Elias Caeiro
Princeton University
Title
Heegner points, Stark-Heegner points and class number problems.
Abstract
The classical class number one problem solved by Heegner, Baker and Stark in the ‘60s states that there are only nine imaginary quadratic fields of class number one. In this talk, I will explain how Heegner points and the Gross–Zagier formula may be used to approach class number problems for imaginary quadratic field. In particular, we are able to determine all negative discriminants of class number at most 3 geometrically without Goldfeld’s analytic method. Assuming Darmon’s conjectural real multiplication theory, this method can be adapted to determine all real quadratic fields of Richaud–Degert type which have class number one, giving a conditional improvement on work of Biró and Lapkova. This is joint work with Henri Darmon and Jingxuan Geng.