# L-series and isogenies of abelian varieties

## TL;DR

This paper explores the relationship between L-series, isogenies, and abelian varieties over number fields, extending classical results by incorporating Dirichlet character twists and field isomorphisms.

## Contribution

It establishes a new criterion for isogeny of abelian varieties over different number fields based on matching L-series twisted by Dirichlet characters and isomorphism of Dirichlet character groups.

## Key findings

- Two abelian varieties are isogenous over an isomorphic number field if their twisted L-series match.
- The isogeny criterion involves the isomorphism of Dirichlet character groups of the base fields.
- Classical results over $\,\mathbb{Q}$ are extended to more general number fields.

## 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.