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

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.