Remarks on a theorem of Pink in presence of bad reduction

TL;DR

This paper advances the Mumford-Tate conjecture by extending Pink's theorem to certain abelian varieties with bad reduction, utilizing quadratic pairs and group classification techniques.

Contribution

It introduces new cases for the Mumford-Tate conjecture by applying quadratic pair classification and generalizes Hall's theorem on Tate module representations.

Findings

Proves new cases of the Mumford-Tate conjecture.

Generalizes Hall's theorem on Tate modules.

Utilizes classification of quadratic pairs in the proof.

Abstract

In this note we prove new cases of the Mumford-Tate conjecture by extending a theorem of Richard Pink for abelian varieties without nontrivial endomorphisms and with bad semistable reduction. We use quadratic pairs introduced by J.G.Thompson in the seventies, an important tool in the program of classifying all simple finite groups. Proof of our main result applies the classification of the quadratic pairs as described by Premet and Suprunenko. Along the way we reprove and generalize a theorem of Chris Hall on the image of Tate module representation of abelian variety as above, to all possible values of its toric dimension.