Remark on nondegeneracy of simple abelian varieties with many endomorphisms

TL;DR

This paper explores how the nondegeneracy property of simple abelian varieties over algebraic closures of rationals relates to their reductions, establishing conditions under which nondegeneracy is preserved.

Contribution

It proves that, under certain assumptions, nondegeneracy of a simple abelian variety over an algebraic closure implies nondegeneracy of its reduction.

Findings

Nondegeneracy of $A$ implies nondegeneracy of $A_0$ under specific conditions.

Establishes a link between properties of abelian varieties over different fields.

Provides theoretical insights into the structure of abelian varieties with many endomorphisms.

Abstract

We investigate a relationship between nondegeneracy of a simple abelian variety $A$ over an algebraic closure of $\mb{Q}$ and of its reduction $A_0$. We prove that under some assumptions, nondegeneracy of $A$ implies nondegeneracy of $A_0$.