The splitting of reductions of an abelian variety

TL;DR

This paper investigates how reductions of an absolutely simple abelian variety over a number field decompose up to isogeny at most places, assuming the Mumford-Tate conjecture, and relates this to a conjecture by Murty and Patankar.

Contribution

It proves that for most places, the reduction is isogenous to a power of an absolutely simple abelian variety, under the Mumford-Tate conjecture, supporting a conjecture of Murty and Patankar.

Findings

Reductions are isogenous to a power of an absolutely simple abelian variety at most places.

Provides a description of the Galois extension generated by Weil numbers of reductions.

Supports the conjecture of Murty and Patankar under certain assumptions.

Abstract

Consider an absolutely simple abelian variety A defined over a number field K. For most places v of K, we study how the reduction A_v of A modulo v splits up to isogeny. Assuming the Mumford-Tate conjecture for A and possibly increasing K, we will show that A_v is isogenous to the m-th power of an absolutely simple abelian variety for all places v of K away from a set of density 0, where m is an integer depending only on the endomorphism ring End(A_Kbar). This proves many cases, and supplies justification, for a conjecture of Murty and Patankar. Under the same assumptions, we will also describe the Galois extension of Q generated by the Weil numbers of A_v for most v.