Henri Darmon: Explicit class field theory and orthogonal groups (NTWS 189)
Channel: Number Theory Web Seminar
Date: 19/11/2023

Abstract: Essentially all abelian extensions of the rational numbers or of a quadratic imaginary field can be generated by special values of the exponential function  or of the modular j-function at explicit arguments in the ground field. Describing the mathematical objects which could play the role of trigonometric and modular functions in generating class fields of more general base fields is the stated goal of explicit class field theory. Around 5 years ago Jan Vonk and I proposed  a framework in which class fields of real quadratic fields can  be generated from the special values of certain “rigid meromorphic cocycles” at real quadratic arguments. Without delving into the details of this framework, I will present some simple concrete consequences of it in settings where the base field is totally real, and explain how they can be proved. The more general statements rest on (but do not require the full force of) the notion of rigid meromorphic cocycles for orthogonal groups of signature (r,r) described in joint work with  Lennart Gehrmann and Mike Lipnowski, and are also inspired by the  calculations in Romain Branchereau’s PhD thesis. (Joint with Jan Vonk)

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, November 16, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 17, 2023 (12am CST, 3am AEDT, 5am NZDT)