Speaker: Noam Elkies (Harvard)
Title: Ranks of elliptic curves and surfaces
Abstract: An elliptic curve E over a field K of characteristic zero is a curve of the form y2 = P(x) = x3 + Ax + B where P has distinct roots; more structurally (and in arbitrary characteristic), it is an algebraic curve of genus 1 over K with a rational point. By theorems of Mordell and Weil, the set E(K) of rational points (solutions in K of y2 = P(x), together with a "point at infinity") forms an abelian group, which is finitely generated in many important cases including K = Q and K = k(t) (assuming in the latter case that E is "nonconstant", that is, not K-isomorphic to an elliptic curve over the ground field k). The rank of E over K is then the rank of this abelian group E(K). In these two cases K = Q and K = k(t), the torsion groups of E(K) are completely understood, but the rank still holds many mysteries. For instance, it is not yet known whether the rank can be arbitrarily large for K = Q or K = C.
In the first part of the talk we review the geometry and arithmetic of elliptic curves, explain how nonconstant curves of large rank over K(t) arise naturally in the study of curves of large rank over K, and announce new rank records for elliptic curves over Q(t) and Q. In the second part, we outline how the arithmetic of a curve E over K(t) is related to the geometry of E as an algebraic surface (an elliptic surface) over K, and how the geometry of K3 surfaces and the arithmetic of certain modular curves gave rise to the new records.