Duke Math. J. 171 (8), 1649-1747, (1 June 2022) DOI: 10.1215/00127094-2021-0036
Javier Fresán, Claude Sabbah, Jeng-Daw Yu
KEYWORDS: L-functions, Kloosterman sums, Galois representations, potential automorphy, ℓ-adic sheaves, exponential motives, irregular Hodge filtration, Fourier transform, connections with irregular singularities, D-modules, mixed Hodge modules, 11G40, 11F80, 11L05, 14F40, 32S35, 32S40
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.