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Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces

Louis Soares

TL;DR

The paper addresses explicit spectral gaps for Laplacians on Hecke congruence covers X_0(p) of thick arithmetic Schottky surfaces X = Γ\H^2, proving uniform gaps for a density-one set of primes under GRH for quadratic L-functions. It builds a framework based on the Venkov–Zograf induction formula to relate new resonances to zeros of twisted Selberg zeta functions Z_Γ(s, λ_p^0), and then leverages refined transfer operators L_{τ,s,ρ} together with Jensen's formula and Hilbert–Schmidt norm estimates to count zeros. A key technical achievement is a sharp average bound over primes for the Hilbert–Schmidt norms of the refined operators, which reduces the zero-counting problem to efficient character-sum estimates under GRH. The result yields an explicit lower bound on the spectrum in a band close to the top of the critical strip for almost all primes and demonstrates how refined transfer-operator techniques can produce explicit spectral gaps for thin, yet thick, hyperbolic surfaces, with potential implications for dynamics and arithmetic on infinite-area surfaces.

Abstract

Let $Γ$ be a Schottky subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and let $X=Γ\backslash \mathbb{H}^2$ be the associated hyperbolic surface. Conditional on the generalized Riemann hypothesis for quadratic $L$-functions, we establish a uniform and explicit spectral gap for the Laplacian on the Hecke congruence covers $ X_0(p) = Γ_0(p)\backslash \mathbb{H}^2$ of $X$ for "almost" all primes $p$, provided the limit set of $Γ$ is thick enough.

Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces

TL;DR

The paper addresses explicit spectral gaps for Laplacians on Hecke congruence covers X_0(p) of thick arithmetic Schottky surfaces X = Γ\H^2, proving uniform gaps for a density-one set of primes under GRH for quadratic L-functions. It builds a framework based on the Venkov–Zograf induction formula to relate new resonances to zeros of twisted Selberg zeta functions Z_Γ(s, λ_p^0), and then leverages refined transfer operators L_{τ,s,ρ} together with Jensen's formula and Hilbert–Schmidt norm estimates to count zeros. A key technical achievement is a sharp average bound over primes for the Hilbert–Schmidt norms of the refined operators, which reduces the zero-counting problem to efficient character-sum estimates under GRH. The result yields an explicit lower bound on the spectrum in a band close to the top of the critical strip for almost all primes and demonstrates how refined transfer-operator techniques can produce explicit spectral gaps for thin, yet thick, hyperbolic surfaces, with potential implications for dynamics and arithmetic on infinite-area surfaces.

Abstract

Let be a Schottky subgroup of and let be the associated hyperbolic surface. Conditional on the generalized Riemann hypothesis for quadratic -functions, we establish a uniform and explicit spectral gap for the Laplacian on the Hecke congruence covers of for "almost" all primes , provided the limit set of is thick enough.
Paper Structure (24 sections, 16 theorems, 165 equations, 2 figures)

This paper contains 24 sections, 16 theorems, 165 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spectral gaps for congruence covers and main result
  3. Thick arithmetic Schottky groups
  4. Outline of proof
  5. Organization of the paper
  6. Notation
  7. Acknowledgments
  8. Preliminaries
  9. Hyperbolic geometry
  10. Hyperbolic surfaces and Fuchsian groups
  11. Spectral theory of infinite-area hyperbolic surfaces
  12. Twisted Selberg zeta function
  13. Venkov--Zograf induction formula
  14. Schottky groups
  15. Combinatorial notation for words
  16. ...and 9 more sections

Key Result

Theorem 1.1

For every finitely generated subgroup $\Gamma\subset \mathrm{SL}_2(\mathbb{Z})$ with $\delta > \frac{5}{6}$ and for every large enough prime $p$ we have where for any two multisets $A$ and $B$ we write $A \stackrel{\text{m}}{=} B$ if and only if the multiplicities of all elements are the same on both sides.

Figures (2)

  • Figure 1: Distribution of resonances for infinite-area $\Gamma\backslash\mathbb{H}^2$ in the case $\delta > \frac{1}{2}$
  • Figure 2: A configuration of Schottky disks and isometries with $m=3$

Theorems & Definitions (26)

  • Theorem 1.1: Gamburd Gamburd1
  • Theorem 1.2: Bourgain--Gamburd--Sarnak BGS
  • Theorem 1.3: Calderón--Magee caldmagee2023
  • Theorem 1.4: Main theorem
  • Lemma 2.1
  • Lemma 2.2: Basic distortion estimates
  • Lemma 2.3: Estimates for $Z(\tau)$ and $Y(\tau)$
  • proof
  • Lemma 2.4: Hilbert--Schmidt norm
  • proof
  • ...and 16 more