Finiteness Theorems for Abelian Varieties over Number Fields

Abstract

Let K be a finite extension of ℚ, A an abelian variety defined over K, π = Gal(K̄/K) the absolute Galois group of K, and l a prime number. Then π acts on the (so-called) Tate module

$$ {T_l}(A) = \mathop{{\lim }}\limits_{{\mathop{ \leftarrow }\limits_n }} \,A[{l^n}](\overline K ) $$

The goal of this chapter is to give a proof of the following results:

• (a) The representation of π on ${T_l}(A){ \otimes_{{{\mathbb{Z}_l}}}}{\mathbb{Q}_l}$ s is semisimple.

• (b) The map

  $$ {\text{En}}{{\text{d}}_K}(A){ \otimes_{\mathbb{Z}}}{\mathbb{Z}_l} \to {\text{En}}{{\text{d}}_{\pi }}({T_l}(A)) $$

  is an isomorphism.

• (c) Let S be a finite set of places of K, and let d > 0. Then there are only finitely many isomorphism classes of abelian varieties over K with polarizations of degree d which have good reduction outside of S.

An erratum to this chapter is available at http://dx.doi.org/10.1007/978-1-4613-8655-1_16