Vol. 14, No. 6, 2020
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Modular invariants for real quadratic fields and Kloosterman sums

Nickolas Andersen and William D. Duke
Vol. 14 (2020), No. 6, 1537–1575
DOI: 10.2140/ant.2020.14.1537
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.

Keywords
Kloosterman sums, real quadratic fields, modular forms
Mathematical Subject Classification 2010
Primary: 11F37
Secondary: 11L05
Milestones
Received: 1 May 2019
Revised: 9 December 2019
Accepted: 6 February 2020
Published: 30 July 2020
Authors
Nickolas Andersen
Brigham Young University
Provo, UT
United States
William D. Duke
UCLA
Los Angeles, CA
United States