Gebhard Boeckle (University of Duisburg-Essen)

will speak on

Drinfeld modular forms, Galois representations and Hecke characters

Time: 4:00PM
Date: Mon 27th April 2009
Location: Mathematical Sciences Seminar Room [map]

Abstract: Work of Eichler-Shimura and Deligne attaches to any classical Hecke eigenforms
of weight at least two, a strictly compatible systems (SCS) of Galois representations.
Using a cohomological theory of Pink and myself, an analogous construction can
be given for cuspidal Drinfeld Hecke eigenforms. Unlike in the classical situation,
the representations are 1- and not 2-dimensional. This is a consequence of the
Eichler-Shimura relation in characteristic p and is suggested by the generating
function for Hecke operators in characteristic p. The latter had motivated Serre to
ask whether the Hecke eigenvalue system would arise from a Hecke character.
Goss reemphasized the question from the Galois representation perspective.
Adapting to the function field setting a correspondence of Khare between SCS of
mod p Galois representations and Hecke characters, we answer the question in
the affirmative. I shall give an introduction to Drinfeld modular forms, explain the
meaning of the above statements and indicate some recent results on ramification
properties of the Galois representations obtained. It turns out that ramification is also
linked to good non-ordinary reduction of Drinfeld modular curves.

(This talk is part of the Algebra and Number Theory series.)

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