On central \(L\)-derivative values of automorphic forms
type: Item
instance of: scholarly article
title: On central \(L\)-derivative values of automorphic forms
zbMATH Open document ID: 1448.11097
author: Shenhui Liu
published in: Mathematische Zeitschrift
publication date: 2018-04-17T00:00:00Z
review text: 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).
reviewed by: Nils Matthes
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 6860659
zbMATH Keywords: holomorphic Hecke eigenforms
zbMATH Keywords: central \(L\)-derivative values
zbMATH Keywords: mollifiers
zbMATH Keywords: nonvanishing
zbMATH Keywords: Heegner cycles
MaRDI profile type: Publication
full work available at URL: s00209-017-1936-6
OpenAlex ID: W2761988026
cites work: The first moment of cusp form $L$-functions in weight aspect on average
cites work: On the central value of symmetric square \(L\)-functions
cites work: Q4338937
cites work: The critical order of vanishing of automorphic \(L\)-functions with large level
cites work: Q5827353
cites work: Heegner points and derivatives of \(L\)-series
cites work: Heegner points and derivatives of \(L\)-series. II
cites work: Q4830109
cites work: Q4252239
cites work: The non-vanishing of central values of automorphic \(L\)-functions and Landau-Siegel zeros
cites work: Non-vanishing of the symmetric square L -function at the central point
cites work: The analytic rank of \(J_0(q)\) and zeros of automorphic \(L\)-functions.
cites work: A lower bound for the rank of $J_0(q)$
cites work: Mollification of the fourth moment of automorphic \(L\)-functions and arithmetic applications.
cites work: Non-vanishing of high derivatives of automorphic L-functions at the center of the critical strip
cites work: Rankin-Selberg \(L\)-functions in the level aspect.
cites work: A mean square formula for central values of twisted automorphic L-functions
cites work: A density theorem on automorphic $L$-functions and some applications
cites work: Nonvanishing of the central \(L\)-values with large weight
cites work: Non-vanishing of high derivatives of Dirichlet \(L\)-functions at the central point
cites work: Q3579593
cites work: Jacobi forms and a certain space of modular forms
cites work: Nonvanishing of quadratic Dirichlet \(L\)-functions at \(s=\frac{1}{2}\)
cites work: The rank of quotients of \(J_0(N)\)
cites work: Gross-Kohnen-Zagier theorem for higher weight forms
cites work: Heights of Heegner cycles and derivatives of L-series
DOI: 10.1007/S00209-017-1936-6
describes a project that uses: Digital Library of Mathematical Functions
description: scientific article; zbMATH DE number 6860659