I would like to know which books I should read to understand the paper ["The local Langlands correspondence for $\mathrm{GL}\_n$ over $p$-adic fields"](https://www.math.uni-bonn.de/people/scholze/LocalLanglands.pdf) written by Peter Scholze.


I only know mathematics subjects at the master's degree level. I also learned about representation theory of finite groups.


Next year I will probably start my PhD in Lie theory but I'd like to read the above article as a personal project.




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**EDIT:**


I found a good article that shows the necessary prerequisites to get a glimpse of the Langlands Program and I want to share it: [Prerequisites for the Langlands Program.](https://www.math.stonybrook.edu/%7Eaknapp/pdf-files/china1.pdf)



Roadmap to understand the Scholze's proof of the local Langlands correspondence for $\text{GL}_n$ over $p$-adic fields
@user860322 The local Langlands correspondence (LLC) for $\mathrm{GL}\_n$ can be understood (very sloppy) as:


There exists a group for $F$, called the Weil(-Deligne)-Group $W\_F$ (and not only for local fields), that encodes the (admissible) representation theory of $\mathrm{GL}\_n(F)$ for every $n$ in a *very nice* way (whatever it means). I don't think that Scholze's paper is the best way to learn about it. There is a nice survey of Clozel called '*Motives and automorphic representations*', which in my eyes would be much better for you.


But of course, there is no short way to go. If you really want it, I would suggest, you learn:


* Basics in algebraic number theory. You should at very least know, what a local ($p$-adic) field is, and how they are constructed. Neukirch covers it well, but you can also have a look in the first chapter of André Weil's Basic Number Theory - he treats it in its full generality, although it may not be appropiated to read in the beginning.
* (local) Class field Theory, since this is the special case $n=1$ for the LLC. I don't recommend you to read local CFT in Neukirch. Weil's Chapter 3 (or so) covers the character theory of local fields, but his approach is highly unintuitive in my eyes (but maybe you will find it great?). Ramakrishnan/Valenza's Fourier Analysis on Number Fields give a very brief and student-friendly introduction in Chapter 6 (if I am not mistaken).
* In Corvallis, Chapter 3.1., John Tate's Number Theoretical Background introduces the Weil Group $W\_F$, but for this you will need to know a little about cohomology theory (at least for the existence of $W\_F$). It may not be the easiest text to read as well.
* Representation Theory of $\mathrm{GL}(F)$ for $F$ local non-archimedean; the 'main paper' is well.. Bernstein/Zelevinsky, but Prasad/Raghuram have a very nice and readable survey. They also already mention the LLC and even give an ad-hoc definition of $W\_F$ (without cohomology theory).


But honestly, I would first learn about rep.theory of compact groups (maybe the first chapters in Bump's Lie Groups?); I think this is a nice intermediary step from finite to infinite groups. Since $\mathrm{GL}\_n(F)$ is locally compact with nice properties (sometimes also called an $l$-group), f.e. the identity $I\_n$ has a neighborhoud of open compact subgroups, in terms of representation theory, it might be seen as the 'next best case' after the compact one (but on this one I may be horribly wrong hehe).


* Probably, to scratch the surface of the bigger picture, it would be good if you learn what the adeles $\mathbb{A}\_K$ of a global field $K/\mathbb{Q}$ are. You can understand the adeles as an object that contains all the local information together and holds it in a very fashion way. Only thanks to the adeles, many things about classical $L$-functions that appeared to be very technical and come out of nowhere (for instance the Riemann zeta function) suddenly had a very natural explanation.


