Seminar at SMU Delhi

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.