The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things must a person know before they can understand the Langlands program and its geometric analogue?

What are the good books for learning these topics?

Is there any book which can explain the Langlands program to an undergraduate with very few prerequisites?

• Re: the last question, it depends on what exactly you mean by "explain." Jul 2, 2011 at 5:19
• By "explain" I mean understanding the meaning of the Langland conjectures, why they are important and why is it being called the grand unified theory of mathematics.
– ABC
Jul 2, 2011 at 5:31
• Do the standard undergraduate courses include Galois and representation theory? Jul 2, 2011 at 5:51
• I can't give an answer, but I want to mention the book Introduction to the Langlands program by Bernstein, Gelbart et al.
– user325
Jul 2, 2011 at 6:19
• @ABC, Langlands isn't really a grand unified theory of mathematics - that's just something Edward Frenkel said to convey the importance of the work to convey the importance of the program to the interested non-expert. If there is a grand unified theory of mathematics, it's probably (higher) category theory, or something related to that, perhaps the Curry-Howard isomorphism or some deeper collection of theorems about computation and math. Anyway, the Langlands program appears to be extremely deep, and looks very interesting - but I'd hardly call it a "grand unified theory." Jun 21, 2018 at 7:45

I am going to contradict the answers given and say: do not read any introductions to the Langlands program at this stage. Instead, learn the following things first (and take your time over them!) and do lots of exercises:

• Complex representation theory of finite groups, character theory (e.g. the book by Isaacs, or my lecture notes)
• Algebraic number theory, starting with the basic theory of number fields, Dedekind domains, class numbers, and leading up to class field theory (that's a project for at least a year, you can start with any introductory book on Galois theory, then go on to an introductory book on algebraic number theory)
• Some basics on algebraic groups and Lie groups. I suspect that you will need to learn some very basic things about manifolds and about varieties first.
• An introductory course on modular forms.

When you have that covered (two or three years down the line), then you will benefit from reading about the Langlands program. In the meantime, once you have learned representation theory and Galois theory (can be done in one or two months if you are very bright), you should approach a faculty member at your university. He or she will be able to give you a very rough overview of the general Langlands philosophy, so that you very roughly know where you are heading.

All this is not supposed to discourage you, but rather to excite you about all the fascinating things that lie ahead of you, and to warn you not to skip any of the essentials if you really want to appreciate the beauty of the whole edifice.

• I'd partly second this answer, since a large part of what the Langlands program aims to do is organize a large body of seemingly disparate, wildly different phenomena/things. Until those phenomena are already familiar, formulating big conjectures about them can all too easily turn into a ghastly, oppressive exercise in definitions... answering questions that haven't been asked, in terms of things previously unmentioned. In addition to the topics above, the infinite_dimensional repn theory of Lie groups is necessary. and this necessitates some functional analysis. The phenomena come first. Jul 3, 2011 at 16:39
• @Andrew, you are very welcome! Feb 19, 2013 at 21:46
• This is an interesting response, and perhaps an even better question @Norman. It really raises a more fundamental question of the training necessary to get comfortable and competent with pursuing your own questions and interests. There is a certain sense in which the "standard undergraduate curriculum" becomes tediously formulaic and more about processing students than really conveying mathematics and developing, and implicitly finding those who are so interested, the right foundations for the student to learn for themselves. Jan 27, 2014 at 12:40
• Plus analytic number theory : the theory of $\zeta(s)$, the PNT and the corresponding theorems for Dirichlet L-functions and how it becomes much more complicated for Artin, Dedekind, modular forms and elliptic curves L-functions. In other words Langlands program begins with visualizing how L-functions are in the same time : arithmetic, analytic, algebraic geometric, Galois-theoretic and representation theoretic. Oct 14, 2019 at 20:44

I don't think the Langlands program is accessible to the average undergraduate student. In particular, you won't understand a thing if you have no prior knowledge of algebraic number theory (and class field theory). But, once you passed through the requirements spelled out in Alex B's answer (and in this paper), then here's a list of interesting references:

1. I second both Gelbart's Introduction to the Langlands program and the book Introduction to the Langlands program.
2. More advanced is the paper of Knapp called Introduction the Langlands program (yes, another one!), in the Edinburgh Proceedings.
3. In fact, there is a whole website full of different resources related to automorphic forms, automorphic representations, and the Langlands program. You can find it here.

But again, remember this: One does not simply learn what the Langlands program is about.

To get started on this check out this paper:

http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/home.html

That's a good survey article on the Langland's program. It's takes you through the historical perspective to Artin reciprocity and it's implications. If you look that over you'll get a good overview of what you need to know; class-field theory, L-functions, p-adic numbers, adeles, automorphic forms, and group representations. To get a good handle on this stuff check out Frohlich and Taylor:

I have a copy of this book and in my opinion it is accessible to someone with a good course in Number Theory and perhaps two courses in Algebra at the grad level. I don't think an undergrad is going to get there on his own but with guidance from an advisor working in number theory you can do it.

Now I'm no expert in Langlands but (I believe) the Langland's program is a series of conjectures that are basically consequences and generalizations of Artin reciprocity.

Give it a shot. The Langlands program is some deep stuff that took on a role of the same magnitude as Klein's Erlanger program before it. You really can't go wrong getting to the bottom of this one.

Peter Scholze, a young man recently made a Clay Mathematics Institute research fellow, wrote some readable papers on the Langlands correspondence.

http://www.math.uni-bonn.de/people/scholze/

If you read them, please share your experience (if it was the right level / style for you).

A quite late answer, but in case anyone else finds this in a search: the Mathematical Sciences Research Institute in Berkeley has put up a series of videotaped lectures by Dr. Edward Frenkel at UC Berkeley (5.5 hours long) from a workshop he gave in fall 2015 that was recorded for television broadcast in Japan. This introduction was meant to be accessible at the undergraduate level, so some may find it useful.

Read Frenkel's Langlands Coorespondence for Loop Groups, https://math.berkeley.edu/~frenkel/loop.pdf

Standard college algebra is the only prerequisite.

• OP asks for a book to learn the subject. What more of an answer do you expect than a book title and possibly a link to the book? Moreover, if a link should be provided, could it get safer than a link to the author's page?
– HSN
Apr 27, 2016 at 14:19
• "Standard college algebra is the only prerequisite" is nonsense. On p. 1, you are required to understand the properties of local, non-Archimedean fields. On p. 2, you are required to understand the properties of smooth representations of totally disconnected groups. Aug 7, 2018 at 20:49
• @LSpice : wait, you didn't learn it in high school ? Aug 7, 2018 at 21:37
• @NicolasHemelsoet, my high school only studied smooth representations of Lie groups. :-) Aug 7, 2018 at 22:01