Automorphy and irreducibility of some l-adic representations

TL;DR

This paper proves potential automorphy of certain l-adic representations under a purity assumption, using Katz's theory, and shows irreducibility for a positive density of primes in automorphic cases.

Contribution

It introduces a new approach to potential automorphy by replacing irreducibility with purity and applies Katz's theory to construct relevant motives.

Findings

Potential automorphy holds under purity without irreducibility.

Constructs many motives using Katz's theory.

Irreducibility for a positive density of primes in automorphic cases.

Abstract

In this paper we prove that a pure, regular, totally odd, polarizable weakly compatible system of $l$-adic representations is potentially automorphic. The innovation is that we make no irreducibility assumption, but we make a purity assumption instead. For compatible systems coming from geometry, purity is often easier to check than irreducibility. We use Katz's theory of rigid local systems to construct many examples of motives to which our theorem applies. We also show that if $F$ is a CM or totally real field and if $\pi$ is a polarizable, regular algebraic, cuspidal automorphic representation of $GL_n(\A_F)$, then for a positive Dirichlet density set of rational primes $l$, the $l$-adic representations $r_{l,\imath}(\pi)$ associated to $\pi$ are irreducible.