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Stream: theory: mathematics

Topic: Shimura varieties

John Baez (Feb 13 2025 at 16:54):

James Dolan used to occasionally tell me about fun ways of thinking about [[symmetric spaces]] and also [[hermitian symmetric spaces]].  I've given talks about the tenfold way and symmetric spaces. So I have a fondness for such topics. But yesterday I started reading

• James Milne, [[Introduction to Shimura varieties]].

and I learned some basic stuff that's pretty fascinating.  

First, a Shimura variety is a hermitian symmetric domain (where "domain" means it's isomorphic as a variety to an open subset of $\mathbb{C}^n$) modulo a discrete subgroup of isometries.

Second, Shimura varieties are the right way to generalize [[modular curves]] to higher dimensions.  (That makes tons of sense if you notice that the hyperbolic plane is a hermitian symmetric domain, and a modular curve is the hyperbolic plane mod a discrete subgroup of isometries.)

Third, we get automorphic representations from Shimura varieties, so they're important in the Langlands program.

David Corfield (Feb 13 2025 at 18:33):

And they point us up the chromatic levels:

image.png
[[topological automorphic form]]

John Baez (Feb 13 2025 at 20:17):

Great!

By the way, when I said "discrete subgroup of isometries" I should have said "arithmetic discrete subgroup of isometries" - we need to stay in the world of algebraic (or arithmetic?) geometry.

John Baez (Feb 13 2025 at 20:20):

I am extremely fond of quotients of real hyperbolic space by discrete subgroups - like the quotient of 9-dimensional hyperbolic space by the discrete group $\text{PSL}(2,\mathbf{O})$, where $\mathbf{O}$ is the Cayley integral octonions, forming a copy of the $\text{E}_8$ lattice in the octonions.

John Baez (Feb 13 2025 at 20:23):

However I think there are some similar juicy examples connected to complex hyperbolic space, and these are likely to be Shimura varieties.

(I gravitate toward concrete examples - especially 'exotic' examples connected to the octonions, since there's a tiny chance I can find something new there, that the experts in 'unexotic' algebraic geometry might not have already found.)

John Baez (Feb 14 2025 at 17:27):

Well, I was way off in hoping that complex hyperbolic space was a bounded symmetric domain and could thus have quotients that are Shimura varieties. There's a classification of bounded symmetric domains here and the simplest example consists of linear operators $T: \mathbb{C}^n \to \mathbb{C}^n$ of operator norm $\lt 1$. There are two nice octonionic examples.

John Baez (Feb 14 2025 at 17:28):

In case anyone is following from home:

A bounded open set Ω in a complex vector space is said to be a bounded symmetric domain if for every x in Ω, there is an holomorphic map $\sigma : \Omega \to \Omega$ with $\sigma^2 = 1$ for which x is an isolated fixed point. Every Hermitian symmetric space of noncompact type is isomorphic to a bounded symmetric domain.

Every bounded symmetric domain can be treated as an open dense subset of a projective algebraic variety, and then its quotient by a torsion-free arithmetic group acting as isometries is a Shimura variety.

(To even define an arithmetic subgroup of the isometry group, we need to get the variety defined over $\mathbb{Q}$. This seems subtle in general, but I imagine that in the most standard examples it's not hard, if you can figure out at all how to describe a bounded symmetric domain using an algebraic variety, to do it using equations involving just rational numbers.)

John Baez (Feb 14 2025 at 17:48):

The most surprising thing I've learned from Milne's book Introduction to Shimura varieties so far is that every Shimura variety arises as moduli space of Hodge structures.

(Not really a moduli space, since it's not parametrizing isomorphism classes of Hodge structures.   It's actually a kind of parametrized family of Hodge structures.  The buzzword is variation of Hodge structure. Alas, on Wikipedia this definition is given using a cryptic definition of Hodge structure.)

John Baez (Feb 14 2025 at 17:55):

By the way, I'm mainly writing this thread as a kind of notebook as I learn this stuff, but I'd also be very happy to try to answer questions about it, at any level. For more of my life I was absolutely terrified of concepts like 'Shimura variety' or 'Hodge structure', but I've finally decided to confront my fears and actually learn some algebraic geometry. Some of it is hard but some is less hard that I feared. In particular when I finally looked up the definition of 'Hodge structure' I was shocked that people could be making such a fuss over such a simple-minded concept! (In fact it holds a lot of richness... but it's not hard to get the basic idea.)

I'd also be very happy if, deus ex machina, someone appears and tells me interesting things.