Overview

This workshop will revolve around ranks of elliptic curves, with special emphasis on the recent theorem of Manjul Bhargava, Chris Skinner and Wei Zhang that a positive proportion of elliptic curves over the field of rational numbers have rank zero and a positive proportion have rank one (and furthermore, that the Shafarevich-Tate groups of these elliptic curves are finite, and that their ranks agree with the order of vanishing of the associated Hasse-Weil L-function at the central point, as predicted by the Birch and Swinnerton-Dyer conjecture). This theorem represents a symbolic landmark, and its proof combines many of the fundamental advances on elliptic curves and the Birch and Swinnerton-Dyer conjecture achieved over the last several decades, growing out of the fundamental work of Gross-Zagier and Kolyvagin on Heegner points, and, more recently:

a) Bhargava's revolutionary program for counting arithmetic objects, notably, his work with Arul Shankar counting small order elements in Selmer groups;

b) the breakthroughs towards the Iwasawa main conjecture growing out of the work of Chris Skinner and Eric Urban and its extension by Xin Wan, and the resulting "converse of Kolyvagin"-type theorems described in work of Skinner, Venerucci, and Zhang.

The workshop may also discuss other aspects of the phenomenology of ranks, including analogies with similar questions in the function field setting, heuristic models for their behaviour and the status of the Birch and Swinnerton-Dyer conjecture for elliptic curves of rank strictly bigger than 1.