On central \(L\)-derivative values of automorphic forms
Let \(k\) be an odd integer, \(f(z)=\sum_{n\geq 1}a_ne^{2\pi inz}\) a cuspidal normalized Hecke eigenform of weight \(2k\) and level one, and \(f_{K_D}\) its base change to \(K_D=\mathbb Q(\sqrt{D})\), where \(D<0\) is a fundamental discriminant. Furthermore, let \(L(s,f)=\sum_{n\geq 1}\frac{\lambda_f(n)}{n^s}\) be the \(L\)-function of \(f\), analytically continued to the complex plane, where \(\lambda_f(n)=\frac{a_n}{n^{k-1/2}}\) are the normalized Hecke eigenvalues of \(f\), and \(L(s,f_{K_D})=L(s,f)L(s,f\otimes\chi_D)\), where \(L(s,f\otimes \chi_D)\) is the twist of \(L(s,f)\) by the Kronecker symbol \(\chi_D=\left(\frac{D}{\cdot}\right)\). The main result of the present paper (Theorem 1) is an asymptotic formula for averages of the first and second moments of \(L'\left(\frac 12,f\right)\) and for the first moment of \(L'\left(\frac 12,f_{K_D}\right)\) over all cuspidal normalized Hecke eigenforms \(f\) of weight \(2k\), as \(k\to \infty\). It is further shown that for sufficiently large odd \(k\) a positive proportion of all normalized Hecke eigenforms \(f\) as above satisfies \(L'\left(\frac 12,f\right)\neq 0\). This is similar to a previous result of \textit{W. Luo} who studied non-vanishing of \(L(\frac{1}{2},f)\), for even \(k\) [Adv. Math. 285, 220--234 (2015; Zbl 1385.11030)]. However, in the present case extra difficulties arise from the presence of additional poles in the test function of the approximate functional equation for \(L'\left(\frac 12,f\right)^2\). Finally, applications to averages of height pairings between Heegner cycles on the modular curve \(X_0(1)\) are also given (see Corollaries 1 and 2).
scientific article; zbMATH DE number 6860659
scientific article