Potential automorphy of certain non self-dual 3-dimensional Galois representations

TL;DR

This paper proves that certain 3-dimensional Galois representations associated with a family of algebraic surfaces are potentially automorphic, ensuring their L-functions have desirable analytic properties.

Contribution

It demonstrates potential automorphy for a new class of non self-dual 3-dimensional Galois representations using recent automorphy lifting techniques.

Findings

Compatible families are potentially automorphic for all parameters z

L-functions have analytic continuation and satisfy functional equations

Advances in automorphy lifting are applied to non self-dual cases

Abstract

In a series of papers, van Geemen and Top have defined a family of surfaces $S_z$ indexed by a nonzero integer parameter $z$, and a compatible family of 3-dimensional Galois representations over $\Q(i)$ attached to each surface. In this note we use recent advancements in potential automorphy and automorphy lifting to show that these compatible families are potentially automophic for all values of $z$, and hence that their L-functions have analytic continuation and a functional equation.