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Quadratic relations between Bessel moments

Javier Fresán, Claude Sabbah and Jeng-Daw Yu

Vol. 17 (2023), No. 3, 541–602

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

0I 0(t)iK 0(t)kit2j1 d t(i,j = 1, ,(k 1)2),

where k 1 is a fixed integer and I0 and K0 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.

Kloosterman connection, period pairing, quadratic relations, Bessel moments
Mathematical Subject Classification
Primary: 32G20, 34M35
Received: 11 June 2020
Revised: 21 June 2021
Accepted: 10 May 2022
Published: 12 April 2023
Javier Fresán
Centre de Mathématiques Laurent Schwartz
CNRS, École polytechnique
Institut Polytechnique de Paris
Claude Sabbah
Centre de Mathématiques Laurent Schwartz
CNRS, École polytechnique
Institut Polytechnique de Paris
Jeng-Daw Yu
Department of Mathematics
National Taiwan University

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