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Symmetric power $ L$-functions for families of generalized Kloosterman sums


Authors: C. Douglas Haessig and Steven Sperber
Journal: Trans. Amer. Math. Soc. 369 (2017), 1459-1493
MSC (2010): Primary 11L05, 14D10, 14F30, 14G15
DOI: https://doi.org/10.1090/tran/6720
Published electronically: June 20, 2016
MathSciNet review: 3572279
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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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Additional Information

C. Douglas Haessig
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
Email: chaessig@math.rochester.edu

Steven Sperber
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: sperber@math.umn.edu

DOI: https://doi.org/10.1090/tran/6720
Received by editor(s): May 2, 2014
Received by editor(s) in revised form: February 20, 2015
Published electronically: June 20, 2016
Article copyright: © Copyright 2016 American Mathematical Society