Period maps, variations of p-adic Hodge structure, and Newton stratifications (research announcement)

Transcription

1 Period maps, variations of p-adic Hodge structure, and Newton stratifications (research announcement) David Hansen January 28, Let X be a connected complex manifold, and let H =(H Z, Fil H Z Z O X ) be a rank n variation of Hodge structure over X. Let X be the GLn (Z)-covering of X parametrizing trivializations Z n HZ,andletGr h =GL n (C)/P h (C) be the Grassmannian parametrizing flags in C n with successive graded pieces having ranks h = {h i =rankfil i /Fil i+1 }. Then we have a natural period morphism π : X Grh of complex manifolds, equivariant for the natural GL n (Z)-actions on the source and target. In this note we sketch an analogous p-adic construction and some of its consequences for Shimura varieties. We came to these ideas while trying to understand Ana Caraiani and Peter Scholze s recent paper. Precisely, if X is a rigid analytic space and V is a Q p -local system of rank n on X, weconstruct aspacet riv V/X parametrizing trivializations of V. When X is smooth and V is de Rham, we construct a canonical p-adic period morphism from T riv V/X to a de Rham affine Grassmannian. We d like to say that T riv V/X is a GL n (Q p )-torsor over X, butunfortunatelyit snotclearwhether T riv V/X is even a reasonable adic space. The correct setting here seems to be that of diamonds: we prove that T riv V/X is naturally a diamond, and that its natural map to the diamond X associated with X is a pro-étale GL n (Q p )-torsor. The necessity of changing to a looser category of geometric objects here seems roughly analogous to the fact that, in the complex analytic situation above, X X is defined only in the complex analytic category even when X begins its life as a projective variety. We also remark that the target of the period morphism outoft riv V/X only exists as a diamond in general; analogously, the open subspace of Gr h parametrizing Hodge structures (and through which π : X Gr h factors) is a non-algebraic complex manifold. Torsors for Q p -local systems, and period maps We start with a very brief recollection on Q p -local systems. Department of Mathematics, Columbia University, 2990 Broadway, New York NY 10027; hansen@math.columbia.edu 1

2 Proposition 1.1. Let X be a locally Noetherian adic space, or a perfectoid space, or a diamond. Then we have a natural category Q p Loc(X) of Q p -local systems on X; these are (certain) sheaves of Q p -vector spaces on the pro-étale site X proet which are locally free of finite rank. When X is locally Noetherian or perfectoid, with associated diamond X, there is a natural equivalence Q p Loc(X) = Q p Loc(X ). These arise quite naturally in geometry: if f : Y X is a proper smooth morphism of locally Noetherian adic spaces over Q p,thenanyr i f proet Q p is naturally a Q p -local system on X (Gabber, Kedlaya-Liu, Scholze-Weinstein). We also remark that if X is a variety over Q p,thenanylisseq p - sheaf on X et induces a Q p -local system on X ad. Our first result is the following (not very difficult) theorem. Theorem 1.2. Let D be any diamond, and let V be a Q p -local system on D of constant rank n. Then the functor T riv V/D :Perf /D Sets {f : T D} Isom QpLoc(T )(Q p n,f V) is representable by a diamond pro-étale over D, andthenaturalmapt riv V/D Dis a pro-étale GL n (Q p )-torsor. The functor from rank n Q p -local systems on D to pro-étale GL n (Q p )-torsors over D given by V T riv V/D is an equivalence of categories, with essential inverse given by D D GLn(Q p) Q n p. Next we recall the definition of a de Rham Q p -local system on a smooth rigid analytic space X, following the notation and terminology in Scholze s p-adic Hodge theory paper. This seems to be the correct p-adic analogue of a variation of (Q or Z )Hodge structures on a complex manifold. Definition 1.3. Let E/Q p be a discretely valued nonarchimedean field with perfect residue field of characteristic p, andletx be a smooth rigid analytic space over Spa E. Letν : X proet X et be the usual projection of sites, and let B + dr, B dr, OB + dr, OB dr be the usual period sheaves on X proet. Given a Q p -local system V on X proet,define D dr (V) =ν (V Qp OB dr ). This is a locally free O Xet -module of finite rank, equipped with a decreasing exhaustive separated filtration by O Xet -local-direct summands and with an integrable connection satisfying Griffiths transversality. There is a natural injective map α dr : ν D dr (V) ν O Xet OB dr V Qp OB dr of OB dr -modules compatible with all structures. We say V is de Rham if α dr is an isomorphism. Maintain the setup of the previous definition, and assume V is de Rham. Then we get two B + dr -lattices in the B dr-local system V Qp B dr :onegivenbym:= V Qp B + dr and the other given by the image of ( ν =0 D dr (V) ν O OB+ Xet dr) under the isomorphism α =0 dr : ( ν D dr (V) ν O Xet OB dr ) =0 V Qp B dr 2

3 induced by α dr.callthissecondonem 0.WhenFil 0 D dr (V) =D dr (V), wehavem 0 M. More generally, let h i =rank OX gr i D dr (V) =rank OX (Fil i D dr /Fil i+1 D dr ) be the Hodge numbers of V, andletµ V : G m GL n be the cocharacter in which the weight i appears with multiplicity h i.weremarkthattheh i s are constant on connected components of X, and thus µ V is as well. Proposition 1.4. The relative positions of M 0 and M inside V Qp B dr are given by µ V. With the setup as above, we now have the following theorem. Theorem 1.5. Let X be a smooth rigid analytic space over Spa E as above, and let V be a de Rham Q p -local system on X of rank n. Assume that X is connected, or more generally that the Hodge cocharacter µ V is constant. Let Gr GLn,µ V be the open Schubert cell in the (ind-)diamond Gr GLn. Then we have a natural GL n (Q p )-equivariant period morphism of diamonds over Spd E. π dr : T riv V/X Gr GLn,µ V Corollary 1.6. Let X be a connected rigid analytic space over Spa E as above, and let V be a de Rham Q p -local system on X of rank n with constant Hodge cocharacter µ V. Then V induces a natural Newton stratification of X into locally closed subsets X = X b b B(GL n/q p,µ V) indexed by Newton polygons lying above the Hodge polygon of V and with matching endpoints. Proof. Pulling back the Newton strata of Gr GLn,µ V defined by Caraiani-Scholze 1 under π dr,weget astratificationof T riv V/X by GL n (Q p )-stable locally closed subsets, which descend under the identifications T riv V/X /GL n (Q p ) = X = X to a stratification of X. Remark. The subsets X b are determined by their rank one points: x X lies in a given X b if and only if its unique rank one generization x lies in X b. Also, it s not clear whether the closure of any X b is a union of X b s, so the term stratification is being used in a loose sense here. Next we compare this generic fiber Newton stratification with a suitable special fiber Newton stratification, in a situation where the latter exists in a reasonably canonical way. More precisely, let f : Y X be a smooth proper morphism of connected flat p-adic formal schemes over Spf O E, with associated morphism of adic generic fibers f : Y X over Spa E. Assume X is smooth. Let M = R i f crys (O/W )[ 1 p ] be the ith relative crystalline cohomology, and let V = Ri f proet Q p be the ith relative p-adic étale cohomology. Then, on the one hand, the Frobenius action on the specializations of M at geometric points of X gives a natural stratification X = X b. b B(GL n/q p,µ V) 1 Warning: Here and in what follows, Caraiani-Scholze use B(G,µ 1 ) everywhere that we use B(G,µ), but everything matches after noting that the two are in bijection viab b 1. 3

4 In this context, the fact that the Newton polygon lies over the Hodgepolygonisafamoustheorem of Mazur. On the other hand, the previous corollary applied to V gives a decomposition X = b B(GL n/q p,µ V) X b with the same indexing set. Theorem 1.7. The stratification of X associated with V agrees on rank one points with the pullback (under the specialization map s : X X ) of the stratification of X associated with M. We sketch the proof. Let x =Spa(K, O K ) X be any rank one point, and choose a geometric point x =Spa(C, O C ) lying over it, with corresponding point x :SpfO C X. Letκ be the residue field of K, givingapoints =Specκ X. Letk = O C /m C = O C /m C be the residue field of O C, so we get a natural geometric point s :Speck X over s. Notethat s = s = x as points in the topological space X, andthatthespecializationmaps sends x to this point. Set L = W (k)[ 1 p ] and A = W (O C ), sothesurjectiono C k induces a surjection A[ 1 p ] L. Specializing the comparison map α dr at x (and noting that V x = H i et(y x, Q p )), we get a pair (T,Ξ) consisting of a finite free Z p -module T = H i et(y x, Z p ) /tors V x together with a B + dr (C)- lattice 2 Ξ=M 0,x = H i dr(y x ) K B + dr (C) T Z p B dr (C). We call such pairs Fargues pairs, in recognition of the following theorem: Theorem 1.8 (Fargues). The following categories are naturally equivalent: i) Pairs (T,Ξ) as above. ii) Breuil-Kisin modules over A := W (O C ). iii) Shtukas over Spa C with one paw at C. Applying this theorem to the particular Fargues pair associated with x above, we get a Breuil- Kisin module M over A; specializingm[ 1 p ] along A[ 1 p ] L gives a ϕ-isocrystal M 0 over L, whose Newton polygon turns out (after unwinding the definition of the Newton strata of Gr GLn,µ V and the equivalences in Fargues s theorem) to record exactly which stratum of X contains x. Onthe other hand, Bhatt-Morrow-Scholze have recently defined a remarkable cohomology functor HA i ( ) on smooth proper formal schemes over Spf O C taking values in Breuil-Kisin modules over A; combining their work with Fargues s theorem, we get an identification M[ 1 p ] = HA i (Y x)[ 1 p ]. Since Bhatt- Morrow-Scholze s work also gives a canonical identification HA i (Y x) A L = Hcrys i (Y s/w (k))[ 1 p ] compatible with Frobenius, we deduce that the ϕ-isocrystal M 0 coincides with the ith rational crystalline cohomology of Y s.sincethenewtonpolygonofthelatterdeterminesthestratum of X containing s = s(x), we redone. With applications in mind, we also consider the following situation. Let G be a reductive group over Q p,sog = G(Q p ) is a locally profinite group. Then for X as in Proposition 1.1, we define G-local systems on X as exact additive tensor functors V :Rep(G) Q p Loc(X) (W, ρ) V W in the obvious way. 3 This generality is not artificial; for example, polarized Q p -local systems correspond to GSp 2n -orgo n -local systems, and we get even more general examples from Shimura 2 One has to be a bit careful in the following expression since K B dr (C) doesn t quite make sense unless x is aclassicalrigidanalyticpoint;hencethequotationmarks. 3 Here of course Rep(G) denotes the category of pairs (W, ρ) with W afinite-dimensionalq p-vector space and ρ : G GL(W ) amorphismofalgebraicgroupsoverq p. 4

5 varieties (as we ll see below). When X is a diamond, the natural functor T riv V/X is a pro-étale G- torsor over X, andweagainprovethattheassociationv T riv V/X is an equivalence of categories, with essential inverse given by sending a pro-etale G-torsor X over X to the G-local system V( X) :Rep(G) Q p Loc(X) (W, ρ) X G,ρ W. When X is a smooth rigid analytic space, we say a G-local system V is de Rham if the associated Q p -local systems V W are de Rham for all (W, ρ) Rep(G). Using the fact that the comparison isomorphism α dr is compatible with direct sums, tensor products, and subquotients, the Tannakian formalism (plus a little more) gives a Hodge cocharacter µ V : G m,e G E such that ρ µ V measures the Hodge filtration on D dr (V W ) for any (W, ρ). We then have the following generalization of Theorem 1.5. Theorem 1.9. Let V be a de Rham G = G(Q p )-local system on a smooth rigid analytic space X, with constant Hodge cocharacter µ V.ThenwehaveanaturalG-equivariant period morphism of diamonds over Spd E. π dr : T riv V/X Gr G,µV We again get a Newton stratification of X, indexednowbythekottwitzsetb(g,µ V ) (which explains our use of this notation in the GL n -setting earlier). Some applications to Shimura varieties Let (G,X) be a Shimura datum, with associated Hodge cocharacter µ and reflex field E. Forany sufficiently small open subgroup K G(A f ),theassociatedshimuravarietysh K is a smooth quasiprojective variety over E. Choose a prime p of E lying over p, andlets K =(Sh K E E p ) ad be the associated rigid analytic space over Spa E p. Let S K = S K be the associated diamond over Spd E p.finally,setg = G(Q p ) as before. Proposition For any sufficiently small K p G(A p f ),thereisadiamonds K p over Spd E p with an action of G such that S K p = lim S K p K p Kp G-equivariantly and compatibly with changing K p (or even with changing the Shimura data). When (G,X) is of Hodge type, Scholze constructed a perfectoid Shimura variety, i.e. aperfectoid space S K p with continuous G-action such that S K p lim Kp S K p K p as adic spaces. Whenever such an S K p exists, we necessarily have S K p = S K for formal reasons. p However, unlike the difficult construction of S K p,theproofofproposition1.10isessentially trivial once the theory of diamonds is set up: the inverse limit of any projective system of diamonds with finite étale transition maps is a diamond. We also point out that knowledge of S K p is strictly weaker than knowledge of S K p:theconstructionofthelatterobjectalsogivesveryrichinformation about the existence of certain affinoid coverings, formal models, compactifications, etc., and S K p 5

6 doesn t a priori have the same applications to p-adic automorphic forms as S K p. In any case, it seems likely that S K p always exists (and X. Shen has recently constructed such perfectoid Shimura varieties when (G,X) is of abelian type), but this is probably out of reach in general. Therefore, it might be surprising that the following result is within reach. Theorem There is a natural G-equivariant period morphism π HT : S K p Fl G,µ of diamonds over Spd E p,compatiblewithchangingk p and functorial in morphisms of arbitrary Shimura data. When (G,X) is of Hodge type (or more generally, of abelian type), this is the morphism of diamonds associated with the refined Hodge-Tate period map of Caraiani-Scholze. π HT : S K p Fl G,µ We briefly sketch the construction of π HT. For any fixed level K p,themaps K p S Kp K p is naturally a pro-étale K p -torsor, so the pushout S Kp K p := S K p Kp G is naturally a pro-étale G-torsor over S K p K p,compatiblywithvaryingk p K p. Note that via the G-action on S K p,wegetacanonicalsplitting S Kp K p = SK p K p \G which is G-equivariant for the diagonal G-action on the right-hand side. By our previous equivalence of categories, S Kp K p gives rise to a G-local system V over S Kp K p,orequivalentlyovers Kp K p,which we call the tautological G-local system. Thecrucialingredientisthen: Theorem 1.12 (R. Liu-X. Zhu). The tautological G-local system V over S Kp K p Hodge cocharacter µ, for any Shimura variety. is de Rham with They prove, surprisingly, that if the stalk of a Q p -local system (or G-local system) V on a connected rigid analytic space is de Rham at one classical point (in the classical sense of Fontaine), then V is de Rham. For a Shimura variety, this allows one to check the de Rham property at special points, where everything is explicit. Thanks to this result, we can apply Theorem 1.9 to get a G-equivariant morphism of diamonds π dr : S Kp K p Gr G,µ over Spd E p.onechecksbyhand,usingtheaforementionedsplitting,that π dr descends along the G-equivariant projection S Kp K p S K p to a G-equivariant morphism π dr : S K p Gr G,µ independent of the auxiliary choice of K p.finally,sinceµ is minuscule, a result of Caraiani-Scholze gives a canonical identification Gr G,µ = Fl G,µ,andwedefineπ HT as the composite of π dr with this isomorphism. 6

7 We remark that when S K p is the diamond of a perfectoid Shimura variety S K p,anymorphism of diamonds arises uniquely from a morphism f : S K p = S K p Fl G,µ f : S K p Fl G,µ of adic spaces. In particular, the proof of Theorem 1.11 actually gives a new construction of the refined Hodge-Tate period map for perfectoid Shimura varieties of Hodge type (or abelian type); this proof is of course related to Caraiani-Scholze s proof, butthedetailsareratherdifferent. Again Gr G,µ = Fl G,µ has a Newton stratification by G-invariant locally closed subsets, indexed by B(G,µ), sopullingbackunderπ HT and descending gives a generic fiber Newton stratification of S Kp K p as before. Suppose now that (G,X) is a Shimura datum of Hodge type, with p>2 and G/Q p unramified. Let K p G be hyperspecial, so by work of Kisin, Sh Kp K p E E p has (among other things) a good integral model SK p K p over O Ep and the special fiber of SK p K p has a natural Newton stratification. Let S Kp K p be the formal scheme given as the p-adic completion of SK p K p, so the associated rigid analytic space identifies with the locus of good reduction S gd K p K p S Kp K p. Again we have a specialization map s : S gd K p K p S Kp K p = S K p K p OEp F q. Theorem The generic fiber Newton strata of S gd K p K p coincide on rank one points with the pullback under s of the special fiber Newton strata. The proof naturally combines (the ideas in the proof of) Theorem 1.7 with the Tannakian formalism and some unwinding of Kisin s construction. When the Sh K s are compact PEL Shimura varieties of type A or C, this is a result of Caraiani-Scholze (cf. Section 4.3 of their paper, especially Lemma and the diagram immediately following). 7