Wushi Goldring: Beyond L and C: D and M-algebraic automorphic representations
Inaugural "Wednesday Stockholm Zoom Seminar"
Time: Wed 2020-04-22 13.15 - 14.15
Participating: Wushi Goldring, Stockholms universitet
Since the pioneering work of Langlands on his program, the consensus has been that the Langlands correspondence over number fields can only apply to automorphic representations which satisfy a certain integrality condition at the archimedean component (i.e. over R). What is more, when this integrality condition is violated in a sufficiently strong way, it is believed (but wide open) that even the Satake parameters of the automorphic representation will be transcendental numbers (e.g. for Maass forms of Laplace eigenvalue 1/4). Building on that, as well as later developments by Clozel, Gross and others, Buzzard-Gee introduced two integrality conditions called L and C-algebraic.
Motivated by the problem of understanding when the Satake parameters of an automorphic representation should be algebraic, we shall introduce a new integrality condition which we call "D-algebraic" ("D" for "difference") and its complement will be called M-algebraic ("M" for "mixed"). Using known and conjectural cases of Langlands functoriality, we will give examples of D (resp. M)-algebraic representations. We show that:
(i) The algebraicity of the Satake parameters for D algebraic representations is reducible to the L-algebraic case if one assumes functoriality.
(ii) By contrast, the algebraicity of the Satake paramers in the M-algebraic case is not reducible to the L-algebraic case via "forward functoriality" (but we give examples where it is nevertheless known because the representation is a functorial image of something D-algebraic "backward functoriality").
This is the first "Wednesday Stockholm Zoom seminar" which will be at 13:15 on Wednesdays. People can attend using Zoom. The meeting ID will be the same each week.