InĀ The Moduli Space of Elliptic Curves we discussed how to construct a space whose points correspond to isomorphism classes of elliptic curves over . This space is given by the quotient of the upper half-plane by the special linear group . **Shimura varieties** kind of generalize this idea. In some cases their points may correspond to isomorphism classes of **abelian varieties** over , 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 . Therefore, the moduli space of elliptic curves over can be expressed as

.

If we wanted to parametrize “level structures” as well, we could replace with a **congruence subgroup** , a subgroup which contains the matrices in which reduce to an identity matrix when we mod out b some natural number which is greater than . Now we obtain a moduli space of elliptic curves over together with a basis of their -torsion:

We could similarly consider the subgroup , the subgroup of containing elements that reduce to an upper-triangular matrix mod , to parametrize elliptic curves over together with a cyclic -subgroup, or , the subgroup of which contains elements that reduce to an upper-triangular matrix with on every diagonal entry mod , to parametrize elliptic curves over together with a point of order . These give us

and

Let us discuss some important properties of these moduli spaces, which will help us generalize them. The space , 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** , , and . 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 , 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 ” are the differential forms on a modular curve.

These modular curves are equipped with **Hecke operators**, and for every not equal to . These are operators on modular forms, but may also be thought of in terms of **Hecke correspondences**. We recall that elliptic curves over are lattices in . Take such a lattice . The -th Hecke correspondence is a sum over all the index sublattices of . 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

where is a semisimple algebraic group over , is a maximal compact subgroup of , and is an **arithmetic subgroup**, which means that it is intersection with has finite index in both and . A theorem of Margulis says that, with a handful of exceptions, 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 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 is an algebraic variety. This is what Shimura varieties accomplish.

To motivate this better, we discuss the idea of Hodge structures. Let be an -dimensional real vector space. A (real) **Hodge structure** on is a decomposition of its complexification as follows:

such that is the complex conjugate of . The set of pairs for which is nonzero is called the **type** of the Hodge structure. Letting , the decomposition is called the **weight decomposition**. An **integral Hodge structure** is a -module together with a Hodge structure on such that the weight decomposition is defined over . A **rational Hodge structure** is defined similarly but with a finite-dimensional vector space over .

An example of a Hodge structure is given by the singular cohomology of a smooth projective variety over :

In particular for an abelian variety , the integral Hodge structure of type given by the first singular cohomology gives an integral Hodge structure of type on its dual, the first singular homology . Specifying such an integral Hodge structure of type on is also the same as specifying a complex structure on . In fact, the category of integral Hodge structures of type is equivalent to the category of complex tori.

Let be the group . 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 on finite-dimensional real vector spaces. This lets us redefine Hodge structures as a pair where is a finite-dimensional real vector space and is a map from to .

We have earlier stated that the category of integral Hodge structures of type 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 denote the Hodge structure on of type and define and analogously. A **polarization** on a real Hodge structure of weight is a morphism of Hodge structures from to such that the bilinear form defined by 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 its first singular homology defines an equivalence of categories between the category of abelian varieties over and the category of polarizable integral Hodge structures of type .

A **Shimura datum** is a pair where is a connected reductive group over , and is a conjugacy class of homomorphisms from to , satisfying the following conditions:

- The composition of any with the adjoint action of on its Lie algebra induces a Hodge structure of type on .
- For any , is a Cartan involution on .
- has no factor defined over whose real points form a compact group.

Let be a Shimura datum. For a compact open subgroup of where is the finite adeles (the restricted product of completions of over all finite places, see also Adeles and Ideles), the **Shimura variety** is the double quotient

The introduction of adeles serves the purpose of keeping track of the level structures all at once. The space is a disjoint union of locally symmetric spaces of the form , where is a connected component of and is an arithmetic subgroup of . By the Baily-Borel theorem, it is an algebraic variety. Taking the inverse limit of over compact open subgroups gives us the **Shimura variety at infinite level** .

Let us now look at some examples. Let , and let be the conjugacy class of the map

There is a -equivariant bijective map from to that sends to . Then the Shimura varieties are disjoint copies of modular curves and the Shimura variety at infinite level classifies isogeny classes of elliptic curves with full level structure.

Let’s look at another example. Let be a -dimensional symplectic space over with symplectic form . Let be the **group of symplectic similitudes** , i.e. for a -algebra

where is called the similitude character. Let be a complex structure on compatible with the symplectic form and let be the conjugacy class of the map that sends to the linear transformation . Then the conjugacy class is the set of complex structures polarized by . The Shimura varieties are called **Siegel modular varieties** and they parametrize isogeny classes of -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 -adic Hodge theory come into play, where we can further relate the -adic analogue of Hodge structures and Galois representations. The study of Shimura varieties is a very fascinating aspect of modern arithmetic geometry.

References:

Reciprocity Laws and Galois Representations: Recent Breakthroughs by Jared Weinstein

Perfectoid Shimura Varieties by Ana Caraiani