Quadratic points on non-split Cartan modular curves

Abstract

In this paper, we study quadratic points on the non-split Cartan modular curves [Formula: see text], for [Formula: see text] and [Formula: see text]. Recently, Siksek proved that all quadratic points on [Formula: see text] arise as pullbacks of rational points on [Formula: see text]. Using similar techniques for [Formula: see text], and employing a version of Chabauty for symmetric powers of curves for [Formula: see text], we show that the same holds for [Formula: see text] and [Formula: see text]. As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular.