Tao on Zhang

Terry Tao has offered a detailed summary of a series of talks by Shou-wu Zhang on rational points on curves. The existence of rational points turns out to depend critically on the genus of the curve, which is a topological and complex-analytic property of the curve.

The summary of each talk provides the flavor of the subject (which is very technical):

  1. Overview of rational points on curves
  2. Gross-Zagier formula and Birch and Swinnerton-Dyer conjecture
  3. Triple L-series and effective Mordell conjecture