Hodge numbers of motives attached to Kloosterman and Airy moments

TL;DR

This paper extends the computation of Hodge numbers for motives related to Kloosterman and Airy moments, revealing new cases with larger Hodge numbers and generalizing previous results for higher dimensions.

Contribution

It provides a method to compute irregular Hodge numbers for motives associated with Kloosterman sums in higher dimensions and for general parameters, expanding prior work limited to specific cases.

Findings

Hodge numbers are 0 or 1 for n=1 case.

Hodge numbers can be greater than 1 for higher n.

Explicit computations for n=2 and general n with gcd(k,n+1)=1.

Abstract

Fres\'an, 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.