Hodge theory of Kloosterman connections

TL;DR

This paper constructs motives linked to Kloosterman sums, proves their L-functions have meromorphic continuation and functional equations, and computes their Hodge numbers, revealing potential automorphy.

Contribution

It introduces a new construction of motives associated with Kloosterman sums and analyzes their Hodge structures and automorphy properties.

Findings

L-functions extend meromorphically and satisfy functional equations

Hodge numbers are all zero or one

Motives are potential automorphic

Abstract

We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their L-functions extend meromorphically to the complex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be "classical", we compute their Hodge numbers by means of the irregular Hodge filtration on their realizations as exponential mixed Hodge structures. We show that all Hodge numbers are either zero or one, which implies potential automorphy thanks to recent results of Patrikis and Taylor.