +7 ·d5 ≡ 0 mod 3 and 5 ·d3 +7 ·d5 ≡ 0 mod 4

ord5 ≡ d3 mod 3, mod 4.

The latter two congruences reduce to

d2 ≡ 1 mod 3, mod 4 if d3 6≡ 0 mod 3, mod 4

The latter congruences are obvious, q.e.d.

Definition 4.12 (Modular invariant). The modular invariant is the automorphicfunction defined on H as


j := 1728 ·g3

2

∆= 1728 ·

g32

g32−27g2

3.

Proposition 4.11 shows: j is a well-defined meromorphic function because itsdenominator ∆ is not constant. The proposition shows in addition, that j has a poleof order = 1 at the point ∞ with residue = 1.

4.2 The algebra of modular forms

Continuing with the investigation started in Proposition 4.3 we show: Modularforms of weight k are those meromorphic differential forms on the modular curvewhich are multiples of a certain divisor which depends on k. Applying once morethe Riemann-Roch theorem we draw several conclusions from this result:

• We derive a formula for the dimension of the vector spaces Mk(Γ ).

• We show: The modular invariant j is a biholomorphic map between the modularcurve and the Riemann surface P1(C).

• We show: Any complex elliptic curve arises from a complex torus. More pre-cisely, the elliptic curve is parametrized by the two elliptic functions ℘Λ and ℘ ′

Λ

defined on a suitable torus C/Λ , see Theorem 4.18.

We continue with the notation Γ = SL(2,Z). By denoting the vector space ofmodular forms of weight k, we will often skip Γ and set Mk := Mk(Γ ).

We denote by p :H∗→X the canonical projection to the modular curve, skippingthe ∗-superscript used in the notation from Theorem 2.34. We introduce the notation

[τ] = p(τ).

In addition we use the floor function: For x ∈ R

bxc := maxn ∈ Z : n≤ x.

Theorem 4.13 (Modular forms are differential forms on the modular curve).Denote by Γ := SL(2,Z) the modular group and by X :=H∗\Γ its modular curve.Consider an even integer k ∈ Z.

4.2 The algebra of modular forms 105

1. Differential forms: The map of vector spaces

α : Mk(Γ )→ H0(

X ,MX ⊗Ω⊗k/2X

), f 7→ ω f := p∗( f ·dτ

k/2),

is injective.

2. Divisor: Consider the divisor

Dk :=k2([∞])+

⌊k4

⌋([i])+

⌊k3

⌋([ρ]) ∈ Div(X).

The attachmentU 7→ΩΓ ,k(U), U ⊂ X open,

with

ΩΓ ,k(U) :=

ω ∈ H0(U,MX ⊗Ω⊗k/2X ) : ω 6= 0 and (ω)≥−Dk|U

∪0

is a line bundle ΩΓ ,k on X. The image of the map α from part 1) is

α(Mk(Γ )) = H0(X ,ΩΓ ,k).

If K ∈ Div(X) denotes the canonical divisor of X, then

ΩΓ ,k ' O(k/2)K+Dk.

Proof. 1. If f ∈Mk(Γ ) then Proposition 4.3 implies that the modular group leavesinvariant the differential form f ·dτk/2 on H. Hence the latter defines a meromor-phic form

α( f ) := ω f := p∗( f ·dτk/2) ∈ H0

(X ,MX ⊗Ω

⊗k/2X

).

2. The sheaf ΩΓ ,k is well-defined. The definition of the divisor Dk results from thefollowing relation between the order of f at τ ∈H∗ and the order of ω f at [τ] ∈ X :

ordτ f =

ord[∞] ω f +(k/2) τ = ∞

hτ ·ord[τ] ω f +(k/2) · (hτ −1) τ ∈H

Here hτ denotes the period of τ , see Definition 2.26. To prove the relationfor τ ∈ ∞, i, ρ we use the local form of

p : H∗→ X

according to Theorem 2.34:


• For t ∈H∪∞ set

q(t) :=

e2πit t 6= ∞

0 t = ∞

Thenq := e2πit

is a local parameter around [∞] ∈ X satisfying

dq = 2πiq dt or dt =dq

2πiq.

Hence

ω f = p∗( f (t) ·dtk/2) = f (q) ·dqk/2

(2πiq)k/2

withf (q(t) := f (t).

Determining the order of the right-hand side at q = 0 and using the definition

ord∞ f := ord0 f ,

showsord[∞] ω f = ord∞ f − (k/2).

• For τ ∈ i, ρ choose a local parameter t on H around τ such that the isotropygroup Γτ operates by multiplication by e-th root of unity, e = hτ . Then

q = te

is a local parameter on X around [τ]. Denote by N the order of ω f at [τ], suchthat

ω f (q) = u(q) ·qN ·dqk/2, u([τ]) 6= 0.

One hasdq = e · te−1 ·dt

dqk/2 = t(k/2)(e−1) · ek/2 ·dtk/2 and qN = te·N .

Hencef (t) ·dτ

k/2 = p∗(ω f (q)

)= p∗

(u(q) ·qN ·dqk/2

)=

= u(q(t)) · ek/2 · tN·e+k/2(e−1) ·dtk/2,

and

ordτ f = N · e+(k/2) · (e−1) = hτ ·ord[τ] ω f +(k/2) · (hτ −1).

In particular:


ord[τ] ω f = ordτ f , ordτ f = ord[τ] ω f , τ ∈D \i, ρ,

ord[∞] ω f = ord∞ f − (1/2)k, ord∞ f = ord[∞] ω f +(k/2)

ord[i] ω f = (1/2) ·ordi f − (1/4)k, ordi f = 2 ·ord[i] ω f +(k/2)

ord[ρ] ω f = (1/3) ·ordρ f − (1/3)k, ·ordρ f = 3 ·ord[ρ] ω f + k.

As a consequence

α(Mk(Γ ))=

ω ∈ H0

(X ,MX ⊗Ω

⊗k/2X

): (ω)≥

− k2

([∞])−

⌊k4

⌋([i])−

⌊k3

⌋([ρ])

.

According to Proposition 2.8 the sheaf Ω 1X is isomorphic to the subsheaf OK ⊂MX .

ThereforeΩ⊗k/2 ' O(k/2)K ⊂MX

andΩΓ ,k ' O(k/2)K+Dk

⊂MX .

The map α is injective because p is a branched covering. The map α is surjective:For ω ∈ H0(X ,ΩΓ ,k) the pull back

p∗(ω) = f dτk/2

defines a modular form f ∈Mk(Γ ) with α( f ) = ω , q.e.d.

Theorem 4.13 explains the name modular form: Modular forms are differentialforms. The formulas in the proof of part 2) show: While a modular form f on H∗ isholomorphic when considered as a function, its representing differential form ω f onthe modular curve X may have poles. Actually, it is the divisor of ω f which encodesthe information about f .

Corollary 4.14 (Dimension of Mk(Γ )). The vector spaces Mk(Γ ) of modular formsand Sk(Γ ) of cusp forms of weight k have the following dimensions:

dim M0(Γ ) = 1, and for even k < 4 : dim Mk(Γ ) = 0.

For even k ≥ 4 holds

dim Mk(Γ ) =

bk/12c k ≡ 2 mod 12bk/12c+1 otherwise

Proof. We compute for X ' P1 the dimension

H0(X ,ΩΓ ,k)


by the Theorem of Riemann-Roch according to Corollary 2.15. We claim

deg(−DΓ ,k +K)< 0 :

The divisorDΓ ,k := (k/2)K +Dk ∈ Div(X)

has degree

deg DΓ ,k =−k+(k/2)+ bk/4c+ bk/3c=−(k/2)+ bk/4c+ bk/3c.

Hencedeg(−DΓ ,k +K) = (k/2)−2−bk/4c−bk/3c ≤ −1

using−bk/4c<−(k/4)+1 and −bk/3c<−(k/3)+1.

Therefore Corollary 2.15 implies for all even k ∈ Z

dim Mk(Γ ) = dim H0(X ,ΩΓ ,k) = 1−0+deg DΓ ,k = 1− (k/2)+ bk/4c+ bk/3c.

We consider the different values of k mod12 separately:

i) k ≡ 0 mod 12:

1− (k/2)+(k/4)+(k/3) = 1+(k/12) = 1+

⌊k

12

⌋.

For the remaining values of k mod 12 we make the ansatz

k = α ·12+β

ii) β = 2:

1− (k/2)+ bk/4c+ bk/3c= 1−6α−1+3α +4α = 1−1+α = α = bk/12c

iii) β = 4:

1− (k/2)+ bk/4c+ bk/3c= 1−6α−2+3α +1+4α +1 = 1+α = 1+ bk/12c

iv) β = 6:

1−6α−3+1+3α +1+4α2 = 1+α = 1+ bk/12c

and similarly for the two remaining cases β = 8 and β = 10, q.e.d.

In particular


Mk = 0,k < 0, dim M0 = 1

dim M2 = 0, dim M4 = dim M6 = dim M8 = dim M10 = 1, dim M12 = 2.

The non-zero vector spaces in the last line are spanned by the Eisenstein series G4, G6and their products. Only M12 contains a non-zero cusp form, namely the discrimi-nant modular form ∆ .

Corollary 4.15 (Weight formula for modular forms). Consider an even integer k ≥ 4and a modular form f ∈Mk(Γ ). Then the sum of the orders of f satisfies the follow-ing equation:

ord∞ f +12

ordi f +13

ordρ f +∑τ

′ ordτ f =k

12

Here the symbol ∑′ denotes the summation over all points τ ∈D \i, ρ.

Proof. We recall the relation between the order of f ∈Mk(Γ ) at a point τ and theorder of the differential form ω f ∈H0(X ,MX⊗Ω k/2) at the point p∗(τ) establishedduring the proof of Theorem 4.13:

ord∞ f = ord[∞] ω f

ordi f = 2 ord[i] ω f +k2

ordρ f = 3 ord[ρ] ω f + k

ordτ f = ord[τ] ω f , τ ∈D \i, ρ.

The modular curve X has genus g = 0 according to Theorem 2.34. The linebundle ΩX

⊗k/2 has degree(2g−2)(k/2) =−k

according to Corollary 2.16. Therefore

deg(ω f ) =−k.

ord∞ f +12

ordi f +13

ordρ f +∑τ

′ ordτ f =

=

(ord[∞] ω f +

k2

)+

(ord[i] ω f +

k4

)+

(ord[ρ] ω f +

k3

)+∑

τ

′ord[τ] ω f =

=−k+k2+

k4+

k3=

k12, q.e.d.

Note: A non-zero modular form f ∈ M4 has zeros exactly along the orbit of ρ .While a non-zero modular form f ∈ M6 has zeros exactly along the orbit of i. Inboth cases, each zero has order = 1.


Corollary 4.16 (Modular curve and modular invariant).

1. The discriminant form ∆ has no zeros in H.

2. The modular invariant is a bihomorphic map j : X '−→ P(C). It attains the distin-guished values

j(i) = 1728 and j(ρ) = 0.

3. All Fourier coefficents (bn)n≥−1 of the modular invariant

j(q) =∞

∑n=−1

bn ·qn

are integers, and b−1 = 1.

Proof. i) According to Proposition 4.11 the discrriminant modular form ∆ has azero of order 1 at the point ∞, i.e.

ord∞ ∆ = 1.

According to Corollary 4.15 the modular form ∆ of weight k = 12 has no furtherzeros.

ii) Part i) implies that the modular invariant has a single pole at ∞, and this polehas order 1. We consider j as holomorphic map on the quotient X = Γ \H

j : X → P(C).

According to Proposition 2.4 the modular invariant j is surjective and assumes allvalues with the same multiplicity. Therefore j is biholomorphic. Corollary 4.15 im-plies G4(ρ) = 0 and G6(i) = 0. The definition

j = 1728 ·g2

3

g23−27g32

implies the values of j at the distinguished points.

iii) To compute the Fourier expansion of

j = 26 ·33 ·g3

2

∆

we combine from Proposition 4.11 and from its proof two formulas. Firstly, theformula

g2 =22

3·π4 ·E4


which implies

g32 =

26

33 ·π12 ·E3

4 ,

and secondly the formula

∆

(2π)12 =∞

∑n=1

an ·qn ∈ Zq.

We obtain

j = 26 ·33 ·g3

2

∆= 26 ·33 ·

26

33 ·π12

∆=

1∞

∑n=1

an ·qn=

1q·

1

1+∞

∑n=2

an ·qn−1

and1

1+∞

∑n=2

an ·qn−1∈ Zq, q.e.d.

Corollary 4.17 (The field of automorphic functions).

1. If an automorphic form f ∈ A0(Γ ) is holomorphic on H with Fourier series

f (τ) = ∑n≥−k

cn ·qn

for a suitable k ∈ N, then a poylnomial P(X) ∈ C[X ] of degree k exists with

• f = P( j),

• and all coefficients of P(X) are Z-linear combinations of the Fourier coeffi-cients c−k, ...,c−1,c0.

2. The field of automorphic functions is

A0(Γ ) = C( j).

Proof. 1. The proof goes by induction on k and uses the weight formula fromCorollary 4.15: If k= 0 than f is a modular function, hence equals the constant c0.For the induction step k−1 7→ k consider the function

g(τ) := f (τ)− c−k · j(τ)k.

The function g has at most a pole of order k− 1 at ∞. Therefore the inductionassumption applies to g.


2. Denote by

A := x1, ...,xn ⊂H∪∞ and B = y1, ...,yn ⊂H∪∞

the respective zero set and pole set of f , counted with multiplicity. Then

g :=n

∏ν=1

j− j(xν)

j− j(yν)∈ A0(Γ )

has the same zero set and pole set as f . The quotient f/g has no poles and zeroson H∪∞. According to part 1) a constant a ∈ C exists with

a =fg

orf = a ·g ∈ C( j), q.e.d.

We now prove the converse of Theorem 3.20.

Theorem 4.18 (Uniformization of elliptic curves E/C by tori). For any ellipticcurve E defined over C and equipped with the analytic structure from Ean ⊂ Pn(C)exists a torus T = C/Λ and a biholomorphic map

Φ : T '−→ Ean.

Proof. Assume that E is defined by the homogenization of the Weierstrass polyno-mial

F(X ,Y ) := Y 2− (4 ·X3−A ·X−B) ∈ C[X ,Y ].

Making the substitution Y = 2 ·Y ′ and applying Proposition 3.11 shows for the dis-criminant

∆F = A3−27B2 6= 0.

Note the signature of the different coefficients of F when applying the proposition.

We search for a latticeΛ = Zω1 +Zω2

with lattice constants GΛ ,4 and GΛ ,6 satisfiying

A = 60 ·GΛ , 4 and B = 140 ·GΛ , 6.

The restriction of the modular invariant

j|H : H→ C, j(τ) := 1728 ·g3

2(τ)

g32(τ)−27g2

3(τ),

is surjective according to Corollary 4.16. Hence a number τ ∈H exists with


j(τ) = 1728 ·A3

∆F.

If g2(τ) = 0 then τ = ρ according to Corollary 4.15. We choose

Λ := Λρ .

Similarly, if g3(τ) = 0 then τ = i, and we choose

Λ := Λi.

Otherwise, clearing the denominators implies(A

g2(τ)

)3

=

(B

g3(τ)

)2

We choose ω1 ∈ C with

Ag2(τ)

=

(1

ω1

)4

andB

g3(τ)=

(1

ω1

)6

.

This choice is possible because we may replace ω1 by iω1. In addition, we set

ω2 := τ ·ω1, i.e. τ = ω2/ω1.

Then (ω1,ω2) is a positively oriented basis of the lattice

Λ := Zω1 +Zω2.

Its lattice constants are

GΛ ,4 =

(1

ω1

)4

·G4(τ) =

(1

ω1

)4

· (1/60) ·g2(τ) = (1/60) ·A

and similarlyGΛ ,6 = (1/140) ·B.

Therefore, according to Theorem 3.20

Φ(C/Λ) = Ean, q.e.d.

Lemma 4.19 (Exact sequence for cusp forms). Denote by

ε : Mk→ C, f 7→ f (∞),

the evaluation with kernel Sk, and for k ≥ 4 by


µ∆ : Mk−12→Mk, f 7→ f ·∆ ,

the multiplication with the discriminant modular form. Then for k ≥ 4 the sequenceof vector space morphisms is exact

0→Mk−12µ∆−→Mk

ε−→ C→ 0.

In particular for k ≥ 4:Mk = Sk⊕C ·Gk,

and multiplication by the discriminant modular form ∆ defines a vector space iso-morphism

µ∆ : Mk−12'−→ Sk.

Proof. The Eisenstein series Gk ∈Mk satisfies Gk(∞) = 2 ·ζ (k) 6= 0. Therefore ε issurjective. If f ∈Mk with ε( f ) = 0, then the quotient

g :=f∆

is holomorphic on H, because ∆ has no zero on H due to Corollary 4.16. Inaddition g is holomorphic at the point ∞ because ∆ has a zero of order 1 at ∞ ac-cording to Proposition 4.11. Hence g ∈Mk−12. Apparently the map µ∆ is injective,q.e.d.

Theorem 4.20 (Algebra of modular forms). The associative algebra

M∗(Γ ) :=

(⊕k≥0

M2k(Γ ),+, ·

)

of modular forms of the module group Γ is freely generated by the two Eisensteinseries G4 and G6:

M = C[G4,G6]' C[T1,T2] (Algebra of polynomials in two indeterminates) .

The subalgebraS∗(Γ ) :=

⊕k≥0

S2k(Γ )⊂M∗(Γ )

is an ideal.

Proof. i) Generated by G4 and G6: We give a proof by induction on k ≥ 4. Anyinteger k ≥ 4 decomposes as

k = 4 ·α +6 ·β

with non-negative integers α,β ∈ Z. Hence

Gα4 ·G

β

6 ∈Mk

4.3 Hecke operators 115

and ε(Gα4 ·G

β

6 ) 6= 0. Consider an arbitrary element f ∈Mk. For a suitable λ ∈ C

ε( f −λ ·Gα4 ·G

β

6 ) = ( f −λ ·Gα4 ·G

β

6 )(∞) = 0.

According to Lemma 4.19 an element g ∈Mk−12 exists with

f −λ ·Gα4 ·G

β

6 = ∆ ·g.

The claim follows by applying the induction assumption to g.

ii) Freely generated: The generators G4 and G6 are algebraically independent.Assume the existence of a polynomial P ∈ C[T1,T2] with

P(G4,G6) = 0.

We may assume that all monomials of P(G4,G6) have the same weight. The case

P(G4,G6) = a ·Gm4 +G6 ·P′(G4,G6), a ∈ C∗,

is excluded because G4(i) 6= 0, but G6(i) = 0. Analogously the case

P(G4,G6) = b ·Gm6 +G4 ·P′(G4,G6) b ∈ C∗,

is excluded because G6(ρ) 6= 0, but G4(ρ) = 0. Hence G4 ·G6 divides P(G4,G6),i.e.

P(G4,G6) = G4G6 ·P′(G4,G6) = 0.

Iteration of the argument for P′(G4,G6) = 0 proves P = 0 by descending induction.

iii) Ideal of cusp forms: According to Lemma 4.19 any cusp form is a multiple of ∆ ,q.e.d.

4.3 Hecke operators

Hecke operators are linear maps on modular forms. Hecke operators average thebehaviour of modular forms with respect to certain sets of matrices Γm⊂GL(2,R)+.These sets generalize Γ1 := Γ when considered a set.

Definition 4.21 (Integral matrices with positive determinant). For any m ∈ N∗consider the set of matrices

Γm := M ∈M(2×2,Z) : det M = m ⊂ GL(2,R)+

and its subset


Γm,prim :=(

a bc d

)∈ Γm : (a,b,c,d) = 1

of primitive matrices. We denote by

Φm : Γ ×Γm→ Γm,(A,M) 7→ A ·M

the canonical Γ -left operation and by

Φm,prim := Φm|Γm,prim : Γ ×Γm,prim −→ Γm,prim

its restriction. The respective orbits sets are denoted

Γ \Γm and Γ \Γm,prim.

The subset Γm,prim ⊂ Γm is stable under the Γ -left operation Φm. For the proofconsider any

A =

(α β

γ δ

)∈ Γ and M =

(a bc d

)∈ Γ

∗m,prim.

The idealI := (αa+βc,αa+βd,γa+δc,γb+δd)⊂ Z

contains the element

δ · (αa+βc)+(−β ) · (γa+δc) = (αδ −βγ)a+(δβ −βδ )c = a

and similar for the other generators of (a,b,c,d). Therefore

I = (a,b,c,d) = (1)

andΦm(A,M) = A ·M ∈ Γm,prim.

Lemma 4.22 (Orbit sets of left-operation). Consider a positive integer m ∈ N∗.

1. A bijective map exists

Γ \Γm'−→V (m) :=

(a b0 d

)∈ Γm : 0≤ b < d

to a subset of upper triangular matrices. In particular: The orbit set of Φm hascardinality σ1(m).

2. A bijective map exists

Γ \Γm,prim'−→V (m, prim) :=

(a b0 d

)∈V (m) : (a,b,d) = 1

.

In particular: The orbit set of Φm,prim has cardinality


ψ(m) := m · ∏p|m

p prime

(1+(1/p))

Proof. 1. We define a mapp : Γm→V (m),

which attaches to any M ∈ Γm a matrix p(M) belonging to the same orbit:

i) Consider a fixed but arbitrary element

M =

(a bc d

)∈ Γm.

We construct a matrix A ∈ Γ such that A ·M ∈V (m):

First, in order to remove the entry c of M we choose coprime integers γ,δ ∈ Zwith

γ ·a+δ · c = 0.

In order to obtain a matrix with det = 1 we choose integers α,β ∈ Z with

α ·δ −β · γ = 1.

Such a choice is possible because γ and δ are coprime. Set

A1 :=(

α β

γ δ

)∈ Γ .

Then

A1 ·M =

(a′ b′

0 d′

)and

det (A1 ·M) = det M = a′ ·d′ = m.

In particular d′ 6= 0 and possibly after replacing A1 by −A1 even a′, d′ > 0.

Secondly, because T ∈ Γ , for any k ∈ Z also

A2(k) := T k ·A1 ∈ Γ .

Then

A2(k) ·M =

(1 k0 1

)·(

a′ b′

0 d′

)=

(a′ b′+ k ·d′0 d′

).

We choose k0 ∈ Z such that

0≤ b′+ k0 ·d′ < d′

and set


A := A2(k0).

ii) The construction from part i) determines uniquely the element

A ·M ∈V (m) :

Assume two matrices Ai ∈ Γ such that

Ai ·M =: Mi ∈V (m), i = 1,2.

Then

M = A−11 ·M1 = A−1

2 ·M2 or A21 ·M1 = M2, A21 := A2 ·A−11 .

Set

A21 =:(

α β

γ δ

)and Mi :=

(ai bi0 di

), i = 1,2.

From (α β

γ δ

)·(

a1 b10 d1

)=

(a2 b20 d2

)follows:

• γ ·a1 = 0, hence γ = 0.• From d2 = δ ·d1 follows δ > 0. Together with α ·δ = 1 it implies α = δ = 1.• As a consequence a1 = a2, d1 = d2, and b1 +β ·d1 = d2.• Because 0≤ b1,b2 < d1 = d2 we get b1 = b2 and β = 0.

As a consequence

A21 =

(1 00 1

)or A1 = A2.

iii) Part i) and ii) show that

p : Γm→V (m), M 7→ A ·M,

is well-defined.

iv) The map p is surjectice because V (m)⊂ Γm. In addition,

p(M1) = p(M2) ⇐⇒ ∃ A ∈ Γ with A ·M1 = M2

⇐⇒ M1 and M2 on the same orbit of Φm.

As a consequence p induces the bijective map

Γ \Γm'−→V (m), M 7→ p(M).

v) The value card V (m) follows at once from the explicit representation of theelements of V (m).


2. The representation of the orbit set Γ \Γm,prim follows from part 1) by restriction.For the cardinality of V (m, prim) see [23, Chap. 5, §1].

Corollary 4.23 (Representatives of the orbit set). For any γ ∈ Γ :

1. The setV (m) · γ ⊂ Γm

is a complete set of representatives of the orbit set Γ \Γm, i.e. with respect to thecanonical projection

Γm→ Γ \Γm, M 7→ [M],

we have the equality of sets

Γ \Γm = [M] : M ∈V (m)= [M · γ] : M ∈V (m).

2. The setV (m, prim) · γ ⊂ Γm,prim

is a complete set of representatives of the orbit set Γ \Γm,prim.

Proof. Consider a fixed, but arbitrary γ ∈Γ . We show: For arbitrary but fixed M ∈V (m)exist an element

M1 ∈V (m)

and a matrix A ∈ Γ such thatA ·M = M1 · γ.

According to Lemma 4.22 a unique element A ∈ Γ exists with

A · (M · γ−1) ∈V (m).

We defineM1 := A ·M · γ−1 ∈V (m).

ThenM1 · γ := A ·M.

Hence M1 · γ lies on the same orbit as M, q.e.d.

Definition 4.24 (Hecke operators of the modular group). For any positive integer m≥ 1the m-th Hecke operator is the C-linear map on the algebra of modular forms

Tm : M∗(Γ )→M∗(Γ )

which is defined on Mk(Γ ), k ∈ N, as


Mk(Γ )→Mk(Γ ), f 7→ Tm( f ) := mk−1 · ∑Γ \Γm

f [M]k.

Here the summation is over an arbitrary complete set of representatives M ∈ Γmof Γ \Γm.

Apparently, for m = 1:

T1 = id : Mk(Γ )→Mk(Γ ).

Note that the value Tm( f ) of the m-th Hecke operator in Definition 4.24 does notdepend on the choice of a fixed representative M: For any γ ∈ Γ

f [γ ·M]k = ( f [γ]k)[M]k = f [M]k

due to Remark 4.4.

Theorem 4.25 (Fourier coefficients of Hecke transforms).

1. The Hecke operators from Definition 4.24 are well-defined, i.e.

f ∈Mk(Γ ) =⇒ Tm( f ) ∈Mk(Γ ).

2. For a modular form f ∈Mk(Γ ) with Fourier expansion

f (τ) =∞

∑n=0

an ·qn, q = e2πi·τ ,

its Hecke transform Tm( f ) ∈Mk(Γ ) has the Fourier series

Tm( f )(τ) =∞

∑n=0

∑r>0

r|(m,n)

rk−1 ·a(mn/r2)

qn.

3. The ideal of cusp forms is stable under all Hecke operators, i.e.

Tm(S∗(Γ ))⊂ S∗(Γ ).

Proof. 1. To prove that Tm( f ) is a modular form of the same weight as f we haveto verify the defining properties from Definition 4.2:

• Appararently Tm( f ) is holomorphic on H.

• Weakly modularity: Consider a fixed but arbitrary matrix γ ∈ Γ . We replacethe summation over the orbit set V (m) by the summation over the set of rep-resentatives V (m) · γ , see Corollary 4.23:


Tm( f )[γ]k = mk−1∑

M∈V (m)

( f [M]k)[γ]k = mk−1 · ∑M∈V (m)

( f [M · γ]k) =

= mk−1 · ∑N∈V (m)·γ

f [N]k = Tm( f ).

• Holomorphy of Tm( f ) at the point ∞: We have to compute the Fourier seriesof Tm f . During the computation we will use the following results:

i) If

M =

(a b0 d

)∈V (m)

then

exp(2πin ·M(τ)) = exp

(2πin ·

aτ +bd

)= e(2πin·(aτ/d) · e2πin·(b/d)

ii)

∑0≤b<d

e2πib·(n/d) =

d d|n0 otherwise

The first case follows from e2πib·(n/d) = 1 for each of the d summandswith 0≤ b < d. The second case follows from the formula of the finite ge-ometric series.

We now use the representation of the orbit set Γ \Γm from Lemma 4.22 andinsert the Fourier series of f . We obtain

Tm( f )(τ) = mk−1 · ∑M∈V (m)

f [M]k(τ) = mk−1 · ∑M∈V (m)

h(M,τ)−k · f (M(τ)) =

mk−1∑

a,d>0ad=m

1dk ·

d−1

∑b=0

f (M(τ)) =

= mk−1∑

a,d>0ad=m

1dk ·

d−1

∑b=0

(∞

∑n=0

an · e2πin·(aτ/d) · e2πin·(b/d))

)

= mk−1∑d>0d|m

1dk ·

d−1

∑b=0

(∞

∑n=0

an · e2πin·(mτ/d2) · e2πib·(n/d))

)

= mk−1∑d>0d|m

1dk ·

(∞

∑n=0

an ·q(nm/d2) ·d−1

∑b=0

e2πib·(n/d))

)


= ∑d>0

d|(m,n)

(md

)k−1(∞

∑n=0

an ·q(nm/d2)

)

Now we make the substitution

(d,n) 7→ (t = (mn)/d2,r = m/d)

obtaining

n =d2 · t

m=

(dm

)2

tm =tmr2 .

Hence

Tm( f )(τ) =∞

∑t=0

∑r>0

r|(m,t)

rk−1 ·a(tm)/r2

q t

with the inner sum being finite. Apparently, the last Fourier series has nocoefficients with negative index. Hence Tm( f ) is holomorphic at 0 ∈ ∆ .

2. For the proof see the Fourier series from the previous part.

3. According to the previous part of the theorem, for a modular form f ∈Mk(Γ ) theFourier expansion of Tm( f ) is

Tm( f )(τ) = a0 ·σk−1(m)+am ·q+O(2).

A cusp form satisfies a0 = 0, hence

Tm( f )(τ) = am ·q+O(2), q.e.d.

The following corollary specializes the general formula from Theorem 4.25. Itgives the formula for the Fourier coefficients of the Hecke transform for prime in-dices.

Corollary 4.26 (Fourier coefficients with prime index of Hecke transforms).Consider a modular form f ∈Mk(Γ ) with Fourier coefficients

(a(n) := an)n≥0.

For arbitrary but fixed m≥ 1 denote by

(b(n) := bn)n≥0

the Fourier coefficients of its Hecke transform Tm( f ). Then for any prime p


b(p) =

a(mp) p - ma(mp)+ pk−1 ·a(m/p) p|m

Proof. The claim follows directly from the general formula in Theorem 4.25:

• If p|m then r|(p,m) implies r = 1 or r = p; note n = p.• If p - m then r|(p,m) implies r = 1, q.e.d.

Corollary 4.27 (Simultaneous eigenform of all Hecke operators). The discrimi-nant modular form

∆ ∈ S12(Γ )

is a simultaneous eigenform, i.e. eigenvector, of all Hecke operators (Tm)m≥1. TheFourier coefficients of the normalized discriminant modular form

∆

(2π)12 ·∞

∑m=1

τm ·qm

are the corresponding eigenvalues, i.e. for all m ∈ N

Tm(∆) = τm ·∆ .

Proof. According to Corollary 4.14 and Lemma 4.19 the vector space S12(Γ ) ofcusp forms of weight k = 12 is 1-dimensional. It has the discriminant modularform ∆ from Definition 4.10 as a basis. Hence ∆ is an eigenvector of each Heckeoperator Tm, m≥ 1. To calculate the corresponding eigenvalue we recall the Fourierexpansion of the normalized discriminant modular form from Proposition 4.11:

∆(τ)

(2π)12 =∞

∑n=1

τn ·qn, τ1 = 1.

According to Theorem 4.25 the Fourier expansion of Tm

(1

(2π)12 ·∆

)is

Tm

(1

(2π)12 ·∆

)(τ) = τm ·q+O(2).

We obtainTm(∆) = τm ·∆ , q.e.d.

Remark 4.28 (Algebra of Hecke operators).


1. The algebra of all Hecke operators

H (Γ ) := Tm : M∗(Γ )→M∗(Γ )|m≥ 1,+,)

is Abelian.

2. On the vector spaces Sk(Γ ),k ≥ 12 even, exists a Hermitian scalar product, thePetersson scalar product,

<−,−>: Sk(Γ )×Sk(Γ )→ C

such that all Hecke operators become Hermitian operators.

In particular, for each arbitrary but fixed k ∈ N the vector space Sk(Γ ) of cuspshas a basis, such that each element from the basis is a simultaneous eigenform forall Hecke operators (Tm)m∈N. For a proof consider [20, Kap. IV, § 2, §3].

Chapter 5Application to Number Theory

5.1 The τ-function of Ramanujan

Definition 5.1 (Ramanujan function). The Fourier coefficients (τn)n ∈ N of thenormalized discriminant form

∆(τ)

(2π)12 =∞

∑n=1

τn ·qn, q = e2πi·τ ,

define the Ramanujan function

τ : N→ Z,n 7→ τ(n) := τn.

According to Proposition 4.11 all Fourier coefficients are integers τ(n) ∈ Z.

Computing the first coefficients τ(n) up to n = 6 gives

∆(τ)

(2π)12 = q−24q2 +252q3−1.472q4 +4.830q5−6.048q6 +O(7)

From these values one verifies at once the product formula

τ(2) · τ(3) =−24 ·252 =−6.048 = τ(6),

and as a particular case of the recursion formula

τ(p2) = τ(p)2− p11 · τ(p)

the formula

τ(22) =−1.472 = τ(2)2−211 = (−24)2−2.048.

125

126 5 Application to Number Theory

From explicit computation of the first 30 values of τ(n) Ramanujan [29]conjectured in 1916 several generalizations. Figure 5.1 shows with equations (103)and (104) two of Ramanujan’s conjectures from his original paper.

5.1 The τ-function of Ramanujan 127

Fig. 5.1 Ramanujan’s conjecture on τ(n) from [29]

The conjecture was proved inter alia one year later by Mordell [26]. Figure 5.2shows Mordell’s introduction from his original paper.


Fig. 5.2 Mordell’s introduction to [26]

Today Ramanujan’s product formula and recursion formula are implied by Hecketheory:

Proposition 5.2 (Product and recursion formula of the Ramanujan function).The Ramanujan function satisfies:

1. Product formula: For all integers m,n≥ 1

τ(m) · τ(n) = ∑r|(m,n)

r11 · τ(mn/r2).

5.2 Complex multiplication and imaginary quadratic number fields 129

In particular, for coprime m and n:

τ(m) · τ(n) = τ(m ·n).

2. Recursion formula: For all primes p and integers k > 1

τ(p) · τ(pk) = τ(pk+1)+ p11 · τ(pk−1).

Proof. 1. Consider an arbitrary, but fixed m ∈ N. Corollary 4.27 shows:

Tm

(∆

(2π)12

)= τ(m) ·

∆

(2π)12

The explicit formula for the Fourier coefficients of the Hecke transform fromTheorem 4.25 implies for all n ∈ N:

∑r|(m,n)

r11 · τ(mn/r2) = τ(m) · τ(n).

For coprime integers m,n we have (m,n) = 1 and the summation reduces to thesingle term for r = 1:

τ(m ·n) = τ(m) · τ(n).

2. We set m = pk and n = p in the product formula from part i). Due to (pk, p) = pwe now have two summands, referring to the indices r = 1 and r = p. We obtain

τ(pk+1)+ p11 · τ(pk−1) = τ(pk) · τ(p), q.e.d.

Remark 5.3 (Growth estimation of the Ramanujan function). The final conjecture ofRamanujan on the τ-function is the growth estimation (104) from 5.1. This conjec-ture lies much deeper than his other conjectures. In the form

|τ(p)| ≤ 2 · p11/2, p prime,

Deligne proved the growth condition of the τ-function as an implication of his proofof the Weil conjectures in 1974.

Note the typo of the factor 2 in Ramanujan’s original equation (104).

5.2 Complex multiplication and imaginary quadratic numberfields

The present section introduces the subclass of tori T with complex multiplication,i.e. with a non-trivial endomorphism ring End(T ). Complex multiplication is a sin-


gular phenomenon in the theory of tori. Tori with complex multplication providea bridge from complex analysis to number theory because their lattice satisfies anarithmetic condition: The generator τ of the normalized lattice Λτ of T generates animaginary quadratic number field K = Q[τ], and End(T ) becomes an order in OK ,see Proposition 5.9.

In the opposite direction, for a given imaginary quadratic number field K thenumber of isomorphism classes of tori T with End(R)=OK equals the class numberof K, see Theorem 5.12.

Proposition 5.4 (Holomorphic maps between tori). Consider a holomorphic map

F : C/Λ1→ C/Λ2

between two tori with F(0) = 0. Then a number α ∈C exists such that the followingdiagram commutes

C C

C/Λ1 C/Λ2

µα

p1 p2

F

Here µα denotes the multiplication by α and pi, i = 1,2, denote the canonicalprojections. In particular, any holomorphic map between tori which respects theneutral element is a group homomorphism.

Proof. The holomorphic map F lifts to a holomorphic map F between the universalcoverings, satisfying F(0) = 0 and for all z ∈ C,ω ∈Λ1,

F(z+ω)− F(z) ∈Λ2.

The lattice Λ2 is discrete and F is continous. Therefore

F(z+ω)− F(z)

does not depend on z. As a consequence for all z ∈ C

F ′(z+ω)− F ′(z) = 0.

Hence F ′ is a holomorphic and doubly periodic function. Therefore F is a constanta ∈ C and for all z ∈ C

F(z) = a · z, q.e.d.

Apparently α has to satisfyα ·Γ1 ⊂ Γ2.

Note that the multiplier α in Proposition 5.4 satisfies


α ·Γ1 ⊂ Γ2.

Definition 5.5 (Endomorphisms of a torus). Any holomorphic map

F : (T,0)−→ (T,0)

of a torus (T,0) with F(0) = 0 is an endomorphism of (T,0).

Proposition 5.6 (Endomorphism ring). The set of endomorphisms of a torus T = C/Γ

constitutes in a canonical way a ring End(T ), the endomorphism ring of T . The mapfrom Proposition 5.4

End(T )−→ α ∈ C : α ·Γ ⊂ Γ ,F 7→ µα ,

is bijective.

Because each torus T is an Abelian group its endomorphism ring End(T ) com-prises at least the group Z:

Z⊂ End(T )⊂ C.

Which intermediate rings are possible?

The case End(T ) =Z is the general case, justifying a distinguished name for toriwith additional endomorphisms:

Definition 5.7 (Complex multiplication). A torus T = C/Γ has complex multipli-cation if Z ( End(T ).

Complex multiplication is an exceptional phenomenon. Its existence depends onan arithmetic property of the lattice of the corresponding torus.

Remark 5.8 (Number field). A number field K is an extension field

Q⊂ K ⊂ C

of finite degree[K : Q] =: n < ∞.

1. Any number field K is an algebraic extension of Q generated by a single element

K =Q(ω).


2. The Galois group Gal(K/Q) of K has order n. It operates on K as a group ofautomorphisms. An element x ∈ K has the trace

Tr(x) := ∑σ∈Gal(K/Q)

σ(x) ∈Q,

and the normN(x) := ∏

σ∈Gal(K/Q)

σ(x) ∈Q.

3. The subring OK ⊂ K of algebraic integers

OK := x ∈ K : f (x) = 0 for a normed polynomial f ∈ Z[X ]

is a lattice of rank = n

OK =n⊕

i=1

Zωi.

Its quotient field isQ(OK) = K.

For all x ∈ OKTr(x) ∈ Z and N(x) ∈ Z.

4. An integral basis of K is a Q-basis (ωi)i=1,...,n of K such that

OK = Zω1 + ...+Zωn.

In particular, all elements of an integral basis are algebraic integers.

The discriminant δK of K is the determinant of an integral basis (ωi)i=1,...,n, i.e.

δK := det(Tr(ωi ·ω j)1≤i, j≤n) ∈ Z.

5. An order of K is a subring O ⊂ OK which is an n-dimensional lattice.

6. A quadratic number field is a field K =Q(√

d) with a squarefree integer d ∈ Z.If d < 0 then K is a quadratic imaginary number field..

Consider an imaginary quadratic number field K =Q(√−d) with d > 0. Its ring

of integers isOK = Z+Zω

with

ω =

√−d d 6≡ 3 mod 4

1+√−d

2d ≡ 3 mod 4

Its discriminant is


δK =

−4d d 6≡ 3 mod 4

−d d ≡ 3 mod 4

Examples are

• d = 1: Number field K =Q(i); ω = i, δK =−4

• d = 3: Number field K =Q(ρ),ρ := e2πi/3; ω = 1+ρ, δK =−3.

Proposition 5.9 (Complex multiplication and imaginary quadratic number fields).A torus T = C/Λτ with a normalized lattice

Λτ = Z ·1+Z · τ, τ ∈H,

has complex multiplication if and only τ belongs to an imaginary quadratic numberfield K. In this case the ring of endomorphisms End(T ) is an order of K.

Proof. i) Assume that T has complex multiplication: Then exists α ∈ C∗ \Z with

α ·Λτ ⊂Λτ .

We obtain a matrix

γ =

(a bc d

)∈M(2×2,Z)

with

α ·(

1τ

)= γ ·

(1τ

).

We obtainα = a+b · τ, α · τ = c+d · τ,

which implies b 6= 0, because α /∈ Z, and

b · τ2 +(a−d) · τ− c = 0.

Therefore τ ∈H belongs to an imaginary quadratic number field.

ii) Assume that τ ∈ H belongs to an imaginary quadratic number field. Thenintegers a,b,c ∈ Z exist such that

a · τ2 +b · τ + c = 0.

Because τ /∈H we have a 6= 0. From

(aτ) ·(

1τ

)=

(0 a−c −b

)·(

1τ

).


follows aτ ∈ End(T ), and T has complex multiplication,

iii) Assume that T has complex multiplication and consider α ∈ End(T ). Aliketo the proof of part i) we obtain a matrix

γ =

(a bc d

)∈M(2×2,Z)

with

α ·(

1τ

)= γ ·

(1τ

).

Therefore α is an eigenvalue of γ and satisfies the integral equation

α2− (d +a) ·α− cd = 0.

As a consequence End(T )⊂ OK .

As submodule of the lattice OK also End(T ) is a lattice. Because

End(T )) Z

the lattice End(T ) has rank = 2. Hence End(T ) is an order, q.e.d.

Example 5.10 (Tori with complex multiplication). The modular group has exactlytwo elliptic points modulo its fundamental domain, see Theorem 2.31:

1. Elliptic point i: The point i ∈H belongs to the imaginary quadratic number field

K :=Q(√−d),d = 1.

The torus T = C/Λi with the normalized lattice

Λi = Z+Z · i

has as endomorphisms the multiplication with arbitrary elements

α ∈Λi.

Hence its endomorphism ring is

End(T ) = Λi,

which equals the ring OK of algebraic integers from K. Recall from Corollary 4.16

j(i) = 1728 = 26 ·33 ∈ Z.

2. Elliptic point ρ = e2πi/3: The point ρ ∈ H belongs to the imaginary quadraticnumber field


K =Q(√−d), d = 3.

The torus T = C/Λρ with the normalized lattice

Λρ = Z+Z ·ρ


α ∈Λρ .

Hence its endomorphism ring is

End(T ) = Λρ .

The result is similar to the first example: End(T ) is the ring OK of algebraicintegers from K. Recall from Corollary 4.16

j(ρ) = 0 ∈ Z.

Also the element 2ρ ∈H belongs to K. The torus

T = C/Λ2ρ

with the normalized lattice

Λ2ρ = Z+Z ·2ρ


α ∈Λ2ρ .

For the proof one uses the equation

ρ2 +ρ +1 = 0.

The endomorphism ring isEnd(T ) = O

withO = Λ2ρ ( OK

an order of K. Notej(2ρ) = j(ρ) = 0 ∈ Z.

Remark 5.11 (Fractional ideals in number fields). Consider a number field K ofdegree [K : Q] = n.

1. A fractional ideal of K is an OK-submodule a ⊂ K for which there exists anelement d ∈ OK ,a 6= 0, with


d ·a⊂ OK .

Hence fractional ideals are the subsets of the form

a=ad⊂ K

with an ideal a ⊂ OK . The element d is named a common denominator of a.Fractional ideals generated by a single element α ∈ K

a= (α) := OK ·α

are named principal fractional ideal.

2. The fractional ideals of K are the finitely generated OK-submodules of K. As aconsequence: Any fractional ideal of K is a lattice of rank = n.

3. For two fractionals ideals a, b ⊂ K the product a · b is the fractional ideal gen-erated by all products a · b, a ∈ a, b ∈ b. Any fractional ideal a 6= 0 has awell-defined fractional ideal a−1 such that

a ·a−1 = (1) = OK .

The fractional ideal a−1 is named the inverse of a.

4. With respect to multiplication the set of fractional ideals is an Abelian group J(K).Its quotient by the subgroup P(K) of principal fractional ideals is named the idealclass group of K

Cl(K) := J(K)/P(K).

We recall from Remark 2.7 the exact sequence of divisor classes on a compactRiemann surface X

1→ C∗→M (X)→ Div0(X)→Cl0(X)→ 0.

For a number field K we have the analogue

1→ K∗→ P(K)→ J(K)→Cl(K)→ 1.

With respect to this analogue fractional ideals correspond to divisors or line bun-dles of degree = 0, and principal fractional ideals correspond to principal divi-sors, i.e. divisors of meromorphic functions.

A fundamental theorem from algebraic number theory states the finiteness of theclass number

hK := ord Cl(K).

Examples:


• For K =Q(i) we haveOK = Z[i] = Z+Z i,

which is a principal ideal domain. Hence hK = 1.

• For K =Q(ρ) we have

OK = Z[ρ] = Z+Z ρ

with hK = 1.

There are exactly 9 imaginary quadratic number fields K =Q(√−d) with hK = 1.

They belong to the values

d ∈ 1,2,3,7,11,19,43,67,163.

We now investigate the consequences of Proposition 5.9 for an imaginary quadraticnumber field K, in particular for its class group Cl(K).

Theorem 5.12 (Ideal classes and endomorphisms of elliptic curves). Consider aquadratic imaginary number field K and denote by OK its ring of integers.

Then a bijection exists between the ideal class group Cl(K) and the set of iso-morphism classes of complex elliptic curves E with End(E)'OK . The bijection isinduced by the attachement

a ∈ J(K) 7→ E = C/a with End(E) = OK .

Its inverse is induced by the attachement

E = C/Λ with End(E) = OK 7→Λ ∈ J(K).

Proof. i) Consider a fractional ideal a ∈ J(K). Due to Remark 5.11 the ideal a⊂ Cis a lattice and defines the torus E = C/a. According to Proposition 2.24 the torusdepends only on the ideal class [a] ∈Cl(K). Proposition 5.6 implies

End(E) = α ∈ C : α ·a⊂ a.

As a consequence OK ⊂ End(E) because a is a fractional ideal.

For the opposite inclusion consider a number α ∈ End(E). Then one checks: Therepresentation as lattice

a= Zω1 +Zω2 ⊂ K

implies α ∈K. Hence End(E)⊂K. We are left to prove, that all elements α ∈ End(E)are integral. Consider a Z-basis (ω1,ω2) of a. The equation


a ·(

ω1ω2

)=

(a bc d

)·(

ω1ω2

)implies that α is an eigenvalue of the matrix(

a bc d

)∈ GL(2,Z).

Therefore α ∈K satisfies a quadratic equation with integer coefficients, i.e. α ∈OK .

ii) Consider an elliptic curve E = C/Λ . We may assume that the lattice Λ isnormalized

Λ = Z+Zτ.

By assumption End(E) = OK . We claim Λ ⊂ K: If OK = Z+Zω then

ω ·(

1τ

)=

(a bc d

)·(

1τ

).

Hence ω is an eigenvalue of(a bc d

)∈M(2×2,Z) with eigenvector

(1τ

).

We obtain an equationω = a+b · τ.

It impliesτ ∈Q(ω) = K.

The assumptionEnd(E) = α ∈ C : αΛ ⊂Λ= OK

implies that Λ is even a fractional ideal of K.

iii) The two constructions from part i) and ii) are inverse, q.e.d.

Remark 5.13 (Ideal class group of orders). Theorem 5.20 is a statement about frac-tional ideals in K with respect to the ring OK . The ideal class group Cl(K) refers tothe maximal order OK of K.

The theorem generalizes literally to the case of fractional ideals with respect toan arbitrary order

O ⊂OK ⊂ K.

The general case sets in a bijective relation the ideal classes with respect to anorder O of K and the isomorphism classes of elliptic curves E with End(E) = O .

5.3 Applications of the modular polynomial 139

5.3 Applications of the modular polynomial

The present section takes up the study of imaginary quadratic number fields Kfrom the viewpoint of modular forms. The modular invariant j links both do-mains: Theorem 5.18 proves that the value j(z) ∈ C is integral over Z for anynon-rational z ∈ K∩H. If

OK = Z+Z ·ω ⊂ K

then[Q( j(ω)) : Q]

equals the class number of K, see Theorem 5.20 for a lower bound. The main tool toderive the necessary properties of the modular invariant is the modular polynomialfrom Definition 5.14.

We take up the investigation of the integral matrices Γm from Section 4.3, in par-ticular Definition 4.21 of the set Γm,prim of primitive matrices and its set V (m, prim)of representatives with respect to the canonical left Γ -operation.

Definition 5.14 (Class invariants and modular polynomial). Consider a positiveinteger m ∈ N and the complete set of representatives of Γ \Γm,prim

V (m, prim) = M1, ...,Mψ(m).

Each matrix Mµ ∈V (m, prim) induces an automorphism of the upper half-plane

αµ : H→H, τ 7→Mµ(τ).

The functionsjµ := j αµ ∈M (H), µ = 1, ...,ψ(m),

are the class invariants of Γ \Γm. The modular polynomial of order m is the polyno-mial

Fm(X) :=ψ(m)

∏µ=1

(X− jµ) ∈M (H)[X ].

Note. The modular invariant is constant along the orbits of the operation Φ .Therefore the value of jµ in Definition 5.14 does not depend on representing a cosetfrom Γ \Γm,prim by the matrix M ∈V (m, prim) or by the matrix γ ·M,γ ∈ Γ .

Apparently, the roots of the modular polynomial are exactly the class invariants

jµ , µ = 1, ...,ψ(m),

corresponding to the Γ -left operation on Γm,prim. If


σµ , µ = 1, ...,ψ(m),

denotes the µ-th elementary symmetric function in ψ(m) variables, then up to signthe functions

σµ( j1, ..., jψ(m)) ∈M (H)

are the coefficients of the modular polynomial Fm(X). E.g.,

(−1)ψ(m)σψ(m) = j1 · ... · jψ(m)

is the constant term and

(−1) ·σ1( j1, ..., jψ(m)) =−( j1 + ...+ jψ(m))

is the coefficient of the term Xψ(m)−1.

The modular polynomial Fm(X) has coefficients from the field of meromorphicfunctions M (H). The next Theorem 5.15 investigates these meromorphic functions.The theorem shows that the coefficients are Z-linear combinations of the modularinvariant j, i.e. the coefficients of the modular polynomial belong to the smallersubring

Z[ j]⊂M (H).

Theorem 5.15 (The modular polynomial). For all positive integers m ∈ N

Fm(X) :=ψ(m)

∏µ=1

(X− jµ) ∈ Z[ j][X ].

Proof. We choose an arbitrary but fixed index µ = 1, ...,ψ(m) and consider themeromorphic function

c = cµ := σµ( j1, ..., jψ(m)) ∈M (H).

i) Weakly modular: According to Corollary 4.23 right multiplication with anelement γ ∈ Γ permutes the orbits of the Γ -left operation Φm

[M1], ..., [Mψ(m)]= [M1 · γ], ..., [Mψ(m) · γ].

Hence right multiplication does not change the value of the elementary symmetricfunctions: For all j = 1, ...,ψ(m)

σµ( j1, ..., jψ(m)) = σµ( j1 γ), ..., jψ(m) γ)).

Therefore c is weakly modular.

ii) Holomorphic on H: The modular invariant is holomorphic on H according toCorollary 4.16. Therefore also the coefficient c is holomorphic.


iii) Meromorphic at ∞: The coefficient c is weakly modular according to part i).Hence c has a Fourier expansion

c(τ) = ∑ν∈Z

aν ·qν , q = e2πiτ .

We have to show that the singularity q = 0 is a pole of c. For the proof we estimatethe growth of c: According to Corollary 4.16 the modular invariant has the Fourierexpansion

j(τ) =1q+P(q)

with P ∈ Zq a power series with integer coefficients. For a matrix

α =

(a b0 d

)∈V (m, prim)

the Fourier series of the class invariant j α derives from the Fourier series of j bysubstituting the variable q = e2πiτ by

e2πiα(τ) = e2πi(b/d) ·q(a/d) = ζmab · (q1/m)a2

.

Hereζm := e2πi/m,

denotes a primitive m-th root of unity. Therefore

|( j α)(τ)| ≤1|q|a/d +O(1)

for a single class invariant, and a constant K exists with

|c(τ)| ≤ K ·1|q|a/d

for small q ∈ ∆ \0.

iv) Integrality c ∈ OK [ j]: Due to the part already proved, Corollary 4.17 impliesthat c is a polynomial with complex coefficients in the modular invariant, i.e.

c ∈ C[ j].

We show that evenc ∈ OK [ j]

withK :=Q(ζm)

the m-th cyclotomic number field. For the proof we recall the substitution of q by


ζmab · (q1/m)a2

.

As a consequence the class invariant j α has a Fourier expansion with respectto q1/m, namely

j(α(τ)) =1

(q1/m)a2 ·ζmab +P(qa/d ·ζm

ab),

which shows that the Fourier coefficents of c belong to K. According to Corollary 4.17the coefficients of the polynomial c ∈ C[ j] are Z-linear combinations of the Fouriercoefficients of c. Hence the coefficients are algebraic integers from

Z+Z ·ζm = OK .

v) c ∈ Z[ j]: The operation of the Galois group

Gal(K/Q)' (Z/mZ)∗.

extends to an operation on the set j1, ..., jψ(m) of class invariants

Gal(K/Q)× j1, ..., jψ(m)→ j1, ..., jψ(m), (χ, jµ) 7→ jµχ :

Assume χ ∈Gal(K/Q) maps χ(ζm) = ζme with (e,m) = 1. If jµ is the class invari-

ant of the matrix

M =

(a b0 d

)then we define jµ

χ as the class invariant of the matrix

Mχ =

(a bχ

0 d

)with

bχ ≡ be mod d.

Because the function c is fixed under the operation of Gal(K/Q), its Fourier coeffi-cients must belong to the fixed field Q. Due to part iv) the Fourier coefficients evenbelong to

Q∩OK = Z, q.e.d.

We now evaluate the modular polynomial Fm(X) ∈ Z[ j][X ] at X = j, i.e. weconsider the polynomial in j

Φm( j) := Fm( j, j) =ψ(m)

∏µ=1

( j− jµ) ∈ Z[ j].

Lemma 5.16 (Leading coefficient of the modular polynomial). For squarefree m≥ 2the polynomial


Φm( j) ∈ Z[ j]

has leading coefficient ±1.

Proof. The functionΦm( j) : H−→ C

is an automorphic function. Consider its Fourier expansion

Φm( j)(τ) =ck

qk +ck−1

qk−1 + higher terms

Because j has a pole at τ = ∞ of order = 1, the coefficient ck of the Fourier series isalso the leading coefficient of Fm as polynomial in j. The coefficient is the productof the highest negative coefficients of the factors

j− jµ , µ = 1, ...,ψ(m).

For arbitrary but fixed µ the representation

j(τ) =1q+P(q)

implies

jµ(τ) = j

(aτ +b

d

)=

e−2πi(b/d)

qa/d +P(

qa/d · e2πi(b/d))

with jµ corresponding to the matrix(a b0 d

)∈V (m, prim).

Hence the highest negative coefficient in the Fourier expansion of j− jµ is1 a < de−2πi(b/d) a > d

Note that the case a = d is excluded because m = a ·d is squarefree. In any case thehighest negative coefficent of

j− jµ

is a root of unity, and the same holds for the coefficient ck of the product

Φ =ψ(m)

∏µ=1

( j− jµ).

Because ck ∈ Z we have ck =±1, q.e.d.


According to Theorem 5.15 the modular polynomial

Fm( j,X) =ψ(m)

∏µ=1

(X− jµ) ∈ Z[ j,X ]

is a polynomial in the two variables j and X with integer coefficients. We indicateits polynomial dependency on the two variables by the notation

Fm( j,X) =N

∑n=0

(∑

r+s=ncrs · jr ·X s

), crs ∈ Z.

The following Proposition 5.17 shows that Fm(t,X) is symmetric in both variables,i.e. the coefficients satisfy

crs = csr.

As a consequence, the highest exponent of j in Fm( j,X) is ψ(m).

Proposition 5.17 (Symmetry of the modular polynomial). The modular polyno-mial

Fm( j,X) ∈ Z[ j,X ]

is symmetric with respect to j and X, i.e.

Fm( j,X) = Fm(X , j).

Proof. We set R = Z[ j] with quotient field

K := Q(R) =Q( j).

i) Irreducibility of Fm(X) ∈ R[X ] over the base field K: The polynomial

Fm(X) =ψ(m)

∏µ=1

(X− jµ(τ)) ∈ R[X ]

has the rootsjµ , µ = 1, ...,ψ(m).

To prove that Fm is irreducible, we show that the Galois group

Gal(L/K)

of the splitting field L=K( j1, ..., jψ(m)) of Fm(X) acts transitively on the roots. Thenno root belongs to the fixed field K. For the proof set

jµ = j αµ

with matrices


αµ ∈V (m, prim), µ = 1, ...,ψ(m),

auch thatαµ : µ = 1, ...,ψ(m) ⊂V (m, prim)

is a complete set of representatives of Γ \Γm. According to Corollary 4.23 forany γ ∈ Γ the same holds for the set

αµ · γ : µ = 1, ...,ψ(m) ⊂ Γm.

Hence the Γ -orbit of any matrix (αµ · γ) equals the Γ -orbit of a suitable matrix αµ ′ .Therefore the functions

( j (αµ · γ))µ=1,...,ψ(m)

are a permutation of the functions

( j α)µ=1,...,ψ(m).

As a consequence, for any γ ∈ Γ the map

L−→ L, f 7→ f γ

is an element of the Galois group Gal(L/K). For each pair (ν ,µ) exist elements γ ′,γ ∈ Γ

withαµ = γ

′ ·αν · γ.

Therefore Gal(L/K) operates transitively on the roots of Fm(X), and the latter poly-nomial is irreducible over K.

ii) Common root of Fm( j,X) and Fm(X , j): We set

f (X) := Fm( j,X) and g(X) := Fm(X , j).

For any matrix α ∈V (m, prim) the function

xα := j α

is a root of the polynomial f (X). In particular,

α :=(

1 00 m

)∈V (m, prim)

satisfies for all τ ∈H

f (xα)(τ) = Fm( j(τ),( j α)(τ)) = Fm( j(τ), j(τ/m)) = 0.

The latter equation can be also interpreted: For all τ ∈H

0 = Fm( j(m · τ), j(τ)) = Fm(( j β )(τ), j(τ)) = g(xβ )(τ)


with the matrix

β :=(

m 00 1

)∈V (m, prim).

But j β is also a zero of f (X) because β = αµ for a suitable µ ∈ 1, ...,ψ(m).

We now apply the basic fact about the minimal polynomial of algebraic fieldextensions: Both polynomials

f (X), g(X) ∈ K[X ]

have the common zeroxβ := j β .

The polynomial f (X) is irreducible, see part i), with leading coefficient = 1. There-fore f (X) is the minimal polynomial of the field extension k⊂ K[xβ ], and a polyno-mial h(X) ∈ K[X ] exists with

g(X) = h(X) · f (X).

According to Theorem 5.15 both polynomials f (X) and g(X) have their coefficientsalready in the ring R. The ring R is factorial as a ring of polynomials over the fac-torial ring Z. The Lemma of Gauss implies that also the polynomial h[X ] has allcoefficients from R. Set

H( j,X) := h(X) ∈ R[X ] = Z[ j][X ].

ThenFm(X , j) = H( j,X) ·Fm( j,X) ∈ Z[ j,X ].

iii) Invariance: The last equation from part ii), considered as an equation of polyno-mials in two variables, is invariant when interchanging the role of j and X . Hencealso

Fm( j,X) = H(X , j) ·Fm(X , j).

ThereforeFm( j,X) = H(X , j) ·H( j,X) ·Fm( j,X),

H(X , j) ·H( j,X) = 1,

andH(X , j) = H( j,X) =±1.

The case H(X , j) =−1 would imply H( j, j) =−H( j, j) and

H( j, j) = 0,

contradicting the fact that Φm( j) = Fm( j, j) has leading coefficient ±1 according toLemma 5.16. Therefore H(X , j) = 1 and


Fm( j,X) = Fm(X , j), q.e.d.

E.g. the modular polynomial F2( j,X) ∈ Z[ j,X ] with ψ(2) = 3 is

F2( j,X) = (X3 + j3)+1.488(X2 j+X j2)−162.000(X2 + j2)−X2 j2+

+8.748.000.000(X + j)+40.773.375(X + j)−157.464.000.000.000.

Theorem 5.18 (Integrality of the modular invariant). For any imaginary quadraticirrational number τ ∈H the value j(τ) ∈ C is an algebraic integer.

Proof. Consider an imaginary quadratic field

K =Q(√−d)

with d ∈ Z, d > 0 squarefree. Set

OK = Z+Z · z.

i) j(z) is integral over Z: Any algebraic integer ξ ∈OK has norm N(ξ )∈Z. Choose

ξ :=

√−d d > 1

1+ i d = 1

Then N(ξ ) ∈ Z is squarefree. Multiplication with ξ defines an endomorphism

OK −→ OK , x 7→ ξ · x.

Hence a matrix

α =

(a bc d

)∈M(2×2,Z)

exists with

ξ ·(

z1

)= α ·

(z1

)The last equation shows:

• The matrix α has the two complex eigenvalues ξ and ξ . Hence

m := N(ξ ) = det α ∈ Z

is squarefree, which implies in particular that α is primitive: α ∈ Γm,prim.


• The generator z ∈ OK satisfies

z =az+bcz+d

= α(z).

Because a matrixαµ ∈V (m, prim), µ ∈ 1, ...,ψ(m),

exists with the same Γ -orbit as α . W.l.o.g.

µ = 1.

Thenj(z) = j(α(z)) = j(α1(z)) = j1(z),

which implies: The function

Φm( j) =ψ(m)

∏µ=1

( j− jµ) ∈ Z[ j]

has the zero z ∈H:Φm( j)(z) = 0.

We know

Φm( j) =N

∑ν=0

aν · jν ∈ Z[ j]

with integer coefficients aν . Due to Lemma 5.16 the leading coefficient satsfies aN =±1.Hence

0 = Φm( j)(z) =N

∑ν=0

aν · j(z)ν

proves that j(z) is integral over Z.

ii) j(τ) is integral over Z[ j(z)]: The element

τ ∈Q+Q · z

has a representation

τ = β (z) :=az+b

dwith a primitive matrix

β =

(a b0 d

)∈ Γm,prim, m := a ·d.

Alike to the final argument from part i) the Γ -orbit of β passes through a matrix αµ ∈V (p, prim),which proves

j(τ) = j(β (z)) = j(αµ(z)) = jµ(z).


Theorem 5.15 implies that jµ is integral over the ring Z[ j]. Hence j(τ) = jµ(z) isintegral over Z[ j(z)].

iii) j(τ) is integral over Z: When combining part i) and part ii) the transitivity ofintegral dependence proves the integrality of j(τ) over Z, q.e.d.

Lemma 5.19 (Operation of the Galois group). Consider an elliptic curve

E ⊂ P2(C)

defined as the variety of a homogeneous polynomial F(X0,X1,X2) ∈ C[X0,X1,X2].

1. Any element σ ∈ Gal(C,Q) defines the conjugate elliptic curve

Eσ :=V (Fσ )

withFσ ∈ C[X0,X1,X1]

the conjugate polynomial.

2. The assignmentE 7→ Eσ

extends in a functorial way to morphisms: If

φ : E1 −→ E2

is a regular map between two elliptic curves in P2(C), then σ defines in a canon-ical way a regular map

φσ : E1

σ −→ E2σ

such thatidσ = id and (φ2 φ1)

σ = φ2σ φ1

σ .

3. The j-invariant transforms according to

j(Fσ ) = σ( jF).

Proof. IfF(X0,X1,X2) = ∑

ν

aν ·Xν

thenFσ (X0,X1,X2) := ∑

ν

σ(aν) ·Xν .

The discriminant of the defining Weierstrass polynomial transforms as

∆(Fσ ) = σ(∆F) 6= 0,


and analogously transforms the j-invariant. Also regular maps transform with σ

because they are locally the quotient of two polynomials, q.e.d.

The class number of an imaginary quadratic number field K relates to the valueof the modular function j at the generator z ∈H of its ring OK of integers.

Theorem 5.20 (Lower bound of the class number). Consider an imaginary quadraticnumber field K and denote by

OK = Z+Z · z, z ∈H,

its ring of integers. Then the class number hK satisfies the estimate

[Q( j(z)) : Q]≤ hK .

Note that j(ω) ∈ C is an algebraic integer due to Theorem 5.18. Theorem 5.20holds even in the strong form

[Q( j(z)) : Q] = hK .

Proof. During the proof we will often switch between tori and elliptic curves, usingimplicitly Theorem 3.20 and Theorem 4.18.

If the extension field Q( j(ω)) has degree

m := [Q( j(z)) : Q]

then m automorphisms exist

id = σ1,σ2, ...,σm ∈ Gal(C/Q)

such that the conjugates

σ1( j(z)), ...,σm( j(z)) ∈ C

are pairwise distinct.

We consider the normalized lattice Λz := OK and the torus T = C/Λz. We denoteby E the elliptic curve corresponding to T . Due to Lemma 5.19 forany σ ∈ Gal(C/Q) the conjugate elliptic curve Eσ has the j-invariant σ( j(z)) andthe same ring of endomorphisms

End(Eσ )' End(E).

Because the elements of the family (σµ( j(z))µ=1,...,m are pairwise distinct, theelliptic curves

(Eσµ )µ=1,...,m


with these j-invariants are pairwise non-isomorphic. Theorem 5.12 implies that thecorresponding ideal classes are pairwise distinct. Their number is bounded by theclass number

hK = card Cl(K).

Hence[Q( j(z) : Q] = m≤ hK , q.e.d.

Corollary 5.21 (Integer values of the modular invariant). Consider an imaginaryquadratic number field K with class number hK = 1. If

OK = Z+Zz

then j(z) ∈ Z.

Proof. The proof follows from Theorem 5.18 and Theorem 5.20.

Chapter 6Outlook to Congruence Subgroups

6.1 Modular forms of congruence subgroups

The theory of modular forms Mk(Γ ) of the modular group Γ = SL(2,Z) has beenstarted in Chapter 4. The whole theory generalizes to the congruence subgroupsof Γ . These groups are defined by arithmetic properties. In the present section weconsider one type of congruence subgroups.

Definition 6.1 (Operation of the conguence subgroups Γ0(N)). Consider a posi-tive integer N ∈ N∗ and the residue map

Z→ Z/NZ.

1. The congruence subgroup of upper triangular matrices of level N is

Γ0(N) :=(

a bc d

)∈ SL(2,Z) : c≡ 0 mod N

2. The canonical left-action of the modular group from Definition 2.26 on the ex-tended half-plane H∗ restricts to a left-operation

ΦN : Γ0(N)×H∗→H∗.

The orbits of the points from Q∪∞ are the cusps of Γ0(N).

Due to the possibility of several cusps Definition 4.2 for modular forms general-izes as follows:

Definition 6.2 (Modular forms of the congruence subgroup Γ0(N)). Consider acongruence subgroup Γ0(N) with a positive integer N ∈ N∗.

153

154 6 Outlook to Congruence Subgroups

1. A modular form of weight k ∈ Z, k ∈ Z, with respect to Γ0(N) is a function

f : H→ C

with the following properties:

• The function f is holomorphic.

• For all γ ∈ Γ0(N) holds f [γ]k = f .

• For all γ ∈ Γ the function f [γ]k is holomorphic at ∞, i.e. it expands into aFourier series with non-negative indices.

If in addition for all γ ∈ Γ the Fourier series of f [γ]k has no constant term,then f is a cusp form of weight k with respect to Γ0(N).

The symbolsMk(Γ0(N)) and Sk(Γ0(N))⊂Mk(Γ0(N))

denote the complex vector space of modular forms of weight k and level N withrespect to the congruence subgroup Γ0(N) and its subspace of cusp forms.

2. For each pair of positive integers (d,M) with (dM)|N on has an injective map

jd,M : Sk(Γ0(M)) → Sk(Γ0(N)), f (τ) 7→ f (d · τ).

The image ∑(d,M)

(dM)|N, M 6=N

im jd,M

⊂ Sk(Γ0(N))

is named the subspace of old forms, and its orthogonal complement with re-spect to the Petersson scalar product the subspace Snew

k (Γ0(N)) of newformsof Sk(Γ0(N)).

Remark 6.3 (Modular curve). Consider a congruence subgroup Γ (N) with a positiveinteger N ∈ N∗.

The orbit space of the operation ΦN is a compact Riemann surface, named themodular curve X0(N) of Γ0(N). For even k ∈ Z a suitable line bundle

ΩΓ0(N),k ∈ Pic(X0(N))

and a canonical isomorphism exist

Mk(Γ0(N))'−→ H0(X0(N),ΩΓ0(N),k).

6.1 Modular forms of congruence subgroups 155

The injection Γ0(N) → Γ of the groups induces a branched covering of the corre-sponding modular curves

X0(N)→ X .

Example 6.4 (Congruence subgroups Γ0(2) and Γ0(4)).

1. The congruence subgroup Γ0(4)⊂ Γ has index [Γ : Γ0(4)] = 6. The right neben-classes Γ0(4) · γ are represented by the elements(

1 00 1

),

(0 −11 0

),

(1 01 1

),

(0 −11 2

),

(0 −11 3

),

(0 −11 4

).

2. The congruence subgroup Γ0(4) has three cusps, namely

• Orbit of ∞ ∈ H∗:

pq∈Q : (p,q) = 1 and (4,q) = 4

∪ ∞, q divisible

by 4.

• Orbit of 1/2∈H∗:

pq∈Q : (p,q) = 1 and (4,q) = 2

, q even, not divisible

by 4,

• Orbit of 0 ∈H∗:

pq∈Q : (p,q) = 1 and (4,q) = 1

, q odd.

3. The modular curve X0(4) has genus g = 0, hence X0(4)' P1. The modular curveis a 6-fold branched covering

X0(4)→ X ' P1

4. For even k the dimension of the space of modular forms is

dim Mk(Γ0(4)) =

(3/2) · k k ≥ 21 k = 00 k < 0

and the subspace of cusp forms has dimension

dim Sk(Γ0(4)) =

3 ·

(k2−1

)k ≥ 4

0 k ≤ 2

All formulas stated above are the particular case N = 4 of general formulas whichinvoke the Euler function


Φ(N) = card (Z/NZ)∗

and the number of orbits with non-trivial isotropy group, in particular the numberof cusps. cf. [7].

See PARI-file Congruence_subgroup_20:

Fig. 6.1 Modular forms of Γ0(2)

Figure 6.1 shows: S12(Γ0(2)) contains 2 two old forms. Both result from the cuspform ∆ ∈ S12(Γ ) embedded via

j1,1(∆)(τ) := ∆(τ)

and viaj2,1(∆)(τ) := ∆(2 · τ).

6.1 Modular forms of congruence subgroups 157

Fig. 6.2 Modular forms of Γ0(4)

Analogously Figure 6.2 shows:

• S8(Γ0(4)) contains 2 two old forms. Both result from the cusp form ∆8 ∈ S8(Γ0(2))embedded via

j1,2(∆8)(τ) := ∆8(τ)

and viaj2,2(∆8)(τ) := ∆8(2 · τ).

• S10(Γ0(4)) contains 2 two old forms. Both result from the cusp form ∆10 ∈ S10(Γ0(2))embedded via

j1,2(∆10)(τ) := ∆10(τ)

and viaj2,2(∆10)(τ) := ∆10(2 · τ).

• S12(Γ0(4)) contains 3 two old forms. They result from the cusp form ∆ ∈ S12(Γ )embedded via

j1,1(∆)(τ) := ∆(τ),

viaj2,1(∆)(τ) := ∆(2 · τ)

and viaj4,1(∆)(τ) := ∆(4 · τ).


6.2 The four squares theorem

The four squares theorem answers the question: How many possibilities exist todecompose a positive integer as the sum of four integer squares?

Fermat recalls in a letter from 1659 to a friend those results from arithmeticwhich he considers his most important contribution to this field. One of these resultsis the claim that any natural number can be written as the sum of four squares. Thefirst prove of this theorem has been given by Lagrange in 1770. The more refinedversion of Theorem 6.8 is due to Jacobi in 1834. For the history of these issues see[30].

Definition 6.5 (Representing integers as sum of squares). The number of possi-bilities to represent a non-negative integer n as the sum of k integer squares definesthe arithmetic function

r : N∗×N∗→ N∗

withr(n,k) := card v = (v1, ...,vk) ∈ Zk : n = v2

1 + ...+ v2k.

Note that r(n,k) counts separately k-tuples with positive and negative signs, anddistinguishes also k-tuples with the same components, but in different order.

Definition 6.6 (Generating function θ ). For arbitrary, but fixed k ∈ N∗ the gener-ating function of the family r(n,k)n∈N from Definition 6.5 is denoted

θ(τ,k) :=∞

∑n=0

r(n,k) ·qn, τ ∈H, q = e2πiτ .

Proposition 6.7 (Generating function as modular form). The generating function θ(−,4)is a modular form of weight 2 of the congruence subgroup Γ0(4), i.e.

θ(−,4) ∈M2(Γ0(4)).

Its Fourier series starts

θ(τ,4) = 1+8q+O(2), q = e2πiτ .

Theorem 6.8 (Four squares theorem). For any n ∈ N, n≥ 1, there are

r(n,4) = 8 ·∑d|n4-d

d

6.3 Modularity of elliptic curves 159

possibilities to represent the non-negative integer n as the sum of 4 integer squares.In particular for 4 - n

r(n,4) = 8 ·σ1(n) = 8 ·∑d|n

d.

Proof. Computation within M2(Γ0(4)):

i) Dimension: dim M2(Γ0(4)) = 2 according to Example 6.4.

ii) For each N ≥ 2 the Eisenstein series

G2,N(τ) := G2(τ)−N ·G2(Nτ)

is a modular form of the congruence subgroup Γ0(N) of weight 2, i.e.

G2,N(τ) ∈M2(Γ0(N)).

The two Eisenstein series G2,2 and G2,4 have the Fourier expansion

G2,2(τ) =−π2

3

1+24 ·∞

∑n=1

( ∑d|n

d odd

d) qn

=−π2

3(1+24q+O(2))

and

G2,4(τ) =−π2

(1+8 ·

∞

∑n=1

(∑

d|n, 4-nd

)qn

)=−π

2(1+8q+O(2)).

Comparing terms of order 0 and 1 in the Fourier expansions shows that bothEisenstein series are linearly independent. Hence the family (G2,2,G2,4) is a basisof M2(Γ0(4)). In particular, it generates θ(−,4). Comparing Fourier coefficients oforder 0 and 1 shows

θ(−,4) =−1

π2 G2,4.

Equating on both sides the Fourier coefficients of order n ∈ N∗ shows

r(n,4) = 8 · ∑d|n, 4-d

d, q.e.d.

6.3 Modularity of elliptic curves

Corollary 4.27 represents the discriminant modular form ∆ as simultaneous eigen-form of all Hecke operators (Tm)m≥1 acting on the complex vector space


V := S12(Γ ).

The corresponding eigenvalues are the Fourier coefficients of ∆ . We now translatethe law of quadratic reciprocity into a similar shape, following the preface from [7].

Remark 6.9 (Reduction of quadrics modulo primes).

1. Consider an arbitrary, but fixed integer d 6= 0 and the quadratic polynomial

F(X) := X2−d ∈ Z[X ]

which defines the affine quadric Q/Q

x ∈ C : F(x) = 0.

For each prime p we consider the reduction mod p

Fp(X) := X2−dp ∈ Fp[X ]

and the set Q(Fp) of points of Q with coordinates from Fp.

Analogously to Remark 3.23 for elliptic curves we define

ap(Q) := card Q(Fp)−1.

The value 1 in the definition is the average number of solutions mod p. Appar-ently, ap(Q) equals the Legendre symbol

ap(Q) =

(dp

).

The values extend to an(Q), n ∈ N, and satisfy the product rule

anm(Q) = an(Q) ·am(Q).

Due to the Law of Quadratic Reciprocity the Legendre symbol depends only onthe residue class p mod 4d.

2. We now represent the sequence (ap(Q))p prime as a sequence of eigenvalues ofa simultaneous eigenvector for a suitable sequence of linear endomorphisms(Tp)p prime. Our aim is to formulate an analogue of the modular group, of modu-lar functions and of Hecke operators:

SetN = NQ := 4|d|,

and consider the multiplicative group

G := (Z/NZ)∗,

6.3 Modularity of elliptic curves 161

and the complex vector space of functions defined on G

V := f : G→ C.

Eventually, as an analogue to the Hecke operators we introduce the family oflinear endomorphisms (Tp)p prime of V defined as

(Tp f )(n) :=

f (p ·n) p - N0 p | N

for n ∈ G. We claim: All endomorphism (Tp)p prime have a simultaneous eigen-vector, namely the distinguished function

fQ : G→ C, f (n) := an(Q).

Note that fQ is well-defined on the residue classes of G because the Legendresymbol depends on the residue class p mod 4d. Computing the eigenvalues weget for all n ∈ G

Tp( fQ)(n) =

fQ(p ·n) = apn(Q) = ap(Q) ·an(Q) p - N

0 p | N

As a consequence, for all primes p ∈ Z

Tp( fQ) = ap(Q) · fQ.

Remark 6.10 (Shimura-Taniyama-Weil conjecture). Consider an elliptic curve E/Qdefined over Q. We introduced in Remark 3.23 for any prime p the value

ap(E) := 1+ p− card E(Fp),

the difference between the average number of points of E with components from Fpand the actual number of points. The Shimura-Taniyama-Weil conjecture relatesthese numbers to the eigenvalues of the Hecke operators acting on a suitable cuspform.

The proof of the Shimura-Taniyama-Weil conjecture is due to Wiles, Taylor,Breuil, Conrad, Diamond as a statement about modular forms:

Theorem 6.11 (Modularity Theorem). For any elliptic curve E/Q denote by NE ∈ Nthe conductor of E. Then a newform

fE ∈ S2(Γ0(NE))

with Fourier coefficients (an)n≥1 exists, such that for all primes p ∈ Z


ap(E) = ap.

The conductor NE of E has the same prime factors as the discriminant of E.The exponent of a prime p in NE depends on the type of the unstable reduction ofE mod p.

For the modularity theorem cf. [7, Theor. 8.8.1]. This textbook does not provethe theorem. But it provides an excellent introduction to the problem and highlightsdifferent views onto the theorem. In joint work with his coworkers, one of the au-thors of [7] proved the final step of the modularity theorem.

For the context of Theorem 6.11 see also the textbook on elliptic curves [35, App. C, § 16].For its proof see [5] and [6].

References

References for these notes are grouped as

• Modular Forms: [3] [5] [6] [9] [19] [20] [22] [24] [26] [27] [28] [29] [?] [33][34] [39]

• Complex Analysis: [1] [2] [10] [14] [17]

• Algebraic Geometry: [12] [15] [31] [35] [36]

• Number Theory: [4] [11] [13] [16] [25] [30]

1. Ahlfors, Lars: Complex Analysis. 2nd edition, McGraw-Hill, Tokyo (1966)2. Ben-Zvi, David, D.: Moduli Spaces. https://www.ma.utexas.edu/users/benzvi/math/pcm0178.pdf3. Chenevier, Gaetan: Introduction aux Formes Moulaires. Internet (2015)4. Cohen, Henri: A Course in Computational Algebraic Number Theory. Springer, Berlin (1993)5. Cornell, Gary; Silvermann, Joseph H.; Stevens, Glenn (Eds.): Modular Forms and Fermat’s

Last Theorem. Springer, New York (1997)6. Darmon, Henri: A proof of the full Shimura-Taniyama-Weil conjecture is announced. Notices

Americ. Math. Soc. 46,11 (1999), p. 1397-14017. Diamond, Fred; Shurman, Jerry: A First Course in Modular Forms. Springer, New York

(2005)8. Dugundji, James: Topology. Allyn and Bacon, Boston (1966)9. Eichler, Martin; Zagier, Don: On the Zeros of the Weierstrass ℘-Function. Mathematische

Annalen 258, (1982), p. 399-40710. Forster, Otto: Riemannsche Flachen. Springer, Berlin (1977)11. Forster, Otto: Algorithmische Zahlentheorie. 2. Auflage Springer Spektrum (2015)12. Fulton, William: Algebraic Curves. An Introduction to Algebraic Geometry. The Ben-

jamin/Cummings Publishing Company (1969)13. Gross, Benedict: The arithmetic of elliptic curves - An update. Arabian Journal of Science

and Engineering 1 (2009), p. 95-103http://www.math.harvard.edu/˜gross/preprints/ell2.pdf

14. Gunning, Robert C.: Lectures on Riemann Surfaces. Princeton University Press (1966)15. Hartshorne, Robin: Algebraic Geometry. Springer, New York (1977)16. Ireland, Kenneth; Rosen, Michael: A Classical Introduction to Modern Number Theory.

Springer, New York (1982)

163

164 References

17. Janich, Klaus: Funktionentheorie. 6. Auflage Springer (2011)18. Knapp, Anthony: Elliptic Curves. Princeton University Press. Princeton (1992)19. Kilford, Lloyd James: Modular forms: a classical and computational introduction. Imperial

College Press, 2nd ed. (2015)20. Koecher, Max; Krieg, Aloys: Elliptische Funktionen und Modulformen. 2. Aufl. Springer,

Berlin (2007)21. Lang, Serge: Algebra. Addison-Wesley, 7th printing (1977)22. Lang, Serge: SL(2,R). Springer, New York (1985)23. Lang, Serge: Elliptic Functions. 2nd edition Springer, Berlin (1987)24. Lang, Serge: Introduction to Modular Forms. Springer, Berlin (1995)25. Marcus, Daniel A.: Number Fields. Springer, New York (1977)26. Mordell, Louis Joel: On Mr. Ramanujan’s empirical expansion of modular functions. Proc.

Cambridge Philosophical Society, 19 (1917), p. 117-12427. Ono, Ken: The last words of a genius. Notices Americ. Math. Soc. 57,11 (2010), p.1410-141928. Pantchichkine, Alexeı: Formes Modulaire et Courbes Elliptiques. https://www-fourier.ujf-

grenoble.fr/ panchish/06ensl.pdf29. Ramanujan, Srinivasa: On certain arithmetical functions. Trans. Cambridge Philosophical So-

ciety. 22(9) (1916), p. 159-18430. Scharlau, Winfried; Opolka Hans: Von Fermat bis Minkowski. Eine Vorlesung uber Zahlen-

theorie. Springer, Berlin (1980)31. Serre, Jean-Pierre: Geometrie Algebrique et Geometrie Analytique. Annales de l’Institut

Fourier. IV (1955-56), p. 1-4232. Borel, A.; Chowla, S.; Herz, C.; Iwasawa, K.; Serre, J-P.: Seminar on Complex Multiplication.

Lecture Notes in Mathematics 21. Springer, Berlin (1966)33. Serre, Jean-Pierre: A Course in Arithmetic. Springer, New York (1973)34. Shimura, Goto: Introduction to the Arithmetic Theory of Automorphic Functions. Iwanami

Shoten and Princeton University Press, o.O. (1971)35. Silvermann, Joseph: The Arithmetic of Elliptic Curves. Springer, New York (1986)36. Silvermann, Joseph; Tate, John: Rational points on Elliptic Curves. Springer, New York

(1992)37. Szpiro, Lucien: Seminaire sur les pinceaux arithmetiques: La conjecture de Mordell.

Asterisque 127. Societe Mathematiques de France, Paris (1985)38. van der Geer, Gerard: Siegel Modular Forms. arXiv:math/0605346 [math.AG]39. Zagier, Don: https://www.youtube.com/watch?v=zKt5L0ggZ3o (2015)

Index

L-series, 86ζ -function, 86ap(E), 85j-invariant, 69

automorphic form, 91automorphic function, 91automorphy factor, 93

base point, 30branch point, 22

canonical divisor, 28class number, 136compact Riemann surface

genus, 26complex multipliction, 131congruence subgroup, 153cusp

of congruence subgroup, 153

degreeof a divisor, 22of a holomorphic map, 22of a plane curve, 63

discrete subgroup, 4discriminant form, 102discriminant of a Weierstrass polynomial, 68divisor, 22doubly periodic, 5dual line bundle, 27dualizing sheaf, 27

Eisenstein series, 97elliptic curve, 63elliptic function, 5elliptic point, 36

Fourier coefficients, 90fractional ideal, 135fundamental domain, 42

group of divisor classes, 23

imaginary quadratic number field, 132

K-valued point, 63

lattice, 5lattice

similar, 33local ring, 62

meromorphic function, 20minimal Weierstrass polynomial, 69modular form, 91modular function, 91modular group

operation on upper half-plane, 36modular invariant, 103Modularity theorem, 161

newform, 154non-singular, 62normalized Eisenstein series, 98normalized lattice, 34number field, 61

orbit space, 36order, 132

perfect field, 61period, 3period

group, 4

165

166 Index

period lattice, 5period parallelogram, 5Picard group, 25positively oriented basis, 33principal divisor, 22

Ramanujan function, 125ramification order, 29ramification point, 22rational function, 64reduction

additive, 83bad reduction, 83good reduction, 83

regular local ring, 62residue field, 62Riemann ζ -function, 98Riemann surface, 20

semistable, 83Serre duality, 27

Shimura-Taniyama-Weil conjecture, 161smooth, 62split-reduction, 83stable, 83

theoremRiemann-Hurwitz, 29Riemann-Roch, 27

torus, 20

unstable, 83upper half-plane, 20

very ample line bundle, 31

weakly modular, 91Weierstrass

℘-function, 8Weierstrass polynomial, 66Weierstrass polynomial in short form, 68

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