We continue to develop the analytic Langlands program for curves over local fields initiated in our earlier papers, following a suggestion of Langlands and a work of Teschner. Namely, we study the Hecke operators which we introduced in those papers in the case of a projective line with parabolic structures at finitely many points for the group (PGL_2). We establish most of our conjectures in this case.
Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature (-1). For each (nin {mathbf {N}}), let (X_{n}) be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or (X_{n}) is an eigenvalue of the associated Laplacian operator (Delta _{X}) or (Delta _{X_{n}}). We say that an eigenvalue of (X_{n}) is new if it occurs with greater multiplicity than in X. We prove that for any (varepsilon >0), with probability tending to 1 as (nrightarrow infty ), there are no new eigenvalues of (X_{n}) below (frac{3}{16}-varepsilon ). We conjecture that the same result holds with (frac{3}{16}) replaced by (frac{1}{4}).
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