Quadratic Points on Non-Split Cartan Modular Curves

TL;DR

This paper investigates quadratic points on specific non-split Cartan modular curves for primes 7, 11, and 13, demonstrating that all such points are pullbacks of rational points and deriving modularity results for elliptic curves over quadratic fields.

Contribution

It extends techniques to show all quadratic points on these curves are pullbacks, establishing modularity of certain elliptic curves over quadratic fields.

Findings

Quadratic points on $X_{ns}(7)$ are pullbacks of rational points.

All quadratic points on $X_{ns}(11)$ and $X_{ns}(13)$ are also pullbacks.

Certain elliptic curves over quadratic fields are proven to be modular.

Abstract

In this paper we study quadratic points on the non-split Cartan modular curves $X_{ns}(p)$, for $p = 7, 11,$ and $13$. Recently, Siksek proved that all quadratic points on $X_{ns}(7)$ arise as pullbacks of rational points on $X_{ns}^+(7)$. Using similar techniques for $p=11$, and employing a version of Chabauty for symmetric powers of curves for $p=13$, we show that the same holds for $X_{ns}(11)$ and $X_{ns}(13)$. As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular.