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 representatins 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 $p=2$.