The Artin Conjecture for some S_5-extensions

TL;DR

This paper proves new cases of Artin's conjecture for Galois representations with image S_5 over Q, demonstrating automorphy under specific local conditions, including a case verified for the smallest discriminant quintic field.

Contribution

It establishes automorphy of certain S_5 Galois representations, including a verified example for the smallest discriminant quintic field, advancing understanding of Artin's conjecture.

Findings

Proved automorphy for S_5 Galois representations with specific local conditions.

Verified Artin's conjecture for the Galois closure of the smallest discriminant quintic field.

Connected automorphy results to non-vanishing of Dedekind zeta functions.

Abstract

We establish some new cases of Artin's conjecture. Our results apply to Galois representations over $\Q$ with image $S_5$ satisfying certain local hypotheses, the most important of which is that complex conjugation is conjugate to $(12)(34)$. In fact, we prove the stronger claim conjectured by Langlands that these representations are automorphic. For the irreducible representations of dimensions 4 and 6, our result follows from known 2-dimensional cases of Artin's conjecture (proved by Sasaki) as well as the functorial properties of the Asai transfer proved by Ramakrishnan. For the irreducible representations of dimension 5, we encounter the problem of descending an automorphic form from a quadratic extension compatibly with the Galois representation. This problem is partly solved by working instead with a four dimensional representation of some central extension of $S_5$. Our modularity results in this case are contingent on the non-vanishing of a certain Dedekind zeta function on the real line in the critical strip. A result of Booker show that one can (in principle) explicitly verify this non-vanishing, and with Booker's help we give an example, verifying Artin's conjecture for representations coming from the (Galois closure) of the quintic field $K$ of smallest discriminant (1609).