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arxiv: 2305.02228 · v2 · submitted 2023-05-03 · 🧮 math.SP · math.DG· math.NT

Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces

Louis Soares This is my paper

Pith reviewed 2026-05-24 08:54 UTC · model grok-4.3

classification 🧮 math.SP math.DGmath.NT
keywords spectral gapSchottky surfacesHecke congruence coversLaplacianhyperbolic surfacesgeneralized Riemann hypothesisarithmetic groups
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The pith

Conditional on GRH for quadratic L-functions, Hecke covers of Schottky surfaces have uniform explicit spectral gaps for almost all primes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to show that, assuming the generalized Riemann hypothesis for quadratic L-functions, the Hecke congruence covers of an arithmetic Schottky hyperbolic surface have a uniform and explicit positive lower bound on the first eigenvalue of the Laplacian, for almost all primes. This bound holds when the limit set of the Schottky group is sufficiently thick. A reader would care because such spectral gaps determine how quickly waves or particles mix on the surface and control the geometry of the covers. The result is uniform in the prime, making it applicable to infinite families of surfaces.

Core claim

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)∖H² of X for 'almost' all primes p, provided the limit set of Γ is thick enough.

What carries the argument

The Hecke congruence covers X_0(p) of the Schottky surface X, combined with the thickness condition on the limit set of Γ.

Load-bearing premise

The generalized Riemann hypothesis for quadratic L-functions holds.

What would settle it

A Schottky group with thick limit set and a prime p where the first Laplacian eigenvalue on X_0(p) falls below the explicit bound despite the GRH being true.

Figures

Figures reproduced from arXiv: 2305.02228 by Louis Soares.

Figure 1
Figure 1. Figure 1: Distribution of resonances for infinite-area Γ\H2 in the case δ > 1 2 2.4. Twisted Selberg zeta function. Given a finitely generated Fuchsian group Γ < PSL2(R), the set of prime periodic geodesics on X = Γ\H2 is bijective to the set [Γ]prim of Γ-conjugacy classes of primitive hyperbolic ele￾ments in Γ. We denote by ℓ(γ) the length of the geodesic corresponding to the conjugacy class [γ] ∈ [Γ]prim. The Selb… view at source ↗
Figure 2
Figure 2. Figure 2: A configuration of Schottky disks and isometries with m = 3 Throughout the rest of this paper, Γ is a non-elementary Schottky group with Schottky data D1, . . . , D2m and γ1, . . . , γ2m as above. This assumption will not be repeated in the sequel. 2.7. Combinatorial notation for words. Let Γ be a Schottky group as in §2.6. We will follow the combinatorial notation of Dyatlov–Zworski [8] for indexing eleme… view at source ↗
read the original 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. 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) = \Gamma_0(p)\backslash \mathbb{H}^2$ of $X$ for "almost" all primes $p$, provided the limit set of $\Gamma$ is thick enough.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper claims that, conditional on the generalized Riemann hypothesis for quadratic L-functions, there is a uniform and explicit spectral gap for the Laplacian on the Hecke congruence covers X_0(p) = Γ_0(p)∖H² of an arithmetic Schottky surface X = Γ∖H² for almost all primes p, provided the limit set of Γ is thick enough.

Significance. If the result holds, it offers an explicit spectral gap under GRH for these covers, which is a valuable contribution to spectral geometry of hyperbolic surfaces. The explicit and uniform nature strengthens the result compared to non-explicit bounds. The conditional framing is appropriate and transparent.

minor comments (1)
  1. [Abstract / §1] The abstract and introduction use the phrase 'almost all primes p' without specifying the density or exceptional set size; a precise statement of the main theorem (likely in §1 or the statement of Theorem X) would clarify the quantitative strength of the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their supportive summary, positive assessment of significance, and recommendation of minor revision. The report lists no specific major comments.

Circularity Check

0 steps flagged

No significant circularity; result is conditional on external GRH

full rationale

The paper's central claim is explicitly conditional on the generalized Riemann hypothesis for quadratic L-functions (an external, unproven conjecture) together with a thickness hypothesis on the limit set of Γ. The abstract states both prerequisites upfront, and the result is framed as holding for almost all primes p only under those assumptions. No load-bearing step reduces by construction to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain whose cited result itself collapses to the target claim. The derivation therefore remains non-circular even though it is conditional.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the external GRH assumption for quadratic L-functions and the geometric thickness condition on the limit set. No free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Generalized Riemann hypothesis for quadratic L-functions
    The spectral gap result is stated as conditional on this hypothesis.

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Reference graph

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