Perturbation of scattering poles for hyperbolic surfaces and central values of \(L\)-series
A scattering pole is a pole of the analytic continuation of the determinant of the scattering matrix to \(\Re(s)<1/2\). From \textit{R. S. Phillips} and \textit{P. Sarnak} [J. Am. Math. Soc. 5, 1-32 (1992; Zbl 0743.30039)] it is known that embedded eigenvalues become scattering poles under perturbation. Here under certain hypotheses on the Eisenstein series, integral formulas for the first variation of the weighted mean of the scattering pole are given. Assume the discrete group is \(\Gamma _0 (q)\), for prime \(q\), and the perturbations are generated by a holomorphic cusp form \(g\) of weight 2 for this group, assuming that \(g\) is a Hecke eigenform with real coefficients and eigenvalue \(\varepsilon _q \) for the Hecke involution. Theorem 1.5 relates the nonvanishing of the \(L\)-series of \(g\) at \(s = 1\) to the movement of the scattering poles away from \(\Re(s)=1/4\). In particular, it says that if \(L(g,1) \neq 0\), the scattering poles move. Examples of \(q\) satisfying the condition are \(q=11, 17, 19, 67, 73\). Theorem 1.9 assumes the Riemann hypothesis and a further condition on at least one of the 2 holomorphic cusp forms involved to show that a positive proportion of the scattering poles move off the line \(\Re(s)=1/4\).
scientific article; zbMATH DE number 1463081
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