Symmetric power 𝐿-functions for families of generalized Kloosterman sums

Haessig, C. Douglas; Sperber, Steven

doi:10.1090/tran/6720

Symmetric power $L$-functions for families of generalized Kloosterman sums
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by C. Douglas Haessig and Steven Sperber PDF

Trans. Amer. Math. Soc. 369 (2017), 1459-1493 Request permission

Abstract:

We construct relative $p$-adic cohomology for a family of toric exponential sums fibered over the torus. The family under consideration here generalizes the classical Kloosterman sums. Under natural hypotheses such as quasi-homogeneity and nondegeneracy, this cohomology, just as in the absolute case, is acyclic except in the top dimension. Our construction gives us sufficiently sharp estimates for the action of Frobenius on relative cohomology so that we may obtain properties of $L$-functions constructed by taking a suitable Euler product (over the family) of local factors using linear algebra operations (such as taking the $k$-th symmetric power or other such operations) on the reciprocal zeros and poles of the $L$-functions of each fiber.

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