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

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

2008-12-30

arxiv.org/abs/0812.4994v1

Mathematics

Number Theory

23 pages

Scientific paper

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.

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