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arXiv · 1801.08174

Modular invariants for real quadratic fields and Kloosterman sums

Nickolas Andersen·William Duke
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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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Nickolas Andersen, William Duke. 2018-01-24. Modular invariants for real quadratic fields and Kloosterman sums. https://doi.org/10.2140/ant.2020.14.1537

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