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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" 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.

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.

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    $\begingroup$ What is your motivation for wanting to read this? I'm having to read between the lines here to guess your background, but it sounds like you aren't even in a position to understand the statements of the results in this paper. There are probably better choices of papers at the intersection of number theory and representation theory given where you are. $\endgroup$
    – Robert
    Commented Jun 30, 2022 at 21:22
  • $\begingroup$ @Robert I want to learn about Langlands Program. Nowadays my main goal is to learn about differential geometry and lie theory. But I want to dedicate at least half an hour every day in order to understand a little bit about Langlands' program. $\endgroup$
    – user860322
    Commented Jun 30, 2022 at 22:02
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    $\begingroup$ Why do you want to learn about the Langlands Program? Here's one initial thing: do you already know basic algebraic number theory (say at the level of Neukirch) and class field theory? If you don't, then you should focus on learning that stuff first. $\endgroup$
    – Robert
    Commented Jun 30, 2022 at 22:09
  • $\begingroup$ @Robert I started to learn these subjects. I also studied a little about valuations in a small project with a professor. $\endgroup$
    – user860322
    Commented Jun 30, 2022 at 22:25
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    $\begingroup$ I'd think it might be hard to even understand why the local Langlands conjecture would be interesting, without understanding more global things. Starting with something like quadratic reciprocity, where the rhetorical question is "why should quadratic symbols have conductors?" :) $\endgroup$
    – paul garrett
    Commented Jun 30, 2022 at 23:40

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@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.
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  • $\begingroup$ Thank you very much! Could you recommend complementary books for those who wish to learn more about the Langlands Program? $\endgroup$
    – user860322
    Commented Jul 2, 2022 at 15:03
  • $\begingroup$ @user860322 Ok, maybe first of all, what I told you is about the "local Langlands Correspondence". This is just a part of the Langlands Program. If you wish to learn about the Langlands Program, there is a book of Kudla, Cogdell and co. called "An Introduction to the Langlands Program", see here: amazon.com/Introduction-Langlands-Program-Joseph-Bernstein/dp/… . On the one hand, it is "very short" (relative to what a solid introduction would really need), but on the other hand, it may not be the best literature for beginners. $\endgroup$
    – Maty Mangoo
    Commented Jul 3, 2022 at 10:20
  • $\begingroup$ @user860322 I can maybe add just an advise for the future: - certain things in Maths have their 'natural order', but also - every individual has its own way to learn. What I want to say with the previous is that.. you have to find by yourself what is the best way to learn new things. But in my opinion, things should be explained in their natural order, such that the approach and the appearing objects make sense. Take a look here math.stackexchange.com/questions/48981/… , I very much agree with the answer, but well.. it is just my opinion. $\endgroup$
    – Maty Mangoo
    Commented Jul 3, 2022 at 10:35

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