Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 1327–1359 | Cite as

On central L-derivative values of automorphic forms

  • Shenhui Liu


We study the average and nonvanishing of the central L-derivative values of L(sf) and \(L(s,f_{K_{\scriptscriptstyle D}})\) for f in an orthogonal Hecke eigenbasis \(\mathcal {H}_{2k}\) of weight 2k cusp forms of level 1 for large odd k. Here \(f_{K_{\scriptscriptstyle D}}\) is the base change of f to an imaginary quadratic field \(K_{\scriptscriptstyle D}=\mathbb {Q}(\sqrt{D})\) with fundamental discriminant D. We prove asymptotic formulas for the first and second moments of \(L'(\frac{1}{2},f)\), as well as the first moment of \(L'(\frac{1}{2},f_{K_{\scriptscriptstyle D}})\), over \(\mathcal {H}_{2k}\) as odd \(k\rightarrow \infty \). Further, we employ mollifiers to establish that for sufficiently large k there are positive proportion of Hecke eigenforms f in \(\mathcal {H}_{2k}\) with \(L'(\frac{1}{2},f)\ne 0\). We also give applications of our results to Heegner cycles of high weights of the modular curve \(X_0(1)\).


Holomorphic Hecke eigenforms Central L-derivative values Mollifiers Nonvanishing Heegner cycles 

Mathematics Subject Classification

11F11 11F67 



The author thanks Professor Wenzhi Luo for suggesting the investigation of central L-derivative values and for his valuable advice and constant support, and Professor Hui Xue for answering many questions regarding Heegner cycles of high weights. The author also thanks Yongxiao Lin and Professor Sheng-Chi Liu for their encouragement and helpful conversations. Finally, the author is grateful to the referee for valuable comments.


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© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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