We will explain a surprising relationship between the contributions to cohomology of automorphic forms and higher algebraic cycles. We will focus on three examples:
(1) Hilbert modular forms: we conjecture that the cohomology associated with weight one Hilbert modular forms is related to Stark units for the associated Artin representation.
(2) Bianchi modular forms: Kartik Prasanna and Akshay Venkatesh conjecturally relate the cohomology certain arithmetic groups to higher algebraic cycles on a product of elliptic curves over imaginary quadratic fields.
(3) Siegel modular forms: in recent work with Kartik Prasanna, we conjecture that the cohomology associated with weight two Siegel modular forms is related to higher algebraic cycles on a product of abelian surfaces over Q.
We will discuss some evidence to support these conjectures.