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 kth 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.

Quick remark about newforms:
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.

Recall the following theorem, due to Hilbert. Let {k} be a field, and {S=k[x_0,...,x_n]} considered as a graded ring, and take {M} any finitely generated graded {S}-module. Then there is a free resolution of {M} of length at most {n+2}. This theorem is important because applying it in the case {M=S/I} for {I} a homogeneous ideal (i.e. to a projective variety), it tells us that the Hilbert polynomial exists and gives an explicit bound on its degree.

I had always assumed this theorem was in some sense fairly deep or hard, but in Chan’s scheme theory course today we saw that it’s actually a small amount of some fairly basic homological algebra.

Step 1: Do the case M=k, where each {x_i} acts as the zero homomorphism. I would rather not do the details of this in a blog post, but it isn’t very hard, and you if you try to write down a resolution that’s as efficient as possible, you should end up with a resolution of length {n+2}, which usually seems to be given the intimidating name `the Koszul complex’.

Step 2: Let {M} now be arbitrary, and consider a minimal free resolution {... F_n \rightarrow F_{n-1} \rightarrow ... \rightarrow F_0 \rightarrow M \rightarrow 0} (in the sense that the first map has the smallest possible number of generators, then the second one does, and so on). By minimality, for all {k\geq 1} a basis element of {F_k} must be mapped to an element with no constant term in any coefficient. Indeed, if {e \in F_{k+1}} is mapped to {(p_1,...,p_l) \in F_k}, each {p_i} is homogeneous (since the maps are maps of graded modules) and if some {p_i} is constant, since {k} is a field and {(p_1,...,p_l)} is in the kernel of {F_k \rightarrow F_{k-1}}, we see that the basis vector {e_i} maps to an element that is expressible as a linear combination of the image of the other {e_j}s, contradicting minimality. What this observation buys us is that if we take such a complex and tensor it with the {S}-module {k}, we get a complex

\displaystyle ... \rightarrow F_n \otimes_S k \rightarrow ... \rightarrow F_1 \otimes_S k \rightarrow F_0 \otimes_S k \rightarrow M \otimes_S k \rightarrow 0,
and all of the maps save the first one are the zero map.

Step 3: Use the fact that {A \otimes B = B \otimes A}, or more generally that {Tor_i(A,B) = Tor_i(B,A)}. This really is the key ingredient. Knowing that the above maps are zero, we see that the rank of {F_j} is just the rank of {Tor_j^S(M,k)}. And this is the rank of {Tor_j^S(k,M)}, which can be computed by as the homology of the complex obtained by tensoring the Koszul complex with {M}. In particular, we see that it vanishes for {i>n+1}, so Hilbert’s theorem is proved.

In this post, in advance of a talk I am giving tomorrow as part of the Harvard Mazur Torsion seminar we shall give an overview of Mazur and Tate’s proof that there are no elliptic curves /{{\mathbb Q}} with a rational 13-torsion point. [WARNING: I apologise that this post is probably currently littered with mistakes, and certainly with abuses of notation, having been written fairly hastily. I will try to come back and make it better later.]

The theory of modular curves had been well-developed at the time of the paper, so there is a smooth affine curve over {{\mathbb Z}[1/13]} out there called {Y_1(13)} whose rational points are precisely isomorphism classes of elliptic curves over {{\mathbb Q}} together with a rational point of order 13. The problem is therefore reduced to proving some curve has no rational points. It turns out to be nicer to work with the compactification {X_1(13)}, which has 12 extra points (`cusps’), six of which are rational.

For quite a few numbers that aren’t 13, one is rather lucky and finds that either {X_1(N)} is an elliptic curve (so write down its equation, do 2-descent to show it has rank 0, and work out that all the rational points are cusps) or more likely one finds that some other closely related curve like {X_0(N)} or a quotient of this by an Atkin-Lehner involution is an elliptic curve of rank 0, and thinking about fibres of maps like {X_1(N) \rightarrow X_0(N)} can again find all the rational points.

However, 13 is unlucky for some, and indeed it is unlucky for us. The curve {X=X_1(13)} we are interested in has genus 2, but the only logical thing to map it to, {X_0(13)}, has genus 0 and a rational point, hence loads of rational points. We must therefore attack the curve {X_1(13)} directly in some way.

One natural thing to do is to consider its Jacobian {J=Jac(X_1(13))}, a 2-dimensional abelian variety whose geometric points correspond to elements of the 0-Picard group of {X(k^{sep})}. In particular it is equipped with, for any choice {P_0} of rational cusp, an embedding defined over {{\mathbb Q}} {X \hookrightarrow J} given by {P \mapsto (P)-(P_0) \in Div(X)}. In particular, if we can show that the Jacobian has Mordell-Weil rank zero and analyse its rational torsion we’ll be done.

The first progress here was made by Ogg, who took one of the other rational cusps and discovered that it was a torsion point of order 19, and the subgroup it generates intersects the curve {X} precisely at the rational cusps. Given our initial bad luck, this is nothing short of a miracle! Since {X} and hence {J} has good reduction at 2, the Riemann hypothesis tells us that {|J({\mathbb Q})_{tor}| \leq |J(\mathbb{F}_2)| \leq (1+\sqrt{2})^4 < 38}, so after Ogg’s work, we know that the only thing left to rule out is a point of infinite order in {J({\mathbb Q})}.

If you have an elliptic curve, the standard trick for computing its Mordell-Weil rank is to take your favourite isogeny {\pi} (usually one whose kernel is a well-understood Galois module) and analyse the cokernel of the map {\pi: E(K) \rightarrow E(K)} by interpreting it as a subgroup of the first Galois cohomology group {H^1(G_K, (Ker \pi)(\bar{K}))}. There is no reason this shouldn’t work for abelian varieties, but finding an isogeny whose kernel we can understand is somewhat harder: for elliptic curves the multiplication by 2 map will usually do nicely, because the 2-torsion can be read off from an equation in Weierstrass form, but we have no such trick here.

What we do have here however are a bewildering family of interesting automorphisms of {J}. In particular, we have the diamond operators {\gamma_m: (E,P) \mapsto (E,mP)} (for {m \in ({\mathbb Z}/13{\mathbb Z})^*/\{\pm\}}), and the Atkin-Lehner involutions {\tau_{\zeta}} which map {(E,P)} to the pair {(E/<P>, [Q])} where {Q} is some 13-torsion point of {E} such that it pairs with {P} under the Weil pairing to give {\zeta}. The Galois group {G_{\mathbb Q}} commutes with the former and acts on the latter in the obvious way ({\sigma(\tau_{\zeta}) = \tau_{\sigma(\zeta)}}). These operators generate a subgroup {\Delta \subset End(J)} isomorphic to the dihedral group, and with the noted Galois action they satisfy {\sigma(g.x) = \sigma(g)\sigma(x)}.

Of great interest to us is the subring {{\mathbb Z}[\gamma_2]}. One can easily verify that {\gamma_2} acts with order 6 on the cusps of {J}, and since {J} is simple (there aren’t many elliptic curves with good reduction at 2 and a 19-torsion point) one deduces that {{\mathbb Z}[\gamma_2] \cong {\mathbb Z}[X]/(X^2-X+1)}, i.e. it’s the Eisenstein integers, a rather nice quadratic PID. We also have a rational subgroup {V(1) \subset J[19]}, so it’s natural to consider trying to find a 19-isogeny, and we now have one staring us in the face: {19} factors into two distinct primes {19=\pi \bar{\pi}} in {{\mathbb Z}[\gamma_2]}, giving an idempotent decomposition

\displaystyle J[19] = J[\pi] \oplus J[\bar{\pi}].

Relabelling as necessary we may assume {V(1) \subset J[\bar{\pi}]}. One can then show that any Atkin-Lehner involution swaps these two eigenspaces and maps {V(1)} onto a subset {V(\gamma) \subset J[\pi]} with a faithful action of {Gal({\mathbb Q}(\zeta_{13})^+/{\mathbb Q})}. Moreover, by considering the {\gamma_2} action on the Weil pairing one shows that the self-duality of {J[19]} (all Jacobians admit a principal polarisation) induces a Cartier duality between {J[\pi]} and {J[\bar{\pi}]}, and so one ends up with a short exact sequence of {G_{\mathbb Q}}-modules:

\displaystyle 0 \rightarrow V(\gamma) \rightarrow J[\pi] \rightarrow \mu_{19} \rightarrow 0.

We are now in good shape to attempt a {\pi}-descent. We already know the Mordell-Weil theorem for abelian varieties, so {J({\mathbb Q})} is a finitely generated {{\mathbb Z}}-module, in particular a finitely generated {{\mathbb Z}[\gamma_2]}-module. Since this is a PID, the structure theorem tells us that if we can show {\pi: J({\mathbb Q}) \rightarrow J({\mathbb Q})} is surjective, then it has rank 0. We will therefore want to consider some kind of Galois cohomology which will give us a small {H^1(J[\pi])} (our dream would be if this could vanish, but that’s maybe rather unrealistic).

Recall that Galois cohomology is just the étale cohomology of abelian sheaves on the spectrum of a field. If instead of allowing our sections to only be defined on a field (an `arithmetic point’) we spread them out across something like {Spec({\mathbb Z}[1/13])}, it figures that maybe the cohomology groups will get smaller. However, we must be careful: we are being led to study the cohomology of 19-primary finite flat group schemes over a base where 19 is not invertible. In particular, we will later need a Kummer exact sequence to analyse {\mu_{19}}, and this fails to be exact as a sequence of étale sheaves. The solution is instead to consider the fppf topology which is finer than the étale topology but still coarser than the fpqc topology (so all schemes are fppf sheaves).

Let’s do that. Recall that {J} was defined over {{\mathbb Z}[1/13]} so we get a naturally defined exact sequence

\displaystyle 0 \rightarrow J[\pi] \rightarrow J \rightarrow J \rightarrow 0,

which can be viewed as a sequence of fppf sheaves on {{\mathbb Z}[1/13]}. Taking global sections, we get a long exact sequence containing a chunk:

\displaystyle J({\mathbb Q}) \rightarrow J({\mathbb Q}) \rightarrow H^1({\mathbb Z}[1/13], J[\pi]),

where the first map is the one we wish to show is surjective.

We secretly know, as with 2-descent, that all the juicy stuff (say, the data in the image of the second map) probably happens at the bad primes, which in this case is just the prime 13. It’s therefore maybe natural to base change to {{\mathbb Q}_{13}} and analyse the situation there.

In fact, the situation there is rather nice: it’s not too hard to prove that {\pi: J({\mathbb Q}_{13}) \rightarrow J({\mathbb Q}_{13})} is in fact an isomorphism. By taking a Neron model (best possible smooth – but not necessarily proper – group scheme whose generic fibre returns {J}) {A/{\mathbb Z}_{13}} for {J}, in particular with {A({\mathbb Z}_{13}) = J({\mathbb Q}_{13})}. We can take the reduction, and notice that {\pi} acts invertibly on the (pro-13) kernel of reduction, so by the snake lemma it suffices to check that {\pi} is injective: in other words, that there are no nontrivial {G_{{\mathbb Q}_{13}}}-invariants of {J[\pi]}. Given our explicit description of this Galois module, this boils down to the fact that {13} doesn’t split completely in {{\mathbb Q}(\zeta_{13})^+/{\mathbb Q}} (it’s totally ramified) or {{\mathbb Q}(\zeta_{19})/{\mathbb Q}} ({13 \not\equiv 1 \mod 19}).

How do we transfer this information back up to {{\mathbb Z}[1/13]}? Well, if we can prove that {H^1({\mathbb Z}[1/13], J[\pi]) \rightarrow H^1({\mathbb Q}_{13}, J[\pi])} is injective then that will be enough (draw a diagram, and note that by what we just showed {J({\mathbb Q}_{13}) \rightarrow H^1({\mathbb Q}_{13}, J[\pi])} is the zero map). At this point it seems worthwhile to want to reconstruct our explicit description of {J[\pi]} and verify that this map is injective on each piece. I.e. recall we proved earlier that the generic fibre of {J[\pi]} is an extension of a module which becomes the trivial {{\mathbb Z}/19{\mathbb Z}} over {{\mathbb Q}(\zeta_{13})^+} by the Galois module {\mu_{19}}. But by a general theorem in a paper by Oort and Tate concerning finite flat group schemes, since 19 isn’t very ramified in {{\mathbb Q}}, we can identify 19-primary finite flat group schemes by their generic fibre. So taking the Zariski closure, we get a short exact sequence

\displaystyle 0 \rightarrow E \rightarrow J[\pi] \rightarrow \mu_{19} \rightarrow 0,

where {E} is some finite flat group scheme whose base change to {{\mathbb Z}[1/13, \zeta_{13}]} is isomorphic to the constant group scheme {{\mathbb Z}/19{\mathbb Z}}.

So we now are reduced to showing that the induced maps on the first cohomology of these pieces when we base change to {{\mathbb Q}_{13}} are injective. Firstly, in fact we can show that {H^1({\mathbb Z}[1/13], E) = 0}. A nonzero element of this group is a nontrivial {E}-torsor on {{\mathbb Z}[1/13]}, which under base change corresponds to a nontrivial {{\mathbb Z}/19{\mathbb Z}}-torsor on {{\mathbb Z}[1/13, \zeta_{13}]}. But the class number of {{\mathbb Q}(\zeta_{13})} has been computed and is in fact 1, so in particular this field doesn’t admit any nontrivial degree 19 covers unramified away from 13.

Finally, we must prove that {H^1({\mathbb Z}[1/13], \mu_{19}) \rightarrow H^1({\mathbb Q}_{13}, \mu_{19})} is injective. Since we had the foresight to use fppf cohomology, we get the Kummer exact sequence

\displaystyle 0 \rightarrow \mu_{19} \rightarrow \mathcal{O}^* \rightarrow \mathcal{O}^* \rightarrow 0.

Taking the long exact sequence of cohomology, and noting that both of these rings are PIDs, so have trivial picard group ({=H^1(\mathcal{O}^*)} by a generalisation of Hilbert 90), we get isomorphisms of these cohomology groups with {\mathcal{O}^*/\mathcal{O}^{*19}}, which one can check explicitly have an induced injection.

And that’s it, we read off that our original {\pi} was surjective, and so we’re done: there are no {{\mathbb Q}}-torsion points of order 13 on elliptic curves over {{\mathbb Q}}.

The large gap in my blog activity over the past months is due largely to my writing time being taken up working on the following extended essay on the basic results in the theory of l-adic cohomology and their applications in proving the Weil conjectures and the theory of Galois representations. It was submitted for Part III of the Mathematical Tripos, the results for which are now known, so publishing it feels appropriate.

Though much of the essay has been drawn from the literature, there are parts which have been restructured or entirely constructed from scratch, and the formality of the presentation should not be mistaken as something that has been peer-reviewed or even very well-checked. However, I hope some people will find it useful and constructive feedback is always appreciated.

Also, if any other (vaguely) current Part III students have published essays online or want to upload theirs, I might consider appending links to this blog post. It would be nice to have a collection of such essays: those which I have read are excellent introductions to more specialist areas pitched at a good level for me and could therefore probably be useful for others.

Etale Cohomology and Galois Representations – Tom Lovering

 
Friends’ Essays

The Local Langlands Correspondences – Remy van Dobben de Bruyn
Etale Cohomology and Galois Representations (with an emphasis on descent theory) – Low Zhen Lin
p-Adic Uniformisation – Chris Williams
D-modules – Chris Elliott
Multiplicative Coalescence – Dominic Yeo
Tutte’s Flow Conjectures – Laszlo Lovasz

In this post we shall investigate an example which I am hoping will clarify some of the more subtle and potentially confusing behaviour in the theory of local fields (and its interaction with that of global fields).

Given a finite extension {L/K} of number fields, a prime {v} of {K} can extend to several different primes of {L}. If {L/K} is Galois, it is sufficiently symmetric that even though there may be several primes, {Gal(L/K)} acts transitively on them and each {\sigma \in Gal(L/K)} gives an isometry {L_{(w)} \rightarrow L_{(\sigma(w))}} (where we write {L_{(w)}} to mean the field {L} equipped with the norm coming from {w}). In particular, one can deduce that the ramification and inertia degree of each of these primes {w|v} are the same. However, if {L/K} is not Galois, the primes {w_1,...,w_k} over {v} could behave very differently from each other, and in general we do have the formula

\displaystyle e_1f_1+...+e_kf_k = [L:K]
but the {e_i} and {f_i} can vary as {i} varies.

Contrast this situation with that of local fields. If {L/K} is a finite extension of local fields, then each prime {v} of {K} extends to a unique prime of {L}, and {ef=[L:K]}. In particular, let us think about what happens if we take a global field {K} and complete it at a prime (obtaining something I’ll write as {K_v}). What happens to an extension {L} of {K}? If {L/K} is Galois, then we saw above that all the primes looked fairly similar, and the completions at all primes are isomorphic via the isometries we noted above.

When {L/K} is not Galois, things become complicated, and different completions of {L} are no longer (generally) isomorphic. Though it shouldn’t be if we think clearly enough about these things, this can be a source of great confusion. We are accustomed to writing things like {K(\sqrt[3]{2})} and assuming it means a well-defined thing (at least up to isomorphism of field extensions), but alas this no longer is so.

The easiest way to see this is probably to consider completion at the infinite place. It is true that as abstract fields (letting {\omega} be a 3rd root of unity) {{\mathbb Q}(\sqrt[3]{2}) \cong {\mathbb Q}(\omega \sqrt[3]{2})}, but if we complete with respect to the obvious archimedean norm (considering these things embedded in {{\mathbb C}}) one of the fields we get is {{\mathbb R}} and the other {{\mathbb C}}. Thinking about this more functorially, we had an abstract field {{\mathbb Q}[X]/(X^3-2)} and because it isn’t Galois, it is allowed to have different-looking archimedean places, and in fact it does: it has one real embedding and one pair of complex embeddings.

Exactly the same issue is a problem at the finite primes, except now life is in general much more complicated because the parameters {e} and {f} can vary so freely, so it is even more important to be very careful. Let us recycle the example above, but look at what happens at the prime {5}. It does not ramify, and splits into two primes, of inertial degrees 1 and 2. Why is this happening? Well, exactly as at the infinite place, there is one embedding into {{\mathbb Q}_5}, and a conjugate pair of embeddings into {{\mathbb Q}_{25}} (where by conjugate I mean there is an element of {G_{{\mathbb Q}_5}} that exchanges them). This corresponds to the fact that {X^3-2 = (X-3)(X^2+3X-1)} modulo {5}, and by Hensel’s lemma this factorisation lifts to a factorisation of {X^3-2} in {{\mathbb Q}_5}.

So what is the careful way to work out what is going on? As the above examples illustrate, if {L/K} an extension of number fields, {L=K(\alpha)} with {g} denoting the minimal polynomial of {\alpha} and {v} is a place of {K}, then the distinct places of {L} extending {v} are precisely the number of ways to embed {L} into an extension of {K_v}, modulo the action of {G_{K_v}}. In other words, the number of ways to realise {\alpha} (a generic root of {g}) as an element of an extension of {K_v}, modulo the {G_{K_v}} action. So actually this is just the number of irreducible factors of the polynomial {g} considered as an element of {K_v}.

Let’s write this more algebraically. Recall that {L=K[X]/(g(X))}, so

\displaystyle L\otimes_K K_v = K_v[X]/(g(X)) = \prod K_v[X](g_i(X)),
where {g=g_1...g_n}, where the {g_i} are distinct irreducible elements of {K_v[X]}. Note (being careful not to confuse what happens here with what happens with ramified primes when you pass to the residue field) that they are always actual irreducible polynomials, since {g} has distinct roots in {\bar{K}}, and of course each factor is a finite extension of a local field so gives a unique local field up to isomorphism (because primes of local fields extend uniquely).

So the upshot is that to see the different possible ways to localise your extension of global fields (i.e. the different places), it is essential to factorise the minimal polynomial of a generator of the global field considered as a polynomial in the appropriate local field (or at least find an irreducible factor), and specify which factor you are using (and of course by basic Galois theory this is equivalent to giving a {G_{K_v}}-orbit of embeddings {L \hookrightarrow \bar{K}_v}).

We finish with a short example, which I found somewhat confusing until I started writing this post. Consider the extension {{\mathbb Q}(\sqrt[3]{10})/{\mathbb Q}}. Let us try to compute the power of {3} dividing its discriminant. This is a scenario fairly similar to our examples above, except of course 3 is now ramifying. By Hensel’s lemma (and {10 \cong 1 \mod 9}), there is some {\alpha} a cube root of 10 in {{\mathbb Q}_3}, but there is no third root of unity, so in {{\mathbb Q}_3} we get {X^3-10 = (X-\alpha)(X^2 + \alpha X + \alpha^2)}. The only contribution to the different can come from the second factor {f(X)=X^2 + \alpha X + \alpha^2}, whose roots {\beta} generate the local ring of integers. Indeed a direct computation shows that {f'(\beta)} is a uniformiser, which tells us that the local different has valuation 1, and therefore (since the inertia degree is 1) only one power of {3} divides the discriminant of {{\mathbb Q}(\sqrt[3]{10})/{\mathbb Q}}. I suppose in general the moral is that it’s probably a bad idea to ever write anything like {{\mathbb Q}_3(\sqrt[3]{10})} unless it’s at least Galois.