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

TL;DR

This paper establishes a Weil bound for half-integral weight Kloosterman sums associated with even lattices, generalizing previous identities and providing new bounds in the context of the Weil representation.

Contribution

It proves a Weil bound for half-integral weight Kloosterman sums using a novel identity relating divisor sums to exponential sums, extending Kohnen's identity.

Findings

Established a Weil bound for half-integral weight Kloosterman sums

Derived a new identity linking divisor sums to exponential sums

Generalized Kohnen's identity for plus space Kloosterman sums

Abstract

Let $L$ be an even lattice of odd rank with discriminant group $L'/L$, and let $\alpha,\beta \in L'/L$. 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.