In this reasonably sketchy post [and I’ll include the warning that I’m fairly new to lots of the content, so there may be errors] I want to outline the main example of Deligne’s `Travaux de Shimura’ paper, namely his consideration of the reductive group {G=GSp_{2n}} together with a nice class of Hodge structures, and the interpretation of the resulting Shimura variety as parameterising {n}-dimensional abelian varieties together with a principal polarisation and level structure. This interpretation allows one to take a coarse moduli space for such things – already known to be defined over {{\mathbb Q}} – as a model for the Shimura variety. Note that we don’t worry about trying to encorporate the extra data of endomorphism structures, though it isn’t much harder to include them in the picture.

Let’s just recall quickly the general setup of Shimura varieties. The idea is that we want to take a reductive group {G} over {{\mathbb Q}} and study the adelic quotient {G(\mathbb{A})/G({\mathbb Q})}. For example, if {G=Res_{E/{\mathbb Q}}(\mathbb{G}_{m,E})}, we recover the idele class group of {E}. In general I guess this space is supposed to be cool because it has an obvious action by {G(\mathbb{A})}: it has `loads and loads of symmetries’, but it’s also a nice global arithmetic object (you can tell I don’t really understand why these should cool, except that the idele class group, modular forms and their various generalisations all live inside this picture in a nice way).

In general however I guess this quotient is too outrageously big to work with as a whole, so we introduce some other gadgets to chop it down a bit. Let {\mathbb{S}} denote the Weil restriction of {{\mathbb C}^*} from {{\mathbb C}} to {{\mathbb R}}, so it’s an algebraic group whose {{\mathbb R}}-points are {{\mathbb C}^*} and whose {{\mathbb C}}-points are two copies thereof. A Hodge structure on {G} is a map of real algebraic groups {\mathbb{S} \rightarrow G_{\mathbb R}}. The reason for this name is that for any representation {G \rightarrow GL(V)} of {G}, one can base change to {{\mathbb C}}, and the eigenspaces for {{\mathbb C}* \times {\mathbb C}*} give rise to a Hodge bigradation on {V_{\mathbb C}}. Such a homomorphism has a natural conjugation action by {G({\mathbb R})}, and we denote the centraliser for this by {K_\infty} (so {K_\infty \backslash G({\mathbb R})} is precisely the conjugacy class {\mathfrak{X}} of {h}). We can now define (usually only for {h} satisfying certain hypotheses) our Shimura variety

\displaystyle Sh_{\mathbb C}(G,h) = K_\infty \backslash G(\mathbb{A}) / G({\mathbb Q}) = (\mathfrak{X} \times G(\mathbb{A}_f)) / G({\mathbb Q}).

This is still in general rather large (the ring of finite adeles is a big profinite object). We therefore tend to also take some open compact subgroup {K \subset G(\mathbb{A}_f)} and quotient out by that to form {Sh_{\mathbb C}(G,h,K)}. This normally (at least in nice circumstances) reduces our massive set to being some nice complex analytic space (the complex analytic structure comes from {\mathfrak{X}}). Indeed, a theorem of Borel and Baily tells us that under certain fairly general conditions this can be realised as a quasiprojective complex algebraic variety. We can then recover the whole Shimura variety as an inverse limit of these.

However, to extract interesting arithmetic information (in Deligne’s paper, explicit reciprocity laws, but I’d guess in modern applications we’re more interested in higher dimensional Galois representations), the goal is to construct models for such things over fields smaller than the complex numbers (hopefully number fields). A model over a field {E} is defined to be a scheme {M_E(G,h)} which still has the juicy {G(\mathbb{A}_f)}-action, and whose base change to {{\mathbb C}} is isomorphic to {Sh_{\mathbb C}(G,h)}. It turns out there is a natural field {E(G,h)} one should hope to find models over (the field of definition of a certain cocharacter over {{\mathbb C}} coming through {h}). Also, if {G} is abelian the associated Shimura varieties are zero-dimensional, so the construction of models just comes down to studying a Galois action, where existence and uniqueness is clear. For general {G} one can then use maps from abelian groups to impose conditions which would be satisfied by a canonical model, which then one can show is, if it exists, unique and in some sense functorial.

Now we have the general framework, let’s crack on with our example. We will take {G=GSp_{2n}}, the group of symplectic similitudes on some fixed vector space {V_{\mathbb Q}} equipped with a symplectic form {\psi}. For {h} we take the most obvious possible Hodge structure you can choose if you try to define it explicitly. To be more precise, I think it can be characterised (with a bit of work: many thanks to Yihang Zhu for explaining many of the details to me earlier tonight) as the unique conjugacy class of all Hodge structures induced by a choice of complex structure {J} on {V_{\mathbb R}} (it will have Hodge type {(-1,0)+(0,-1)} corresponding to the {i,-i} eigenspaces of {J}) with the property that {\psi(x,Jx)} is either positive or negative definite (let’s call these `nice’ complex structures).

So given these data we can form a Shimura variety (and let’s also fix {K} an open compact in {G(\mathbb{A}_f)}). What are the points of this Shimura variety? They are elements of {\mathfrak{X} \times G(\mathbb{A}_f)} modulo the actions of {G({\mathbb Q})} and {K}. Elements of {\mathfrak{X}} correspond to nice complex structures and elements of {G(\mathbb{A}_f)} to symplectic automorpisms {\alpha} of {V \otimes \mathbb{A}_f}. What about the equivalence relation? We need only consider classes of {\alpha} modulo {K}. To encorporate the {G({\mathbb Q})} action requires a slightly subtler idea: we think of these objects being intrinsic objects on some other vector space {H_{\mathbb Q}} which happens to be isomorphic to {V_{\mathbb Q}}, and this data is fixed under an obvious {G({\mathbb Q})} of automorphisms.

The upshot is that the points of our Shimura variety correspond to {{\mathbb Q}}-vector spaces {H_{\mathbb Q}} which possess an isomorphism to {V_{\mathbb Q}} inducing on them a symplectic form (also abusively called {\psi}), together with a nice complex structure {J} on {H_{\mathbb R}} and a {K}-orbit of symplectic {\mathbb{A}_f}-linear isomorphisms {k: H_{\mathbb Q} \otimes \mathbb{A}_f \rightarrow V_{\mathbb Q} \otimes \mathbb{A}_f}. Note also that the datum of a symplectic form on {H_{\mathbb Q}} is only fixed up to rescaling by {{\mathbb Q}^*} (this is the difference between {Sp} and {GSp}).

And now the miracle. Suppose we have such a datum. Fix any integer lattice {\Lambda} in {H_{\mathbb Q}} and viewing {H_{\mathbb R}} together with {J} as a complex {n}-dimensional vector space, we can form the complex torus {(H_{\mathbb R}, J)/\Lambda}. Furthermore, by niceness of {J} this admits a Riemann form (after freely rescaling the symplectic form by an appropriate element of {{\mathbb Q}^*}, which is totally acceptable), so in fact it is a polarised abelian variety {A}. Finally, we note that {H_{\mathbb Q} \otimes \mathbb{A}_f} is just the (isogeny) Tate module {\hat{V}(A)} of {A}, so {k} can be interpreted as a level structure in a nice familiar way.

Let’s take stock. Once one works out the extent to which data is kept track of, we get the following. Note that we chose a lattice, and also our polarisation was defined over {{\mathbb Q}} and up to {{\mathbb Q}^*}-action, and we had an isogeny Tate module. The easiest moduli interpretation is therefore at the level of isogeny classes: and we get our points corresponding to the data:

An abelian variety {A} considered up to isogeny.
An `isogeny-polarisation’ of {A} ({{\mathbb Q}}-bilinear form on {H_1(A,{\mathbb Q})} with definiteness property), modulo {{\mathbb Q}^*}-rescaling.
An isomorphism {V \otimes \mathbb{A}_f \rightarrow \hat{V}(A)}, considered modulo {K}-action.

It is easy to see how to define the inverse map (from these data to those we previously identified with our Shimura variety). Just associate {A \mapsto H_1(A,{\mathbb Q})} with the obvious Hodge structure, and the level structure and polarisations correspond in a clear way.

Note that it is possible, by fixing a lattice in {V} and making some fiddly but not all so difficult arguments, to go one step further and interpret these data as honest abelian varieties together with a principal polarisation and a level structure (but it’s getting late, so maybe I will omit these details for now).

Once we are here, we have the awesome fact that people have already constructed moduli spaces for these objects, and these moduli spaces are defined over {{\mathbb Q}}, so we automatically get rational models for these Shimura varieties. We started with a priori very analytic but highly symmetric objects, and discovered that they are also equipped with an action by the absolute Galois group of {{\mathbb Q}} which must surely have an intimate and mysterious relationship with the wealth of pre-existing symmetries.

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