The Weil bound for generalized Kloosterman sums of half-integral weight

Abstract

Let $L$ be an even lattice of odd rank with discriminant group $L^{\prime }/L$, and let $\alpha ,\beta \in L^{\prime }/L$. We prove the Weil bound for the Kloosterman sums $S_{\alpha ,\beta }(m,n,c)$ of half-integral weight for the Weil Representation attached to $L$. We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen's identity for plus space Kloosterman sums with the theta multiplier system.

22 pages