Seminar at SMU Delhi
March 5, 2015 (Thursday) ,
3:30 PM at Webinar
Tata Institute of Fundamental Research, Mumbai
Reductions of Galois Representations
Abstract of Talk
The Galois representations we are interested in are certain
representations of the Galois group of the field of p-adic
numbers into invertible two by two matrices over a finite
field. They arise by reducing certain characteristic zero
representations.An open problem is whether these reductions
are reducible or irreducible.
The answer depends on the size of two quantities, the weight,
an integer greater than 2, and the slope, a non-negative
rational number. When the slope is 0, the reductions are all
reducible, independent of the weight.A common misconception
in the subject is that when the slope is positivethe reduction
is always irreducible. There are now several results for small
slopes which show that this is not true.
In general, the complexity of the problem grows dramatically
as the slope (and weight) increase. We now have a nearly
complete understanding of the reduction for slopes less than 2
and all weights, due to the work of several people, and most
recently due to the work of the speaker and S. Bhattacharya.
In the two talks, I shall explain some results and ideas that
go into the proof. In particular, I shall introduce the
underlying building (tree) and mention how certain spaces of
functions are related to the problem via the Local Langlands
Correspondence. I shall also mention some elementary related
problems in the modular representation theory of general
linear groups which are of independent interest.