Split reductions of simple abelian varieties

TL;DR

This paper investigates how the reduction properties of simple abelian varieties over number fields relate to their endomorphism rings, showing that commutativity leads to simplicity in reductions, while noncommutativity causes reducibility.

Contribution

It establishes a link between the structure of the endomorphism ring of an abelian variety and the behavior of its reductions at various primes, providing new criteria for simplicity and reducibility.

Findings

Commutative endomorphism rings imply almost all reductions are absolutely simple.

Noncommutative endomorphism rings lead to reducible reductions at a positive density of primes.

The results connect algebraic endomorphism structures with arithmetic reduction behavior.

Abstract

Consider an absolutely simple abelian variety X over a number field K. If the absolute endomorphism ring of X is commutative and satisfies certain parity conditions, then the reduction X_p is absolutely simple for almost all p. Conversely, if the absolute endomorphism ring of X is noncommutative, then X_p is reducible for p in a set of positive density.