Type A Images of Galois Representations and Maximality

TL;DR

This paper proves that for large primes, the Galois images of compatible systems from algebraic varieties are maximally large within their Zariski closures, specifically of type A in the Cartan-Killing classification.

Contribution

It establishes maximality of type A Galois images in the Zariski closure for large primes in compatible systems from algebraic varieties.

Findings

Galois images are maximal of type A for sufficiently large primes.

The result applies to compatible systems arising from étale cohomology.

Maximality is in the sense of Zariski closure within GL(n).

Abstract

Given a compatible system of n-dimension l-adic Galois representations arising from \'etale cohomology of any complete, non-singular variety, we prove that for sufficiently large prime l, type A Galois image (in the Cartan-Killing classification) is in some sense maximal in its Zariski closure in GL(n).