On Heegner Points for primes of additive reduction ramifying in the base field

TL;DR

This paper develops a method to construct Heegner points for elliptic curves with primes of additive reduction ramifying in the base field, extending to Darmon points for real quadratic fields.

Contribution

It introduces a new construction technique for Heegner points at primes of additive ramification, broadening the scope of explicit point construction on elliptic curves.

Findings

Method successfully constructs Heegner points in new ramification scenarios

Extends techniques to Darmon points for real quadratic fields

Provides explicit examples and theoretical framework

Abstract

Let $E$ be a rational elliptic curve, and $K$ be an imaginary quadratic field. In this article we give a method to construct Heegner points when $E$ has a prime bigger than $3$ of additive reduction ramifying in the field $K$. The ideas apply to more general contexts, like constructing Darmon points attached to real quadratic fields which is presented in the appendix.