Determining cusp forms by central values of Rankin-Selberg \(L\)-functions (Q708266)

It has been an interesting and important subject to determine modular forms by central values of \(L\)-functions. Let \(H_k(1)\) denote a Hecke basis of the space of holomorphic cusp forms of weight \(k\) for \(\mathrm{SL}_2(\mathbb Z)\). In [Math. Ann. 345, No. 4, 843--857 (2009; Zbl 1234.11065)] \textit{S. Ganguly, J. Hoffstein} and \textit{J. Sengupta} proved that if \(g\in H_l(1)\) and \(g'\in H_{l'}(1)\) with \[ L(1/2, f\otimes g)=L(1/2, f\otimes g') \] for all \(f\in H_k (1)\) for infinitely many \(k\), then \(l=l'\) and \(g=g'\). Now let \(g\) and \(g'\) be normalized Hecke Maass cusp forms for \(SL_2(\mathbb Z)\) with Laplace eigenvalues \(\frac{1}{4}+\nu^2\) and \(\frac{1}{4}+{\nu'}^2\) respectively. In this paper under review the author proves that if \[ L(1/2, f\otimes g)=L(1/2, f\otimes g') \] for all \(f\in H_k (1)\) for infinitely many \(k\), then \(\nu=\nu'\) and \(g=g'\).