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The non-vanishing of Rankin-Selberg zeta-functions at special points. (English) Zbl 0595.10025

The Selberg trace formula and related topics, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 53, 51-95 (1986).
[For the entire collection see Zbl 0583.00006.]
To a real analytic cusp form f of weight zero and a holomorphic cusp form Q of even weight 2k, both for the congruence subgroup \(\Gamma_ 0(q)\), one may associate the Rankin-Selberg zeta-function \(L_ f\), for Re s large given by \[ L_ f(s)=\sum^{\infty}_{n=1}q(n) a(n) n^{-s}, \] with q(n) and a(n) the Fourier coefficients of Q and f. If \(s_ f(1-s_ f)\) is the eigenvalue on f of the Laplacian, one may ask whether \(L_ f(s_ f)\) is zero or not. In the case of \(2k=4\), R. S. Phillips and P. Sarnak, Invent. Math. 80, 339-364 (1985; Zbl 0558.10017)], related the vanishing of \(L_ f(s_ f)\) to the persistence of f as a cusp form under changes of the metric in the direction specified by Q.
In this paper this vanishing is studied asymptotically for those f which are automorphic for the full modular group. Q is fixed and f runs through an orthonormal basis \(\{f_ j\}\) of the cuspidal spectrum for PSL(2,\({\mathbb{Z}})\) in weight zero. Take \(s_ j=s_{f_ j}\), with Im \(s_ j>0\). Let A(T) be the number of \(f_ j\) with Im \(s_ j\leq T\) and \(L_{f_ j}(s_ j)\neq 0\). The main result is \[ A(T)\quad \gg \quad T (\log T)^{-2} ; \] it may be compared with the known result \(\#\{j: Im s_ j\leq T\}\sim T^ 2/12. \)
Conditional to two assumptions on the Fourier coefficients of real analytic cusp forms, including the Ramanujan-Petersson conjecture, it is shown that \[ A(T)\quad \gg \quad T^{9/8} (\log T)^{-18}. \] The proofs are complicated. Involved in it are approximate functional equations for Rankin-Selberg zeta-functions and the spectral theory of modular forms.
Reviewer: R.W.Bruggeman

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F11 Holomorphic modular forms of integral weight