CHAT - Darmon, Modular functions and explicit class field theory

Slides are available on the CHAT series website: https://chiyunhsu.github.io/activitie...

The idea of CHAT series is to invite established professors (in Number Theory) to talk about either (1) their math career in general (2) a theorem they proved or a theory they developed, explained from a personal and historical perspective, like how they came up with the problem, how the ideas came into place, or what the theorem/theory meant to us, etc.

Date: May 24, 2021
Speaker: Henri Darmon (McGill)
Title: Modular functions and explicit class field theory: private reminiscences and public confessions
Abstract: The problem of constructing class fields of number fields from explicit values of modular functions has its roots in the theory of cyclotomic fields and the theory of complex multiplication. The latter theory acquired a renewed currency in the second half of the 20th century through its connections to the arithmetic of elliptic curves, manifested in the work of Coates--Wiles, Rubin, Gross--Zagier, and Kolyvagin.

I will give a personal account of my path towards a (slightly) better understanding of explicit class field theory for real quadratic fields and its applications to elliptic curves, taking advantage of the CHAT format to focus on the misconceptions, false starts, and dead ends that have marked my roundabout and tortuous, but also very enjoyable, mathematical journey so far.