Maleeha Khawaja

Low degree points on curves

Faltings (1983) proved that a nice curve C/Q of genus at least 2 has only finitely many points defined over any given number field. Suppose C/Q has infinitely many algebraic points of a given degree. For example, if C is hyperelliptic then C has infinitely many quadratic points. Let L be a number field. We define the set of L-new points on C to be the set of points on C defined over L, but not over any strictly smaller extension. Fix a hyperelliptic curve C/Q. We will discuss joint work with S. Siksek which provides sufficient conditions for the set of L-new points on C to be empty for 100% of quadratic fields, when ordered by absolute discriminant. We will discuss analogous results for cubic points on the Fermat quartic and cubic points on the unit equation, with the help of a powerful counting theorem of Bhargava, Taniguchi and Thorne (2024).