Characterizing Abelian Varieties by the Reductions of the Mordell-Weil Group

TL;DR

This paper shows that the sizes of the reductions of the Mordell-Weil group at various primes uniquely determine the isogeny class of an abelian variety over a number field, under certain rank conditions.

Contribution

It establishes that the reduction sizes of the Mordell-Weil group encode enough information to identify the isogeny class of an abelian variety, extending Faltings' results.

Findings

Reduction sizes determine the isogeny class.

The result applies when all non-trivial subvarieties have positive rank.

Provides an analogue to Faltings' 1983 theorem.

Abstract

Let $A$ be an abelian variety defined over a number field $K$. If $\mathfrak{p}$ is a prime of $K$ of good reduction for $A$, let $A(K)_\mathfrak{p}$ denote the image of the Mordell-Weil group via reduction modulo $\mathfrak{p}$. We prove in particular that the size of $A(K)_\mathfrak{p}$, by varying $\mathfrak{p}$, encodes enough information to determine the $K$-isogeny class of $A$, provided that the following necessary condition is satisfied: $B(K)$ has positive rank for every non-trivial abelian subvariety $B$ of $A$. This is the analogue to a result by Faltings of 1983 considering instead the Hasse-Weil zeta function of the special fibers $A_\mathfrak{p}$.