Galois representations attached to moments of Kloosterman sums and conjectures of Evans

TL;DR

This paper demonstrates that moments of Kloosterman sums across all primes are related to geometric Galois representations and proves conjectures connecting these moments to modular form coefficients.

Contribution

It establishes a geometric Galois representation framework for Kloosterman sum moments and proves Evans' conjectures linking moments to modular forms.

Findings

Moments of Kloosterman sums arise from geometric Galois representations.

Bounds on ramification of these Galois representations are provided.

Proofs of Evans' conjectures for seventh and eighth symmetric power moments.

Abstract

Kloosterman sums for a finite field arise as Frobenius trace functions of certain local systems defined over $\Gm$. The moments of Kloosterman sums calculate the Frobenius traces on the cohomology of tensor powers (or symmetric powers, exterior powers, etc.) of these local systems. We show that when $p$ ranges over all primes, the moments of the corresponding Kloosterman sums for $\mathbb{F}_{p}$ arise as Frobenius traces on a continuous $\ell$-adic Galois representation that comes from geometry. We also give bounds on the ramification of these Galois representations. All this is done in the generality of Kloosterman sheaves attached to reductive groups introduced in \cite{HNY}. As an application, we give proofs of conjectures of R. Evans (\cite{Evans0}, \cite{Evans}) expressing the seventh and eighth symmetric power moments of the classical Kloosterman sum in terms of Fourier coefficients of explicit modular forms. The proof for the eighth symmetric power moment conjecture relies on the computation done in Appendix B by C.Vincent.