Short talk: Motivic action conjectures

A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. A recent series of conjectures proposes an arithmetic explanation: a hidden degree-shifting action of a certain motivic cohomology group. We will give an overview of these conjectures, focusing on the examples of $GL_2$ over Q and over quadratic fields, and $GSp_4$ over Q.