Today I’ve found myself mainly thinking about the interaction between Frobenius and the filtration on crystalline cohomology, referring to Mazur’s classic paper http://projecteuclid.org/download/pdf_1/euclid.bams/1183533965 as well as Ogus’ paper on “Griffiths Transversality in Crystalline Cohomology” and thought I would record a summary here before I forget everything. Of course, both of these papers are from the 1970s, so if there has been any significant advance or later examples of interest that any readers know of I’d be extremely interested to hear.

Firstly the setup. Let $X_0/k$ be a smooth projective variety over a perfect field of characteristic $p$, and let us also suppose we have a smooth proper lift $X/W(k)$. We can form the crystalline cohomology and have comparisons $M:= H^i_{crys}(X_0/W(k)) =H^i_{crys}(X/W(k)) = H^i_{dR}(X/W(k))$. Let us assume these are free modules and in fact also that the Hodge cohomology groups $H^q(X, \Omega^p_{X/W(k)})$ are free. Then we have the further relation that $M/pM = H^i_{dR}(X_0/k)$.

What structures are in play? Algebraic de Rham cohomology comes with a Hodge filtration $F^k \subset M$ (which depends on the lift $X$, though of course mod $p$ it does not), and crystalline cohomology is equipped with a semilinear Frobenius $\Phi: \sigma^*M \rightarrow M$ (which does not depend on the lift).

The fundamental relationship between these two structures is given by a theorem of Mazur, which implies under the freeness assumptions we have made that the Frobenius determines the mod $p$ Hodge filtration.

Theorem (Mazur): The reduction mod $p$ of $\Phi^{-1}(p^jM)$ is precisely the reduction of $F^j$.

(Note for the statement that Frobenius induces an isomorphism $\Phi^*M/p\Phi^*M = H^i_{dR}(X_0^{(p)}/k) \rightarrow H^i_{dR}(X_0/k)=M/pM$ preserving the Hodge filtration.)

One immediate consequence of this is the conjecture of Katz relating the slopes of Frobenius to the shape of the Hodge filtration.

Corollary: For the crystalline cohomology of $X_0/W(k)$, the Newton polygon lies above the Hodge polygon.

We do not explain what this means here, except to remark that Mazur’s theorem allows you to find bases in which the matrix for $\Phi$ has columns divisible by powers of $p$ of widths given by the Hodge numbers. The statement, which is about relating these numbers to valuations of eigenvalues, is then just an easy result in linear algebra.

The next obvious question is what can we say about the Hodge filtration $F^k$ coming from our smooth lift? The above result tells us that

$\Phi(\sigma^*F^k) \subset p^kM + p\Phi(\sigma^*M)$.

Mazur was also able to prove that $\Phi(\sigma^*F^k) \subset p^{v(k)} M$, where $v(k) := \text{sup}_{l \geq k} v_p(p^l/l!)$.

Question (Mazur): Is it in fact the case that $\Phi(\sigma^*F^k) \subset p^k M$? If so, we say $M$ is strongly divisible, and this statement is equivalent to Frobenius inducing an isomorphism $\sum_i p^{-i} \sigma^*F^i \rightarrow M$.

Note that whenever the Hodge filtration has length shorter than $p$, this is immediate from Mazur’s second inequality, since $v(k)=k$ in this case for all nonempty pieces of filtration. For example, for the cohomology of a curve or $H^1$ of an abelian variety we do get something strongly divisible.

The reason for the distance between Mazur’s estimate and the notion of strong divisibility is the phenomenon of Griffiths transversality, which was investigated by Ogus in the crystalline context following Griffiths’ work in classical Hodge theory.

Classically suppose you have a variety $X/\mathbb{C}[[t]]$ and you want to study its de Rham cohomology. Using the Gauss-Manin connection, one can identify the cohomology group itself with the cohomology of the constant family $X'$ defined by the fibre at $t=0$. However, the Hodge filtrations will not agree, but are allowed to vary within the confines imposed by Griffiths transversality. Explicitly, one can show that

$F_{X}^k \subset \sum_{i \geq 0} t^i/i! F_{X'}^{k-i}$.

In the crystalline situation, given two smooth lifts $X,X'$, one obtains a similar formula,

$F_{X}^k \subset \sum_{i \geq 0} p^i/i! F_{X'}^{k-i}$.

However, note that the denominators start to cancel off the powers of $p$, once $i \geq p$, and this is exactly why Mazur can only get the estimate involving $v(k)$. For a more general statement in the same vein see Ogus’ “corollary 2.5.”

These are all inequalities, and it is natural to ask if they are “strict”. In other words, does Griffiths transversality really happen (do these filtrations vary), and to what extent? One obvious example to bear in mind is abelian varieties, where every possible lift of the Hodge filtration corresponds to a lift (and determines it uniquely: this is Grothendieck-Messing theory). If, as the filtration lengths increase, there is enough freedom for the lifts to vary widely within the constraints imposed by transversality, then one would expect a counterexample to strong divisibility.

Ogus manages precisely this, and his example is as follows. Suppose $p>2$, and consider the hypersurface $X'/W(k)$ given by

$X': X_0^{p+2} + \dots X_{p+1}^{p+2} + pX_0\dots X_{p+1} = 0$.

Then $\Phi(\sigma^*(F^p_{X'}(H^p_{crys}(X_0/W(k)))))$ fails to lie in $p^p H^p_{crys}(X_0/W(k))$.

This is achieved by comparing to the Fermat hypersurface which also visibly lifts the special fibre. One can show that in fact the cohomology of this is strongly divisible, exploiting the fact the group actions in play give an explicit decomposition of the cohomology, making it easy to control. Comparing the filtration coming from this to the filtration coming from $X'$, using a careful deformation-theoretic study of these differences, Ogus deduces that the latter fails to be strongly divisible.

This blog post is a place to collect information about the Harvard number theory learning seminar spring 2014. We meet Mondays 4:15-6pm (actual times: if you run on Harvard time, we meet at 4:08) in SC 507.

After a brief intense seminar on Scholze’s thesis and the MSRI conference, we are currently reading the paper of Matt Emerton on Local-Global compatibility for GL_2(Q) in the p-adic Langlands correspondence.

While nominally being organised by myself, this is now really being run by Erick Knight, to whom all angry complaints and difficult questions about the mathematical content and organisation should be addressed. I am still prepared to accept compliments, lavish gifts, etc. and still exercise the right to write dates in the format DD/MM.

Schedule of Talks

(2 introductory talks by Erick on the general p-adic Langlands program)

24/03 – Introduction, Completed cohomology and promodular representations (Rong)

07/04 – p-adic Langlands done correctly (Yihang)

14/04 – Completed Hecke Algebras 1 (Koji)

21/04 – Completed Hecke Algebras 2 (Bao?)

28/04 – Proof of Main Theorem (Yunqing)

04/05 – Application to the Fontaine-Mazur Conjecture (Tom)

In a short article appearing on the arxiv today Cuntz and Deninger seem to give a new simple construction of the ring of Witt vectors. Seems like good news for people like me who use them on a daily basis but would probably need to go looking for a copy of Serre’s “Local Fields” if asked for the details on how to actually construct them. This post represents my processing the definition appearing in their paper.

Firstly, recall that for any perfect $\mathbb{F}_p$-algebra $R$, a strict $p$-ring for $R$ is a ring $A$ in which $p$ is not a zero-divisor, Hausdorff and complete for the $p$-adic topology, and with $A/p \cong R$. The key fact about these rings is the following.

Proposition: For every perfect $\mathbb{F}_p$-algebra $R$, a strict $p$-ring $W(R)$ exists, and is unique up to unique isomorphism as a ring over $R$ (in fact $R \mapsto W(R)$ is a functor).

This is of course well-known. The new construction goes as follows. View $R$ as a monoid (remember multiplication and the identity element), and form the free ring $\mathbb{Z}R$ thereon. This has elements $\sum_i n_i[r_i]$ with addition as a “free $\mathbb{Z}$-module” and multiplication coming from the monoid structure. For example, this means that weirdly $[0]\not= 0$ yet (but $[1]=1$). There is also (induced from the identity on $R$ as a monoid) a natural ring map $\mathbb{Z}R \rightarrow R$. Let $I$ be its kernel, and form the $I$-adic completion $A$. It turns out (and the proof in their paper is pretty short) that this is a strict $p$-ring and so canonically isomorphic to $W(R)$.

One interesting feature of this construction is that of course you can run it for $R$ not perfect, where the Witt construction also gives you something but which is non-canonical (a Cohen ring’). Apparently this construction in general gives a different one from the Witt construction (so it is genuinely different: not a clever repackaging).

In this post I want to advertise Serre’s lovely note, following the opening section which gives a simple definition of p-adic modular forms (via q-expansions) and uses this and some elementary congruences to construct the classical p-adic L-functions. Serre’s writing is excellent so I give only a brief account and let the interested reader consult the master.

The definition goes as follows. Fix a level (in fact, let’s just take level 1), and a prime $p$ (which we will take to be odd, but the case $p=2$ appears no harder). By taking q-expansions, one can consider the algebra of classical modular forms of level 1 (all weights and indeed mixed weights) as a subalgebra $\mathcal{M} \subseteq \mathbb{C}[[q]].$ In fact, flat base change and the construction of models for modular curves implies one can actually do this on the integral and rational levels in such a way as loses no information: one is naturally led to study $\mathcal{M}_\mathbb{Z} \subseteq \mathbb{Z}[[q]]$ and $\mathcal{M}_\mathbb{Q} = \mathcal{M}_\mathbb{Z} \otimes \mathbb{Q}$. But now note that $\mathbb{Z}[[q]]\otimes \mathbb{Q} \subset \mathbb{Z}_p[[q]][1/p]$, a subring of $\mathbb{Q}_p[[q]]$ which can be equipped with the structure of a Banach algebra via the ‘sup norm’ (the distance between $\sum a_n q^n$ and $\sum b_n q^n$ is the largest value of $|a_n-b_n|$).

It thus makes sense to consider the closure of $\mathcal{M}_\mathbb{Q}$ with respect to this topology, and it is this which Serre defines to be the algebra $\mathcal{M}_p$ of $p$-adic modular forms. In other words, a p-adic modular form is just a power series with p-adic coefficients which can be approximated uniformly coefficient-wise by the q-expansions of classical modular forms.

Pure classical modular forms have a weight $k$. What about p-adic modular forms? One crucial computation in Serre’s paper that makes the theory work says that if two modular forms are congruent modulo a large power of $p$ then their weights are also congruent modulo a large power of $p$ and also modulo $p-1$. From this one can easily attach a weight in the group $X = \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}_p$ to a p-adic modular form (and since we are working at level 1, all weights arising will be even (so the subgroup $2\mathbb{Z}/(p-1)\mathbb{Z}$ on the first factor is all that is hit). We should remark that this computation has as an input the classical theorem of Clausen-von Stadt on the denominators of Bernoulli numbers.

Another interesting consequence of this which Serre notes is the following. Suppose we have a power series $f= a_0+a_1q+a_2q^2+...$ and we would like to show it’s a p-adic modular form. Maybe we can write it as a putative limit of some modular forms $f^i=a^i_0+a^i_1q+a^i_2q^2+...$, except we only know that for all $n \geq 1$ the coefficients $a_n^i \rightarrow a_n$ converge (but know nothing about the constant coefficients). Suppose further we also know the weights $k^i$ converge to some nonzero $k \in X$.

Well then the weight zero modular form $a_0$ and $f$ cannot be too close p-adically, where the length scale implicit in the phrase ‘too close’ is given with reference to one of the modular forms in question, so one gets a bound of the form $sup_{n \geq 1} |a_n| \geq C |a_0|$, where $C$ is some constant depending on how far $k$ is from zero. Applying this to our sequence $f^i$ we learn in particular that the $a^i_0$ lie in a closed bounded subset of $\mathbb{Q}_p$, so there is a convergent subsequence, and passing to the corresponding subsequence of modular forms we deduce that $f$ is a p-adic modular form after all.

One could complain that this is a silly theorem: when will we have that all but the constant coefficient is known to converge? Recall the classical weight $k$ Eisenstein series, whose q-expansions are given in terms of the Bernoulli numbers $B_k = -k \zeta(1-k)$ and the “sum of $(k-1)$th powers of all divisors of $n$” function $\sigma_{k-1}(n)$:

$G_k = -\frac{B_k}{2k} + \sum_{n\geq 1} \sigma_{k-1}(n)q^n$.

If we take a p-adically convergent sequence $k_1,k_2,... \rightarrow k \in X$ of weights that also tends to infinity in the archimedean metric, then one sees explicitly that p-adically:

$\sigma_{k_i-1}(n) = \sum_{d|n} d^{k_i-1} \rightarrow \sum_{d|n, p\not|d} d^{k-1} =: \sigma^*_{k-1}(n)$.

Therefore the sequence $G_{k_i}$ of Eisenstein series fit exactly in the situation of the above theorem, and we deduce that there is a p-adic Eisenstein series of any weight $k \in X$, which is well-defined because $\sigma^*_{k-1}(n)$ depends only on $k$. In particular, we deduce the existence of a well-defined continuous p-adic function $\zeta_p:X-\{1\} \rightarrow \mathbb{Q}_p$ such that $\zeta_p(k) = \frac{1}{2} \zeta(1-k)$ for all $k \in \mathbb{Z}_{\geq 2}$. One can check that this is exactly the p-adic zeta function constructed classically by Kubota-Leopoldt and featuring on the analytic side of the main conjecture of Iwasawa theory.

In this post I want to briefly record the following nice recipe for predicting the dimension of a Galois representation you expect to extract from a Shimura variety (and the local factors you expect to see). More precisely, let’s suppose I have a cohomological cuspidal representation $\pi$ of a reductive group $G/F$. Then Langlands would conjecture the existence of a map

$Gal(\bar{F}/F) \rightarrow ^LG(\bar{\mathbb{Q}_l})$.

However, it’s not clear the L-group necessarily has a preferred representation, and ultimately this is what we will construct in the cohomology of our Shimura variety. To see where this extra information comes from, recall that the Shimura variety is defined by both the group $G$ and a map $h:\mathbb{S} \rightarrow G_\mathbb{R}$. This latter datum can be made into a Hodge cocharacter $\mu: \mathbb{G}_m \rightarrow G$. But a cocharacter of $G$ is a character of its Langlands dual, i.e. a weight, and it is possible to take the irreducible algebraic representation with this as highest weight, and extend it to the whole L-group.

Composing the map predicted by Langlands reciprocity with this representation gives the Galois representation we might hope to find in the cohomology of our Shimura variety. As a nice bonus, the cocharacters floating around are miniscule, so computing the dimensions of these irreducible reps is really easy (the weight spaces are multiplicity free and just Weyl-translates of the highest weight).

This is a quick note to record some thoughts following from Toby Gee’s first lecture of his course at the Arizona Winter School, where he observes that quadratic reciprocity is a completely immediate consequence of basic algebraic number theory. I feel rather silly for never having noticed this before, and hope I don’t insult the reader by providing a post on it.

That quadratic reciprocity follows immediately from class field theory is standard, and for the rational numbers class field theory can be decomposed into the irreducibility of cyclotomic polynomials (reciprocity laws for cyclotomic extensions) and the Kronecker-Weber theorem (cyclotomic extensions fill out all the abelian extensions). Of these, I would only consider the second to be hard’.

The key point that makes quadratic reciprocity strictly easier than class field theory is that for $p$ an odd prime, the quadratic extensions $K_p := \mathbb{Q}(\sqrt{(-1)^{(p-1)/2} p})/\mathbb{Q}$ are the unique quadratic extensions ramified only at $p$. They therefore obviously satisfy the Kronecker-Weber theorem, since $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ has a degree 2 subextension which is ramified only at $p$ and thus equal to $K_p$.

We can easily make this more explicit. Consider the quadratic character

$\chi: G_\mathbb{Q} \rightarrow Gal(K_p/\mathbb{Q}) = \{ \pm 1 \}$.

By definition (more or less), for $q \not= p$ odd, this character is unramified and $\chi(Frob_q) = \left(\frac{(-1)^{(p-1)/2} p}{q}\right)$.

On the other hand, by our previous observation (“Kronecker Weber” in this special case), and the standard isomorphism between $Gal(\mathbb{Q}(\zeta_p)/\mathbb{Q})$ and $(\mathbb{Z}/p\mathbb{Z})^*$ (“irreducibility of cyclotomic poly”: here just Eisenstein’s theorem), we obtain the factorisation:

$\chi: G_\mathbb{Q} \rightarrow (\mathbb{Z}/p\mathbb{Z})^* \rightarrow \{\pm 1 \}$,

where $Frob_q$ is mapped to the class of $q$ modulo $p$. Equipped with this description (and recalling that $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic), it is clear that

$\chi(Frob_q) = \left(\frac{q}{p}\right)$.

Comparing the two expressions obtained, we recover the classical quadratic reciprocity law. One can also handle $q=2$ the same way with a small amount of care (over the correct way to interpret the first expression: Hensel’s lemma doesn’t give a direct comparison with a Legendre symbol in this case).

This blog post is where I will put a talk list and notes from the graduate student seminar I am organising this semester at Harvard on attaching Galois representations to automorphic representations. I will try to keep it updated reasonably often, and any comments (either by email or left on the blog, whichever is more convenient) would be strongly appreciated.

The list of talks in the seminar (happening from 4:30-6:00 in Science Centre room 232) is as follows:

04/02. Modular forms and Galois representations – Rong,

11/02. Modular forms as automorphic representations – Cheng Chiang,

18/02. Hilbert modular forms and the Jacquet-Langlands correspondence – Yihang

25/02. Shimura varieties and some local systems – Ananth

04/03. Eichler-Shimura relations on Shimura curves – Me

25/03. Langlands-Kottwitz for the Modular Curve – Me

08/04. Langlands-Kottwitz for Unitary Groups I – Yihang

15/04. Langlands-Kottwitz for Unitary Groups II – Me

01/05. The Trace Formula and GL_n I – Me

06/05. The Trace Formula and GL_n II – Chao

13/05. Extracting the n-dimensional Galois representation – Carl

20/05. What is currently known in general? – Jack Thorne (tbc).

My notes are here (last updated 27/03/13).

In this post I want to briefly summarize the following interesting paper of Milne and Suh http://www.jmilne.org/math/articles/2009c.pdf in which they give a general method for constructing connected Shimura varieties with the property that one can apply an automorphism of $\mathbb{C}$ to their defining equations and obtain a variety with a different fundamental group.

On the one hand, since (most) automorphisms of the complex numbers are extremely weird and certainly not continuous, perhaps this should not be surprising. On the other hand, the theory of etale cohomology (or, if you prefer, GAGA) implies that the cohomology of these spaces must always be the same, and the theory of the etale fundamental group implies that the profinite completion of the fundamental groups are the same. In light of these results, I find that one can obtain different honest fundamental groups rather surprising. I should probably mention that I think the first such example is credited to Serre, but I haven’t yet tracked down the relevant article, and Milne-Suh’s construction is also interesting for other reasons.

Working with connected Shimura varieties has, at the outset, two obvious advantages for tackling this problem. Firstly, a connected Shimura variety is something of the form $\Gamma \backslash X$, where $X$ is a symmetric hermitian domain and $\Gamma$ a torsion free subgroup of some algebraic group. The geometry of this situation is very closely related to the theory of the algebraic group in which $\Gamma$ resides, and $X$ is simply connected, so the fundamental group is simply $\Gamma$ itself. Secondly, Milne (following important work of Shimura, Deligne, Kazhdan, …) has published some wonderful detailed results on how connected Shimura varieties behave under automorphisms of $\mathbb{C}$, so by quoting some of these we can get to our result more quickly (but should note that in the case where our variety is of abelian type the corresponding results follow from Deligne’s theory, and in the PEL case from the classical theory of complex multiplication and moduli spaces).

There are two key ingredients.

Input 1: Margulis super-rigidity

Let $F$ be a totally real field, and $H,H'$ absolutely simple simply connected algebraic groups over $F$, and $H_*,H'_*$ their Weil restrictions down to $\mathbb{Q}$. Suppose we have lattices of $H_*(\mathbb{Q})$ and $latex H’_*(Q)$ that are isomorphic as groups. Then the principle of Margulis super-rigidity says that $H'$ is isomorphic to some twist of $H$ by a field automorphism of $F$.

In particular, suppose we were in the situation where the base changes of $H,H'$ to the finite adeles of $F$ were isomorphic, and fix $K$ an open compact subgroup (of both groups under this isomorphism). If the lattices cut out in the rational points of each group by $K$ were isomorphic, it would force $H' \cong \sigma H$ for $\sigma$ an automorphism of $F$. This is significant for us, because these lattices will be the fundamental groups of our pairs of conjugate Shimura varieties, and the result we have just stated says that we can guarantee they fail to be isomorphic by simply arranging for $H'$ to fail to be a conjugate of $H$ by an automorphism of $F$.

Input 2: Information about conjugates of connected Shimura varieties

Here is where I’d say the meat of the argument takes place. Let us say that a variety is of “type $(H,X)$” if it’s of the form $\Gamma \backslash X$ for $X$ a symmetric hermitian domain and $\Gamma$ a congruence subgroup of $H(F)$.

Main Theorem: Let $H$ be a simply connected group, and fix $V$ a variety of type $(H,X)$, and $\tau \in Aut(\mathbb{C})$. Then $\tau V$ is a variety of type $(H',X')$ and we can relate $H$ and $H'$ locally by the relations:

• At finite places, $H'_v \cong H_v$.
• At infinite places, $H'_v \cong H_{\tau v}$.

The proof of this statement goes roughly as follows. Firstly, Milne’s papers on conjugates of Shimura varieties give a description of $H'$ as a twist of $H$ by a torsor of a proalgebraic group $S^0_F$ which is Tannaka dual to the category of CM-motives over $\mathbb{C}$ with $F$-endomorphisms. In our situation (for a given Shimura datum) the map involved factors through an algebraic quotient, which implies it factors through the Mumford-Tate group of an actual CM abelian variety $A$ with endomorphisms by $F$. With this description, one can reduce the study of the torsor to that of a certain set of isomorphisms between Betti cohomology groups of $A$ and $\tau A$. At each place of $F$, one can then compute it explicitly via the canonical comparison isomorphisms to l-adic and de Rham cohomology, finding in particular that it is trivial at the finite places and deducing the required relation at the infinite places.

Payoff: Suppose we can find a group $H/F$  absolutely simple simply connected  associated with a symmetric hermitian domain $X$, and $\tau \in Aut(\mathbb{C})$ such that whenever $H'/F$ satisfies the relations of the main theorem it fails to be isomorphic to an $Aut(F)$-conjugate of $H$. Note that the isomorphisms (from the main theorem) at all the finite places give an isomorphism over the finite adeles, and it isn’t too hard to show that given $V=\Gamma \backslash X$, the corresponding open compact of $H(\mathbb{A}_{F,f})$ is identified with that of $H'$ corresponding to $\Gamma'$ with $\tau V = \Gamma'\backslash X'$. We therefore deduce from Margulis super-rigidity that $V$ and $\tau V$ have different fundamental groups.

This gives us a rather general lovely recipe for constructing such varieties, and I would advise reading the original paper to get an idea of the different kinds of things one can do. For the sake of completeness I will just sketch an example. Fix your favourite totally real field $F \not= \mathbb{Q}$ with no automorphisms (e.g. any cubic non-Galois field, but there exist lots of examples of all degrees at least 3). Now take the group of norm 1 elements of your favourite quaternion algebra over $F$ which split (say) at exactly one infinite place. Now take some automorphism of $\mathbb{C}$ which moves this place somewhere else. These visibly give data required by the ‘payoff’ and thus the connected Shimura varieties obtained will have the desired property.

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.

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.