A finiteness property of abelian varieties with potentially ordinary good reduction

Abstract

A g-dimensional abelian variety A/F over a number field F is of GL(2)-type if End(A/F ) := End(A/F ) ⊗Z Q contains a field E of degree g. We call such a field E an endomorphism field of A. We say that an F -simple abelian variety A/F over a number field F of dimension g is non-CM if End (A/F ×F Q) does not contain any semi-simple commutative algebra of degree 2g over Q (cf. [ACM]). If an abelian variety A/F of GL(2)-type is F -simple, D = End (A/F ) is a division algebra with a positive involution α → α∗. Since D has a maximal commutative subfield stable under ∗, we may assume that its endomorphism field E is totally real or a CM field. The Galois action on the Tate module of an F -simple abelian variety A/F of GL(2)-type with endomorphism field E produces a two-dimensional strictly compatible system of Galois representations ρA = {ρλ : Gal(Q/F ) → GL2(Eλ)}λ indexed by primes λ of E. Thus we have its L-function L(s, ρA). Two F -simple abelian varieties A and B = Aχ are twist equivalent if L(s, ρB) = L(s, ρA ⊗ χ) for a finite order character χ : Gal(Q/F ) → Q. Note that the dimension is possibly unbounded over a twist equivalent class. Since Tate’s conjecture has been proven by Faltings for abelian varieties, one could formulate this equivalence by insisting that the two abelian varieties share a simple component over an abelian extension of F . In this more geometric context, Q-simple abelian varieties have been studied in depth as Q-simple factors of modular Jacobians by Ribet (for example, see [R] and his papers quoted there). However we adopt an analytic definition of twistequivalence using their L-function as it can also be applied to rank 2 Q-motives (for which the Tate conjecture is still unknown). Here the identity of two L-functions