Wushi Goldring (Stockholm University)
My talk concerns the algebraic properties of automorphic representations. These infinite-dimensional representations of reductive groups over number fields are defined using harmonic analysis. For every prime p, they admit p-adic analogues of Laplacian eigenvalues called Hecke eigenvalues. One of the main mysteries of the Langlands Program is that some automorphic representations have algebraic Hecke eigenvalues while others have transcendental ones. For some, the algebraicity follows from the geometry of Shimura varieties and/or locally symmetric spaces, while for others there are conjectures predicting either algebraicity or transcendence. But there are also instances where it is unclear whether to expect algebraic or transcendental eigenvalues.
I will discuss when Langlands Functoriality, another central theme of the Langlands Program, can be used to reduce the algebraicity for a representation π of a group G to that of some other representation π' of some other group G' for which algebraicity is known for geometric reasons. Via difficult dictionaries, this translates into much more elementary problems in group theory. In the negative direction, we give several group-theoretic obstructions to the existence of π'. In particular, this gives a conceptual explanation for why π' doesn't exist when π arises from non-holomorphic analogues of modular forms called Maass forms. In the positive direction, we exhibit new cases of algebraicity of Hecke eigenvalues for automorphic representations for which no direct link to geometry is known. For some of these, we also associate the Galois representations predicted by the Langlands correspondence.
