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. 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.