Elliptic Curves, Torsors, and L-functions

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The award provides funding to support participation of primarily junior researchers (postdocs and graduate students) in the meeting consisting of the workshop ``Elliptic Curves, Torsors and L-functions" and the lectures by B.H. Gross on the ranks of elliptic curves in the Distinguished Lecture Series ``Virginia Mathematics Lectures." The workshop and the lectures will take place March 24-29 at the University of Virginia. The lectures by Gross will include a survey of the Birch--Swinnerton-Dyer conjecture as well as very recent results on the average rank for all elliptic curves over the field of rational numbers and the arithmetic of hyperelliptic curves. The lectures in the workshop will complement Gross's lectures by expanding into related areas of arithmetic geometry and the arithmetic theory of algebraic groups. The meeting is sponsored in part by the Institute of Mathematical Sciences at the UVA Mathematics Department. The website is http://www.people.virginia.edu/~lww8k/workshop/workshop.html.The rank is the most important but least understood invariant of elliptic curves over Q. The recent work of M. Bhargava, B.H. Gross, B. Poonen and others on `average' ranks marked a breakthrough in the field by introducing the viewpoint of `arithmetic statistics'; that is, studying the distribution of such arithmetic invariants. The general methods of arithmetic statistics are applicable to many adjacent areas of arithmetic geometry. The objective of the planned workshop at UVA is precisely to cater to the intense interest in these methods from those working on related areas, by bringing together experts on this topic with both senior and junior researchers from adjacent areas of arithmetic geometry.