What is a Shimura variety and why should I care about them?

Question

Shimura varieties have come up tangentially in talks with some of my advisors. My vague understanding is that they are "things that behave like moduli spaces of abelian varieties having some additional structure". I am familiar with the modular curves $X(\Gamma)$ for $\Gamma$ a congruence subgroup of $SL_2(\mathbb{Z})$, and I know they are examples of Shimura varieties. In particular $X_0(N)$ is the moduli space of elliptic curves equipped with a cyclic $N$-isogeny, etc.

I have tried to understand the Wikipedia definition of a Shimura variety, but it is pretty unintelligible to me. However, I can identify some features that are analogous to the modular curves situation, for example Wikipedia's construction involves a double coset space, similarly to constructing $X(\Gamma)$. (That said, the definition uses a lot of Lie theory and algebraic group theory, topics I am less familiar with than number theory or algebraic geometry.)

Perhaps someone could explain why Shimura varieties are important to number theory and how they fit in to problems we care about. I would really appreciate if someone could explain how the relationship between modular curves and their algebraic geometry, modular forms, and elliptic curves generalizes. Another vague intuition I have is that the modular curves are very "special", in that they have a complex-analytic-geometry interpretation too. You get this very nice, concrete realization and I would be very surprised if something similar happened if you looked at the moduli space of elliptic curves with $N$-cyclic isogeny over $\mathbb{Q}_p$, for example. How does this idea carry over when thinking about Shimura varieties? Is there a modularity theorem for Shimura varieties?

Thanks so much!

Answer 1

This sounds a lot more appropriate for MO than MSE, and, to be honest, as I don’t know very much about Shimura varieties, I’d like an authoritative answer myself. I think the broad idea of what follows is mostly correct, but since you know algebraic geometry and number theory, you’re aware that there can be a gigantic cliff of technical work between such an explanation and any kind of formal definition.

My understanding is that Shimura varieties are basically one higher-dimensional generalization of modular curves. The big picture should be as follows: we want to relate automorphic forms (invariant subspaces of “something like” $L^2(G(A_K))$ where $A_K$ is the ring of adeles of a number field $K$ and $G$ some reductive group, typically $GL_n$, or $SL_n$, or a symplectic, orthogonal, or unitary group) to Galois representations.

Why do we want that? For starters, because of the $L$-functions. It’s unclear that Galois $L$-functions have an analytic continuation or a functional equation, and it’s difficult to attach a meaning to central $L$-values (like in BSD or Bloch-Kato, even in the case of rank zero) – but these are somewhat clearer for an automorphic $L$-function.

How do we do that? Look at the case of modular curves. They’re algebraic varieties with Hecke correspondences. So we can “split” their cohomology according to Hecke eigenspaces. The coherent cohomology (I guess) yields the automorphic side. To get the Galois representations, we use the étale cohomology, which requires an algebrization of the modular curve – and also has Hecke eigenspaces.

But what do we do for representations that are not bidimensional? The Langlands conjectures predict that they should be associated to automorphic forms for bigger groups.

But given an automorphic form $F$, how can we construct an associated Galois representation? Well, our best bet (and by that, I mean: one of the only things we know) is to find some algebraic variety with a Hecke action such that the system of eigenvalues of $F$ appears in this Hecke action (in the coherent cohomology). Then we can take the corresponding eigenspace of the étale cohomology. We see here a non-trivial interplay of what’s supposed to be a real-analytic structure (in the general case) with an actual algebraic structure.

Perhaps the easiest examples of Shimura varieties are, in fact, the moduli spaces for abelian varieties of fixed dimension with polarization and level structures, generalizing the case of $X_0(N)$ to higher-dimensional abelian varieties.

(Note that there isn’t really a good algebraic moduli space of elliptic curves with an $N$-isogeny over $\mathbb{Q}_p$ – unless that would just be $X_0(N)(\mathbb{Q}_p)$).

The algebraicity of Shimura varieties also lets one consider them as $p$-adic analytic spaces (of some sort), and this is useful to consider arguments based on $p$-adic deformations (read: methods that basically started with FLT and have been expanded since) or generalizations of Hida theory.