Chow motives, L-functions, and powers of algebraic Hecke characters

The Langlands and Fontaine–Mazur conjectures in number theory describe when an automorphic representation f arises geometrically, meaning that there is a smooth projective variety X, or more generally a Chow motive M in the cohomology of X, such that there is an equality of L-functions L(M,s) = L(f,s). We explicitly describe how to produce such a variety X and Chow motive M in the case of powers of certain automorphic representations, called algebraic Hecke characters. This is joint work with J. Lang.