Special \(L\)-values of Rankin-Selberg convolutions
\textit{H. M. Stark} has recently proved that one single value \(L(E,1)\) (if \(\neq 0\)) determines the isogeny class of a modular elliptic curve \(E\) [Prog. Math. 139, 737--743 (1996; Zbl 0870.11036)]. \textit{K. Luo} and \textit{D. Ramakrishnan} [Invent. Math. 130, 371--398 (1997; Zbl 0905.11024); Determination of modular elliptic curves by Heegner points, Olga Taussky-Todd: In memoriam. Cambridge, MA: International Press. Pac. J. Math., Spec. Issue, 251--258 (1998; Zbl 1012.11052)] proved that the twisted central \(L\)-values \(L(f\otimes \chi_d,\frac 12)\) (or \(L'(f\otimes \chi_d,\frac 12)\)) determine a cuspidal normalized newform \(f\) uniquely, where \(d\) ranges over all fundamental discriminants and \(\chi_d(\cdot)= (\frac d\cdot)\). The author demonstrates the following result: Let \(f\in S_{2k}^{\text{new}}(N)\), \(g\in S_{2k'}^{\text{new}}(N)\) be two normalized newforms. Fix \(s_0\in [\frac 12, 1]\), and let \(c\) be a constant and \(l\) be a positive integer. If there exist infinitely many primes \(p\) such that \(L(f\otimes h,s_0)= cL(g\otimes h,s_0)\) for all normalized newforms \(h\in S_{2l}^{\text{new}}(p)\), then \(f=g\).
scientific article; zbMATH DE number 1337012