Roadmap to Carayol-Deligne-Langlands

Question

Having begun self-study of Fermat's Last Theorem a few years ago, I have only recently begun to understand and appreciate the theorem of Carayol-Deligne-Langlands on local-global compatibility for cusp forms. There seems to be remarkably little written about the subject despite how powerful and useful the theorem is, and without a knowledge of French or an expert advisor in Representation Theory/Arithmetic Geometry, I am not sure how to go about learning the theorem from a place resembling the "beginning."

I am looking for a roadmap which will take me from knowledge of basic Algebraic Number Theory, Elliptic Curve/Abelian Variety Theory, Scheme Theory, and Modular Forms knowledge all the way through the proof of the theorem in question, in all cases. I do not mind if the roadmap is severely extensive (even unreasonably so). The more detail and concrete references the answer includes, the better. In addition to references for detailed study of the necessary background and the proof, I also welcome any helpful survey articles or notes on the topic.

I also am wondering if it happens to be true that we have local-global compatibility "at p," since Carayol-Deligne-Langlands only provides compatibility at all primes $\ell \neq p$. If this has been answered, resources to learn more about its proof are also welcome.

Answer 1

An elementary introduction is available in Litt - Local–global compatibility and applications to the arithmetic of modular curves.

Historically, the first proof dealing with the simplest case came out in Deligne’s letter to Piatetski-Shapiro (the completed argument appeared in Brylinski’s appendice). Deligne’s method was generalized to the full $\operatorname{GL}_2$ case by Carayol and later to arbitrary $\operatorname{GL}_n$ by Harris–Taylor in their book.

As far as I know, the state-of-the-art results start from Caraiani‘s thesis Local-global compatibility and the action of monodromy on nearby cycles (see also her second paper, which extends the results to the case $\ell=p$).