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A unified approach to Rohrlich-type divisor sums

Daeyeol Jeon, Soon-Yi Kang, Chang Heon Kim, Toshiki Matsusaka

Abstract

We propose a systematic method for analyzing Rohrlich-type divisor sums for arbitrary congruence subgroups $Γ_0(N)$. Our main theorem unifies various results from the literature, and its significance is illustrated through the following five applications: (1) the valence formula, (2) a natural generalization of classical Rohrlich's formula to level $N$, (3) an explicit version of the theorem by Bringmann-Kane-Löbrich-Ono-Rolen, (4) an extension of the generalized Rohrlich formula proposed by Bringmann-Kane, and (5) an alternative proof of the decomposition formula for twisted traces of CM values of weight 0 Eisenstein series.

A unified approach to Rohrlich-type divisor sums

Abstract

We propose a systematic method for analyzing Rohrlich-type divisor sums for arbitrary congruence subgroups . Our main theorem unifies various results from the literature, and its significance is illustrated through the following five applications: (1) the valence formula, (2) a natural generalization of classical Rohrlich's formula to level , (3) an explicit version of the theorem by Bringmann-Kane-Löbrich-Ono-Rolen, (4) an extension of the generalized Rohrlich formula proposed by Bringmann-Kane, and (5) an alternative proof of the decomposition formula for twisted traces of CM values of weight 0 Eisenstein series.

Paper Structure

This paper contains 14 sections, 10 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Regularized Petersson inner product
  3. Punctured fundamental domain
  4. Auxiliary calculations using Stokes' theorem
  5. Proof of \ref{['thm:main']}
  6. At singularities on $\mathbb{H}$
  7. At cusps
  8. Applications
  9. Maass--Poincaré series
  10. Valence formula
  11. Rohrlich's formula of level $N$
  12. Explicit version of Bringmann--Kane--Löbrich--Ono--Rolen's theorem
  13. Bringmann--Kane's formula of level $N$
  14. Twisted traces of CM values of the Eisenstein series

Key Result

Theorem 1.1

For a meromorphic modular form $f$ of weight $k$ and a smooth modular form $g$ of weight $2$ (without singularities on $\mathbb{H}$) on $\Gamma_0(N)$, we have that if the limits $C_{1, \rho}(f,g), C_{2, \rho}(f,g)$ defined by exist. Here, we set $\omega_z \coloneqq \#\overline{\Gamma_0(N)}_z$ and $\overline{\Gamma_0(N)} \coloneqq \Gamma_0(N)/\{\pm I\}$.

Figures (1)

  • Figure 1: A fundamental domain $\mathcal{F}_0^*(4)$ and neighborhoods for a finite set $\{z_1, z_2, z_3\}$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Example 1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Corollary 4.3: CohenStromberg2017
  • ...and 16 more