Three lectures on elliptic surfaces and curves of high rank
| dc.creator | Elkies, Noam D. | |
| dc.date | 2007-09-18 | |
| dc.date.accessioned | 2026-07-07T08:30:32Z | |
| dc.date.available | 2026-07-07T08:30:32Z | |
| dc.description | 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. | |
| dc.description | 14 pages; extended abstract for an invited lecture series at the 7/2007 Oberwolfach workshop ``Explicit Methods in Number Theory'', expanded from my lecture notes for publication in the Oberwolfach Reports | |
| dc.identifier | https://arxiv.org/abs/0709.2908 | |
| dc.identifier | http://arxiv.org/abs/0709.2908 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/138265 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14J27; 11G05 | |
| dc.title | Three lectures on elliptic surfaces and curves of high rank | |
| dc.type | text |