L Functions associated to automorphic representations

We start with a reductive group \(G\) over some number field \(K\) with automorphic representation \(\pi\) of \(G(\mathbb{A}_F)\).

We would like to build an L function associated to this representation, but in fact doing this is something which to my knowledge we don’t know how to do for basically any group beyond dimension 1.

First, the good. We can build the local factor of the L function at all but finitely many places. The idea is the following.

Theorem: There exist a finite set of places S such that for all places \(\mathfrak{p}\) outside of S, \(G_{\mathfrak{p}}\) is unramified.

This follows from a collection of results. First is a combination of generic smoothness on \(\mathcal{G}\) over \(\mathcal{O}_K\) which tells us that we have \(X\) over \(\mathcal{O}_K^S\) smooth, thus the fibers are smooth and thus by a result of Bruhat and Tits our fibers, namely \(G_\mathfrak{p}\) are unramified.

In this situation we have a very nice cartan Decomposition which is something like KTK which will eventually imply that the local hecke algebra is commutative.

We now discuss somme local representation theory.