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 $$X_0(1)$$ .