algebraic and arithmetic geometry

Algebraic and Arithmetic Geometry

Lectures by Caucher BirkarNotes by Tony Feng

Lent 2014

Contents

1 Introduction 71.1 Rational points on varieties . . . . . . . . . . . . . . . . . . . . . . . 71.2 Dimension one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 General strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Heights 112.1 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Heights on projective space . . . . . . . . . . . . . . . . . . . . . . . 132.3 The Weil height machine on varieties . . . . . . . . . . . . . . . . . . 162.4 Height theory over function fields . . . . . . . . . . . . . . . . . . . . 21

3 Arakelov Geometry 253.1 Hermitian metrics on line bundles . . . . . . . . . . . . . . . . . . . . 253.2 Line bundles on number fields . . . . . . . . . . . . . . . . . . . . . . 26

4 Abelian Varieties 294.1 Abelian varieties over algebraically closed fields . . . . . . . . . . . . 294.2 Abelian varieties over number fields . . . . . . . . . . . . . . . . . . . 314.3 Torsion points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Weak Mordell-Weil . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 The Brauer-Manin Obstruction 375.1 Chevalley’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 Tsen’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3 Brauer-Severi varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 395.4 The Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.5 The Brauer-Manin obstruction . . . . . . . . . . . . . . . . . . . . . 42

6 Birational Geomery 456.1 Potential density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2 Kodaira dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.3 Surface fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3

CONTENTS

7 Statistics of Rational Points 517.1 Counting on projective space . . . . . . . . . . . . . . . . . . . . . . 517.2 Counting on abelian varieties . . . . . . . . . . . . . . . . . . . . . . 537.3 Jacobians of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

8 Zeta functions 558.1 The Hasse-Weil zeta function . . . . . . . . . . . . . . . . . . . . . . 558.2 The Frobenius Morphism . . . . . . . . . . . . . . . . . . . . . . . . 588.3 The Weil conjectures for curves . . . . . . . . . . . . . . . . . . . . . 588.4 Zeta functions of arithmetic schemes . . . . . . . . . . . . . . . . . . 61

4

Disclaimer

These are lecture notes originally “live-TEXed” for a course offered by Caucher Birkarat Cambridge University in Lent 2014. They have been very lightly edited by me. Itake full responsibility for all errors.

5

CONTENTS

6

Chapter 1

Introduction

1.1 Rational points on varieties

Algebraic geometry is about studying solutions of polynomials. Consider polynomi-als

F1, . . . , Fm ∈ k[x1, x2, . . .].

Classically, mathematicians were most interested in the case k = Q, but they werenaturally led to expand considerations to number fields, finite fields, function fields,etc. Diophantine geomery is the study of algebraic geometry over these special fields.

Example 1.1.1. Consider the polynomial F (x, y) = x2 + y2 ∈ Q[x, y]. Over Q, theonly solution is (0, 0). What happens if we extend our domain field? In this case,we get no additional points over R, but we do get lots of points over C.Example 1.1.2. Consider the polynomial F (x, y, z) = x2+y2−7z2 ∈ Q[x, y, z] cuttingout a variety. What are its rational points?

We claim that over Q, the only solution is (0, 0, 0). Why? Suppose that (a, b, c)is a rational solution. By clearing denominators, we may assume that a, b, c ∈ Z.Reducing mod 7, we must have a2 + b2 ≡ 0 (mod 7). The only squares in F7 are0, 1, 4, so it must be the case that 7 | a and 7 | b. Then 7 | c as well, and we are done(e.g. by induction).Example 1.1.3. Consider F (x, y, z) = x5+y5−7z5 ∈ Q[x, y, z]. It is an open problemwhether or not this has nontrivial positive rational solutions.Example 1.1.4. Consider F (x, y, z) = x2 + y2− z2 ∈ Q[x, y, z]. All rational solutionscan be described explicitly. This is explained by the geometric fact that F cuts outP1Q in the projective plane.

General setup. Let k be a field, X ⊂ Pnk a projective variety over k. ThenX = V (F1, . . . , Fm), where F1, . . . , Fm ∈ k[t0, . . . , tn]. We should think of X as ascheme, not just a set of solutions. For any field extension k′/k, we can consider thek′ points of X, X(k′). We are interested in the following questions:

7

CHAPTER 1. INTRODUCTION

1. Is X(k′) empty?

2. Is X(k′) is dense (in the Zariski topology)?

3. If X(k′) is dense, how are its points distributed?

1.2 Dimension one

Let X be a smooth, projective variety of dimension one over a number field K.There is a very important invariant associated to X, which is the genus. This canbe defined as

g = h0(ωX) = h1(OX).

Let’s consider some possibilities for low genus.

1. g = 0. Either X(K) = ∅ or it is infinite, and infinite - in fact, isomorphic toP1(K).

2. g = 1. X is an elliptic curve, and X(K) forms a finitely generated abeliangroup (this is the Mordell-Weil theorem).

3. g ≥ 2. Then X(K) is finite (a deep theorem of Faltings).

Note that if g = 0, ω−1X is ample. If g = 1, then ωX is trivial, and if g ≥ 2, then

ωX is ample. So we see that the geometric properties of X seem to determine itsarithmetic. This is a general philosophy.

1.3 Higher dimensions

The situation is much more complicated. There are a few kinds of varieties withextra structure that makes them more tractable:

1. Varieties “close to projective space.” For instance, rational varieties are thosewhich (by definition) are birational to projective space.

2. Abelian varieties. This form the nicest case, and the Mordell-Weil theoremoffers some control of X(K). The next few lectuers will be devoted to provingMordell-Weil.

Here are a few conjectures that express the philosophy “geometry determinesarithmetic.”

Conjecture 1.3.1 (Lang). Let X be a smooth, projective variety over a number fieldK. If ωX is ample, then X(K) is not dense.

Note that this implies Faltings’s theorem in the case of a curve.

8

1.4. GENERAL STRATEGIES

Conjecture 1.3.2 (Campana). If ωX ' ΩX or ωX is anti-ample, then X(K ′) isdense for some finite extension K ′/K.

Example 1.3.3. If X = V (F ) is a hypersurface in Pnk , then ωX is ample if and onlyif d > n+ 1, trivial if d = n+ 1, and anti-ample if d < n+ 1.

1.4 General strategies

It is difficult to provide a uniform theory for rational points on varieties. Here aresome general strategies/remarks that can be useful. Let X be a smooth projectivevariety over Q. We are interested in X(Q), which can be difficult to understand.

• One can study X(R), which is a smooth manifold, by using the techniques ofdifferential geometry. One can also look at X(Fp). This is descibed by theWeil conjectures. Sometimes, piecing this data together can give informationabout X(Q) (the local-global principle).

• There is a natural action of Gal(Q/Q) on X(Q). From this we get Galoisrepresentations, and we can try to think of X(Q) as the fixed part.

9

CHAPTER 1. INTRODUCTION

10

Chapter 2

Heights

2.1 Valuations

Definition 2.1.1. Let k be a field. An absolute value is a function | · | : k → R≥0 suchthat

• |x| = 0 ⇐⇒ x = 0,

• |xy| = |x||y|,

• |x+ y| ≤ |x|+ |y|.

If the stronger condition |x+ y| ≤ max|x|, |y| holds, then we say that the absolutevalue is non-Archimedean. Otherwise, it is Archimedean.

Example 2.1.2. For K = Q, we have the usual absolute value

|n|∞ = maxx,−x.

This is archimedean. There are also non-archimedean absolute values, constructedas follows. Fix any prime p ∈ Z. Then for x = pk ab , where p - ab, we set |x| = p−k.We let

MQ = places of Q = | · |∞, | · |p : p prime.

If K is a number field, then there are n = [K : Q] embeddings σ : K → C. Foreach embedding σ, we get a different absolute value (the pullback of the complexabsolute value). If σ(K) 6⊂ R, then σ and σ are distinct embeddings with | · |σ = | · |σ.

Let R = OK be the ring of integers of K. Pick a non-zero prime ideal p ⊂ R.Since OK is a Dedekind domain, p is a maximal ideal. Moreover, Rp is a discretevaluation ring, and its valuation extends to an order function

ordp : K∗ → Z

11

CHAPTER 2. HEIGHTS

Explicitly, if x is a uniformizer for Rp, then any x ∈ K∗ may be written as tmu whereu ∈ R∗p, and then ordp(x) = m. Now let (p) = Z ∩ P and let e = ordp(p). Define

|x|p =

0 x = 0

p− ordp(p)/e x 6= 0.

We denote by MK the places of K, which include both archimedean and non-archimedean places. For v ∈MK , the distance function

d(x, y) := |x− y|v

makes K into a metric space. We let Kv to be the completion of K with respectto this metric. If v is archimedean corresponding to d : K → C, then Kv = R orC, depending on whether or not the corresponding embedding lands in R. If v isnonarchimedean, corresponding to some prime ideal p ⊂ R, then R ⊂ Rp ⊂ K. Thecompletion is defined algebraically as

Rp = lim←−Rp/pm

and Kv = Frac(Rp). This is equivalent to the analytic completion.The algebraic geometry motivation is as follows: if X is a scheme and p ∈ X a

point, then a Zariski open neighborhood of p is very coarse. In contrast, Spec OX,pbehaves like an “analytic neighborhood,” capturing local data.Definition 2.1.3. Let K ⊂ K ′ be number fields. Suppose that w ∈MK , v ∈MK′ aretwo places such that

| · |v|K = | · |w.

Then the local degree of v over w is [K ′v : Kw].If R is the ring of integers of K and R′ the ring of integers of K ′, then the

geometric picture is that we have a map π : SpecR′ → SpecR, sending v 7→ w. Thisinduces a pullback map π∗ on divisors, with the property that

deg π∗w =∑vi 7→w

[K ′vi : Kw].

This leads to the following result.

Proposition 2.1.4. If v1, . . . , vr ∈MK′ are the places of K ′ lying over a place w ofK, then

[K ′ : K] =

r∑i=1

[K ′vi : Kw].

Definition 2.1.5. Let K be a number field, v ∈ MK , lying over w ∈ MQ. Letnv = [Kv : Qw]. Then we define the normalized absolute value

||x||v = |x|nvv .

12

2.2. HEIGHTS ON PROJECTIVE SPACE

It turns out that nv = ef , where f = [kv : Fp] is residue extension degree. Soanother way of writing the normalized absolute value is as

||x||v = (#kv)− ordv(x).

Proposition 2.1.6 (Product formula). Let K be a number field. Then for all non-zero x ∈ K, ∏

v∈MK

||x||v = 1.

Proof sketch. Fix w ∈MQ and let v1, . . . , vr ∈MK be the places lying over w. Thenwe have

r∏i=1

||x||vi = ||NK/Qx||w.

The result then follows from the product formula for Q, which is trivial.

Remark 2.1.7. This is analogous to the geometric fact that if π : X → Y is a mapof schemes, then π∗ div(f) = div(NX/Y (f))

2.2 Heights on projective space

Our goal is to define a height, which is a kind of measure of complexity, on the pointsof a variety.

Definition 2.2.1. Let K be a number field, x = [a0 : . . . : an] ∈ Pn(K). We definethe naïve height of x to be

HK(x) =∏

v∈MK

max||ai||v.

By the product formula, this is well-defined. From this it is easy to see thatHK(x) ≥ 1, because we can assume that ai = 1 for some i.

Example 2.2.2. Consider K = Q. Let x = [a0 : . . . , an] ∈ Pn(Q). By scalingappropriately, we can assume that the ai are integers, with no common factor. Thenfor each non-archimedean place v, max||ai||v = 1. So at the end of the day, wehave HQ(x) = max |ai|∞. Thus we see that the height is very simple over Q.

Theorem 2.2.3. Let K be a number field, x = [a0 : . . . : an] ∈ Pn(K). Let R bethe ring of integers of K, and assume without loss of generality that all ai ∈ R. LetI = (a0, . . . , an). Then we have∏

v∈MKv-∞

max||αi|v =1

NK/Q(I).

13

CHAPTER 2. HEIGHTS

Proof. For any nonarchimedean v ∈MK ,

max||ai||v = max

1

(#kv)ordv(ai)

=

1

(#kv)min ordv(ai)=

1

(#kv)ordv(I).

Now, we have a unique prime factorization for ideals, I = pm11 . . . pmr

r . So∏p

1

(#kv)ordv(I)=∏p

1

#(R/pmrr )

=1

#(R/I).

Of course, #(R/I) = NK/Q(I).

Example 2.2.4. Consider K = Q, x = [1 : 2] ∈ P1(Q). Then the ideal is I = (1, 2) =R, so N(I) = 1 and HQ(x) = 2. The only contribution to the height is from theArchimedean places, which we already knew. More generally, this will be true forany field with class number 1.

Example 2.2.5. Consider K = Q(i), x = [1 : 2] ∈ P1(K). Again we have N(I) = 1.For a complex place v, we have ||2||v = |2|2 = 4 so that HK(x) = 2. In particular,we see that the height of a point depends on the field we are considering.

Theorem 2.2.6. Let K ⊂ K ′ be an extension of number fields, x ∈ Pn(K). Thenwe have

HK′(x) = HK(x)[K′:K].

Proof. This is a simple application of the degree formula

[K ′ : K] =∑v|w

nv.

We have

HK′(x) =∏

v∈MK′

max||ai||v

=∏

w∈MK

∏v|w

max|ai|nvv

=∏

w∈MK

∏v|w

max|ai|nvw

=∏

w∈MK

max|ai|∑

v|w nv/nw

w

=

∏w∈MK

max|ai|nww

[K′:K]

.

14

2.2. HEIGHTS ON PROJECTIVE SPACE

This computation motivates the definition:

Definition 2.2.7 (Absolute height). If K is a number field and x ∈ Pn(Q), then wedefine the absolute height of x to be

H(x) = HK(x)1/[K:Q]

where K is any number field over which x is defined. By Theorem 2.2.6, this iswell-defined.

We also defineh(x) := logH(x)

which is an additive version of the height. Finally, for any a ∈ K we define h(a) :=h(1 : a).

Theorem 2.2.8 (Northcott). Let B,D ∈ N. Then the set

x = [a0 : . . . : an] ∈ Pn(Q) : H(x) ≤ B, [Q(x) : Q] ≤ D

is finite, where Q(x) = Q(ai/aj).

Proof. We can assume that aj = 1 for some j. Then for all i and all v ∈ MK , wehave

B ≥ max||ai||v ≥ max||ai||v, 1.

Taking the product over v, we get H(x) ≥ H(ai) for any individual i. Now, it sufficesto show that there are finitely many possibilites for the ai, i.e. for all 1 ≤ ` ≤ D andall B,D ∈ N,

#a ∈ Q | H(a) ≤ B, [Q(a) : Q] ≤ ` <∞.

Let K = Q(a) and let b1 = a, . . . , b` be the n conjugates of a, i.e. the roots of theminimal polynomial for a over Q,

f(t) = (t− b1) . . . (t− b`).

To show that there are finitely many possibilities for the a’s, it suffices to show thatthere are finitely many possibilities for f . For that, it suffices in turn to show thatthere are finitely many possibilities for the coefficients of f . (This is basically areduction to the case K = Q.)

First of all, note that there are only finitely many coefficients, since the degree ofa is bounded. The coefficient of tr is (−1)rsr(a), where sr(a) is the rth elementarysymmetric polynomial in b1, . . . , bd. We want to give a bound for coefficients.

|sr(a)|v = |∑

bi1 . . . bir |v ≤∑|bi1 . . . bir |v ≤ C(r, v) max|bi|rv.

The constant C(r, v) can be taken to be(`r

)if v is Archimedean, and 1 otherwise.

So thenmax|s0(a)|nv

v , . . . , |s`(a)|nvv ≤ C(r, v)nv max|bi|nv

v , 1`.

15

CHAPTER 2. HEIGHTS

Taking the product over all v ∈MK , we get an inequality of the form

HK([s0(a) : . . . : s`(a)]) ≤ C∏

HK(bi)`.

Since the bi are conjugates, H(bi) = H(b0) = H(a) for all i.This implies that the heights of the si(a) are bounded, and it is easy to check

that there are finitely many rational numbers with bounded height.

Example 2.2.9. If K is a number field, x = [a0 : . . . : an] ∈ Pn(K). Then H(x) =1 ⇐⇒ all the ai are roots of unity.

2.3 The Weil height machine on varieties

We now extend the notion of height to more general settings.Let X be a projective variety over Q and ϕ : X → PnQ be a morphism. Set

hϕ(x) = h(ϕ(x)).

Theorem 2.3.1 (Weil). Let X be smooth and projective over Q. Then we have ahomomorphism of groups

Pic(X)→ H(X) :=functions f : X(Q)→ Rbounded functions

denoted L 7→ hL, such that

1. if ϕ : X → Pn is a morphism and L = ϕ∗H, where H is a hyperplane, thenhL = hϕ, and

2. if π : Y → X is a morphism of smooth, projective varieties, then the pullbackdiagram commutes:

Pic(X)π∗ //

Pic(Y )

H(X)

π∗ // H(Y )

We work towards this result, first establishing some useful technical estimates.

Proposition 2.3.2. Let ϕ : Pn 99K Pm be a rational map of degree d, with base locusZ.

1. If x ∈ Pn \ Z(Q),h(ϕ(x)) ≤ dh(x) + cϕ.

16

2.3. THE WEIL HEIGHT MACHINE ON VARIETIES

2. If X ⊂ Pn is a closed subvariety disjoint from Z, and x ∈ X(Q), then

h(ϕ(x)) = dh(x) + c′ϕ,X .

Proof. Pick x ∈ Pn \Z(Q), say x = [a0 : . . . : an]. Take a number field K over whichconstants relevant to situation, in particular x and ϕ (but also things will will beintroduced later) are defined.

For any place v ∈MK , define

|x|v = max|ai|v|F |v = max|coefficients of F |v

ε(v, r) =

r v | ∞1 v -∞.

By the triangle inequality,

|ϕi(x)|v ≤ ε(v,(n+ d

n

))|ϕi|v|x|dv.

Then|ϕ(x)|v = max

i|ϕi(x)|v ≤ ε(v,

(n+ d

n

)) max|ϕi|v|x|dv.

Now raising everything to the nv[K:Q] power and taking the product over all v (so

everything is finally well-defined by the product rule), we get

H(ϕ(x)) ≤(n+ d

n

)H(ϕ)H(x)d

The reason that we get the constant(n+dn

)with exponent 1 comes from the degree

formula applied to the archimedean places, since the constant is 1 for the non-archimedean places. Taking the logarithm, we get

h(ϕ(x)) ≤ dh(x) + cϕ. (2.1)

To prove (2), let IX = 〈f1, . . . , fr〉 be the ideal of X in Pn. We assumed thatX ∩ Z = ∅, so there exists ` > 0 such that 〈t0, . . . , tn〉` ⊂ 〈ϕ0, . . . , ϕm, f1, . . . , fr〉.

In particular, for all j we can write

t`j =∑

ψi,jϕi +∑

gi,jfi.

We can assume that the ψi,j are all homogeneous of the degree ` − d. Evaluatingthis at the point x, we get

x`j =∑i,j

ψi,j(x)ϕi(x).

17

CHAPTER 2. HEIGHTS

Taking absolute values, we obtain

|xj |`v ≤ ε(v,m+ 1) maxi,j|ψi,j(x)|vmax

i|ϕi(x)|v

≤ ε(v,m+ 1)ε(v,

(`− d+ n

n

)) max

i,j|ψi,j |v|x|`−dv max|ϕi(x)|v.

Raising everything to the nv/[K : Q] power and taking the product over v ∈ MK ,we get

H(x)` ≤ C ′′H(x)`−dH(ϕ(x)).

Taking the logarithm we find that

dh(x) ≤ C ′ + h(ϕ(x)).

Combining this with the earlier inequality (2.1), we deduce the result.

Lemma 2.3.3. Let X be a variety over Q and ϕ,ψ be two morphisms to projectivespace such that

ϕ∗H ∼ ψ∗H ′.

Then hϕ = hψ in H(X).

Proof. Let ϕ : X → Pn and ψ : X → Pm be two maps such that ϕ∗H ∼ ψ∗H ′ ∼ D.Then we can write in coordinates ϕ = [f0 : . . . : fn] and ψ = [g0 : . . . : gm] withfi, gi ∈ H0(X,D).

Now, H0(X,D) is finitely dimensional over Q, so we can pick a basis h0, . . . , hNfor it. This defines a morphism X → PN (in other words, the morphism defined bythe complete linear system |D|) and ϕ,ψ factor through it:

Xα //

ϕ

PN

β

Pn

Xα //

ψ

PN

γ

Pm

Then γ and β are rational maps of degree one (they are projections), so Lemma2.3.2 implies that hα and hβ differ by a bounded function, and similarly for hα andhγ .

Lemma 2.3.4. Suppose that s : Pn×Pm → PN=(n+1)(m+1)−1 is the Segre embedding,i.e. s([a0 : . . . : an], [b0 : . . . : bm]) = [aibj ], then h(s(x, y)) = h(x) + (y).

Proof. Easy exercise.

Lemma 2.3.5. Let νd : Pn → PN be the Veronese embedding. Then h(νd(x)) =dh(x).

Proof. Easy exercise.

18

2.3. THE WEIL HEIGHT MACHINE ON VARIETIES

Proof of Weil’s Theorem 2.3.1. If L ∈ Pic(X) is basepoint-free (or equivalenty, OX(L)is generated by global sections), then L ∼ ϕ∗H for some morphism ϕ : X → Pn,then we put hL := hϕ. This is well-defined by Lemma 2.3.3.

Now, any L ∈ Pic(X) can be written as the difference of two basepoint-freedivisors. In fact, any divisor is the difference of two very ample divisors by standardresults on ample divisors. So we can write L = M ′−M ′′ whereM ′,M ′′ are basepoint-free. Extend by linearity: hL = hM ′ − hM ′′ .

Is this well-defined? If L = M ′ −M ′′ = R′ − R”, then M ′ + R′′ = M ′′ + R′.Then hM ′+R′′ corresponds to the Segre embedding of |M ′| and |R′′|, and similarly forhM ′′+R′ . Lemma 2.3.4 shows that hM ′′+R′ = hM ′ +hR′′ and also hM ′′+R′ = hM ′+R′′ .Using Lemma 2.3.3 again, we see that this is indeed well-defined.

The second statement is obvious from the observation that for basepoint-freedivisors L on X, we have π∗hL = hπ∗L.

For a line bundle L, denote by

B(L) =⋂

D∈|L|

D

its base locus.

Theorem 2.3.6 (Weil). With the same notation as above, we have:

1. For any representative hL (from its class in H(X)), there exists a constant Csuch that for all x ∈ X \Bs(L)(Q), we have hL(x) ≥ C.

2. Assume that L is ample. For any representative hL and B ∈ N, we have

#x ∈ X(Q) : hL(x) ≤ B <∞.

The first assertion is an analogue of the fact that the intersection number of adivisor and a curve away from its base locus is non-negative. (We shall see soon thatthe height can be thought of as analogous to an intersection product.)

Proof. If B(L) = X, i.e. H0(X,L) = 0, then there is nothing to prove. Otherwise,there is an effective divisor D ∼ L. Replacing L with D, we may assume that L iseffective. Then we can write L as the difference of two basepoint-free divisors M ′

and M ′′, so that L = M ′ −M ′′. We may also assume that hL = h′ϕ − h′′ϕ whereϕ′ : X → Pn′ and ϕ′′ : X → Pn′′ correspond to M ′ and M ′′.

Since M ′′ and L are effective, we have an inclusion

H0(X,M ′′) ⊂ H0(X,M ′).

19

CHAPTER 2. HEIGHTS

If ϕ′, ϕ′′ are the maps to projective space corresponding to M ′,M ′′, then we get acommutative diagram

Xϕ′ //

ϕ′′

Pn′

π

Pm′

and we can take π to be a projection.It is easy to check from the definition of heights that

hL(x) = hϕ′(x)− hϕ′′(x) = h(ϕ′(x))− h(ϕ′′(x)) ≥ 0 for x ∈ X \M ′(Q).

Varying hM ′ and hL in their equivalence classes changes this only by a boundedamount, which establishes (1).

For the second part, note that since L is ample some high multiple of L is veryample. Since we know the effect on the (logarithmic) height of composing with aVeronese embedding is just multiplication by a scalar, we may assume that L isvery ample, hence gives an embedding of X in Pn, and then the result follows fromNorthcott’s Theorem.

Definition 2.3.7. Let X be a smooth projective variety over Q, L a divisor on X, andϕ : X → X a morphism such that ϕ∗L ∼ αL for some α > 1. Define the canonicalheight of L by

h(x) = limm→∞

hL(ϕm(x))

αm.

Theorem 2.3.8. A canonical height function h : X(Q) → R is well-defined (inde-pendent of the choice of representative hL). Moreover,

1. h = hL in H(X)

2. h(ϕ(x)) = αh(x).

Proof. We show that h is well-defined. First note that we have

|hL(ϕ(x))− αhL(x)| = |hαL(x)− αhL(x)| < C

for some C independent of x. Therefore,∣∣∣∣hL(ϕ(x))

α− hL(x)

∣∣∣∣ < C

α.

We show that the sequence in the limit is a Cauchy sequence.∣∣∣∣hL(ϕn(x))

αn− hL(ϕm(x))

αm

∣∣∣∣ ≤ n∑i=m+1

∣∣∣∣hL(ϕi(x))

αi− hL(ϕi−1(x))

αi−1

∣∣∣∣<

n∑α=m+1

C

αi.

20

2.4. HEIGHT THEORY OVER FUNCTION FIELDS

Since α > 1, and any two choices of hL differ by an absolutely bounded constant, itis also clear that the limit is independent of the choice of hL. Also, property (2) isevident.

Exercise 2.3.9. With the same notation/asumptions, assume further that L is ample.Then h ≥ 0. Show that for all x ∈ X(Q),

h(x) = 0 ⇐⇒ ϕm(x) = ϕn(x) for some m 6= n.

2.4 Height theory over function fields

There is a strong analogy between number fields and function fields of curves, andwe can develop the theory of heights over function fields as well.

Let k be an algebraically closed field of characteristic 0, C a smooth projectivecurve over k, and K = k(C). Let X be a smooth projective variety over X (definedover K).

Let L be a divisor on X. There is a projective variety X/k and a morphismX → C whose generic fiber is X. This really takes the apparatus of scheme theoryto understand properly, but we can do it in an ad hoc manner in this case. X comesequipped with a projective structure morphism to SpecK, so we have an embedding

X ⊂ PnK //

PnC

SpecK // C

We can then let X be the closure of X in PnC . We will assume that X is smooth.This choice is always possible over characteristic zero, by the theorem on resolutionof singularities.

We can take the closure of L in X, getting a divisor∑miLi. A point x ∈ X(K)

corresponds toSpecK //

%%

X

SpecK

Since C is smooth and projective, this is equivalent to a C-point of X,

C //

X

C

which we denote by x. We then define

hL(x) := deg x∗L.

21

CHAPTER 2. HEIGHTS

Now, this definition depends on our choice of model, X. However, it turns out tobe well-defined up to a bounded function. If x ∈ X(K ′) where K ′/K is a finiteextension, then we define hL(x) = deg x∗L

[K′:K] , consider instead the diagram

C ′x //

X

C

Lemma 2.4.1. If X → C and X ′ → C are models as above, then hL−h′L is boundedon X(K).

Proof. We can choose a smooth model X ′′ dominating both X and X′. For x ∈

X(K), we get C-points of X,X ′, X ′′.

Cx′′ //

x

x′

**

X′′

λ

α~~ β

X

X′′

C

By definition,

hL(x) = deg x∗L = deg(x′′)∗α∗L

h′L(x) = deg(x′)∗L′= deg(x′′)∗β∗L

′

Let G = α∗L − β∗L′. Since X → C and X′ → C are isomorphisms over the

generic point of C (both have generic fiber X), G restricts to 0 on the generic point ofC. In other words, G is mapped to a finite set of points by λ : X

′′ → C. Therefore,there is an effective divisor D on C such that −λ∗D ≤ G ≤ λ∗D. Then the pullbackof G via x′′ has degree between −degD and degD.

Theorem 2.4.2. There is a homomorphism

Pic(X)→ HZ(X) =functions X(k)→ Zbounded functions

such that

1. (functoriality) If Y is smooth and projective over K and π : Y → X, then thediagram

Pic(X) //

Pic(Y )

HZ(X) // HZ(Y )

22

2.4. HEIGHT THEORY OVER FUNCTION FIELDS

commutes, and

2. If X = PnK and L is a hypersurface, then hL = h in HZ(X), where h is definedas follows: if x = [f0 : . . . : fn] ∈ PnK , then

h(x) =∑c∈C

maxi− ordc(fi).

3. If L ≥ 0 and x /∈ L, then hL(x) ≥ 0.

Proof. First we need to show that Pic(X) → HZ(X) is well-defined, i.e. if L ∼ 0then hL is bounded. If L ∼ 0, then we may write L = div(f) for some rationalfunction f on X. But f can also be thought of as a rational function f on X, andL − div(f) is union of finitely many vertical fibers over C since it is not supportedat the generic point. As in the proof of Lemma 2.4.1, that implies

deg x∗(L− div f) is a bounded function of x.

But deg x∗(L− div f) = hL(x)− deg div(x∗f), and deg div(x∗f) = 0.

1. If X → C and Y → C are models of X,Y we can choose Y so that we have acommutative diagram

Y

// X

Y // X

Then it becomes clear that π∗hL = hπ∗L.

2. It suffices to show the result when L is the divisor class represented by ahyperplane. Let Hi be the hyperplane cut out by ti. Now pick x ∈ X(K) andconsider the diagram

C //

X

C

We can assume that the image of C is not contained in H i, which is the locuscut by ti on X. Let Di = x∗H i. Then

Di −Dj = x∗(H i −Hj) = x∗(div(ti/tj)) = div(fi/fj) = div(fi)− div(fj).

Then for any c ∈ C, we have

miniordcDi − ordcDj = min

iordc(fi)− ordc(fj). (2.2)

23

CHAPTER 2. HEIGHTS

Now, mini ordcDi = 0 because there is no base locus, so summing the lefthand side of (2.2) over all c gives −degDj . On the right hand side, we get∑

c ordc(fj) = 0 because fj is a rational function, so we conclude that

−hL(x) = −degDj =∑c

mini

ordc(fi)

if L = Hj .

24

Chapter 3

Arakelov Geometry

In this section we will see a glimpse of Arakelov theory, which gives an intersection-theoretic interpretation of the height in number field settings analogous to that inthe function field setting.

3.1 Hermitian metrics on line bundles

Let M be a complex manifold and L a locally free sheaf on M .

Definition 3.1.1. A hermitian metric on L is a map

〈−,−〉 : L ×L → C0 := sheaf of C-valued continuous functions.

such that

• 〈af + bg, h〉 = a〈f, h〉+ b〈g, h〉 for all f, g, h ∈ L (U) and a, b ∈ OM (U).

• 〈f, h〉 = 〈h, f〉.

• ||f || =√〈f, f〉 > 0 if f is nowhere vanishing.

If rank L = 1, then the norm determines the inner product.

Example 3.1.2. If L = OM , then we can define

〈f, g〉 = fg.

Example 3.1.3. If M = Pn and L = O(1), then L is generated by global sectionst0, . . . , tn. We can then define the norm on the level of global sections. We set

||f ||(x) =|f(x)|

maxxi

A priori neither f(x) nor xi is well-defined, but the ratio is homogeneous of degree0 and hence well-defined.

25

CHAPTER 3. ARAKELOV GEOMETRY

Pullback. If ϕ : M ′ → M is a holomorphic map and we have a metric on L overM , then we get a metric on ϕ∗L by

〈ϕ∗f, ϕ∗g〉 = 〈f, g〉 ϕ.

In particular, if x → M is the inclusion of a point, then ϕ∗L is a vector spacewith a hermitian form.

3.2 Line bundles on number fields

Let K be a number field, X a smooth projective variety over K, and R the ring ofintegers of K. Let C = SpecR and X → C be a model of X (a projective morphismwhose generic fiber is X). Let L be a locally free sheaf on X.

For every rational point σ : K → C, we get a “fiber over σ,” Xσ := X ×C SpecC.

Xσ//

X //

X

SpecC // SpecK // C

Since X is smooth over K, Xσ is a smooth variety over SpecC. That means thatit can be considered as a smooth manifold over C. Let Lσ be the pullback of Lto Xσ. C is obviously not a proper curve, but we should view the infinite places asgiving a “compactification” of it.

A Hermitian metric on L can be thought of as a collection of metrics on Lσ

with a compatibility condition for conjugate embeddings:

〈fσ, gσ〉(x) = 〈fσ, gσ〉(x)

where x ∈ Xσ is the image of x ∈ Xσ under the isomorphism Xσ → Xσ induced bycomplex conjugation.

Suppose that rank L = 1 and X = SpecK, so X = C. We want to define thenotion of deg L . Normally this only makes sense for proper curves, but we recallthat we think of R as being “nearly proper” because of the infinite places.

Since C = SpecR, L corresponds to the R-module L = H0(C,L ). Since L islocally free of rank one, L is flat (in fact, projective) over C.

Definition 3.2.1. Assume that L has a Hermitian metric (or equivalently, norm).Pick some non-zero ` ∈ L , and define

deg L = log #(L/R`)−∑

σ:K→Clog ||`||σ.

This seems like a wacky definition at first, but it is actually analogous to aformulation of the usual definition in terms of taking a global section (in this case

26

3.2. LINE BUNDLES ON NUMBER FIELDS

`) and add up the orders of vanishing. The quantity #(L/R`) can be thoughtof as

∏p |`|− ordp(`)p , and the second term in the definition can be thought of as a

contribution from the infinite places.

Theorem 3.2.2. deg L does not depend on the choice of `.

Proof. Pick 0 6= `, `′ ∈ L. Then there exist a, a′ ∈ R such that a` = a′`′. So wereduce to the case where `′ = a`. By the product formula, we have∏

v∈MK

||a||v = 1 for all a ∈ K.

Taking logarithms, we get (by some computations done earlier)

0 = − log #(R/a) +∑v|∞

log ||a||v.

Therefore, it suffices to show that log #(R/a`) = log #(R/`)+log #(R/a). For that,we invoke the short exact sequence

0→ R`/Ra`→ L/Ra`→ L/R`→ 0

and the observation that R`/Ra` ∼= R/a.

Theorem 3.2.3. With the same setup as before, let L be an invertible sheaf on Xwith a given metric. Let L = L |X = OX(D). Then

hL = hD in H(X)

where

hL (x) =deg x∗L

[K ′ : K]if x ∈ X(K ′).

First, some general remarks are in order about constructing metrics from givenones. Suppose M is a complex manifold and L1 and L2 are invertible sheaves withgiven metrics. Then:

• there exists a metric on L1 ⊗L2 such that

||f ⊗ g||L1⊗L2 = ||f ||L1 ||g||L2 .

• there exists a metric on L −12 such that ||f ||L−1

2= 1||f ||L

• If M is compact, any two metrics on L1 are equivalent: there exist constantsc1, c2 > 0 such that

c1||f || ≤ ||f ||′ ≤ c2||f ||.

27

CHAPTER 3. ARAKELOV GEOMETRY

Proof. Using the above remarks, we can see that if N1 and N2 are invertible sheavesonM then deg(N1⊗N2) = deg N1+deg N1 and deg N1

−1= −deg N1. By tensoring

with a large very ample divisor, we can assume that L is very ample. So we reduceto the caseX = Pn, X = PnC , L = OX(1) with the metric give in our earlier example.

Pick x = [a0 : . . . : an] ∈ X(k). We can assume that ai ∈ R. Pulling back Lto C = SpecR, we get a line bundle L which is just the ideal (a0, . . . , an). Foreach σ : K → C, we get a morphism SpecC → PnC = Xσ whose image is thepoint [σ(a0) : . . . : σ(an)]. Assume without loss of generality that a0 6= 0. We thencompute

deg x∗L = log #L/(Ra0)−∑σ

log ||a0||σ

By the usual arguments, log #L/(Ra0) = log ||a0|| − log #(R/L). Furthermore,

||a0||σ = log |σ(a0)|+ log max|σ(ai)|.

Now, log #(R/Ra0) −∑

σ log |σ(a0)| is the degree of the trivial line bundle, whichis zero. Cancelling this from deg x∗L , we are left with

− log #(R/L) +∑σ

log max||ai||σ.

This is precisely hD(x), up to a bounded function.

28

Chapter 4

Abelian Varieties

4.1 Abelian varieties over algebraically closed fields

Let k be an algebraically closed field.

Definition 4.1.1. A quasiprojective group variety X/k is a quasiprojective varietywhose closed points have a group structure.

A better definition is that there are morphisms X×X m−→ X, X i−→ X, Spec k e−→X, satisfying the usual axioms.

Example 4.1.2. The additive group Ga has the underlying variety A1. The multi-plicative group Gm has the underlying variety A1 \ 0.Definition 4.1.3. An abelian variety is a projective group variety.

Example 4.1.4. Abelian varieties of dimension 1 are elliptic curves. The group lawon elliptic curves is well-known. On the other hand, the canonical bundle on anabelian variety must be trivial, which for curves forces g = 1.

Remark 4.1.5. It is well-known that if k = C, any abelian variety A is a complextorus, i.e. its analytification is A = Cn/Λ where Λ is a lattice.

Lemma 4.1.6 (Rigidity). Assume that X,Y, Z are varieties with X projective. Letϕ : X × Y → Z be a morphism. If there exists y0 ∈ Y such that ϕ|X×y0 is constant,then ϕ|X×y is constant for all y ∈ Y .

If there exists y0 ∈ Y and x0 ∈ X such that ϕ|x0×Y and ϕ|X×y0 are constant,then ϕ is constant.

Proof. Exercise.

Definition 4.1.7. Let A,B be abelian varieties. A morphism ϕ : A → B is a grouphomomorphism which is also a morphism of varieties.

There are two distinguished kinds of morphisms from an abelian variety to itself:

29

CHAPTER 4. ABELIAN VARIETIES

• A translation is a morphism of the form ψ(x) = x+ a for all x ∈ A.

• There is a multiplication by n morphism πn : A→ A sending x 7→ [n]x.

Theorem 4.1.8. Suppose that ϕ : A→ B is a morphism of abelian varieties. Thenϕ is a composition of a translation and a homomorphism.

Proof. Composing with a translation, we may assume that ϕ(0A) = 0B. Considerthe morphism ψ : A × A → B defined by ψ(x, x′) = ϕ(x + x′) − ϕ(x) − ϕ(x′). Bythe group axioms, ψ|A×0A = 0B and ψ|0A×A = 0B, so ψ is constant, hence ϕ isa homomorphism.

Theorem 4.1.9 (Theorem of the cube). Suppose that X,Y, Z are varieties withX,Y projective. Assume that L is an invertible sheaf on X × Y × Z. Suppose thatthere exist x0 ∈ X, y0 ∈ Y, z0 ∈ Z such that

L |x0×Y×Z ,LX×y0×Z ,L |X×Y×z0 are all trivial.

Then L is trivial.

Proof. The proof is actually rather difficult, requiring the theorem on cohomologyand base change. See Mumford’s book, page 55.

Theorem 4.1.10 (Theorem of the cube for abelian varieties). Let A be an abelianvariety, L a divisor on A. For I ⊂ 1, 2, 3 let ϕI : A × A × A → A be given byϕI(x1, x2, x3) =

∑i∈I xi. Then

ϕ∗123L− ϕ∗12L− ϕ∗13L− ϕ23∗ + ϕ∗1L+ ϕ∗2 + ϕ∗3L is trivial.

Proof. Let D be the divisor associated to the huge line bundle that we want to showis trivial. Consider D|A×A×0. Well,

ϕ123|A×A×0 = ϕ12|A×A×0ϕ13|A×A×0 = ϕ1|A×A×0ϕ23|A×A×0 = ϕ2|A×A×0

Since restriction commutes with pullbacks, we see that D|A×A×0 ∼ 0. Now we canapply the theorem of the cube to get the result.

Corollary 4.1.11 (Mumford’s formula). Let A be an abelian variety, L a divisoron A, and πn : A→ A be the multiplication-by-n map. Then

π∗nL ∼n2 + n

2L+

n2 − n2

π∗−1L.

30

4.2. ABELIAN VARIETIES OVER NUMBER FIELDS

Proof. Consider the morphism Aπn,π1,π−1−−−−−−→ A × A × A. Pull back the big trivial

divisor from the theorem of the cube, to get

π∗nL− π∗n+1L− π∗n−1L+ π∗nL+ π∗1L+ π∗−1L = 0.

Then apply induction to get the result.

Remark 4.1.12. If π∗−1L = L (we say that L is symmetric), then Corollary 4.1.11implies that

π∗nL ∼ n2L.

On the other hand, if π∗−1L = −L (we say that L is antisymmetric), then Corollary4.1.11 implies that

π∗1L ∼ nL.

4.2 Abelian varieties over number fields

We say that A is an abelian variety over a number field if 0 ∈ A(K) and all thestructure morphisms are defined over K.

Consider the morphism π2 : A → A (multiplication by 2). If L is a symmetricdivisor/line bundle, then π∗2L ∼ 4L. Recall that we defined a canonical height fordivisors and morphisms with this property:

h(x) = limn→∞

hL(π2n(x))

4n

Recall that h = hL in H(A).

Remark 4.2.1. Any group variety in characteristic 0 is smooth, by generic smooth-ness. We are implicitly using that here.

Theorem 4.2.2. The canonical height h satisfies the following properties:

1. h(mx) = m2h(x),

2. h(x+ y) + h(x− y) = 2h(x) + 2h(y)

3. Define the pairing 〈, 〉 : A(K)×A(K)→ R by

〈x, y〉 =h(x+ y)− h(x)− h(y)

2.

Then this pairing is bilinear.

4. If L is ample, then for any c the set x ∈ A(K) | h(x) ≤ c is finite.

Remark 4.2.3. This says that the height is actually a quadratic function: note thath(x) = 〈x, x〉.

31


Proof. 1. If πr : A→ A is multiplication by r, then

hL(πr(x)) = hπ∗rL(x) +O(1)

= hr2L(x) +O(1)

= r2hL(x) +O(1)

By definiton, the canonical height is

h(mx) = limn→∞

m2hL(π2n(x)) +O(1)

4n= m2h(x).

2. This comes from the Theorem of the cube. Consider the morphism A× A →A×A×A corresponding to (x, y) 7→ (x, y,−y). The theorem of the cube saysthat

ϕ∗123L− ϕ∗12L− ϕ∗23L− ϕ∗13L+ ϕ∗1L+ ϕ∗2L+ ϕ∗3L ∼ 0.

Note that ϕ123 θ(x, y) = x, ϕ12 θ(x, y) = x+y, ϕ23 θ = 0, etc. Using these,we can compute the pullback of the above line bundle to A×A, and we get

hL(x)− hL(x+ y)− hL(x− y) + hL(x) + hL(y) + hL(−y) = O(1).

Since L is symmetric, hL(−y) = hL(y), and we get that

2hL(x) + 2hL(y) = hL(x+ y) + hL(x− y) +O(1).

Then taking the limit for the canonical height, we get what we want.

3. By a similar calculation with the theorem of the cube,

h(x+ y + z)− h(x+ y)− h(x+ z)− h(y + z) + h(x) + h(y) + h(z) = 0.

This implies that〈x+ y, z〉 = 〈x, z〉+ 〈y, z〉.

4. This is immediate from Theorem 2.3.6.

Theorem 4.2.4 (Mordell-Weil). If A is an abelian variety over a number field K,then A(K) is a finitely generated abelian group.

First we prove:

Theorem 4.2.5 (Weak Mordell-Weil). If A is an abelian variety over a number fieldK, and m ≥ 2, then A(K)/mA(K) is finite.

32

4.3. TORSION POINTS

This is not hard if one admits some algebraic machinery (apply Galois cohomol-ogy to

0→ A[m]→ Am−→ A→ 0

and use general facts about Galois cohomology).Obviously, Mordell-Weil implies the weak Mordell-Weil. We will show how to

deduce the other direction.

Proof. Take an ample symmetric divisor L on A, and let h be the associated canonicalheight (as above). There exist y1, . . . , yr ∈ A(K) such that their classes hit allelements of A(K)/mA(K). Let c = maxih(yi). Let S = x ∈ A(K) | h(x) ≤ c.Now S is finite because h = hL in H(A), and then we can apply Northcott’s theorem.

Pick x ∈ A(K) with h(x) > c. Then there exists yi and x′ such that x−yi = mx′.We have

h(x− yi) ≤ h(x+ yi) + h(x− yi) = 2h(x) + 2h(yi)

(h is non-negative because we chose L to be ample) and

h(x− yi) = m2h(x′).

Therefore, m2h(x′) ≤ 2h(x) + 2c, which implies h(x′) < h(x). Therefore, we havewritten x in terms of some yi and a point with strictly smaller height. The theoremthen follows by “infinite descent.”

4.3 Torsion points

Let A be an abelian variety over a number field K. A point x ∈ A(K) is called anm-torsion point if mx = 0. (This is the usual notion of torsion in a group.) We letA(K)tors be the set (group) of all torsion elements of A, or more generally A(F )torsfor the torsion points defined over F .

The set of m-torsion points of A is the kernel of πm. In particular, it is a closedsubscheme of A.

Lemma 4.3.1. The m-torsion subgroup πm is a finite subgroup of A.

Proof. If not, then it contains a projective curve C. Then (π∗mL) · C = 0, since thisis just the degree of the pullback via πm of L to C, and πm restricts to a constantfunction on C. On the other hand, π∗mL ∼ m2L, which is ample if L is, and anample divisor intersects any curve positively, which is a contradiction.

Corollary 4.3.2. The map πm : A→ A is surjective.

Proof. This follows from the fact that πm has zero-dimensional fibers, so its imageis full-dimensional (hence dense by irreducibility of A), and is proper.

33


By Mordell-Weil, A(K)tors is finite. We can prove this directly: if x ∈ A(K) ism-torsion, then mx = 0 so h(mx) = h(0) = m2h(x), so h(x) < h(0) if m 6= 1. Thenwe can apply Northcott’s theorem to see that there can only be finitely many suchpoints (height bounded by a constant).

4.4 Weak Mordell-Weil

We have reduced the Mordell-Weil theorem to the weak Mordell-Weil theorem.Since the A(K)m is finite, there is a finite extension K ′/K such that A(K)m ⊂

A(K ′). The Mordell-Weil theorem for K ′ implies that for K, so it suffices to showthe Mordell-Weil theorem for K ′. Therefore, in proving the Mordell-Weil theoremwe may assume that A(K)m ⊂ A(K).

Lemma 4.4.1. Let A be an abelian variety over a number field K such that A(K)m ⊂A(K) for m ≥ 2. Then there exists a finite Galois extension K ⊂ L such that ify ∈ A(K) and my ∈ A(K) then y ∈ A(L).

Let’s see how this implies the Weak Mordell-Weil theorem. Let G = Gal(L/K).By definition, this is the group of automorphisms of L fixing K, so it acts on A(L).We then define a map α : A(K)→ Hom(G,A(K)m) as follows. For x ∈ A(K), picky such that my = x. Now sending g 7→ g(y)− y. There are a few details to check tosee that this makes sense:

• g(y)− y is indeed an m-torsion point, since multiplying by m shows mg(y)−my = g(my)−my = g(x)− x = 0 since x ∈ A(K).

• This is well-defined since we have assumed that K contains all the m-torsionpoints. Indeed, any two choices of y differ by an m-torsion point, which g fixes.

• We should check that this is indeed a homomorphism. If g, g′ ∈ G, theng′g 7→ g′g(y) − y = g′(g(y) − y) + g′(y) − y. Since g(y) − y is an m-torsionpoint, it is fixed by g′, so this is just equal to g(y)− y + g′(y)− y.

Furthermore, it is obvious that α is a homomorphism. Note that kerα = mA(K),since α(x) = 0 ⇐⇒ g(y) = y for all g ⇐⇒ y ∈ A(K).

Therefore, A(K)/mA(K) → Hom(G,A(K)m). Since G and A(K)m are bothfinite, the latter is finite so A(K)/mA(K) must be finite as well.

Proof of Lemma 4.4.1. We will just give a sketch.

1. Pick y ∈ A(K) such that my ∈ A(K). Let K(y) be the field generated by y,i.e. the residue field of the point y. Then we claim that K(y)/K is Galois ofexponent m. Indeed, any two choices of y differ by an m-torsion point, whichis defined over A(K), so that K(y) = K(g(y)) for any g ∈ Gal(K/K).

34

4.4. WEAK MORDELL-WEIL

Now we define a map Gal(K/K) → A(K) sending g 7→ g(y) − y. Thekernel is Gal(K/K(y)), so we have a injection Gal(K(y)/K) → A(K)m ∼=(Z/mZ)2 dimA. We haven’t proved the last isomorphism, but it is easy to seein characteristic zero from the “Lefschetz principle” (any algebraic geometricstatement over C implies the same for any algebraically closed field of charac-teristic 0). Over C, it is obvious from the complex torus description of abelianvarieties as Cg/Λ.

2. We can extendA to an abelian scheme B over some open subset U = SpecRS ⊂SpecR, where R is the ring of integers of K.

A

// B

SpecK // U

We require B → SpecRS to be smooth and projective. Suppose that y ∈ A(K)satisfies my ∈ A(K). This corresponds to a diagram

SpecK(y)

// A //

πm

B

πm

SpecK // A //

B

SpecR // U

We can “spread” the morphism SpecK(y)→ A to a morphism Uy → B, whereUy is the inverse image of U under SpecRy → SpecR. In characteristic zero,πm is automatically étale since it is smooth of relative dimension 0. Localizingfurther if necessary, we may assume that the multiplication-by-m map is alsoétale on B also Uy → U is étale.

3. There are finitely many possibilities for Uy → U - this is a fact from algebraicnumber theory (Hermite’s theorem). Note that it is related to the finitenessof the S-class group, which class field theory tells us is the Galois group of themaximal extension unramified outside S. Hence there are only finitely manypossible extensions K ⊂ K(y), implying that the compositum of all the K(y)is only a finite Galois extension.

35


36

Chapter 5

The Brauer-Manin Obstruction

5.1 Chevalley’s Theorem

Theorem 5.1.1 (Chevalley-Warning). Let P1, . . . , Pr ∈ Fq[t0, . . . , tn] be polyno-mials. Assume that Pi(0, . . . , 0) = 0 for all i, and

∑di := Pi ≤ n. If Y =

V (P1, . . . , Pr) ⊂ An+1(Fq), then #Y ≡ 0 (mod p). In particular, #Y ≥ 2.

Proof. Note that for a ∈ Fq, we have aq−1 = 0 if a = 0 and 1 otherwise. In otherwords, the indicator function of F ∗q ⊂ Fq is a 7→ aq−1.

Define φ =∏i(1−P

q−1i ). Then for (a0, . . . , an) ∈ An+1(Fq), we have φ(a0, . . . , an) =

1 if and only if (a0, . . . , an) ∈ Y and is 0 otherwise. Therefore,

#Y ≡∑

(a0,...,an)∈An+1(Fq)

φ(a0, . . . , an) (mod p).

So it suffices to show that the right hand side is 0 (mod p). Let’s focus on a monomialtm00 . . . tmn

n appearing on the right hand side above, which necessarily has total degree≤ n(q − 1). The contribution from such a monomial is

∑(a0,...,an)

am00 . . . amn

n =∏i

∑a∈Fq

ami

.

Since∑mi ≤ n(q− 1), there is some j such that q− 1 - mj or mj = 0. If q− 1 - mj ,

then∑

a∈Fqamj ≡ 0 (mod p).

If the Pi are homogeneous, then Theorem 5.1.1 implies that they cut out analgebraic set X in Pn with a rational point.

Example 5.1.2. If X is smooth, IX = 〈P1, . . . , Pr〉 and dimX = n − r (i.e. X is acomplete intersection), then X is a Fano variety (i.e. −KX is ample). It is a generalfact that if X is a Fano variety, then X(Fq) 6= ∅. In fact, this is true even if X isrationally connected (a much more general condition).

37

CHAPTER 5. THE BRAUER-MANIN OBSTRUCTION

5.2 Tsen’s Theorem

We now consider a similar setup over the function field of a curve.

Theorem 5.2.1 (Tsen). Let k be an algebraically closed field, K the function fieldof a curve C/k, and P1, . . . , Pr ∈ K[t0, . . . , tn] be homogeneous of degree degPi = disuch that

∑di ≤ n. Then if Y = V (Pi) ⊂ An+1(K), #Y ≥ 2.

In particular, if the Pi are homogeneous then this is saying that the correspondingprojective variety has a rational point.

Proof. We’ll show the result with a weaker estimate.By assumption, the transcendence degree of K/k is 1. Therefore, there exists

s ∈ K such that K/k(s) is a finite extension. Let R be the integral closure of k[s]inside K. Then R is a finite k[s]-module by commutative algebra. Therefore, thereexist elements α1, . . . , αa ∈ R such that R is generated by the αi as a module overK[s]. We can write αiαj =

∑gi,j,`α`. Let d′ = maxdeg gi,j,`.

The Pi are polynomials over K = Frac(R), and by clearing denominators we mayassume that their coefficients lie in R. Now for a multi-index I = (m0, . . . ,mn) wedenote (as usual) tI = tm0

0 . . . tmnn . Then we can express

Pi =∑I

βi,ItI bi,I ∈ R.

By the earlier comments, we can write

βi,I =∑`

fi,I,`α` fi,j,` ∈ k[s].

Let d′′ = maxdeg fi,I,`.If hi,j ∈ k[s] is of degree ≤ d′′′, then

Pi(∑`

h0,`α`, . . . ,∑

hn,`α`) =∑

ei,`α` (5.1)

where the ei,` ∈ k[s] are determined by the hij and Pi. From all the bounds we have,we can get bounds on the degree of the ei,`, e.g.

deg ei,` ≤ (d′′′ + d′)di + d′′.

Now we can treat the coefficients of all hij as points in Pa(n+1)(d′′′+1)−1(k). So thecondition that (5.1) is 0 involves a bunch of equations over k (one for each coefficient),specifically

r∑i=1

((d′′′ + d′)di + d′′ + 1) ≤ (d′′′ + d′)n+ rd′′ + r.

38

5.3. BRAUER-SEVERI VARIETIES

Considering these as equations on Pa(n+1)(d′′′+1)−1, we see that if d′′′ is sufficientlylarge then the projective space has much larger dimension than the number of equa-tions, so there must be some solutions. That corresponds to choices of the hij solving(5.1).

Remark 5.2.2. If the Pi cut out a smooth complete intersection, then it is Fano. Itis a fact due to Graborr-Harris-Star that if X is a Fano variety over such a field K,then X(K) 6= ∅. Again, the statement holds even for X rationally connected, whichis stronger.

Example 5.2.3. The equation t20 + t21 + t22 has no solutions in P2(Q). This is a Fanovariety with no rational points overK, so the result certainly doesn’t hold for numberfields.

Conjecture 5.2.4. If X is a Fano variety over some number field K, then thereexists a finite extension K ⊂ K ′ is dense in X (in the Zariski topology).

5.3 Brauer-Severi varieties

Let k be any field. A very basic comment is that if ϕ : X → Y is a morphism definedover k and x ∈ X(k) then ϕ(x) ∈ Y (k). The cleanest way of looking at this is tothink of it in terms of the functor of points: a point X(k) is the same as a morphismSpec k → X, and composing with ϕ gives a morphism Spec k → Y .

Lemma 5.3.1. Suppose that ϕ : X 99K Y is a rational map defined over k, wherethere exists a smooth x ∈ X(k) and Y is projective. Then Y (k) 6= ∅.

Proof. Let π : X ′ → X be the blowup of X at x. Since X is smooth at x. Let E bethe exceptional divisor, which is isomorphic to PdimX−1

k . Now X ′ is smooth near thegeneric point of E, and Y is projective. The obvious rational map ϕ′ : X ′ → Y isdefined near the generic point of E. Thus we get an induced rational map E 99K Y .Applying induction, we reduce to the case dimX = 1, and ϕ will be regular at anysmooth point (since the image is projective).

Corollary 5.3.2. If ϕ : X 99K Y is a birational map (defined over k) and X,Y aresmooth and projective, then X(k) 6= ∅ ⇐⇒ Y (k) 6= ∅.

Corollary 5.3.3. If X 99K Pn is birational (over k) and X is smooth and projective,then X(k) 6= ∅.

Definition 5.3.4. A variety X/K is called a Brauer-Severi variety if Xk ' Pnk.

Example 5.3.5. Suppose k = Q and X = V (t20 + t21 + t22) ⊂ P2Q. This is evidently not

isomorphic to P1Q, but it is after base-changing to the algebraic closure.

Theorem 5.3.6 (Châtelet). Suppose that X is a Brauer-Severi variety over k. ThenX(k) 6= ∅ ⇐⇒ X ' Pnk .

39


Proof. One direction is obvious, so suppose that X is Brauer-Severi with a rationalpoint x ∈ X(k). Since X is geometrically smooth, X is certainly smooth over k.In particular, x is a smooth point. That implies that if π : Y → X is the blowupof X at x, then the exceptional divisor E is isomorphic to Pn−1

k . Then we haveY ⊂ X × Pn−1

k .

Y ⊂ Pn−1k

π

ψ

$$X Pn−1

k

The projection defines a map Y → Pn−1k . If L is the hyperplane divisor on Pn−1

k ,then π∗ψ

∗L is a divisor on X, which after base change to k becomes the class ofthe hyperplane divisor (by our understanding of the situation over an algebraicallyclosed field). Therefore, it is base-point free already, and defines a map X → Pnk .We know that this is an isomorphism after base-change to k, so it is already anisomorphism.

Definition 5.3.7. Let X be a variety defined over a number field K. We say that Xsatisfies the Hasse principle if X(K) 6= ∅ ⇐⇒ X(Kv) 6= ∅ for all v ∈MK .

Example 5.3.8. If K = Q, this is saying that X has a rational point over Q if andonly if it has one in R and mod p for all p.

It is a (very nontrivial) theorem that the Hasse principle holds for Brauer-Severivarieties. It also holds for smooth projective hypersurfaces of degree 2, but fails forsome elliptic curves and some projective hypersurfaces of degree 3.

Example 5.3.9. We present a variety which violates the Hasse principle.Let K = Q and X = V (t20− 5t21− t3t4, t20− 5t22− (t3 + t4)(t3 + 2t4)) ⊂ P4. This is

a smooth surface in P4. (In fact, it is a Fano variety.) Let x1 = (1 : 0 :√−1 : 1 : 1),

x2 = (5 : 5 :√

5 : 10 : −10), x3 = (0 : 0 :√−5 : 5 : 0), and x4 = (0 : 5 : 2

√−15 :

−25 : 5). At least one of these is defined over each Qp. More precisely, if v 6= (2),then one of x1, x2, x3 ∈ X(Qv) and x4 ∈ X(Q2). It also clearly has real points.

We claim that X(Q) = ∅. Suppose for the sake of contradiction that (a : b :c : d : e) ∈ X(Q). We can assume that d, e ∈ Z and (d, e) = 1. If p is a prime≡ ±2 (mod 5), then we claim that ordp(de) is even. If not, then there are integersα, β, γ, δ, ε such that

δε = α2 − 5β2, ordp(δε) is odd, and p - αβ.

Then α2 ≡ 5β2(p) =⇒ 5 is a square in Fp. By quadratic reciprocity, this forcep ≡ ±1 (mod p).

Continuing in this way, we get d ≡ ±1 (mod 5) and e ≡ ±1 (mod 5). Similarly,we can show that d+ e ≡ ±1 (mod 5) and d+ 2e ≡ ±1 (mod 5).

40

5.4. THE BRAUER GROUP

5.4 The Brauer group

The Brauer group is a natural abelian group associated to field k.

Definition 5.4.1. A central simple algebra (CSA) over k is a finite dimensional, as-sociative k-algebra with center k with no non-zero proper two-sided ideals.

Example 5.4.2. The trivial CSA over k is A = k.

Example 5.4.3. Mn(k), the algebra of n × n matrices, is a CSA over k. The onlynon-obvious conditions are the “central” and “simple.” The “central” is familiar, andthe “simple” is not too difficult to check either.

Example 5.4.4. Let k be a field and a, b ∈ K \ 0. Let (a, b)k be the associative k-algebra generated by s, t such that s2 = a, t2 = b, ts = −st. A = k⊕k〈s〉⊕k〈t〉⊕k〈st〉.For obvious reasons, this is called a “quaternion algebra.” If char k 6= 2, this is acentral simple algebra.

A special class of central simple algebras consists of divison algebras.

Definition 5.4.5. A central division algebra (CDA) over k is a CSA over k such thatall non-zero elements are invertible.

Example 5.4.6. If k = R, then the usual quaternions H = (−1,−1)R is a CDA/k.

Example 5.4.7. If k is any field of characteristic not 2, then (a2, b)k is not a CDA/k.Indeed, (s+ a)(s− a) = 0 so s+ a is not invertible.

Theorem 5.4.8 (Wedderburn). If A is a cental simple algebra / k, then there existsa unique CDA D / k such that A 'Mn(D).

Remark 5.4.9. If A and B are CSAs / k then A⊗kB is a CSA / k. If A is a CSA / k,then one can form a CSA algebra Aop (this comes from the natural way of viewinga ring as a category), in which a · b is b · a in A. It is a fact that A⊗k Aop 'Mn(k)for some n. The idea of this identification is define a map A ⊗ Aop → Homk(A,A)sending a⊗b 7→ (α 7→ aαb). The kernel is a two-sided ideal, hence trivial. Surjectivityfollows from dimension counting.

Definition 5.4.10. Let k be a field. If A,B are CSAs over k, we say that A and Bare equivalent if there exists a CDA D and integers m,n such that A 'Mn(D) andB ∼= Mm(D).

Now the Brauer group of k, denoted Br(K), is defined to be the set of equiva-lence classes of central simple algebras, with product [A] · [B] = [A ⊗k B], inverse[A]−1 = [Aop], and identity k.

Example 5.4.11. If k is algebraically closed, then Br(k) = 0. Indeed, we claim thatany CDA over k is k. If α ∈ D \k, then k ⊂ k[α] ⊂ D. Since D is finite-dimensional,so is k[α]. Moreover, this is an integral domain, hence a finite field extension of k.Since k is algebraically closed that means k ∼= k[α].

41


Example 5.4.12. There is a result of Frobenius which says that the there are onlytwo CDAs / R: R,H. (There are three division algebras, the third being C - butthis is not central over R). Therefore, Br(R) = Z/2Z.Example 5.4.13. Br(Fq) = 0. The reason is that if D is a CDA/k, then D is finite.Another result of Wedderburn implies that a finite CDA is a field. That forcesD = Fq, and then the centrality forces q = p.

5.5 The Brauer-Manin obstruction

Theorem 5.5.1. There is an isomorphism invp : Br(Qp) = Q/Z. Moreover, thereexists an exact sequence

0→ Br(Q)→⊕v∈MK

Br(Qv)∑

invv−−−−→ Q/Z→ 0.

There exists a similar exact sequence for any number field.More generally, you can actually define Brauer groups for schemes. Suppose X

is a scheme.

Definition 5.5.2. An Azumaya algebra over X is a locally free OX -algebra A of finiterank if for all x ∈ X, A(x) is a central simple algebra over κ(x).

This is like a family of CSAs over the points of X. Now if A and B is an Azumayaalgebra over X, then A ⊗OX

B is an Azumaya algebra over X. We can similarlydefine the “opposite” of an Azumaya algebra. We can define an equivalence relationby A ∼ B if there exist locally free sheaves E and F such that

A⊗Hom(E , E) ' B ⊗Hom(F ,F).

In addition, one checks that A ⊗ Aop is equivalent to OX . The Brauer group of Xis the group given by the above equivalence classes.

Suppose that X is a variety over k. We have a natural morhpism

SpecK → X

where K is the function field of X. So we have a natural homomorphism Br(X)→Br(K). If X is smooth over k, it turns out that this is injective.

Remark 5.5.3. If k is a field, then Br(k) is equivalently defined as

• H2ét(Spec k,Gm), or

• Br(K) = H2(G,Ks \ 0) where G = Gal(ks/k). The first statement is basi-cally a repackaging of this one.

• Br(K) is the “group of Brauer-Severi varieties over k”.

42

5.5. THE BRAUER-MANIN OBSTRUCTION

Suppose that X is a projective variety over k. Let AK be the ring of adeles fork. The adelic points of X are X(Ak), and any point can be written as a sequence of(xv ∈ X(kv)) such that all but finitely many of the coordinates are integers (in theirrespective fields).

For each v ∈ Mk, and for each [A] ∈ Br(k), we obtain a map X(kv) → Br(kv)by pullback. Indeed, an element of Br(X) is an Azumaya algebra on X, and we canpull it back via xv ∈ X(kv) to get some kv-algebra, which is a CSA by construction.Therefore, for each A) ∈ Br(X) this fits into a commutative diagram

X(k) //

X(Ak)

0 // Br(k) //

⊕v∈Mk

Br(Kv) // Q/Z // 0

The fact that this map is actually well defined is nontrivial (why does an Azumayaalgebra restrict to the trivial CSA in Br(kv) for all but finitely many v?).

So we see that a necessary condition for (xv) ∈ X(Ak) to come from a globalsolution (i.e. from X(k)) is that (xv) 7→ 0 in Q/Z. For any B ⊂ Br(X), we define

X(AK)B = (xv) ∈ X(AK) | (xv) 7→ 0 ∈ Q/Z for all [A] ∈ B.

Definition 5.5.4. Note that we always have X(K) ⊂ X(AK)Br(X) ⊂ X(AK). IfX(AK) 6= ∅ but X(AK)Br(X) = ∅, then we say that there is a Brauer-Manin obstruc-tion to the Hasse principle.

Example 5.5.5. Recall that X = V (t20− 5t21− t3t4, t20− 5t22− (t3 + t4)(t3 + 2t4)) ⊂ P4Q

is a counterexample to the Hasse principle (it has local solutions, but no globalsolutions). We did this by elementary, explicit arguments. What if we try to find aBrauer-Manin obstruction?

Let K be the function field of X and recall that we have an injection Br(X) →Br(K). Let [A1], . . . , [Ar] ∈ Br(K). Then for each i there is an open subset Ui ⊂ Xsuch that the functions and elements involved in a multiplication table for Ai aredefined and non-zero on Ui ⊂ X. The Ai determine an Azumaya algebra over Ui, ifUi is small enough. If X =

⋃Ui and the AUi agree on overlaps (in particular, they

are the same class in Br(K)), then they glue together to define an Azumaya algebraA/X.

In our particular example, consider the following quaternion algebras over K:

(5,t3

t3 + t4)K , (5,

t4t3 + t4

)K , (5,t3

t3 + 2t4)K , (5,

54

t3 + 2t4)K .

A relatively easy calculation reveals that these are all isomorphic as CSAs over K.This list can be extended to a list that determines an Azumaya algebra on X.

We will try to describe the mapsX(Qv)→ Br(Qv)→ Q/Z. This is a case-by-caseanalysis. Suppose v is archimedean (the standard real place of Q), so Qv = R. Pick

43


xv ∈ X(R). If f = t3t3+t4

is regular at xv and non-zero, so A(xv) = (5, f(xv))Qv=R istrivial in Br(R) because 5 is a square. In particular, its image is zero in Q/Z. Toshow that the map is trivial for all xv, either extend the list of algebras or use somecontinuity argument.

Now suppose v is a nonarchimedean place. Assume p 6= 2 and 5 is a squarein Qp. The same arguments as in the case Qv = R show that X(Qp) → Q/Z iszero. Suppose p 6= 2, 5 but 5 is not a square in Qp. Then 5 is not a square in Fp,by Hensel’s Lemma. If xv = (a0 : . . . : a4) ∈ X(Qp), ai ∈ Zp, then by looking atthe equations defining X we can assume that a3 6= 0 (mod p) or a4 6= 0 (mod p).Then A(xv) = (5, b)Qp where b ∈ Zp \ 0. By class field theory, that means thatinvv([A(xv)]) = 1

2 .The case p = 2 is similar, and we find that the invariant map is trivial again.Finally, we consider the case p = 5. Considering the equations of X mod 5, we

see that

(a3 − a4)(a3 − 2a4) ≡ 0 (mod 5) and ± a0 ≡ a3 ≡ a4 (mod 5).

Now if xv ∈ X(Q5), invv([A(xv)]) = (5, 3)Q2 = 12 again by class field theory (since

a3a3+a4

≡ 3 (mod 5)). This gives a Brauer-Manin obstruction.

44

Chapter 6

Birational Geomery

6.1 Potential density

For this section, we will assume that all varieties are geometrically irreducible.

Definition 6.1.1. If X is a variety over a field k, e say that the k-rational pointsare dense if X(k) is dense in X (in the Zariski topology). We say that the rationalpoints are potentially dense if X(k′) is dense for some finite extension k′/k.

Remark 6.1.2. Density is hard to analyze; potential density is easier.

Example 6.1.3. Suppose X is a Brauer-Severi variety over k. If X(k) 6= ∅, thenX ' Pnk so X(k) is dense if k is a number field, and X(k) is not dense if k is afinite field. (In fact, we never get density for finite fields for positive-dimensionalvarieties.) So any Brauer-Severi variety over any number field is potentially dense.

Example 6.1.4. Suppose that X is a smooth projective variety of dimension 1. Ifg(X) = 0, then X is a Brauer-Severi variety.

If g(X) = 1, then X(k) may be finite or infinite (i.e. not dense or dense). Butpotentially density always holds.

If g(X) ≥ 2, and we are over a number field, then potential density never holdssince X(k) is always finite (Faltings).

Example 6.1.5. Suppose X = C × Y where C is a smooth projective curve of genusg ≥ 2 and Y is a smooth projective variety over K, a number field. Then potentialdensity fails for X because it does for C.

Example 6.1.6. Suppose X is an abelian variety of dimension > 1 over k a numberfield. Let Γ be the torsion subgroup of X(k). In trying to show that potential densityholds for X, we can pass to a finite extension. Then we may assume that there existsa smooth projective curve C ⊂ X defined over k with g(C) ≥ 2. By the Manin-Mumford conjecture (proved by Raynaud), Γ ∩ C(k) is finite. Pick x ∈ C \ Γ(k).Then G = Zx ⊂ X(k) is infinite.

45

CHAPTER 6. BIRATIONAL GEOMERY

First assume that A is simple (i.e. has no nontivial abelian subvarieties). LetY = G in the Zariski topology. By a result of Faltings, Y is a finitte union of abeliansubvarieties, so Y = X.

If X is not simple, then by the theory of abelian varieties X is a product X =X ′ ×X ′′ of abelian varieties. Then we apply induction. If X ′ or X ′′ has dimensionone, G shows that X ′(k) or X ′′(k) is dense (since it’s infinite). If X has dimensionone, then we get potential density for X ×X, hence potential density for X.

6.2 Kodaira dimension

Suppose X is a smooth projective variety over a field k. Let KX be the canonicaldivisor of X.

Definition 6.2.1. We define the Kodaira dimension of X as follows: if h0(X,mKX) =0 for all m > 0, then define κ(X) = −∞. If h0(X,mKx) > 0 for some m > 0, letκ(X) be the largest ` such that

0 < lim supm→∞

h0(X,mKX)

m`<∞.

In other words,h0(X,mKX) grows like m`. It is a fact that this ` exists. Thinkof h0(X,mKX) as being like a polynomial, and the Kodaira dimension as beingthe degree. (Aside: χ(mKX) is a polynomial, and if we have some sort of positivitycondition onKX that kills the higher cohomology, then it is a polynomial form 0).It turns out that κ(X) ∈ −∞, 0, 1, . . . ,dimX.Definition 6.2.2. If κ(X) = dimX, we say that X is of “general type.”

Example 6.2.3. If X is a curve, then

κ(X) = −∞ ⇐⇒ g(X) = 0

κ(X) = 0 ⇐⇒ g(X) = 1

κ(X) = 1 ⇐⇒ g(x) ≥ 2

Conjecture 6.2.4. If X is a smooth, projective variety over k a number field. Thenif κ(X) = dimX, potential density fails. If κ(x) = 0, then potential density holds.

We know that if X is Fano (−KX is ample, so κ(X) = −∞), then potentialdensity holds.

6.3 Surface fibrations

We assume that X is a smooth projective surface over k an algebraically closed field.

Definition 6.3.1. We say that a curve E ⊂ X is a (−1)-curve if E ' P1 and E2 =E · E = −1.

46

6.3. SURFACE FIBRATIONS

If E is a (−1)-curve, then Castelnuovo’s theorem says that E is the exceptionaldivisor of a blowup X π1−→ X1. Repeating, we obtain a sequence

X → X1 → . . .→ Xn = Y

where Y has no −1 curves.

The classification of surfaces.

• If κ(X) = κ(Y ) = −∞ then Y ' P2 or a conic fibration Y → C where thegeneral fiber P1 and C is a smooth projective curve.

• If κ(X) = κ(Y ) = 0 then Y is an abelian variety or a K3 surface (the canonicalbundle is trivial) or a finite quotient of one of these.

• If κ(X) = κ(Y ) = 1 then there exists an elliptic fibration X → C (a map to asmooth projective curve whose general fiber is an elliptic curve).

• If κ(X) = κ(Y ) = 2 then X and Y are of general type.

Note that κ(X) = κ(Y ) since the Kodaira dimension is a birational invariant, so thiscovers all cases.

If k is not algebraically closed, then the same holds after replacing by a finiteextension (another reason why it’s nicer to consider potential density).

Example 6.3.2. Let X be a smooth projective surface over a number field k, withκ(X) = −∞. We can assume that X = P2 or there exists a conic bundle X → C.The caseX = P2 is clear, so we consider the other case. After passing to an extension,we get a morphism to a smooth projective curve whose fibers are smooth projectivecurves of genus 0 (in particular, the generic fiber is such.) By a variant of Tsen’sTheorem applied to the generic fiber (thinking of the generic fiber as a curve overthe function field), there is a section

C //

X

C

This lets us lift rational points from C to X. There are several cases to consider:

• if g(C) = 0, then we can assume that C ' P1 (after passing to a finite exten-sion). For almost all s ∈ C(k), the fiber Xs over s is a smooth projective curveof genus 0. Now, using ϕ we get Xs(k) 6= ∅, so |Xs(k)| is infinite. Puttingthese all together, we get a dense set of rational points for X.

• if g(C) = 1, the same argument applies after passing to a finite extension toobtain infinitely many rational points for C.

47


• if g(C) ≥ 2, then potential density fails for X because it fails for C.

In view of the classification of surfaces, it is natural to ask the following: supposeϕ : X → C is an elliptic fibration over a number field K, where C is a smoothprojective curve and X is a smooth projective surface. Does potential density holdfor X?

Example 6.3.3. Let’s thinking about the triviale example: X = E × C where E isan elliptic curve over k and C is a smooth projective curve over k. Let ϕ : X → Cdenote the projection. If g(C) = 0 or 1, then potential density holds because it doesfor both factors. If g(C) ≥ 2, then potential density fails because it fails for C.

Before going on, let’s remind ourselves of a fact (cf. Hindry-Silverman, “Dio-phantine geometry” exercise C7):

Theorem 6.3.4 (Chevalley-Weil). Suppose π : X ′ → X is an étale morphim betweensmooth (or just normal) projective varieties over k a number field. Then there existsa finite extension k′/k such that π−1(X(k)) ⊂ X ′(k′).

This is analogous to the “monodromy theorem” in topology, which says that ifX ′ → X is a covering space, then any path in X lifts to X ′.

Example 6.3.5. Suppose k is a number field, E is an elliptic curve over k, and C isa smooth projective curve. Assume that E(k) contains a nontrivial 2-torsion pointα. Define σ : E → E by σ(e) = e + α. This is an involution, since α is 2-torison.Assume that we also have an involution on C, say τ : C → C, and suppose thequotient C/〈τ〉 is P1. (In other words, C is hyperelliptic: any two degree two mapto P1 gives such an involution.)

Now we can define an involution on µ : E ×C → E ×C by (e, c) 7→ (σ(e), τ(c)).This has no fixed points (since translation on E already doesn’t). Therefore, themap E × C → E × C/〈µ〉 is étale.

E × C //

C

E × C/〈µ〉 = X

ϕ // C/〈τ〉 = P1

• If g(C) = 0, then potential density for E × C implies it for X. (Potentialdensity is always preserved under dominant morphisms.) [κ(X) = −∞]

• If g(C) = 1, then potential density for E × C implies it for X, by the sameargument. [κ(X) = 0]

• If g(C) ≥ 2, then potential density fails for C, hence fails for E × C, hencefails for X by Chevalley-Weil. [κ(X) = 1]

48

6.3. SURFACE FIBRATIONS

Remark 6.3.6. If X → Y is a morphism and Y is of general type, then Lang’sconjecture says that potential density fails for Y . Therefore, it fails for X as well.You might think that having a map to a variety of general type is the only obstructionto potential density. However, in the above example with g(C) ≥ 2, you can showthat there is no surjective map X → Y where Y is of general type (X is an ellipticfibration over P1, so you can’t have such a map to a curve of genus g ≥ 2.)

General methods for elliptic fibrations. Suppose there exists a section

C

// X

ϕ

C

and a multisectionM

// X

ϕ

C

where M is a smooth projective curve mapping surjectively to C. The existence ofa section says that the general fiber has a rational point, and is therefore an ellipticcurve (over k). Suppose Suppose C(k) is dense and M(k) is dense. (We can thinkof the section as the 0-section, picking out the identity along each fiber and makingX an abelian scheme over C.)

Each x ∈M(k) gives a rational point on some fiber of ϕ. If most of these pointsare non-torsion, then we get potential density for X (since we thus get infinitelymany points on these fibers). However, this non-torsion condition is kind of weird -we should explore what kind of nice geometric conditions imply this.

Lemma 6.3.7. If there exists c ∈ C such that ϕ is smooth over c, and M → C isnot smooth over c, then the non-torsion condition holds on ϕ−1(c).

Proof sketch. Here’s a sketch of the proof. For any m ∈ N, there exists a divisorDm ⊂ X such that torsion points of order m are all inside Dm.

First assume that there existsm such thatM ⊂ Dm. Then ifXC → CC,MC, (Dm)Cdenote the base-changes to C, there exists a sequence c1, c2, . . .→ c such thatM → Cis étale over ci. Since M → C is not étale, the degree of the map is at least 2. Overci, we get at least two points pi, p′i in MC both converging to the same point. Butwe also have that pi − p′i is m-torsion for each i, which is a contradiction.

On the other hand, if M 6⊂ Dm for any m then most of the points in M(k)are non-torsion by Merel’s theorem (for any fixed number field k, there exists m0

depending (only) on k such that any elliptic curve over k has no torsion points oforder m ≥ m0).

49


Suppose that you have a multisection M , but no section C. We can base changeby M to a family over M with a section.

M ×C M //

''

M ×C X //

X

M // C

Then you can apply techniques that we have already discussed.

50

Chapter 7

Statistics of Rational Points

7.1 Counting on projective space

Let k be a number field, X ⊂ Pnk a projective variety over k. Let Hk be the usualheight function on Pnk . Define a counting function on X(k):

N(X(k), B) = #x ∈ X(k) | Hk(x) ≤ B.

The general question is what it looks like, e.g. does it grow like a polynomial?It is difficult to say anything in general, but for special classes of varieties we can

say something.

Theorem 7.1.1. Assume X ⊂ Pnk is a smooth projective curve such that X(k) 6= ∅.Then there are constants a, b depending on k and the embedding such that

N(X(k), B) =

aBb + o(Bb) g = 0

a logbB + o(logbB) g = 1

a+ o(1) g ≥ 2

Proof. If g(X) = 0, then X ' P1k, and the result will follow from the next theorem.

If g(X) = 1, then X is an elliptic curve, and the result will follow from nextlecture.

If g(X) ≥ 2, then the result follows from Falting’s theorem.

Theorem 7.1.2 (Schanuel). If X = Pnk , then there exists a constant a depending onk, n such that

N(X(k), B) = aBn+1 + o(Bn+1).

Moreover,

a =hR

ωζK(n+ 1)

(2r1(2π)r2√

D

)n+1

(n+ 1)r1+r2−1.

51

CHAPTER 7. STATISTICS OF RATIONAL POINTS

Remark 7.1.3. Here we use the standard notation of algebraic number theory: h isthe class number of Ok, R is the regulator, ω is the number of roots of unity, ζk isthe (Dedekind) zeta function of k, r1 is the number of real embeddings, r2 is thenumber of conjugate pairs of complex embeddings, and D is the absolute value ofdiscriminant of K/Q.

Proof. We treat only K = Q, where we have h = 1, R = 1, ω2, r1 = 1, r2 = 0, D = 1.If x = (a0, . . . , an) ∈ An+1(Z), put |x| = max|ai| and gcd(x) = gcd(a0, . . . , an).

Define functions

M(B) = #x ∈ An+1(Z) | x 6= 0, |x| ≤ B.

Also setM(B, d) = #x ∈ An+1(Z) | gcd(x) = d, |x| ≤ B.

There are various relations among these, e.g. M(B, d) = M(B/d, 1) because if x iscounted in M(B, d), then dividing through all coordinates by d gives an element ofM(B/d, 1).

SinceM(B) =

∑d∈N

M(B, d) =∑d∈N

M(B/d, 1)

the Möbius inversion formula says

M(B, 1) =∑d∈N

µ(d)M(B/d).

NowM(B) = #a ∈ Z | |a| ≤ Bn+1 − 1 = (2bBc+ 1)n+1 − 1.

If we plug this in to the Möbius inversion formula, we get

M(B, 1) =∑d∈N

µ(d)[(2bB/dc+ 1)n+1 − 1

]=

∑1≤d≤B

µ(d)

[(2B

d

)n+1

+O

((B

d

)n)]

= (2B)n+1 1

ζ(n+ 1)+O(Bn+1)

because ∑d∈N

µ(d)

dn+1=

1

ζ(n+ 1).

Finally, note that N(Pn(Q), B) = 12M(B, 1), since any point in Pn(Q) has a unique

representative with integer coordinates whose gcd is 1, except up to multiplyingeverything by −1.

52

7.2. COUNTING ON ABELIAN VARIETIES

Example 7.1.4. Let X = Pnk and consider the l-tuple Segre embedding X ⊂ Pmk (herem =

(n+``

)− 1).

The previous theorem says that the counting function for X(k) with respect tothe embedding X = Pnk is N(X(k), B) = aBn+1 + o(Bn+1).

Under the Segre embedding, we get instead N(X(k), B) = aBn+1` + o(B

n+1` ) as

B →∞. The degree is smaller, which is to be expected because the coordinates are“bigger.” How can we see this?

Recall that HPn

k (x) =∏v∈Mk

max||ai||v and

HPm

k (x) =∏v∈Mk

maxj||φj(x)||v

where the φj are a basis for `-degree polynomials, so this is exactly HPn

k (x)`. There-fore,

N(X(k), B) = #X(k) | HPn

k (x) ≤ B = N(X(k), B1/`).

Remark 7.1.5. We could take a height functionHL with respect to some ample divisorand we can define a counting function N(X(k), B, L) = #x ∈ X(k) | HL(x) ≤ B.The most interesting case is when X is a Fano variety, and we have the naturalchoice L = −KX . For instance, this applies to hypersurfaces X ⊂ Pn of degree ≤ n.

7.2 Counting on abelian varieties

Let X ⊂ Pnk be an abelian variety over k a number field. Suppose that Γ is asubgroup of X(k), and let r = rank Γ.

Theorem 7.2.1. Let L be the hyperplane divisor on Pnk , so L|X is a symmetricample divisor. Then

N(Γ, B) = N(X(k), B)|Γ = a logr/2(B) + o(logr/2B).

Proof. Let h be the canonical height function associated to L|X . Recall that

hPn = h+ bounded function.

It is enough to show that N ′(Γ, B) = aBr/2 + o(Br/2), where

N ′(Γ, B) = #x ∈ Γ | h(x) ≤ B.

This canonical height is nice to work with: recall in particular we proved that it is aquadratic form. Tensoring with R, we get a quadratic form h : Γ ⊗Z R → R. (Thisis non-trivial, but left as an exercise.) Now let Λ ⊂ Rr be the image of Γ (it will bea lattice). It then suffices to show that

#λ ∈ Λ | q(λ) ≤ B = aB1/2 + o(B1/2).

This is more or less obvious: the number of lattice points inside a region of radius√B is about Br/2. Details are left as an exercise.

53

CHAPTER 7. STATISTICS OF RATIONAL POINTS

7.3 Jacobians of curves

How do you get abelian varieties (of dimension ≥ 1)? The easiest way is to take theJacobian of a curve.

Suppose X is a smooth projective curve over field k which is geometrically irre-ducible. Then Pic(Xk) is an abelian group, and there is an exact sequence

0→ Pic0(Xk)→ Pic(Xk)deg−−→ Z→ 0.

So Pic0(X) is also an abelian group. It turns out that Pic0(Xk) is a variety, andthis group structure makes it into an abelian variety. This is denoted Jk. We havea morphism

Xk → Jk

sending y to the class of y − x, for any fixed point x ∈ Xk.

• If g(X) = 0, then Xk = P1kand Pic0(Xk) = 0.

• If g(X) = 1, then Pic0(X) ' X.

• If g(X) ≥ 2, then Xk → Jk is an embedding.

It is always the case that dim Jk = g(X).If X(k) 6= ∅, then there is an abelian variety J over k and a morphism X → J

whose base change to the algebraic closure recovers Xk → Jk. From now aone, weassume that g(X) ≥ 2 and k is a number field, so we get an embedding X(k) ⊂ J(k).By the theorem we just proved,

N(X(k), B) ≤ a(logB)r/2 where r = rankJ(k).

This seems silly, since Faltings proved that X has finitely many points. But that re-sult is very difficult. Mumford proved a stronger estimate N(X(k), B) ≤ c log logB,and this was further developed to give an alternate proof of Faltings’ Theorem.

Now assume that k = C. We have an exact sequence

0→ Z→ C exp−−→ C∗ → 0.

There is an associated short exact sequence of sheaves,

0→ Z→ OX → O∗X → 0.

The long exact sequence of cohomology then gives

0→ H1(ZX)→ H1(OX)→ H1(O∗X)→ H2(ZX)→ . . .

and H1(O∗X) ∼= Pic(X), the boundary homomorphism being the degree (or firstChern class). The kernel is Pic0(X), which by exactness is Pic0(X) ∼= H1(OX)

H1(X,Z)=

Cg(X)

H1(X,Z). This is a complex torus.

54

Chapter 8

Zeta functions

8.1 The Hasse-Weil zeta function

Definition 8.1.1. Let X be a variety over Fq. Let Nm = #X(Fmq ) for m ≥ 1. TheHasse-Weil zeta function of X is defined to be

ZX(t) := exp

( ∞∑m=1

Nm

mtm

)∈ Q[[t]].

Example 8.1.2. Suppose X = AnFq. Then Nm = #X(Fqm) = qmn. So

ZX(t) = exp

( ∞∑m=1

qmn

mtm

)

= exp

( ∞∑m=1

(qnt)m

m

)= exp (− log(1− qnt))

=1

1− qnt.

Example 8.1.3. SayX = AnFqand Y is a variety over Fq. We try to compute ZX×Y (t).

Well,

#(X × Y )(Fqm) = #X(Fqm)#Y (Fqm).

55

CHAPTER 8. ZETA FUNCTIONS

So

ZX×Y (t) = exp

∑m≥1

#X(Fqm)#Y (Fqm)

mtm

= exp

∑m≥1

#Y (Fqm))

m(qnt)m

= ZY (qnt).

Lemma 8.1.4. Suppose X is a variety over Fq and Y ⊂ X is a closed subset,U = X \ Y . Then

ZX(t) = ZY (t)ZU (t).

Proof. #X(Fqm) = #Y (Fqm) + #U(Fqm). When we take the exponential, the addi-tion is turned into multiplication:

ZX(t) = exp

(∑ #Y (Fqm) + #U(Fqm)

mtm)

= ZY (t)ZU (t).

Example 8.1.5. Let X = PnFq. Let Y be a hyperplane, e.g. V (t0). Then Y ' Pn−1

Fq

and U ' AnFqm, so

ZX(t) = ZY (t)ZU (t) = ZY (t)1

1− qnt,

so we see (e.g. by induction) that

ZX(t) =n∏k=1

1

(1− t)(1− qt) . . . (1− qnt).

Theorem 8.1.6. Let X be a variety over Fq. Then we have the formula

ZX(t) =∏

x closed ∈X

1

1− tdeg(x).

Proof. Suppose ar is the number of closed points x ∈ X having degree r. Considerthe base change

XFqm//

X

SpecFqm // SpecFq.

Now, there is a natural bijection X(Fqm) ↔ XFqm(Fqm) by the definition of base

change. If x′ ∈ XFqm(Fqm) and x is the image of x′ in X, then deg x | m, since

Fq ⊂ κ(x) ⊂ κ(x′) = Fqm .

56

8.1. THE HASSE-WEIL ZETA FUNCTION

If deg x = r, then Aut(Fqr/Fq) acts on x′ if x′ ∈ XFmq

(Fqr). That is, the base-change of x = SpecFqr has degree r in XFqm

. We conclude that

Nm = #X(Fqm) = #XFqm(Fqm) =

∑r|m

rar.

Now, consider

∑m≥1

Nm

mtm =

∑m≥1

∑r|m rar

mtm

=∑r≥1

ar∑`≥1

1

`tr`

=∑r≥1

ar log1

1− tr

=∑r≥1

log

(1

1− tr

)ar

Corollary 8.1.7. If X is a variety over Fq, then ZX(t) ∈ Z[[t]].

Theorem 8.1.8 (Weil conjecture). Let X be a smooth projective variety, which isgeometrically irreducible. Then the zeta function ZX(t) satisfies:

1. ZX(t) ∈ Q(t) is a rational function.

2. ZX(t) satisfies the functional equation

ZX

(1

qnt

)= ±qne/2teZX(t) for some e.

3. (“Riemann Hypothesis”) We have

ZX(t) =P1(t)P3(t) . . . P2n−1(t)

P0(t) . . . P2n(t)

where Pi(t) ∈ Z[t], P0(t) = 1− t, P2n(t) = 1− qnt, and for 1 ≤ i ≤ 2n− 1,

Pi(t) =∏

(1− αi,jt)

where αi,j are algebraic integers with |αi,j | = qi/2.

57


8.2 The Frobenius Morphism

Let F = Fq. Then we can define a Frobenius homomrophism F : F→ F by sendinga 7→ aq. The key point is that

Fqm = a ∈ F | Fm(a) = a.

Suppose X is variety over Fq ⊂ PnFq. Then Fm induces a morphism PnF → PnF. Since

Fq is fixed by Fm, this is an endomorphism of X. That gives an interpretation

Nm = #X(Fqm) = #x ∈ X | Fm(x) = x.

The motivation for the Weil conjectures is an analogy to the Lefschetz Fixed PointTheorem, applied to the Frobenius.

8.3 The Weil conjectures for curves

We assume throughout that X is a smooth projective curve over Fq. Recall that adivisor on X is an expresison of the form

D =∑

diDi

where the di ∈ Z and the Di are closed points on X. The degree of D is∑di degDi,

where degDi := [κ(Di) : Fq]. (By the Nullstellensatz, degDi = 1 if we are workingover an algebraically closed field.) So this gives a homomorphism Pic(X)→ Z.

Fix L ∈ Pic(X). Define the linear system of L to be

|L| = D ≥ 0 | D ∼ L.

This is precisely the projectivization of H0(X,L) := H0(X,O(L)). So

#|L| = qh0(L) − 1

q − 1.

By Riemann-Roch,h0(L) = degL+ 1− g + h1(L).

By Serre duality, h1(L) = h0(K−L), so in particular h1(L) vanishes if degL > 2g−2.Define Picr(X) = L ∈ Pic(X) | degL = r. It is easy to see that there is a

bijection Pic0(X) ↔ Pic1(X) if Picr(X) 6= ∅ (so the Picr(X) are Pic0(X)-torsors).It is also a nontrivial fact that Picr(X) is nonempty for all r.

Let J be the Jacobian of X. An important property of J is that there is a naturalbijection

J(Fq)↔ Pic0(X)

(Really, this should be by definition, i.e. the Jacobian should be defined as thescheme that represents the Picard functor). This implies that # Picr(X) = #J(Fq).

58

8.3. THE WEIL CONJECTURES FOR CURVES

Theorem 8.3.1. The rationality statement in the Weil conjectures holds if dimX =1.

Proof. Recall that we proved an “Euler product” factorization

ZX(t) =∏

x closed ∈X

1

1− tdeg x.

Expanding this out as a formal power series, this is

ZX(t) =∏

x closed ∈X

∑m≥0

tm deg x

=∑D≥0

tdegD

=∑D≥0

∑degD=d

td

=∑D≥0

∑0≤d=degD≤2g−2

td +∑D≥0

∑d=degD≥2g−1

td

Now we group divisors into linear systems. Let α = #J(Fq) = # Pic0(X). The firstsum is a polynomial because there are only finitely many non-empty linear systems ofbounded degree, and the second sum is nice because Riemann-Roch tells us exactlywhat h0(D) is, namely d+ 1− g, which in turn tells us #|D|. So we can rewrite

ZX(t) =

2g−2∑r=0

∑L∈Picr(X)

qh0(L) − 1

q − 1

tr +∑

r≥2g−1

∑L∈Picr(X)

qr+1−g − 1

q − 1

tr

= polynomial in t+ geometric series

=P (t)

(1− t)(1− qt).

Theorem 8.3.2. The functional equation holds for dimX = 1. More precisely,

ZX

(1

qt

)= q1−gt2−2gZX(t).

Proof. If g = 0, this follows from the formula we computed earlier for projectivespace. Henceforth, we assume that g ≥ 1.

We can write ZX(t) = S1 + S2, where

S1 =

2g−2∑r=0

∑L∈Picr(X)

qh0(L)

q − 1

tr

59


(this is very similar to the polynomial term we obtained above, but without the −1in the numerator) and

S2 =

2g−2∑r=0

∑L∈Picr(X)

−1

q − 1

tr +∑

r≥2g−1

αqr+1−g − 1

q − 1tr

= − α

(q − 1)(1− t)+

αqgt2g−1

(q − 1)(1− qt).

We will actually check the functional equation for S1 and S2 separately. For S2, wejust have to plug stuff in to our formula:

S2

(1

qt

)=

αqt

(q − 1)(1− qt)− αq1−gt2−2g

(q − 1)(1− t)= q1−gt2−2gS2(t).

For S1, we use the fact that L 7→ KX − L gives a bijection on the set

Λ = L ∈ Pic(X) | 0 ≤ degL ≤ 2g − 2.

By Serre duality and Riemann-Roch, h0(L) + g − 1− degL = h0(K − L). So

S1

(1

qt

)=

2g−2∑r=0

∑L∈Pic0(X)

qh0(L)

q − 1

1

qrtr

=

2g−2∑r=0

∑L∈Pic0(X)

qh0(K−L)

q − 1

1

(qr)2g−2−r

=

2g−2∑r=0

∑L∈Pic0(X)

qh0(L)−(r+1−g)

q − 1

1

(qr)2g−2−r

= q1−gt2−2g

∑L∈Pic0(X)

qh0(L)

q − 1

tr

= q1−gt2−2gS1(t).

Remark 8.3.3. Recall that ZX(t) = P (t)(1−t)(1−qt) . By definition of ZX(t), ZX(0) = 1.

We can write

P (t) =

2g∏j=1

(1− αjt)

60

8.4. ZETA FUNCTIONS OF ARITHMETIC SCHEMES

where the αj might a priori be 0, since it follows from the proof that

degP ≤ 2g.

By the functional equation,

ZX

(1

qt

)= q1−gt2−2gZX(t)

so2g∏j=1

(qt− αj) = qg2g∏j=1

(1− α− jt)

and since ZX(0) = 1 =⇒ P (0) = 1, none of the αj are zero. Therefore, we in facthave degP = 2g.

8.4 Zeta functions of arithmetic schemes

Definition 8.4.1. An arithmetic scheme is a scheme of finite type over Z.

Lemma 8.4.2. Suppose X is an arithmetic scheme. Then x ∈ X is a closed pointif and only if κ(x) is finite.

Proof. Pick x ∈ X, and suppose that κ(x) is finite. Let y be the image of x in SpecZso the residue field of y is finite. Then y = (p) for some prime p. So x lies in thefiber of X over (p)

Xp//

X

SpecFp // SpecZ

Now, Xp is a scheme of finite type over Fp. Then x is a closed point in Xp, sinceotherwise κ(x) would be the function field of a positive-dimensional variety. SinceXp → X is a closed embedding, x is closed in X.

Conversely, suppose that x is a closed point. With the same notation as before,if y is closed then the result is obvious. If y is not closed, then it is the generic point(0) of SpecZ. Shrinking, we can assume that X = SpecB and there exist someSpecA ⊂ SpecZ such that B is finitely generated as an A-algebra. Algebraically,we have a diagram

κ(x) Boo

Q

OO

A

OO

oo

Now κ(x) is finitely generated over B (since it is closed), hence over A. But it isalso a field extension of Q, which makes this impossible.

61


Definition 8.4.3. Let X be an arithmetic scheme. For any closed x ∈ X, let N(x) =#κ(x). Define the zeta function

ζX(s) =∏

x closed ∈X

1

1−N(x)−s.

Remark 8.4.4. One could think of this as a product over all points with the conven-tion that ∞−s = 0.

Since X is of finite type over Z, for any n ∈ N the set

#x ∈ X | N(x) ≤ n is finite.

This implies that we can expand ζX(s) as a Dirichlet series∑

n≥1anns that converges

for Rep s 0.Example 8.4.5. Let X = SpecZ. Then ζX(s) is the Riemann zeta function

ζ(s) =∏p

(1− p−s)−1 =∑n

n−s.

More generally, if K is a number field and X = SpecOK , then ζX(s) is the Dedekindζ function.Remark 8.4.6. If X is an arithmetic scheme, then ζX(s) = ζXred(x) (the closed pointsare the same). If the set of closed points can be written as a union closed points insubschemes Xi, where the Xi are closed or open, then

ζX(s) =∏i

ζXi(s).

Example 8.4.7. Let X be an arithmetic scheme such that X → SpecZ maps X toone point (p) ∈ SpecZ. Then X is a scheme over Fp. For simplicity, assume that Xis a variety over Fp. Then

ζX(s) =∏

x closed ∈X

1

1−N(x)−s.

Now, N(x) = pdeg x. So this concides with ZX(p−s), where ZX is the Hasse-Weilzeta function.Example 8.4.8. Suppose X = AnZ. Then

ζX(s) =∏p

ζXp(s) =∏p

ZXp(p−s).

and we already computed the Hasse-Weil zeta function forXp = AFp . The conclusionis that

ZX(s) =∏p

1

1− ps−n.

62

8.4. ZETA FUNCTIONS OF ARITHMETIC SCHEMES

This is just a translated version of the Riemann ζ function.By using our earlier comment, we can compute the zeta function for projective

space by using the decomposition PnZ = AnZ ∪ PnZ.

Theorem 8.4.9. Let X be an arithmetic scheme. Varying s ∈ C, ζX(s) is absolutelyconvergent (as a product) if Rep s > dimx.

Proof. If X = X1 ∪ X2, then it suffices to show that the result holds for two outof three of (X1, X2, X). Then one can reduce to the case X = SpecB, where Bis a finitely A algebra and SpecA ⊂ SpecZ is an open subset. Using Noethernormalization, B is a finite extension of A[x1, . . . , xn], which reduces to the case ofaffine space.

It is a thorem that ζX(s) can be meromorphically continued to Rep s > n− 12 .

Fact. Assume X is integral and let K be the function field of X, i.e. K = κ(η),where η is the generic point of X. If K has characteristic 0 (i.e. X → SpecZ isdominant) then ζX(s) has a simple pole at s = n. If K has characteristic p 6= 0,then ζX(s) has simple poles at the complex numbers n+ 2πi m

log q where m ∈ Z andq is the largest power of p such that Fq ⊂ K (i.e. the algebraic closure of Fp in K,or the “field of constants” of K).

Conjecture. ζX(s) has a meromorphic continuation to the entire complex plane.(Possibly with some regularity assumptions on X.)

This is known in certain cases, e.g. dimX = 1, An,Pn, or positive characteristic(this last case following from the Weil conjectures).

Conjecture. Assuming X is regular, the zeta function ζX(s) in the region 0 <Rep s ≤ n has zeros only on the line Rep s = 1

2 ,32 , . . . and poles only on the vertical

lines 1, 2, . . . , n.

Again, in positive characteristic this is implied by the Weil conjectures.

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algebraic and arithmetic geometry

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