On the nonvanishing of Rankin-Selberg \(L\)-functions
\textit{R. S. Phillips} and \textit{P. Sarnak} [Invent. Math. 80, 339-364 (1985; Zbl 0558.10017)] have shown that if \(u_ i\) is a Maaß cusp form for the congruence group \(\Gamma_ 0 (N)\) and \(Q\) is a holomorphic cusp form of weight 4, the cusp form is annihilated under quasi-conformal deformation determined by \(Q\) if the Rankin-Selberg \(L\)-function \(L(Q \otimes u_ i,s)\) does not vanish at \(s=s_ j=1/2+it_ j\) corresponding to the eigenvalue \(\lambda_ j=s_ j(1-s_ j)\) of the non-Euclidean Laplacian on \(u_ j\). Here the author proves that \[ \#\{j \mid t_ j \leq T,\;L(Q \otimes u_ j,s) \neq 0\} \gg T^{2- \varepsilon}. \] \vskip3mm So, assuming that the multiplicity of the eigenvalue \(\lambda_ j\) is bounded by \(t^ \varepsilon_ j\), for generic \(\Gamma \subset SL(2,\mathbb{R})\), the determinant of the scattering matrix is a meromorphic function of order 2. This contrasts with the case of congruence subgroups when the order is 1. The paper makes use of [\textit{J.-M. Deshouillers} and \textit{H. Iwaniec}, Contemp. Math. 53, 51-95 (1986; Zbl 0595.10025)]. The Deshouillers and Iwaniec paper contains a nonvanishing result with \(2-\varepsilon\) replaced by \(1-\varepsilon\).
scientific article; zbMATH DE number 179197