# 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

**Lecturer: **
Wushi Goldring, Stockholms universitet

**Location: **
Zoom: stockholmuniversity, Meeting ID: 68944808319

### Abstract

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").

### Additional Information

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.