Lectures on rational points on curves

118 Pages·2006·0.784 MB·English

by  Bjorn Poonen

Lectures on rational points on curves
March 5, 2006 version
Bjorn Poonen
Department of Mathematics, University of California, Berkeley, CA
94720-3840, USA
E-mail address: [email protected]
URL: http://math.berkeley.edu/~poonen
This research was supported by NSF grant DMS-0301280 and the Miller Institute for Basic
Research in Science.
(cid:13)c Bjorn Poonen 2006.
Contents
Chapter 0. Introduction 1
0.1. Notation 1
Chapter 1. Varieties over perfect fields 3
1.1. Affine varieties 3
1.2. Projective varieties 5
1.3. Base extension 7
1.4. Irreducibility 7
1.5. Morphisms and rational maps 8
1.6. Dimension 10
1.7. Smooth varieties 11
1.8. Valuations and ramification 13
1.9. Divisor groups and Picard groups 14
1.10. Twists 16
1.11. Group varieties 17
1.12. Torsors 19
Exercises 21
Chapter 2. Curves 23
2.1. Smooth projective models 23
2.2. Divisor groups and Picard groups of curves 24
2.3. Differentials 25
2.4. The Riemann-Roch theorem 26
2.5. The Hurwitz formula 28
2.6. The analogy between number fields and function fields 31
2.7. Genus-0 curves 35
2.8. Hyperelliptic curves 36
2.9. Genus formulas 39
2.10. The moduli space of curves 41
2.11. Describing all curves of low genus 43
iii
Exercises 45
Chapter 3. The Weil conjectures 49
3.1. Some examples 49
3.2. The Weil conjectures 50
3.3. The case of curves 51
3.4. Zeta functions 52
3.5. The Weil conjectures in terms of zeta functions 53
3.6. Characteristic polynomials 54
3.7. Computing the zeta function of a curve 54
Exercises 55
Chapter 4. Abelian varieties 57
4.1. Abelian varieties over arbitrary fields 57
4.2. Abelian varieties over finite fields 62
4.3. Abelian varieties over C 68
4.4. Abelian varieties over finite extensions of Q 70
p
4.5. Cohomology of the Kummer sequence for an abelian variety 72
4.6. Abelian varieties over number fields 73
Exercises 76
Chapter 5. Jacobian varieties 79
5.1. The Picard functor and the definition of the Jacobian 79
5.2. Basic properties of the Jacobian 80
5.3. The Jacobian as Albanese variety 81
5.4. Jacobians over finite fields 82
5.5. Jacobians over C 83
Exercises 84
Chapter 6. 2-descent on hyperelliptic Jacobians 87
6.1. 2-torsion of hyperelliptic Jacobians 87
6.2. Galois cohomology of J[2] 90
6.3. The x−T map 91
6.4. The 2-Selmer group 92
Exercises 92
´
Chapter 7. Etale covers and general descent 93
7.1. Definition of ´etale 93
iv
7.2. Constructions of ´etale covers 93
7.3. Galois ´etale covers 97
7.4. Descent using Galois ´etale covers: an example 98
7.5. Descent using Galois ´etale covers: general theory 101
7.6. The Chevalley-Weil theorem 102
Exercises 102
Chapter 8. The method of Chabauty and Coleman 105
Chapter 9. The Mordell-Weil sieve 107
Acknowledgements 109
Bibliography 111
v

CHAPTER 0
Introduction
The main goal of these notes is to explain how to answer questions like “What are the
rational number solutions to x4 +y4 = 17?”
(cid:127)
Warning 0.0.1. These notes are not yet finished, and what is written is only a first
draft. Read them at your own risk! If you see ♣♣♣ Bjorn: [], that means that it hasn’t been
written or fixed up yet.
0.1. Notation
Let chark denote the characteristic of a field k. By a global field k, we mean a number
field (finite extension of Q) or a global function field (finite extension of F (t) for some finite
q
field F ). A local field is the completion k of a global field k with respect to some place v.
q v
1

CHAPTER 1
Varieties over perfect fields
Let k be a perfect field. Fix an algebraic closure k. Let G = G be the profinite group
k
Gal(k/k).
Remark 1.0.1. Much of what we say remains true for imperfect fields k if we replace k
by a separable closure ks of k and define G = Gal(ks/k). At various points in these notes,
k
we will indicate how things are different over imperfect fields.
We will keep our terminology compatible with the language of schemes, even though we
try to keep the exposition down-to-earth whenever possible. Our exposition in this chapter
is intended not as a thorough introduction to algebraic geometry, but rather as a review,
with particular attention to issues that arise when working over perfect fields that are not
algebraically closed.
Definition 1.0.2. Officially, a k-variety is a separated scheme X of finite type over k.
Do not worry if you do not know what this means: for most of our purposes, it is enough
to know its set X(L) of L-rational points for each field extension L ⊇ k, and this set can be
defined in down-to-earth terms. For most of these notes, it will be OK to think of a k-variety
as the set X(k) in some ambient affine or projective space.
(cid:127)
Warning 1.0.3. Our varieties need not be irreducible or reduced: see Section 1.4.
1.1. Affine varieties
If n ∈ Z , define n-dimensional affine space over k as
≥0
An = An := Speck[t ,...,t ].
k 1 n
IfyoudonotknowwhatSpecmeans, justrememberthatAn(L) = Ln foreachfieldextension
L ⊇ k, or even for any k-algebra L.
Definition 1.1.1. An affine k-variety is a subscheme X = Speck[t ,...,t ]/I ⊆ An for
1 n k
some n ∈ Z and some ideal I ⊆ k[t ,...,t ]. Affine k-varieties are also called closed k-
≥0 1 n
subvarieties of An. Concretely, if f~ = (f ,...,f ) is a sequence of generators of the ideal I,
1 m
3
and L is any k-algebra, then
X(L) = {~a ∈ Ln : f~(~a) = 0}.
The k-algebra k[t ,...,t ]/I is called the affine coordinate ring of X.
1 n
Two affine varieties in An are considered the same if they are defined using the same
k
k, the same n, and the same ideal I of k[t ,...,t ]. We will sometimes use variable names
1 n
other than t ,...,t .
1 n
Examples 1.1.2.
(1) Let X be the R-variety x2 + y2 = −1 in A2; i.e. X = SpecR[x,y]/(x2 + y2 + 1).
R
Let Y be the R-variety 1 = 0 in A2. Then X(R) = Y(R), but X and Y are not the
R
same since the ideals (x2+y2+1) and (1) of R[x,y] are not the same. Another way
to see that X and Y are different is to observe that X(C) is nonempty, while Y(C)
is empty.
(2) Let X be as above, and let Z be the C-variety x2 +y2 = −1 in A2. Then X and
C
Z are different. A variety always comes equipped with a ground field. But see
Section 1.3.
(3) LetS betheC-varietySpecC[x]/(x2)inA1,andletT betheC-varietySpecC[x]/(x)
C
in A1. Then S and T are different, since the ideals (x2) and (x) of C[x] are different,
C
even though S(L) = T(L) for every field extension L ⊇ C. We can see the difference
by taking L = k[(cid:15)]/((cid:15)2).
Suppose X and X0 are two affine k-varieties in An. It follows from the Nullstellensatz
that X(k) = X0(k) if and only if the corresponding ideals of k[t ,...,t ] have the same
1 n
radical. If one is willing to restrict attention to affine varieties X defined by radical ideals,
and one does not care about preserving the field k as part of the data defining X, then one
n
can use the subset X(k) of k as a stand-in for X. Thus we have some reconciliation with the
elementary approach to algebraic geometry in which affine varieties are defined as subsets of
n
k .
Definition 1.1.3. A hypersurface in An is a closed subvariety defined by a single nonzero
polynomial.
Definition 1.1.4. An affine variety is reduced if its defining ideal I is radical, or equiv-
alently if its affine coordinate ring is reduced, meaning that it has no nonzero nilpotent
elements.
4