In this talk we will discuss an arithmetic analogue of the gonality of a nice curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite.  By work of Faltings, Harris-Silverman and Abramovich-Harris, it is understood when this invariant is 1, 2, or 3; by work of Debarre-Fahlaoui these criteria do not generalize.  We will focus on scenarios under which we can guarantee that this invariant is actually equal to the gonality using the auxiliary geometry of a surface containing the curve.

This is joint work with Geoffrey Smith.