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Over the past two years we have improved several of the (Mordell-Weil) rankrecords 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 independentpoints P_i in E(Q) (the previous record was 24, by R.Martin and W.McMillen2000), and the first example of a curve over Q with Mordell-Weil groupisomorphic with (Z/2Z) x Z^18 (the previous rank record for a curve with a2-torsion point was 15, by Dujella 2002). In these lectures we give some of thebackground, theory, and computational tools that led to these new records andrelated applications.  I Context and overview: the theorems of Mordell(-Weil) and Mazur; the rankproblem; the approaches of Neron--Shioda and Mestre; elliptic surfaces andNeron specialization; fields other than Q.  II Elliptic surfaces and K3 surfaces: the Mordell-Weil and Neron-Severigroups; K3 surfaces of high Neron-Severi rank and their moduli; an elliptic K3surface over Q of Mordell-Weil rank 17. Some other applications of K3 surfacesof high rank and their moduli.  III Computational issues, techniques, and results: slices of Niemeierlattices; finding and transforming models of K3 surfaces of high rank;searching for good specializations. Summary of new rank records for ellipticcurves.

Three lectures on elliptic surfaces and curves of high rank