Function fields of algebraic curves share many properties with number fields, such as the field of rational numbers ℚ. An example of a function field is the field of rational functions k(t) over a field k. A consequence is that elliptic curves E over number fields and function fields behave similarly. In both cases one has the Mordell–Weil Theorem that states that the group of points on E is finitely generated, and it is a computational challenge to find points that generate this group, and to compute its rank.

However, one difference is that in the function field case, one can consider an elliptic curve over k(t) as a surface over k. In this talk I will discuss how the theory of surfaces can help with determining the group of points on E.