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
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.

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Keywords
Kloosterman sums, real quadratic fields, modular forms
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