6
$\begingroup$

This might seems like a bit of philosophical question and so maybe if I keep reading a bit more, I might get my answer. But, I ask nonetheless.

In my attempt to study Shimura varieties, I came across the definition of a locally symmetric space associated to connected reductive group $G/\mathbb{Q}$ as follows:

Let $A_G\subset G$ denote the maximal $\mathbb{Q}$-split torus in the center of $G$ and $A_\infty=A_G(\mathbb{R})$. Let $K_{\infty}\subset G(\mathbb{R})$ denote a maximal compact subgroup. If, for any topological group $\mathfrak{G}$, the connected component of the identity is denoted by $\mathfrak{G}^\circ$, then we define $$X=G(\mathbb{R})/A^\circ_\infty K_\infty^\circ.$$ Now, let $\Gamma\subset G(\mathbb{Q})$ be a subgroup such that $\Gamma\cap G(\mathbb{Q})\cap \operatorname{GL}_n(\mathbb{Z})$ has fintie index in both $\Gamma$ and $G(\mathbb{Q})\cap \operatorname{GL}_n(\mathbb{Z})$ for some faithful representation $G\hookrightarrow \operatorname{GL}_n$. The locally symmetric space is defined as $$ X(\Gamma)=\Gamma\backslash X. $$

My question is basically, why?

Intuitively what is this trying to achieve? As in, why are we quotienting out by $K_\infty^\circ$ and $A_\infty^\circ$ (maybe this is the least we need to do to ensure some kind of compactness-like result, or better cohomology groups)?

I do have some familiarity with moduli spaces of elliptic curves with level structures and I know that we can get examples using this construction (which helps construct Galois representations using certain modular forms), but when I see this construction, it just looks like generalising without any end goal in mind (which is obviously not true and why I said I should probably keep reading rather than ask this question — but I cannot help myself).

I am sure the question sounds vague, but I just don't have an image in mind when I think of these locally symmetric spaces and it would greatly help it someone could help fix that.

$\endgroup$
10
  • 1
    $\begingroup$ Some Shimura varieties (those of abelian type) classify abelian varieties with additional structure. See: Deligne, Pierre Travaux de Shimura. (French) Séminaire Bourbaki, 23ème année (1970/1971), Exp. No. 389, pp. 123–165. Lecture Notes in Math., Vol. 244, Springer, Berlin, 1971. $\endgroup$ May 8, 2023 at 10:51
  • 1
    $\begingroup$ Are you asking for a (motivating) reason for defining symmetric space, locally symmetric space, or both? $\endgroup$ May 8, 2023 at 11:16
  • 1
    $\begingroup$ @MarsaultChabat yes, I am trying to find some motivation so that I can create a better picture in my head. For instance, when someone sees a redictive group, I don't think it is very natural to think that we should try and understand the geoemetry of this quotient. Or is it natural? $\endgroup$ May 8, 2023 at 12:47
  • 1
    $\begingroup$ I'm far from an expert, but I think it is helpful to start with examples, which have rich and beautiful geometries. In the simplest case: $G=SL_2$, then $X= SL_2(\mathbb{R})/SO_2$ is the upper half plane. Then $X(\Gamma)$ would include modular curves. In the next simplest case, $G$ is the Weil restriction of $SL_2$ of a real quadratic field, and $X(\Gamma)$ would be a Hilbert modular surface.... $\endgroup$ May 8, 2023 at 13:56
  • 1
    $\begingroup$ This is not the most general definition of a locally symmetric space. $\endgroup$ May 8, 2023 at 14:36

1 Answer 1

12
$\begingroup$

There is a very natural, intrinsic definition of a "symmetric space", as a manifold (Riemannian or Hermitian) with an extra symmetry of a certain prescribed type. It is then a theorem, not a definition, that all such objects have the form $G(\mathbb{R}) / A_\infty^\circ K_\infty^\circ$ for a reductive group $G$. You can find this perspective, for instance, in Milne's "Shimura Varieties and Moduli".

From this perspective, defining a symmetric space as $G(\mathbb{R}) / A_\infty^\circ K_\infty^\circ$ is rather misleading. It's common to define it in this way in number-theory texts, because it avoids wading through large amounts of quite intricate (aka: beautiful) differential geometry before getting to the number-theoretic parts of the theory; but it makes the motivation highly non-obvious.

(Edit: Just to clarify, this is about symmetric spaces, not locally symmetric spaces. A locally symmetric space is the quotient of a symmetric space by a discrete subgroup of its automorphism group $G$, so once you understand the motivation for symmetric spaces, the motivation for locally symmetric spaces should hopefully be clear.)

$\endgroup$
1
  • $\begingroup$ This is a very helpful answer. Thank you very much. $\endgroup$ May 8, 2023 at 17:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.