Shimura Varieties

In The Moduli Space of Elliptic Curves we discussed how to construct a space whose points correspond to isomorphism classes of elliptic curves over $\mathbb{C}$ . This space is given by the quotient of the upper half-plane by the special linear group $\text{SL}_{2}(\mathbb{Z})$ . Shimura varieties kind of generalize this idea. In some cases their points may correspond to isomorphism classes of abelian varieties over $\mathbb{C}$ , which are higher-dimensional generalizations of elliptic curves in that they are projective varieties whose points form a group, possibly with some additional information.

Using the orbit-stabilizer theorem of group theory, the upper half-plane can also be expressed as the quotient $\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$ . Therefore, the moduli space of elliptic curves over $\mathbb{C}$ can be expressed as

$\displaystyle \text{SL}_{2}(\mathbb{Z})\backslash\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$ .

If we wanted to parametrize “level structures” as well, we could replace $\text{SL}_{2}(\mathbb{Z})$ with a congruence subgroup $\Gamma(N)$ , a subgroup which contains the matrices in $\text{SL}_{2}(\mathbb{Z})$ which reduce to an identity matrix when we mod out b some natural number $N$ which is greater than $1$ . Now we obtain a moduli space of elliptic curves over $\mathbb{C}$ together with a basis of their $N$ -torsion:

$Y(N)=\Gamma(N)\backslash\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$

We could similarly consider the subgroup $\Gamma_{0}(N)$ , the subgroup of $\text{SL}_{2}(\mathbb{Z})$ containing elements that reduce to an upper-triangular matrix mod $N$ , to parametrize elliptic curves over $\mathbb{C}$ together with a cyclic $N$ -subgroup, or $\Gamma_{1}(N)$ , the subgroup of $\text{SL}_{2}(\mathbb{Z})$ which contains elements that reduce to an upper-triangular matrix with $1$ on every diagonal entry mod $N$ , to parametrize elliptic curves over $\mathbb{C}$ together with a point of order $N$ . These give us

$Y_{0}(N)=\Gamma_{0}(N)\backslash\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$

and

$Y_{1}(N)=\Gamma_{1}(N)\backslash\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$

Let us discuss some important properties of these moduli spaces, which will help us generalize them. The space $\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$ , i.e. the upper-half plane, is an example of a Riemannian symmetric space. This means it is a Riemannian manifold whose group of automorphisms act transitively – in layperson’s terms, every point looks like every other point – and every point has an associated involution fixing only that point in its neighborhood.

These moduli spaces almost form smooth projective curves, but they have missing points called “cusps” that do not correspond to an isomorphism class of elliptic curves but rather to a “degeneration” of such. We can fill in these cusps to “compactify” these moduli spaces, and we get modular curves $X(N)$ , $X_{0}(N)$ , and $X_{1}(N)$ . On these modular curves live cusp forms, which are modular forms satisfying certain conditions at the cusps. Traditionally these modular forms are defined as functions on the upper-half plane satisfying certain conditions under the action of $\text{SL}_{2}(\mathbb{Z})$ , but when they are cusp forms we may also think of them as sections of line bundles on these modular curves. In particular the cusp forms of “weight $2$ ” are the differential forms on a modular curve.

These modular curves are equipped with Hecke operators, $T_{p}$ and $\langle p\rangle$ for every $p$ not equal to $N$ . These are operators on modular forms, but may also be thought of in terms of Hecke correspondences. We recall that elliptic curves over $\mathbb{C}$ are lattices in $\mathbb{C}$ . Take such a lattice $\Lambda$ . The $p$ -th Hecke correspondence is a sum over all the index $p$ sublattices of $\Lambda$ . It is a multivalued function from the modular curve to itself, but the better way to think of such a multivalued function is as a correspondence, a curve inside the product of the modular curve with itself.

With these properties as our guide, let us now proceed to generalize these concepts. One generalization is through the concept of an arithmetic manifold. This is a double coset space

$\Gamma\backslash G(\mathbb{R})/K$

where $G$ is a semisimple algebraic group over $\mathbb{Q}$ , $K$ is a maximal compact subgroup of $G(\mathbb{R})$ , and $\Gamma$ is an arithmetic subgroup, which means that it is intersection with $G(\mathbb{Z})$ has finite index in both $\Gamma$ and $G(\mathbb{Z})$ . A theorem of Margulis says that, with a handful of exceptions, $G(\mathbb{R})/K$ is a Riemannian symmetric space. Arithmetic manifolds are equipped with Hecke correspondences as well.

Arithmetic manifolds can be difficult to study. However, in certain cases, they form algebraic varieties, in which case we can use the methods of algebraic geometry to study them. For this to happen, the Riemannian symmetric space $G(\mathbb{R})/K$ must have a complex structure compatible with its Riemannian structure, which makes it into a Hermitian symmetric space. The Baily-Borel theorem guarantees that the quotient of a Hermitian symmetric space by an arithmetic subgroup of $G(\mathbb{Q})$ is an algebraic variety. This is what Shimura varieties accomplish.

To motivate this better, we discuss the idea of Hodge structures. Let $V$ be an $n$ -dimensional real vector space. A (real) Hodge structure on $V$ is a decomposition of its complexification $V\otimes\mathbb{C}$ as follows:

$\displaystyle V\otimes\mathbb{C}=\bigoplus_{p,q} V^{p,q}$

such that $V^{q,p}$ is the complex conjugate of $V^{p,q}$ . The set of pairs $(p,q)$ for which $V^{p,q}$ is nonzero is called the type of the Hodge structure. Letting $V_{n}=\bigoplus_{p+q=n} V^{p,q}$ , the decomposition $V=\bigoplus_{n} V_{n}$ is called the weight decomposition. An integral Hodge structure is a $\mathbb{Z}$ -module $V$ together with a Hodge structure on $V_{\mathbb{R}}$ such that the weight decomposition is defined over $\mathbb{Q}$ . A rational Hodge structure is defined similarly but with $V$ a finite-dimensional vector space over $\mathbb{Q}$ .

An example of a Hodge structure is given by the singular cohomology of a smooth projective variety over $\mathbb{C}$ :

$\displaystyle H^{n}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=\bigoplus_{i+j=n}H^{j}(X,\Omega_{X/\mathbb{C}}^{i})$

In particular for an abelian variety $A$ , the integral Hodge structure of type $(1,0),(0,1)$ given by the first singular cohomology $H^{1}(A(\mathbb{C}),\mathbb{Z})$ gives an integral Hodge structure of type $(-1,0),(0,-1)$ on its dual, the first singular homology $H_{1}(A(\mathbb{C}),\mathbb{Z})$ . Specifying such an integral Hodge structure of type $(-1,0),(0,-1)$ on $H_{1}(A(\mathbb{C}),\mathbb{Z})$ is also the same as specifying a complex structure on $H_{1}(A(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}} \mathbb{R}$ . In fact, the category of integral Hodge structures of type $(-1,0),(0,-1)$ is equivalent to the category of complex tori.

Let $\mathbb{S}$ be the group $\text{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{\text{m}}$ . It is the Tannakian group for Hodge structures on finite-dimensional real vector spaces, which basically means that the category of Hodge structures on finite-dimensional real vector spaces are equivalent to the category of representations of $\mathbb{S}$ on finite-dimensional real vector spaces. This lets us redefine Hodge structures as a pair $(V,h)$ where $V$ is a finite-dimensional real vector space and $h$ is a map from $\mathbb{S}$ to $\text{GL}(V)$ .

We have earlier stated that the category of integral Hodge structures of type $(-1,0),(0,-1)$ is equivalent to the category of complex tori. However, not all complex tori are abelian varieties. To obtain an equivalence between some category of Hodge structures and abelian varieties, we therefore need a notion of polarizable Hodge structures. We let $\mathbb{R}(n)$ denote the Hodge structure on $\mathbb{R}$ of type $(-n,-n)$ and define $\mathbb{Q}(n)$ and $\mathbb{Z}(n)$ analogously. A polarization on a real Hodge structure $V$ of weight $n$ is a morphism $\Psi$ of Hodge structures from $V\times V$ to $\mathbb{R}(-n)$ such that the bilinear form defined by $(u,v)\mapsto \Psi(u,h(i)v)$ is symmetric and positive semidefinite.

A polarizable Hodge structure is a Hodge structure that can be equipped with a polarization, and it turns out that the functor that assigns to an abelian variety $A$ its first singular homology $H_{1}(X,\mathbb{Z})$ defines an equivalence of categories between the category of abelian varieties over $\mathbb{C}$ and the category of polarizable integral Hodge structures of type $(-1,0),(0,-1)$ .

A Shimura datum is a pair $(G,X)$ where $G$ is a connected reductive group over $\mathbb{Q}$ , and $X$ is a $G(\mathbb{R})$ conjugacy class of homomorphisms from $\mathbb{S}$ to $G$ , satisfying the following conditions:

The composition of any $h\in X$ with the adjoint action of $G(\mathbb{R})$ on its Lie algebra $\mathfrak{g}$ induces a Hodge structure of type $(-1,1)(0,0)(1,-1)$ on $\mathfrak{g}$ .
For any $h\in X$ , $h(i)$ is a Cartan involution on $G(\mathbb{R})^{\text{ad}}$ .
$G^{\text{ad}}$ has no factor defined over $\mathbb{Q}$ whose real points form a compact group.

Let $(G,X)$ be a Shimura datum. For $K$ a compact open subgroup of $G(\mathbb{A}_{f})$ where $\mathbb{A}_{f}$ is the finite adeles (the restricted product of completions of $\mathbb{Q}$ over all finite places, see also Adeles and Ideles), the Shimura variety $\text{Sh}_{K}(G,X)$ is the double quotient

$\displaystyle G(\mathbb{Q})\backslash (X\times G(\mathbb{A}_{f})/K)$

The introduction of adeles serves the purpose of keeping track of the level structures all at once. The space $\text{Sh}_{K}(G,X)$ is a disjoint union of locally symmetric spaces of the form $\Gamma\backslash X^{+}$ , where $X^{+}$ is a connected component of $X$ and $\Gamma$ is an arithmetic subgroup of $G(\mathbb{Q})^{+}$ . By the Baily-Borel theorem, it is an algebraic variety. Taking the inverse limit of over compact open subgroups $K$ gives us the Shimura variety at infinite level $\text{Sh}(G,X)$ .

Let us now look at some examples. Let $G=\text{GL}_{2}$ , and let $X$ be the conjugacy class of the map

$\displaystyle h:a+bi\to\left(\begin{array}{cc}a&b\\ -b&a\end{array}\right)$

There is a $G(\mathbb{R})$ -equivariant bijective map from $X$ to $\mathbb{C}\setminus \mathbb{R}$ that sends $h$ to $i$ . Then the Shimura varieties $\text{Sh}_{K}(G,X)$ are disjoint copies of modular curves and the Shimura variety at infinite level $\text{Sh}(G,X)$ classifies isogeny classes of elliptic curves with full level structure.

Let’s look at another example. Let $V$ be a $2n$ -dimensional symplectic space over $\mathbb{Q}$ with symplectic form $\psi$ . Let $G$ be the group of symplectic similitudes $\text{GSp}_{2n}$ , i.e. for $k$ a $\mathbb{Q}$ -algebra

$\displaystyle G(k)=\lbrace g\in \text{GL}(V\otimes k)\vert \psi(gu,gv)=\nu(g)\psi(u,v)\rbrace$

where $\nu:G\to k^{\times}$ is called the similitude character. Let $J$ be a complex structure on $V_{\mathbb{R}}$ compatible with the symplectic form $\psi$ and let $X$ be the conjugacy class of the map $h$ that sends $a+bi$ to the linear transformation $v\mapsto av+bJv$ . Then the conjugacy class $X$ is the set of complex structures polarized by $\pm\psi$ . The Shimura varieties $Sh_{K}(G,X)$ are called Siegel modular varieties and they parametrize isogeny classes of $n$ -dimensional principally polarized abelian varieties with level structure.

There are many other kinds of Shimura varieties, which parametrize abelian varieties with other kinds of extra structure. Just like modular curves, Shimura varieties also have many interesting aspects, from Galois representations (related to their having Hecke correspondences), to certain special points related to the theory of complex multiplication, to special cycles with height pairings generalizing results such as the Gross-Zagier formula in the study of special values of L-functions and their derivatives. There is also an analogous local theory; in this case, ideas from $p$ -adic Hodge theory come into play, where we can further relate the $p$ -adic analogue of Hodge structures and Galois representations. The study of Shimura varieties is a very fascinating aspect of modern arithmetic geometry.

References:

Shimura variety on Wikipedia

Reciprocity Laws and Galois Representations: Recent Breakthroughs by Jared Weinstein

Perfectoid Shimura Varieties by Ana Caraiani

Introduction to Shimura Varieties by J.S. Milne

Lecture Notes for Advanced Number Theory by Jared Weinstein

5 thoughts on “Shimura Varieties”

AntiCyberLic

April 18, 2020 at 7:22 pm

> We can fill in these cusps to “compactify” these moduli spaces, and we get modular curves X(N), X_{0}(N), and X_{1}(N). On these modular curves live cusp forms, which are modular forms satisfying certain conditions at the cusps. Traditionally these modular forms are defined as functions on the upper-half plane satisfying certain conditions under the action of \text{SL}_{2}(\mathbb{Z}), but when they are cusp forms we may also think of them as sections of line bundles on these modular curves.

I believe the usual terminology is that on the Y(N)’s you get the “meromorphic modular forms”, and then extending holomorphically to the cusps of X(N) gives you the notion of “(holomorphic) modular forms”, and this is what is normally meant by “modular forms”. Then cusp forms are those modular forms which vanish at the cusps. Hecke operators act on modular forms, and map cusp forms to cusp forms. Just wanted to clarify 🙂

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