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