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- W1877851746 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." @default.
- W1877851746 created "2016-06-24" @default.
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- W1877851746 date "2007-09-18" @default.
- W1877851746 modified "2023-09-27" @default.
- W1877851746 title "Three lectures on elliptic surfaces and curves of high rank" @default.
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