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.

arXiv commentv2: 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

References in corpus (9)

• E_1-degeneration of the irregular Hodge filtration (with an appendix by Morihiko Saito)
• Hodge theory of the middle convolution
• Automorphy and irreducibility of some l-adic representations
• On the irregular Hodge filtration of exponentially twisted mixed Hodge modules
• Twistor property of GKZ-hypergeometric systems
• Galois representations attached to moments of Kloosterman sums and conjectures of Evans
• Quadratic relations between Bessel moments
• Irregular Hodge theory
• L-Functions for Symmetric Products of Kloosterman Sums

Cited by in corpus (10)

• Quadratic relations between Bessel moments
• $\mathbb{Q}$-linear dependence of certain Bessel moments
• Zeta functions of Z_p-towers of curves
• Hodge numbers of motives attached to Kloosterman and Airy moments
• Transformations of Hypergeometric Motives
• Hodge properties of Airy moments
• Irregular Hodge numbers for rigid $G_2$-connections
• Irregular Hodge filtration of hypergeometric differential equations
• On the Periods of Twisted Moments of the Kloosterman Connection
• Singularities of functions: a global point of view