Hodge theory of Kloosterman connections
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.
Keywords
Cite
@article{arxiv.1810.06454,
title = {Hodge theory of Kloosterman connections},
author = {Javier Fresán and Claude Sabbah and Jeng-Daw Yu},
journal= {arXiv preprint arXiv:1810.06454},
year = {2022}
}
Comments
v2: Results and exposition improved. v3: Substantial revision after referee reports. The main construction now takes place in the setting of exponential mixed Hodge structures. We partially reorganized the material and removed one of the appendices. Some statements have been strengthened, several proofs simplified, and a few mistakes corrected. v5: Final published version