Functional equations of $L$-functions for symmetric products of the Kloosterman sheaf

Abstract

We determine the (arithmetic) local monodromy at 0 and at ∞ of the Kloosterman sheaf using local Fourier transformations and Laumon's stationary phase principle. We then calculate e-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. © 2010 American Mathematical Society.