Nonvanishing of \(L\)-values and the Weyl law. (English) Zbl 1003.11019

Let \(\Gamma \) be a Fuchsian group of the first kind, and let \(N_{\Gamma }(T)\) be the counting function (in terms of \(t\)) for the eigenvalues \(1/4+t^2\) of the hyperbolic Laplacian of the surface \(\Gamma \backslash \mathbb{H}\), where \(\mathbb{H}\) is the upper half-plane. The Weyl law is the claim \[ N_{\Gamma }(T) \sim (4\pi)^{-1}\mu (\Gamma \backslash \mathbb{H})T^2, \] where \(\mu \) denotes the hyperbolic area. It is shown that this is false for generic hyperbolic surfaces under the assumption that the eigenvalue multiplicities of the Laplacian on \(\Gamma _0(p)\backslash \mathbb{H}\), for a prime \(p\), are bounded. Owing to the work of R. Phillips and P. Sarnak [Invent. Math. 80, 339-364 (1985; Zbl 0558.10017)], this assertion can be reduced to non-vanishing of “critical” values of certain Rankin-Selberg zeta-functions. In a previous paper [Duke Math. J. 69, 411-425 (1993; Zbl 0789.11032)], the author had proved such a result, which is however slightly too weak for the present purpose. This is now sharpened so as to show the non-vanishing property in a positive proportion of cases.


11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11M41 Other Dirichlet series and zeta functions
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