# Shimura Varieties: motivation

This will be the first in a series of posts discussing Shimura varieties. In particular, we will focus here on a sort of broad motivation for the subject—why Shimura varieties are a natural thing to study and, in particular, what they give us.

# A desideratum

The topic of Shimura varieties is one of notorious difficulty to learn. There are four interconnected reasons why this is true. The first is that to even define a Shimura variety, in any guise, requires an exceptional amount of technical sophistication and care. Secondly, Shimura varieties is a subject whose fruits are only harvestable after a considerable amount of buildup, and even then, the fruits are particularly high in the tree. Third, Shimura varieties have the beautiful, but daunting, property that they have a huge range of descriptions, most of which look entirely different. And, lastly, there is a serious dearth of accessible literature on the subject—to my knowledge, the only article which is simultaneously in English, semi-comprehensive, and didactic (meaning not couched in a more difficult manuscript), are Milne’s notes on the subject.

These first two reasons make it hard to stay motivated when learning about Shimura varieties—the objects are so hard to define, and it’s so non-obvious what they do. The second two topics make cross-referencing very difficult; even if one can find two good articles on the subject, they may be approached from two entirely different perspectives with nary a word about the connections between the approaches.

For this reason, it is particularly important, if one has an interest in learning about Shimura varieties, to have an upfront ‘overall perspective’. To understand broadly why Shimura varieties are a natural object to consider, are of sufficient importance, and how the various parts of the subject fit together. This is the goal of this post. In all likelihood I will not be able to perfectly summarize these various considerations (of the reasons, not the least being that I am not an expert). But, for the sake of the reader, and more honestly for my own sake, I will try.

# General motivation

As mentioned in the last paragraph, Shimura varieties are objects with many faces. One of the leading causes for this multiple-personality-disorder is that Shimura varieties fulfill many roles. Once one has a zen enough understanding of the subjects involved, one sees that the various objects that Shimura varieties touch are, despite first appearances, not unrelated. But, this requires a sensei’s eye.

So, before we get into the details of any of these various guises of Shimura varieties, let us first, for the sake of overview, list them now. Shimura varieties are, amongst other things, objects of the following form/serving the following goal:

1. Shimura varieties are highly symmetrical objects which admit extremely rich actions of various Lie groups (and their discrete subgroups).
2. Shimura varieties are moduli spaces of abelian varieties, perhaps with extra structure.
3. Shimura varieties are moduli spaces of Hodge structures. Or, in fancier terms, moduli spaces of motives.
4. Shimura varieties are objects used (conjecturally) to realize the global Langlands correspondence.

Now, in some amount of rigor, Shimura varieties associated to a (reductive) group $G/\mathbb{Q}$ are going to be systems of varieties $S_K$ over some number field $E$, indexed by (special) compact open subgroups $K\subseteq G(\mathbb{A}_\mathbb{Q}^\infty)$ (with $\mathbb{A}^\infty_\mathbb{Q}$ being the finite adeles) for $G$ some algebraic group over $\mathbb{Q}$. We now want to explore the relationship between this imprecise definition and the numbered topics above.

It will turn out that modular curves are the simplest ‘interesting’ Shimura variety. This is convenient since modular curves are something most students of number theory, at any reasonable level of knowledge, have seen a fair amount of. For this reason, we shall try to understand these notions above is the simplified context of modular curves.

# Symmetric spaces

## General setup

We begin first with a discussion of Shimura varieties from the point of view of ‘highly symmetrical spaces’. So, to this end, let us tentatively call a geometric object (by which we roughly mean a topological space with extra structure) $X$ a symmetric space if the following two conditions hold:

1. The automorphism group $\mathrm{Aut}(X)$ acts transitively on $X$.
2. There exists a point $p\in X$ which possesses a symmetry $s_p\in\mathrm{Aut}(X)$. A symmetry $s_p$ at $p$ is an involutive automorphism fixing $p$, and $p$ being the only fixed point in a neighborhood of $p$.

Note that from 1. the existence of a symmetry at $s_p$ at $p$ guarantees the existence of a symmetry $s_q$ at $q$ for all $q\in X$. Indeed, suppose that $g\in\mathrm{Aut}(X)$ satisfies $g(p)=q$ and let $s_q$ be $gs_p g^{-1}$.

Now, these spaces are, as we said before, ‘highly symmetric’. In general, a space may have just one type of symmetry—maybe there is one point $p$ with a symmetry at $p$. But, as we showed above, the homogeneity of symmetric spaces (the fact that their automorphism groups act transitively) guarantees a veritable well-spring of symmetries—in particular, one for each point of the space.

Now, it will turn out that we are going to be interested, mostly, in symmetric spaces on geometric objects which are Hermitian manifolds. Without going into too much detail here, a Hermitian manifold is a complex analytic version of a Riemannian manifold. One way of making this rigorous is that a Hermitian manifold $X$ is a Riemannian manifold $(M,g)$ together with an integrable almost complex structure $J$ which acts by isometries.

It will turn out in this case that the automorphism group $\mathrm{Aut}(X)$ of such an object carries the unique structure of a manifold (with underlying topology the compact-open topology) so that its connected component $G:=\mathrm{Aut}(X)^\circ$ is a Lie group. It will then turn out that $G$ still acts transitively on $X$, making $X$ into a homogenous space, and thus showing that symmetric spaces do have rich actions by Lie groups.

Now, besides being interesting objects in their own right, there is a general reason why symmetric spaces are a natural object of study, especially from the point-of-view of representation theory or, said differently, harmonic analysis. Namely, modern harmonic analysis seeks to study objects with large sets of symmetries, usually discrete symmetries. It turns out that, not shockingly, functions on symmetric spaces have enough symmetries to be interesting from the point of view of harmonic analysis. A concrete example of this is the creation of discrete series for semisimple Lie groups.

We will be able to show that there is a classification of (Hermitian symmetric spaces) corresponding roughly to the classification of semi-simple Lie groups, making these objects not so inconceivably complicated.

Broad relationship to Shimura varieties: Thinking of Shimura varieties as a system of varieties $S_K$ one can then show that the analytification of these varieties $S_K$, which are complex manifolds, are quotients of symmetric spaces by discrete subgroups.

## The case of modular curves

One perspective on modular curves, most likely one’s first introduction to the subject, is that modular curves are quotients of the upper half-plane $\mathbb{H}$ by discrete subgroups $\Gamma$ of $\mathrm{SL}_2(\mathbb{R})$. So, to put this into the perspective of symmetric spaces, let us first discuss the upper half-plane as such an object.

Recall that we define the upper half-plane $\mathbb{H}$ to be the set of complex numbers with positive imaginary part. This inherits from $\mathbb{C}$ the structure of an open Riemann surface. Moreover, this space has the natural structure of a Hermitian manifold with metric given by

$\displaystyle g=\frac{dxdy}{y^2}$

a metric with everywhere negative curvature.

Now, we claim that the upper half-plane equipped with this metric $g$ is a Hermitian symmetric space. Indeed, one can check that the automorphism group of the pair $(\mathbb{H},g)$ is $\mathrm{PSL}_2(\mathbb{R})$ with the action being given by fractional linear transformations:

$\displaystyle \begin{pmatrix}a & b\\ c & d\end{pmatrix}z:= \frac{az+b}{cz+d}$

It’s then clear that this acts transitively on $\mathbb{H}$, and moreover it has a symmetry at $i$ given by

$\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}$

proving that, indeed, $(\mathbb{H},g)$ is a Hermitian symmetric space. Note then, that we can show it’s isomorphic to $\mathrm{PSL}_2(\mathbb{R})/\mathrm{SO}_2(\mathbb{R})$—this is because $\mathrm{PSL}_2(\mathbb{R})$ acts transitively on $\mathbb{H}$ and $\mathrm{SO}_2(\mathbb{R})$ is the stabilizer of $i$.

So, now as stated before, modular curves are going to be quotients of $\mathbb{H}$ by certain discrete subgroups of $\mathrm{PSL}_2(\mathbb{R})$. Let us discuss one of the simplest cases, which is when takes $\Gamma$ to be the subgroup

$\Gamma(N)=\ker\left(\mathrm{SL}_2(\mathbb{Z})\to\mathrm{SL}_2(\mathbb{Z}/N\mathbb{Z})\right)$

to which we associated the modular curve

$Y(N):=\mathbb{H}/\Gamma(N)$

Now, these spaces $Y(N)$ naturally have the structure of a Riemann surface. Moreover, one can show that they have a natural compactification

$X(N)=\left(\mathbb{H}\cup\mathbb{P}^1(\mathbb{Q})\right)/\Gamma(N)$

which still carries the structure of a Riemann surface.

But, then, by the Riemann existence theorem we know that $X(N)$ has the structure of an algebraic curve $X(N)/\mathbb{C}$ for which $X(N)$, as a Riemann surface, is its analytification. Thus, $Y(N)$ carries the structure of an affine algebraic curve for which $Y(N)$ as a Riemann surface, is its analytification. Moreover, it’s a classical result that these curves actually admit models over $\mathbb{Q}(\zeta_n)$.

This shows some features of the general relationship between Shimura varieties and symmetric spaces. The Shimura variety $Y(N)$, a curve over $\mathbb{Q}$, has analytification isomorphic to the quotient of a Hermitian symmetric space by a discrete subgroup of its automorphism group.

Also, we see here a connection to harmonic analysis since on these quotients $Y(N)$ is where one usually thinks about the highly-symmetric functions that are modular forms. The generalization of this to more general hermitian symmetric spaces is the theory of automorphic forms which, roughly, are just functions which admit a large group of (discrete) symmetries.

Moreover, this process by which one shows that quotients of certain symmetric spaces by certain discrete subgroups have an algebraic structure follows a very similar pattern to what happened above and is a simple manifestation of the very general, very powerful theorem of Baily-Borel which is what one applies in the general theory of Shimura varieties.

# Moduli spaces of abelian varieties

Now, we would like to discuss Shimura varieties from the perspective of moduli spaces of abelian varieties with extra structure. In particular, we would like to discuss the so-called PEL type Shimura varieties. The PEL is an acronym for polarization, endomorphism, and level structure since, after all, these are what PEL type Shimura varieties aim to classify.

Now, the general definition of a PEL type Shimura variety is extraordinarily complicated coming not only from the amount of structure they attempt to classify (they classify $5$-tuples of data), but also from the complicated conditions the data needs to satisfy to make the moduli problems have reasonable moduli spaces (either fine or reasonably rich coarse moduli spaces).

So, for this reason, we will suffice ourselves with an illustrative example instead of the general case. Namely, let us fix integers $g$, $d$, and $N$ and consider the following moduli problem:

$\mathcal{A}_{d,N}:\mathsf{Set}/\mathbb{Z}[\frac{1}{N}]\to \mathsf{Set}$

given by

\mathcal{A}_{d,N}(T)=\left\{(A,\phi_N,\lambda):\begin{aligned}&(1)\, A/T\text{ is an abelian scheme of dimension }g\\ &(2)\,\phi_N:\left(\mathbb{Z}/N\mathbb{Z}\right)^{2g}\xrightarrow{\approx}A[N]\text{ is an isomorphism of group schemes}\\ &(3) \,\lambda\text{ is a polarization of degree }d^2\end{aligned}\right\}/\text{iso.}

One call this the moduli of Mumford data of level $N$ and degree $d$. One can then show that $\mathcal{A}_{d,N}$ always has a coarse moduli space $A_{d,N}$ which is fine for $N\geqslant 3$, and which is smooth over $\mathbb{Z}[\frac{1}{dN}]$. One can go one step-further and consider the moduli functor $\mathcal{A}_{d,N}^J$, where $J$ is a symplectic matrix in $\mathrm{GL}_g(\mathbb{Z}/N\mathbb{Z})$, and demand that the isomorphism $\phi_N$ carry the Weil pairing induced by $\lambda$ to the pairing on $(\mathbb{Z}/N\mathbb{Z})^{2g}$ induced by $J$—more rigorously, the pairing of $\phi_N(x)$ and $\phi_N(y)$ is $\zeta_N^{x^\top J y}$. Such moduli problems are similarly representable by schemes $A_{d,n}^J$

This is an example of PEL type moduli problem where we have ‘P’ data (the polarization $\lambda$) and ‘L’ data (the level isomorphism $\phi_N$) but no endomorphism data. Just to give a sense of what type of endomorphism data one might ask for, one could consider abelian schemes $A/T$ similar to the above case, but with a fixed embedding $\mathcal{O}_K\hookrightarrow \mathrm{End}(A)$ for $K/\mathbb{Q}$ a totally real field. These lead to the so-called Hilbert-Blumenthal modular varieties.

Now, the arithmetic importance of this space is obvious. One has reduced the study of all abelian varieties with some level structure to the study of a reasonable family of geometric objects $A_{d,N}$. And, the arithmetic importance of this is clear since abelian varieties are amongst the most important arithmetic geometrical spaces as they include, of course, elliptic curves.

Broad relation to Shimura varieties: All PEL type Shimura varieties are Shimura varieties, the individual moduli schemes being the $S_K$ for varying $K$ (the parameter $K$ corresponding to the level data we choose to focus on).

So, to reiterate, not all Shimura varieties are of this form—the PEL type just form an understandable, and important, subscase of the general theory.

## The case of modular curves

Now, the relationship to modular curves is a classical one. One begins by showing that, $Y(N)$ (as defined above) ‘parameterizes’ elliptic curves $E$ over $\mathbb{C}$ together with a $\mathbb{Z}/n\mathbb{Z}$-basis $(a,b)$ for $E[N]$ such that under the Weil pairing this basis goes to $\zeta_n:=e^{\frac{2\pi i}{n}}$. We call this data, for reasons to soon be made clear, $\Gamma(N)$-level data.

Of course, here the notion of ‘parameterize’ is intentionally vague. Namely, one associates to any $\tau\in\mathbb{H}$ the pair

$\displaystyle \left(\mathbb{C}/\Lambda_\tau,\left(\frac{1}{N},\frac{\tau}{N}\right)\right)$

where $\Lambda_\tau$ is the lattice $\mathbb{Z}+\mathbb{Z}\tau$, and $\displaystyle \left(\frac{1}{N},\frac{\tau}{N}\right)$ denotes the specified basis for the $N$-torsion in the elliptic curve $\mathbb{C}/\Lambda_\tau$ which can be identified with $\frac{1}{N}\Lambda/\Lambda$. Note that, indeed, under the Weil pairing, the pair $\displaystyle \left(\frac{1}{N},\frac{\tau}{N}\right)$ goes to $\zeta_n$.

Then, by ‘parameterize’ we mean that this association from $\mathbb{H}$ to elliptic curves with $\Gamma(N)$-level data is surjective and two elements $\tau$ and $\tau'$ give rise to isomorphic data (isomorphisms being isomorphisms preserving the level data) if and only if they differ by the action of $\Gamma(N)$. Thus, at least as sets, we can identify the isomorphism classes of elliptic curves with $\Gamma(N)$-level data with $\mathbb{H}/\Gamma(N)$.

This association, besides being practically nice (having explicit descriptions of all elliptic curves with $\Gamma(N)$-level data), is conceptually important. Namely, this association has, somewhat magically, given the set of elliptic curves with level $\Gamma(N)$-data a notion of geometry—the structure of an open Riemann surface. This opens us up to a new avenue of attack on the study of elliptic curves (with level $\Gamma(N)$-data). In particular, as mentioned above, we might study the geometry of their parameter space $Y(N)$ and hope to deduce from this interesting facts about the objects they parameterize.

Now, while the above is nice from a classical point of view, many things are left entirely unclear and unmotivated from such a treatment. For example, the bijection between $Y(N)$ and elliptic curves over $\mathbb{C}$ with $\Gamma(N)$-level data, while nice, is somewhat ‘random’. Who’s to say that we couldn’t have created a ‘natural bijection’ between the set of elliptic curves with $\Gamma(N)$-data and another random complex manifold $X$? Why is the geometry we put on this set of elliptic curves with level data the ‘right geometry’? Moreover, why should we expect, in any shape or form, this complex analytic theory to tell us something about number theory—especially something as deep as Fermat’s Last Theorem? And, on a less grandiose scale, why do the Hecke operators ‘make sense’?

All of these questions are answered by the realization that the Riemann surfaces we considered above are really shadows of the much greater, more arithmetic theory we discussed in the last section. The key is, in fact, to ask for much more than an object parameterizing an elliptic curve over $\mathbb{C}$ (with level data) but, instead, to ask for an object parameterizing elliptic curves over, well, (almost) anything—to look for moduli schemes.

This theory is predicted, in some sense, by a throw-away comment made in the section on modular curves as quotients of symmetric spaces. Namely, since the $X(N)$ are compact Riemann surfaces it follows from the Riemann existence theorem that $X(N)$ is $X(N)^\mathrm{an}$ for some smooth projective curve which we also denote $X(N)/\mathbb{C}$. Thus, a natural question one might ask is, well, does the curve $X(N)$ admit a model over $\mathbb{Q}$? If so, how can we describe this object?

And so enters the Grothendieck perspective on the notion of space. Namely, if we want to understand the geometry of an object which tells us about elliptic curves over $\mathbb{C}$ or $\mathbb{Q}$, we really need to understand what this object tells us about elliptic curves over any scheme. Namely, what we really want to do is write down a moduli functor

$\mathcal{M}(N):\mathsf{Sch}/\mathbb{Q}\to\mathsf{Set}$

which roughly has the following description:

$\mathcal{M}(N)(T)=\left\{\text{elliptic curves }E/T\text{ with }\Gamma(N)\text{-level data}\right\}/\text{iso.}$

(where, here, level data means something more sophisticated) then any object $Y(N)$ representing this functor will come with an inherent geometry. More succinctly, even though $Y(N)(\mathbb{C})$ is canonically the set of isomorphism classes of elliptic curves over $\mathbb{C}$ (with $\Gamma(N)$-level data) its geometry only comes from specifying $Y(N)(T)$ for all $T$.

Extended remark: A natural question after reading the above might be the following. Why not just consider a functor/space classifying all elliptic curves or, thinking about the last section, all abelian schemes? Why introduce the notion of level structure/polarization structure/endomorphism structure at all if, really, it’s abelian varieties that we care about? Well, the answer to this is a subtle one. But, intuitively, it’s because the problem of classifying elliptic curves (or abelian varieties), with no extra structure, is simply not ‘nice enough’ to access by classical algebraic geometry.

Said less cryptically: the functor associating to a scheme $T$ the isomorphism classes of elliptic curves over $T$ is not representable. There is no inherent scheme whose geometry gives us perfect information about the functor (i.e. represents it). Even the best approximating scheme to this functor (the coarse moduli space) is just the line $\mathbb{A}^1_j$ (called the $j$-line), which has no interesting geometry nor does it carry any interesting families of elliptic curves.

The problem, more technically, is that the functor of isomorphism classes of elliptic curves is not a sheaf on the (small) étale site of $\mathbb{Q}$—for a finite separable extension $L/\mathbb{Q}$ the map

$\{\text{elliptic curves}/\mathbb{Q}\}/\text{iso.}\to \{\text{elliptic curves}/L\}/\text{iso.}:E\mapsto E\times_{\mathbb{Q}}L$

is not injective (just think of quadratic twists!), a necessary condition for sheaves. It turns out that this is caused by the existence of automorphisms of elliptic curves (this is in line with the theory of twists).

One can fix this problem by not identifying isomorphic elliptic curves. Instead, one can think of a ‘functor’ which associates to $T$ the category of all elliptic curves over $T$ and morphisms being isomorphisms—in essence we’ve pulled apart isomorphism classes, but by adding in the morphisms(which are isomorphisms) we haven’t actually forgotten the data of which elliptic curves are isomorphic. This process, as one can see with a little thought, totally, utterly, and in the most obvious way, has eliminated non-trivial automorphisms. But, by doing so we have given up working with honest-to-god functors into $\mathsf{Set}$ and instead are working with functors valued in categories (more specifically, groupoids). Consequently we have no hope of accessing this by scheme theory since all their functors are $\mathsf{Set}$-valued.

Thus, we have entered into the realm of algebraic stacks. And, indeed, the functor $\mathcal{M}_{1,1}$ associating to any $T$ the groupoid of elliptic curves over $T$ (in the way described above) is actually a Deligne-Mumford stack. Now, stacks are geometric objects just like schemes. So, one might think that we could deduce interesting geometric results by studying the stack $\mathcal{M}_{1,1}$. This is not unreasonable and people have done this (this is the approach taken in this article).

But, for our purposes, and in general, the geometry will be much more reasonable/manageable if we stick to the land of schemes. This is achieved by adding level data which, in most cases, ‘rigidifies’ the objects enough to eliminate automorphisms, and thus give them a fighting chance of giving representable functors or, at least, having rich approximating schemes (i.e. coarse moduli spaces).

So, it turns out that these functors $\mathcal{M}(N)$ are well-behaved enough to allow interesting scheme-theoretic study. Namely, for all $N$ the functor $\mathcal{M}(N)$ has a coarse moduli space $Y(N)$. And, for $N\geqslant 3$, the scheme $Y(N)$ is actually a fine moduli space. This is not at all shocking. Indeed, one can easily see that $\mathcal{M}(N)$ is nothing but $\mathcal{A}_{1,N}^J$ from the last section, where $J$ corresponds to the element $1\in\mathrm{Mat}_1(\mathbb{Z}/N\mathbb{Z})$. So, these spaces are endowed with a canonical, inherent geometry coming from this moduli description.

One now sees the arithmetic content of the objects $Y(N)$. For example, a weak version of Mazur’s torsion theorem says that $Y(N)$ has no $\mathbb{Q}$-points for $N\geqslant 2$. Moreover, it is this sort of perspective, to attack the arithmetic of the modular curves, instead of individual elliptic curves, which has proven successful in similar situations. This is especially true since the moduli spaces $X(N)$ carry much more symmetry (the Hecke symmetry) than individiual elliptic curves.

Remark: Of course, if one is willing to define slightly different moduli problems, in particular the moduli problem associated to $Y_1(N)$, then one can actually state Mazur’s torsion theorem literally in terms of points on moduli spaces.

# Moduli spaces of Hodge structures

Now, if one wants to expand their net to capture other types of Shimura varieties more general than the PEL type mentioned in the last section, one has to leave the comfort of moduli spaces of abelian varieties (with extra structure). But, one might hope that there is another larger, natural class of Shimura varieties which still parameterizes geometric objects containing, as a proper subset, the theory of abelian varieties. This is the case (roughly) for the theory of Shimura varieties of Hodge type which (again roughly) parameterize Hodge structures.

So, let us begin by recalling the definition of a Hodge structure. Namely, let us fix a subring $A\subseteq\mathbb{C}$, and define an $A$Hodge structure on the finitely generated free $A$-module $M$ to be a decomposition

$\displaystyle M\otimes_A\mathbb{C}\cong\bigoplus_{(p,q)\in\mathbb{Z}^2}M^{p,q}$

where $M^{p,q}$ are $\mathbb{C}$-subspaces of $M\otimes_A \mathbb{C}$ and $M^{q,p}=\overline{M^{p,q}}$ for all $(p,q)$. We say that $M$ is pure of weight $n$ if $M^{p,q}=0$ for $p+q\ne n$. More generally, we define the type of $M$ to be the set of $(p,q)\in\mathbb{Z}^2$ with $M^{p,q}\ne 0$.

The most natural example, and the one which informs most of the theory of Hodge structure, comes from complex algebraic geometry. Indeed, let us suppose that $X$ is a compact Kahler manifold, then for all $n\geqslant 0$ the Hodge spectral sequence degenerates giving us the Hodge decomposition of $X$

$\displaystyle H^n_\mathrm{sing}(X,\mathbb{C})=\bigoplus_{p+q=n, p,q\geqslant 0}H^{p,q}$

where $H^{p,q}$ is isomorphic to $H^q(X,\Omega^p_\mathrm{hol.})$(the reversal of $(p,q)$ is related to the reversal in the Hodge spectral sequence) and such that Hodge symmetry holds (i.e. that $H^{q,p}=\overline{H^{p,q}}$). Thus, we see that this gives the $\mathbb{Z}$-module $H^n_\mathrm{sing}(X,\mathbb{Z})$ the structure of a pure $\mathbb{Z}$-Hodge structure of weight $n$.

Now, the general importance of Hodge structures, beyond the obvious importance in literal Hodge theory, is that they are, in some sense, a shadow of the theory of ‘motives’. I will not (at least now) define what this rigorously means, but suffice it to say that motives are supposed to be geometric objects which encapsulate a sort of universal cohomology theory (see this nice article of Mazur).

To make this much more concrete, let us give the simplest possible example of a Hodge structure. Namely, we claim that Hodge structures of type $\{(-1,0),(0,-1)\}$ on a real vector space $V$ correspond to complex structures (i.e. $\mathbb{R}$-algebra maps $\mathbb{C}\to\mathrm{End}_{\mathbb{R}}(V)$). Indeed, to a complex structure (which corresponds to a matrix $J$ with $J^2=-1$) we can associate the decomposition

$V\otimes_\mathbb{R}\mathbb{C}=V^{-1,0}\oplus V^{0,-1}$

where $V^{-1,0}$ is the $i$-eigenspace of $J$ acting on $V_\mathbb{C}:=V\otimes_\mathbb{R}\mathbb{C}$ and $V^{0,-1}$ is the $-i$-eigenspace (note that $J^2=-1$ implies that on $V_\mathbb{C}$ $J$ is diagonalizable with eigenvalues $\pm i$). Conversely, suppose that we have a Hodge structure on $V$ of type $\{(-1,0),(0,-1)\}$. Then, consider the element $J_\mathbb{C}$ of $\mathrm{GL}(V_\mathbb{C})$ which acts by multiplication by $i$ on $V^{-1,0}$ and $-i$ on $V^{0,-1}$. Note then that $J_\mathbb{C}$ commutes with complex conjugation, and so restricts to a map on $V$.

Remark: The odd sign convention, choosing $\{(-1,0),(0,-1)\}$ instead of $\{(1,0),(0,1)\}$, is related to Deligne’s sign conventions. They are though, in essence, just sign conventions, and shouldn’t cause the reader too much worry, except to be reminded that it’s important to stay consistent. The rough reason for this choice is that we normalize our sign conventions so that cohomology has positive degrees which, as one proceeds in the theory, forces the complex structures to have negative degrees.

But, it’s still unclear how this relates to any of the mathematics we’ve been talking about in the previous section. The most obvious tie-in is the following:

$\left\{\text{Abelian varieties }/\mathbb{C}\right\}\longleftrightarrow\left\{\begin{matrix}\text{polarizable }\mathbb{Z}\text{-Hodge structures}\\ \text{of type }\{(-1,0),(0,=1)\}\end{matrix}\right\}$

Let us not say what it means for a Hodge structure to be polarizable (it’s related to the existence of a Riemann form), but, instead, describe what the map is. To an abelian variety $A/\mathbb{C}$ (thought about as a complex Lie group) we associate the homology group $H_1(A,\mathbb{Z})$ which is a Hodge structure of type $\{(-1,0),(0,-1)\}$ coming from the Hodge decomposition of its dual $H^1_\mathrm{sing}(X,\mathbb{Z})$ (the polarization, again, is coming from the Riemann form). Conversely, suppose that $M$ is a polarizable $\mathbb{Z}$-Hodge structure. Then, we can associate the complex torus:

$(M\otimes_\mathbb{Z}\mathbb{R})/M$

where we use the fact we have a Hodge structure of type $\{(-1,0),(0,-1)\}$ to deduce that $M\otimes_\mathbb{Z}\mathbb{R}$ does, in fact, have a complex structure (the algebraicity of this torus follows from the existence of a polarization/Riemann form).

Broad relationship to Shimura varieties: There will be a certain class of Shimura varieties (a subset of those of ‘Hodge type’) which are moduli spaces of Hodge structures. But, more generally, almost all of the defining ‘Deligne axioms’ for a Shimura variety are created with an eye towards making something like a ‘moduli space of Hodge structures.’

## The case of modular curves

To relate modular curves to moduli spaces of Hodge structures, we begin again with the observation that, as we proved in the last section, complex structures on the real vector space $V$ correspond to Hodge structures of type $\{(-1,0),(0,-1)\}$.

So, what does this have to do with modular curves? Let us think about how we usually try and parameterize elliptic curves. We usually consider the space $\mathbb{C}/\Lambda$ as we fix $\mathbb{C}$ and vary $\Lambda$. What happens if we turn this on its head? Namely, think of the lattice $\mathbb{Z}^2\subseteq\mathbb{R}^2$ as being fixed, and vary the complex structure on $\mathbb{R}^2$—we again have parametrized elliptic curves. Said differently, we have fixed the real Lie group $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$ and are varying the complex structures on $\mathbb{R}^2$ to get all the complex structures on $S^1\times S^1$.

So, we have seen how parameterizing Hodge structures is essentially what $Y(1)$ does, and, with some extra work (that we won’t do right now) one can soup this up to deal with level structures as well. The rough idea being that extra ‘Hodge type’ data can be added in by focusing not only on Hodge structures but ‘Hodge tensors’. This will encode, for example, endomorphisms. Thus, we see that there is actually a fairly concrete and explicit way in which $Y(1)$ and (plausibly) $Y(N)$.

Remark: One thing worth noticing in our discussion above was that absence of the phrase polarizable. Namely, we gave an equivalence of categories between between abelian varieties and polarizable Hodge structures of type $\{(-1,0),(0,-1)\}$. So, when we restrict to dimension $1$ abelian varieties, we expect to have to say the phrase ‘polarizable’ as well. This is actually not the case since all dimension $2$ $\mathbb{Z}$-Hodge structures are polarizable. This is related to the fact that all compact Riemann surfaces are algebraic (although much, much simpler)!

# Realizations of the global Langlands correspondence

This is, for me, the most tantalizing use of Shimura varieties. But, of course, it’s also the one which needs, by far, the most buildup to state correctly. So, this is the one where I am going to be the most handwavey.

The global Langlands correspondence is part of a triangle of deep conjectures attempting to relate three of the main branches of modern mathematics: number theory, algebraic geometry, and (harmonic) analysis. It tries to do so using the unifying subject of representation theory. The triangle looks roughly something like the following:

$\begin{matrix} & & \{\text{automorphic forms}\} & & \\ & \nearrow & & \nwarrow & \\ \{\text{Galois representations}\} & & \longrightarrow & & \{\text{algebraic varieties over }\mathbb{Q}\}\end{matrix}$

where automorphic forms is the (harmonic) analysis, Galois representations is the number theory, and algebraic varieties is the algebraic geometry. The arrows are supposed to be representing some sort of ‘equivalence’.

The arrow connecting Galois representations and algebraic varieties is something along the line of the Fontaine-Mazur conjecture. The arrow connecting Galois representations and automorphic forms is something like the Langlands conjecture, which is what we’d like to focus on.

Very broadly, one might like, for various algebraic groups $G/\mathbb{Q}$ to relate representations of $G(\mathbb{A}_f)$ with representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/E)$ for some number field $E$. In other words, we want to represent ‘analytic representations’ with ‘number theoretic ones’—opening both sides to techniques not previously thought possible.

Now, if we want to connect these representations, the first obvious step might be to even have them show up to the same party. Namely, we’d like to find a space on which both $G(\mathbb{A}_\mathbb{Q}^\infty)$ and $\mathrm{Gal}(\overline{\mathbb{Q}}/E)$ act linearly and commutatively. In particular, we want a reasonable space on which $G(\mathbb{A}_\mathbb{Q}^\infty)\times \mathrm{Gal}(\overline{Q}/E)$ acts.

So, how does one usually produce representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/E)$? Perhaps the most fruitful way of producing such representations is by using geometry—by using étale cohomology. Specifically, if $X/E$ is some smooth variety, then $H^i_{\mathrm{\acute{e}t}}(X_{\overline{\mathbb{Q}}},\overline{\mathbb{Q}_\ell})$ is a continuous $\overline{\mathbb{Q}_\ell}$-representation of $\mathrm{Gal}(\overline{\mathbb{Q}}/E)$ and representations of this form make up all ‘reasonable’ Galois representations (cf. again, the Fontaine-Mazur conjecture).

But, how do we get representations of $G(\mathbb{A}_\mathbb{Q}^\infty)$ on such geometric objects? There is a brilliant, but simple way of doing just that. Namely, what if instead of just one $X/E$ we had a family $S_K/E$ of algebraic varieties indexed by (some) open compact subgroups $K\subseteq G(\mathbb{A}_\mathbb{Q}^\infty)$ and for which $g\in G(\mathbb{A}_\mathbb{Q})$ carries an isomorphism over $E$:

$S_K\xrightarrow{g}S_{gKg^{-1}}$

Then, the direct limit

$\displaystyle \varinjlim_K H^i_{\mathrm{\acute{e}t}}((S_K)_{\overline{\mathbb{Q}}},\overline{\mathbb{Q}_\ell})$

will carry an action of $G(\mathbb{A}_\mathbb{Q}^\infty)\times\mathrm{Gal}(\overline{\mathbb{Q}}/E)$ as desired.

Moreover, if we had an understanding of $S_K^\mathrm{an}$ well, then we could choose an isomorphism $\overline{\mathbb{Q}_\ell}$ and transform the above into the group

$\displaystyle \varinjlim_K H^i_\mathrm{sing}(S_K^\mathrm{an},\mathbb{C})$

which, with our understanding of the analytification, one might be able to understand well. In particular, if $S_K^\mathrm{an}$ was a quotient of a symmetric space, then we could hopefully relate this cohomology group to objects defined in harmonic analysis—in particular, we could use Matsushima’s formula.

Broad relation to Shimura varieties: The notion of Shimura varieties will allow us to associate to $G$ this group $S_K$ of varieties with the desired properties (including the analytification being quotients of symmetric spaces by discrete subgroups).

## The case of modular curves

Explaining what happens here for modular curves would take us too far afield to make any sense. So, let us delay this discussion until later where we will discuss it at great length.