L-series and isogenies of abelian varieties

Abstract

Faltings's isogeny theorem states that two abelian varieties are isogenous over a number field precisely when the characteristic polynomials of the reductions at almost all prime ideals of the number field agree. This implies that two abelian varieties over $\mathbb{Q}$ with the same $L$-series are necessarily isogenous, but this is false over a general number field. Let $A$ and $A'$ be two abelian varieties, defined over number fields $K$ and $K'$ respectively. Our main result is that $A$ and $A'$ are isogenous after a suitable isomorphism between $K$ and $K'$ if and only if the Dirichlet character groups of $K$ and $K'$ are isomorphic and the $L$-series of $A$ and $A'$ twisted by the Dirichlet characters match.