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A Hyperbolic Analogue of the Rademacher Symbol

Toshiki Matsusaka

TL;DR

The paper develops a hyperbolic analogue of the Rademacher symbol by constructing and analyzing Ψ_γ(σ) for hyperbolic elements γ, σ in SL_2(Z). It leverages weight-2 Eisenstein series E_γ(z,s) and their s→0 limits to connect to harmonic, polar harmonic, and locally harmonic Maass forms, introducing the holomorphic F_γ(z) and the hyperbolic Dedekind symbol Φ_γ(σ). Two explicit formulas for Ψ_γ(σ) are derived: a geometric counting form expressing Ψ_γ(σ) via intersections of geodesics with vertical lines, and a continued-fraction formula Ψ_γ(σ) = -2( Σ min(a_i,b_j) − ψ_γ(σ) ) that is computable from the continued fractions of γ and σ. The results tie Ψ_γ(σ) to modular-knot linking numbers, provide finite computable sums, and extend cocycle relations for the hyperbolic Dedekind symbol, enriching the arithmetic and geometric understanding of hyperbolic analogues in the theory of modular forms.

Abstract

One of the most famous results of Dedekind is the transformation law of $\log Δ(z)$. After a half-century, Rademacher modified Dedekind's result and introduced an $\mathrm{SL}_2(\mathbb{Z})$-conjugacy class invariant (integer-valued) function $Ψ(γ)$ called the Rademacher symbol. Inspired by Ghys' work on modular knots, Duke-Imamoglu-Tóth (2017) constructed a hyperbolic analogue of the symbol. In this article, we study their hyperbolic analogue of the Rademacher symbol $Ψ_γ(σ)$ and provide its two types of explicit formulas by comparing it with the classical Rademacher symbol. In association with it, we contrastively show Kronecker limit type formulas of the parabolic, elliptic, and hyperbolic Eisenstein series. These limits give harmonic, polar harmonic, and locally harmonic Maass forms of weight 2.

A Hyperbolic Analogue of the Rademacher Symbol

TL;DR

The paper develops a hyperbolic analogue of the Rademacher symbol by constructing and analyzing Ψ_γ(σ) for hyperbolic elements γ, σ in SL_2(Z). It leverages weight-2 Eisenstein series E_γ(z,s) and their s→0 limits to connect to harmonic, polar harmonic, and locally harmonic Maass forms, introducing the holomorphic F_γ(z) and the hyperbolic Dedekind symbol Φ_γ(σ). Two explicit formulas for Ψ_γ(σ) are derived: a geometric counting form expressing Ψ_γ(σ) via intersections of geodesics with vertical lines, and a continued-fraction formula Ψ_γ(σ) = -2( Σ min(a_i,b_j) − ψ_γ(σ) ) that is computable from the continued fractions of γ and σ. The results tie Ψ_γ(σ) to modular-knot linking numbers, provide finite computable sums, and extend cocycle relations for the hyperbolic Dedekind symbol, enriching the arithmetic and geometric understanding of hyperbolic analogues in the theory of modular forms.

Abstract

One of the most famous results of Dedekind is the transformation law of . After a half-century, Rademacher modified Dedekind's result and introduced an -conjugacy class invariant (integer-valued) function called the Rademacher symbol. Inspired by Ghys' work on modular knots, Duke-Imamoglu-Tóth (2017) constructed a hyperbolic analogue of the symbol. In this article, we study their hyperbolic analogue of the Rademacher symbol and provide its two types of explicit formulas by comparing it with the classical Rademacher symbol. In association with it, we contrastively show Kronecker limit type formulas of the parabolic, elliptic, and hyperbolic Eisenstein series. These limits give harmonic, polar harmonic, and locally harmonic Maass forms of weight 2.
Paper Structure (14 sections, 40 theorems, 253 equations)

This paper contains 14 sections, 40 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. Eisenstein series of weight $2$
  3. Parabolic case
  4. Elliptic case
  5. Hyperbolic case
  6. Hyperbolic Rademacher symbol
  7. Definitions
  8. First explicit formula for $\Psi_\gamma(\sigma)$
  9. Hyperbolic Dedekind symbol
  10. Cocycle conditions
  11. Second explicit formula for $\Psi_\gamma(\sigma)$
  12. Continued fractions
  13. Hooley's estimate
  14. Eisenstein series of weight $0$

Key Result

Theorem 1.1

For a primitive hyperbolic element $\gamma = \left(\right) \in \Gamma$ with $c > 0$ and $a+d > 2$, we define the hyperbolic Eisenstein series $E_\gamma(z,s)$. Then the limit of $E_\gamma(z,s)$ as $s \to 0^+$ exists, and we have for some holomorphic function $F_\gamma(z)$. The definitions of some notations are given in s3-3. This is the $i\infty$-component of a locally harmonic Maass form of weigh

Theorems & Definitions (70)

  • Theorem 1.1: \ref{['limit-formula-for-hyperbolic']}
  • Theorem 1.2: \ref{['main-theorem4']}
  • Theorem 1.3: \ref{['main-theorem1']}
  • Theorem 1.4: \ref{['explicit-psi']}
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Corollary 2.5
  • ...and 60 more