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A conjecture of Evans on sums of Kloosterman sums


Authors: Evan P. Dummit, Adam W. Goldberg and Alexander R. Perry
Journal: Proc. Amer. Math. Soc. 138 (2010), 3047-3056
MSC (2010): Primary 11L05; Secondary 33C20
DOI: https://doi.org/10.1090/S0002-9939-10-10486-9
Published electronically: May 4, 2010
MathSciNet review: 2653929
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Abstract: In a recent paper, Evans relates twisted Kloosterman sheaf sums to Gaussian hypergeometric functions, and he formulates a number of conjectures relating certain twisted Kloosterman sheaf sums to the coefficients of modular forms. Here we prove one of his conjectures for a fourth order twisted Kloosterman sheaf sum $ T_n$ of the quadratic character on $ \mathbf{F}_p^\times$. In the course of the proof we develop reductions for twisted moments of Kloosterman sums and apply these in the end to derive a congruence relation for $ T_n$ with generalized Apéry numbers.


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Additional Information

Evan P. Dummit
Affiliation: Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, Wisconsin 53706-1388
Email: Dummit@math.wisc.edu

Adam W. Goldberg
Affiliation: 617 Logan Lane, Danville, California 94526
Email: AdamWGoldberg@gmail.com

Alexander R. Perry
Affiliation: Department of Mathematics, 4517 Lerner Hall, Columbia University, 2920 Broadway, New York, New York 10027-8343
Email: arp2125@columbia.edu

DOI: https://doi.org/10.1090/S0002-9939-10-10486-9
Received by editor(s): July 24, 2009
Received by editor(s) in revised form: July 27, 2009
Published electronically: May 4, 2010
Communicated by: Jim Haglund
Article copyright: © Copyright 2010 Evan Dummit, Adam Goldberg, Alexander Perry