Quadratic relations between Bessel moments
Journal
Alg. Number Th. 17 (2023) 541-602
Journal Volume
17
Journal Issue
3
Date Issued
2020-06-04
Author(s)
Abstract
Motivated by the computation of certain Feynman amplitudes, Broadhurst and
Roberts recently conjectured and checked numerically to high precision a set of
remarkable quadratic relations between the Bessel moments \[ \int_0^\infty
I_0(t)^i K_0(t)^{k-i}t^{2j-1}\,\mathrm{d}t \qquad (i, j=1, \ldots, \lfloor
(k-1)/2\rfloor), \] where $k \geq 1$ is a fixed integer and $I_0$ and $K_0$
denote the modified Bessel functions. In this paper, 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 in
arXiv:2005.11525, 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.
Subjects
Bessel moments | Kloosterman connection | period pairing | quadratic relations; Mathematics - Algebraic Geometry; Mathematics - Algebraic Geometry; Mathematics - Number Theory
Description
40 pages, 1 figure, 1 table. V2: Comparison of periods now takes
place in the setting of exponential mixed Hodge structures. Add an appendix
by the second author for necessary tools. Change format. 61 pages, 1 figure,
1 table
place in the setting of exponential mixed Hodge structures. Add an appendix
by the second author for necessary tools. Change format. 61 pages, 1 figure,
1 table
Type
journal article