LOCAL INDECOMPOSABILITY OF TATE MODULES OF NON CM ABELIAN VARIETIES WITH REAL MULTIPLICATION

OF NON CM ABELIAN VARIETIES WITH REAL MULTIPLICATION

HARUZO HIDA

Abstract. Indecomposability ofp-adic Tate modules over thep-inertia group for non CM (par- tiallyp-ordinary) abelian varieties with real multiplication is proven under unramifiedness ofpin the base field and in the multiplication field.

Contents

1. Deformation space of an AVRM 3

2. Hilbert modular Shimura variety 5

3. Eigen differentials 7

4. Eigen coordinates 8

5. Local indecomposability of an AVRM 14

6. CM periods and proportionality constants 16

7. Density of partially ordinary primes 18

References 18

As quoted by Ghate–Vatsal in [GV] Question 1 (and answered there affirmatively to a good extent), it is a fundamental problem posed by R. Greenberg to decide indecomposability of an elliptic modularp-adic Galois representation restricted to the decomposition group atpin the ordinary case (as long as the representation is not of CM type). Jointly with B. Balasubramanyam, they have generalized the result to the cases of Hilbert modular forms for primespsplitting completely in the totally real base field (see [GV1]). There are some good applications of their results (see for example [E] and [G]). We can ask the same question forp-adic Galois representations arising from the Tate module of an abelian varietyAwith real multiplication defined over a number fieldk; so, End(A/k) is the integer ringO of a totally real field of degree dimA. This concerns the p-adic Tate module TpAfor a primep|(p) ofO as a module over the decomposition groupDp at a placeP overpof k.

Suppose thatAhas good reduction moduloP. If the finite flat group schemeA[p] ofp-torsion points is local-local at P, as is well known, the Dp-module A[p](Q_p) or TpA⊗^OpFp is often irreducible (as proven in [ALR] IV-38, §A.2.2 for elliptic curves, see also the generalization of Serre’s modulo pmodularity conjecture for totally real k; in particular, §3.1 of [BDJ]). Anyway in this paper, we limit ourselves to the case whereA[p](Fp)6={0}. This assumption does not necessarily mean that Ahas ordinary good reduction atp.

We answer this question affirmatively in Theorem 5.1 for an abelian variety with sufficiently many real multiplication but without complex multiplication (over Q) if the endomorphism totally real fieldF and the fieldkof definition are unramified atpoverQ. Since simple factors of the jacobian of a Shimura curve are often of this type, we should be able to answer the original question by Greenberg. Indeed, Bin Zhao [Z] has just removed our assumption of unramifiedness ofpin the base field k and in the multiplication field F and has proved unconditionally indecomposability for the

Date: December 30, 2012.

2010Mathematics Subject Classification. primary 14G35, 11G15, 11G18, 11F80 secondary 11G10, 14L05.

Key words and phrases. abelian variety, Tate module, real/complex multiplication, Galois representation, deformation.

The author is partially supported by the NSF grants DMS 0753991 and DMS 0854949.

1

nearly ordinary Galois representation of each weight 2 Hilbert cusp form at least over an odd degree totally real base field. If the base field has even degree, he had to assume square-integrability at a finite place or the existence of an abelian variety associated to the cusp form.

Since a CM abelian variety with ordinary good reduction atphas semi-simple Tate module over Dp, we may regard thep-local indecomposability of thep-adic Tate module as a characterization of non CM/CM abelian varieties via localp-inertia action. The importance of such characterization is emphasized in [H11a], and other examples of characterization are listed there.

Throughout this paper, we fix an algebraic closureQinCand an algebraic closureQ_v ofQv(the v-adic field orRforv=∞) for each rational placev. We write Cv for the completion ofQ_v under v. Therefore, for eachv, we fix a field embeddingiv :Q,→Cv (here i∞ coincides with the original inclusionQ⊂C) and identify allCvwithCso that we have the following commutative diagram for eachv:

Q Q

iv



y^{∩ ∩}

 y^i∞ Cv C.

Here is a sketch of our proof of indecomposability (and an outline of the paper), assuming for simplicity that the abelian varietyAover a number fieldk⊂Qhas multiplication by the integer ring Oof a real quadratic fieldF; so, 2 = [F :Q] = dimAand End⁰_F(A_/Q) = End⁰_F(A/k×^kQ_/Q) =F(this means that A does not have complex multiplication), where “End⁰_F(?)” denotes the total quotient ring of theO-linear endomorphism algebra EndO(?) of the object “?” inside. Such abelian varieties are parameterized by Hilbert modular varieties whose theory we recall in Section 2 to the extent we need (explicitly or implicitly).

LetOdenote the integer ring ofk. We denote primes ofkby upper case Gothic letters and those ofFby lower case Gothic letters, and the corresponding Roman (lower case) character is the residual characteristic of a prime denoted by a Gothic character. Suppose further for simplicity thatAhas ordinary good reduction at primesPandLwithp6=l. On the contrary to what we want to prove, suppose thatA[p^∞] for a primep abovephas a semi-simple Tate module over Dp = Gal(Q_p/kP).

Then the p-adic Serre–Tate coordinate tp(A) of Ais equal to 1. If p is a unique prime over (p) in O, this identitytp(A) = 1 forces Ato be a canonical lift having complex multiplication. Thus we assume (p) =pp⁰ in F with p 6=p⁰. By the Serre–Tate theory, for the Witt vector ring Wp with coefficients in an algebraic closureFpofFp, the deformation space of the reductionAP=A⊗^OPFp

(forFP:=O/Pregarded as sitting inFp) is isomorphic toSb/Wp=Sbp/Wp:=Gbm⊗^ZpOp=Sbp×Sbp⁰

withSbp :=Gbm⊗^ZpOp, andtp is the coordinate of the multiplicative formal groupGbm⊗^ZpOp=Gbm

(as Op = Zp). In other words, Gbm/Wp is the formal completion ofGm = Spec(Wp[tp, t⁻¹p ]) along the origin (i.e., the identity) of its reduction modulop. We recall in Section 1 some details of the construction of Serre–Tate coordinates.

The fieldM = End⁰_F(A_P/Fp) is a CM quadratic extension ofF. We have two Serre–Tate coordi- natestpandtp⁰. SinceAdoes not have CM, we must havetp⁰(A)6= 1 (as the origin ofSbcorresponds to the canonical lift ofAPwhich has complex multiplication by the CM fieldM). By the universality ofS,b M^×acts onSbnaturally by an isogeny action. Then by the reciprocity law ofSbin [H10]§3.3, we havetp◦α=t^α_p^σ(1−c) for an embeddingσ:M ,→Qinducing the placep ofF after composing with ip. Herecdenotes the generator of Gal(M/F). For thel-adic placelinduced by il◦σ, we have the Serre–Tate coordinatetlof the deformation spaceSbl=Gbm⊗^ZpOlofAL. Now the abelian varietyA gives rise to ak-pointxof the Hilbert modular Shimura varietySh/Q forGL(2)/F. Looking at the k-tangent spaceT anxatx∈ShK(k) (for a neat open compact subgroupK⊂GL2(F_A^(∞)) maximal atpandl), the Lie algebrasLie(bSp) andLie(Sbl) are related to (theσ-part of)T anxvia appropriate scalar extension; therefore, the action of M^× on Lie(Sbp) can be transferred to Lie(Sbl) via T anx. We will see in Section 4 thattp (resp. tl) is the exponential of the corresponding linear coordinate τp (resp. τl) ofLie(Sbp) (resp. Lie(Sbl)). From this, we conclude in Theorem 4.5 thattlalso satisfies tl◦α=t^α_l^σ(1−c) forα∈M^×; so, End⁰_F(A_L/Fl) must be isomorphic toM. In Section 3, we describe

the canonical O-action on ΩSh/k, T anx and on Lie(Sb), and in Section 4, we study the M-action (extending theO-action) on the Serre–Tate coordinates (and its linear version).

Because of x ∈ Sh(k) ⊂ Sh(Cp) = Sh(Cl) (under our identification: Cp = Cl), the formal completionOb^Sh,x along the points x∈Sh(Cp) =Sh(Cl) canonically contains OSbp and OSbl. Thus inOb^Sh,x, we can compareτp = log_p◦tp and τl = log_l◦tl for thev-adic logarithm log_v. Since they both satisfy τ?◦α =α^σ(1−c)τ?, they are proportional; so, τl(A) = 0 and hence tl(A) = 1, which implies the Tate moduleTlAof A[l^∞] is also semi-simple under Dl. As is well known (cf. [O] 2.7 and Section 7 in the text), split ordinary primes for an abelian surface have Dirichlet density 1 for k(replacingkby its finite extension if necessary), and using non-CM property of thep-adic Galois representation on the Tate module TpA (combined with Chebotarev density), we can find l such that M 6∼= End⁰F(A_L/Fl), getting a contradiction (whose details will be given in Section 5). In the text, we fill details of this argument and generalize this to totally real fieldsF with [F :Q]≥2, to primesp with residual degree ≥1 and to abelian varietiesAwhich may not be ordinary at P(i.e., only partially ordinary atP).

As a byproduct of our proof, we get (in Theorem 4.5) the proportionality ofτpandτl. Though it is irrelevant to the proof of the main theorem, we will compute in Section 6 explicitly the proportionality constant as the ratio of the square of the Katz p-adic and l-adic CM periods. In the above sketch of the proof, we used heavily existence of many primes ofkat which the given abelian varietyA/k

has good (partially) ordinary reduction. We give a brief account of the fact that such primes has density 1 among primes of (the Galois closure of)kin Section 7.

As is clear from this sketch, there is much room for generalization of our argument to abelian varieties with less multiplication. We hope to come back to the study of local indecomposability for general abelian varieties in near future.

The author thanks Brian Conrad for some help of identifying an optimal reference in rigid analytic geometry for quotes here.

1. Deformation space of an AVRM

Pick a rational primesp. LetF be a totally real field with integer ringOin whichpis unramified.

Writed= [F :Q]. Let (A, λ, θ)/k be an abelian variety with real multiplication by O (in short an AVRM by O) over a number field k ⊂ Q with integer ring O in which p is unramified. Here θ : O ,→ End(A/k) is a ring homomorphism sending the identity to the identity making Lie(A) a free O⊗^Zk-module of rank 1, and λ : A → ^tA for the dual abelian scheme ^tA = Pic⁰_A/k is an O-linear polarization of degree prime top. Thus thep-adic Tate module TpA is free of rank 2 over Op=O⊗^ZZp, and dimA= [F :Q]. Let P={x∈O:|ip(x)|^p<1}be the prime of kinduced by ip :Q,→Cp. LetFp =FP be an algebraic closure ofFP =O/P, and writeW =Wp for the ring of Witt vectors with coefficients inFp. We identifyW with the subring of Cp which is the p-adic completion of the ring of integers in the maximal unramified extension ofQp inQp⊂Cp. Thusip

embeds O in W, and we regard O_P ⊂W. We put W = W^p =i⁻¹_p (Wp)⊂ Qwhich is a discrete valuation ring with residue fieldFp (and is a strict henselization ofZ(p)=Q∩Zp).

Let OP be the P-adic completion of O, and suppose that (A, λ, θ)/k extends to the triple (A, λ, θ)/OP for an abelian scheme A/OP. Thus we can think of the special fiber (A0, λ0, θ0)_/Fp = (A, λ, θ)×Op Fp. Let CLW denote the category of complete local W-algebras with residue field Fp. We consider the fiber category D^/W (overCLW) of deformations over Rof (A0, λ0, θ0)_/Fp over R∈CLW. Objects ofD^/W are triples (A⁰, λ⁰, θ⁰)/RsatisfyingιA⁰ : (A⁰, λ⁰, θ⁰)×^RFp∼= (A0, λ0, θ0)_/Fp. A morphismφ: (A⁰, λ⁰, θ⁰)/R→(A⁰⁰, λ⁰⁰, θ⁰⁰)/Ris a morphism of AVRM’sφ:A⁰→A⁰⁰such that

(1) the special fiberφ0ofφsatisfiesιA⁰⁰◦φ0◦ι⁻¹_A0 = idA0 for the identity map idA0 ofA0; (2) λ⁰ =^tφ◦λ⁰⁰◦φ;

(3) φ◦θ⁰(a) =θ⁰⁰(a)◦φfor alla∈O.

We suppose that

(NLL) the reductionA0:=AP=A⊗^OPFp has nontrivialp-torsionFp-points.

So, we have a prime idealp|pofOsuch that thep-adic Barsotti–Tate groupA[p^∞] is ordinary, fitting into the following connected-´etale exact sequence of Barsotti–TateOp-modules over W:

0→µp^∞⊗^ZpOp→A[p^∞]/W →Fp/Op→0.

Let Σ^ord= Σ^ord_p be the set of primes pinO over psuch thatA[p^∞]/W is not connected. Then we put Σ^ll= Σ^ll_p =

p|p:p6∈Σ^ord . We may regard Σ^? as a set of field embeddingsσ:F ,→Qmade up ofσwith{α∈O:|ip(σ(α))|^p <1}=p∈Σ^?.

We now want to relate D^/W with the following fiber category DEF over CLW of quadruples (A0, D,Λ, )/R, whereDis a Barsotti–TateO-module overR, Λ :D→^tDis anO-linear isomorphism of Barsotti–Tate groups into the Cartier dual^tDofDand:D0∼=A0[p^∞] is anO-linear isomorphism of Barsotti–TateO-modules for the special fiberD0 =D⊗^Spec(W)Spec(Fp). We requireto satisfy the following commutativity:

D0

−−−−→ A0[p^∞]

Λ0

 y

 y^λ0

tD0 ←−−−−_t

tA0[p^∞], where Λ0is the special fiber of Λ. A morphism

φ: (A0, D,Λ, )→(A⁰₀, D⁰,Λ⁰, ⁰)

ofDEF is a pair of O-linear morphismsφBT :D→D⁰ in HomBT_/W(D, D⁰) andφD :A0 →A⁰₀in HomGSCH_/Fp(A0, A⁰₀) making the following two diagrams commutative:

A0[p^∞] −−−−→^φD A⁰₀[p^∞]

x



x

⁰ D0 −−−−→

φBT

D⁰₀

and

D0 φBT

−−−−→ D⁰₀

Λ

 y

 y^Λ0

tD0 ←−−−−tφBT

tD⁰₀.

HereGSCH(resp. BT) stands for the fiber category of group schemes (resp. Barsotti–Tate groups) overCLW. We have a natural functor: D^/W →DEF given by

A7→(A0=A⊗^W Fp, A[p^∞], λ|^A[p∞], ι) for the canonical isomorphismι:A[p^∞]⊗^W Fp∼=A0[p^∞].

Theorem 1.1 (Serre–Tate). The above functor: D^/W →DEF is a canonical equivalence of cate- gories.

A proof of this is given in [K] though the input ofθ and λis not there (see [K1] Lemma 1.11.6 or [Ra]§1.7 for how to modify the argument incorporating endomorphisms and a polarization).

We can splitDEF into the product of the following fiber categories: DEF^ord and DEF^ll over CLW. Here “ord” stands for ordinary Barsotti–Tate groups and “ll” stands for local-local Barsotti–

Tate groups. The objects of DEF^? is given by (A0, D^?,Λ^?, ^?)/R such that D^? is a Barsotti-Tate O-module deforming A0[p^∞]^? = Q

p∈Σ^?A0[p^∞] for ? = ord, ll and Λ^? : D^? ∼= ^tD^? is an O-linear isomorphism and ^? : D0^? ∼=A0[p^∞]^?. Thus D^ll is local-local (i.e., D^ll and ^tD^ll both connected).

The quadruple (A0, D^?,Λ^?, ^?) is supposed to satisfy the same compatibility of diagrams defining DEF. Since by O-linearity, we have a canonical decomposition D = D^ord ×D^ll for an object

(A0, D,Λ, )/R of DEF with the maximal ordinary Barsotti–Tate O-module D^ord in D and the local-local complementD^ll ofD^ord inD so thatD=D^ord×D^ll, we get a decomposition

DEF 3(A0, D,Λ, )7→ {(A0, D^ord,Λ|^Dord, |^Dord),(A0, D^ll,Λ|^Dll, ^ll)} ∈DEF^ord×DEF^ll which induces the equivalence betweenDEF andDEF^ord×DEF^ll.

We consider the following deformation functorDb :CL/W →SET S given by D(R) =b {(A, λ, θ)/R|(A, λ, θ)/R∈ D}/∼=.

To describe the deformation space representingD, we writeb OΞ=Q

l∈ΞOlfor the Ξ-adic completion ofO for any finite set of prime ideals Ξ ofO. Then we putFΞ=OΞ⊗^OF and O(Ξ)=F ∩OΞin FΞ.

Proposition 1.2. Let the notation be as above. Suppose that pis unramified in F/Qand in k/Q.

Then the deformation functor Db is representable by the smooth formal scheme Sb/W which is a product of two smooth formal schemes: Sb^ord isomorphic to Gbm⊗^ZpO_Σ^ord =Q

p∈Σ^ordGbm⊗^Zp Op

and Sb^ll, whereSb^ll ∼= Spf(W[[T1, . . . , Tm]]) (non-canonically) for m= rankZpOΣ^ll, and each factor Sb^? represents the formal stackDEF^?.

Proof. As is well known, the formal stackDEF overW is represented by a formal schemeSbsmooth overW of relative dimension equal to rankZpOp. Indeed, adding anO-linear prime-to-padelic level structure η^(p) : (Ob^(p))²∼=T^(p)A= lim

←−^p-NA0[N](Fp) to (A0, λ0, θ0), the quadruple (A0, λ0, θ0, η^(p)) (called a test object) gives rise to anFp-point x0 of the Hilbert modular Shimura varietySh^(p)_/W = Sh^(p)_/Z

(p)×W of prime toplevel. We recall in Section 2 the definition of the Shimura variety. Then Sb/W can be identified with the formal completion of Sh^(p)_/W alongx0 (and the Shimura variety is smooth overZ(p) of relative dimension equal to rankZpOp= [F :Q]). For the part DEF^ord, by the theory of Serre–Tate, once we identifyA0[p^∞]^ord with (µp^∞⊗^ZpO_Σ^ord)×(F_Σ^ord/O_Σ^ord) by a level p-structure

ηp: (µp^∞⊗^ZpO_Σ^ord)×(F_Σ^ord/O_Σ^ord)∼=A0[p^∞]^ord,

the space Sb^ord is isomorphic to Gbm⊗^Zp OΣ^ord whose origin corresponds to the unique canonical lift ofA0[p^∞]^ord with complex multiplication (which may be different from the startingA[p^∞]^ord).

Since DEF ≈ DEF^ord ×DEF^ll (equivalence of categories), we have a product decomposition Sb = Sb^ord×Sb^ll. Since Sb and Sb^ord are formally smooth, Sb^ll must be formally smooth. Then by

dimension computation, we have dimW Sb^ll =mas above.

We have Sbp^ord ∼= Gbm⊗^ZpOΣ^ord = Q

p∈Σ^ordSb_p^ord for Sb_p^ord = Gbm⊗^ZpOp. Thus Sb_p^ord is the Op- factor of the formal completion ofGm⊗^ZO= Spec(W[t^ξ]ξ∈O) along the origin corresponding to the maximal ideal (p, t−1). Here we used the fact that for the cocharacter groupLof a split torusT/W, we have a canonical isomorphism: T ∼= Spec(W[t^l]l∈L) =Gm⊗^ZLfor the group algebra W[L]∼= W[t^l]l∈L with the canonical variable t of Gm. Since Gbm⊗^ZO = Gbm⊗^ZOp (Op = lim

←−ⁿO/pⁿO), Gbm⊗^ZpOp has variabletp=t⊗1p for the idempotent 1p ofOp inOp=Q

p⁰|pOp⁰. We calltp the Serre–Tate (multiplicative) coordinate ofSb_p^ord.

2. Hilbert modular Shimura variety

Here is a more detailed definition of AVRM byO. WriteO^∗ = HomZ(O,Z) (the Z-linear dual), which can be identified with the different inverse of F/Q by the trace pairing. A triple (A, λ, θ)/S

with an abelian scheme A over a scheme S with real multiplication is called an AVRM byO if it satisfies the following four conditions:

(rm1) θ=θA:O ,→End(A/S) is an embedding of algebras taking the identity to the identity.

(rm2) λis anO-linear symmetric isogenyλ:A→^tAinduced by an ample line bundle fiber-by-fiber geometrically (see [GIT] 6.2). Identifying^tA withA⊗^Ocfor a fractional idealc ofF, λis called ac-polarizationofA. Here λis calledsymmetricifλ=^tλ.

(rm3) The image ofθAis stable under the Rosati involution on the endomorphism algebra End⁰(A):

α7→α^∗=λ⁻¹◦^tα◦λ.

(rm4) AsO⊗^ZO^S-modules, we have an isomorphism: Lie(A)∼=O⊗^ZO^S(⇔π∗(ΩA/S)∼=O^∗⊗^ZO^S forπ:A→S) locally under the Zariski topology ofS, where the sheafLie(A) of Lie algebras ofA(i.e., the pull-back of the tangent bundle overA/S by the 0-section) is anO-module by the action induced fromθ.

An AVRM A defined over a fieldκ(potentially) has complex multiplication (or has CM) if the algebra End⁰_F(A×^κκ) = EndO(A×^κκ)⊗^OQ(of F-linear endomorphisms ofA×^κκdefined over κ) for an algebraic closureκof κis not equal toF (and ifκ⊂Qand Ahas CM, End⁰_F(A×^κκ) is a quadratic extension ofF). An AVRMAover an integral domainrhas CM ifA×^rκhas CM for the quotient fieldκofr. An AVRMA/r is said to be non CMif End⁰_F(A×^rκ) =F.

In the rest of the paper, we denote by Ξ a finite set of rational primes unramified inF containing the fixed prime p, and we put Z(Ξ) = T

l∈Ξ(Q∩Zl) ⊂ Q (the localization at Ξ of Z). We write O(Ξ) for the ringO⊗^ZZ(Ξ)⊂F and let O_(Ξ)+^× be the group of totally positive units inO(Ξ). Let A^(Ξ) be the fiber category overZ(Ξ)-schemes made of abelian schemes with real multiplication by O up to prime-to-Ξ isogenies. Thus the objects of A^(Ξ) are triples (A, λ, θA)/S over aZ(Ξ)-scheme S, andλis the set {aλ:=λ◦θA(a)}a∈O^×_(Ξ)+ for a polarization λwith polarization ideal prime to Ξ. The morphisms: (A, λ, θA)/S →(A⁰, λ⁰, θ⁰_A)/S of A^(Ξ) are given by the elementsφ: A→A⁰ in HomGSCH/Z(Ξ)(A, A⁰)⊗^ZZ(Ξ) compatible with the data (λ, θA) and (λ⁰, θA⁰). HereGSCH/Rstands for the category of group schemes over a ringR. We consider the Hilbert modular Shimura variety Sh^(Ξ)_/Z

(Ξ) which is known to represent the stack made out of (A, λ, θA, η^(Ξ)) for a prime-to-ΞO-linear level structure η^(Ξ): (F_A^(Ξ,∞))²∼=T^(Ξ)A⊗^ZZ(Ξ) sending the determinant pairing (a, b)7→det(a, b) fora, b∈(F_A^(Ξ,∞))² to the pairing onT^(Ξ)Ainduced by the polarization up to similitude factors in (F_A^(Ξ,∞))^×, where T^(Ξ)A is the prime-to-Ξ Tate module ofA,FAis the adele ring of F andF_A^(Ξ,∞) is its prime-to-Ξ part given by{x∈FA|xl=x∞= 0 for alll∈Ξ}. Strictly speaking, to getSh^(Ξ)_Z(l) (l ∈ Ξ), for Ξ^(l) := Ξ\ {l}, we need to actually identify the Ξ^(l)-part of the Tate module T_Ξ^(l)A with O_Ξ²(l) (without specifying an isomorphism; see [H06] §1 for more details). Then the schemes {Sh/Z_(l)}^l∈Ξglue intoSh^(Ξ)_/Z

(Ξ). The pro-schemeSh^(Ξ) is smooth overZ(Ξ) (cf. [PAF] Section 4.2).

LetG= ResF /QGL(2) with center Z/Q. Under the actionη^(Ξ) 7→η^(Ξ)◦g, eachg ∈G(A^(Ξ),∞) acts on Sh^(Ξ) as an automorphism. LetW^Ξ=T

l∈ΞW^l (inside Q) which is a henselian semi-local Dedekind domain with localization at each maximal idealmlof residual characteristiclequal toW^l. We fix a non CM test object (A, λ, θ, η^(Ξ))/WΞ satisfying (NLL). Let x∈Sh^(Ξ)(W^Ξ) be the point representing this test object.

Supposep∈Ξ. By a theorem of Tate (cf. [ABV] Appendix I), the reduction (A0, λ0, θ0, η^(Ξ)₀ )_/Fp= (A, λ, θ, η^(Ξ))/WΞ×^WΞFp has complex multiplication by a fieldM = End⁰_F(A_0/Fp) (F-linear endo- morphism algebra). RegardingM ⊂Qas an F-algebra, we write eθP :M ∼= End⁰_F(A_0/Fp) for the identification (so, θeP|^F = θ0). Let R = EndO(A_0/Fp) which is an order of M. The quadruple (A0, λ0, θ0, η^(Ξ)₀ )_/F gives rise to a point x0 ∈ Sh^(Ξ)(Fp). Let V be the geometrically irreducible component of Sh^(Ξ)_/WΞ containingx; so, V_/Fp =V ×^WΞFp is geometrically irreducible and contains x0. By the smoothness of Sh^(Ξ) over Z(Ξ) and the existence of the smooth projective toroidal compactification ofSh^(Ξ),V_/Fp is geometrically connected.

By the global reciprocity law of Shimura (cf. [Sh] II), we know

Theorem 2.1. Let the notation and the assumption be as above. Supposep∈Ξwhich is unramified in F/Q. Let κ=Fp or Q. For the geometrically connected component V/κ ⊂Sh^(Ξ)_/Z

(Ξ) ×^Z(Ξ) κ, the scheme automorphism groupAut(V/κ)⊂G(A^(Ξ),∞) is given by the semi-direct product of the field

automorphism groupAut(F/Q)and

E^(Ξ)= {g∈G(A^(Ξ))|det(g)∈O^×_(Ξ)+F_∞+^× } Z(Z(Ξ))G(R)+

={g∈G(A^(Ξ),∞)|det(g)∈O^×_(Ξ)+} Z(Z(Ξ)) ,

whereG(R)+ is the identity connected component of G(R) and “· · ·” indicates closure under adelic topology. Here Aut(F/Q) acts on E^(Ξ) through its action on G(A). For the above point x0 ∈ Sh^(Ξ)(Fp), the stabilizer of x0 inE^(Ξ) is the image of the morphismρb:R^×_(Ξ)→G(A^(Ξ),∞)given by θeP(α)◦η₀^(Ξ)=η^(Ξ)₀ ◦ρ(α)b forα∈R^×_(Ξ)= (R⊗^ZZ(Ξ))^×⊂M^×.

By this theorem, the stabilizerT inE= Aut(V_/Fp) ofx0 is the image ofR^×_(Ξ) for the orderRin the integer ringo ofM under the morphism ρb: TM = ResR/ZGm →Gof group schemes defined overA^(Ξ),∞ (so, it is a conjugate of an anisotropic torus inGunder an element inG(A^(Ξ),∞)). It would appear later that we do not use this theorem, but it will be implicitly used to assure that our action ofTM(Z(Ξ)) preserves each geometrically irreducible component ofSh^(Ξ) as its action factors throughE^(Ξ).

Proof. The first assertion is proven by T. Miyake in [M] forκ=Q(following the method of Shimura in [Sh] II) via the well known identification of the automorphism group of a bounded symmetric domain with the corresponding Lie group (e.g., [DLS] Chapter IV); in our Hilbert modular case, Aut(H^I) (H={z∈C|Im(z)>0}) has identity connected component isomorphic to P SL2(R)^I for I = Homfield(F,Q). See [PAF] Theorem 4.14 for another algebraic proof for κ=Q, and the case where κ= Fp is treated in [H06]. Ifg ∈ E^(Ξ) stabilizes x0, by universality, we have a prime-to-Ξ F-linear endo-isogeny

α: (A0, λ0, θ0, η^(Ξ)₀ )_/Fp∼= (A0, λ0, θ0, η^(Ξ)₀ ◦g)_/Fp.

In other words,αcan be regarded as an element ofR^×_(Ξ). SinceθeP(α)◦η^(Ξ)=η^(Ξ)◦ρ(α), the matrixb

ρ(α) coincides withb g in the quotientE^(Ξ).

Lemma 2.2. Suppose that the AVRM (A0, λ)_/Fp is defined over a finite field Fq ⊂ Fp and p ∈ Σ^ord. Then the algebraM = End⁰_F(A_0/Fp)is a CM quadratic extensionM overF generated by the Frobenius endomorphism over Fq. In particular, the prime p in F splits intoPP inR for primes P 6=P.

By this lemma, we haveRp=o_p=RP×R_P ∼=Op×Op for the integer ringoofM.

Proof. By definition,Mp is isomorphic toF-linear endomorphism algebra End⁰_F(A0[p^∞]/Fq), which is therefore a semi-simple quadratic extension of Fp. ThusM is commutative, and the embedding restricted toM ⊂Mpfactors through EndF(A0/Fq) and its image contains theq-th power Frobenius mapφ. SinceF[φ]/F is a CM field because of the positivity and non-triviality of Rosati involution c ofλ0, we must haveM =F[φ] which is a CM quadratic extension ofF.

Note that φ and its dual φ^c = λ⁻¹◦^tφ◦λ satisfy φφ^c = φ^cφ = q. Since φ is invertible on A0(Fp)⊃A0[p^∞] 6= 0, a=φ+φ^c ∈ O is prime top. Thus the minimal polynomialX²−aX+q of φ over F satisfies X²−aX+q ≡ X(X −a) modp with (a modp) 6= 0; so, p must split in

R⊃O[φ].

We often chooseP so thatP containsφ; so, A0[P] is connected.

3. Eigen differentials

We keep the notation introduced in Theorem 2.1 in the rest of the paper. For an open compact subgroup K of G(A^(Ξ),∞), we write VK for the image of V under V ⊂ Sh^(Ξ) Sh^(Ξ)_K for the

geometric quotient Sh^(Ξ)_K := Sh^(Ξ)/K (thus V is the pro-variety V = lim

←−^KVK in the sense of Grothendieck [EGA] IV.8 or [EAI]§5.1.4). We takeK sufficiently small so thatKis neat; i.e.,

ΓgKg⁻¹ ={α∈gKg⁻¹∩G(Q)|det(α) is totally positive}

does not have torsion modulo its center (this holds true if K is inside the principal congruence subgroup of levelN ≥3). Under neatness, VK is smooth overW^Ξ.

Let π: (A, θ_A)→ VK be the universal AVRM over VK. Then ω =π∗Ω_A/VK has an O-action viaθ_A and by (rm4), is locally free of rank 1 overO⊗^ZO^VK (which is a module over O⊗^ZW^Ξ).

Since O⊗^ZW^Ξ = WΞ^I by sending a⊗b to ϕ(a)b for ϕ ∈I = Homfield(F,Q) (as primes in Ξ are unramified inO), we can accordingly splitω=L

ϕ∈Iω^ϕfor a line bundleω^ϕon which theO-action factors throughϕ. By the Kodaira-Spencer map (see [K]§3.6), ΩVK/WΞ is canonically isomorphic to L

ϕω^2ϕforω^2ϕ=ω^ϕ⊗^O_VKω^ϕ (see [K1] 1.0.21). Thus the cotangent space ΩVK/W has a canonical O^×_(Ξ)-action. A section T of the structure sheaf of a Zariski (resp. complex analytic, p-adic rigid analytic) neighborhood Dx (of x) is called “belonging to ϕ” if dT ∈ H⁰(Dx,(ω^2ϕ)^∗) for the line bundle (ω^2ϕ)^∗corresponding to the algebraic bundleω^2ϕ(by GAGA theorems; see Example 2.4.6 in [C]). Here we use∗=anin the complex analytic case and in the rigid analytic case (and we place nothing in the algebraic case; if we need to specify the placev, we write∗=v-an). IfT is a part of the coordinate system belonging toϕ(at v∈Ξ∪ {∞}), we callT a ϕ-coordinate(atv).

Recall M = End⁰_F(A0)⊃EndO(A0) =R. The field M is a CM quadratic extension ofF, and let TM = ResR/ZGm. Recall the morphism ρb: TM → G(A^(Ξ),∞) of algebraic groups over A^(Ξ),∞

given byθeP(α)◦η₀^(Ξ)=η^(Ξ)₀ ◦ρ(α). The pointb x0∈V(Fp) is a fixed point ofρ(Tb M(Z(Ξ))), and let U be a Zariski open neighborhood ofx0. Suppose that we have a non-constantf∈Fp(V) vanishing at x0 and a p-adically open subgroup T of TM(Z(Ξ)) such that f ◦ρ(α) =b λαf with a constant λα ∈ Fp for all α∈ T. Let X^◦ ={u∈ U|f(u) = 0} andX be the Zariski closure of X^◦. Since f is non-constant, dimX =d−1. Then, by definition,X is stable under the action of the image T ⊂ E^(Ξ) of ρ(Tb M(Z(Ξ))). Supposing that x0 is in the ordinary locus, by [H10] Proposition 3.8, X =V if dimX =d−1>0. This is impossible, so such a nontrivialf does not exist.

Suppose that we have a flat integral covering space U /Ue whose image containsx0on which the action of T extends. If f◦ρ(α) =b λαf with a constant λα ∈ Fp for all α ∈ T for f ∈ Fp(Ue), we consider the image X^◦ ⊂ U of {u ∈ Ue|f(u) = 0} and its Zariski closure X. By the same argument, we haveX=V ifd−1>0. Thus even inFp(Ue), such anf 6= 0 does not exist. Thus if 06=f ∈Ob^V,x0 satisfies this property, f is transcendental overO^V,x0 if [F :Q]≥2. We will not use this transcendence result in the rest of the paper.

4. Eigen coordinates

Let V be the geometrically irreducible component ofSh^(Ξ)_/WΞ as in Theorem 2.1 containing the pointx∈V(W^Ξ), and take a neat open compact subgroupKofG(A^(Ξ),∞). As before, letx0be the image of xinV(Fp) (assuming p∈Ξ). RecallVK defined in Section 3 which is the geometrically irreducible component of Sh^(Ξ)_K containing the projection of x. Let O be the stalk O^VK/WΞ,x ⊂ O^V/WΞ,x =OSh^(Ξ),x at theW^Ξ-pointx∈V(W^Ξ). The W^Ξ-algebra O is semi-local (of finite type), and its maximal idealml,x over (l)⊂ W^Ξ forl∈Ξ is unique. SinceV /VK is ´etale, we may identify theP-adic localization-completion ofOand that ofO^V,xfor any prime idealP ofO^V,x. We consider the primeP =Px given by the kernel of O → W^Ξ induced by evaluation atx. Then P is a prime ideal inside the Jacobson radical ofO (i.e., the intersection of all the maximal ideals{ml,x}^l∈Ξ of O indexed by the residual characteristic l ∈Ξ ofml,x). LetK be the field of fraction ofW^Ξ, and consider the localization completionOb^P ∼=K[[T1, . . . , Td]] fordvariablesTj. We can extend scalar to Cv forv=l,∞byK ⊂Q−→^iv Cv and considerO^v=O ⊗^WΞCv. Then the evaluation atx∈V(Cv) gives rise to aCv-algebra homomorphismO^v →Cvwhose kernel, we denote byPv; so,P =Pv∩ O. Then we localize O^v at Pv and write the localization as O^v,Pv. We then complete itPv-adically,

gettingOb^v,Pv = lim

←−ⁿO^v,Pv/P_vⁿO^v,Pv. We then have a commutative diagram of inclusions O −−−−→_⊂ Ob^P =K[[T1, . . . , Td]]

∩



y ^iv

 y^∩ Ob^v,Pv Cv[[T1, . . . , Td]].

As described above, we study coordinates around a point on Sb^ord. The coordinate can be a formal function on Sb^ord or a rigid analytic function defined on a rigid open neighborhood of the point in the associated p-adic rigid analytic space denoted by (Sb^ord)^p-an. Here is what we mean by the rigid analytic space associated to Sb^ord. A functor associating to a formal scheme Spf(A) a rigid analytic space is studied by Raynaud and Berthelot. The theory of Raynaud is explained in [BL], and one can find a good account of that of Berthelot in [dJ] Section 7. As a special case of his theory, Berthelot associates, to Spf(Wp[[(T)]]/(f1(T), . . . , fm(T))) for a finite set of variable (T) = (T1, . . . , Tn) and a finite set of power seriesfj(T)∈W[[(T)]] :=W[[T1, T2, . . . , Tn]], the rigid analytic space X^rig given by the zero set of {fj}^j in the open unit disk Dⁿ in the n-dimensional affine space over the field of fraction F rac(W) ofW. Here we take the base-change ofX^rig toCp

(see [NAA] §9.3.6), and write it as X_/C^p-an_p. This construction appears very naive, but as exposed in [dJ], this functor has intrinsic meaning and functorial construction, and it extends naturally the results of Raynaud to formal schemes not necessarily of finite type over W. For pϕ ∈ Σ^ord associated to ip ◦ϕ : F → Cp, we extend ip◦ϕ to a projection Wp⊗^ZO =W_p^I → Cp (which is still denoted by ip ◦ϕ), and we put τϕ = τϕ,p = log_p◦ip◦ϕ◦tpϕ for the Serre–Tate coordinate tpϕ onSb_p^ord_ϕ =Gbm⊗^ZpOpϕ. Then a convergent power series in Cp{{τϕ}}ϕ∈Σ^ord is a rigid analytic function well defined on (Sb^ord)^p-an, where Cp{{τϕ}} is made up of power series P

n=(nϕ)≥0anτⁿ inCp[[τϕ]]_ϕ∈Σ^ord with lim|n|→∞|an|^p = 0 for|n|=P

ϕnϕ. The GAGA type theorems associating an analytic sheaf F^p-an (againF^p-an is the base change to Cp of F^rig in [dJ]) to a formal coherent sheafFare given in this setting in [dJ] §7.1.11, and the analytic/formal sheaf of K¨ahler differential is discussed in [dJ]§7.1.12. In particular, for a rigid pointx∈X^p-an coming from a smooth formal pointx∈X(W) in the generic fiber, the formal cotangent space ΩX/W(x)⊗^W Cp and the analytic cotangent space ΩX^p-an/Cp(x) at xare identical.

For anl-adic Barsotti–TateO-moduleB defined over a ringC, we write EndO(B/C) for the ring ofO-linear endomorphisms ofBdefined over a ringCand put End⁰_F(B/C) = EndO(B/C)⊗^ZQ. For a finite extensionkofQinQwith integer ringO, putW^k =W^Ξ∩k.

Lemma 4.1. Let l ∈ Ξ be a prime. Let E/F be a quadratic extension with TE = ResE/QGm. Suppose that (A, λ, θ, η^(Ξ)) is the test object sitting over x∈ V(W^Ξ) such that the image of x in VK (for a neat open compact subgroup Kof G(A^(Ξ),∞))falls inVK(W^k)for a finite extensionk/Q.

ForL =W^k∩m_l,x, let(AL, λL, θL, η_L^(Ξ)) be its reduction modulo m_l,x sitting over xL ∈V(Fl) (so AL is defined overFL=O/L). Suppose that we have a level l-structureηl: (µl^∞⊗^ZlOl)∼=A[l^∞]^◦ defined over W^l for a primel∈Σ^ord_l such thatηl⊕^tη⁻¹_l : (µl^∞ ⊗^ZlOl)⊕(Fl/Ol)∼=A[l^∞] overWl

as Barsotti–Tate O-modules, where A[l^∞]^◦ is the connected component of the Barsotti–Tate group A[l^∞]/Wl and ^tη⁻¹_l is the dual inverse ofηlunder the Cartier duality induced by the polarization λ.

If we have embeddingsρb:TE(Q)→G(A^(Ξ),∞)and E3α7→αq ∈End⁰_F(AL[q^∞]/FL)for all primes q outside Ξ satisfying η^(Ξ)_L,q◦ρ(α) =b αq◦η^(Ξ)_L,q for all q 6∈ Ξ for the q-component η^(Ξ)q of the level structureη^(Ξ), then

(1) E is a CM field isomorphic toEnd⁰_F(A_L/Fl),

(2) ρ(rb ^×_(Ξ)) (r(Ξ)=r⊗^ZZ(Ξ))fixes xL∈V(Fl)for an order rofE maximal atl,

(3) we have an F-linear embedding θeL :E ,→End⁰_F(A_L/Fl) such that eθL(α) induces αq in the endomorphism algebraEnd⁰_F(AL[q^∞]) for allq6∈Ξ,

(4) there exist an extension of σ:F ,→Q toE (still denoted byσ) and a l-adic rigid analytic σ-coordinateτσ =τσ,l of the rigid analytic space associated to Sb_l^ord centered at xin S(Wb l)

such thatτσ(x) = 0and τσ◦ρ(α) =b il(α^σ(1−c))τσ for eachσinducing the place lonF after composing il.

Moreover the coordinateτσ as in (4)above is unique up to scalar multiples.

We call a σ-coordinate satisfying (4) above aρ-eigenb σ-coordinate.

Proof. The test object (AL, λL, θL) is defined over the finite fieldFL =O/L. LetR⁰ = EndO(AL/FL).

Since Σ^ord_l 6=∅, by a theorem of Tate (see [ABV] Appendix I, Theorem 3),R⁰ is an order of a semi- simple quadratic extension M⁰ of F (generated by the Frobenius endomorphism over FL) whose simple components are all CM fields; so, M⁰ must be a CM quadratic extension (as it cannot be isomorphic toF×F). By another theorem of Tate ([ABV] Appendix I, Theorem 2), we have

EndO(AL[q^∞]/FL) = End_{O[Gal(Fl/FL)]}(TqAL) =R⁰⊗^ZZq=:R⁰_q.

SinceE ,→End⁰_F(AL[q^∞]/FL) byα7→αq, we haveEq =E⊗^ZQq ∼=M_q⁰ for all primes qoutside Ξ, and we conclude to haveθeL:E ∼=M⁰; so,E is a CM field. Putr=θe⁻¹_L (R⁰)⊂E, which is an order ofE maximal atlby Lemma 2.2 applied tol. Note thatEl∼=Fl×Fl. SinceθeL(α) forα∈r^×_(Ξ) is a prime-to-Ξ isogeny sending (AL, λL, η^(Ξ)_L )_/Fl to (AL, λL,θeL(α)◦η^(Ξ)_L )_/Fl = (AL, λL, η_L^(Ξ)◦ρ(α))b _/Fl, b

ρ(α) fixes xL by the universality ofSh^(Ξ)_/F

l.

By Serre–Tate theory, we have the Serre–Tate coordinatet :=tl of the l-component Gbm⊗^ZlOl

of Sb_l^ord such that an O-linear automorphism a ∈ AutO(Gbm⊗Ol/Wl) = O_l^× acts by t 7→ t^a. As seen in [H10] Lemma 3.3,α∈End⁰_F(AL) =E acts onSb_l^ord by universality, and this action induces a homomorphism : r^×_(Ξ) → AutO(bGm⊗Ol/Wl) = O^×_l sending α ∈ E^× to α^σ(1−c) ∈ Fl under an extension σ : E ,→ Q of σ : F ,→ Q inducing l and the projection of El to one of its simple components isomorphic toFl. Though the proof of [H10] Lemma 3.3 is given for ordinary AL, the same argument applied toAL[l^∞] (in place ofAL[l^∞] there) works equally well.

Let log_l: 1 +lWl→Wlbe thel-adic logarithm map. Writeσlfor the projection ofWl⊗^ZO⊂W_l^I toWlinduced byσ:O ,→Q. Then we get a standardσ-coordinateτσ as the following composition:

Gbm⊗^ZlOl(Wl) = (1 +lWl)⊗^ZlOl log_l⊗1

−−−−→Wl⊗^ZlOl σl

−→Wl.

We can perform the same procedure for other primeslϕ∈Σ^ord_l , and get a coordinateτϕ forϕ∈I associated tolϕsatisfying the same property above replacingσbyϕ. The associated rigid analytic space (Sb^ord_l )^l-an has by definition the coordinates {τϕ}ϕ∈Σ^ord_l (here Σ^ord_l is regarded as a set of field embeddings of F into Cl). Note here tσ(x) = 1 and τσ(x) = 0 (by the splitting A[l^∞] ∼= (µl^∞⊗^ZlOl)⊕(Fl/Ol) overWl) but forϕnot associated tol,τϕ(x) may not be equal to 0. Then any other rigid analytic coordinatef centered atx(i.e.,f(x) = 0) is anyway a power series convergent on an open subset of the rigid analytic space (Sb_l^ord)^l-an associated to Sb_l^ord. Therefore it has power series expansionf(τ) =P

νaντ^ν forτ^ν =Q

ϕ∈Σ^ord_l τϕ^νϕ. Iff◦ρ(α) =b il(α^σ(1−c))f, by equating the power series expansion, we find thatf is a constant multiple ofτσ. If v = ∞, we put G∞ = AutF(H^I) (the complex analytic automorphism group inducing O- linear map on ΩH^I/C); so, G∞ = P SL2(FR) = Q

ϕ∈IP SL2(R) according to the decomposition FR=F⊗^QR=Q

ϕ∈IRfor which the projection to theϕ-factor inducesi∞◦ϕ:F →R(e.g., [DLS]

Chapter IV). As is well known, the stabilizer ofx∈V(W^Ξ) in G∞is given by an anisotropic torus inG∞. Write the torus asT^∞,x=T^x=Q

ϕ∈IT^ϕ,x forT^ϕ,x∼=C^×/R^×. We put Σ^ord_∞ :=I.

Ifv=pand Σ^ord_p 6=∅, we writeGp= AutF(Sb^ord) (the group of formal automorphisms overWp

compatible with O_(Ξ)^× -action on Ω_Sbord/Wp). Note that Sb^ord ∼=Gbm⊗^ZpOΣ^ord as seen in Section 1.

We fix once and for all the identification Sb^ord =Gbm⊗^ZpOΣ^ord. This is tantamount to fixing an isomorphism µp^∞ ⊗^ZpOp ∼= A0[p^∞]^◦ for each p ∈ Σ^ord_p ; so, as before, we write tp for the Serre–

Tate (multiplicative) coordinate of Sb^ord_p . We have AutSCH(Gm) = AutSCH(Spec(Wp[t, t⁻¹])) = Gm(W[t, t⁻¹]) = W[t, t⁻¹]^× (the scheme automorphism group). Similarly, writing t = 1 +T, we have Aut(Gbm) = Aut(Spf(Wp[[T]])) =Gbm(W[[T]]) = 1 +m_W[[T]] for the maximal idealm_W[[T]] of