## 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.## References

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## Additional Information

**C. Douglas Haessig**- Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
- MR Author ID: 727731
- Email: chaessig@math.rochester.edu
**Steven Sperber**- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- MR Author ID: 165470
- Email: sperber@math.umn.edu
- Received by editor(s): May 2, 2014
- Received by editor(s) in revised form: February 20, 2015
- Published electronically: June 20, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 1459-1493 - MSC (2010): Primary 11L05, 14D10, 14F30, 14G15
- DOI: https://doi.org/10.1090/tran/6720
- MathSciNet review: 3572279