Motivic Action Conjecture for Doi-Naganuma Lifts

A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. In a series of papers, Venkatesh and his collaborators proposed an arithmetic reason for this: a hidden degree-shifting action of a certain motivic cohomology group. In this talk, we will explain the setup of this conjecture for coherent cohomology of Hilbert modular forms over ℂ, and give some recent result supporting it. This is joint work with Alex Horawa.