Hodge numbers of motives attached to Kloosterman and Airy moments

Abstract

Fresán, Sabbah, and Yu constructed motives ${\mathrm{M}}_{n+1}^k(\mathrm{Kl})$ over $\mathbb{Q}$ encoding symmetric power moments of Kloosterman sums in $n$ variables. When $n=1$, they use the irregular Hodge filtration on the exponential mixed Hodge structure associated with ${\mathrm{M}}_2^k(\mathrm{Kl})$ to compute the Hodge numbers of ${\mathrm{M}}_2^k(\mathrm{Kl})$, which turn out to be either $0$ or $1$. In this article, I explain how to compute the (irregular) Hodge numbers of ${\mathrm{M}}_{n+1}^k(\mathrm{Kl})$ for $n=2$ or for general values of $n$ such that $\gcd (k,n+1)=1$. I will also discuss related motives attached to Airy moments constructed by Sabbah and Yu. In particular, the computation shows that there are Hodge numbers bigger than $1$ in most cases.

final version, to appear in Crelle's Journal

References in corpus (3)

• Quadratic relations between periods of connections
• L-Functions for Symmetric Products of Kloosterman Sums
• $L$-functions of Kloosterman sheaves

Cited by in corpus (2)

• Singularities of functions: a global point of view
• Irregular Hodge filtration of hypergeometric differential equations