Finiteness Theorems for Abelian Varieties over Number Fields

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.