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Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces

Part of: Discontinuous groups and automorphic forms Partial differential equations on manifolds; differential operators Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  26 September 2025

Louis Soares Open the ORCID record for Louis Soares [Opens in a new window]
Louis Soares*
Affiliation:
Independent Researcher, Arbon, Switzerland
*
Email: louis.soares@gmx.ch
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Abstract

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and let $X=\Gamma \backslash {\mathbb{H}}^2$ be the associated hyperbolic surface. We consider the family of Hecke congruence coverings of $X$, which we denote as usual by $ X_0(q) = \Gamma _0(q)\backslash {\mathbb{H}}^2$. Conditional on the Lindelöf Hypothesis for quadratic L-functions, we establish a uniform and explicit spectral gap for the Laplacian on $ X_0(q)$ for “almost” all prime levels $q$. Assuming the generalized Riemann hypothesis for quadratic $L$-functions, we obtain an even larger spectral gap.

MSC classification

Primary: 58J50: Spectral problems; spectral geometry; scattering theory
Secondary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas 11F06: Structure of modular groups and generalizations; arithmetic groups

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Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 41
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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