Number Theory Seminar

Speaker: Wushi Goldring (Stockholm)

Title: Propagating algebraicity via functoriality

Abstract: We attempt -- and almost entirely succeed -- to classify the
automorphic representations π of reductive groups over number
fields, for which the algebraicity of the Hecke eigenvalues (or Satake
parameters) is reducible via Langlands functoriality to cases already
known for algebro-geometric reasons (if π is singular, the only
current option being via the coherent cohomology of Shimura
varieties). Historically, an important case -- for which algebraicity
is still wide open -- is that of Maass forms of Galois-type:
non-holomorphic eigenfunctions of the hyperbolic Laplacian on the
upper half-plane with eigenvalue 1/4. Also, using functoriality to
reduce to the cohomology of Shimura varieties has been applied
successfully by many people to the construction of automorphic Galois
representations, in cases where the algebraicity of the Hecke
eigenvalues was already known (usually via the Betti cohomology of a
locally symmetric space). However, as far as we know, no new case of
algebraicity of Hecke eigenvalues had previously been established by
reducing via functoriality to Shimura varieties (where we transfer π
"forward" to a larger group; looking at "special" π which descend to
a smaller group "going backward" is a different matter).

In the positive direction, we give several examples of non-holomorphic
automorphic forms which are superficially similar to Maass forms, but
whose algebraicity does reduce to the coherent cohomology of Shimura
varieties via either known or open cases of functoriality; in the
known cases of functoriality we are also able to attach Galois
representations. We also introduce new notions of "D" and
"M-algebraic", which generalize/refine the "L" and "C" of Buzzard-Gee
and the "W-algebraic" of Patrikis, and we give examples of cuspidal
\pi which are "farther" from L-algebraic than previously considered
yet still have algebraic Hecke eigenvalues, answering a question of
Patrikis. In the negative direction, we give a conceptual,
group-theoretic explanation for why Maass forms and many other forms
are not reducible to known cases via functoriality (so the sign error
found by Taylor in a previous attempt on Maass forms was not
coincidental, but rather necessary). There remains only a sliver of
cases where it is still perhaps unclear whether or not reduction via
functoriality is possible.

The talk will take place on Zoom. If you would like to receive the Zoom link and are not part of our current NT Seminar mailing list, please contact the organizer.