# Serious introduction to the Langlands program for nonspecialist

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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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. – Tom Lovering Jul 22 '14 at 22:11

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 $GL_2$.