## BCN Spring16 Workshop on Number Theory & K-theory

### Stark points and Hida-Rankin p-adic L-functions

Daniele Casazza

#### Abstract

Let $E$ be an elliptic curve defined over $\mathbb Q$ and let $K\mid \mathbb Q$ be a quadratic imaginary field such that $(E,K)$ satisfies the Heegner hypothesis. Let $\psi: G_K\to \mathbb \C$ be a ring class character associated to the order $\mathcal O_c \subset K$ of conductor $c \in \mathbb Z_{\geq 1}$. The works of Gross-Zagier and Zhang provide a connection between the Heegner point associated to $\psi$ and the first derivative of the function $L(E,\psi,s)$ at the point $s=1$.

Using Hida-Rankin's method and assuming some mild hypothesis we are able to prove a $p$-adic formula which is an analogus to that of Gross-Zagier and Zhang. This formula is totally explicit and it can be recast in the more general framework of the elliptic Stark conjecture formulated by Darmon, Lauder and Rotger. Our formula confirms this conjecture and gives an alternative method for computing Heegner points.