Wushi Goldring: Are automorphic representations algebraic?

An integral part of the quest for algebraicity in the Langlands Program is to show that automorphic representations – initially defined by real and complex analysis – have deep algebro-geometric properties.

The first half of my talk – aimed at graduate students – will motivate our approach to these conjectures via the classical work of Deligne and Serre on modular forms and the long-standing open problem of showing that Maass forms of eigenvalue 1/4 have algebraic Hecke eigenvalues. The second will illustrate two advances on algebraicity from two different directions. From geometry: Joint work with Jean-Stefan Koskivirta on nondegenerate limits of discrete series, see arXiv:1507.05032 . From representation theory: New applications of Langlands Functoriality to algebraicity.