Functional Equations of $L$-Functions for Symmetric Products of the Kloosterman Sheaf

TL;DR

This paper computes the local monodromy and epsilon-factors of symmetric product L-functions of the Kloosterman sheaf, establishing their functional equations and proving a conjecture on the signs of constants.

Contribution

It determines the local monodromy and epsilon-factors for symmetric products of the Kloosterman sheaf, leading to explicit functional equations and confirming Evans' conjecture.

Findings

Calculated local monodromy at 0 and infinity

Derived epsilon-factors for symmetric products

Proved functional equations and Evans' conjecture

Abstract

We determine the (arithmetic) local monodromy at 0 and at $\infty$ of the Kloosterman sheaf using local Fourier transformations and Laumon's stationary phase principle. We then calculate $\epsilon$-factors for symmetric products of the Kloosterman sheaf. Using Laumon's product formula, we get functional equations of $L$-functions for these symmetric products, and prove a conjecture of Evans on signs of constants of functional equations.