the tate conjectures for product and quotient varieties
TRANSCRIPT
THE TATE CONJECTURES FOR PRODUCTAND QUOTIENT VARIETIES
by
RACHID EJOUAMAI
A thesis submitted to the
Department of Mathematics and Statistics
in conformity with the requirements for
the degree of Doctor of Philosophy
Queen’s University
Kingston, Ontario, Canada
September 2013
Copyright c© Rachid Ejouamai, 2013
Abstract
This thesis extends Tate’s conjectures from the smooth case to quotient varieties. It
shows that two of those conjectures hold for quotient varieties if they hold for smooth
projective varieties. We also consider arbitrary product of modular curves and show
that the three conjectures of Tate (in codimension 1) hold for this product. Finally we
look at quotients of the surface V = X1(N)×X1(N) and prove that Tate’s conjectures
are satisfied for those quotients.
i
Acknowledgements
First of all, I want to express my gratitude to my advisor, Professor Ernst Kani, who
gave me another chance to complete my studies after I had to leave years ago due to a
medical condition. I would like to thank him also for introducing me to this beautiful
topic, for his support and encouragement, for spending a big amount of time teaching
me various things in mathematics, for being generous with his ideas, and for his wise
instructions. Finally, I would like to think him for being extremely patient with me
during the past years.
I would like to thank Professor Mike Roth, who made a suggestion during the time
of my comprehensive exam, which helped with a great deal in this thesis. I would also
like to think him for giving me the permission to use one of his unpublished results,
and for allowing me to have a copy of his preprint.
I would like to thank Professor Ram Murty, for giving me the chance to speak in
the number theory seminar in three occasions, and for making helpful comments an
suggestions during my talks.
I would like to thank the department staff, especially the graduate secretary,
Jennifer Read for her helpfulness, kindness and competence.
I would like to think my family and my friends for their moral support.
ii
Statement of Originality
The main results of this thesis which constitute original work are
Theorem 3.2.1, Theorem 3.3.2, Theorem 3.5.1, Theorem 3.5.3, Theorem 4.1.10,
Theorem 4.3.1 Theorem 5.3.2, Theorem 5.3.4 and Theorem 5.3.6.
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Contents
Abstract i
Acknowledgements ii
Statement of Originality iii
Chapter 1 Introduction 1
1.1 Tate’s Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 An overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . 4
Chapter 2 Preliminaries 7
2.1 `- adic Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Cohomology with compact support . . . . . . . . . . . . . . . 8
2.1.3 Tate Twists . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.4 Cup product and Cohomology ring . . . . . . . . . . . . . . . 10
2.1.5 The fundamental class and the trace map . . . . . . . . . . . 11
2.1.6 Poincare Duality Theorem . . . . . . . . . . . . . . . . . . . . 13
2.2 Algebraic Cycles and the Cycle map . . . . . . . . . . . . . . . . . . 14
2.2.1 Algebraic Cycles and Numerical Equivalence . . . . . . . . . . 14
2.2.2 Flat pullback of cycles . . . . . . . . . . . . . . . . . . . . . . 14
2.2.3 The Cycle map . . . . . . . . . . . . . . . . . . . . . . . . . . 15
iv
2.3 L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 The finite field case . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.3 The global case . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.4 Compatible systems of `-adic modules . . . . . . . . . . . . . 18
2.4 Tate’s conjectures in the smooth projective case . . . . . . . . . . . . 21
Chapter 3 Quotient Varieties 23
3.1 The `-adic cohomology of a quotient variety . . . . . . . . . . . . . . 23
3.2 Poincare duality for quotient varieties . . . . . . . . . . . . . . . . . . 24
3.3 Cycles on a quotient variety and the cycle map . . . . . . . . . . . . 28
3.3.1 Cycles on a quotient variety . . . . . . . . . . . . . . . . . . . 28
3.3.2 Intersection product on a quotient variety . . . . . . . . . . . 29
3.3.3 Cycle map on a quotient variety . . . . . . . . . . . . . . . . . 30
3.4 The L-series of a quotient variety . . . . . . . . . . . . . . . . . . . . 33
3.5 Tate Conjectures for a Quotient Variety . . . . . . . . . . . . . . . . 33
Chapter 4 The Tate Conjectures for a Product of Modular Curves 39
4.1 Generalities on products of curves . . . . . . . . . . . . . . . . . . . . 39
4.1.1 Conjecture T 1 for products of curves . . . . . . . . . . . . . . 39
4.1.2 Conjecture TL1 for products of algebraic curves . . . . . . . . 40
4.2 Modular forms and modular abelian varieties . . . . . . . . . . . . . . 45
4.2.1 Modular forms and the Shimura construction . . . . . . . . . 45
4.2.2 Rankin-Selberg Convolution . . . . . . . . . . . . . . . . . . . 47
4.3 Tate’s conjecture for a product of modular curves . . . . . . . . . . . 48
4.3.1 Proof of the first identity . . . . . . . . . . . . . . . . . . . . . 49
4.3.2 Proof of the second identity . . . . . . . . . . . . . . . . . . . 51
Chapter 5 Quotients of Products of Modular Curves 53
5.1 Endomorphisms of Abelian Varieties and `-adic Cohomology . . . . . 53
5.2 Cohomology of modular Abelian varieties and the Hecke algebra . . . 55
5.3 Tate Conjectures for Quotients of a Product of Modular Curves . . . 58
v
5.3.1 Conjecture T 1 for quotients of a product of curves . . . . . . . 58
5.3.2 Conjecture TL1 for quotients of a product of two modular curves 58
Bibliography 70
vi
Chapter 1
Introduction
1.1 Tate’s Conjectures
In arithmetic geometry, one of the most basic but unsolved problems is the following.
Given an algebraic variety V defined over Q, determine the nature of the set V (Q)
of rational points of V . For example, if n ∈ Z is a given integer, does the elliptic
curve y2 = x3 + n2x have infinitely many rational solutions? This is a hard question,
but thanks to the theory of L-functions and the conjecture of Swinnerton-Dyer, an
explicit but conjectural answers to this question can be given, as is explained in
Koblitz’s book [17].
A natural generalization of the above basic problem is to determine the nature of
the set of the algebraic subvarieties of V that are defined over Q. A related question
is: how large is the group of rational cycles (i.e. formal sums of algebraic subvarieties)
up to numerical equivalence?
In 1963, J. Tate [33] formulated conjectural answers to the previous question. He
made some conjectures which relates cycles on a smooth projective variety to `-adic
cohomology groups. One of those conjectures, for instance, states that
the rank of the group of rational cycles up to numerical equivalence is
equal to the order of some pole of a certain L-series attached to the
cohomology group.
1
Such a conjecture is important because one does not know in general the size of the
group of cycles up to numerical equivalence, and this conjecture provides an analytic
tool to compute it. Tate was partially motivated by the Birch and Swinerton-Dyer
conjecture, which also gives an analytic method to compute the rank of the Mordell-
Weil group of an elliptic curve. In the same framework, there is another interesting
conjecture of Tate, which states that
the space of rational cycles up to numerical equivalence tensored with Q`
is isomorphic to the subspace of Galois invariants of the `-adic
cohomology space via a map called the cycle map.
Before one can even state this last conjecture, one has to assume another preliminary
conjecture regarding the cycle map. It states
the cycle map is injective, its kernel does not depend on `, and the
kernel coincides with the group of cycles which are numerically
equivalent to zero.
We refer to the preliminary conjecture as Conjecture Pr, to the conjecture on L-series
as TL and to the conjecture on Galois invariant cycles as T .
1.2 This Thesis
1.2.1 An overview
The main goal of this thesis is to demonstrate that Tate’s conjectures can be naturally
extended to quotient varieties by finite groups, which are not smooth in general. We
will prove the following result (cf. Theorems 3.5.1 and 3.5.3)
Theorem Let X be a smooth projective variety over a finitely generated field K of
characteristic zero, and let G be a finite group acting on X by K-automorphisms
a) If conjecture Pr holds for X, then it also holds for Y = X/G.
b) If conjecture T holds for X, then it also holds for Y = X/G.
2
Such a theorem is very important, as it extends Tate’s conjectures to quotients.
Especially, it gives results about the quotient variety Y = X/G whenever those
conjectures are known to be true for X. The proof of this theorem rests on the
validity of the Poincare duality theorem for quotient varieties. Note that this duality
theorem is well known for the smooth projective case, but that it is false in general for
singular varieties. Once established, the Poincare duality theorem allows us to define
a cycle map in the same way it is defined in the smooth projective case. Proving the
previous theorem also requires a non obvious functorial property of the cycle map.
As for the conjecture TL, one cannot hope that it will descend from X to X/G
like conjecture Pr and conjecture T . The main obstacle is that few things are known
about the L-series associated to a smooth projective variety. For instance, the ana-
lytic continuation for such L-series is still an open problem, and experts such as Serre
believe that it will only follow from another big conjecture in arithmetic geometry
which states: the L-series attached to smooth projective varieties can be expressed in
terms of L-series coming from automorphic forms. However, we will prove conjecture
TL for certain quotients of the product X1(N)×X1(N). This example at least makes
it legitimate to make the following conjecture.
Conjecture If Y = X/G is a quotient variety with X is smooth and projective, then
conjecture TL holds for Y as stated for the smooth projective case.
It should be mentioned that Jannsen [10] extended Tate’s conjectures to all (pos-
sibly singular) varieties by using homology instead of cohomology. However, his work
is highly technical since it involves his theory of mixed motives and deep facts about
K-theory. He then proved that his (extended) conjectures follow from Tate’s con-
jectures under certain extra hypotheses. However, without considerable work, it is
not clear whether his results cover the case of quotient varieties and, even if they do,
whether his results in this case are the same as those proved in these thesis. (Perhaps
one needs the Poincare duality theorem for quotient varieties to be able to compare
these results.) At any rate, it seems harder to extract them from his work rather than
to prove them directly as in this thesis.
3
1.2.2 Thesis organization
This thesis is organized as follows.
Chapter 2
This chapter is about general material that we will need in the subsequent chapters,
especially the basic properties of `-adic cohomology and algebraic cycles. We will
state the Poincare duality theorem for smooth varieties and then use it to define the
cycle map as explained in Katz’s paper [16].
Chapter 3
In this chapter we start by defining what is a quotient variety, and then we state an
important theorem on the `-adic cohomology for quotient varieties due to Dr. Roth
(Theorem 3.1.1). This theorem is useful in many occasions in this thesis, especially
in the proof of the Poincare duality theorem for quotient varieties. I should mention
that the idea of establishing a Poincare duality for quotient varieties was suggested to
me by Dr. Roth. Next we state and prove the Poincare duality theorem for quotient
varieties (Theorem 3.2.1). The proof of this theorem also requires an important
lemma (Lemma 3.2.2) which will also be used in other occasions.
In the subsequent section we discuss cycles on quotient varieties and cite a result
from Fulton’s book which identifies cycles on quotient varieties. We then give Ful-
ton’s definition of the intersection product on a quotient variety to define numerical
equivalence. Then we prove a functorial property of the cycle map on quotient vari-
eties (Theorem 3.3.2). This result is the main tool to prove that Tate’s conjectures
descend to quotient varieties.
Now for the L-series on quotient varieties, we state an important theorem due to
Dr. Kani (Theorem 3.4.1). This theorem is crucial in the last chapter. In fact, the
L-series on a quotient variety cannot be defined without this result.
The main results of this chapter are Theorems 3.5.1 and 3.5.3, which were already
mentioned in Subsection 1.2.1. They are stated and proved in the last section of
Chapter 3.
4
Chapter 4
In this chapter we focus on products of curves. In the first part of this chapter we
reduce conjecture TL1 (TL1 is TL for codimension 1) on a product of curves into
a statement about the simple components of the Jacobians of the curves (Theorem
4.1.10). In order to do this we use standard results, such as the Kunneth formula, and
the isomorphism between the first cohomology of a curve with that of its Jacobian.
In the second part of this chapter we give a proof of conjecture TL1 for an arbitrary
product of modular curves. This uses the first part of Chapter 4 together with
standard results on modular forms such as the Shimura construction and the theorems
of Rankin and V. Kumar Murty about the Rankin convolution.
Here we should mention that the result was already known for products of two
modular curves, more precisely for a product of the form X1(N)×X1(M). Apparently
this was proven in a unpublished work of J. Tunnell; cf. Murty [23], Section 7. Also
Ogg has proved similar results for a product of “modular” elliptic curves; cf. Ogg
[25]. More recently, Ramakrishnan was able to prove the conjecture TL for any
codimension for 4-fold products of the form X0(N1) × X0(N2) × X0(N3) × X0(N4).
cf. Ramakrishnan [26].
Chapter 5
In this chapter, we consider quotients of the product surface X1(N)×X1(N) by any
subgroup H ⊂ (Z/NZ)××(Z/NZ)× which acts via the diamond operators. We denote
this quotient by YH . The main result of this chapter is to prove conjecture TL1 for
YH . We start this chapter by reviewing the action of the algebra of endomorphisms of
an abelian variety on its `- adic cohomology (Theorem 5.1.1 and the discussion that
follow). Next we apply this to modular abelian varieties and we make the connection
with the Hecke algebra. Then we discuss the Galois action on the cohomology group
and state Shimura’s Theorem which determines the characteristic polynomial with
coefficients in the Hecke algebra (Theorem 5.2.1). We also state a result due to Ribet
regarding the components of the `-adic cohomology (Theorem 5.2.3). In the last
section we state and prove the main result of chapter 5 (Theorem 5.3.2). The proof
5
uses Theorem 5.3.4, Theorem 5.3.6 and other standard facts.
6
Chapter 2
Preliminaries
2.1 `- adic Cohomology
2.1.1 Basic Properties
Let X be a scheme which is separated and of finite type over a field K, let K denote
a separable closure of K, and let X be the base extension of X to K. Then for any
integers N ≥ 1 and k ≥ 0, one has cohomology groups: Hk(Xet, Z/NZ); cf. Katz [16],
p.22 or Milne [19], ch III. These are abelian groups killed by N and are endowed with
a left action of the absolute Galois group GK = AutK(K); this action is induced from
the right action of GK on X. Note that if f : X → Y is a morphism of K-schemes,
then we have an induced morphism:
f ∗ : Hk(Y et, Z/NZ)→ Hk(Xet, Z/NZ).
This follows from Milne [19] Remark III .1.6(c) and the fact that f ∗(Z/NZ) = Z/NZ.
It is immediate that f ∗ is GK-equivariant. As is explained in Milne [19], pp. 171-172,
the canonical pairing
Γ(Xet, Z/NZ)× Γ(Xet, Z/NZ)→ Γ(Xet, Z/NZ⊗ Z/NZ) = Γ(Xet, Z/NZ).
induces canonical pairing called the cup product
∪i,j : H i(Xet, Z/NZ)×Hj(Xet, Z/NZ)→ H i+j(Xet, Z/NZ),
7
for all i, j. It follows immediately from Milne [19], Proposition V.1.16, that these are
GK-equivariant and are functorial with respect to K-morphisms f : X → Y .
It is a fundamental (but difficult) theorem that the Hk(Xet, Z/NZ) are finite
groups if N is invertible in K; cf. Katz [16] p.22. In addition one has that
Hk(Xet, Z/NZ) = 0 for k > 2 dim(X). (1)
We now fix a prime ` which is invertible in K, and put
Hk(Xet, Z`) = lim←−n
Hk(Xet, Z/`nZ).
It is immediate that Hk(Xet, Z`) is a Z`-module with a continuous GK action. More-
over, it follows from Milne [19], Lemma V.1.11 and the previous finiteness assertion
that Hk(Xet, Z`) is a finitely generated Z`-module. In particular,
Hk` (X) := Hk(Xet, Z`)⊗Q`
is a finite dimensional Q`-vector space. It thus follows from the previous discussion
that Hk` (X) satisfies the following properties:
1. Finiteness and Vanishing: If ` is different from the characteristic of K
then: Hk` (X) is a finite dimensional vector space over Q` and it is zero for
k > 2 dim(X).
2. Galois Structure: The absolute Galois group GK = AutK(K) acts continu-
ously on Hk` (X).
3. Functoriality: If f : X → Y is a K-morphism of schemes then we have a GK-
equivariant morphism on cohomologies: f ∗ : Hk` (Y )→ Hk
` (X).
2.1.2 Cohomology with compact support
As before, let X be a separated scheme of finite type over K. By Nagata [24], there
exists an embedding j : X ↪→ X ′, where X ′/K is proper. Put
Hkc (Xet, Z/NZ) := Hk(X ′
et, j!(Z/NZ)).
8
where j!(Z/NZ) is defined as on p.76 of Milne [19]. It turns out (cf. Milne [19], p.277)
that this group does not depend on the choice of the pair (X ′, j), so in particular
Hkc (Xet, Z/NZ) = Hk(Xet, Z/NZ)
if X/K is proper.
If ` is a prime, then we have the `-adic cohomology group with compact support:
Hk`,c(X) = lim←−
n
Hkc (Xet, Z/`nZ)⊗Q`
These are again finite dimensional Q`-vector spaces with a continuous GK-action and
vanish for k > 2 dim(X); cf. Katz [16], p.22.
If f : X → Y is a morphism of K-schemes, then in general we do not have an
induced map: f ∗ : Hk`,c(Y )→ Hk
`,c(X) unless f is proper; cf. Milne [19], p.229.
We conclude this subsection with the following proposition:
Proposition 2.1.1. (a) Hk`,c(X) coincides with the ordinary `- adic cohomology when
X/K is proper.
(b) If X is irreducible of dimension d and U is an open dense subset then one has
an isomorphism: fU : H2d`,c(U)→ H2d
`,c(X).
Proof. The first statement follows from the previous discussion. For the second state-
ment, we observe that the proof of Lemma VI.11.3. in Milne [19] shows that we have
isomorphisms:
jU,N : H2dc (U, Z/NZ)→ H2d
c (X, Z/NZ)
for each N . Using the compatibility of these isomorphisms as N vary and taking
inverse limit over N = `n we get an isomorphism on the projective limits. Finally
tensoring with Q` gives the second statement.
2.1.3 Tate Twists
Since we will consider Tate twists of cohomology groups later, it is convenient to
introduce them first more generally.
9
For n ≥ 1, let µ(`n) denote the group of `n-th roots of the unity in K. The Galois
group acts on each µ(`n) via the mod `n cyclotomic character: χ`n : GK → (Z/`nZ)×.
Moreover, the groups µ(`n) form a projective system, and one defines the Tate module
T`(µ) to be the inverse limit T`(µ) = lim←−(µ(`n)); this is a free Z` module of rank one,
on which the Galois group GK acts via the `-adic cyclotomic character χ` : GK → Z×` .
Now let V`(µ) = T`(µ)⊗Q`; this is a one dimensional vector space over Q`, on which
the Galois group acts by acting on the first factor. For m ∈ Z we define the m-th
Tate twist of V`(µ) by:
V`(µ)(m) =
{(V`(µ))⊗m, m ≥ 0;
HomQ`((V`(µ))⊗−m, Q`), m < 0.
Now if H is a Q`- vector space with a GK action, we define its m-th Tate twist by:
H(m) = H ⊗ V`(µ)(m). It is clear from the definition that Tate twists satisfy the
identity: H(m + n) = H(m)(n) for all m, n ∈ Z.
2.1.4 Cup product and Cohomology ring
It follows from the discussion of sections 2.1.1 and 2.1.3 that for any integers k, j, m, n
one has Galois equivariant pairings:
∪k,j : Hk` (X)(n)×Hj
` (X)(m) −→ Hk+j` (X)(n + m)
with the property that if f : X → Y is a K-morphism of schemes then we have a
commutative diagram
Hk` (Y )(n)×Hj
` (Y )(m)f∗×f∗ //
∪k,j
��
Hk` (X)(n)×Hj
` (X)(m)
∪k,j
��
Hk+j` (Y )(n + m)
f∗// Hk+j
` (X)(n + m)
As is mentioned in Milne [19] p.268, the cup product makes the direct sum:
H∗` (X) =
⊕k
Hk` (X)([k/2])
10
into a graded ring, called the cohomology ring. It is clear from the property of the
cup product above that a morphism of schemes f : X → Y induces a homomorphism
of graded rings:
f ∗ : H∗` (Y )→ H∗
` (X).
2.1.5 The fundamental class and the trace map
In this subsection X will be a smooth irreducible variety over an algebraically closed
field K except at the end.
Relative Cohomology
Let X be an algebraic variety and Z a closed subvariety of X, one can define coho-
mology groups (functors) HkZ(X,−) called cohomologies with support in Z; cf. Milne
[19], p.91. It can be shown that for Z a complete subvariety there is a canonical
morphism (natural transformation):
fZ : HkZ(X,−)→ Hk
c (X,−),
where, as before, Hkc (X,−) is the cohomology with compact support; cf. Milne
[19], Proposition I.1.29. In particular one has morphisms fZ,N : HkZ(X, Z/NZ) →
Hkc (X, Z/NZ) for each integer N which are compatible as N varies. Thus, letting
N = `n for a prime ` and taking inverse limits, we get Z`-linear homomorphism:
fZ,` : HkZ(X, Z`)→ Hk
c (X, Z`).
Now tensoring with Q` gives a morphism:
fZ,` : HkZ(X, Q`)→ Hk
c (X, Q`).
Smooth pairs
As is defined in Milne [19] Chapter VI, 5. A smooth K-pair (Z,X) of codimension
n is a closed immersion of smooth K-schemes i : Z ↪→ X with Z has codimension n.
A morphism of smooth K-pairs f : (Z ′, X ′) → (Z,X) is a morphism of K-schemes
f : X ′ → X such that: Z ′ = Z ×X X ′.
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Fundamental class
As is explained in Milne [19], to any smooth K-pair (Z,X) of pure codimension
n one can associate a cohomology class: SZ/X,N ∈ H2nZ (X, Z/NZ)(n), N prime to
the characteristic of K, which is uniquely determined by the four properties listed
in Theorem VI.6.1. of Milne [19]. By varying N = `r, we thus obtain an element
SZ/X,` ∈ H2nZ (X, Q`)(n).
Proposition 2.1.2. If f : (Z ′, X ′)→ (Z,X) is a morphism of smooth K-pairs, then
under the map : f ∗ : H2nZ (X, Z/NZ)(n)→ H2n
Z′ (X ′, Z/NZ)(n) we have f ∗(SZ/X,N) =
SZ′/X′,N for all integers N prime to the characteristic of K, and hence f ∗(SZ/X,`) =
SZ′/X′,`, if ` 6= char(K).
Proof. For the first statement see Milne [19] part c) of Theorem VI.6.1. The second
statement follows from the first by taking inverse limits.
Let X be smooth of dimension d, and let P ∈ X be a closed point, then (P, X) is a
smooth pair and so we have an associated cohomology class SP/X ∈ H2dP (X, Q`)(d).
Recall that we have a canonical morphism:
fP,` : H2dP (X, Q`)(d)→ H2d
c (X, Q`)(d)
We let clX(P ) = fP,`(SP/X). It turns out that clX(P ) does not depend on the partic-
ular closed point P ∈ X; cf. Milne [19] Theorem VI.11.1.(a)
Now if X is an arbitrary variety over K, let U = Xns be the smooth part of X.
Since U is a dense open subset of X, we can use the isomorphism fU of Proposition
2.1.1 and we define clX(P ) = fU(clU(P )).
Trace Map
Theorem 2.1.3. If X/K is a variety, then there exists a unique isomorphism
tX : H2dc (X, Q`)(d)→ Q`
such that tX(clX(P )) = 1, for all P ∈ Xns.
12
Proof. For X smooth this follows from Milne [19] Theorem VI.11.1.(a) by taking
inverse limits.
Now for X non smooth, let U = Xns be the smooth part of X. By applying the
smooth case to U there is a unique map tU : H2d`,c(U)(d)→ Q` such that tU(clU(P )) =
1. Now recall from Proposition 2.1.1 that we have a canonical isomorphism: fU :
H2d`,c(U)(d)→ H2d
`,c(X)(d). We define the trace map for X by
tX = tU ◦ fU−1 : H2d
`,c(X)→ Q`.
which is an isomorphism, moreover, for P ∈ U = Xns we have tX(clX(P )) = (tU ◦fU
−1)(clX(P )) = (tU◦fU−1)(fU(clU(P ))) = tU(clU(P )) = 1. This proves the existence,
the uniqueness is trivial because tX is an isomorphism.
2.1.6 Poincare Duality Theorem
Let ϕX denote the composition of the cup product pairing
∪X : Hk` (X)(m)×H2d−k
` (X)(d−m) −→ H2d` (X)(d)
with the trace map:
tX : H2d` (X)(d)→ Q`.
The following is the Poincare duality theorem:
Theorem 2.1.4. Suppose that ` is different from the characteristic of K, and that
X/K is a smooth proper and geometrically irreducible variety of dimension d. Then
the pairing
ϕX = tX ◦ ∪X : Hk` (X)(m)×H2d−k
` (X)(d−m) −→ Q`
defines a perfect duality.
Proof. This follows from Corollary VI.11.2 of Milne [19], again by taking inverse limits
and noticing that cohomology with compact support coincides with the ordinary
cohomology because X is proper.
13
2.2 Algebraic Cycles and the Cycle map
2.2.1 Algebraic Cycles and Numerical Equivalence
Let X be a smooth projective variety of dimension d over an algebraically closed field
K. A prime cycle of codimension k on X is an irreducible subvariety V of X of
codimension k. A cycle C of codimension k on X is an element of the free abelian
group generated by the prime cycles that is, a formal linear combination of the form:
C =∑
nj · [Vj], where the nj’s are integers all zero except finitely many, and the Vj
’s are prime cycles of codimension k. We denote by Zk(X) the group of cycles of
codimension k.
For any i ≤ d, one has an intersection pairing
Zi(X)× Zd−i(X)→ Z
which associates to a pair (V, W ) an integer denoted by V.W ; cf Fulton [7] Ch. 8.3.
We say that a cycle V ∈ Zi(X) is numerically equivalent to zero if for any ir-
reducible element D ∈ Zd−i(X) we have V.D = 0. Cycles which are numerically
equivalent to zero form a subgroup of Zi(X) denoted by Zi0(X). One denotes the
quotient group Zi(X)/Zi0(X) by Zi(X); it is called the group of cycles up to numer-
ical equivalence.
2.2.2 Flat pullback of cycles
Let f : X → Y be a flat morphism of smooth varieties. For any irreducible subvariety
V of Y of codimension k set f ∗(V ) = [f−1(V )] =∑
nj · [Wj] where the Wj’s are the
irreducible components of f−1(V ) and the nj’s are their multiplicities (see Fulton [7],
page 15 for the definition of nj). We extend f ∗ by linearity to get a morphism of
groups of cycles
f ∗ : Zk(Y )→ Zk(X).
This map is called the flat pullback of cycles, see Fulton, [7] page 18.
14
2.2.3 The Cycle map
For any 0 ≤ k ≤ d, one has a cycle map
cyc : Zk(X)→ H2k` (X)(k)
This map is defined by means of the Poincare duality theorem as follows. If i : Z ↪→X is an irreducible closed subvariety of codimension k, then we have a restriction
morphism on cohomologies
i∗ : H2(d−k)` (X)→ H
2(d−k)` (Z).
Twisting by (d− k), this yields the following morphism
i∗(d−k) : H2(d−k)` (X)(d− k)→ H
2(d−k)` (Z)(d− k).
Composing i∗(d−k) with the trace map
tZ : H2(d−k)` (Z)(d− k)→ Q`,
we get the map
tZ ◦ i∗(d−k) : H2(d−k)` (X)(d− k)→ Q`.
By Poincare duality there exists a unique element ηZ ∈ H2k` (X)(k) such that:
ϕX(ηZ , ·) = tZ ◦ i∗(d−k)
where ϕX is the Poincare pairing. We define:
cyc(Z) = ηZ
Now that the image of a prime cycle is defined, one extends cyc by linearity to any
cycle.
15
2.3 L-functions
2.3.1 The finite field case
Here we assume that our variety X is defined over K = Fp. Let ` be a prime different
from p and let H i`(X) be the i-th `-adic cohomology group. The Galois group GFp acts
continuously on H i`(X). Let Fp ∈ GFp denote the Frobenius element, i.e, Fpx = xp
for all x ∈ Fp. We define the following polynomial:
FX,i,`(t) := det((1− tF−1p )|H i
`(X)).
The following result is due to Deligne; cf. [1] Theorem I.6.
Theorem 2.3.1. If X is smooth and projective, then we have the following:
1. The polynomial FX,i,`(t) has coefficients in Z and is independent of `.
2. If λ is a root of FX,i,`(t), then |λ| = pi/2.
Thus, one can attach to H i`(X) an L-series defined as follows
Li(X, s) = FX,i,`(t)(p−s).
2.3.2 Zeta function
Let X be any variety over Fp. The zeta function of X is defined by
Z(X, t) = exp
(∑n≥1
νn(X)
n· tn)
,
where νn(X) is the number of points with coordinates in X(Fpn). From its definition,
the object Z(X, t) belongs to Q((t)). Now the following theorem of Grothendieck
shows that it belongs to Q(t), cf. Milne [19], Theorem VI.13.1.
Theorem 2.3.2. Assume that X is proper of dimension d over Fp, and let ` 6= p.
With the same notations as in the previous subsection we have the identity
Z(X, t) =2d∏i=0
FX,i,`(t)(−1)(i+1)
.
16
2.3.3 The global case
Here we assume that our variety X is smooth and projective over Q. Let H i`(X) be
the i-th `-adic cohomology group. The Galois group GQ acts continuously on H i`(X).
Let p be a rational prime different from `, and fix a prime ideal P in the ring Z of all
algebraic integers, such that p ∈ P. Let DP and IP denote the decomposition and
inertia subgroups of GQ with respect to P. We say that H i`(X) is unramified at p if
IP acts trivially on H i`(X). For such a prime we define the following polynomial
Fp,i,`(X)(t) = det(1− tF−1p |H i
`(X))
This polynomial has coefficients in Q` and depend only on the conjugacy class of Fp.
Let Z(p) denotes the localization of Z at the prime ideal (p). We say that X has
good reduction at p if there exists a smooth and projective scheme X over Z(p) with
generic fiber X0 ' X. The special fiber Xp of X is a reduction of X modulo p; it
is smooth and projective over Fp. One considers as in the previous subsection the
polynomial
FXp,i,`(t) := det(1− tF−1p |H i
`(Xp)).
Using Theorem 2.3.1 this polynomial has coefficients in Z and is independent of `.
We have the following
Proposition 2.3.3. Assume that X is smooth and projective and that X has good
reduction Xp at p. If P is a prime of Z over p and ` 6= p, then there is an isomorphism
of DP-modules
H i`(X) ' H i
`(Xp),
where DP acts on H i`(Xp) via DP/IP ' GFp. In particular, H i
`(X) is unramified at p
and we have the equality
Fp,i,`(X)(t) = FXp,i,`(t),
so Fp,i,`(X)(t) has integral coefficients and is independent of `.
Proof. This is explained in Tate’s article [33], page 108.
17
Remark 2.3.4. It turns out that for X smooth and projective there are only fi-
nitely many primes p such that X has bad reduction at p; cf. Hindry-Silverman [9],
Proposition A.9.1.6.
2.3.4 Compatible systems of `-adic modules
Let M = (M`)` be a family of GQ-modules, where ` run over all rational primes,
and each M` is a Q`-vector space. We say that (M`)` is a strictly compatible system
of weight w, if there exists a finite set of primes S and an integer w such that the
following conditions are satisfied
1. For any ` and any p /∈ S ∪ {`} the module M` is unramified at p.
2. For any ` and any p /∈ S ∪ {`} the polynomial Fp,`(t) := det((1 − tF−1p )|M`)
has coefficients in Q and is independent of `. We can thus drop the ` from the
notation and denote it by Fp(t).
3. For any p /∈ S the roots of Fp(t) in C have absolute value pw2 .
It is immediate that if the conditions 1,2 and 3 are satisfied by two finite sets of
primes S1 and S2, then they are also satisfied by their intersection S1∩S2. We denote
by SM the smallest set of primes such that the three conditions above hold.
One can associate an L-series to any such system M as follows. For any p /∈ SM
define
L(M, s, p) = Fp(p−s).
Now define
L(M, s) =∏
p/∈SM
L(M, s, p)−1.
Proposition 2.3.5. If M is a strictly compatible system of weight w, then the L-
series L(M, s) converges absolutely for <(s) > 1 + w2.
Proof. For any p /∈ SM we can write
L(M, s, p)−1 =∏
1≤i≤m
1
1− λp,i · p−s,
18
where m = dimQ`(M`), ` is any prime different from p and λp,i are the roots of Fp,`(t)
in C. Taking absolute value we have
|L(M, s, p)−1| =∏
1≤i≤m
1
|1− λp,i · p−s|
Now using the facts that |1− |λp,i||p−s|| ≤ |1− λp,i · p−s| and |λp,i| = pw2 we get
|L(M, s, p)−1| ≤ 1
|1− pw2−<(s)|m
,
and so
|L(M, s)| ≤∏
p/∈SM
1
|1− pw2−<(s)|m
.
Now if <(s) > w2
+ 1 then |1− pw2−<(s)| = 1− p
w2−<(s) > 0, hence
|L(M, s)| ≤∏
p/∈SM
1
|1− pw2−<(s)|m
=∏
p/∈SM
1
(1− pw2−<(s))m
=∏
p∈SM
(1− pw2−<(s))m · ζ(<(s)− w
2)m,
which converges for <(s) > w2
+ 1.
For any finite set of primes S containing SM let LS(M, s) =∏
p/∈S L(M, s, p)−1.
We then have the following
Proposition 2.3.6. 1. LS(M, s) converges for <(s) > 1 + w2.
2. If w 6= −1, then L(M, s) has a pole of order n at s = w2
+ 1 if and only if
LS(M, s) has a pole of order n at s = w2
+ 1.
Proof. We have
L(M, s) = LS(M, s) ·∏
p/∈S\SM
L(M, s, p)−1
hence statement 1 follows. To prove statement 2), we need only to show that the
factor∏
p/∈S\SML(M, s, p)−1 has no poles or zeros at s = w
2+ 1. In fact we will prove
that for any p 6∈ SM the function s 7→ L(M, s, p) has no poles or zeros at s = w2
+ 1.
We have by definition L(M, s, p) = Fp(p−s), since the functions s 7→ p−s is analytic
19
on the whole complex plane then so is s 7→ L(M, s, p) (as composition of two analytic
functions) thus it has no poles at all. Now if s 7→ L(M, s, p) has a zero at s = w2
+ 1
then p−w2−1 will be a root for the polynomial Fp(t), hence by the weight condition we
would have p−w2−1 = p
w2 , which is false unless w = −1.
Let M and N be strictly compatible systems with exceptional sets SM and SN
and associated L-series L(M, s) and L(N, s). We say that L(M, s) and L(N, s) are
equal up to finitely many Euler factors and we write L(M, s) ∼ L(N, s), if there
exists a finite set of primes S containing SM and SN such that∏
p/∈S L(M, s, p)−1 =∏p/∈S L(N, s, p)−1.
In the following proposition we summarize some standard properties of strict com-
patible systems.
Proposition 2.3.7. 1. The system {Q`(r)}` with r ∈ Z is strictly compatible of
weight −2r. Moreover the L-series associated with this system is ζ(s+ r) where
ζ is the Riemann zeta function.
2. If M = {M`}` and N = {N`}` are two strictly compatible systems of the same
weight w, then the direct sum M ⊕N = {M` ⊕N`}` is also strictly compatible
of weight w. Moreover L(M ⊕N, s) ∼ L(M, s)L(N, s).
3. If M = {M`}` and N = {N`}` are two strictly compatible systems of weights
wM and wN then the tensor product M ⊗N = {M`⊗N`}` is strictly compatible
of weight wM +wN . In particular for any r ∈ Z the Tate twist M(r) = {M`(r)}`is a strictly compatible system.
4. If M = {M`}` is a strict compatible system of weight w then the dual M∨ =
{M`∨}` is strictly compatible of weight −w.
The main result of this subsection is the following result essentially due to Deligne.
Theorem 2.3.8. Let X be a smooth projective geometrically irreducible variety of
dimension d over Q. For any integer 0 ≤ k ≤ 2d the system of `-adic cohomology
groups {Hk` (X)}` is a strictly compatible system of weight k.
20
Proof. This follows from Theorem 2.3.1, Proposition 2.3.3 and Remark 2.3.4.
Using the previous theorem, one can associate an L-series to the system {Hk` (X)}`.
This L-series we denote by
Lk(X, s) := L(Hk` (X), s).
2.4 Tate’s conjectures in the smooth projective case
Let X be a smooth projective variety over a finitely generated field K and let
cyci : Zi(X)→ H2i` (X)(i)
be the cycle map defined earlier, and let Zi0,`(X) be the kernel of cyci. In his funda-
mental article, Tate [33] made the following preliminary conjecture
Conjecture 2.4.1. 1. The subgroup Zi0,`(X) does not depend on `,
2. Zi0,`(X) consists exactly of the cycles which are numerically equivalent to zero,
3. The quotient group Zi(X) = Zi(X)/Zi0,`(X) is finitely generated and
cyci⊗1 : Zi(X)⊗Z Q` → H2i` (X)(i)
is injective.
Remark 2.4.2. Statements 1 and 3 are known to be true in characteristic zero. For
i = 1, all of the statements 1, 2 and 3 are true in all characteristics. See Tate’s article
[33] pages 97, 98 for a sketch of the proof.
With the previous conjecture assumed to be true, Tate made some important
conjectures relating rational algebraic cycles to cohomology and L-series. We restrict
ourselves to K = Q and we start with the conjecture relating rational cycles to
cohomology. First, let Zi(X) be the subgroup of Zi(X) generated by the classes of
algebraic cycles which are defined over Q, that is,
Zi(X) = Zi(X)GQ = {ξ ∈ Zi(X) : σξ = ξ for all σ ∈ GQ}
21
Conjecture 2.4.3. The injective map in Conjecture 2.4.1 part 3) induces an isomor-
phism
cyci⊗1 : Zi(X)⊗Z Q` → H2i` (X)(i)GQ ,
or equivalently the following identity holds
dimQ`(Zi(X)⊗Z Q`) = dimQ`
H2i` (X)(i)GQ .
The other interesting conjecture relating Zi(X) to the L-series L2i(X, s) is the
following
Conjecture 2.4.4. The L-series L2i(X, s) has a pole at s = 1 + i of order equals to
− dimQ`(Zi(X)⊗Z Q`).
22
Chapter 3
Quotient Varieties
Let X be a smooth projective variety over a field K, G a finite group acting faithfully
on X by automorphisms. Then there exists a variety Y (cf. [8] Expose V, or [22]
Corollary 1.10) and a finite surjective map: π : X → Y of degree equals the order of
G, such that for any σ ∈ G we have: π ◦ σ = π. Moreover the variety Y is universal
for this property. In other words, if Z is an algebraic variety over K and if h : X → Z
is a map such that the diagram:
Xσ //
h @@@
@@@@
X
h~~~~~~
~~~
Z
is commutative for all σ ∈ G, then there is a unique map f : Y → Z such that
h = f ◦π. The pair (Y, π) is uniquely determined by this universal property up to an
isomorphism. It is called the quotient of X by G, and is denoted by X/G.
3.1 The `-adic cohomology of a quotient variety
It is a natural question to ask how the etale cohomology groups of a quotient variety
Y = X/G are related to those of X. Let Y = X/G be a quotient variety. Since the
group G acts on X, then it acts on Hk` (X) by functoriality. Now since π ◦ σ = π for
23
all σ in G, the image of the map: π∗ : Hk` (Y ) → Hk
` (X) is contained in Hk` (X)G. A
more precise result is the following:
Theorem 3.1.1 (Roth [29]). With the same notations as in the previous discussion
the map
π∗ : Hk` (Y )→ Hk
` (X)G
is an isomorphism.
3.2 Poincare duality for quotient varieties
The aim of this subsection is to formulate and prove a Poincare duality theorem for
quotient varieties over a field of characteristic zero. Let Y = X/G be a quotient
variety with X smooth and projective over a field of characteristic zero, and let
π : X → Y be the corresponding projection map. Moreover, let
ϕY : Hk` (Y )(m)×H2d−k
` (Y )(d−m) −→ Q`
be the pairing obtained by composing the cup product
∪Y : Hk` (Y )(m)×H2d−k
` (Y )(d−m) −→ H2d` (Y )(d)
and the trace map
tY : H2d` (Y )(d)→ Q`.
We have the following
Theorem 3.2.1. The Poincare pairing
ϕY : Hk` (Y )(m)×H2d−k
` (Y )(d−m)→ Q`
is a perfect duality. Moreover, we have a commutative diagram
Hk` (Y )(m)×H2d−k
` (Y )(d−m)π∗×π∗ //
ϕY
��
Hk` (X)(m)×H2d−k
` (X)(d−m)
ϕX
��Q` ×|G|
// Q`
24
where ϕX is the Poincare pairing for X.
We need first to prove the following result.
Lemma 3.2.2. If π : X → Y is a finite morphism of degree n between two vari-
eties of dimension d over a field of characteristic zero, then the following diagram is
commutative
H2d`,c(Y )(d) π∗ //
tY
��
H2d`,c(X)(d)
tX
��Q` ×n
// Q`
Proof. First, assume that X and Y are smooth and pick any unramified point p in
Y , so |π−1(p)| = n. Set π−1(p) = {q1, q2, ..., qn}. Then we have a morphism of smooth
pairs; π : (X, {q1, ...qn})→ (Y, p). Hence by Proposition 2.1.2 we have:
π∗(Sp/Y ) = S{q1,...,qn}/X =∑
1≤i≤n
Sqi/X
Hence it follows that
π∗(clY (p)) =∑
1≤i≤n
clX(qi)
Now by Theorem 2.1.3 clX(qi) does not depend on the particular points qi and depends
only on X, hence we get the equation:
π∗(clY (p)) = n · π∗(clX(q))
where q is any point in the set π−1(p). Next, applying the trace map tX to this
equation yields
tX(π∗(clY (p))) = n · tX(clX(q)) = n · 1 = n.
and thus
tX ◦ π∗ = n · tY .
This proves the result when X and Y are smooth.
Now, for an arbitrary X and Y let U = Y ns denote the subset of Y obtained by
removing all of the singular points in Y , and let V = (π−1(U) ∩Xns) ⊂ X. Both U
25
and V are open dense subsets and we have a finite morphism of degree n, π : V → U .
Thus the first case applies to this situation and we have a commutative diagram:
H2d`,c(U)(d) π∗ //
tU
��
H2d`,c(V )(d)
tV
��Q` ×n
// Q`
Now consider the isomorphisms fU : H2d`,c(U)(d)→ H2d
`,c(Y )(d) and fV : H2d`,c(V )(d)→
H2d`,c(X)(d). Since we also have tY = tU ◦fU
−1 and tX = tV ◦fV−1, therefore composing
the previous diagram with the maps fU−1 and fV
−1 we get tX ◦ π∗ = n · tY which is
the desired result.
Now we give a proof of the previous theorem.
Proof. of Theorem 3.2.1
Recall the commutative diagram from Section 2.1.4
Hk` (Y )(m)×H2d−k
` (Y )(d−m)π∗×π∗ //
∪Y
��
Hk` (X)(m)×H2d−k
` (X)(d−m)
∪X
��
H2d` (Y )(d)
π∗// H2d
` (X)(d)
Composing this diagram with that of Lemma 3.2.2 gives the following diagram:
Hk` (Y )(m)×H2d−k
` (Y )(d−m)π∗×π∗ //
ϕY
��
Hk` (X)(m)×H2d−k
` (X)(d−m)
ϕX
��Q` ×|G|
// Q`
This proves the second assertion of the theorem.
It remains to show that ϕY is non-degenerate. Since by Theorem 3.1.1 π∗ × π∗ is
injective with image Hk` (X)(m)
G × H2d−k` (X)(d−m)
G, it suffices to prove that the
restriction of ϕX (or equivalently the restriction of ∪X) to the subspace Hk` (X)(m)G×
H2d−k` (X)(d−m)G is non degenerate. Since ∪X is already non degenerate in the whole
space, it suffices to show that if an element α in Hk` (X)(m)G is such that α∪X β = 0
26
for all elements β in H2d−k` (X)(d − m)G, then α ∪X b = 0 for all elements b in
H2d−k` (X)(d−m) and that the same is true by interchanging the roles of α and β. To
do this, we first notice that from the diagram in Section 2.1.4 we have the following
identity:
g∗.a ∪X g∗.b = g∗.(a ∪X b)
which holds for all pairs (a, b) and all group elements g.
Now consider the element εG of Q`[G] given by :
εG =1
|G|∑g∈G
g
It is a general fact that for any Q`[G]-module V we have: V G = εGV .
Now let b in H2d−k` (X)(d − m). Then εG.b ∈ H2d−k
` (X)(d − m)G, hence by our
assumption about α we have
α ∪X εGb = 0 (2)
on the other hand, by using the linearity we have
α ∪X εGb =1
|G|∑g∈G
α ∪X g∗.b
Now using the facts that g∗.a ∪X g∗.b = g∗.(a ∪X b) and g∗α = α for all g we get
α ∪X εGb = εG(α ∪X b) (3)
hence from (2) and (3) we have
εG(α ∪X b) = 0 (4)
Now εG acts as 1 in the space H2d` (X)(d). Indeed we know that H2d
` (X)(d) is a one di-
mensional space, and we know also by Roth’s theorem that H2d` (X)(d)G ' H2d
` (Y )(d).
Thus, H2d` (X)(d)G is also a one dimensional space, and hence H2d
` (X)(d)G = H2d` (X)(d),
which means that G acts trivially on H2d` (X)(d) therefore εG acts as 1 in H2d
` (X)(d),
this together with equation (4) imply that α ∪X b = 0 as desired. Similarly one can
prove the same thing by interchanging the roles of Hk` (X)(m) and H2d−k
` (X)(d−m).
27
3.3 Cycles on a quotient variety and the cycle map
3.3.1 Cycles on a quotient variety
Let Y = X/G with X smooth and projective over a characteristic zero field K.
How does the group of i- cycles on Y = X/G relate to that on X. First take base
extensions X and Y over the algebraic closure of K. Then the group G acts on
X, and Y is the resulting quotient variety. Moreover the group G naturally acts on
the group of i-cycles Zi(X). Unfortunately one does not have a flat pullback π∗ in
this situation because the map π : X → Y is not flat in general. However there is
an alternative definition of π∗ given by Fulton [7] Example 1.7.6 page 20. Fulton’s
definition specialized to the case of char(K) = 0 is as follows:
For any cycle C in Zi(X) define the inertia group at C by
IC = {σ ∈ G : σ|C = idC}.
Now let [V ] be an irreducible cycle in Zi(Y ) and for any irreducible component C of
π−1(V ), let eC = |IC |. We define:
π∗([V ]) =∑
eC [C]
the sum is over all irreducible components C of π−1(V ). Note that since the field has
characteristic zero then the inseparability degree in Fulton’s definition is 1.
With this definition of π∗ one has an injective map
π∗ : Zi(Y )→ Zi(X)G
which becomes an isomorphism after tensoring with Q.
π∗ ⊗ 1 : Zi(Y )⊗Q→ Zi(X)G ⊗Q
Since the map π : X → Y is proper, one also have a push forward map
π∗ : Zi(X)→ Zi(Y ).
28
π∗ is defined as follows: π∗(V ) = deg(V/W )W where W = π(V ) and deg(V/W ) is
the degree of the finite map π|V : V → W . We have
π∗ ◦ π∗ = ×|G|; (5)
cf. Fulton [7], Example 1.7.6 page 20.
3.3.2 Intersection product on a quotient variety
In general there is no good intersection theory for singular varieties, but in the case of
a quotient variety Y = X/G one can define an intersection map in Y inherited from
that of X provided that X is smooth. If a ∈ Zi(Y ) and b ∈ Zd−i(Y ) one defines a · bby the following formula (cf. Fulton [7] Example 8.3.12 pages 142-143)
a · b =1
|G|(π∗(a) · π∗(b))
In contrast with the smooth case, a · b is not necessarily an integer but rather an
element of 1|G|Z. Thus the formula above defines an intersection mapping
Zi(Y )⊗Q× Zd−i(Y )⊗Q→ Q
Moreover, this mapping is compatible with that of X, namely, we have a projection
formula: for a ∈ Zi(Y ) and β ∈ Zi(X) we have
π∗(a) · β = a · π∗(β);
cf. Fulton [7] Example 8.3.12 pages 142-143. Despite the apparent dependence on X,
it is asserted in Fulton [7] Examples 16.1.3 and 17.4.10 that the intersection map on
Y is independent of X.
We say that a cycle a ∈ Zi(Y ) is numerically equivalent to zero, if for any cycle
b ∈ Zd−i(Y ) we have a·b = 0. We denote the subgroup of cycles numerically equivalent
to zero by Zi0(Y ). We have the following
Proposition 3.3.1. With the notations as in the previous discussion the following
holds
(π∗)−1(Zi0(X)) = Zi
0(Y )
29
Proof. Let a ∈ Zi0(Y ). We want to show that a ∈ (π∗)−1(Zi
0(X)) or equivalently
that π∗(a) ∈ Zi0(X). Let β ∈ Zd−i(X). Using the projection formula we have
π∗(a) · β = a · π∗(β). But a ∈ Zi0(Y ), hence a · π∗(β) = 0, thus π∗(a) · β = 0, which
means π∗(a) ∈ Zi0(X).
Conversely, if a ∈ (π∗)−1(Zi0(X)) then π∗(a) ∈ Zi
0(X). We want to show that
a ∈ Zi0(Y ). Let b ∈ Zd−i(Y ). Applying the projection formula again we have
π∗(a) · π∗(b) = a · π∗(π∗(b))
Now since π∗(a) ∈ Zi0(X) we have π∗(a) · π∗(b) = 0 thus,
a · π∗(π∗(b)) = 0.
Now using the fact that π∗(π∗(b)) = |G| · b the previous equation becomes
|G|(a · b) = 0
hence a · b = 0, which means a ∈ Zi0(Y ).
3.3.3 Cycle map on a quotient variety
Using Theorem 3.2.1(Poincare duality) one can define a cycle map for quotient va-
rieties in the same way as smooth varieties, see Subsection 2.2.3. The main result
about this cycle map is the following:
Theorem 3.3.2. If Y = X/G is a quotient variety of dimension d, then the following
diagram commutes
Zi(Y )cycY //
π∗
��
H2i` (Y )(i)
π∗
��
Zi(X) cycX
// H2i` (X)(i)
30
Proof. Let Z ⊂ Y be an irreducible i-cocycle and let W ⊂ π−1(Z) be an irreducible
component of π−1(Z), then we have the following diagram on varieties
W� � iW //
πW
��
X
π
��Z
� �
i// Y
where πW is the restriction of π to W and i and iW are the closed immersions. Hence
by functoriality we get the following commutative diagram on cohomology groups:
H2(d−i)` (Y )(d− i)
i∗ //
π∗
��
H2(d−i)` (Z)(d− i)
π∗W��
H2(d−i)` (X)(d− i)
i∗W
// H2(d−i)` (W )(d− i)
Composing this diagram with the trace maps:
tW : H2(d−i)` (W )(d− i)→ Q`
and
tZ : H2(d−i)` (Z)(d− i)→ Q`
and using Lemma 3.2.2 we get the following commutative diagram:
H2(d−i)` (Y )(d− i)
π∗ //
tZ◦i∗
��
H2(d−i)` (X)(d− i)
tW ◦i∗W��
Q` ×λW
// Q`
i.e, λW ·(tZ◦i∗) = (tW ◦i∗W )◦π∗, where λW is the degree of the finite map: πW : W → Z.
Now by definition of the cycle map we have
tZ ◦ i∗ = ϕY (cycY (Z), ·)
and
tW ◦ i∗W = ϕX(cycX(W ), ·)
31
where ϕX and ϕY are the Poincare pairings, and so we have
λW · ϕY (cycY (Z), ·) = ϕX(cycX(W ), ·) ◦ π∗ (6)
Multiplying (6) by the ramification index eW and summing over all irreducible com-
ponents W of π−1(Z) we get
(∑W |Z
eW · λW ) · ϕY (cycY (Z), ·) =∑W |Z
eW · ϕX(cycX(W ), ·) ◦ π∗ (7)
Using linearity, the right hand side of (7) becomes ϕX(cycX(∑
W |Z eW ·W ), π∗(·)),and using the identity
∑W |Z eW · λW = |G| which follows from equation (5) we get
|G| · ϕY (cycY (Z), ·) = ϕX(cycX(∑W |Z
eW ·W ), π∗(·)) (8)
Now by Fulton’s definition of π∗ we have π∗(Z) =∑
W |Z eW ·W , hence (8) becomes
|G| · ϕY (cycY (Z), ·) = ϕX(cycX(π∗(Z)), π∗(·)) (9)
Recall from Theorem 3.2.1 we have the following commutative diagram:
H2i` (Y )(i)×H
2(d−i)` (Y )(d− i)
π∗×π∗ //
ϕY
��
H2i` (X)(i)×H
2(d−i)` (X)(d− i)
ϕX
��Q` ×|G|
// Q`
In particular we have the identity
ϕX(π∗(a), π∗(·)) = |G|ϕY (a, ·)
which holds for any a ∈ H2i` (Y )(i). Taking a = cycY (Z) we get
|G| · ϕY (cycY (Z), ·) = ϕX(π∗(cycY (Z)), π∗(·)). (10)
Now comparing (9) and (10) we get
ϕX(π∗(cycY (Z)), π∗(·)) = ϕX(cycX(π∗(Z)), π∗(·))
Finally, using the non-degeneracy of ϕX we get
cycX(π∗(Z)) = π∗(cycY (Z)),
which is the desired result.
32
3.4 The L-series of a quotient variety
Let Y = X/G be a quotient variety. For 0 ≤ i ≤ 2 dim(Y ) the absolute Galois group
GQ acts on the cohomology groups H i`(Y ) which are finite dimensional vector spaces
over Q`. Let p be an unramified prime H i`(Y ). Then we define as in section 2.3 the
following polynomial:
Pp(Y )(t) = det((1− tF−1p |H i
`(Y ))
We have the following unpublished result due to Kani [11], which generalizes Theorem
2.3.8 to quotient varieties.
Theorem 3.4.1 (Kani). If Y = X/G is a quotient variety, with X is smooth pro-
jective and geometrically irreducible over Q, then the system {H i`(Y )}` is strictly
compatible of weight i.
3.5 Tate Conjectures for a Quotient Variety
In this section we will use previous results to prove that some of the Tate’s conjectures
descend naturally to a quotient variety if they hold for the cover. More precisely let
Y = X/G be a quotient variety with X is a smooth projective variety over a finitely
generated field K of characteristic zero. We have the following result.
Theorem 3.5.1. If Conjecture 2.4.1 holds for X, then it also holds for Y .
Proof. By Theorem 3.3.2 we have the following commutative diagram
Zi(Y )cycY //
π∗
��
H2i` (Y )(i)
π∗
��
Zi(X) cycX
// H2i` (X)(i)
Let Zi0,`(Y ) = ker(cycY ) and Zi
0,`(X) = ker(cycX), assume that part 1 of Conjecture
2.4.1 holds for X, that is, Zi0,`(X) is independent of `. We want to prove that Zi
0,`(Y )
is independent of ` as well. From the previous diagram we get easily the identity
33
Zi0,`(Y ) = π∗−1(Zi
0,`(X)). Now since π∗ and Zi0,`(X) are independent of ` then so is
Zi0,`(Y ).
Now assume that part 2 of 2.4.1 holds for X; that is, Zi0,`(X) = Zi
0(X). From
Proposition 3.3.1 we know that Zi0(Y ) = π∗−1(Zi
0(X)) and we just saw that Zi0,`(Y ) =
π∗−1(Zi0,`(X)), so, Zi
0,`(Y ) = Zi0(Y ).
Let Zi(X) = Zi(X)/Zi0(X) and Zi(Y ) = Zi(X)/Zi
0(Y ). Now assume that part
3 of Conjecture 2.4.1 holds for X, that is, Zi(X) is finitely generated and the map
cycX ⊗1 : Zi(X) ⊗ Q` → H2i` (X)(i) is injective. We want to prove that the same is
true for Y . Since Zi0(Y ) = π∗−1(Zi
0(X)), the map π∗ : Zi(Y ) → Zi(X) induces an
injective map of Z-modules
π∗ : Zi(Y )→ Zi(X),
hence Zi(Y ) is finitely generated as a submodule of a finitely generated Z-module.
To show that the map; cycY ⊗1 : Zi(Y )⊗Q` → H2i` (Y )(i) is injective, we notice first
that the commutative diagram above induces the following one
Zi(Y )⊗Q`
cycY ⊗1 //
π∗⊗1��
H2i` (Y )(i)
π∗
��
Zi(X)⊗Q` cycX ⊗1// H2i
` (X)(i)
that is, π∗ ◦ (cycY ⊗1) = (cycX ⊗1) ◦ (π∗⊗ 1). Now let ξ such that (cycY ⊗1)(ξ) = 0;
Then (cycX ⊗1)◦(π∗⊗1)(ξ) = 0, but cycX ⊗1 and π∗⊗1 are both injective, therefore
ξ = 0 and hence cycY ⊗1 is injective.
As a direct consequence of the previous result and what is known for the smooth
case we have the following result.
Theorem 3.5.2. If Y = X/G is a quotient variety, with X smooth and projective
over K then parts 1 and 3 of Conjecture 2.4.1 are true for Y . If in addition i = 1,
then part 2 of Conjecture 2.4.1 is also true for Y .
Proof. This is a direct consequence of Theorem 3.5.1 and Remark 2.4.2.
34
Let X be such that Conjecture 2.4.1 holds for X and let Y = X/G. The next
result shows that Conjecture 2.4.3 also descends from X to Y .
Theorem 3.5.3. If Conjecture 2.4.3 holds for X, then it also hold for Y .
Before we give a proof of this theorem, we need a lemma that explains the behavior
of the cycle map with respect to automorphisms.
Lemma 3.5.4. Let V be any variety which satisfies the Poincare duality theorem, for
example V is either smooth or a quotient variety in characteristic zero. Let f : V → V
be an automorphism of schemes, and let f ∗ : Zk(V ) → Zk(V ) be the flat pullback
defined in Subsection 2.2.2. Moreover, let f ∗ : H2k` (V )(k)→ H2k
` (V )(k) be the induced
linear automorphism. Then the following diagram commutes
Zk(V )cyc //
f∗
��
H2k` (V )(k)
f∗
��
Zk(V ) cyc// H2k
` (V )(k)
Proof. Let Z ∈ Zk(V ) be an irreducible cycle, let W = f−1(Z), and let fW be the
restriction of f to W . Then we have the following commutative diagram on varieties
W� � iW //
fW
��
V
f
��Z
� �
iZ// V
where iZ and iW are the closed immersions. Hence by functoriality we get the following
diagram on cohomology groups
H2(d−i)` (V )(d− i)
i∗Z //
f∗
��
H2(d−i)` (Z)(d− i)
f∗W��
H2(d−i)` (V )(d− i)
i∗W
// H2(d−i)` (W )(d− i)
composing this diagram with the trace maps
tW : H2(d−i)` (W )(d− i)→ Q`
35
and
tZ : H2(d−i)` (Z)(d− i)→ Q`
and using lemma 3.2.2 we get the following commutative diagram
H2(d−i)` (V )(d− i)
f∗ //
tZ◦i∗Z��
H2(d−i)` (V )(d− i)
tW ◦i∗W��
Q` ×1// Q`
i.e. (tZ ◦ i∗Z) = (tW ◦ i∗W ) ◦ f ∗. Now by definition of the cycle map we have:
tZ ◦ i∗Z = ϕV (cycY (Z), ·)
and
tW ◦ i∗W = ϕV (cycV (W ), ·)
where ϕV is the Poincare pairings. Hence we have
ϕV (cycV (Z), ·) = ϕV (cycV (W ), ·) ◦ f ∗ = ϕV (cycV (W ), f∗(·)) (11)
Now by definition of the flat pullback in Subsection 2.2.2, we have f ∗(Z) = f−1(Z) =
W (observe that f is an automorphism). Thus (11) becomes the following
ϕV (cycV (Z), ·) = ϕV (cycV (f ∗(Z)), f∗(·)) (12)
Now using the diagram in Subsection 2.1.4 and Lemma 3.2.2, we have the following
commutative diagram
H2i` (V )(i)×H
2(d−i)` (V )(d− i)
f∗×f∗ //
ϕV
��
H2i` (V )(i)×H
2(d−i)` (V )(d− i)
ϕV
��Q` ×1
// Q`
In particular we have the identity
ϕV (f ∗(a), f∗(·)) = ϕV (a, ·)
which holds for any a ∈ H2i` (V )(i). Taking a = cycY (Z) we get:
ϕV (cycV (Z), ·) = ϕV (f ∗(cycV (Z)), f∗(·)) (13)
36
Comparing (12) and (13) we get
ϕV (f ∗(cycV (Z)), f∗(·)) = ϕV (cycV (f ∗(Z)), f∗(·))
Now using the non-degeneracy of ϕV we get
f ∗(cycV (Z)) = cycV (f ∗(Z))
which is the desired result.
Proof. of Theorem 3.5.3. Here, we work under the assumption that the preliminary
conjecture is valid for X. In particular, we are assuming that cycX ⊗1 is injective.
From the proof of Theorem 3.5.1 we have the diagram
Zi(Y )⊗Q`
cycY ⊗1 //
π∗⊗1��
H2i` (Y )(i)
π∗
��
Zi(X)⊗Q` cycX ⊗1// H2i
` (X)(i)
Let Zi(X) = Zi(X)GQ and Zi(Y ) = Zi(Y )GQ . By Lemma 3.5.4 both maps cycX ⊗1 :
Zi(X)⊗Q` → H2i` (X)(i) and cycY ⊗1 : Zi(Y )⊗Q` → H2i
` (Y )(i) are Galois equivari-
ant. Thus taking Galois invariance in the previous diagram, we get the following
commutative diagram
Zi(Y )⊗Q`
cycY ⊗1 //
π∗⊗1��
H2i` (Y )(i)GQ
π∗
��
Zi(X)⊗Q` cycX ⊗1// H2i
` (X)(i)GQ
Assume that the map cycX ⊗1 is surjective (hence bijective), we want to show that
cycY ⊗1 is surjective as well. Let ξ ∈ H2i` (Y )(i)GQ , we want to find C ∈ Zi(Y )⊗Q`
such that (cycY ⊗1)(C) = ξ. Since cycX ⊗1 is surjective (hence bijective), there is a
unique D ∈ Zi(X)⊗Q` such that (cycX ⊗1)(D) = π∗(ξ). Now, π∗(ξ) ∈ [H2i` (X)(i)GQ ]G
and by Lemma 3.5.4, the map cycX ⊗1 is G- equivariant, thus D ∈ Zi(X)G ⊗ Q`.
Now Im(π∗ ⊗ 1) = Zi(X)G ⊗ Q`, hence there exists C ∈ Zi(Y ) ⊗ Q` such that
(π∗ ⊗ 1)(C) = D. Thus
[(cycX ⊗1) ◦ (π∗ ⊗ 1)](C) = π∗(ξ)
37
Now using the commutativity of the diagram, namely,
(cycX ⊗1) ◦ (π∗ ⊗ 1) = π∗ ◦ (cycY ⊗1) we get
[π∗ ◦ (cycY ⊗1)](C) = π∗(ξ).
Since π∗ is injective, we get
(cycY ⊗1)(C) = ξ.
This completes the proof.
The previous results show that some theorems and conjectures extends to quotient
varieties from the smooth case. Now it is natural to ask if the same is true for
Conjecture 2.4.4. In other words, can we prove Conjecture 2.4.4 for Y = X/G if
we assume it to be true for X. Unfortunately there is no apparent reason that this
descendence strategy will work here too, unless there is a deep relationship between
the L-series of X and that of Y . All what we know is that the L-series of Y is a factor
of the L-series of X and we do not have an interpretation of their ratio, especially
of the order of its pole, for it does not come from a variety. However, we will prove
Conjecture 2.4.4 in Chapter 4 for a special case of quotient varieties, namely that of
a quotient of a product of two modular curves. We will make use of the advanced
machinery of modular forms and abelian varieties associated with them, and even in
this particular case we will prove it directly for Y and not by descending it from X
to Y . Nevertheless this special case makes it legitimate to extend Conjecture 2.4.4 to
quotient varieties.
Let Y = X/G be a quotient variety with X is smooth and projective over Q. By
Theorem 3.4.1, we know that {H i`(Y )}` is a strictly compatible system of weight i.
Therefore one can associate to it an L-series;
Li(Y, s) := L({H i`(Y )}`, s).
We have the following
Conjecture 3.5.5. If Y = X/G is a quotient variety with X smooth projective and
geometrically irreducible over Q, then the L-series L2i(Y, s) has a pole at s = 1 + i of
order equals to − dimQ`(Zi(Y )⊗Z Q`).
38
Chapter 4
The Tate Conjectures for a
Product of Modular Curves
4.1 Generalities on products of curves
In this section we will focus on varieties of the form V = X1 ×X2 × · · · · ×Xr where
Xi/Q are smooth geometrically irreducible projective curves. In all what follows, a
curve means a smooth geometrically irreducible projective curve over Q. We will refer
to Conjecture 2.4.3 for i = 1 as Conjecture T 1 and we refer to Conjecture 2.4.4 for
i = 1 as Conjecture TL1.
4.1.1 Conjecture T 1 for products of curves
We have the following well known result, which is essentially due to Faltings.
Proposition 4.1.1. With the notations as above, Conjecture T 1 holds for V , in
particular, we have the following identity
dimQ(Z1(V )⊗Q) = dimQ`(H2
` (V )(1))GQ
Proof. It is shown in Tate’s article [34] Theorem 5.2 which is a consequence of Faltings
theorem on abelian varieties, that if Conjecture T 1 holds for two varieties X and Y
then it holds for their product X × Y . Also we know that Conjecture T 1 is trivial
39
for curves (because Z1(X) ' Z and H2` (X)(1) ' Q` by the trace map isomorphism),
hence the previous result follows by induction on r.
4.1.2 Conjecture TL1 for products of algebraic curves
The purpose of this subsection is to show that proving Conjecture TL1 for a product
of curves reduces to proving it for a product of two curves. We will also prove that
this conjecture is equivalent to a conjecture about the Jacobian varieties of the curves.
First we notice that by Proposition 4.1.1, Conjecture TL1 becomes the following
Conjecture 4.1.2. Let L2(V, s) denotes the L-series associated with the cohomology
group H2` (V ) then we have the following:
− ords=2 L2(V, s) = dimQ`(H2
` (V )(1))GQ
Now we cite some facts that we will use in this subsection
Fact 1-The Kunneth Formula
Let X and Y be algebraic varieties over a field K. Then the cohomology group of
X × Y is given by an isomorphism of Galois modules:
Hk` (X × Y ) '
⊕i+j=k
(H i`(X)⊗Hj
` (Y ))
cf. Milne [19] Theorem 8.21 and Remark 8.24.
Fact 2- Cohomology of a Curve and its Jacobian
Let X be a projective smooth curve over a field K and let JX denotes its Jacobian
variety. Assume further that X has a K-rational point P . Then the natural map
fP : X → JX induces an isomorphism on cohomology groups:
f ∗P : H1` (JX)
∼→ H1` (X)
cf. Milne [20], Corollary 9.6.
40
Proposition 4.1.3. To prove Conjecture 4.1.2 for an arbitrary product of curves
V = X1 × X2 × · · · · ×Xr, r ≥ 2, it suffices to prove it for any sub product of two
curves Xi ×Xj, 1 ≤ i, j ≤ r.
Before giving a proof to this result we need some lemmas.
Lemma 4.1.4. Let V = X1×X2× · · · · ×Xr, where each Xi/Q is a curve. Then we
have the following isomorphism
H2` (V ) ' Q`(−1)r ⊕ [⊕i<j(H
1` (Xi)⊗H1
` (Xj))]
Proof. We proceed by induction on r. For r = 2, the Kunneth formula yields
H2` (V ) ' (H0
` (X1)⊗H2` (X2))⊕ (H2
` (X1)⊗H0` (X2))⊕ (H1
` (X1)⊗H1` (X2))
We know that H0` (Xi) ' Q`. Also, by the trace map isomorphism we have H2
` (Xi)(1) 'Q`, hence H2
` (Xi) ' Q`(−1). Thus, replacing these in the previous isomorphism we
get
H2` (V ) = Q`(−1)2 ⊕ (H1
` (X1)⊗H1` (X2))
Now assume that the result holds for r ≥ 2 and let us prove it for r + 1. Write
V = X1 × V1, where V1 = X2 × · · · · ×Xr+1. Using the Kunneth formula we have
H2` (V ) = (H0
` (X1)⊗H2` (V1))⊕ (H2
` (X1)⊗H0` (V1))⊕ (H1
` (X1)⊗H1` (V1))
= Q`(−1)⊕H2` (V1)⊕ (H1
` (X1)⊗H1` (V1))
Now using the induction hypothesis together with the easy fact that H1` (V1) =
⊕2≤i≤r+1H1` (Xi) we get the desired result.
Lemma 4.1.5. We have ords=2(L(Q`(−1), s)) = −1.
Proof. Indeed we have
L(Q`(−1), s)) =∏
p
1
1− p−s+1= ζ(s− 1)
and we know that ζ(s− 1) has a simple pole at s = 2.
41
Lemma 4.1.6. If X1 and X2 are two curves, then Conjecture 4.1.2 for V = X1×X2
is equivalent to the following identity
− ords=2 L(H1` (X1)⊗H1
` (X2), s) = dimQ`(H1
` (X1)(1)⊗H1` (X2))
GQ
Proof. Using Lemma 4.1.4 we have
H2` (V ) ' Q`(−1)2 ⊕ (H1
` (X1)⊗H1` (X2)), (14)
hence
L1(V, s) ∼ L(Q`(−1), s))2 · L(H1` (X1)⊗H1
` (X2), s),
and hence
ords=2 L2(V, s) = ords=2 L(Q`(−1), s))2 + ords=2 L(H1` (X1)⊗H1
` (X2), s).
Now by Lemma 4.1.5 we get
ords=2 L2(V, s) = −2 + ords=2 L(H1` (X1)⊗H1
` (X2), s). (15)
On an other hand twisting (14) by Q`(1) we get
H2` (V )(1) ' Q`
2 ⊕ (H1` (X1)(1)⊗H1
` (X2)).
Taking GQ invariance we get
H2` (V )(1)GQ ' Q`
2 ⊕ (H1` (X1)(1)⊗H1
` (X2))GQ .
Thus
dimQ`H2
` (V )(1)GQ = 2 + dimQ`(H1
` (X1)(1)⊗H1` (X2))
GQ , (16)
hence by (16) and (15) we see that Conjecture 4.1.2 is equivalent to the identity
ords=2 L(H1` (X1)⊗H1
` (X2), s) = dimQ`(H1
` (X1)(1)⊗H1` (X2))
GQ .
42
Now we give a proof of Proposition 4.1.3.
Proof. of Proposition 4.1.3. By Lemma 4.1.4 we have
H2` (V ) = Q`(−1)r ⊕ [⊕i<j(H
1` (Xi)⊗H1
` (Xj))] (17)
hence
ords=2 L(H2` (V ), s) = ords=2 L(Q`(−1), s)r +
∑i<j
ords=2 L(H1` (Xi)⊗H1
` (Xj), s)
= −r +∑i<j
ords=2 L(H1` (Xi)⊗H1
` (Xj), s) (18)
Now we are assuming that Conjecture 4.1.2 is true for the products Xi × Xj hence
by Lemma 4.1.6 we have
− ords=2 L(H1` (Xi)⊗H1
` (Xj), s) = dimQ`(H1
` (Xi)(1)⊗H1` (Xj))
GQ
hence (18) becomes
− ords=2 L(H2` (V ), s) = r +
∑i<j
dimQ`(H1
` (Xi)(1)⊗H1` (Xj))
GQ (19)
On an other hand, twisting (17) by Q`(1) and taking GQ invariance, we see that
the right hand side of equation (19) equals dimQ`H2
` (V )(1)GQ which completes the
proof.
Remark 4.1.7. If X1 and X2 are smooth projective curves over Q with a Q-rational
point, then conjecture 4.1.2 for the product X1 ×X2 is equivalent to the following
dimQ`(H1
` (JX1)(1)⊗H1` (JX2))
GQ = − ords=2 L(H1` (JX1)⊗H1
` (JX2), s)
where JX1 and JX2 are the Jacobian varieties of X1 and X2. This follows from Lemma
4.1.6 and Fact-2 above.
The previous remark suggests the following conjecture:
43
Conjecture 4.1.8. If A, B are abelian varieties over Q then we have the following
dimQ`(H1
` (A)(1)⊗H1` (B))GQ = − ords=2 L(H1
` (A)⊗H1` (B), s).
We end this section with the following result.
Proposition 4.1.9. Let A, B be abelian varieties, and let A ∼∏
i Aini and B ∼∏
j Bjmj be the isogeny decompositions of A and B into simple factors. If Conjecture
4.1.8 holds for all pairs Ai, Bj then it also holds for A, B.
Proof. Using the Kunneth formula we have
H1` (A) = ⊕iH
1` (Ai)
ni , and H1` (B) = ⊕jH
1` (Bi)
mj
hence
H1` (A)⊗H1
` (B) = ⊕i,j(H1` (Ai)⊗H1
` (Bj))ni·mj . (20)
From this it follows that
ords=2 L(H1` (A)⊗H1
` (B), s) =∑i,j
ni ·mj · ords=2 L(H1` (Ai)⊗H1
` (Bj), s). (21)
On the other hand, tensoring (20) with Q`(1) and taking GQ invariance we get
dimQ`(H1
` (A)(1)⊗H1` (B))GQ =
∑i,j
ni ·mj · dimQ`(H1
` (Ai)(1)⊗H1` (Bj))
GQ , (22)
hence it follows from equations (21) and (22) that if Conjecture 4.1.8 holds for each
pair Ai, Bj then it also holds for A, B.
As a conclusion we proved the following
Theorem 4.1.10. Let V = X1 × X2 × · · · · ×Xr, r ≥ 2 where Xi/Q are smooth
projective curves with a rational point. Conjecture 4.1.2 for V is a consequence of the
following statement:
For any pair of curves Xi, Xj and any simple component A of the jacobian JXi
and any simple component B of the jacobian JXjwe have the following
dimQ`(H1
` (A)(1)⊗H1` (B))GQ = − ords=2 L(H1
` (A)⊗H1` (B), s)
Proof. Combine Proposition 4.1.3, Remark 4.1.7 and Proposition 4.1.9.
44
4.2 Modular forms and modular abelian varieties
4.2.1 Modular forms and the Shimura construction
The following is a brief review of some notions and facts about modular forms. Details
can be found in [3] Chapters 5 and 6.
Let Γ be a congruence subgroup of level N with Γ1(N) ≤ Γ ≤ Γ0(N), and let
S2(Γ) denote the space of Γ-cusp forms of weight 2. Let TN,Q ⊂ EndC(S2(Γ)) denote
the Hecke algebra of level N generated over Q by the operators Tn and 〈n〉 for all n.
(Here, as in [3], p. 178, 〈n〉 = 0 if (n,N) > 1.) Equivalently, TN,Q is generated over
Q by Tp and 〈p〉 for all primes p.
A cusp form f =∑
n≥1 cf (n)qn ∈ S2(Γ) is a normalized newform for TN,Q if the
following two conditions are satisfied
1. f is an eigenfunction for TN,Q,
2. c1(f) = 1.
3. f is an element of the space S2(Γ)new.
See [3] Section 5.6. for the definition of S2(Γ)new.
It turns out that for such an f we have Tnf = cf (n)f and 〈n〉f = χf (n)f where
χf is the nebentypus of f .
Consider the the eigenvalue map νf : TN,Q → C, associated with f . We have
νf (Tn) = cf (n) and νf (〈n〉) = χf (n). Let Kf denotes the image of νf , since TN,Q is
a finitely generated Q-algebra (cf. [2], p. 40) we conclude that Kf is a finite field
extension of Q. For any embedding σ of Kf into C one defines a conjugate to f given
by fσ =∑
n≥1 σcf (n)qn. It turns out that fσ is also a newform. cf. [3], Theorem
6.5.4.
For f as above, we write as usual
L(f, s) =∑n≥1
cf (n)n−s.
For our purpose it is also useful to consider, for an integer M with N |M the L-series
LM(f, s) =∑
(n,M)=1
cf (n)n−s
45
which agrees with L(f, s) up to finitely many Euler factors. Note that
LM(f, s) = L(f(M), s)
where f(M) =∑
(n,M)=1 cf (n)qn is a modular form and is in fact a TfM -eigenform for
some level M which is a multiple of M ; cf. Ribet [27], Proposition 2.4. Using some
identities which are satisfied by the coefficients of f we find that LM(f, s) can be
written as an Euler product as follows
LM(f, s) =∏p-M
(1− cf (p)p−s + χf (p)p1−2s)−1.
Using the estimate |cf (n)| � n12+ε which is due to Eichler and Shimura, it is easy to
see that L(f, s) converges absolutely for <(s) > 2. We also have the following result;
cf. [3] Theorem 5.10.2.
Theorem 4.2.1. The L-series L(f, s) admits an analytic continuation to the whole
complex plane.
Shimura associated with such an f an abelian variety Af defined over Q which
is unique up to Q-isogeny and which satisfies the properties listed in the following
theorem cf. [3] Section 6.6 and section 9.5.
Theorem 4.2.2. 1. dim Af = [Kf : Q],
2. There exists an injective map Kf ↪→ End0(Af ).
3. If Fp is the Frobenius element at p (p coprime to `N) then
det(1− tF−1p |H1
` (Af )) =∏σ
(1− σcf (p)t + σχf (p)pt2)
In particular, the L-series L2(Af , s) equals the product∏
σ L(fσ, s) up to finitely
many Euler factors.
Part (3) of the previous theorem can be found in [2], page 47.
Next we give a result due to Ribet concerning the modular abelian varieties Af ;
cf. [27] Corollary 4.2.
46
Theorem 4.2.3. We have End0Q(Af ) ' Kf , in particular Af is simple.
Here as usual, End0Q(Af ) = EndQ(Af ) ⊗Z Q, where EndQ(Af ) is the algebra of
endomorphisms of Af which are defined over Q.
Let Γ be a modular group of level N as in the beginning of this section, and let
JΓ/Q be the Jacobian variety of the modular curve XΓ/Q. The next result gives the
isogeny decomposition of JΓ.
Theorem 4.2.4. We have an isogeny decomposition
JΓ ∼∏f
Anf
f
where the product runs over all normalized newforms of weight 2 (of all levels) on Γ
up to Galois equivalence, and the nf ’s are some appropriate integers.
Proof. See Ribet [27] for Γ = Γ1(N) and Kani [15], Eq (13), for the general case.
4.2.2 Rankin-Selberg Convolution
Let f =∑
cf (n)qn and g =∑
cg(n)qn be two newforms of levels dividing some
integer N . The L-series of f and g can be written in an Euler product forms
LN(f, s) =∏p6|N
(1− cf (p)p−s + χf (p)p1−2s)−1
LN(g, s) =∏p6|N
(1− cg(p)p−s + χg(p)p1−2s)−1
For any p not dividing N we can rewrite the Euler factors at p as follows
1− cf (p)p−s + χf (p)p1−2s = (1− αpp−s)(1− α′
pp−s)
1− cg(p)p−s + χg(p)p1−2s = (1− βpp−s)(1− β′
pp−s),
where αp + α′p = cf (p), βp + β′
p = cg(p), αpα′p = pχf (p), βpβ
′p = pχg(p) and |αp| =
|α′p| = |βp| = |β′
p| = p12 . One defines the Rankin convolution of L(f, s) and L(g, s)
away from the level as follows
47
L(f ⊗ g, s) =∏p6|N
(1− αpβpp−s)−1(1− αpβ
′pp
−s)−1(1− α′pβpp
−s)−1(1− α′pβ
′pp
−s)−1
The main result about Rankin convolution due to Rankin and Murty is the fol-
lowing
Theorem 4.2.5. 1. The L-series L(f ⊗ g, s) converges absolutely for <(s) > 2, it
has an analytic continuation to the entire complex plan with at most a simple
pole at s = 2. Moreover the pole occurs if and only if g = f c where f c is the
complex conjugate of f .
2. Assume that L(f ⊗ g, s) is holomorphic at s = 2, then L(f ⊗ g, 2) 6= 0.
Proof. See [23] Proposition 6.3 for 2 and [23] page 499 for 1. Note that the hypothesis
of [23] are satisfied here because f(N) and g(N) are eigenforms of some level N .
4.3 Tate’s conjecture for a product of modular curves
In this section we give a proof of Tate’s conjecture for an arbitrary product of modular
curves. Let Γ1, Γ2, ..., Γr be congruence subgroups of levels N1, N2, ..., Nr such that
for each i we have Γ1(Ni) ≤ Γi ≤ Γ0(Ni), and consider the r-fold product
V = XΓ1 ×XΓ2 × · · · · ×XΓr
where XΓiis the modular curve associated with Γi.
Theorem 4.3.1. With the same notations as above, conjecture TL1 is true for V .
For any i let JΓidenotes the Jacobian variety of the curve XΓi
. By Theorem 4.2.4
one has an isogeny decomposition
JΓi∼∏f
Anf
f
Thus by Theorem 4.1.10 we are reduced to prove the following
48
Theorem 4.3.2. If f and g are newforms of levels dividing some N , then
dimQ`(H1
` (Af )(1)⊗H1` (Ag))
GQ = − ords=2 L(H1` (Af )⊗H1
` (Ag), s).
Moreover, both sides of the previous identity are equal to the following[Kf : Q] if f and g are Galois conjugates ,
0 otherwise .
4.3.1 Proof of the first identity
Proposition 4.3.3. We have
L(H1` (Af )⊗H1
` (Ag), s) ∼∏σ,τ
L(fσ ⊗ gτ , s)
where σ and τ run over the embeddings of Kf and Kg in C respectively.
Proof. From Theorem 4.2.2 part 3, we have the identities
det(1− tF−1p |H1
` (Af )) =∏σ
(1− σcf (p)t + σχf (p)pt2)
and
det(1− tF−1p |H1
` (Ag)) =∏τ
(1− τcg(p)t + τχg(p)pt2).
These identities are satisfied whenever p 6 |`NfNg, where Nf and Ng are the levels of
f and g. Now for any σ and τ we have
1− σcf (p)t + σχf (p)pt2 = (1− α1,σ(p)t)(1− α2,σ(p)t)
and
1− τcg(p)t + τχg(p)pt2 = (1− β1,τ (p)t)(1− β2,τ (p)t).
hence the eigenvalues of F−1p |H1
` (Af ) are {αi,σ(p) : i = 1, 2 and σ ∈ HomQ−alg(Kf , C)},and the eigenvalues of F−1
p |H1` (Ag) are {βj,τ (p) : j = 1, 2 and τ ∈ HomQ−alg(Kg, C)}.
Therefore the eigenvalues of F−1p |H1
` (Af )⊗H1` (Ag) are {αi,σ(p)βj,τ (p)/i, j = 1, 2 and σ ∈
HomQ−alg(Kf , C) and τ ∈ HomQ−alg(Kg, C)}. From this we get
det(1− tF−1p |H1
` (Af )⊗H1` (Ag)) =
∏σ,τ
∏i,j
(1− αi,σ(p)βj,τ (p)t).
49
Now plug in t = p−s. We see that the local L-series of the compatible system
{H1` (Af )⊗H1
` (Ag)}` at p is given by
Lp(H1` (Af )⊗H1
` (Ag), s)−1 =
∏σ,τ
∏i,j
(1− αi,σ(p)βj,τ (p)p−s)−1
Now, by definition in Section 4.2.2, the factor∏
i,j(1− αi,σ(p)βj,τ (p)p−s)−1 is exactly
the local p factor of the Rankin convolution L(fσ ⊗ gτ , s). Therefore taking the
product over all primes not dividing NfNg we get the equation
∏p6|Nf Ng
Lp(H1` (Af )⊗H1
` (Ag), s)−1 =
∏σ,τ
L(fσ ⊗ gτ , s).
This proves the proposition.
Theorem 4.3.4. We have the identity
− ords=2 L(H1` (Af )⊗H1
` (Ag), s) =
[Kf : Q] if f and g are Galois conjugates ,
0 otherwise .
Proof. Using Proposition 3.3.2 we see that
ords=2 L(H1` (Af )⊗H1
` (Ag), s) =∑σ,τ
ords=2 L(fσ ⊗ gτ , s) (23)
Assume that f and g are not Galois conjugates. Then for any pair (σ, τ) we have
(fσ)c 6= gτ , hence by Theorem 4.2.5 the L-series L(fσ⊗gτ , s) is holomorphic at s = 2
and has no zero at s = 2. Thus ords=2 L(H1` (Af ) ⊗ H1
` (Ag), s) =∑
σ,τ 0 = 0. Now
If g and f are Galois conjugates, then Af and Ag are isogenous. Therefore H1` (Af )
and H1` (Ag) are isomorphic Galois modules, and therefore H1
` (Af ) ⊗ H1` (Ag) 'GQ
H1` (Af )⊗H1
` (Af ). Thus, Equation (23) becomes the following
ords=2 L(H1` (Af )⊗H1
` (Ag), s) =∑σ,τ
ords=2 L(fσ ⊗ f τ , s) (24)
which can be rewritten as follows
ords=2 L(H1` (Af )⊗H1
` (Ag), s) =∑τ=cσ
ords=2 L(fσ ⊗ f τ , s) +∑τ 6=cσ
ords=2 L(fσ ⊗ f τ , s)
50
Since for τ 6= cσ, the series L(fσ ⊗ f τ , s) has no zero or pole at s = 2, we have∑τ 6=cσ ords=2 L(fσ ⊗ f τ , s) = 0, and hence
ords=2 L(H1` (Af )⊗H1
` (Ag), s) =∑τ=cσ
ords=2 L(fσ ⊗ f τ , s)
=∑τ=cσ
−1 = −#{(σ, τ) : τ = cσ}
Since σ 7→ cσ is a bijection of HomQ−alg(Kf , C), we have −#{(σ, τ) : τ = cσ} =
−[Kf : Q]. This completes the proof.
4.3.2 Proof of the second identity
Before we give the proof of the second identity of Theorem 4.3.2, we need to state
some results.
Proposition 4.3.5. Let Af denotes the abelian variety associated with the newform
f . We have the following isomorphism of GQ- modules
H1` (Af )(1) 'GQ H1
` (Af )∨
Proof. This is a consequence of the Weil pairing, and the fact that the `-adic coho-
mology is dual to the Tate module; cf. [21], page 132, and [21], Theorem 15.1, part
a.
Next we state Faltings Theorem on abelian varieties.
Theorem 4.3.6. If A, B are abelian varieties over Q, then one has an isomorphism
HomGQ(H1` (A), H1
` (B)) ' HomQ(B, A)⊗Q`
Proof. This follows from Faltings [4], Corollary 1, and the fact that the `-adic coho-
mology is dual to the Tate module. See also Tate [34], p 72.
Now we prove the second identity of Theorem 4.3.2.
51
Proof. By Proposition 4.3.5 we have an isomorphism
H1` (Af )(1)⊗H1
` (Ag) ' H1` (Af )
∨ ⊗H1` (Ag)
' Hom(H1` (Af ), H
1` (Ag)).
Thus, taking GQ-invariance we get the isomorphism
(H1` (Af )(1)⊗H1
` (Ag))GQ ' HomGQ(H1
` (Af ), H1` (Ag)).
Now using Theorem 4.3.6, the right hand side of the previous isomorphism is in turn
isomorphic to HomQ(Ag, Af )⊗Q`. Thus, taking dimension in both sides we have the
equality
dimQ`(H1
` (Af )(1)⊗H1` (Ag))
GQ = dimQ`(HomQ(Ag, Af )⊗Q`)
Now since Af and Ag are simple, any non zero element of HomQ(Ag, Af ) will be an
isogeny. However, a result due to Ribet states that Af and Ag are isogenous if and
only if f and g are Galois conjugates; cf. Kani, [15], Theorem 16. Therefore
HomQ(Ag, Af ) '
EndQ(Af ) if f and g are Galois conjugates ,
0 otherwise .
Using this and the fact that dimQ(EndQ(Af )) = [Kf : Q] we get
dimQ`(H1
` (Af )(1)⊗H1` (Ag))
GQ =
[Kf : Q] if f and g are Galois conjugates ,
0 otherwise .
which is the desired second identity of Theorem 4.3.2.
52
Chapter 5
Quotients of Products of Modular
Curves
5.1 Endomorphisms of Abelian Varieties and `-
adic Cohomology
Let F be a field, let F denote its separable closure, and let A/F be an abelian
variety. Set End0F (A) := EndF (A) ⊗Z Q, where EndF (A) is the Z-algebra of F -
endomorphisms of A. In this section we study in some details the action of End0F (A)
on the first cohomology group H1` (A). For this it is useful to first define the notion
of the Tate module associated with an abelian variety. For any integer N , let A[N ]
denotes the subgroup of A(F ), consisting of all elements of order dividing N . It
turns out that for all integers N which are coprime to the characteristic of F , the
group A[N ] is a free Z/NZ-module of rank 2g (cf. Milne [21], Remark 8.4, p.116),
where g = dim(A). Moreover, the group A[N ] has a natural action by the Galois
group Gal(F/F ). Let ` be a prime different from the characteristic of F , then for all
integers n, multiplication by ` define surjective homomorphisms
[`] : A[`n+1]→ A[`n]
53
These surjective maps make {A[`n]}n≥1 into a projective system. The Tate module
is by definition the inverse limit associated with this system
T`(A) := lim←−n
A[`n]
One may view T`(A) as the set of sequences {αn}n≥1 satisfying the following two
conditions
1. αn ∈ A[`n] for all n and,
2. [`]αn+1 = αn for all n.
Since every A[`n] is free of rank 2g over Z/`nZ, we see that T`(A) is a free Z`-module
of rank 2g. Moreover the Galois group Gal(F/F ) acts on T`(A) as follows
σ{αn}n≥1 = {σαn}n≥1
Taking this one step further, we define
V`(A) = T`(A)⊗Z`Q`.
This is a Q`-vector space of dimension 2g with an action by the Galois group Gal(F/F ).
We have the following result.
Theorem 5.1.1. Let ` be a prime different from the characteristic of F . We have
the following:
1. The natural representation ϕ : End0F (A)⊗Q` → EndQ`
(V`(A)) is injective.
2. For any subfield L of End0F (A) which has the same identity element as End0
F (A),
the space V`(A) is a free L⊗Q Q`- module of rank 2 dim(A)[L:Q]
.
3. H1` (A) is the dual space of V`(A), i.e, H1
` (A) ' V`(A)∨
Proof. See [21] Theorem 12.5 for 1), and [21] Proposition 12.12 for 3) and [21] Theo-
rem 15.1 (a) for 3).
54
Consider the Q`-space VA = H1` (A) ⊗Q`
Q`. It follows from parts 2) and 3) of
the previous theorem that for any subfield L of End0F (A), the space VA is a free
L⊗Q Q`- module of rank 2 dim(A)[L:Q]
. For any embedding σ : L→ Q`, we have an induced
morphism of Q`-algebras
πσ : L⊗Q Q` → Q`.
πσ is uniquely determined by πσ(θ ⊗ 1) = σ(θ). Conversely, any morphism of Q`-
algebras f : L ⊗Q Q` → Q` has the form πσ for some embedding σ : L → Q`. Since
L⊗Q Q` is semisimple we have an isomorphism of Q`-algebras
π : L⊗Q Q` →∏σ
Q`,
which is given by
π(θ ⊗ 1) = (πσ(θ ⊗ 1))σ = (σ(θ))σ.
Consider eσ to be the element of L ⊗Q Q` which maps under π to the projector
(0, ..0, 1, 0.., 0) of the σ factor. Now let
VA,σ = eσVA;
this is just the eigenspace associated with πσ, that is, the subspace of VA given by
VA,σ = {v ∈ VA : Tθ(v) = πσ(θ) · v for all θ ∈ L⊗Q Q`}.
VA,σ has dimension 2 dim(A)[L:Q]
as a Q`-vector space. Moreover we have the following
decomposition
VA = ⊕σVA,σ.
Remark 5.1.2. The space VA,σ can be also defined as a tensor product VA,σ =
VA ⊗L⊗QQ`Q` where Q` is viewed as a L⊗Q Q`-module via the map πσ.
5.2 Cohomology of modular Abelian varieties and
the Hecke algebra
In this section we focus on modular abelian varieties. Recall that from Theorem
4.2.3, we have End0Q(Af ) is a number field isomorphic to Kf , where Kf is the field
55
generated over Q by the Fourier coefficients of the newform f . Thus in view of
Theorem 5.1.1, the space H1` (Af ) is a free Kf ⊗Q`
Q`-module of rank 2 (observe that
[Kf : Q] = dim(Af )). Here in the case of modular abelian varieties, one have a more
concrete interpretation of the action of Kf on H1` (Af ), namely that it comes from the
Hecke algebra. More precisely, let f =∑
n≥1 af (n)qn be the Fourier expansion of the
newform f . The form f is an eigenfunction for the Hecke algebra TN,Q, and satisfies
Tnf = af (n)f and 〈n〉f = χf (n)f.
Consider the the eigenvalue map associated with f ,
νf : TN,Q → C.
We have νf (Tn) = af (n) and νf (〈n〉) = χf (n). Thus Kf = νf (TN,Q) is a quotient of
TN,Q, hence TN,Q acts on H1` (Af ) via the surjective map
νf : TN,Q → Kf .
Now we want to understand the Galois action on H1` (Af ). Since any element of
Kf⊗Q`Q` gives rise to a Galois equivariant endomorphism on H1
` (Af ), we see that the
image of the Galois group GQ inside EndQ`(H1
` (Af )) is contained in the subalgebra
EndKf⊗Q`Q`
(H1` (Af )). Now since H1
` (Af ) is a free Kf ⊗Q`Q`- module, we can define
the characteristic polynomial of Galois transformations with coefficients in Kf⊗Q`Q`,
we are especially interested to the characteristic polynomial of the Frobenius elements
at unramified primes. We have the following result due to Shimura; cf. [3] Page 390.
Theorem 5.2.1. Let Nf denotes the level of the new form f and let p be a prime
not dividing `Nf . If Fp is a Frobenius element at p, then we have the following
det(1− tF−1p |H1
` (Af )) = 1− af (p)t + pχf (p)t2
where af (p) is the pth Fourier coefficient of f and χf is the Dirichlet character asso-
ciated with f .
Remark 5.2.2. The coefficients of the characteristic polynomial in the previous the-
orem has to be seen as linear operators rather than just elements of Kf . In fact af (p)
and χf (p) has to be seen as the Hecke operators Tp and 〈p〉 as acting on H1` (Af ).
56
Recall that (Z/NZ)× is mapped surjectively to the finite subgroup of (TN,Q)×
consisting of diamond operators. The map identifies an element δ ∈ (Z/NZ)× with
the diamond operator 〈δ〉 ∈ (TN,Q)×. Thus δ ∈ (Z/NZ)× acts on H1` (Af ) as 〈δ〉,
which in turn acts via its image νf (〈δ〉) = χf (δ) ∈ K×f .
Let Vf = H1` (Af ) ⊗Q`
Q`. In view of the discussion following Theorem 5.1.1 we
have a decomposition
Vf = ⊕σVf,σ
where σ run over the embedding of Kf in Q`. An element θ ∈ Kf acts on each Vf,σ
by multiplication by σθ. Hence, an element T of the Hecke algebra (TN,Q) acts on
Vf,σ by multiplication by σνf (T ). In particular the group (Z/NZ)× acts on Vf,σ by
the following
δ(ξ) = σνf (〈δ〉)(ξ) = σ(χf (δ)) · ξ. (25)
Now for the Galois action on the space Vf,σ, it follows from Theorem 5.2.1 and
Remark 5.1.2 that the characteristic polynomial of a Frobenius element Fp is given
by
det(1− tF−1p |Vf,σ) = 1− σaf (p)t + pσχf (p)t2.
To end this section we state an important result due to Ribet about the spaces
Vf,σ.
Theorem 5.2.3. The spaces Vf,σ are irreducible GQ-modules and hence
dimQ`EndGQ(Vf,σ) = 1
.
Proof. See Ribet [27] Proposition 4.1 and its proof.
57
5.3 Tate Conjectures for Quotients of a Product
of Modular Curves
5.3.1 Conjecture T 1 for quotients of a product of curves
Let X1, ..., Xr be smooth projective and geometrically irreducible curves over Q, and
consider the product V = X1× ...×Xr. Moreover let G be an arbitrary group acting
on V by automorphisms. We have the following
Theorem 5.3.1. Conjecture T 1 holds for the quotient variety V/G.
Proof. This is an immediate consequence of Proposition 4.1.1 and Theorem 3.5.3.
5.3.2 Conjecture TL1 for quotients of a product of two mod-
ular curves
Let X1(N), X0(N) denote the usual modular curves over Q associated to the modular
groups Γ1(N) and Γ0(N). cf. [3], Section 7.7. We have isomorphisms X1(N)⊗Q C 'Γ1(N)\H and X0(N) ⊗Q C ' Γ0(N)\H where H is the upper half plane and H =
H ∪Q ∪∞ . We have the exact sequence
0→ Γ1(N)→ Γ0(N)→ (Z/NZ)× → 0
where the map Γ0(N)→ (Z/NZ)× sends γ =
(a b
c d
)to d Mod N . Thus we have
an isomorphism
Γ0(N)/Γ1(N) ' (Z/NZ)×
Using this isomorphism, one make the group (Z/NZ)× acts on the Riemann surface
X1(N) as follows
〈δ〉 · (Γ1(N)τ) = Γ1(N)σδ(τ)
where σδ is any element
(a b
c d
)∈ Γ0(N) such that d ≡ δ Mod N . The quotient
of X1(N) by this action is X0(N). Moreover this action is defined over Q. cf. [3],
58
Section 7.9. Now for any two elements 〈δ〉 and 〈δ′〉 in (Z/NZ)× the automorphisms
X1(N)〈δ〉→ X1(N) and X1(N)
〈δ′〉→ X1(N)
define an automorphism on the product surface
X1(N)×X1(N)〈δ〉×〈δ′〉−→ X1(N)×X1(N)
this defines an action of (Z/NZ)× × (Z/NZ)× on X = X1(N) × X1(N). For any
subgroup H of (Z/NZ)× × (Z/NZ)× let YH = X/H denotes the quotient variety of
X by H. Recall that by Theorem 3.4.1, the L-function L2(YH , s) is defined. We have
the following result.
Theorem 5.3.2.
dimQ(Z1(YH)⊗Q) = −ords=2(L2(YH , s)) = dimQ`(H2
` (YH)(1)GQ)
The identity dimQ(Z1(YH) ⊗ Q) = dimQ`(H2
` (YH)(1)GQ) follows from Theorem
5.3.1. In this section we will give a proof to the identity
ords=2(L2(YH , s)) = − dimQ`(H2
` (YH)(1)GQ). (26)
To prove this identity we first apply the Kunneth formula, we have the decompo-
sition
H2` (X1(N)×X1(N)) '
(H0` (X1(N))⊗H2
` (X1(N))⊕ (H2` (X1(N))⊗H0
` (X1(N))⊕ (H1` (X1(N))⊗H1
` (X1(N)))
Next using Theorem 3.1.1 we have
H2` (YH) ' H2
` (X1(N)×X1(N))H
Thus
H2` (YH) '
[(H0` (X1(N))⊗H2
` (X1(N))⊕(H2` (X1(N))⊗H0
` (X1(N))⊕(H1` (X1(N))⊗H1
` (X1(N)))]H
Now in order to evaluate the right hand side of the previous identity, we need first to
review some facts about the Kunneth formula.
59
Let X, Y be varieties over Q. Recall that the Kunneth formula states that for
any integer k we have a decomposition
Hk` (X × Y ) '
⊕i+j=k
(H i`(X)⊗Hj
` (Y )).
This decomposition is compatible with maps, in the sense that if f : X → X and
g : Y → Y are morphisms and f × g : X × Y → X × Y is the product map, then the
induced map (f × g)∗k on the cohomology group Hk` (X × Y ) is identified under the
previous decomposition as follows
(f × g)∗k '⊕
i+j=k
f ∗i ⊗ g∗j (27)
where f ∗i and g∗j are the induced maps on the cohomology groups H i`(X) and Hj
` (Y ).
Applying this to our case we get
H2` (YH) '
(H0` (X1(N))⊗H2
` (X1(N))H⊕(H2` (X1(N))⊗H0
` (X1(N))H⊕(H1` (X1(N))⊗H1
` (X1(N)))H ,
where, in each factor H i`(X1(N)) ⊗Hj
` (X1(N)), i + j = 2, an element 〈δ1〉 × 〈δ2〉 in
H acts as follows
(〈δ1〉 × 〈δ2〉) · (ξ ⊗ η) = 〈δ1〉∗ξ ⊗ 〈δ2〉∗η.
Now (Z/NZ)× acts trivially on the spaces H0` (X1(N)) and H2
` (X1(N)). Indeed by
Theorem 3.1.1 we have
H0` (X1(N))(Z/NZ)× ' H0
` (X0(N)) ' Q`
and
H2` (X1(N))(Z/NZ)× ' H2
` (X0(N)) ' Q`(−1).
Therefore we get the isomorphism
H2` (YH) ' Q`(−1)2 ⊕ (H1
` (X1(N))⊗H1` (X1(N)))H .
60
Thus, by the same argument as in the previous chapter, mainly Lemma 4.1.6, we
conclude that proving Equation (26) is the same as proving the following
−ords=2(L((H1` (X1(N))⊗H1
` (X1(N)))H , s)) = dimQ`
(((H1
` (X1(N))⊗H1` (X1(N))(1))H)GQ
)(28)
Observe that associating an L-series to H1` (X1(N)) ⊗ H1
` (X1(N)))H makes sense,
because of Theorem 3.4.1.
Now we want to decompose the space (H1` (X1(N)) ⊗ H1
` (X1(N)))H into smaller
subspaces. To do this we consider the Jacobian variety J1(N) associated with the
curve X1(N). The cusp at infinity is a Q- rational point in X1(N), therefore, by Fact
2, Subsection 4.1.2 we have an isomorphism of Galois modules
H1` (X1(N)) 'GQ H1
` (J1(N)).
Also using Theorem 4.2.4 we have a decomposition
H1` (J1(N)) 'GQ
⊕f
H1` (Af )
nf ,
where the f ’s run over newforms of some level dividing N up to Galois equivalence,
and the nf are appropriate integers. Thus we get the following(H1
` (X1(N))⊗H1` (X1(N))
)H 'GQ
⊕f,g
((H1
` (Af )⊗H1` (Ag)
)H)nf ng
. (29)
Since we want to associate L-series to the spaces (H1` (Af )⊗H1
` (Ag))H
, we need the
following lemma
Lemma 5.3.3. The system {(H1` (Af )⊗H1
` (Ag))H}` is strictly compatible.
Proof. Consider the product Af×Ag. We have seen that (Z/NZ)× acts on the abelian
varieties Af and Ag as a part of the action of the Hecke algebra. Therefore the group
H ⊂ (Z/NZ)× × (Z/NZ)× acts naturally on Af ×Ag. Consider the quotient variety
(Af × Ag)/H. We know by Theorem 3.1.1 that
H2` ((Af × Ag)/H) ' H2
` (Af × Ag)H
61
Now applying the Kunneth formula together with the compatibility with morphism
as in Equation (27) we have
H2` ((Af × Ag)/H) ' H2
` (Af )H ⊕H2
` (Ag)H ⊕
(H1
` (Af )⊗H1` (Ag)
)H. (30)
Again by Theorem 3.1.1 H2` (Af )
H ' H2` (Af/H) and H2
` (Ag)H ' H2
` (Ag/H). Thus
(30) becomes the following
H2` ((Af × Ag)/H) ' H2
` (Af/H)⊕H2` (Ag/H)⊕
(H1
` (Af )⊗H1` (Ag)
)H(31)
Now by Theorem 3.4.1 the systems {H2` ((Af×Ag)/H)}`, {H2
` (Af/H)}` and {H2` (Ag/H)}`
are all strictly compatible. Thus, we deduce from (31) that the system {(H1` (Af )⊗H1
` (Ag))H}`
is also strictly compatible.
Now we want to determine the L-series of the system {(H1` (Af )⊗H1
` (Ag))H}`.
Consider the space
Wf,g =(H1
` (Af )⊗H1` (Ag)
)⊗Q`
Q`
'(H1
` (Af )⊗Q`Q`
)⊗(H1
` (Ag)⊗Q`Q`
).
We know from Section 5.2 that(H1
` (Af )⊗Q`Q`
)= ⊕σVf,σ and
(H1
` (Ag)⊗Q`Q`
)= ⊕τVg,τ
Thus
Wf,g ' ⊕σ,τ (Vf,σ ⊗ Vg,τ ).
From Section 5.2 we know that an element δ ∈ (Z/NZ)× acts on Vf,σ by multiplication
by σ(χf (δ)), hence an element (δ, λ) ∈ (Z/NZ)×× (Z/NZ)× acts on the space Vf,σ⊗Vg,τ by multiplication by σ(χf (δ)) · τ(χf (λ)). Now we have
WHf,g ' ⊕σ,τ (Vf,σ ⊗ Vg,τ )
H
We want to compute the space (Vf,σ ⊗ Vg,τ )H . Let χf,σ := σχf and χg,τ := τχg, and
consider the product character
χf,σ · χg,τ : (Z/NZ)× × (Z/NZ)× −→ Q×`
62
Let Jf,g,σ,τ be the kernel of χf,σ · χg,τ . By definition of the action of H on Vf,σ ⊗ Vg,τ
we have the following
(Vfσ ⊗ Vgτ )H =
Vfσ ⊗ Vgτ if H ⊂ Jf,g,σ,τ ,
0 otherwise .
Therefore we get the following
WHf,g '
⊕H⊂Jf,g,σ,τ
(Vf,σ ⊗ Vg,τ ) . (32)
Now let i` : Q` ↪→ C be a fixed embedding, and for any embedding σ : Kf ↪→ Q`, let
σ := i` ◦ σ : Kf ↪→ C. We have the following
Theorem 5.3.4. The L-series associated with the strictly compatible system {(H1` (Af )⊗H1
` (Ag))H}`
is given by
L((H1
` (Af )⊗H1` (Ag)
)H, s) ∼
∏H⊂Jf,g,σ,τ
L(f eσ ⊗ geτ , s)
where, σ = i` ◦ σ and i` is any embedding from Q` into C. Moreover, L(f eσ ⊗ geτ , s) is
the Rankin convolution of f eσ and geτ .
Proof. Let p a prime not dividing `N , and let Fp be a Frobenius element at p. Let
Q(t) = det(1− t · F−1p |((H1
` (Af )⊗H1` (Ag))
H)
By Lemma 5.3.3, Q(t) has integral coefficients and is independent of `. Now by
Equation (32) we have
Q(t) = det(1− t · F−1p |WH
f,g) =∏
H⊂Jf,g,σ,τ
det(1− t · F−1p |Vf,σ ⊗ Vg,τ )
We want to decompose Q(t) into linear factors. From the discussion following Theo-
rem 5.2.1 we know that the following holds:
det(1− tF−1p |Vf,σ) = 1− σaf (p)t + pσχf (p)t2,
and
det(1− tF−1p |Vg,τ ) = 1− τag(p)t + pτχg(p)t2.
63
Now for any σ and τ satisfying H ⊂ Jf,g,σ,τ we have
1− σaf (p)t + pσχf (p)t2 = (1− α1,σ(p)t)(1− α2,σ(p)t)
and
1− τag(p)t + pτχg(p)t2 = (1− β1,τ (p)t)(1− β2,τ (p)t).
Thus
Q(t) =∏
H⊂Jf,g,σ,τ
∏i,j∈{1,2}
(1− αi,σ(p)βj,τ (p)t).
Since Q(t) has coefficients in Z, then for any embedding i` : Q` ↪→ C we have
Q(t) =∏
H⊂Jf,g,σ,τ
∏i,j∈{1,2}
(1− i`(αi,σ(p)βj,τ (p))t) ,
and therefore we see that the local p-factor of the L-series associated to {(H1` (Af )⊗H1
` (Ag))H}`
is given by
Q(p−s) =∏
H⊂Jf,g,σ,τ
∏i,j∈{1,2}
(1− i`(αi,σ(p)βj,τ (p))p−s
).
Also it is easy to see that
1− σaf (p)t + pσχf (p)t2 = (1− i`(α1,σ(p))t)(1− i`(α2,σ(p))t)
and
1− τ ag(p)t + pτχg(p)t2 = (1− i`(β1,τ (p))t)(1− i`(β2,τ (p))t).
Therefore it follows by definition in Section 4.2.2 that the factor∏i,j∈{1,2}
(1− i`(αi,σ(p)βj,τ (p))p−s)−1
is exactly the local p-factor of the Rankin convolution L(f eσ⊗geτ , s). Thus, taking the
product over all primes not dividing N , we get our result.
Now we compute the order of the pole of the L-series L(((H1` (Af ))⊗H1
` (Ag))H , s)
at s = 2. We have the following
64
Corollary 5.3.5.
ords=2L(((H1` (Af ))⊗H1
` (Ag))H , s) = −#{(σ, τ) : H ⊂ Jf,g,σ,τ and geτ = f c◦eσ}.
Proof. From Theorem 5.3.4 we have
ords=2L(((H1` (Af ))⊗H1
` (Ag))H , s) =
∑H⊂Jf,g,σ,τ
ords=2L(f eσ ⊗ geτ , s).
Now using Theorem 4.2.5 we see that
ords=2L(((H1` (Af ))⊗H1
` (Ag))H , s) = −#{(σ, τ) : H ⊂ Jf,g,σ,τ and geτ = f c◦eσ}.
where c is the complex conjugation.
Now recall from the beginning of this subsection that proving Theorem 5.3.2 was
reduced to prove Equation (28) which we restate here:
−ords=2L((H1` (X1(N))⊗H1
` (X1(N)))H , s) = dimQ`
((H1
` (X1(N))⊗H1` (X1(N))(1))H
)GQ .
(33)
Now in view of the decomposition in Equation (29) and taking under consideration
Theorem 4.2.5 part 2), we see that it suffices to show that
−ords=2L(((H1` (Af ))⊗H1
` (Ag))H , s) = dimQ`
((H1
` (Af )⊗H1` (Ag)(1))
H)GQ
Now using Corollary 5.3.5, we see that we are reduced to prove the following identity
dimQ`
((H1
` (Af )⊗H1` (Ag)(1))
H)GQ = #{(σ, τ) : H ⊂ Jf,g,σ,τ and geτ = f c◦eσ.} (34)
Now tensoring the space (H1` (Af ) ⊗ H1
` (Ag)(1))H with Q` and using Equation (32)
we have((H1
` (Af )⊗H1` (Ag)(1))H
)⊗Q`
Q` = WHf,g(1) '
⊕H⊂Jf,g,σ,τ
(Vf,σ(1)⊗ Vg,τ ) . (35)
Now taking Galois invariants we have((H1
` (Af )⊗H1` (Ag)(1))H
)GQ ⊗Q`Q` = WH
f,g(1)GQ '
⊕H⊂Jf,g,σ,τ
(Vf,σ(1)⊗ Vg,τ )GQ .
(36)
65
Thus
dimQ`
((H1
` (Af )⊗H1` (Ag)(1))
H)GQ = dimQ`
WHf,g(1)
GQ = (37)
∑H⊂Jf,g,σ,τ
dimQ`(Vf,σ(1)⊗ Vg,τ )
GQ . (38)
Now combining Equations (34) and (38) we see that we need only to prove the fol-
lowing∑H⊂Jf,g,σ,τ
dimQ`(Vf,σ(1)⊗ Vg,τ )
GQ = #{(σ, τ) : H ⊂ Jf,g,σ,τ and geτ = f c◦eσ}. (39)
However, one sees easily that this last identity follows from the following
dimQ`(Vf,σ(1)⊗ Vg,τ )
GQ =
1 if geτ = f c◦eσ,
0 otherwise.(40)
The following theorem gives a proof of Equation (40) and completes the proof of
Theorem 5.3.2.
Theorem 5.3.6. The following are equivalent
1. The space (Vf,σ(1)⊗ Vg,τ )GQ is non-zero.
2. For any embedding i` : Q` ↪→ C, we have geτ = f c◦eσ.
Moreover, if the above conditions hold, then
dimQ`(Vf,σ(1)⊗ Vg,τ )
GQ = 1.
We need the following easy lemma.
Lemma 5.3.7. Let V be a two dimensional vector space over a field K, and let G be a
group acting on V . If g ∈ G is represented with a matrix T =
(a b
c d
)with respect
to some basis {v1, v2}, then on the dual space V ∨, g is represented with the matrix
δ−1 ·
(d −c
−b a
)with respect to the dual basis {v∗1, v∗2}, where δ is the determinant
of T .
66
Proof. We know that G acts on V ∨ by the role g(ϕ) = ϕ◦g−1. Hence, if T =
(a b
c d
)is the matrix representing g on V with respect to a basis {v1, v2}, then the matrix
representing g on V ∨ with respect to the dual basis {v∗1, v∗2} is the transpose of the
invese of T , that is (T−1)t = δ−1 ·
(d −c
−b a
), with δ = det(T ).
Now we give a proof of Theorem 5.3.6.
Proof. of Theorem 5.3.6.
Let i` : Q` ↪→ C be an embedding. Assuming that (Vf,σ(1)⊗ Vg,τ )GQ is non zero, we
want to show that geτ = f c◦eσ. We have an isomorphism of Galois modules
Vf,σ(1)⊗ Vg,τ 'GQ Hom(Vf,σ(1)∨, Vg,τ ).
Taking GQ invariance we get an isomorphism
(Vf,σ(1)⊗ Vg,τ )GQ ' HomGQ(Vf,σ(1)∨, Vg,τ )
We know from Theorem 5.2.3 that Vf,σ is simple, therefore Vf,σ(1) is simple, and
hence Vf,σ(1)∨ is simple. Thus any non-zero element of HomGQ(Vf,σ(1)∨, Vg,τ ) will be
an isomorphism of Galois modules. As we are assuming that HomGQ(Vf,σ(1)∨, Vg,τ )
is non-zero, we conclude that Vf,σ(1)∨ and Vg,τ are isomorphic Galois modules. Thus
for any prime p not dividing `N we have equality of traces
Tr(F−1p |Vf,σ(1)∨) = Tr(F−1
p |Vg,τ ) (41)
We know from Section 5.2 that det(1 − tF−1p |Vf,σ) = 1 − σaf (p)t + pσχf (p)t2 and
det(1− tF−1p |Vf,τ ) = 1− τag(p)t + pτχg(p)t2. Hence Tr(F−1
p |Vg,τ ) = τag(p).
We want to compute Tr(F−1p |Vf,σ(1)∨). We have Tr(F−1
p |Vf,σ) = σaf (p) and
det(F−1p |Vf,σ) = pσχf (p), therefore Tr(F−1
p |Vf,σ(1)) = p−1σaf (p) and det(F−1p |Vf,σ(1)) =
p−1σχf (p). Thus, using Lemma 5.3.7 we get Tr(F−1p |Vf,σ(1)∨) = (σχf (p))−1 · σaf (p).
Therefore, Equation (41) becomes the following
(σχf (p))−1 · σaf (p) = τag(p) (42)
67
Applying the embedding i` : Q` ↪→ C to Equation (42) we get
(σχf (p))−1 · σaf (p) = τ ag(p) (43)
which can be rewritten as
(χf eσ(p))−1 · af eσ(p) = ageτ (p), (44)
where f eσ and geτ are the conjugates of f and g with respect to the embeddings
σ : Kf ↪→ C and τ : Kg ↪→ C. Combining Equation (44) with the well known identity
χf eσ(p) · af eσ(p) = af eσ(p) (cf. Ribet [28] page 21), we get af eσ(p) = ageτ (p), or again
afc◦eσ(p) = ageτ (p). (45)
Since the previous identity holds for all primes p not dividing `N , it follows that
f c◦eσ = geτ .
Conversely, assume that the identity f c◦eσ = geτ holds for an embedding i` : Q` ↪→C. We want to prove that the space (Vf,σ(1) ⊗ Vg,τ )
GQ is non-zero. By what we
did earlier, this is the same as proving that HomGQ(Vf,σ(1)∨, Vg,τ ) is non-zero, and
this is the same as proving that Vf,σ(1)∨ and Vg,τ are isomorphic as Galois modules.
Now since both spaces are simple (hence semisimple), then by Serre [31] Lemma
in page I-11, it suffices to show that for any γ ∈ GQ we have equality of traces
Tr(γ|Vf,σ(1)∨) = Tr(γ|Vg,τ ). Now by the Cebotarev density theorem (cf. Serre [31]
Corollary 2 part (a) page I-8.) and the fact that the maps γ 7→ Tr(γ|Vf,σ(1)∨) and
γ 7→ Tr(γ|Vg,τ ) are continuous, it suffices to show that the identity
Tr(F−1p |Vf,σ(1)∨) = Tr(F−1
p |Vg,τ )
holds for all primes p not dividing `N . To do this, it suffices to show that
i`(Tr(F−1p |Vf,σ(1)∨)) = i`(Tr(F−1
p |Vg,τ ))
for any embedding i` : Q` ↪→ C. But we computed both sides of the last identity
earlier, we found that i`(Tr(F−1p |Vf,σ(1)∨)) = afc◦eσ(p) and i`(Tr(F−1
p |Vg,τ )) = ageτ (p).
Thus the identity we want to prove is just the following:
afc◦eσ(p) = ageτ (p).
68
This is true, since we are assuming that f c◦eσ = geτ .
Now we prove the last assertion of the previous theorem, i.e., that under one of
the equivalent conditions (1) or (2), we have dimQ`(Vf,σ(1) ⊗ Vg,τ )
GQ = 1. Assume
(1) holds, we have dimQ`(Vf,σ(1)⊗ Vg,τ )
GQ = dimQ`HomGQ(Vf,σ(1)∨, Vg,τ ). Now since
HomGQ(Vf,σ(1)∨, Vg,τ ) is non zero, then as before Vf,σ(1)∨ and Vg,τ are isomorphic
Galois modules. Thus we have an isomorphism HomGQ(Vf,σ(1)∨, Vg,τ ) ' EndGQ(Vg,τ ),
and EndGQ(Vg,τ ) is one dimensional by Theorem 5.2.3. This completes the proof.
69
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