Colloque des sciences mathématiques du Québec

16 octobre 2020 de 15 h 00 à 16 h 00 (heure de Montréal/Miami) Réunion Zoom

Trigonometric functions and modular symbols

Colloque présenté par Nicolas Bergeron (École normale supérieure (Paris), France)

Conférence de la Chaire Aisenstadt

Semestre thématique : Théorie des nombres - Cohomologie en arithmétique

Résumé :

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.

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