The Saito-Kurokawa lifting and Darmon points

Abstract

Let $E_{/_\Q}$ be an elliptic curve of conductor $Np$ with $p\nmid N$ and let $f$ be its associated newform of weight 2. Denote by $f_{\infty }$ the $p$-adic Hida family passing though $f$, and by $F_{\infty }$ its $\hat{I}›$-adic Saito-Kurokawa lift. The $p$-adic family $F_{\infty }$ of Siegel modular forms admits a formal Fourier expansion, from which we can define a family of normalized Fourier coefficients $\{{\tilde{A}}_T(k){\}}_T$ indexed by positive definite symmetric half-integral matrices $T$ of size $2\times 2$. We relate explicitly certain global points on $E$ (coming from the theory of Stark-Heegner points) with the values of these Fourier coefficients and of their $p$-adic derivatives, evaluated at weight $k=2$.

14 pages. Title changed