Download this article
Download this article For screen
For printing
Recent Issues

Volume 18
Issue 6, 1039–1219
Issue 5, 847–1038
Issue 4, 631–846
Issue 3, 409–629
Issue 2, 209–408
Issue 1, 1–208

Volume 17, 12 issues

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
 
Other MSP Journals
Quadratic relations between Bessel moments

Javier Fresán, Claude Sabbah and Jeng-Daw Yu

Vol. 17 (2023), No. 3, 541–602
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

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.

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

Open Access made possible by participating institutions via Subscribe to Open.