Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces

Let Γ be a Schottky subgroup of SL₂(Z) and let X=Γ H² be the associated hyperbolic surface. We consider the family of Hecke congruence coverings of X , which we denote as usual by X₀(q) = Γ ₀(q) H² . Conditional on the Lindelöf Hypothesis for quadratic L-functions, we establish a uniform and explicit spectral gap for the Laplacian on X₀(q) for “almost” all prime levels q . Assuming the generalized Riemann hypothesis for quadratic L -functions, we obtain an even larger spectral gap.