Reductions of abelian varieties of generalized Mumford type

TL;DR

This paper investigates the reduction behavior of certain simple abelian varieties over number fields, analyzing their monodromy groups, reduction places, and confirming aspects of a broader conjecture on their splitting properties.

Contribution

It provides new insights into the reduction patterns and Dirichlet density of places for abelian varieties of generalized Mumford type, confirming a special case of a conjecture.

Findings

Reduction behavior of abelian varieties with minimal monodromy groups analyzed

Dirichlet density of reduction places computed and characterized

Confirmation of a special case of a broader splitting conjecture

Abstract

We study the special fibers of a certain class of absolutely simple abelian varieties over number fields with endomorphism rings $\bz$ and possessing $l$-adic monodromy groups of the least possible rank. We also study the Dirichlet density of the places at which the possible reductions occur and confirm a special case of a broader conjecture for the splitting of reductions of abelian varieties over number fields.