A conjecture of Evans on sums of Kloosterman sums

Dummit, Evan; Goldberg, Adam; Perry, Alexander

doi:10.1090/S0002-9939-10-10486-9

A conjecture of Evans on sums of Kloosterman sums
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by Evan P. Dummit, Adam W. Goldberg and Alexander R. Perry PDF

Proc. Amer. Math. Soc. 138 (2010), 3047-3056

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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