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I recently became interested in the Langlands program and hope to learn more.

For context, I am an analytic number theorist but have some light background in algebraic number theory and modular forms. In particular from Neukirch's Algebraic Number Theory, Diamond-Shurman's A First Course in Modular Forms, and Shimura's Modular Forms: Basics and Beyond. Aside from these, I have very little in way of prerequisites except the usual ones one would expect of any analytic number theorist.

What are the best introductions to the Langlands program for someone with limited prerequisites?

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  • $\begingroup$ Ed Frenkel's introduction to the Geometric Langlands programme includes a cursory overview of the "classical" Langlands programme which you might find useful. In terms of understanding anything properly, I think there is just too much out there to learn and you'll have to narrow down the question a bit first. $\endgroup$
    – Tom Lovering
    Commented Jul 22, 2014 at 22:11

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Gelbart's AMS article Introduction to the Langlands program is a fairly standard place to start, for a broad overview of what the global conjectures look like and why we're interested in them. There's also a book by the same title by Bernstein, Gelbart et al, which gives more detail. There are also some notes by Knapp that I don't seem to have anymore, but you should be able to find them easily enough. There's an awful lot of different topics that all converge in the "Langlands program" -- to understand "everything" you're going to have to have good command of a pretty intimidating list of topics -- so the best thing to do (at least, what I found to be the best) is try and get a broad overview while taking a lot for granted, and then learn more about the things that particularly interest you. However, if you by "Langlands program" you really just mean "Langlands reciprocity for Galois representations and the local correspondence" then those references will be just fine.

Once you've got a basic understanding of what the conjectures say (or at least the global conjecture for $GL_2$) the answer depends on what you mean by a "serious" introduction. If you mean some level of formality, but skipping over lots of technical details and proofs then the two references above should be fine. If you want to actually understand things at a technical level (and be able to follow the proofs where they exist) then you've got a lot of work to do. I don't really know much about the global conjectures at a technical level, but for the local conjectures a good start is Bushnell-Henniart's Local langlands conjecture for $GL_2$.

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