In what I hope will become a series of posts, I want to think about the following question (to which, at the time of writing of this post, I have no idea of the complete answer).

Question: Given a Shimura datum $(G,\mathfrak{X})$ and level $K \subset G(\mathbb{A}_f)$, giving rise to a Shimura variety $M=M(G,\mathfrak{X},K)$ defined over the number field $E$, and given $\mathfrak{p}$ some prime of $E$, when does $M$ have good reduction at $\mathfrak{p}$?

In this post we sketch the significance of this question in the theory of (nice) automorphic forms.

Motivation from arithmetic geometry
In the theory of abelian varieties, recall the Neron-Ogg-Shaferevich criterion which tells us that (assuming $l \not= p$) an abelian variety has good reduction if and only if its $l$-adic cohomology is unramified as a Galois representation. For more general proper algebraic varieties over a number field/local field, only one direction of this theorem survives. Suppose $X/K$ has good reduction: i.e. (let’s base change to temporarily assume $K$ a local field if necessary and let $k$ be the residue field) it admits a smooth proper model $\mathcal{X}/\mathcal{O}_{K}$. Then for any lisse etale sheaf $\mathcal{F}$ on $\mathcal{X}$, the proper smooth base change theorem furnishes us with a canonical isomorphism between the cohomology groups

$H^i(X_{\bar{K}}, \mathcal{F}) \cong H^i(\mathcal{X}_{\bar{k}}, \mathcal{F})$.

Since this is canonical, it commutes with Galois action, which implies in particular that as a Galois representation $H^i(X_{\bar{K}}, \mathcal{F})$ is unramified.

Similarly, at $l=p$ there is a result that if a variety has good reduction, its cohomology is crystalline. By judicious choice of $l$ (or using two different primes $l$) it is often possible to avoid thinking about this.

In the land of arithmetic geometry one can therefore note the following theorem.

Fix $X$ smooth and proper over a number field $E$.

Consider the sets:
$S_{bad} = \{$ primes of $E$ where $X$ has bad reduction $\}$
$S_{galois} = \{$ primes of $E$ where part of the cohomology of $X$ is ramified/not crystalline$\}$.

Theorem: $S_{galois} \subseteq S_{bad}$.

Remark: Neron-Ogg Shaferevich implies equality for abelian varieties, though in general equality does not hold (for example, there exist curves with bad reduction but whose Jacobian has good reduction).

A whistlestop tour of the theory of “nice” automorphic forms
Let us now return to Shimura varieties (and a longish sketchy digression into the theory of automorphic forms based loosely on notes by Teruyoshi Yoshida). Suppose we have some automorphic representation of $G$ geometrically defined according to the following recipe. Take $V$ an appropriate $G(\mathbb{A}_f)$-equivariant vector bundle on $M(G,\mathfrak{X})$, extending to a bundle on an appropriate compactification at each level. Then at each level $K$, we define $\mathcal{A}_V(K)$ to be the sections of this bundle over $M(G,\mathfrak{X},K)$. It is also clear we can associate $K' \subset K$ to $\mathcal{A}_V(K') \supset \mathcal{A}_V(K)$, so we can consider the union $\mathcal{A}_V$: the (infinite-dimensional) space of “automorphic forms of weight $V$“.

Crucial example: Taking $G=GL_2$ (and the choice of $\mathfrak{X}$ corresponding to putting a complex structure on $\mathbb{R}^2$), the Shimura variety with sufficiently fine level structure admits an interpretation as the moduli space of elliptic curves together with a level structure defined on torsion points. Taking $V$ to be the $k$th tensor power of the line bundle corresponding to the relative cotangent space of the universal elliptic curve, we recover $\mathcal{A}_V$ as the space of modular forms of weight $k$ (with all levels considered simultaneously).

Now, $\mathcal{A}_V$ is a (large) representation of $G(\mathbb{A}_f)$, and it satisfies some rather juicy properties.

– It’s smooth: equal to the union of its invariants by open compact subgroups (by definition). This is very useful, as the category of smooth representations of $G(\mathbb{A}_f)$ is abelian: we can talk about irreduciblity, subrepresentations, quotients, etc.
– It’s admissible: the space of invariants by any open compact is finite dimensional (since spaces of sections of vector bundles over a complete space are finite dimensional).

We now say that a representation is automorphic of weight $V$ if it is an irreducible subrepresentation of $\mathcal{A}_V$. There are a distinguished subset of these representations consisting of forms satisfying a vanishing property at each unipotent radical of a parabolic subgroup of $G$, which we shall refer to as cuspidal representations. These objects are those of a kind to which the Langlands philosophy would have us attach a Galois representation. WARNING: usually automorphic representations are given with components at infinite places, but we systematically suppress these here.

Some readers may be rather more familiar in the case of $GL_2$ of taking a newform (a particularly nice cusp form: it’s a Hecke eigenform and doesn’t come from some lower level) and attaching a Galois representation to this. It turns out (at least for $GL_2$) that one can make a correspondence between newforms of weight $k$ and automorphic representations of weight $k$.

Given a newform $f$, we get an obvious cuspidal representation $G(\mathbb{A}_f)f$ (that was easy! :)).

Conversely (and here I don’t know how general we can make the argument: we need multiplicity 1 and other facts about representations splitting up nicely into local factors), given a cuspidal representation $\pi$ of $GL_2$, there is a largest “unipotent mod p” subgroup $U_1(p^n)$ of $GL_2(\mathbb{Z}_p)$ which fixes precisely a one-dimensional subspace of the local factor $\pi_p$. In fact for almost all primes, we will have $U=GL_2(\mathbb{Z}_p)$, so these glue into a canonical (“unipotent mod N for N minimal”) open compact subgroup $U_1(N)$ of $GL_2(\mathbb{A}_f)$ with $\pi^{U_1(N)}$ a 1-dimensional space of automorphic forms. Picking a generator for this vector space (say, one determined by a normalisation condition of the q-expansion at our favourite cusp), we recover our newform.

One might also like to know that the nebentypus of the newform is precisely the Dirichlet character corresponding to the central character of our cuspidal representation.

Back to the main story:

Now, given an automorphic representation, it is true (at least for a sufficiently nice group like $GL_n$) that we can break it up as a restricted tensor product $\pi = \otimes_p \pi_p$, where for almost all $p$ we have $\pi_p^{G(\mathbb{Z}_p)} \not= 0$. At primes where such a fixed vector exists, we say the representation is unramified, and we can define the (finite) set of all other primes:

$S_{autom} = \{$rational primes where $\pi$ is ramified $\}$.

Of course, with more work and complications which I don’t have time to work through, one could consider automorphic forms over an arbitrary number field $E$ and then rather than considering rational primes it would be sensible to consider primes of that number field.

Now, the Langlands philosophy predicts that we should be able to attach a Galois representation to a cuspidal representation, and that in this context we should have $S_{autom} = S_{galois}$.

In many cases, since we are dealing with automorphic forms which already live on algebraic varieties, it is possible to realise the Langlands correspondence in the cohomology of our Shimura variety. Matsushima’s formula gives, roughly speaking (and ignoring a cornucopia of serious issues):

$H^*(M(G,\mathfrak{X}), V) \cong \bigoplus_{\pi \text{ cuspidal of level } V} (\pi \otimes W_\pi)$ (where $W_\pi$ is a Galois representation attached to $\pi$).

And now the punchline…
And now finally we can start to see why our question about reduction of the Shimura variety at level $K$ is relevant. If we have a cuspidal representation $\pi$ that is unramified at a prime $p$, then for a sufficiently high level $K$ which may be kept prime to $p$ (still containing a whole hyperspecial subgroup at $p$), we can find the Galois representation attached to $\pi$ in the (finite-dimensional) cohomology group $H^*(M(G,\mathfrak{X}, K), V)$.

It follows that if we can show $M(G,\mathfrak{X}, K)$ has good reduction at $p$ (for $K$ sufficiently small but prime to $p$), in other words that $S_{autom} \supseteq S_{bad}(M)$, then by proper smooth base change we establish one direction of the result predicted by Langlands, namely that

$S_{galois} \subseteq S_{autom}$.

Let’s take stock. In light of our above discussion it seems sensible to refine our question to the following conjecture (which could be completely naive and wrong – I still don’t really know).

Possibly Naive Conjecture: Consider the Shimura variety $M=M(G,\mathfrak{X},K)$ defined over the number field $E$, and let $\mathfrak{p}$ be a prime of $E$. If $K$ is sufficiently small but contains a hyperspecial maximal subgroup at $\mathfrak{p}$, then $M$ has good reduction at $\mathfrak{p}$.

In this post we sketched the following consequence (though at stages we possibly needed $G=GL_n$).

Consequence: Let $\pi$ be an automorphic form arising from a vector bundle on $M$. Then any unramified prime of $\pi$ is also an unramified prime of the associated Galois representation.

Let us remark that in some sense the author would guess this should be the “easy” direction of the conjecture $S_{galois} = S_{autom}$. Ribet’s famous result on level-lowering for modular forms (a crucial ingredient in the proof of Fermat’s last theorem) is an example of something which goes the other way: it takes a Galois representation coming from a modular form, but with fewer primes ramifying than ramify in the modular form, and deduces the existence of a corresponding modular form of appropriately lower level (i.e. an automorphic representation ramifying at fewer primes). Also, true results in this direction at least hint that the properties “good reduction” and “unramified cohomology” might be closer for Shimura varieties than general varieties.

That took longer than I was planning, and I should probably stop here. If anyone wants to post comments clarifying some of the things which I left vague or passed to a special case mainly out of ignorance, that would be very useful. Next time I shall consider at least the example of modular curves, and at most the example of arbitrary PEL varieties. I shall also (perhaps slightly superflously to the task in hand) try to give a survey of the beautiful deformation theory of abelian varieties in characteristic p.