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A route towards understanding Shimura varieties?

2011-10-08T01:55:07Z

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.)

Tensor and Hom objects for finite flat group schemes

2011-04-13T18:14:24Z

Is the category of finite flat group schemes equipped with "tensor products" and Hom-objects, encoding bilinear maps? I'm aware that the Cartier dual is $Hom(\mathbb{G}, \mathbb{G}_m)$, and want to know if this is part of a systematic collection of objects. For example, is there a "free ring scheme on $\mathbb{G}$"?

If so, given two affine group schemes whose underlying rings are free over the base, are there explicit descriptions of the tensor product and Hom objects in terms of the multiplication and comultiplication rules on the original rings?

Over a field, is there a description in terms of the Dieudonne correspondence?

(References, if they exist, would be very much appreciated. Thank you.)

Terminology for weighted projective spaces

2010-09-25T13:52:46Z

(As suggested, this is a repost from math.stackexchange.com.)

For a sequence of positive integers $a_1, \ldots, a_n$ and a base ring $R$ there is a graded ring $R[x_1,\ldots, x_n]$ where $x_i$ is in degree $a_i$. We can then apply Proj and get a scheme, and this is usually called a weighted projective space; if all of the $a_i$ are 1, then the resulting scheme really is projective space.

However, the way that this arises is as the quotient of $\mathbb{A}^n \setminus 0$ by an action of the multiplicative group, given by $(x_1,\ldots,x_n) \simeq (\lambda^{a_1} x_1, \ldots, \lambda^{a_n} x_n)$ for all $\lambda$. This is a "coarse" group quotient.

There is an alternative version where one instead takes the associated quotient stack/orbifold, and this has a number of nice properties (including possession of a line bundle $\mathcal{O}(1)$); this is true more generally of a graded ring.

It seems that most people associate "weighted projective space" with the scheme-theoretic notion. What is the appropriate terminology for the stack-theoretic version of this construction?