TIFR
December 2, 2010
On Local Galois Representations attached to Automorphic Forms: In arithmetic, Galois representations are one of the fundamental objects of interest and they arise quite naturally in several places.
The Galois representations coming from cuspidal automorphic forms on $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ are expected to be irreducible as representations of the absolute Galois group of $\mathbb{Q}$. However, the local representations, obtained by restricting to a decomposition subgroup, can be reducible. In this talk, we will show how a generalized notion of ordinariness for automorphic forms implies the reducibility of such local representations. We also show that non-ordinariness implies irreducibility in certain cases.
When $n = 2$ and $p = 2$, we will also discuss the semisimplicity of local Galois representations attached to ordinary cuspidal eigenforms, following the approach of Ghate-Vatsal for odd primes. This requires proving some new results in Hida theory for the prime $p = 2$.