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The Prime Geodesic Theorem for the Picard Orbifold

Ikuya Kaneko

TL;DR

This work establishes a sharp prime geodesic theorem for the Picard orbifold $\mathrm{PSL}_{2}(\mathbb{Z}[i])\backslash\mathbb{H}^{3}$ with error terms governed by subconvexity in quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. The authors combine an unconditional analysis yielding $\mathcal{E}_{\Gamma}(x) \ll x^{\tfrac{255}{172}+\varepsilon}$ (equivalently $1.483+\varepsilon$) with a conditional bound $\mathcal{E}_{\Gamma}(x) \ll x^{\tfrac{245}{172}+\varepsilon}$ under generalized Lindelöf, by resolving the mean-Lindelöf hypothesis in this setting via a pre-Kuznetsov approach and Landau-type averaging. A central innovation is to integrate a suite of tools—mean-square Fourier-coefficient asymptotics on $\Gamma\backslash\mathbb{H}^{3}$, a 3D Brun–Titchmarsh-type refinement, a limiting-regime exponent-pair framework over $\mathbb{Q}(i)$, a symplectic zero-density bound for quadratic characters, and spectral–arithmetic averaging over lattices—without appealing to a full spectral large sieve. The resulting results interpolate between unconditional subconvexity bounds and the conjectural Lindelöf regime, advancing the understanding of 3D prime geodesic behavior and highlighting the deep connections between automorphic forms, $L$-functions, and hyperbolic geometry. The methods have potential applicability to other Kleinian groups and higher-rank analogues via rapid decay Poincaré series and Kuznetsov-type identities.

Abstract

We establish the prime geodesic theorem for the Picard orbifold $\mathrm{PSL}_{2}(\mathbb{Z}[i]) \backslash \mathbb{H}^{3}$, wherein the error term shrinks proportionally to improvements in the subconvex exponent for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. Our result sheds light on a venerable conjecture by attaining an unconditional exponent of $1.483$ and a conditionally superior exponent of $1.425$ under the generalised Lindelöf hypothesis. The argument synthesises, among other elements, the complete resolution of Koyama's (2001) mean Lindelöf hypothesis over $\mathbb{Q}(i)$, an improved Brun-Titchmarsh-type theorem over short intervals, a bootstrapped multiplicative exponent pair in the limiting regime, and a zero density theorem for the symplectic family of quadratic characters. Notably, despite the theoretical strength of our manifestations towards the mean Lindelöf hypothesis, the fundamental toolbox relies exclusively on the optimal mean square asymptotics for the Fourier coefficients of Maass cusp forms via the pre-Kuznetsov formula.

The Prime Geodesic Theorem for the Picard Orbifold

TL;DR

This work establishes a sharp prime geodesic theorem for the Picard orbifold with error terms governed by subconvexity in quadratic Dirichlet -functions over . The authors combine an unconditional analysis yielding (equivalently ) with a conditional bound under generalized Lindelöf, by resolving the mean-Lindelöf hypothesis in this setting via a pre-Kuznetsov approach and Landau-type averaging. A central innovation is to integrate a suite of tools—mean-square Fourier-coefficient asymptotics on , a 3D Brun–Titchmarsh-type refinement, a limiting-regime exponent-pair framework over , a symplectic zero-density bound for quadratic characters, and spectral–arithmetic averaging over lattices—without appealing to a full spectral large sieve. The resulting results interpolate between unconditional subconvexity bounds and the conjectural Lindelöf regime, advancing the understanding of 3D prime geodesic behavior and highlighting the deep connections between automorphic forms, -functions, and hyperbolic geometry. The methods have potential applicability to other Kleinian groups and higher-rank analogues via rapid decay Poincaré series and Kuznetsov-type identities.

Abstract

We establish the prime geodesic theorem for the Picard orbifold , wherein the error term shrinks proportionally to improvements in the subconvex exponent for quadratic Dirichlet -functions over . Our result sheds light on a venerable conjecture by attaining an unconditional exponent of and a conditionally superior exponent of under the generalised Lindelöf hypothesis. The argument synthesises, among other elements, the complete resolution of Koyama's (2001) mean Lindelöf hypothesis over , an improved Brun-Titchmarsh-type theorem over short intervals, a bootstrapped multiplicative exponent pair in the limiting regime, and a zero density theorem for the symplectic family of quadratic characters. Notably, despite the theoretical strength of our manifestations towards the mean Lindelöf hypothesis, the fundamental toolbox relies exclusively on the optimal mean square asymptotics for the Fourier coefficients of Maass cusp forms via the pre-Kuznetsov formula.
Paper Structure (29 sections, 32 theorems, 221 equations, 2 figures, 1 table)

This paper contains 29 sections, 32 theorems, 221 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Statement of results
  4. Overview of the manoeuvres
  5. Organisation of the paper
  6. Preliminaries
  7. Asymptotic notation
  8. Analytic notation
  9. Arithmetic notation
  10. Algebraic notation
  11. Characters and -functions
  12. Gaussian Zagier -series
  13. Gaussian Kloosterman sums
  14. Mean Lindelöf hypothesis
  15. Prior breakthroughs
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $\Gamma = \mathrm{PSL}_{2}(\mathbb{Z}[i])$. For a generator $D$ of the fundamental discriminant of a quadratic extension of $\mathbb{Q}(i)$, let $\chi_{D} = (\frac{D}{\cdot})$ denote the primitive quadratic character modulo $\mathfrak{d} \unlhd \mathbb{Z}[i]$, and let $\vartheta \in [0, \frac{1} for some arbitrarily large $\vartheta^{\prime} \geq 0$. Then we have for any $\varepsilon > 0$ that

Figures (2)

  • Figure 1: A comparison of \ref{['eq:Balkanova-Frolenkov']}, \ref{['eq:unconditional']}, and \ref{['eq:conditional']} as $\vartheta \in [0, \frac{1}{4})$ varies
  • Figure 2: A comparison between \ref{['eq:Balog-et-al']} and \ref{['eq:zero-density-estimate']} as $\sigma \in [\frac{1}{2}, 1]$ varies

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • ...and 57 more