This will (hopefully) be the first in a series of four posts based off a lecture series given at the Morningside Center of Mathematics in Beijing. The goal overall is to talk about some recent advances in the -adic geometry of Shimura varieties and their integral canonical models. The goal of this (essentially standalone) post more specifically is to give a sort of broad overview for what Shimura varieties and their integral canonical models are, and why one should care about them.
It is a fool’s errand to try and define Shimura varieties in any sort of rigorous way in such a small amount of time, and I will make zero attempt to do so here. For that I can only point to other references, some of which I list here: [Deligne1], [Deligne2], [GenestierNgo], [Hida], [Hoermann], [Kerr], [Lan], [Milne1], [Milne2], [Moonen], [Morel].
Disclaimer: For the sake of exposition, the below will take many non-trivial liberties, most of which are about ideas for which it’s difficult to even make rigorous sense of. For this reason, I caution the reader for taking anything written here too seriously, and to treat it only as vague motivation. I apologize in advance to any experts who take issue with these inaccuracies.
§0 motives and objects with -structure
Below we will frequently make use of the ideas of ‘motives’ and objects with ‘-structure’. We take a sort of primitivist approach to these notions, not trying too seriously to even describe the desiderata they are meant to satisfy. That said, to help orient the unfamiliar reader, let me say a few words below on each.
§0.1 Motives
(I cannot possibly do justice to the idea of motives here, especially in addressing the subtleties for what such a fantastic theory would entail. For that I can only suggest consulting the much more enlightening references [Milne3] and [Baez] (and especially the references discussed in §5 of the latter).)
For the uninitiated, one can think of the category of of -motives on (for a ring and scheme ) as being the receptacle for an ‘optimal’ cohomology theory
,
on the category of smooth projective -schemes. Namely, we expect that should be something like an -linear -category. Roughly this means you can -linearly add morphisms in , take tensor products in (in a way that is -linear although I won’t demand the tensor product is ‘over ‘), and that you can make sense of exact sequences in .
The ‘optimality’ here means that for any other ‘reasonable’ cohomology theory , where is (for simplciitly) an exact -linear -category, there should be a factorization
where is some ‘realization functor’. So, any extra structure that the category could contain (Galois action, filtration, Hodge structure,…) should somehow already be captured in , thus making the ‘richest category of cohomological invariants’.
Again, we are taking a very primitivist perspective here, mostly out of necessity. Motives in any naive sense like I am trying to describe them ‘less than exist’: forgetting their existence, there isn’t even a guess of what the category is (or what properties it should satisfy), especially over the type of general bases that I have allowed here. I just want to mention two cases at which one can gesture towards motivic thinking in the way I mean it here.
Remark: For the sake of some readers’ sanity, let me point out that there do exist precisely defined notions of ‘motives’, e.g., Voevodsky motives. The relationship to the idealized vaugery of motives we mean here is largely conjectural, especially over general bases . So, somehow the ‘rigorously defined’ motives considered in, for example, motivic cohomology, don’t directly facet into our discussion.
§0.1.1 Abelian varieties
The category should carry something like a ‘weight grading’ which, in rough terms, should be related to the degree of cohomology, i.e., which , we are taking. For -motives with coefficients in (i.e., ) there is a very rough idea for what the category of -motives should be: abelian schemes.
Namely, there should something like a fully faithful embedding
where the source is the isogeny category of abelian varieties (i.e., the localization of along the class of isogenies. The map
This idea stands up to some amount of scrutiny via the fact that first cohomology groups of most cohomology theories depend only on the associated Albanese variety:
.
In other words, one roughly has , in line with the ‘optimality’ we would want to satisfy.
Remark: On the other hand, some readers may also see how abjectly absurd the above is in its oversimplicity. The Albanese variety does not exist for arbitrary bases (although it does exist over reduced varieties over an algebraically closed field as in [Serre]). Moreover, when it exists, the Albanese variety is usually not an abelian scheme itself, but a torsor for an abelian scheme. So, at best, one may believe that (isogeny classes of) abelian varieties do fully faithfully embed into the category of -motives with -coefficients. But even more dire is that the fact that factorizes through the Albanese does not always hold for all bases and all cohomology theories. Again this discussion of motives is meant more for flavor and not so much for precision.
Now, one cannot perform linear algebra operations in like tensor product, i.e., it’s not an exact -linear -category, and this property will be quite important when discussing -objects below. But, as does enjoy these properties, one can form the smallest exact -linear -subcategory of containing . This is called the category of abelian motives over , and is denoted .
§0.1.2 Archimedean Hodge theory
It is a classic observation that comes for Archimedean Hodge theory, that one has a functor
,
where is the category of (pure) -Hodge structures. In other words, the singular cohomology of the underlying projective complex manifold , together with the Hodge decomposition of , allows one to build from a -Hodge structure. Now, the theory of motives would dictate that there should be a realization functor
such that . Moreover, one can essentially interpret the Hodge conjecture as saying that is fully faithful.
This extends to the relative situation, where for a smooth -variety one can define the category of variations of -Hodge structures, which are like a holomorphically varying family of objects for . One then has a functor
,
where is the induced morphism of complex manifolds. One also expects from the Hodge conjecture that the realization functor in this relative setting is fully faithful.
Let me emphasize the upshot here. The Hodge conjecture predicts that the category of -motives over a quasi-compact smooth -variety are a full subcategory of (variations of) -Hodge structures. This means that, at least conjecturally, all possible cohomological information that could be contained in a smooth projective -scheme is already contained in its Betti cohomology. Thus, in some sense this shifts our attention from constructing the category of motives in some abstract sense, and instead to singling out the ‘correct’ subcategory of Hodge structures. In particular, this gives a sort of ‘cheating’ way to get at motives: we can instead work with Archimedean Hodge theory as a substitute…at least for smooth -varieties.
To see how this connects with abelian varieties, we recall the following result of Deligne.
Proposition [Deligne3, Rappel (4.4.3)]: Let be a quasi-compact smooth variety over . Then, there is an equivalence of categories
.
Here the target category is the category of variations of -Hodge structures which are
of type –this just means that these are the only pairs of that show up in the Hodge decomposition.
Remark (rem:Deligne-explained): This theorem can be made a lot less daunting in the case , and it further helps explain this polarizability condition. Namely, one can check that if is a finite free -module, then a -Hodge structure of type on it amounts to a complex structure on , in particular the rank of is even, say . One may then consider the quotient which is an abelian variety of dimension …or is it!
In the case of dimension (i.e., ) this truly is the case as all such lattice quotients are elliptic curves. But, for these are, in general, non-algebraizable complex tori. The exact condition needed to guarantee that they algebraizable is that this complex torus possesses a Riemann form (see [Rosen, §3]) which amounts to the claim that the associated Hodge structure is polarizable.
In particular, Deligne’s theorem tells us that gives us a fully faithful embedding of into . Then, we can apply this ‘cheating’ mindset here. Above we defined the full subcategory of abelian motives, i.e., the -linear -category generated by abelian varieties in . By the Hodge conjecture this should be identified with its image in and agree with the -linear -category generated by in . But, we can actually rigorously define this last category!
To underscore this, if we take , for a smooth -variety , to be the full -linear category generated by in , this is a non-conjectural way of defining what ought to be equivalent to category of (true) abelian motives over . But, of course, the category , unlike , is non-conjectural!
§0.2 Objects with -structure
Recall as above that for a ring an exact -linear -category is one where you can -linearly add morphisms, take tensor products, and that you can make sense of exact sequences.
The key example of this is as follows.
Example: Let be a (finite type, affine, smooth) group scheme over . Then, the category of (finite -projective) representations of is an -linear exact -category in the obvious way.
Definition (defn:H-objects):The category of -objects in is the category of exact (i.e., sends exact sequences to exact sequences) -linear -functors
.
Perhaps the most illuminating example of this idea is the following.
Example (e.g., see [Broshi]): Let be a scheme over a DVR . Then, the category of vector bundles on is an exact -linear -category in the obvious way, and identifies with the category of -torsors on .
In particular, I would implore the reader to think of -objects in (at least up to first approximation) as being something like ‘-torsors in ‘.
§1 A dreamland: moduli of motives
We begin by trying to describe Shimura varieties as they ought to be. As is indicated by the notation, such varieties are built using the input of the data and . The first of these, the more important of the two, is the following.
Definition (defn:shim-data):A Shimura datum is a pair consisting of
a reductive group over ,
,
such that … . Its Hodge cocharacter is defined by for any .
A morphism of Shimura data is a morphism of -groups such that , and is a closed embedding if is.
Let me make some comments on some of the structures appearing in this definition:
The group (i.e., viewed as a real algebraic group) is the so-called Deligne torus. It is the Tannakian fundamental group of the Tannakian category of real Hodge structures (with the usual fiber functor). So, is naturally identified with (the notation means ‘isomorphism classes’). In particular, one may view the in a Shimura datum as being an isomorphism class of real Hodge structures equipped with -structure.
One has a natural (up to normalization) identification thus the notation on the right-hand side of makes sense. From the perspective that is an isomorphism class of objects in , one may view as the (conjugacy class of) cocharacter(s) defining the Hodge filtration on these real Hodge structures, whence the name.
We are actually already almost at the naive definition of , and only need one extra piece of notation/terminology.
Definition(defn:reflex-field): The reflex field of the Shimura datum is the field of definition of .
Remark: There is a small subtlety about the phrase ‘field of definition’ for a conjugacy class of cocharacters. Namely, it is possible that the action of on such conjugacy classes fixes as a conjugacy class, but fixes no actual element in . In other words, there needn’t actually be a cocharacter in defined over . That said, this subtlety disappears (see [Kottwitz1, Lemma 1.1.3]) with the mild assumption that is quasi-split (i.e., has a rationally-defined Borel).
So, we are already ready to try and ‘define’ what is. Namely, it is meant to be a smooth quasi-projective -variety, and thus (by Yoneda) to characterize it, it is enough to tell you the value for a smooth quasi-compact -variety .
Expectation (exp:Shim-var): For a smooth quasi-compact -variety ,
To try and explain what the meaning of these extra terms ‘type ‘ and ‘étale -level’ structure means, let us set up some notation. Namely, a -motive on with -structure is an exact -linear -functor . We then obtain a diagram as follows
Here
is the -adic étale realization functor, so the output is an étale -local system,
is the Hodge realization functor, so the output is a variation of -Hodge structure .
We are then defining and to make the diagram commute (i.e., they are -adic and Hodge realization of , respectively).
We can then describe the above-mentioned conditions on as follows:
Type : This means that for every -point of one has that , i.e., that the pullback to of the Hodge realization of is isomorphic to in , i.e., we’re fixing the pointwise isomorphism class of our real Hodge structures to be .
Note that as is defined over , there is an action of on , and so the above condition should be stable under such an action. But, as is defined (roughly) to be the (smallest) field where the isomorphism class of makes sense, this tracks.
Étale -level structure: Here we are fixing a compact open subgroup (where ). Then such a level structure is a global section of the sheaf
.
Remark (rem:lvl-str): Let us tease this out slightly more. Here is the trivial -local system with -structure: it associates to a representations of the constant local system . One may then think of as the sheaf of ‘simultaneous trivalizations’ of the as varies. This naturally has an action of as the automorphisms of as a collection of -local systems with -structure is . Thus, we can take the quotient sheaf , and an étale -level structure is a global section of this.
This may be a bit hard to grok on first read, but the idea is roughly the following. As each is a -local system one can trivialize it (with -structure) if one works ‘very locally’, e.g., over some masive pro-étale local cover . Now one does not expect any particular (family of) trivialization(s) over to descend to (i.e., you don’t expect to be trivial over ). But, descending a -orbit of such trivializations has such a chance because this means that no particular trivialization on the nose descends, but it does up to ‘blurring’ (i.e., up to changing it by the action of ).
The result of this, roughly, is that this allows only a ‘partial trivialization’ of . For example, as is compact open you know (roughly) that it corresponds to a product of compact open subgroups of as varies, and for almost all it must be something like for almost all . At these ‘hyperspecial’ places (more on this later) one can convince themselves that this partial trivialization says nothing: there is no condition on . For other ‘non-hyperspecial’ , a typical example of such a is , in which case one can roughly think that the ‘partial trivialization’ at is not a full trivialization of but a trivialization of (i.e., the reduction modulo of some -lattice in ). This is not a perfectly rigorous explanation, but it should hopefully give a sense for what’s going on.
Expectation (exp:Shim-var) should immediately indicate to you why Shimura varieties are so important in arithmetic geometry: they are moduli spaces for motives, the object of centralmost importance in the linear algebra study of varieties. That said, this expectation is just that, an expectation:
As mentioned in §0, we don’t have any rigorous definition of , and so we’re foiled at the very first step of trying to make our expectation even a rigorous conjecture. But, there is some hope as gives rise to a smooth -variety , one might apply Archimedean Hodge theory to approach things, as indicated in §0.1.2, especially in special (abelian type!) cases.
The hope of using Archimedean Hodge theory to put Expectation (exp:Shim-var) on more firm footing (as mentioned in 1.) is nice, but, in fact, the expectation itself is not nearly good enough to actually work with Shimura varieties as arithmetic-geometric objects. Namely, it doesnt’t cover the case when is smooth over a -adic field, which would be pivotal for studying (the base change of) Shimura varieties over fields like , a cornerstone of our modern approach to arithmetic geometry. In this -adic realm Archimedan Hodge theory can do nothing to help us. But, there is still some hope to work -adically: if Archimedean Hodge theory can’t work in the -adic world what about…-adic Hodge theory? More on this later.
Remark (conditions-on-(G,X)): In our ‘definition’ of Shimura datum, I never actually stated the conditions on they are required to satisfy. The reason for this is that they are opaque. But, let me vaguely say their purpose, and take this up again in the next section.
Namely, there is no a priori reason to believe that the functor described in Expectation (exp:Shim-var) should be representable by a scheme, i.e., that it should be algebraic. For example, our definition involves Archimedean Hodge theory, which is an inherently analytic theory–maybe this space should only exist analytically (in an appropriate sense)? Roughly these extra unstated conditions on are to guarantee that this algebraicity holds.
§2 Harsh reality: the lifecycle of a Shimura variety
The dreamland from the last section must now collide with the grim meat-hook reality of Shimura varieties. Namely, while Expectation (exp:Shim-var) is beautiful it is, as we said, not workable in our current understanding of things. That said, Shimura varieties themselves aren’t conjectural: they are flesh-and-blood mathematical objects. So, what do they actually look like?
Remark: The more astute reader will notice that I am not making any assumptions on below. Technically one needs to assume that is a so-called neat subgroup to really get an algebraic variety. This is a technical point though that I will ignore. It also seems well-known among experts that this is somewhat of an illusory problem, and could be remedied by working with stacks in the appropriate categories at all stages. A similar issue (and solution) holds for the distinction between and , for those that know what that means.
The construction, in steps
Let me describe, in the roughest of possible terms, the creation of Shimura varieties in four steps.
Step 1: real manifold
Shimura varieties start out life as something quite far from a smooth quasi-projective algebraic variety over the number field : they start out as real manifolds.
Before we state this precisely, let us observe that as is a -conjugacy class we may write it as for some subgroup . In fact, the (unstated) conditions on force to be the unique (up to conjugacy) maximal compact subgroup of . Regardless, it’s evident that inherits the structure of a real manifold from .
We then set
Here , and is acting diagonally on the product in the first expression whereas is acting only on the factor of the second expression.
Remark: As indicated by the notation , this is the -points of the eventual . In particular, this looks miles away from Expectation (exp:Shim-var), but is slightly closer than these first appearances.
Namely, if is a -motive with -structure as in Expectation (exp:Shim-var) (written there) then you could imagine considering the pair
.
As we’re over , the étale -level condition is literally just an element of the set
.
Moreover, determines an object of we know that it must belong to, by the Type condition, the isomorphism class . So, it corresponds to an element of .
As we are interested in isomorphism classes of such , the pair . is in fact only well-defined up to isomorphism of -motives with -structure, i.e., up to the of by . Thus, we see that we only get a well-defined class in
.
The sort of faith-based part of this then is that the theory of motives should imply that this association does actually form a bijection between this double-quotient set and the set of isomorphism classes appearing in Expectation (exp:Shim-var).
Step 2: complex manifold
All connected components of are isomorphic, and so we fix one . One may then write
,
where the index set is finite and each is an ‘arithmetic group’ (i.e., the intersection of a compact open subgroup of with ). Up to the finiteness of this index, this is just a simple exercise in writing this double quotient as a union of single quotients.
Thus, to put a complex manifold structure on is tantamount to putting a complex manifold structure on the real manifold . And, this we can specify (and prove) using the (unstated) conditions on in a pleasing way.
Proposition (prop:cmplx-str): There exists a unique complex manifold structure on such that is an object of , i.e., a variation of real Hodge structures.
In other words, we have sort of tailor made to be a period domain (i.e., a moduli space of Hodge structures, see [CMSP]).
Remark: Let me try to make this feel a little less abstract. Fix an element of the conjugacy class . This determines a unique parabolic (e.g., see [Conrad, Theorem 4.1.7]). The associated (partial) flagvariety is the -variety
.
As , this space classifies the parabolics conjugate to . The isomorphism class of this partial flag variety doesn’t depend on the choice of , so we denote it by .
Observe then that we have a natural map of real manifolds
,
sending to the conjugate parabolic . Then, ultimately, Proposition (prop:cmplx-str) amounts to the claim that there exists a unique complex manifold structure on such that the map is holomorphic (for any choice of ).
Step 3: complex variety
Now comes what, to me, is perhaps the most miraculous part of the construction of Shimura varieties (although this is probably a function of my own mathematical weaknesses).
Theorem (Baily–Borel): There exists a unique smooth quasi-projective -variety with underlying complex manifold such that for any other smooth quasi-projective -variety the natural map
is a bijection.
In words this says that there is a unique way to algebraize (to a smooth quasi-projective -variety) such that any holomorphic map , for a smooth quasi-projective -variety , uniquely algebraizes.
One could perhaps view this through the lens of Expectation (exp:Shim-vars) as saying that the (unstated) conditions on force the type of motivic objects being classified to automatically be algebraizable, e.g., compare this to Deligne’s theorem from §0.1.2.
Remark: Let me try to make this ever so slightly less abstract. For a representation of one can build a holomorphic vector bundle on the complex manifold :
,
where is acting diagionally on , where it’s action on is through the inclusion , and still only acts on .
(Such a vector bundle is called an automorphic vector bundle. They in fact, as varies, define an object of , i.e., a -torsor on .)
Take for a -dominant weight such that . One can show that
defines a normal, connected, projective -variety (although almost never smooth!), called the Baily–Borel compactification of the Shimura variety. Moreover, it’s not hard to see that there is a holomorphic map
.
The minor miracle is that this is actually a Zariski open embedding, and so defines a quasi-projective complex variety structure on . Moreover, by a hyperbolicity argument (again related to the unstated conditions on ) this has the desired property as in Baily–Borel’s theorem.
Step 4: descent to reflex field
We now come to the most opaque part of the construction, the one that makes the arithmetic-geometry of Shimura varieties so complicated: the descent from a -variety to a -variety.
Remark: Often one thinks of Shimura varieties as varieties over . From this perspective, the descent to is not part of the definitional structure, but a choice: there could be several models over . Which one do we choose? The one we will describe now is the so-called canonical model. We will see below why this name is justified.
By quasi-projectivity it turns out (see [Milne4]) that giving such a descent is (essentially) equivalent to defining a (‘continuous’) action of the group on .
The definition of this action is ‘done’ in two steps:
Step 4.A:
When is of a very simple type, namely toral type, i.e., for a -torus , one can define this action using class field theory.
Remark: Let me slightly expand on this. As is a point, is a finite disjoint union of copies of . This automatically has a unique model over (something not guaranteed) by taking the corresponding disjoint union of copies of .
We then need to define an action of on this -model. Class field theory gives us a map
.
As is defined over (by definition!) we get a map
,
where this second map is the norm map for . Combining these two maps gives us a map
,
and the action on is now clear.
Step 4.B:
We now bootstrap from Step 4.A. More precisely, we show that there are enough special points, i.e., points in the image of a map of the form
,
for some toral Shimura variety (see Proposition (prop:canonical) below for how such a map is defined). If you have enough such special points, it’s believable that one can uniquely specify an action of using Step 4.A as input.
Remark: The above description of Step 4.B is an immense oversimplification. First, it’s not even clear what enough means. The reflex fields of these special points will (in general) be bigger than , but one is still somehow hoping that you get enough juice out of the special points as you vary them (i.e., the interesection of their reflex fields is close to ).
Secondly, this may help you determine what the action must be, but it doesn’t help you necessarily prove it exists. Step 4.A forces how acts on special points, and having enough such special points might by some ‘continuity argument’ force the action to be unique if it exists, but how do you extend the action beyond the special points?
It is perhaps telling that this last step was not completed until 1983 by Borovoi and Milne (see [Milne5]), nearly 15 years after the introduction of Shimura varieties. Moreover, the ultimate solution is still quite ‘opaque’–one does not ever really get an explicit description of the action of which is (one of) the culprit(s) for the arithmetic mystery of general Shimura varieties.
The output
Let us now try to explicitly describe what the output of the above procedure is: what actual structure do you get out of the theory of Shimura varieties.
The individual Shimura varieties
To begin with, let us state the obvious. For each Shimura datum and each level structure one gets a smooth quasi-projective variety over the reflex field .
The fact that these models are the right ones, thus justifying the name ‘canonical model’, is that they satisfy the following amazing functoriality property.
Proposition (prop:canonical)[Deligne]: Suppose that is a morphism of Shimura data. Then, for any levels with the map of sets
,
descends uniquely to a map of varieties over the compositum of the reflex fields:
.
Remark: On the one hand, this functoriality is quite amazing as the set-theoretic map underlying this map of -varieties is of an Archimedean-Hodge-theoretic flavor. So the fact that they admit arithmetic models is incredible. On the other hand, it’s not that surprising technically. Namely, it is easy to check by hand that this result holds for toral-type Shimura varieties. This then essentially concludes the proof as descending the map is the same as being equivariant for the action of and this action is determined on special points.
From the perspective of Expectation (exp:Shim-var), it’s clear what this map should do. Given a -motive with -structure on , we get a -motive with structure via the rule:
,
i.e., if is a representation of , then is a representation of and so it makes sense to apply to it. This is of type by our assumption that , and assuming it’s easy to see that an étale -level structure on induces an étale -level structure on .
The Hecke action
The second, and in some sense equally important, output of our construction of Shimura varieties is the so-called Hecke action. This takes the form of the existence of two types of special maps between the Shimura varieties for at different levels.
The maps
For any element of and level , one can produce an isomorphism
which on -points is given by
.
One again exploits the canonical model structure to show this map is defined over .
From the perspective of Expectation (exp:Shim-var), the map is the result of using to modify the étale -level structure (to an étale -level structure).
The maps
For any normal containment of levels one builds a a projection map
which on -points is given by
.
and is a Galois cover with Galois group (with a coset acting as ).
Remark: The phrase ‘Galois cover’ often assumes that the source and target is connected. This is not the case here. So, what I really mean is that is a torsor for the constant group scheme associated to the finite group .
From the perspective of Expectation (exp:Shim-var), the map takes an étale -level structure and applies the ‘blurrification’ obtained by taking the associated étale -level structure, which exactly forgets -worth of information.
The action at infinite level
It is often useful to combine the information of the maps and into one package. Namely, we can consider the -scheme
,
with the transition maps given by . As these transition maps are finite (and thus affine), this actually does exist as a scheme over . Moreover, as we are limiting over all levels, one actually sees that each element of gives a morphism of -schemes
.
This defines a continuous action of on in the sense of [Milne6, §II.10].
Given Expectation (exp:Shim-var), one would expect that for a smooth quasi-compact -variety that
From this perspective the map is just modifying this trivialization, and the projection maps
are just sending such a trivialization to its class in .
Hecke correspondences
One often hears of ‘Hecke correspondences’ for Shimura varieties, and so it might be helpful to explain how this facets into the theory. Namely, suppose that we have two different levels and in . Then, there is no direct way to relate them by the maps and , but there is always a relate them via a correspondence. Namely, we obtain the following correspondence
from to , where we have written for simplicity. This is often called a Hecke correspondence.
Remark: The reason that Hecke correspondences are useful is because of their relationship with cohomology, which is of great importance as we will see in §3.2.1 below. Namely, taking in the definition of the Hecke correspondence allows one to build (via standard theory) a morphism of cohomology groups
compatible with -action.
In fact, one can understand this action in another useful way. Namely, the action of action on defines an action of on . Moreover, for any level there is a natural identification
.
From standard representation theory of locally profinite group this endows the cohomology group with an action of the Hecke algebra . As is well-known, this algebra is generated by indicator functions of the form , and the action of this function on the cohomology group agrees with that coming from the Hecke correspondence above.
Of course, this all works for more general coefficients–so-called, automorphic local systems.
[AGHM] Andreatta, F., Goren, E., Howard, B. and Madapusi Pera, K., 2018. Faltings heights of abelian varieties with complex multiplication. Annals of Mathematics, 187(2), pp.391-531.
[FujiwaraKato] Fujiwara, K. and Kato, F., 2006. Rigid geometry and applications. Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math, 45, pp.327-386.
[Milne5] Milne, J.S., 1983. The action of an automorphism of C on a Shimura variety and its special points. In Arithmetic and Geometry: Papers Dedicated to IR Shafarevich on the Occasion of His Sixtieth Birthday Volume I Arithmetic (pp. 239-265). Boston, MA: Birkhäuser Boston.
I have not been able to post of late as I’ve been quite busy working on several projects.
I wanted to make a post though discussing a new draft with my collaborator A. Bertoloni Meli that I’m quite excited about. In it we discuss a method for characterizing the local Langlands conjecture for certain groups as in Scholze’s paper [Sch]. Namely we show that for certain classes of groups an equation like that in the Scholze–Shin conjecture (see [Conjecture 7.1, SS]) is enough to characterize the local Langlands conjecture (for supercuspidal parameters) at least if one is willing to assume that other expected properties of the local Langlands conjecture hold.
The main original idea of this paper is the realization that while the Langlands–Kottwitz–Scholze method only deals with Hecke operators at integral level (e.g. see the introduction to [Sch]) that one can circumvent the difficult questions this raises (e.g. see [Question 7.5,SS]) if one is willing to not only consider the local Langlands conjecture for in isolation, but also the local Langlands conjecture for certain groups closely related to (so-called elliptic hyperendoscopic groups). Another nice byproduct of this approach is that while the Scholze–Shin conjecture is stated as a set of equations for all endoscopic triples for our paper shows that one needs only consider the trivial endoscopic situation (for elliptic hyperendoscopic groups of ).
This paper is closely related to the paper mentioned in this previous post where me and A. Bertoloni Meli discuss the proof of the Scholze–Shin conjecture for unramified unitary groups in the trivial endoscopic triple setting.
References
[Sch] Scholze, Peter. The Local Langlands Correspondence for GL_n over -adic fields, Invent. Math. 192 (2013), no. 3, 663–715.
[SS] Scholze, P. and Shin, S., 2013. On the cohomology of compact unitary group Shimura varieties at ramified split places. Journal of the American Mathematical Society, 26(1), pp.261-294.
While the explanation of my work was the original goal of the notes, they have since evolved into a motivation for the Langlands program in terms of the cohomology of Shimura varieties, as well as explaining some directions that the relationships between Shimura varieties and Langlands has taken in the last few decades (including my own work).
I hope that it’s useful to any reader out there. Part I was mostly written with me, four years ago, in mind. So, in a perfect world someone out there will be in the same headspace as I was, in which case it will (hopefully) be enlightening.
In case you’re wondering the intended level for the reader is probably: 1-3 year graduate student with interest in number theory and/or arithmetic geometry. In particular, for Part I there is an assumption that the reader has some basic knowledge about: Lie groups, algebraic geometry, number theory (e.g. be comfortable with what a Galois representation is), algebraic group theory, and etale cohomology (although this can be black-boxed in the standard way–e.g. all one needs to know is the contents of Section 3 of this set of notes). Part II is mostly written as an introduction to a research topic, and so requires more background.
Enjoy!
PS, feel encouraged to point out any mistakes/improvements that you think are worth mentioning.
EDIT: While these notes might be still useful to read, if one wants a more in-depth explanation of the ideas below see this post and the notes from this post.
——
This will be the first in a series of posts discussing Shimura varieties. In particular, we will focus here on a sort of broad motivation for the subject—why Shimura varieties are a natural thing to study and, in particular, what they give us.
-structure
.
–this just means that these are the only pairs of
that show up in the Hodge decomposition.
of
is the category of exact (i.e., sends exact sequences to exact sequences)
.
a reductive group over
,
is defined by
for any
.
is a morphism
of
, and is a closed embedding if
is.
(i.e.,
viewed as a real algebraic group) is the so-called Deligne torus. It is the Tannakian fundamental group of the Tannakian category
of real Hodge structures (with the usual fiber functor).
is naturally identified with
(the notation
means ‘isomorphism classes’). In particular, one may view the
thus the notation on the right-hand side of
, one may
view as the (conjugacy class of) cocharacter(s) defining the Hodge filtration on these real Hodge structures, whence the name.
of the Shimura datum
is the
-adic étale realization functor, so the output is an étale
-local system,
is the Hodge realization functor, so the output is a variation of
gives rise to a smooth
, one might apply Archimedean Hodge theory to approach things, as indicated in §0.1.2, especially in special (abelian type!) cases.
, a cornerstone of our modern approach to arithmetic geometry.
is an object of
, i.e., a variation of real Hodge structures.
with underlying complex manifold
with
the map of sets
,
.
![[g]_\mathsf{K}](https://s0.wp.com/latex.php?latex=%5Bg%5D_%5Cmathsf%7BK%7D&bg=ffffff&fg=333333&s=0&c=20201002)
![[g]_{\mathsf{K}}\colon \mathrm{Sh}_{\mathsf{K}}(\mathbf{G},X)\xrightarrow{\sim}\mathrm{Sh}_{g^{-1}\mathsf{K}g}(\mathbf{G},X)](https://s0.wp.com/latex.php?latex=%5Bg%5D_%7B%5Cmathsf%7BK%7D%7D%5Ccolon+%5Cmathrm%7BSh%7D_%7B%5Cmathsf%7BK%7D%7D%28%5Cmathbf%7BG%7D%2CX%29%5Cxrightarrow%7B%5Csim%7D%5Cmathrm%7BSh%7D_%7Bg%5E%7B-1%7D%5Cmathsf%7BK%7Dg%7D%28%5Cmathbf%7BG%7D%2CX%29&bg=ffffff&fg=333333&s=0&c=20201002)
![[g]_{\mathsf{K}}\colon \mathbf{G}(\mathbb{Q})(x,g')\mathsf{K}\mapsto \mathbf{G}(\mathbb{Q})(x,g'g)g^{-1}\mathsf{K}g](https://s0.wp.com/latex.php?latex=%5Bg%5D_%7B%5Cmathsf%7BK%7D%7D%5Ccolon+%5Cmathbf%7BG%7D%28%5Cmathbb%7BQ%7D%29%28x%2Cg%27%29%5Cmathsf%7BK%7D%5Cmapsto+%5Cmathbf%7BG%7D%28%5Cmathbb%7BQ%7D%29%28x%2Cg%27g%29g%5E%7B-1%7D%5Cmathsf%7BK%7Dg&bg=ffffff&fg=333333&s=0&c=20201002)
![[g]_{\mathsf{K}}](https://s0.wp.com/latex.php?latex=%5Bg%5D_%7B%5Cmathsf%7BK%7D%7D&bg=ffffff&fg=333333&s=0&c=20201002)
![[g]\colon \mathrm{Sh}(\mathbf{G},X)\to \mathrm{Sh}(\mathbf{G},X)](https://s0.wp.com/latex.php?latex=%5Bg%5D%5Ccolon+%5Cmathrm%7BSh%7D%28%5Cmathbf%7BG%7D%2CX%29%5Cto+%5Cmathrm%7BSh%7D%28%5Cmathbf%7BG%7D%2CX%29&bg=ffffff&fg=333333&s=0&c=20201002)
![[g]](https://s0.wp.com/latex.php?latex=%5Bg%5D&bg=ffffff&fg=333333&s=0&c=20201002)
Hard Arithmetic 
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