On effective determination of modular forms by twists of critical \(L\)-values
Given a holomorphic newform \(f\), one would like to know if sufficiently many twisted \(L\)-values \(L(1/2, f \otimes \chi_d)\) suffice to recover \(f\). The answer is affirmative by a result of \textit{W. Luo} and \textit{D. Ramakrishnan} [Invent. Math. 130, No. 2, 371--398 (1997; Zbl 0905.11024)], and the present paper gives a uniform quantitative statement in terms of the level and the weight: more precisely, if \(f \in S_{2k}(q)\), \(g \in S_{2k'}(q')\) and \(L(1/2, f \otimes \chi_d) = L(1/2, g\otimes \chi_d)\) for all \(d \ll \max(q, q')^{11/2+\varepsilon} \max(k, k')^{4+\varepsilon}\), then \(f=g\). This result is based on two auxiliary results of independent interest: 1) The author derives a uniform asymptotic formula for a mean value roughly of the shape \[ \sum_d \chi_d(l) L(1/2, f \otimes \chi_d) F(d/Y) \] for a prime \(l\) and \(F\) a smooth test function. The main term contains the Hecke eigenvalue \(\lambda_f(l)\). 2) It follows from Rankin-Selberg theory that \(|\lambda_f(l) - \lambda_g(l)|\) cannot always be small if \(f\not=g\). [Reviewer's remark: It could have been mentioned in the paper that double Dirichlet series yield a stronger error in the asymptotic formula for the mean value in the Y-aspect, see \textit{J. Hoffstein} and \textit{W. Luo} [Math. Res. Lett. 4, No. 2-3, 435--442, appendix 443--444 (1997; Zbl 0893.11034)]. It is unclear (and perhaps even unlikely), however, if the uniformity in the level and the weight of \(f\) could be made sufficiently good to improve the main theorem of the paper.]
scientific article; zbMATH DE number 5721509
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