Classicality of overconvergent Hilbert eigenforms: case of quadratic residue degrees

Classicality of overconvergent Hilbert eigenforms:

case of quadratic residue degrees

YICHAOTIAN(*)

ABSTRACT- LetFbe a real quadratic field,pbe a rational prime inert inF, andN4 be an integer coprime top. Consider an overconvergentp-adic Hilbert eigen- formfforFof weight (k1;k2)2Z²and levelG00(N). We prove that if the slope of fis strictly less than minfk1;k2g 2, thenf is a classical Hilbert modular form of levelG00(N)\G0(p).

1. Introduction

We fix a prime numberp>0. A famous theorem of Coleman says that an overconvergent p-adic (elliptic) modular eigenform of small slope is actually classical. More precisely, letN5 be an integer coprime top, and X₁(N)^an be the rigid analytification of the usual modular curve of level G1(N) overQ_p. We denote byX1(N)^an_ordthe ordinary locus ofX1(N)^an. For p5,X1(N)^an_ordis simply the locus whereEp 1(Eisenstein series of weight p 1), the standard lift of the Hasse invariant, has non-zero reduction modulop. For any integerk2Z, Katz [Ka73] defined the space M^y_k(G₁(N)) of overconvergent p-adic modular forms of weight k. An element in M^y_k(G₁(N)) is a section of the modular line bundlev^k defined over a strict neighborhood ofX1(N)^an_ord inX1(N)^an. Moreover, Katz also defined a com- pletely continuous operator Up acting on M^y(G1(N)). There is a natural injection from M_k(G₁(N)\G₀(p)) to M^y_k(G₁(N)), where M_k(G₁(N)\G₀(p)) is the space of classical modular forms of weightkand levelG₁(N)\G₀(p), that is, sections ofv^kover the modular curveX(G₁(N)\G₀(p)). In [Col96],

(*) Indirizzo dell'A.: Morningside Center of Mathematics, Chinese Academy of Science, 55 Zhong Guan Cun East Road, Beijing, 100190, China.

E-mail: yichaot@math.ac.cn

This work was partially supported by a grant DMS-0635607 from the National Science Foundation during the author's stay at the USA.

Coleman proved that iff 2M^y_k(G1(N)) is aUp-eigenvector with eigenvalue apandvp(ap)<k 1, thenfactually lies in Mk(G1(N)\G0(p)). Coleman's original proof for this deep result was achieved by an ingenious dimension counting argument. Later on, Buzzard [Bu03] and Kassaei [Ks06] re- proved Coleman's theorem by an elegant analytic continuation process.

The basic idea of Buzzard-Kassaei was to extend successively the sectionf by the functional equation f ˆ 1

a_pUp(f) to the entire rigid analytic space X(G₁(N)\G₀(p))^an. Actually, Buzzard proved that f can be extended to the union of ordinary locus and the area with supersingular reduction of X(G1(N)\G0(p))^an. Then Kassaei constructed another form g on the complement to Buzzard's area, and showed thatf andgglue together to an analytic section ofv^k overX(G₁(N)\G₀(p))^an. The rigid GAGA theorem [Ab11, 7.6.11] then implies that this is indeed a genuine section ofv^kover the algebraic modular curveX(G₁(N)\G₀(p)). In this process, the theory of canonical subgroups for elliptic curves developed in [Ka73] due to Lubin and Katz plays a fundamental role.

There have been many efforts in generalizing the classical theory on overconvergent p-adic modular forms to other situations. First of all, to generalize overconvergentp-modular forms and theU_p-operator, we need to construct canonical subgroups in more general context. This has been done by many authors. For instance, [KL05], [GK09] consider the Hilbert case, and [AM04], [AG07] treat the general case for abelian varieties, and finally in [Ti10], [Fa09], [Ra09] and [Ha10] the canonical subgroups are constructed for generalp-divisible groups. Using the canonical subgroups, overconvergent p-adic modular forms and the Up-operators can be con- structed similarly in various settings. However, the generalization of Co- leman's classicality criterion need more hard work. As far as I know, this criterion has been generalized in the following cases. In [Col97a], Coleman himself generalized his results to modular forms of higher level at p.

Kassaei considered in [Ks09] the case of modular forms defined over various Shimura curves. In [Sa10], Sasaki generalized it to the case of Hilbert eigenforms whenptotally splits in the totally real field defining the Hilbert-Blumenthal modular variety. Finally, Pilloni proved in [Pi09] the classicality criterion for overconvergent Siegel modular forms of genus 2.

In this paper, we will follow the idea of Buzzard-Kassaei to study over- convergent Hilbert modular forms in the quadratic inert case.

To simplify the notation, let's describe our result in a special but es- sential case. LetFbe a real quadratic number field in whichpis inert, and O_Fbe its ring of integers. We putk'F_p²,W ˆ O_FpandQ_kˆW[1=p]. We

denote byBˆ fb₁;b₂gthe two embeddings ofFintoQ_k. LetN4 be an integer coprime top. We consider the Hilbert-Blumenthal modular variety Xover Spec(W) that classifies prime-to-ppolarized abelian schemesAwith real multiplication byO_F of levelG₀₀(N). LetYbe the moduli space that classifies the same data and together with an (O_F=p)-cyclic subgroup of A[p]. For each pair of integers~kˆ(k1;k2)2Z^B, we have the modular line bundlev^~k overXandY(See 2.2 for its precise definition). For each finite extensionLofQ_k, we put M~k(G00(N)\G0(p);L)ˆH⁰(YL;v^~k), and call it the space of (geometric) Hilbert modular forms of levelG00(N)\G0(p) and weight~kwith coefficients inL. This is a finite dimensional vector space over Lby classical Koecher principle, and the theory of arithmetic com- pactifications of Hilbert-Blumenthal modular varieties [Rap78, Ch90, DP94] implies that it actually descends to a finite flatZ[1=N]-module.

LetXandYbe respectively the completion ofXandYalong their special fibers, andX_rigandY_rigbe their rigid analytic generic fibers aÁ la Raynaud [Ab11, Ch. 4]. We still have a natural forgetful mapp:Y_rig!Xrig. For each

~k2Z^B, we denote still byv^~kthe rigidification of the line bundlev^~k. LetX^ord_rig be the ordinary locus ofXrig, i.e. the locus where the universal rigid Hilbert- Blumenthal abelian varietyArigoverXrighas good ordinary reduction. Then the multiplicative part of the universal finite flat group schemeA_rig[p] de- fines a sections:X^ord_rig !Y_rigof the projectionpoverX^ord_rig. We denote by Y^ord_rig the image ofs, so thatpj_Yord

rig :Y^ord_rig! X^ord_rig is an isomorphism of quasi- compact rigid analytic spaces. For a finite extensionLofQ_k, Kisin and Lai [KL05] defined an overconvergent Hilbert modular form of levelG₀₀(N) and weight~kwith coefficients inLto be a section ofv^~koverX^ord_rig that extends to a strict neighborhood ofX^ord_rig;L. We denote by M_~^y_k(G₀₀(N);L) the space of such forms. This is a direct limit of infinite dimensional Banach spaces overL.

Moreover, the theory of canonical subgroups for Hilbert modular vari- eties allows them to define a completely continuous Up-operator on M_~^y_k(G₀₀(N);L). Note that a relatively weak formulation of the existence of canonical subgroups says that the sections :X^ord_rig !Y^ord_rigextends to a strict neighborhood ofX^ord_rig, or equivalently the isomorphismpj_Y^ord

rig extends to a strict neighborhood. Therefore, there exists a natural injection

M~k(G00(N)\G0(p);L)!M_~^y_k(G00(N);L):

We say an element f in M_~^y_k(G₀₀(N);L) is classical if it lies in the image of this injection. The main result of this paper is the following

THEOREM1.1. Let f 2M_~^y_k(G00(N);L)be a Up-eigenvector with eigen- value ap. I f vp(ap)<minfk1;k2g 2, then f is classical.

Actually, we prove our main Theorem in a slightly more general setting 2.7. Note that our results imply that, in the quadratic inert case, the clas- sical points are Zariski dense in the eigencurve for overconvergent Hilbert modular forms of levelG₀₀(N) constructed in [KL05] (See Theorem 2.9).

Let's indicate the ideas of the proof. First, by rigid GAGA and a rigid version of Koecher principle (Proposition 2.2), we just need to extend f analytically to the entire rigid spaceY_rig. To achieve this, the key point is to understand the dynamics of the Hecke correspondence Up on Y_rig (2.9 ). Three ingredients from the work of Goren and Kassaei [GK09] will be important for us. The first one is the stratification on the special fiberY_k defined by them; the second is their valuation onY_rigvia local parameters;

and the third one is the so-called ``Key Lemma'' [GK09, 2.8.1], which relates the partial Hasse invariants with the certain local parameters ofYk. In this paper, we will interpret their valuation onY_rigin terms of partial degrees (cf. 4.5 ±, 4.2). They are natural refinements in the real multiplication case of the usual degree function, which has been introduced by Fargues [Fa10]

and applied by Pilloni [Pi09] to the analytic continuation ofp-adic Siegel modular forms. Actually, our work originates from an effort to understand the geometric meaning of Goren-Kassaei's valuation. Compared with the totally split case, our difficulty comes from the fact that the p-divisible group associated with a Hilbert-Blumenthal abelian variety (HBAV) with RM byOF is a genuinep-divisible group of dimension 2, so its group law can not be explicitly described by one-variable power series. We overcome this by using Breuil-Kisin modules to compute the partial degrees of thep- torsion of a HBAV. This approach is motivated by the recent work of Hattori [Ha10]. These local computations via Breuil-Kisin modules com- bined with Goren-Kassaei's ``Key Lemma'' will give us enough information to understand the dynamics of the Hecke correspondenceUpexcept the case mentioned in Prop. 4.10 and 5.6. In this exceptional case, we have to study in detail the local moduli of deformations of a superspecial HBAV.

This is achieved in Appendix Bby using Zink's theory on Dieudonne windows [Zi01]. Finally, we can prove that the form f extends to an ad- missible open subset ofY_rigthat contains the tube over the complement to the codimension 2 stratum in Goren-Kassaei's stratification onYk (Prop.

5.9). By a useful trick invented by Pilloni in [Pi09, §7], this allows us to conclude that f extends indeed to the entireY_rig (Prop. 4.9).

This paper is organized as follows. In Section 2, we review the facts that we need on the Hilbert-Blumenthal modular varieties and state the main theorem 2.7 and its consequence on the Zariski density of classical points in the eigencurves for overconvergent Hilbert modular forms. In Section 3, we

perform the computations mentioned above on the (OF=p)-cyclic subgroups of a HBAV over a complete discrete valuation ring via Breuil-Kisin modules.

In particular, we give an alternative proof (Thm. 3.10) for the existence of canonical subgroups in the Hilbert case proven in [GK09]. Section 4 is mainly dedicated to the review on Goren-Kassaei's work, and we provide also an- other proof of their ``Key Lemma'' using Dieudonne theory (Prop. 4.1).

Section 5 is the heart of this work, and it contains a complete proof of Theorem 1.1. Finally, we prove our general main theorem 2.7 in Section 6.

The proof of the general case is a combination of the split case treated by Sasaki [Sa10] and the case in Section 5. We have two appendices. In the first one, we gather some general results on the extension and gluing of sections in rigid geometry. In Appendix B, we study the local deformation space of a superspecialp-divisible group with formal real multiplication byZp^g, where g1 is an integer andZ_p^gis the ring of integers of the unramified extension ofQ_pof degreeg. As a by-product, we see that the local moduli admits some canonical choices of local parameters T₁; ;T_g such that the p-divisible groups corresponding toTiˆ0 admits ``formal complex multiplication'' by Z_p^2g orZp^gZp^g according to the parity of g(cf. Remark B12). Thesep- divisible groups (or those isogenous to them) seem to deserve more study, and should be considered as the canonical lifting (or quasi-canonical lifting) of the superspecialp-divisible group in the formal real multiplication case.

We hope that we can return to the problem in the future.

After I finished a preliminary version of this paper and distributed it among a small circle, Vincent Pilloni showed me a draft of his joint work [PS11b] with Benoit Stroh, where similar results were obtained in- dependently. The influence of the works [Ks06], [GK09] and [Pi09] on this work will be obvious for the reader. I express my hearty gratitude to their authors. I am especially grateful to Christophe Breuil for his careful reading of a preliminary version of this paper, and for his valuable sug- gestions. I also would like to thank Ahmed Abbes, Liang Xiao, Kaiwen Lan and Tong Liu for helpful discussions during the preparation of this paper.

Finally, I would like to thank the anonymous referee who pointed out two errors of earlier version of this paper.

1.1 ±Notation

LetFbe a totally real number field withgˆ[F:Q]>1,OFbe its ring of integers, d_F the different of F. Let p be a fixed prime number un-

ramified inF. For a prime idealpofOLabovep, we putk(p)ˆ OL=pand denote byjk(p)j ˆp^fp the cardinality ofk(p). LetN4 be a fixed integer coprime to p. Let k be a finite subfield of Fp containing all k(p) and a primitive N-th root of unity, WˆW(k) be the ring of Witt vectors with coefficients inkandQ_kˆW[1=p]. LetBbe the set of embeddings ofFinto Q_k. For each prime ideal p of O_F dividing p, let B_pB be the subset consisting of embeddings b such that b ¹(pW)ˆp. So we have Bˆ`

pjpBp. Ifsdenotes the Frobenius on Q_k, thenb7!sb defines a natural cyclic action of Frobenius on eachB_p.

In general, for a finite setI, we denote byjIjits cardinality.

Let C_p be the completion of an algebraic closure ofQ_k. All the finite extensions ofQ_kare understood to be subfields ofCp. We denote byvpthe p-adic valuation onCp, and byj j_p:C_p !R>0 the non-archimedean ab- solute valuejxj_pˆp ^vp(x).

2. Hilbert modular varieties, Hilbert modular forms and the statement of the main theorem

LetS be a scheme. A Hilbert-Blumenthal abelian variety byO_F (or a HBAV for short) overS is an abelian schemeA overS equipped with an embedding of ringsi:OF,!End_S(A) such that Lie(A) is anO_S OF-module locally free of rank 1. IfAis a HBAV overS, the dual ofA, denoted byA^_, has a canonical structure of HBAV overS. We denote byP(A) the fppf-sheaf overS of symmetricO_L-linear homomorphisms of abelian schemesA!A^_, and by P(A)^‡ P(A) the cone consisting of symmetric polarizations.

We fix a positive integerN4 coprime top. Letcbe a fractional ideal ofF prime top, andc^‡cbe the cone of totally positive elements. Con- sider the functor

F c: ALG_W !SETS

which associates to eachW-algebra R the set of isomorphism classes of triples (A;l;c_N) where:

Ais a HBAV over Spec(R);

l is a c-polarization of A, i.e., an O_F-linear homomorphism l:c! P(A) sendingc^‡toP(A)^‡such that the induced map of fppf- sheaves on Spec(R)

A_OFcƒƒƒƒ!^1Al A_OFP(A) !A^_: ax7!al(x)7!l(x)(a) is an isomorphism.

c_N is an embedding of abelian fppf-sheaves of OF-modules m_Nd_F¹,!A[N].

It is well known that this functor is representable by a smooth and quasi-projective schemeXcover Spec(W) of relative dimensiong, which we usually call the c-Hilbert modular variety overW of level G00(N) [Go01, Ch. 4, § 3.1]. By a result of Ribet, the fibers ofXcare geometrically irre- ducible [Go01, Ch. 3 § 6.3].

2.1 ±Definition of Hilbert modular varieties

LetRbe aW-algebra, and (A;l;c_N) be an object inX(R). Anisotropic

(O_F=p)-cyclic subgroup H of A is a closed subgroup scheme HA[p]

which is stable under O_F, free of rank 1 overO_F=pas abelian fppf-sheaf over Spec(R), and isotropic under theg-Weil pairing

A[p]A[p]ƒƒ!^1g A[p]A^_[p]!m_p

induced by ag2 P(A)^‡of degree prime top. So whenAis defined over a perfect field k of characteristic p, a subgroup HA[p] is (O_F=p)- cyclic if and only if its Dieudonne module is a free (k O_F)-module of rank 1. LetG_cbe the functor which associates to eachW-algebraRthe set of isomorphism classes of 4-tuples (A;l;c_N;H), where (A;l;c_N) is an object in Fc(R), andHA[p] is an isotropic (OL=p)-cyclic subgroup of A. The functorGcis representable by a schemeYcover Spec(W). We call Y the c-Hilbert modular variety of level G₀₀(N)\G₀(p). The natural forgetful map (A;l;c;H)7!(A;l;c) defines a morphism of W-schemes p:Yc!Xc, which is finite eÂtale of degree Q

pjp(p^fp‡1) on the generic fibers over Q_k.

Note that, for an object (A;lA;c_A;N;H) in Yc(R), the quotient BˆA=H is naturally equipped with a structure of HBAV. Let f :A!Bbe the natural isogeny and f^t :B!Abe the unique isogeny such that ff^tˆp1_B and f^tf ˆp1_A. Ifl:A_OFc! A^_ is the iso- morphism given byl_A:c! P(A), we define ac-polarization onBbyl_B ˆ 1

p(f^t)l_A:c! P(A)! P(B), where (f^t) :P(A)! P(B) is given by f7!(f^t)^_ff^t. Finally, since H has order prime to N, the isogeny f :A!B induces an isomorphism f :A[N]! B[N]. We define c_B;N as m_Nd_F¹ƒƒ!^cA;N A[N]! B[N]. We get an object (B;l_B;c_B;N) inX_c(R).

Fix a finite setfc1; ;ch^‡gof fractional ideals ofF prime top, which form a set of representatives for the narrow class group Cl^‡_F ofF. We put

Xˆa^h‡

iˆ1

Xci and Y ˆa^h‡

iˆ1

Yci:

We callX(resp.Y) the Hilbert modular varieties of levelG00(N) (resp. of levelG00(N)\G0(p)). In the sequel, an object (A=R;l;c_N;H) inY(R) will be usually omitted as (A;H), if there is no confusions on the polarizationl and the level structurec_N.

2.2 ±Modular line bundles

LetTbe the algebraic group (Res_OF=ZG_m)_ZWoverW, andX(T) be the group of characters ofT. For anyb2BˆEmd_Q(F;Q_k), letx_b2X(T) be the character

T(R)ˆ(R OF)!RˆGm(R) given by ra7!rb(a):

Then (x_b)_b2B form a basis of X(T)! Z^B over Z. For an element (k_b)_b2Bˆ P

b2Bk_bb2Z^B, we denote byx_~kˆ Q

b2Bx^k_b^b the corresponding char- acter ofT.

Let A !X be the universal HBAV over X, and vˆeV¹_A=X where e:X! Ais the unit section ofA. This is a locally freeO_X O_F-module of rank 1, and we have

vˆM

b2B

v_b;

wherev_bis the submodule ofv_A=XwhereOFacts viax_b. For any character

~kˆ(kb)_b2B2Z^B, we define a line bundle v^~k ˆO

b2B

v^k_b ^b:

By abuse of notation, we denote still byv^~kits pull-backs overY viap. DEFINITION 2.1. For a W-algebra R0, we call the elements of H⁰(XR0;v^~k) (resp. H⁰(YR0;v^~k)) (geometric) Hilbert modular forms with coefficients in R0 of weight ~kˆP

b kbb and level G00(N) (resp. of level G00(N)\G0(p)) over R0.

We have the following modular interpretation of (geometric) Hilbert modular forms aÁ la Katz. For each R0-algebra R, we consider 5-tuples (A=R;l;c_N;H;v), where (A=R;l;c_N;H) is an element inY(R) andvis a generator ofv_A=RˆeV¹_A=Ras anR O_F-module. Then a Hilbert modular form f of levelG₀₀(N)\G₀(p) and weight~koverR₀is equivalent to a rule that assigns to eachR₀-algebraR, each 5-tuple (A;l;c_N;H;v) as above, an element f(A;H;v)2Rsatisfying the following properties:

f(A;H;av)ˆx_~k(a) ¹f(A;H;v) fora2(R O_F);

iff:R!R⁰is a homomorphism ofR₀-algebras and (A⁰;l⁰;c⁰_N;H⁰;v⁰) is the base change to R⁰ of (A;l;c_N;H;v), then f(A⁰;H⁰;v⁰)ˆ f(f(A;H;v)).

We have a similar description of Hilbert modular forms of levelG00(N) overR₀, and we leave the details to the reader.

2.3 ±Toroidal compactifications

We recall some well known facts on the toroidal compactifications of Hilbert modular varieties (cf. [Rap78, Ch90, DT04]). Let (c;c^‡) be a prime- to-pfractional ideal ofO_F. AG₀₀(N)-cuspCofX_cis an equivalence class of the following data:

(1) ProjectiveOF-modulesaandbof rank 1.

(2) An isomorphism ofO_F-modulesb ¹a! c.

(3) An exact sequence of projectiveO_F-modules 0!d_F¹a ¹!L!b!0:

(4) An embedding ofOF-modules:

i_C:a ¹d_F¹=Nc ¹d_F¹,!L=NL:

Set M_Cˆabˆb²c, and M_CˆHom_Z(M;Z)'a ¹b ¹d_F¹ˆd_F¹a ²c.

The positivity oncand that ond_Finduce natural positivities onM_CandM_C. For each G00(N)-cusp C, we choose a rational polyhedral cone decom- positionfsag_a2ICofM_C^;‡[ f0gthat is invariant under the natural action by U_F;N² such that the quotient fsag_a2IC=U_F;N² is finite. Here, U_F;N O_F denotes the subgroup of units congruent to 1 moduloN.

We putRˆWh

q^j:j2 1 NM_Ci

, andU_CˆSpec(W[q^j:j2_N¹M_C]). Let U_C,!Ssabe the embedding corresponding tos, andS^saˆSpf(Rsa) denote

the completion ofS_saalongZ_saˆS_sa U_C. LetU_C,!S(fs_ag) be the toric embedding given by fs_ag_a2IC, and S(fs^ _ag) be the completion of S(fs_ag) along S(fs_ag) U_C. SoS(fs^ _ag) has an affine open covering by the S^_sa's.

Put Spec(R⁰_sa)ˆSpec(Rsa)_Ssa U_C with R⁰_saˆRsa

hq^j:j2 1 NM_Ci

. We have a morphism of schemes over Spec(R⁰_sa):

i:b,!Gma ¹d_F¹ˆSpec(R[Xa:a2a])

given by X_a(i(b))ˆq^ab. B y Mumford's construction, we have a semi- abelian scheme Tate(a;b)ˆ(Gma ¹d_F¹)=i(b) over Spec(Rsa) equipped with a natural action of O_F, which is ac-polarized abelian scheme over Spec(R⁰_sa), and degenerates intoG_ma ¹d_F¹ over Z_sa. As explained in [KL05, 1.6.1], the data i_C:a ¹d_F¹=Na ¹d_F¹,!L=NL defines a G₀₀(N)- level structure on Tate(a;b) over Spec(R⁰_sa). Thus, one gets a morphism Spec(Rsa)!Xc.

By ``gluing'' the local charts^SsatoXcalongS^sa_SsaU_C(cf. [Rap78] and [Ch90]), one gets a toroidal compactificationXc,!Xc, and an isomorphism of formal schemes

X^{^}a

C

S(fs^ _ag)=U_F;N² ;

whereX^{^}denotes the completion ofXalong the boundaryX X. There exists a semi-abelian scheme Awith real multiplication by OF over Xc

which extends the universal HBAVAoverXc, and whose restriction to eachS^_sa is Tate(a;b). We putXˆ`

c X_c, where (c;c^‡) runs through a set of prime-to-p representatives of the strict ideal class group of F. The (O_X O_F) torsor v extends to X; hence, for any ~k2Z^B, the line bundlev^~k extends uniquely to a line bundle onX, which we still denote by v^~k.

We define a G00(N)\G0(p)-cusp (C;H) on Y to be a G00(N)-cusp Cˆ(a;b;L;i_C) as above together with an (O_F=p)-cyclic subgroup HL=pL. By choosing a rational polyhedral cone decomposition for each cusp (C;H) compatible with that for X, one generalizes the pre- vious construction to get an toroidal compactificationYofYin the same manner as the Siegel case treated by Stroh [St10a]. ThenY is a proper smooth scheme overW, which containsY as an open dense subscheme.

We have a similar description ofY^{^}in terms of local charts. The natural projection Y!X extends to a morphism Y!X.

2.4 ±Rigid analytic spaces

Let X and Ybe the respectively the formal completions of X and Y along their special fibers. The formal schemeXrepresents the functor that attaches, to each admissiblep-adic formal scheme over Spf(W), the set of polarized HBAV with aG₀₀(N)-level structure; and we have a similar in- terpretation forY. LetXrigandY_rigbe the associated rigid analytic spaces in the sense of Raynaud,X_Q^an_k andY_Q^an_k be the analytic spaces over Q_k as- sociated with the Q_k-schemes X_Qk and Y_Qk. Similarly, we have formal schemes X, Y for toroidal compactifications, and their associated rigid analytic spacesX_rig,Y_rig. Then we have natural inclusions of rigid analytic spaces

X_rigX_Q^an_k X_rig; Y_rig Y_Q^an_k Y_rig:

For any extension of valuation fieldsL=Q_k, we use a subscriptLto denote the base change of a rigid space overQ_ktoL, e.g.X_rig;L,X_Q^an_k;LˆX_L^an, ... For any weight~k2Z^B, by an obvious abuse of notation, we still denote byv^~k the modular line bundles of weight~kon the formal schemesXandY, and on rigid spacesXrig,Y_rig.

PROPOSITION 2.2 [Koecher's Principle]. For any finite extension L overQ_k, we have a commutative diagram of canonical isomorphisms

where the horizontal arrows are natural restriction map, and the vertical arrows are analytification maps.

PROOF. The diagram above is clearly commutative. The top horizontal isomorphism is the classical Koecher principle [Ch90, Thm. 4.3(i)]. The left vertical isomorphism follows from rigid GAGA [Ab11, 7.6.11]. The lower horizontal arrows are clearly injective. To finish the proof, it suffices to show that the restriction map

H⁰(Y_rig;L;v^~k)!H⁰(Y_rig;L;v^~k)

is an isomorphism. Since bothW-formal schemesYandYare admissible, we haveH⁰(Y_rig;L;v^~k)H⁰(Y;v^~k)_WL and similarly for H⁰(Y_rig;L;v^~k).

We are thus reduced to proving the similar statement for formal schemesY andY, then further reduced to showing that restriction map

H⁰(Y_WW=pⁿ;v^~k)!H⁰(Y_WW=pⁿ;v^~k)

is an isomorphism for all n1. By the construction of Y, it suffices to prove that, for everyf2H⁰(Y_WW=pⁿ;v^~k), theq-expansion of f around each cusp has no poles. This follows from the same computation as in

[Ra74, 4.9]. p

2.5 ±Up-operators

Let fc₁; c_h^‡g be the fixed set of prime-to-p representatives of the strict ideal class group Cl^‡_F. Fix a primepofFabovep. For eachci, there exists unique 1jh^‡and a totally positive elementj_i2F^;‡such that c_ipˆ(j_i)c_j. Note that j_i is only determined up to elements of U_F^‡, the group of totally positive units inF; ifpis not inert inF, then there is no canonical choice for such aj_i. We fix such aj_ifor each 1ih^‡.

IfAis a HBAV over aW-algebraR, we have a decomposition of finite and locally free group schemes overR

A[p]ˆY

pjp

A[p];

wherepruns through all the prime ideals ofOFdividingp, andA[p] is the subgroup scheme killed by all a2p. Then A[p] is a group scheme of (O_F=p)-vector spaces of dimension 2. We fix a prime idealpofO_Fabovep, and putk(p)ˆ O_F=p. LetC(p) be the scheme overQ_kwhich represents the functor that attaches to aQ_k-algebraRthe set of isomorphism classes of 5- tuples (A;l;c_N;H;H⁰) where:

(A;l;c_N;H) is an object inY(R);

H⁰A[p] is a closed (OF=p)-cyclic isotropic subgroup scheme such thatH⁰\Hˆ f0g.

We will two maps

…2:1†

given respectively as follows:

p₁(A;l;c_N;H;H⁰)ˆ(A;l;c_N;H)

p2(A;l;c_N;H;H⁰)ˆ(B;lB;c_B;N;(H⁰‡H)=H⁰);

where Bis the quotient abelian scheme A=H⁰, c_B;N is the induced level structure onBsuch that fc_Nˆc_B;Nif f :A!Bdenotes the canonical quotient isogeny. It remains to describe the polarizationl_BonB. Assume that l induces an isomorphism l:AOFci! A^_ for 1ih^‡. Let j_i2F^;‡be the element chosen above such thatc_ipˆ(j_i)c_j for a unique 1jh^‡. Letg:B!AOFp ¹ be the canonical quotient isogeny with kernelA[p]=H⁰B[p]. It is easy to check that the quasi-isogeny

BOFc_j ƒƒƒ!^1ji BOFc_ipƒƒƒ!^g1 AOFc_iƒ!^l A^_ ƒƒ^g!^_ B^_OFp ¹ B^_ is a genuine isogeny, and it gives the desired polarizationl_B. Here, the last arrowB^_!B^__OFp ¹B^_=B^_[p] is the canonical quotient with kernel B[p].

Since we are in characteristic 0, bothp1 andp2 are finite eÂtale of de- gree jk(p)j. Let (A;l;c_N;H;H⁰) be the universal object on C(p), and f:A ! B ˆ A=H⁰ be the canonical isogeny. We have a commutative diagram

where both the left and right squares are cartesian. We have a natural morphism of (O_C(p) O_F)-torsors:

p₂(v_A=YQk)! f₂(V¹_B=C(p))ƒ!^f f₁(V¹_A=C(p)) p₁(v_A=YQk);

which induces a natural morphism of line bundles f :p₂(v^~k) !p₁(v^~k)

for any~k2Z^B. For a finite extensionL=Q_k, we define theU_p-operator on the spaceH⁰(Y_L;v^~k) as the composite

…2:2† Up:H⁰(YL;v^~k)ƒ!^p2 H⁰(C(p)_L;p₂(v^~k))ƒ!^f

H⁰(C(p)_L;p₁(v^~k))ƒƒƒƒ!

jk(p)j1 tr

H⁰(Y_L;v^~k);

where ``tr'' is induced by the trace mapp1p₁(v^~k)!v^~k. Explicitly, ifL⁰=L is a finite extension such that (A;H)ˆ(A;l;c_N;H)2Y(L), and v is a generator ofv_AˆeV¹_A=L0 as an (L⁰ O_F)-module, then we have

(Upf)(A;H;v)ˆ 1 jk(p)j

X

H0A[p]

H0\Hˆ0

f(A=H⁰;(H⁰‡H)=H⁰;p ¹f^(v));

…2:3†

where H⁰ runs over all the O_F-subgroups of A[p] of order jk(p)j with H\H⁰ˆ0, andf^:A=H⁰!Ais the canonical isogeny with kernelA[p]=H⁰. REMARK2.3. As pointed by the referee of this paper, our definition of Up-operators has the obviously disadvantage that it depends on the choices of thej_i's, which are only canonically determined up to elements ofU_F^‡, although it is harmless for the proof of our main theorem. This ambiguity disappears when we restrict to so-called arithmetic Hilbert modular forms, which are the forms giving rise to Galois representations.

Actually, there is a natural action ofU_F^‡onYgiven by [e]:(A;l;c_N;H)7!

(A;el;c_N;H). Let UF;N O_F be the subgroup of units congruent to 1 moduloN. Then the subgroupU_F;N² acts trivially, because the endomorphism u:A!Ainduces an isomorphism of tuples (A;l;c_N;H)'(A;el;c_N;H) ifeˆu² with u2U²_F;N. Thus the action ofU_F^‡ factors through the finite quotientU_F^‡=U_F;N² . We have also an equivariant action ofU_F^‡=U_F;N² on v^~k so that we get a natural action ofU^‡_F=U_F;N² on theH⁰(Y_L;v^~k). The space of arithmetic Hilbert modular formsis defined to be the invariant subspace of H⁰(Y_L;v^~k) under U^‡_F=U_F;N² . Then for those forms, the ambiguity coming from the choices forj_i's will disappear. For more discussion on this issue, see [KL05, 1.11] and [DT04].

2.6 ±Norms

We fix a weight~kˆ P

b2Bkb2Z^Band a finite extensionL=Q_k. LetL⁰=L be a finite extension, and Qˆ(A;H)2Y_rig(L⁰) be a rigid point, i.e. a morphism of formal schemes Q:Spf(O_L⁰)!Y such that Q(A;H)ˆ (A;H), where (A;H) is the universal formal HBAV together with its uni- versal isotropic (OF=p)-cyclic subgroup overY_rig. Letvbe a generator of the free (OL⁰ OF)-modulev_AˆH⁰(A;V¹_A=OL0), andvbbe itsb-component for anyb2B. Then

v^~k ˆ b2Bv^k_b^b

is a basis of theOL⁰-moduleQ(v^~k). For any element f 2Q(v^~k)O_L0 L⁰, we write f ˆf(A;H;v)v^~k with f(A;H;v)2L, and define

jfj ˆ jf(A;H;v)j_p:

For any admissible open subsetV Y_rig;L, and a sectionf 2H⁰(V;v^~k), we define

jfj_V ˆ sup

Q2jVjjf(Q)j 2R_>0[ f1g;

wherejVjdenotes the set of rigid points ofV. IfVis quasi-compact, then

jfj_V 2R>0 by the maximum modulus principle in rigid analysis, and

H⁰(V;v^~k) is anL-Banach space with the normj j_V. IfV is clear from the context, we usually omit the subscriptV from the notation.

2.7 ±Hasse invariants

LetRbe ak-algebra, andAbe a HBAV overR. Letv_A=RˆH⁰(A;V¹_A=R), and Lie(A) be the Lie algebra ofA; so we have Lie(A)ˆHomR(v_A=R;R).

The Verschiebung homomorphism V_A:A^(p) !A induces a map of R- modules HW:Lie(A)^(p) !Lie(A), where Lie(A)^(p) is the base change of Lie(A) via the absolute Frobenius endomorphism F_R:a7!a^p of R.

Equivalently, we have a canonical map h:v_A=R!v⁽_A=R^p) :

Note that v_A=R is a locally free R OF-module of rank 1, and let v_A=Rˆ L

b2Bv_A=R;b, wherev_A=R;bis the direct summand on whichOFacts via the characterx_b. Thus we have a decompositionhˆ L

b2Bh_b, where hb:v_A=R;b!v⁽_A=R;s^p) 1b:

…2:4†

The morphismh_b thus defines a Hilbert modular form (with full level) of weightps ¹b boverk, and we call it theb-partial Hasse invariant.

The product Eˆ Q

b2Bhb is thus a Hilbert modular form of weight (p 1)P

b2Bboverk, called simplythe Hasse invariant. IfAis a HBAV over an algebraically closed field containingk, the Hasse invariantE(A)6ˆ0 if and only if Ais ordinary in the usual sense, i.e., the finite group scheme A[p] is isomorphic tom^gp(Z=pZ)^g.

2.8 ±Ordinary locus

LetX_k,Y_kbe the special fibers ofXandY, andX_k^ordbe the locus where the Hasse invarianthdoes not vanish, or equivalently the open subscheme ofXk parametrizing polarized ordinary HBAV. For a HBAVAover ak- algebraR, the kernel of the Frobenius homomorphism ofAis naturally an (O_F=p)-cyclic isotropic subgroup of A[p]. In other words, the kernel of Frobenius defines a sections:X_k!Y_kof the projectionp:Y_k!X_k. We putY_k^ord ˆs(X_k^ord). In particular,Y_k^ord is isomorphic toX_k^ord.

LetX^ordandY^ordbe respectively the open formal subschemes ofXand Ycorresponding to the open subsetsX_k^ordXk and Y_k^ordYk, and X^ord_rig andY^ord_rig be the associated rigid analytic generic fibers. ThenX^ord_rig andY^ord_rig are respectively quasi-compact admissible open subsets of X_rig andY_rig. Let (A;l;c_N) be the universal formal HBAV over X. Over the ordinary locusX^ord, we have an extension of finite flatO_F-group schemes

0!A[p]^m!A[p]!A[p]^et!0;

whereA[p]^etis eÂtale of orderp^g, andA[p]^mis of multiplicative type and lifts the kernel of Frobenius of AWk. The finite flat subgroup A[p]^m is (OF=p)-cyclic and isotropic for the Weil pairing induced by the polarization l, and it defines thus an section s:X^ord !Y^ord lifting the section s:X_k^ord !Y_k^ordgiven by the kernel of Frobenius. In particular, the natural projectionsY^ord!X^ord andY^ord_rig !X^ord_rig are canonical isomorphisms.

DEFINITION2.4. LetLbe a finite extension ofQ_k.

(i) For~k2Z^B, an element ofH⁰(X^ord_rig;L;v^~k) is called ap-adic Hilbert modular formof levelG₀₀(N) and weight~kwith coefficients inL.

(ii) We say an elementf 2H⁰(X^ord_rig;L;v^~k) isoverconvergentiffextends to a strict neighborhoodV ofX^ord_rig;LinX_rig;L. We put

M_~^y_k(G₀₀(N);L)ˆlim

!V H⁰(V;v^~k)

whereVruns over the strict neighborhoods ofX^ord_rig;LinXrig;L, and we call it the space of overconvergentp-adic Hilbert modular forms of levelG00(N) and weight~k.

REMARK2.5. By the theory of canonical subgroups (cf. [KL05, § 3] and [GK09, Thm. 5.3.1]), the isomorphism of ordinary locip:Y^ord_rig !X^ord_rig ex- tends to a strict neighborhood ofY^ord_rig inY_rig. Therefore, the natural notion

of (overconvergent)p-adic Hilbert modular forms of levelG00(N)\G0(p) is the same as its counterpart of levelG00(N). Hence, we can always consider an elementf 2M_~^y_k(G00(N);L) as a section ofv^~kover a strict neighborhood ofY^ord_rig;L.

We refer the reader to [Be96] for the definition of strict neighborhood.

Here we construct an explicit fundamental system of strict neighborhoods of X^ord_rig;L in Xrig;L by using the Hasse invariant. Let gE^k0 be a lift in H⁰(X;v^{k0(p 1)~1}), where~1ˆ(1; ;1)2Z^B, of k₀-th power of the Hasse invariant E^k0 for some integerk01. The existence of such a lift follows from Koecher's principle and the fact that v^~1 is ample on the minimal compactification X. For any rational number 0<r1, we denote by X^ord_rig;L(r) the admissible open subset ofXrig;L where jgE^k0j r^k0. Since the Hasse invariant is well-defined modulo p, the subset Xrig;L(r) does not depend on the choice of the lift gE^k0 if p ^1=k0 <r1. It is clear that Xrig;L(1)ˆX^ord_rig;L, and the Xrig;L(r)'s form a fundamental system of strict neighborhoods ofX^ord_rig;L inX_rig;L. Hence, we have

M_~^y_k(G00(N);L)ˆ lim

r!1 H⁰(X^ord_rig;L(r);v^~k):

Note that eachH⁰(X^ord_rig;L(r);v^~k) is a Banach space overL, and the natural restriction map

H⁰(X^ord_rig;L(r);v^~k)!H⁰(X^ord_rig;L(r⁰);v^~k)

for 0<r<r⁰<1 is compact [KL05, 2.4.1]. Therefore, M_~^y_k(G00(N);L) is (compact) direct limit of Banach spaces overL.

By Remark 2.5, we have a natural injective map

H⁰(YL;v^~k)'H⁰(Y_rig;L;v^~k)!M_~^y_k(G00(N);L);

where we have used Prop. 2.2 for the first isomorphism. We denote its image by M_~k(G₀₀(N)\G₀(p);L), and call it the space ofclassicalHilbert modular forms.

2.9 ±Up-operators in the rigid setting

For a prime idealpofO_Fabovep, letC(p)^anbe the rigid analytification of the schemeC(p) overQ_k considered in 2.5. Then just asC(p), the rigid spaceC(p)^anrepresents an analogous functor in the rigid analytic setting,

and we still have a universal object (A^an;l;c_N;H;H⁰) overC(p). We have analogous morphismsp1;p2:C(p)^an!Y_Q^an_k. We put

C(p)_rigˆp₁¹(Y_rig)ˆp₂¹(Y_rig):

The rigid analytic space C(p)_rig is the locus ofC(p)^an whereA^an has good reduction, it classifies the objects (A;H;H⁰), where (A;H) is a rigid point of Y_rig, andH⁰A[p] is a group scheme of (OF=p)-vector space of dimension 1 withH\H⁰ˆ0. We have a rigid version of the Hecke correspondence:

given by p₁(A;H;H⁰)ˆ(A;H) and p₂(A;H;H⁰)ˆ(A=H⁰;(H‡H⁰)=H⁰).

We have also a set theoretical Hecke correspondence between the rigid points ofY_rig

U_p:Y_rig!Y_rig …2:5†

Q7!p₂(p₁¹(Q)):

Here, it is an obvious notation for convenience, becauseU_pis not really a morphism of rigid analytic spaces. IfUandVare admissible open subsets of Y_rig such thatU_p(U)V, i.e.p₁¹(U)p₂¹(V). A rigid version of the formula (2.3) defines theUp-operator

Up:H⁰(V;v^~k)ƒ!^p2 H⁰(p₂¹(V);p₂v^~k)ƒ!^f H⁰(p ¹(U);p₁v^~k)ƒƒƒ!^{jk(p)j1 tr} H⁰(U;v^~k):

LEMMA2.6. The ordinary locusY^ord_rig is stable under the Hecke corre- spondence Up, i.e. we havep2(p₁¹(Y^ord_rig))Y^ord_rig.

PROOF. LetLbe a finite extension ofQ_k, (A;H)2Y^ord_rig(L) be a rigid point, i.e.Ais a HBAV overO_Lwith ordinary reduction, andHA[p] is the multiplicative part. We have to show that (A=H⁰;(H‡H⁰)=H⁰) still lies in Y^ord_rig for any isotropic (O_F=p)-subgroupH⁰A[p] withH⁰\Hˆ0. Actu- ally, such a H⁰ is necessarily eÂtale over OL. Therefore, the isogeny A!A=H⁰is eÂtale, and the subgroup (H‡H⁰)=H⁰is the multiplicative part

of the HBAVA=H⁰. p

This easy Lemma implies immediately that aU_p-operator analogous to the classical case can be defined on the spaceH⁰(Y^ord_rig;L;v^~k) for any weight

~k2Z^B and any finite extension L of Q_k. In order to show that over- convergentp-adic Hilbert modular forms are stable underUp, we need to extend canonically the sections :X^ord_rig !Y^ord_rig to a strict neighborhood of X^ord_rig. As already mentioned in Remark 2.5, this is the theory of canonical subgroups, and it has been developed by many authors (cf. for instance [KL05] and [GK09]). The main result of this paper is the following

THEOREM2.7. Let f be an element of M_~^y_k(G₀₀(N);L). Assume that for every prime ideal p of OF above p, we have [k(p):Fp]2 and Up(f)ˆapf with vp(ap)<min

b2Bpfk_bg [k(p):Fp], then f is classical, i.e., f 2M_~k(G₀₀(N)\G₀(p);L).

REMARK2.8. It is reasonable to expect that the theorem is also true without the restriction [k(p):F_p]2. The main obstacle to this general- ization is that the geometry of Y in the higher dimensional case is too complicated, and we don't well understand the dynamics of the U_p-op- erator.

In the reminder of this section, we supposep3, and indicate some consequences of our results on eigencurves for overconvergent Hilbert eigenforms.

2.10 ±Applications to Kisin-Lai's eigencurves

We follow the treatments in [KL05]. LetLbe a finite extension ofQ_k, andRbe a Banach algebra overLwith a submultiplicative normj j, and Z2Rsuch thatjZj<p ^{2p p1}. We fix an integerk0>0 coprime topsuch that thek₀-th power of the Hasse invariant lifts togE^k02H⁰(X;v^{k0(p 1)~1}). For

~k2Z^B, Kisin and Lai defined in [KL05, 4.2.3] the space of overconvergent Hilbert modular forms overRof levelG00(N) and weight~k‡Zto be space

M_~^y_k‡Z(G00(N);R)ˆlim

!V H⁰(V;v^~k)^QkR;

…2:6†

where V runs over a fundamental system of quasi-compact strict neigh- borhoods ofX^ord_rig inX_rig. This space is equipped with an action of the Hecke operators T_a (resp.U_a) for each ideal a O_F coprime topN (resp. not coprime topN). We point out that Kisin-Lai's definition of these operators involves the liftgE^k0t, and ifZˆ0, we come back to the definition (2.4). We denote by T^y_~k‡Z(m_N) the ring of endomorphisms generated by these op-