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Effective estimates for traces of singular moduli

Oscar E. González, Qihang Sun

TL;DR

The paper addresses the problem of estimating traces of singular moduli of the j-function at CM points and extends the Andersen–Duke results by providing an effective asymptotic with explicit error terms. It develops an explicit analysis in Kohnen’s plus space, introducing and bounding plus-space Kloosterman sums and their associated zeta function via projections of projected Poincaré series and inner-product unfolding for Maass cusp forms of half-integral weight. The main contributions are an explicit bound for the error term in the generalized twisted traces Tr_d j_m(z_D) and a corollary that yields computable approximations to these traces with clear dependence on the discriminant, level, and divisor functions. This yields practical, computable estimates for traces of singular moduli and clarifies the trade-off between exponent growth and the achievement of explicit constants, with potential applications to numerical investigations of these traces.

Abstract

Traces of singular moduli can be approximated by exponential sums of quadratic irrationals. Recently Andersen and Duke used theory of Maass forms to estimate generalized twisted traces with power-saving error bounds. We establish an asymptotic formula with effective error bounds for such traces. Our methods depend on an explicit bound for sums of Kloosterman sums on $Γ_0(4)$.

Effective estimates for traces of singular moduli

TL;DR

The paper addresses the problem of estimating traces of singular moduli of the j-function at CM points and extends the Andersen–Duke results by providing an effective asymptotic with explicit error terms. It develops an explicit analysis in Kohnen’s plus space, introducing and bounding plus-space Kloosterman sums and their associated zeta function via projections of projected Poincaré series and inner-product unfolding for Maass cusp forms of half-integral weight. The main contributions are an explicit bound for the error term in the generalized twisted traces Tr_d j_m(z_D) and a corollary that yields computable approximations to these traces with clear dependence on the discriminant, level, and divisor functions. This yields practical, computable estimates for traces of singular moduli and clarifies the trade-off between exponent growth and the achievement of explicit constants, with potential applications to numerical investigations of these traces.

Abstract

Traces of singular moduli can be approximated by exponential sums of quadratic irrationals. Recently Andersen and Duke used theory of Maass forms to estimate generalized twisted traces with power-saving error bounds. We establish an asymptotic formula with effective error bounds for such traces. Our methods depend on an explicit bound for sums of Kloosterman sums on .
Paper Structure (6 sections, 11 theorems, 104 equations)

This paper contains 6 sections, 11 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fourier coefficients of projected Poincaré series
  4. Estimates for the Kloosterman-Selberg zeta function
  5. Perron's formula
  6. Proof of Theorems \ref{['Thm2']} and \ref{['Thm1']}

Key Result

Theorem 1.1

For each negative discriminant $D$, let $d$ be any fundamental discriminant dividing $D$ and let $m\geq 1$. Then for $0<Y\ll\frac{1}{m}$ we have where $\theta \in [0, \frac{7}{64}]$ is an admissible exponent toward the Ramanujan conjecture for Maass cusp forms of integral weight.

Theorems & Definitions (17)

  • Theorem 1.1: ADasymptotic
  • Theorem 1.2
  • Theorem 1.3
  • Remark
  • Corollary 1.4: ADasymptotic, Corollary 2.3
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Lemma 3.1: ADinvariants, Lemmas 5.1 and 5.5
  • Lemma 3.2
  • ...and 7 more