On the splitting fields of generic elements in Zariski dense subgroups

TL;DR

This paper demonstrates that, for a broad class of algebraic groups over finitely generated fields, the combined splitting fields of generic elements encode enough information to determine the group up to isogeny, with few exceptions.

Contribution

It establishes a link between the splitting fields of generic elements and the classification of algebraic groups over finitely generated fields, extending previous understanding.

Findings

The commensurability class of the field of splitting fields determines the algebraic group up to isogeny.

Most cases follow this pattern, with only a few exceptions.

The result applies to Zariski dense subgroups of connected, absolutely almost simple algebraic groups.

Abstract

Let $G$ be a connected, absolutely almost simple, algebraic group defined over a finitely generated, infinite field $K$, and let $\Gamma$ be a Zariski dense subgroup of $G(K)$. We show, apart from some few exceptions, that the commensurability class of the field $\mathcal{F}$ given by the compositum of the splitting fields of characteristic polynomials of generic elements of $\Gamma$ determines the group $G$ upto isogeny over the algebraic closure of $K$.