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Modular representation theory and representations of Galois groups

Mathematics Colloquium

Modular representation theory and representations of Galois groups
Series: Mathematics Colloquium
Location: MATH 501
Presenter: Matthew Emerton, University of Chicago
Modular representation theory studies the actions of groups on vector spaces over finite fields, 
such as the field F_p obtained by considering the integers modulo a prime p.   Such actions are called 
``modular representations'' (because they involve working modulo p).  A basic question in the subject is 
whether such an action can be ``lifted'' to characteristic zero --- concretely, can we find matrices with
integral entries that give a representation of the group on a vector space over (some extension of) the 
rational numbers, whose reductions modulo p recover the original modular representation.
 
This may seem like quite a technical question, and it is!   But it turns out that questions of exactly this type
arise in number theory, in relation to some of the fundamental problems in the Langlands program.   The 
groups of interest are frequently Galois groups; another class of groups of interest consists of matrix groups 
over finite fields, such as GL_n(F_p). 
 
In this talk I will introduce this circle of ideas, with a focus on motivation and key examples.   I also hope to 
discuss some recent progress in the field; this will be joint work with Toby Gee.
(Refreshments will be served.)