Modular invariants for real quadratic fields and Kloosterman sums

Andersen, Nickolas; Duke, William D.

doi:10.2140/ant.2020.14.1537

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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.

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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

Vol. 14, No. 6, 2020