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 $A^{\prime }$ be two abelian varieties, defined over number fields $K$ and $K^{\prime }$ respectively. Our main result is that $A$ and $A^{\prime }$ are isogenous after a suitable isomorphism between $K$ and $K^{\prime }$ if and only if the Dirichlet character groups of $K$ and $K^{\prime }$ are isomorphic and the $L$-series of $A$ and $A^{\prime }$ twisted by the Dirichlet characters match.

21 pages, the notation section of this article overlaps partially with arXiv:1901.06198. Added a proof for CM elliptic curves in the case of quadratic twists