STNB

STNB2019 (33rd edition)

Venkatesh's conjectures on arithmetic groups

Co-ordinators

Daniel Barrera, Xavier Guitart, Marc Masdeu, Santiago Molina Blanco, Victor Rotger Cerda and Carlos de Vera Piquero

Description

Venkatesh has recently formulated a series of conjectures, in collaboration with Galatius, Harris and Prasanna, which aim to explain the presence of the same system of eigenvalues in several cohomological degrees of a bounded symmetric domain. The simplest non-trivial examples of this phenomenon arise in the scenario of classical modular forms of weight 1, and Bianchi modular forms over an imaginary quadratic field. Venkatesh's conjectures explain this phenomenon by means of a graded action of a derived Hecke algebra, which can be recast as a Selmer group.

Talks

1. Introduction and overview. (Santiago Molina Blanco)
2. The derived Hecke algebra and the main conjecture of Venkatesh. (Carlos de Vera Piquero)
3. Derived Hecke operators and weight one modular forms. (Xavier Guitart)
4. A Gross--Zagier trace formula for product of theta series. (Óscar Rivero Salgado)
5. Proof of Venkatesh’s conjecture for dihedral weight one forms. (Óscar Rivero Salgado and Victor Rotger Cerda)