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Modular invariants for real quadratic fields and Kloosterman sums

Nickolas Andersen, William Duke

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.

Modular invariants for real quadratic fields and Kloosterman sums

Abstract

We investigate the asymptotic distribution of integrals of the -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 -functions.

Paper Structure

This paper contains 7 sections, 28 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Geometric invariants and Kloosterman sums
  3. Background
  4. An estimate for coefficients of Maass cusp forms
  5. The Kuznetsov formula for Kohnen's plus space
  6. Proof of Theorem \ref{['thm:main-kloo']}
  7. Young's theorem for even discriminants

Key Result

Theorem 1.1

For each positive fundamental discriminant $D$, let $d$ be any positive fundamental discriminant dividing $D$. Then for each $m\geq 1$ we have

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Remark
  • Theorem 1.3
  • Theorem 1.4
  • Remark
  • Remark
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • ...and 36 more