## 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

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