Emmanuel Lecouturier

Verifying Venkatesh's conjecture on derived Hecke operators in the Bianchi case

Akshay Venkatesh proposed general conjectures relating two things:

"Derived" Hecke operators acting on the singular cohomology of a certain locally symmetric space, and
An action of a motivic cohomology group of an adjoint motive.

The first interesting case - outside the analogous conjecture for weight one forms in a coherent cohomology setting - seems to be the one of parallel weight 2 Bianchi cuspidal eigenforms, where Hecke eigenclasses occur both in cohomological degree 1 and 2. No computations have been done so far to check the validity of the conjecture. We shall give an algorithm to test the conjecture for Bianchi forms over Q(i), and explain one would need some new ideas to actually do the computation in practice.