Three lectures on elliptic surfaces and curves of high rank

TL;DR

This paper discusses recent advances in elliptic curve ranks over Q, including new record examples, theoretical background, and computational techniques involving elliptic surfaces and K3 surfaces.

Contribution

It presents the first examples of elliptic curves with 28 independent points and a Mordell-Weil group isomorphic to (Z/2Z) x Z^18, along with methods to find such high-rank curves.

Findings

First elliptic curve with 28 independent points over Q

Elliptic curve with Mordell-Weil group (Z/2Z) x Z^18

Development of computational tools for high-rank elliptic curves

Abstract

Over the past two years we have improved several of the (Mordell-Weil) rank records for elliptic curves over Q and nonconstant elliptic curves over Q(t). For example, we found the first example of a curve E/Q with 28 independent points P_i in E(Q) (the previous record was 24, by R.Martin and W.McMillen 2000), and the first example of a curve over Q with Mordell-Weil group isomorphic with (Z/2Z) x Z^18 (the previous rank record for a curve with a 2-torsion point was 15, by Dujella 2002). In these lectures we give some of the background, theory, and computational tools that led to these new records and related applications. I Context and overview: the theorems of Mordell(-Weil) and Mazur; the rank problem; the approaches of Neron--Shioda and Mestre; elliptic surfaces and Neron specialization; fields other than Q. II Elliptic surfaces and K3 surfaces: the Mordell-Weil and Neron-Severi groups; K3 surfaces of high Neron-Severi rank and their moduli; an elliptic K3 surface over Q of Mordell-Weil rank 17. Some other applications of K3 surfaces of high rank and their moduli. III Computational issues, techniques, and results: slices of Niemeier lattices; finding and transforming models of K3 surfaces of high rank; searching for good specializations. Summary of new rank records for elliptic curves.