**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 *y*^{2} = *P*(*x*) =
*x*^{3} + *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 *y*^{2} = *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.