The Saito-Kurokawa lifting and Darmon points

TL;DR

This paper explores the relationship between global points on elliptic curves and Fourier coefficients of $p$-adic lifts of modular forms, revealing new connections in the context of Saito-Kurokawa lifts and Darmon points.

Contribution

It explicitly relates Stark-Heegner points on elliptic curves to Fourier coefficients of $ ext{p}$-adic Saito-Kurokawa lifts, advancing understanding of their arithmetic properties.

Findings

Explicit formulas connecting global points and Fourier coefficients.

Identification of $p$-adic derivatives of Fourier coefficients with special points.

Enhanced understanding of the arithmetic of Siegel modular forms.

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 $\Lambda$-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 $\{\widetilde A_T(k)\}_T$ 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 $k=2$.