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the (715), this (217), and (198), that (170), for (151), #shimura (103), #varieties (103), one (97), with (73), are (72), over (68), but (64), geometry (60), can (54), structure (54), hodge (50), type (49), let (48), theory (47), what (46), which (45), https (44), pdf (44), there (44), then (44), more (43), variety (42), from (41), not (41), action (39), canonical (36), such (36), some (36), category (36), algebraic (35), scheme (35), map (35), expectation (35), should (35), models (34), org (34), abelian (34), conjecture (34), namely (34), here (34), theorem (34), example (34), arithmetic (33), motives (33), will (33), smooth (33), integral (32), level (32), shim (31), exp (30), have (29), rigid (29), cohomology (29), group (29), see (29), var (29), has (29), only (29), www (28), remark (28), model (28), adic (27), how (27), sense (27), fact (26), any (26), math (25), these (25), langlands (23), class (23), all (23), they (23), most (23), you (22), points (22), above (22), finite 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e 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 é tale 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 flag variety 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 ...