Determination of cusp forms by central values of Rankin-Selberg \(L\)-functions (Q392981)

Let \(\{u_j\mid j\in\mathbb{N}\}\) be an orthogonal basis of the space of the Maass cusp forms of the group \(\mathrm{SL}(2,\mathbb{Z})\), and let \(\lambda_j(n)\) be the \(n\)-th normalized Fourier coefficient of \(u_j\) and let \[ f(z)= \sum^\infty_{n=1} \lambda_f(n) n^{(k-1)/2} e^{2\pi inz} \] be a normalized holomorphic Hecke eigenform for \(\mathrm{SL}(2,\mathbb{Z})\) of weight \(k\). The author proves that, for any rational prime \(p\), the following asymptotic formula holds: \[ \sum^\infty_{j=1} \omega_j(T) \lambda_j(p) L(f\otimes u_j,1/2)= \lambda_f(p) p^{-1/2} T^2\log T\Biggl(1+ O_{k,p}\Biggl({\log\log T\over\log T}\Biggr)\Biggr) \] as \(T\to\infty\), where \(\omega_j: \mathbb{R}^*_+\to \mathbb{R}^*_+\) is a suitable weight-function and \(L(f\otimes u_j,s)\) is the Rankin convolution \(L\)-function. Making use of that asymptotic formula and the strong multiplicity one theorem, the author concludes that if, for two normalized holomorphic Hecke \(\mathrm{SL}(2,\mathbb{Z})\)-eigenforms \(f_1\) of weight \(k_1\) and \(f_2\) of weight \(k_2\), the relation \[ L(f_1\otimes u_j, 1/2)= L(f_2\otimes u_j, 1/2)\quad\text{for every }j\in\mathbb{N} \] is satisfied, then \(k_1=k_2\) and \(f_1=f_2\).