Symmetric power L-functions for families of generalized Kloosterman sums

C. Douglas Haessig, Steven Sperber

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)1459-1493
Number of pages35
JournalTransactions of the American Mathematical Society
Volume369
Issue number2
DOIs
StatePublished - 2017

Bibliographical note

Publisher Copyright:
© 2016 American Mathematical Society.

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