Elliptic minuscule pairs and splitting abelian varieties

TL;DR

This paper investigates the conditions under which absolutely simple abelian varieties over number fields have simple specializations at a positive density of places, using monodromy and classification of algebraic group pairs.

Contribution

It provides a partial answer to Murty and Patankar's question by classifying pairs (G,V) related to monodromy representations of abelian varieties.

Findings

Classification of pairs (G,V) with maximal torus acting irreducibly

Partial answer to the density of simple specializations

Connection between monodromy and abelian variety properties

Abstract

We partially answer, in terms of monodromy, Murty and Patankar's question: Given an absolutely simple abelian variety over a number field, does it have simple specializations at a set of places of positive Dirichlet density? The answer is based on the classification of pairs (G,V) consisting of a semi-simple algebraic group G over a non-archimedean local field and an absolutely irreducible representation V of G such that G admits a maximal torus acting irreducibly on V.