Rational points on abelian varieties

40 Pages·2006·0.14 MB·English

Rational points on abelian
varieties
Karl Rubin
MSRI, January 17 2006
Abelian varieties
An abelian variety is a connected projective group
variety.
One-dimensional abelian varieties are elliptic
curves, which in characteristic different from 2 and
3 can be defined by Weierstrass equations
2 3
y = x + ax + b
3 2
with a, b ∈ k and 4a + 27b 6= 0.
Karl Rubin, MSRI Introductory Workshop, January 17 2006
Abelian varieties
The jacobian of a curve of genus g is an abelian
variety of dimension g.
An abelian variety over C is a complex torus (but
in dimension greater than one not every complex
torus is an abelian variety).
Karl Rubin, MSRI Introductory Workshop, January 17 2006
Rational points on abelian varieties
If A is an abelian variety defined over a field k, the
k-rational points A(k) form a commutative group.
Basic Problem: Given an abelian variety A over
k, find A(k).
Mordell-Weil Theorem. If k is a number field,
then A(k) is a finitely generated abelian group.
Karl Rubin, MSRI Introductory Workshop, January 17 2006
Overview
We don’t know how to compute A(k) in general,
+
so instead we study A(k)/nA(k) for n ∈ Z .
By the Mordell-Weil theorem,
∼
r
A(k) = A(k) ⊕ Z
tors
for some r ≥ 0. We call r the rank of A(k). Then
∼
r
A(k)/nA(k) = A(k) /n(A(k) ) ⊕ (Z/nZ) .
tors tors
Karl Rubin, MSRI Introductory Workshop, January 17 2006
Overview
∼
r
A(k)/nA(k) = A(k) /n(A(k) ) ⊕ (Z/nZ) .
tors tors
In particular, if we know A(k)/nA(k) and A(k) ,
tors
then we can compute the rank r.
For example, if n = p is prime then
dim A(k)/pA(k) = dim A(k)[p] + rank(A(k)).
F F
p p
Karl Rubin, MSRI Introductory Workshop, January 17 2006
Overview
We don’t know how to compute A(k)/nA(k)
in general either, so we will define an effectively
computable Selmer group S (A/k) containing
n
A(k)/nA(k).
X
The Shafarevich-Tate group (A/k) is the “error
term” (so we hope it’s small)
X
0 → A(k)/nA(k) → S (A/k) → (A/k)[n] → 0
n
X X
where (A/k)[n] is the n-torsion in (A/k).
X
Unfortunately (A/k) is very mysterious. This is
why computing A(k), or A(k)/nA(k), is so difficult.
Karl Rubin, MSRI Introductory Workshop, January 17 2006
Outline of talk
• Kummer theory on abelian varieties (first approx-
imation to the Selmer group, and sketch of proof
of the Mordell-Weil theorem)
• The Selmer group
• Principal homogeneous spaces and the Shafare-
vich-Tate group
Karl Rubin, MSRI Introductory Workshop, January 17 2006
Notation
sep
Let k be a separable closure of k and G =
k
sep
Gal(k /k).
If n is prime to the characteristic of k, let A[n]
sep
denote the kernel of multiplication by n in A(k ).
∼
2 dim(A)
Then A[n] = (Z/nZ) .
1 1
We will abbreviate H (k, A[n]) = H (G , A[n]).
k
Karl Rubin, MSRI Introductory Workshop, January 17 2006
Kummer theory on abelian varieties
Suppose first that A[n] ⊂ A(k). We define a
Kummer map
A(k) → Hom(G , A[n])
k
as follows. For x ∈ A(k),
sep
• choose y ∈ A(k ) such that ny = x,
σ
• map σ ∈ G to y − y ∈ A[n].
k
σ
Since A[n] ⊂ A(k), y − y is independent of
σ
the choice of y and the map σ 7→ y − y is a
homomorphism.
Karl Rubin, MSRI Introductory Workshop, January 17 2006