New Algebraic Points on Curves

Let K be a degree d number field. A famous theorem of Faltings states that a "nice" curve C of genus at least 2 defined over K has only finitely many points defined over K. However, C can of course admit infinitely algebraic points of degree d. For example, a hyperelliptic curve defined over the rationals admits infinitely many quadratic points. In this talk, we will see many examples of curves which have infinitely many points of degree 2 or 3 and, moreover, we shall see the number fields arising from these points are relatively rare.