Quadratic relations between Bessel moments

Fresán, Javier; Sabbah, Claude; Yu, Jeng-Daw

doi:10.2140/ant.2023.17.541

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Quadratic relations between Bessel moments

Javier Fresán, Claude Sabbah and Jeng-Daw Yu

Vol. 17 (2023), No. 3, 541–602

DOI: 10.2140/ant.2023.17.541

Abstract

Motivated by the computation of some Feynman amplitudes, Broadhurst and Roberts recently conjectured and checked numerically to high precision a set of remarkable quadratic relations between the Bessel moments

\int_{0}^{\infty} I_{0} {(t)}^{i} K_{0} {(t)}^{k - i} t^{2 j - 1} d t (i, j = 1, \dots, ⌊ (k - 1) ∕ 2 ⌋),

where $k \geq 1$ is a fixed integer and $I_{0}$ and $K_{0}$ denote the modified Bessel functions. We interpret these integrals and variants thereof as coefficients of the period pairing between middle de Rham cohomology and twisted homology of symmetric powers of the Kloosterman connection. Building on the general framework developed by Fresan, Sabbah and Yu (2020), this enables us to prove quadratic relations of the form suggested by Broadhurst and Roberts, which conjecturally comprise all algebraic relations between these numbers. We also make Deligne’s conjecture explicit, thus explaining many evaluations of critical values of $L$ -functions of symmetric power moments of Kloosterman sums in terms of determinants of Bessel moments.

Keywords

Kloosterman connection, period pairing, quadratic relations, Bessel moments

Mathematical Subject Classification

Primary: 32G20, 34M35

Milestones

Received: 11 June 2020

Revised: 21 June 2021

Accepted: 10 May 2022

Published: 12 April 2023

Authors

Javier Fresán
Centre de Mathématiques Laurent Schwartz CNRS, École polytechnique Institut Polytechnique de Paris France

Claude Sabbah
Centre de Mathématiques Laurent Schwartz CNRS, École polytechnique Institut Polytechnique de Paris France

Jeng-Daw Yu
Department of Mathematics National Taiwan University Taipei Taiwan

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Vol. 17, No. 3, 2023