I'm in the embarrassing situation that I want to ask a question that was already asked, but (for complicated reasons) never answered. I'd like to try with a blank slate.

Shimura varieties show connections to a lot of interesting mathematical subjects. They're a topic of active research and have been of importance in number theory and the Langlands program.

However, the theory has a bit of a reputation: for heavy prerequisites; for a large and difficult-to-penetrate body of literature; for seminar talks that spend a minimum of half an hour getting past the definitions. ("Aren't you assuming that the polarizations are principal here?" "I don't see why that has cocompact center.")

Let's suppose that a poor graduate student doesn't have the best access to the experts, but has gone to lengths to make themselves familiar with "the basics" on modular curves and Shimura curves. There is still a bewildering abundance of new material and new ideas to absorb:

  • Abelian schemes.

  • Reductive algebraic groups and the switch to the adelic perspective.

  • Representation theory and the switch in perspective on modular/automorphic forms.

  • $p$-divisible groups and their various equivalent formulations.

  • Moduli problems and geometric invariant theory.

  • Deformation theory.

  • Polarizations. (Yes, I think this deserves its own bullet point.)

  • (This is a placeholder for any and all major topics that I forgot.)

Obviously there is a lot to learn, and there's no magic way to obtain enlightenment.

But for an outsider, it's not clear where to start, what a good place to read is, what really constitutes the "core" of the subject, or even if one might cobble together a basic education while learning things that will prove useful outside of this specialty.

Is a route from modular curves to Shimura varieties that will help both with understanding the basics of the subject, and with getting an idea of where to learn more?

Thank you (and sorry for the side commentary).

(Often these kinds of questions ask for "roadmaps"; but "roadmap" seems like it presupposes the existence of roads.)


2 Answers 2


I think a great introduction to this subject is given in two articles of J.S. Milne: one from the book James Arthur, David Ellwood, Robert Kottwitz (eds.)-Harmonic Analysis, the Trace Formula, and Shimura Varieties and a shorter version available at his website. Especially the first one does not assume many pre-requisites and goes from modular curves to Shimura varieties (with a view towards the Langlands program).


I think the general wisdom is that Deligne's Travaux de Shimura and Milne's Introduction to Shimura Varieties are the most comprehensive references, with the latter being somewhat lighter on prerequisites (but heavier on examples).

I've heard it suggested by people who work in the area that the best way to learn the theory is via special cases and examples, motivated by focused research problems. I suppose this is true of many things, though.

It might also help to learn about quaternionic Shimura curves first, assuming that you know a bit about modular curves.


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