Henri Darmon: p-adic analysis and Hilbert's twelfth problem (CRM-PIMS-Fields Prize Lecture)

Henri Darmon, Professor of Mathematics at McGill University, presents the following lecture upon receiving the 2017 CRM-PIMS-Fields prize.

Title:
p-adic analysis and Hilbert's twelfth problem

Abstract:
Modular functions play an important role in many aspects of number theory. The theory of complex multiplication, one of the grand achievements of the subject in the 19th century, asserts that the values of modular functions at quadratic imaginary arguments generate (essentially all) abelian extensions of imaginary quadratic fields. Hilbert's twelfth problem concerns the generalisation of this theory to other base fields. I will describe an ongoing work in collaboration with Jan Vonk which identifies a class of functions that seem to play the role of modular functions for real quadratic fields. A key difference with the classical setting is that they are meromorphic functions of a p-adic variable (defined in the framework of "rigid analysis" introduced by Tate) rather than of a complex variable.

Website:

Presentation date and location:
Wednesday, September 13, 2017 - 4:10pm to 5:00pm
Bahen Centre, University of Toronto

Hosted and organized by The Fields Institute. Recorded and edited by Adnan Zuberi.