#### 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.

##### Keywords
Kloosterman sums, real quadratic fields, modular forms
Primary: 11F37
Secondary: 11L05