Quebec Mathematical Sciences Colloquium

October 16, 2020 from 15:00 to 16:00 (Montreal/Miami time) Zoom meeting

Trigonometric functions and modular symbols

Colloquium presented by Nicolas Bergeron (École normale supérieure (Paris), France)

Chaire Aisenstadt Chair Conference

Thematic Semester: Number Theory - Cohomology in Arithmetic

In his fantastic book “Elliptic functions according to Eisenstein and Kronecker”, Weil writes:

“As Eisenstein shows, his method for constructing elliptic functions applies beautifully to the simpler case of the trigonometric functions. Moreover, this case provides […] the simplest proofs for a series of results, originally discovered by Euler.”

The results Weil alludes to are relations between product of trigonometric functions. I will first explain how these relations are quite surprisingly governed by relations between modular symbols (whose elementary theory I will sketch). I will then show how this story fits into a wider picture that relates the topological world of group homology of some linear groups to the algebraic world of trigonometric and elliptic functions. To conclude I will briefly describe a number theoretical application.

This is based on a work-in-progress with Pierre Charollois, Luis Garcia and Akshay Venkatesh.