in this talk we will discuss an arithmetic analogue of the gonality of a 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 well-understood when this invariant is 1, 2, or 3; by work of debarre--fahlaoui these criteria do not generalize to e at least 4. we will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality. this is joint work with geoffrey smith.