Rankin-Selberg \(L\)-functions in the level aspect.
A well-known problem in the theory of automorphic forms is to study the size of \(L\)-values, especially on the critical line. The ``convexity'' bound is obtained by combining the absolute convergence to the right with the functional equation and then employing the Phragmen-Lindelöf method. One seeks estimates that break this convexity bound. For example, \textit{W. Duke, J. Friedlander} and \textit{H. Iwaniec} [Invent. Math. 115, 209--217 (1994; Zbl 0791.11049)] obtained such an estimate for the second moment of the quadratic twists of the family of primitive cuspidal newforms on \(\Gamma_0(q)\) with trivial nebentypus and weight \(k\) (denoted \(S_k^*(q)\)). In this paper the authors give asymptotic estimates for the modulus of the Rankin-Selberg \(L\)-functions (and their derivatives) \(L^{(j)}(f\otimes g,1/2+it)\). Here \(g\) is a fixed modular form on \(\Gamma_1(D)\), \(D\) square-free, which is either a primitive cuspidal holomorphic newform of integral weight or a non-exceptional Maass form of weight zero, and the estimate obtained is valid for all \(f\in S_k^*(q)\) with \(k\geq2\) even and \((q,D)=1\). As a consequence the authors give an unconditional improvement on the trivial bound for the problem of distinguishing primitive cusp forms by their first Fourier coefficients. They also present asymptotic results for the second moments of the Rankin-Selberg values for \(q\) prime and \(k<12\), and use this to prove various nonvanishing results for the critical values \(L(f\otimes g,1/2)\) (resp.\ \(L'(f\otimes g,1/2)\) when the vanishing is to odd order). In addition, they give a linear independence result for certain isotypic components of the images of Heegner divisors under the first Hecke operators. The authors' main result follows from the amplification method of Friedlander and Iwaniec and their previous work [Invent. Math. 142, 95--151 (2000; Zbl 1054.11026)]. To take care of terms arising from the level and nebentypus of \(g\) the authors employ the theory of Atkin-Lehner-Li operators.
scientific article; zbMATH DE number 1820921
scientific article