Quadratic points on non-split Cartan modular curves.

In this paper, we study quadratic points on the non-split Cartan modular curves X n s (p) , for p = 7 , 1 1 , and 1 3. Recently, Siksek proved that all quadratic points on X n s (7) arise as pullbacks of rational points on X n s + (7). Using similar techniques for p = 1 1 , and employing a version of Chabauty for symmetric powers of curves for p = 1 3 , we show that the same holds for X n s (1 1) and X n s (1 3). As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular. [ABSTRACT FROM AUTHOR]