Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces

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.

31 pages. Improved text, incorporated suggestions, fixed several typos, added more references

References in corpus (2)

• Fourier dimension and spectral gaps for hyperbolic surfaces
• Fractal uncertainty for transfer operators