Modular invariants for real quadratic fields and Kloosterman sums

TL;DR

This paper studies the distribution of certain integrals related to real quadratic fields, providing new bounds for Kloosterman sums and applying advanced spectral techniques to improve error estimates.

Contribution

It introduces a novel bound for sums of half-integral weight Kloosterman sums using a specialized Kuznetsov formula and subconvexity estimates.

Findings

Established a uniform bound for Kloosterman sums of half-integral weight.

Derived an improved error term in the asymptotic distribution of $j$-function integrals.

Applied spectral methods to analyze ideal class distributions in real quadratic fields.

Abstract

We investigate the asymptotic distribution of integrals of the $j$-function that are associated to ideal classes in a real quadratic field. To estimate the error term in our asymptotic formula, we prove a bound for sums of Kloosterman sums of half-integral weight that is uniform in every parameter. To establish this estimate we prove a variant of Kuznetsov's formula where the spectral data is restricted to half-integral weight forms in the Kohnen plus space, and we apply Young's hybrid subconvexity estimates for twisted modular $L$-functions.