Classe d'isog\'enie de vari\'et\'es ab\'eliennes pleinement de type GSp

TL;DR

This paper extends Faltings' isogeny criterion for abelian varieties of GSp type, showing that matching prime divisors of point counts over finite fields suffices for isogeny over a number field.

Contribution

It introduces a new criterion for isogeny of GSp-type abelian varieties based on prime divisors of point counts, generalizing previous results for elliptic curves.

Findings

Prime divisors of point counts determine isogeny for GSp-type abelian varieties.

The criterion applies to varieties with endomorphism ring Z and odd dimension.

The proof adapts ideas from Serre, Frey-Jarden, and extends Hall-Perucca's elliptic curve results.

Abstract

Faltings in 1983 proved that a necessary and sufficient condition for two abelian varieties $A$ and $B$ to be isogenous over a number field $K$ is that the local factors of the L-series of $A$ and $B$ are equal for almost all primes of $K$ ; for each such prime this implies that $A$ and $B$ have the same number of points over the residue field. We show in this article that for abelian varieties faithfully of type GSp (a class containing the abelian varieties with endomorphism ring $\mathbb{Z}$ and of odd dimension) `having the same number of points' may be replaced by `the number of points have the same prime divisors' and still gives a sufficient condition for $A$ and $B$ to be $K$-isogenous. The proof is based on ideas of Serre \cite{serreim72} and Frey-Jarden \cite{FJ} and follows closely Hall-Perucca \cite{hallp} who proved the result for elliptic curves.