A conjecture of Evans on sums of Kloosterman sums.

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. [ABSTRACT FROM AUTHOR]