On the Periods of Twisted Moments of the Kloosterman Connection

TL;DR

This paper investigates the algebraic and analytical properties of twisted symmetric powers of the Kloosterman connection, linking period numbers to Bessel moments and establishing rational relations among them.

Contribution

It provides explicit computations of period pairings and interprets period numbers in terms of Bessel moments, revealing new rational relations.

Findings

Computed period pairings for twisted Kloosterman connections.

Expressed period numbers in terms of Bessel moments.

Proved rational and quadratic relations among Bessel moments.

Abstract

This paper aims to study the Betti homology and de Rham cohomology of twisted symmetric powers of the Kloosterman connection of rank two on the torus. We compute the period pairing and, with respect to certain bases, interpret these associated period numbers in terms of the Bessel moments. Via the rational structures on Betti homology and de Rham cohomology, we prove the $\mathbb{Q}$-linear and quadratic relations among these Bessel moments.