On central L-derivative values of automorphic forms

Abstract

We study the average and nonvanishing of the central L-derivative values of L(s, f) 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 $X_0(1)$.