Quadratic Points on Non-Split Cartan Modular Curves

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.

18 pages. To appear in International Journal of Number Theory. Minor corrections and rewriting. Slightly improved sieving step and more computation details included

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