Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces

1. Problem Statement and Context

The Spectral Gap Problem:
The paper addresses the distribution of eigenvalues of the Laplace–Beltrami operator ($\Delta$) on hyperbolic surfaces. Specifically, it investigates the spectral gap—the gap between the first non-zero eigenvalue and the bottom of the spectrum—for congruence covers of infinite-area hyperbolic surfaces.

• Background: For finite-area modular surfaces ($SL_2(\mathbb{Z})\backslash \mathbb{H}^2$), Selberg's famous $3/16$ theorem (and the conjectured $1/4$) establishes a uniform spectral gap.
• The "Thin" Group Challenge: This paper focuses on Schottky subgroups $\Gamma \subset SL_2(\mathbb{Z})$, which are "thin" groups (infinite index). For these groups, the associated surface $X = \Gamma \backslash \mathbb{H}^2$ has infinite area.
  • The spectrum of $\Delta_X$ is sparse. If the Hausdorff dimension $\delta$ of the limit set satisfies $\delta > 1/2$, the smallest eigenvalue is $\lambda_0 = \delta(1-\delta)$.
  • The goal is to determine if the Hecke congruence covers $X_0(p) = \Gamma_0(p) \backslash \mathbb{H}^2$ (where $\Gamma_0(p)$ is defined by $c \equiv 0 \pmod p$) possess a uniform spectral gap above $\lambda_0$.

Previous Limitations:

• Gamburd (2000): Proved a gap of $5/36$ for $\delta > 5/6$.
• Bourgain–Gamburd–Sarnak (2011): Established a gap for any $\delta > 1/2$, but the gap size $c(\Gamma)$ was non-explicit (dependent on the group in a non-computable way).
• Calderón–Magee (2022): Improved the gap for Schottky groups to $\delta/6 + 2/3$ (for $\delta > 4/5$) with an explicit constant, but the bound was still not optimal for large $\delta$.

The Goal:
To establish an explicit, uniform spectral gap for Hecke congruence covers of arithmetic Schottky groups that improves upon previous bounds, specifically targeting a gap size of $5/6\delta$.

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2. Methodology

The proof combines spectral theory, transfer operator theory, and analytic number theory.

A. Spectral Theory and Transfer Operators

1. Twisted Selberg Zeta Functions:
   New eigenvalues on the cover $X_0(p)$ correspond to zeros of the twisted Selberg zeta function $Z_\Gamma(s, \lambda_p^0)$, where $\lambda_p^0$ is the induced representation of the trivial representation on $\Gamma_0(p)$ minus the trivial representation on $\Gamma$.
   $$Z_{\Gamma_0(p)}(s) = Z_\Gamma(s) Z_\Gamma(s, \lambda_p^0)$$
   The zeros of $Z_\Gamma(s, \lambda_p^0)$ in the interval $[1/2, \delta]$ correspond to new eigenvalues.

2. Refined Transfer Operators:
   The author utilizes the Fredholm determinant identity linking the zeta function to transfer operators:
   $$Z_\Gamma(s, \lambda_p^0) = \det(1 - L_{s, \lambda_p^0})$$
   To obtain explicit estimates, the paper employs refined transfer operators $L_{\tau, s, \lambda_p^0}$ introduced by Dyatlov and Zworski. These operators depend on a "resolution parameter" $\tau$.

   • Key Insight: The zeros of the standard transfer operator are also zeros of the refined operator.
   • Jensen's Formula: The number of zeros $N_p(\sigma)$ in an interval is estimated using Jensen's formula, which bounds the zero count by the logarithm of the operator's norm (specifically the Hilbert–Schmidt norm).

B. Number-Theoretic Estimates

The core difficulty lies in estimating the Hilbert–Schmidt norm of the operator $L_{\tau, s, \lambda_p^0}$ summed over primes $p$.

1. Trace Formula: The Hilbert–Schmidt norm involves a sum of traces of the representation $\lambda_p^0(\gamma)$.
   $$\sum_{p \sim x} \| L_{\tau, s, \lambda_p^0} \|_{HS}^2 \sim \sum_{p \sim x} \sum_{\gamma} \text{tr}(\lambda_p^0(\gamma)) \dots$$
2. Kronecker Symbol Reduction:
   Using Lemma 3.6, the trace $\text{tr}(\lambda_p^0(\gamma))$ is shown to be related to the Kronecker symbol (a quadratic character):
   $$\text{tr}(\lambda_p^0(\gamma)) = \left( \frac{\text{tr}(\gamma)^2 - 4}{p} \right)$$
   provided $\gamma \not\equiv \pm I \pmod p$.
3. Generalized Riemann Hypothesis (GRH):
   The sum of these characters over primes is estimated using the GRH for quadratic L-functions.
   • Under GRH, the sum $\sum_{p \le x} \log(p) \left( \frac{d}{p} \right)$ is bounded by $O(x^{1/2+\epsilon})$.
   • This allows the author to bound the sum of the Hilbert–Schmidt norms over a dyadic range of primes.

C. Optimization

The final estimate depends on the choice of the resolution parameter $\tau$. The author optimizes $\tau$ (specifically $\tau \approx x^{-3/2\delta}$) to balance the error terms arising from the operator norms and the number-theoretic bounds.

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3. Key Contributions and Results

Main Theorem (Theorem 1.4)

Let $\Gamma \subset SL_2(\mathbb{Z})$ be a Schottky group with Hausdorff dimension $\delta > 3/4$. Assuming the Generalized Riemann Hypothesis (GRH) for quadratic L-functions, there exists a density-one subset of primes $P$ such that for every $p \in P$:
$$\Omega(X_0(p)) \cap \left[ \frac{5}{6}\delta + \eta, \delta \right] \stackrel{m}{=} \Omega(X) \cap \left[ \frac{5}{6}\delta + \eta, \delta \right]$$
where $\Omega(X)$ denotes the multiset of spectral parameters $s$ corresponding to eigenvalues.

Implications:

• Explicit Gap: The spectral gap is explicitly given as $\frac{5}{6}\delta + \eta$.
• Improvement: This improves upon:
  • Gamburd's $5/36$ (which corresponds to $\delta \approx 1$).
  • Calderón–Magee's $\delta/6 + 2/3$.
  • Since $\frac{5}{6}\delta > \frac{\delta}{6} + \frac{2}{3}$ for $\delta > 1$, the new bound is strictly larger (better) for high-dimensional limit sets.
• Density: The result holds for "almost all" primes. The number of exceptional primes up to $x$ is bounded by $O_\epsilon(x^{1 - \frac{3}{\delta}(\sigma - \frac{5}{6}\delta) + \epsilon})$.

Technical Lemmas

• Lemma 3.6 (Trace Formula): Establishes the precise connection between the trace of the induced representation and the Kronecker symbol $\left( \frac{\text{tr}(\gamma)^2 - 4}{p} \right)$.
• Proposition 3.9: Provides the crucial bound on the sum of squared Hilbert–Schmidt norms over primes, relying on GRH to achieve the $x^{1/2}$ saving.

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4. Significance

1. Explicitness: Unlike previous results for thin groups (e.g., Bourgain–Gamburd–Sarnak), this paper provides an explicit formula for the spectral gap. This is crucial for applications in dynamics and number theory where quantitative bounds are required.
2. Optimality for Schottky Groups: The bound $\frac{5}{6}\delta$ is a significant improvement over previous explicit bounds for Schottky groups, pushing the known spectral gap closer to the theoretical limit of $1/4$ (in terms of $\lambda$) or $\delta$ (in terms of $s$).
3. Methodological Synthesis: The paper successfully synthesizes:
   • Geometric Group Theory: Using Schottky groups and limit sets.
   • Functional Analysis: Using refined transfer operators and Fredholm determinants.
   • Analytic Number Theory: Using character sums over primes and GRH.
4. Conditional Nature: The result is conditional on GRH. The author notes that unconditional bounds for character sums over primes currently have too high a dependency on the conductor to yield useful results in this context, highlighting a bottleneck in current analytic number theory.

Conclusion

Louis Soares provides a breakthrough in the spectral theory of infinite-area hyperbolic surfaces by proving that Hecke congruence covers of arithmetic Schottky surfaces possess a large, explicit spectral gap for almost all primes, assuming GRH. This result refines the understanding of how "thin" groups behave under congruence subgroups and offers a concrete quantitative tool for future research in arithmetic quantum chaos and expander graphs.