-adic modular symbols over totally real elds ymlongo/docs/Balasub... · in X(I) correspond to...

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Λ-adic modular symbols over totally real fields *† Baskar Balasubramanyam and Matteo Longo § Abstract We consider an Hida family of nearly ordinary cusp forms on a quaternion algebra defined over a totally real number field. The aim of this work is to construct a cohomology class with coefficients in a p-adic sheaf over an Iwasawa algebra that can be specialized to cohomology classes attached to classical cusp forms in the given Hida family. Our result extends the work on Greenberg and Stevens on modular symbols attached to ordinary Hida families when the ground field is the field of rational numbers. Contents 1 Introduction 2 2 Hida theory for totally real fields 7 2.1 Cusp forms over quaternion algebras .............. 7 2.2 Hecke operators .......................... 8 2.3 Nearly ordinary Hecke algebras ................. 10 3 Λ-adic modular symbols 13 3.1 Classical modular symbols .................... 13 3.2 Λ-adic modular symbols ..................... 14 3.3 The Control Theorem ....................... 20 3.4 Freeness of W ........................... 22 3.5 Description of Ker(ρ κ ) ...................... 23 3.6 Dimension bounds ........................ 24 3.7 Proof of the Control Theorem .................. 25 * Key words and phrases: Shimura Varieties, Hilbert Modular Forms, Modular Symbols, Hida Families. 2000 Mathematical Subject Classification. 11G18, 11F41, 11F67. California Institute of Technology, Pasadena CA 91125 § Dipartimento di Matematica Pura e Applicata, Via Trieste 63, 35121, Padova, Italy. 1

Transcript of -adic modular symbols over totally real elds ymlongo/docs/Balasub... · in X(I) correspond to...

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Λ-adic modular symbols over totally real fields∗†

Baskar Balasubramanyam‡ and Matteo Longo §

Abstract

We consider an Hida family of nearly ordinary cusp forms on aquaternion algebra defined over a totally real number field. The aimof this work is to construct a cohomology class with coefficients ina p-adic sheaf over an Iwasawa algebra that can be specialized tocohomology classes attached to classical cusp forms in the given Hidafamily. Our result extends the work on Greenberg and Stevens onmodular symbols attached to ordinary Hida families when the groundfield is the field of rational numbers.

Contents

1 Introduction 2

2 Hida theory for totally real fields 72.1 Cusp forms over quaternion algebras . . . . . . . . . . . . . . 72.2 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Nearly ordinary Hecke algebras . . . . . . . . . . . . . . . . . 10

3 Λ-adic modular symbols 133.1 Classical modular symbols . . . . . . . . . . . . . . . . . . . . 133.2 Λ-adic modular symbols . . . . . . . . . . . . . . . . . . . . . 143.3 The Control Theorem . . . . . . . . . . . . . . . . . . . . . . . 203.4 Freeness of W . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Description of Ker(ρκ) . . . . . . . . . . . . . . . . . . . . . . 233.6 Dimension bounds . . . . . . . . . . . . . . . . . . . . . . . . 243.7 Proof of the Control Theorem . . . . . . . . . . . . . . . . . . 25∗Key words and phrases: Shimura Varieties, Hilbert Modular Forms, Modular Symbols,

Hida Families.†2000 Mathematical Subject Classification. 11G18, 11F41, 11F67.‡California Institute of Technology, Pasadena CA 91125§Dipartimento di Matematica Pura e Applicata, Via Trieste 63, 35121, Padova, Italy.

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1 Introduction

Let F/Q be a totally real field of degree d. We will always assume in thispaper that d > 1 because the results that we will present are generalizationsof results already known ([4]) when the base field is the field of rationalnumbers. Denote by r the ring of integers of F . For any abelian groupA, define A := A ⊗Z Z, where Z is the profinite completion of Z. For anyquaternion algebra B/F let G = GB be the linear algebraic group over Qsuch that G(Q) = B×. Let GA denote the adelization of G and denoterespectively by Gf ,G∞ and G∞+ the finite part, the infinite part and theconnected component of G∞ containing the identity. Fix a compact opensubgroup S ⊆ Gf such that U0(n) ⊇ S ⊇ U1(n) for some integral ideal nof F , where U0(n) and U1(n) are congruence groups defined in §2.1. Let pbe a rational prime which is prime to 2n and define S(pα) := S ∩ U(pα)for any non negative integer α, where U(pα) is also defined in §2.1. Fix anembedding ι : Q ↪→ Qp and a finite extension K of Qp containing ι ◦ µ(F )for all archimedean places µ of F . Denote by O the ring of integers of K.

Let I denote the set of embeddings of F into C. Let n, v ∈ Z[I] be fixedweight vectors such that n+2v ≡ 0 mod Zt, where t = (1, ...1) ∈ Z[I] and letk := n+ 2t and w := v + k − t. Following [7], we denote by hn,ord

k,w (S(pα),O)the Hecke algebra over O for the space of p-nearly ordinary Hilbert cuspforms of weight (k, w) and level S(pα). In §2.3 we recall Hida’s constructionof the universal p-odinary Hecke algebra

R := lim←−α

hn,ord2t,t (S(pα),O).

This Hecke algebra is universal in the sense that each nearly ordinary Heckealgebra hn,ord

k,w (S0(pα), ε,K) acting on the K-vector space of cusp forms ofweight (k, w), level S0(pα) and finite order character ε : S0(pα)/S1(pα) → Cis isomorphic to a residue algebra of R. Here S0(pα) := S ∩ U0(pα) andS1(pα) := S ∩ U1(pα). More precisely, let

G := lim←−α

S0(pα)r×/S(pα)r×,

and denote by W the free part of G. Then R has a natural structure ofΛ-algebra, where Λ := O[[G]] is the Iwasawa algebra of G and there areisomorphisms:

RP/PRP∼−→ hn,ord

k,w (S0(pα), ε,K) (1)

for suitable ideals P of Λ, where RP is the localization of R at P . See [7,Theorem 2.4] or §2.3 for details.

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Let Λ := O[[W ]] be the Iwasawa algebra of the free part W of G. Denoteby L := Frac(Λ) the field of fractions of L and by L its algebraic closure.Fix a point

θ ∈ HomO -algebrascont (R, L).

By [7, Theorem 2.4], the image of θ is contained in a finite extension K of L.Let I denote the integral closure of Λ in K. Say that a point

κ ∈ X (I) := HomO−algebrascont (I,Qp)

is arithmetic if its restriction to G has kernel equal to P for some of theideals P appearing in (1). See [7, pages 150-152] or §2.3 for a more precisenotion of arithmetic point. A corollary of [7, Theorem 2.4] states that pointsin X (I) correspond to p-adic Hilbert modular forms and arithmetic pointsκ corresponds to classical p-nearly ordinary Hilbert modular forms fκ ofsuitable weight (k, w), level S0(pα) and character ε. See [7, Corollary 2.5] fordetails. In particular, to any arithmetic point κ is associated a weight (n, v)(or (k, w)), a level S0(pα) and a character ε.

Remark 1.1. Note that the Iwasawa algebra Λ := O[[W ]] ofW is isomorphic tothe formal power series ring in s variables O[[X1, . . . , Xs]], where s = d+1+δFand δF is the non negative integer appearing in Leopoldt’s conjecture. Another way to state [7, Corollary 2.5] is that any f ∈ Sk,w(U1(npα), K) hass-dimensional p-adic deformations over O (in the sense of [7, pages 152-153]).

To discuss the main result of this paper, we need some technical assump-tions. Assume that B is unramified at all finite primes in F . Denote byFA the adele ring of F . By the Strong Approximation Theorem, choosegλ ∈ GL2(FA) for 1 ≤ λ ≤ h(pα) and a suitable integer h(pα) depending onα with (gλ)np = (gλ)∞ = 1 and such that there is the following disjoint uniondecomposition:

GB =

h(pα)∐λ=1

GQgλSG∞+ . (2)

If α = 0 write h for h(1). If B is indefinite, we can assume that the gλ doesnot depend upon α. Define the following arithmetic groups depending on S:

Γλ(pα) := gλS(pα)G∞+g−1λ ∩GB(Q) and Γ

λ(pα) := Γλ(pα)/(Γλ(pα) ∩ F×).

(3)Let IB be the set of infinite places where B is unramified. If |IB| = r, thenG∞ is isomorphic to GL2(R)r ×Hd−r. Since Γλ(pα) is a subgroup of G∞, itis acts on Hr (H is the complex upper half plane).

Assumption 1.2. The groups Γλ

:= Γλ(1) are torsion free for all λ =

1, . . . , h.

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See [6, Lemma 7.1] for conditions under which Assumption 1.2 is veri-fied. In particular, there are infinitely many square-free integers for which

Γλ(U0(N)) is torsion-free for all λ. Under this assumption, for any O[Γλ]-

module E, there is a canonical isomorphism H∗(Γλ\Hr, E) ' H∗(Γλ, E),

where E is the coefficient system associated to E. The technical Assump-tion 1.2 can probably be dispensed with by employing the methods in [6,pp. 351-352, proof of Theorem 3.1] and [7, p. 155, Proposition 3.1], but wekeep it to enlighten the notations and the arguments. See also [3, §7.4] for acomplete reference of this approach.

The aim of this work is to construct a cohomology class with coefficientsin a p-adic space that can be specialized to cohomology classes attached tocusp forms in a Hida family. This is a generalization of a similar construction[4]. The main difference here is that the Iwasawa algebra Λ in this case isisomorphic to a power series ring in at least d+ 1-variables over O, while inthe rational case it is just isomorphic to a power series ring in one variable.

Remark 1.3. In [6] Hida constructs for each weight v as above an ordinaryuniversal Hecke algebra

Rv = hordv (U1(p∞),O)

such that each hn,ordk,w (U1(pα),O) with k parallel to −2v can be obtained as a

residue algebra ofRv. These Hecke algebrasRv are endowed with a structureof O[[X1, . . . , XδF ]]-algebra. The Iwasawa algebra Λ considered here has dmore variables in order to unify these various Hecke algebras as v and thecharacter ε vary.

A consequence of the fact that the Iwasawa algebra considered here isbigger than that in [4] is that the role played by the set of primitive vectors(Z2

p)′ in [4] will be played in this context by the p-adic space

X := NC\GL2(rp) ' lim←−α

S(pα)r×\Sr×,

which has a greater rank. Here N is the standard lower unipotent subgroupof GL2(rp) and C is the closure of e := S∩F embedded diagonally in GL2(rp),where rp := r⊗Z Zp. A similar p-adic space has been defined and studied in[2].

Remark 1.4. In order to justify the definition of X, we note that, in the caseof [4], there is a canonical bijection between the set of primitive vectors (Z2

p)′

and the inverse limit lim←−Γ1(Npm)\Γ1(N), so our definition of X seems to bea correct generalization of the concept of primitive vectors used in [4]. Wewill clarify this point of view in the following lines.

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The action of the Hecke operators Up for prime ideals p | p on X is similarto that considered in [2]. To describe X more precisely, for any prime idealp | p of F , let (r2

p)′ denote the set of primitive vectors of r2

p, i. e. the set ofelements (x, y) ∈ r2

p such that at least one of x and y does not belong to p.Set (r2

p)′ :=

∏p|p(r

2p)′. Denote by e the closure of S ∩ F× in r×p . Then X can

be identified via the map

γ =

(a bc d

)7−→

((a, b), det(γ)

)with e\((r2

p)′ × r×p ). Hence X may be viewed as an analogue of the primitive

vectors appearing in [4]. Elements of X will be denoted by ((x, y), z).Following [4], define DX to be the space of O-valued measures on X. This

space is endowed with Λ and Λ-algebra structures. Let X(S) be the Hilbertmodular variety associated to S and letDX denote the local coefficient systemon X(S) associated to DX . We suppose from now to the end of this sectionthat

r = |IB| ≤ 1

In particular, since d > 1, B is always a division algebra. Further, since weare also assuming that B is unramified at all finite places, we have r = 0 ifd is even and r = 1 if d is odd. Finally, X(S) is always compact in this case.We define the space of Λ-adic modular symbols in this context to be

W := Hr(X(S),DX) = Hrcpt(X(S),DX)

the r-th cohomology of X(S) with coefficients in DX . It follows from [1,Proposition 4.2] that this definition of Λ-adic modular symbols is consistentwith the analogous definition in [4]. The group W is endowed with a structure

of Λ-module.For any ring R, let L(n,R) be the R-module of homogeneous poly-

nomials in 2d variables X = (Xσ)σ∈I , and Y = (Yσ)σ∈I of degree nσ inXσ, Yσ. Denote by γ = ( a bc d ) 7→ γ∗ :=

(d −b−c a

)the main involution of

M2(R) and define a right action of GL2(R) on L(n,A), for any R-algebraA, by (P |γ)(X, Y ) := det(γ)vP ((X, Y )γ∗), where (X, Y )γ∗ is matrix mul-tiplication. Denote by L(n, v, A) the right representation of GL2(R) thusobtained. For any character ε as above, write L(n, v, ε, A) for the ∆0(n)-module L(n, v, A) with the action of ∆0(n) twisted by ε, that is, denoting by|ε this new action: P |εγ := ε(γ)P |γ for γ ∈ ∆0(n).

For any arithmetic point κ of weight (n, v) and character ε, define aspecialization map ρn,v,ε : DX → L(n, v, ε,K) by

µ 7−→ ρn,v,ε(µ) :=

∫X′ε(x)zv(xY − yX)ndµ(x, y, z),

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where X ′ the subset of X consisting of elements ((x, y), z) such that x ∈ r×p .The map ρn,v,ε,K induces a map on the cohomology groups which we againdenote by the same symbol

ρn,v,ε : W −→ Hr(X(S0(pα)),L(n, v, ε,K)

).

Here S0(pα) is the level of the modular form fκ associated to the arithmeticpoint κ and L(n, v, ε,K) is the coefficient system on S0(pα) associated toL(n, v, ε). By following [4] and [2], a corresponding control theorem for thesespecialization maps is stated in Theorem 3.7 and proved in §3.7. More pre-cisely, fix θ : R → L and an arithmetic κ : I → Qp. Let Pκ be the kernel ofθκ := κ ◦ θ and Kκ the image of θκ. Set

Wκ := Hr(S0(pα),L(n, v, ε,Kκ)

)The map ρn,v,ε extends to the localization RPκ of R at Pκ and one obtains amap

ρκ : W⊗Λ RPκ −→Wκ.

For any O[T (p)]-module M , let Mord denote its ordinary part, that is, themaximal subspace of M on which the Hecke operator T (p) acts as a unit. Leth : H → R be the natural map obtained by the action of Hecke operatorson Λ-adic cusp forms. For any arithmetic point κ, let hκ be the compositionof h with the localization morphism R → RPκ . For any H ⊗Λ RPκ-moduleM , let Mhκ denote the subset of M consisting of those m ∈ M such that(T (q)⊗1)m = (1⊗hκ(T (q))) ·m for all prime ideals q in r. If fκ is a classicaleigenform for an arithmetic point κ, let

Wfκκ = {φ ∈Wκ| T0(q)φ = aq(g)φ}

denote the fκ-eigenspace of Wκ, where aq(g) is the eigenvalue of the T0(q)operator on fκ. Hence, ρκ induces a map:

ρκ : (W⊗Λ RPκ)hκ −→Wfκκ .

Theorem 1.5. Fix θ : R → I and let κ : I → Qp be an arithmetic pointof weight (nκ, vκ) and character εκ. Assume that RPκ is a regular local ring.Then the map

ρκ : (W⊗Λ RPκ)hκ/Pκ(W⊗Λ RPκ)hκ →Wfκκ

is an isomorphism.

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A more general statement taking into account the action of the complexconjugation on W and Wκ when r = 1 is given in Theorem 3.7.

Acknowledgments. This paper is part of the Ph.D thesis of the first author.The first author would like to thank sincerely his advisor Prof. F. Diamondfor his help and encouragement. Both authors would like to thank Prof. H.Darmon for suggesting the problem and for his advice and support duringtheir visits to McGill University. Finally, we sincerely thank the referee for hiscarefully reading the manuscript and suggesting many valuable correctionsand improvements.

2 Hida theory for totally real fields

2.1 Cusp forms over quaternion algebras

Let F be a totally real field of degree d > 1 over Q and denote by r its ringof integers. Denote by FA the adele ring of F and by F the ring of finiteadeles. Let I denote the set of embeddings of F into C.

Fix a quaternion algebra B/F which is unramified at all finite places ofF , let IB ⊆ I be the set of archimedean places of F where B is split anddenote by r = |IB| ∈ {0, . . . , d} its cardinality.

Let n, v ∈ Z[I] be fixed weight vectors such that n + 2v ≡ 0 mod Zt,where t = (1, ...1) ∈ Z[I] and let k := n+ 2t and w := v + k− t. Let J ⊂ IBand for any 1 ≤ λ ≤ h(U), denote by Sk,w,J(Γλ(U),B,C) the C-vector spaceof cusp forms which are defined in [6, §2] and set:

Sk,w,J(U,B,C) :=

h(U)∏λ=1

Sk,w,J(Γλ(U),B,C).

If B and B′ are quaternion algebras over F unramified at all finite places,by [6, Theorem 2.1], we know that then there exists and isomorphism

iU : Sk,w,IB(U,B,C)∼−→ Sk,w,IB′ (U,B

′,C).

Also by [7, Theorem 2.2] there exists an isomorphism between Sk,w,J(U,B,C)and Sk,w,IB(U,B,C) for any J ⊂ IB. These isomorphisms also commute withthe Hecke operators.

Let R be any maximal order in B and we fix an identification R =R ⊗ r ∼= M2(r) where r = r ⊗ Z. For any ideal m ⊆ r define the following

open compact semigroups of GL2(F ):

∆0(m) :=

{(a bc d

)∈ GL2(F ) ∩M2(r) : c ≡ 0 (mod m)

},

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∆1(m) :=

{(a bc d

)∈ ∆0(m) : a ≡ 1 (mod m)

},

U0(m) :={γ ∈ ∆0(m) : det(γq) ∈ r×q , for all prime ideals q ⊆ r

}, (4)

U1(m) := U0(m) ∩∆1(m), (5)

U(m) :=

{(a bc d

)∈ U1(m) : dq − 1 ∈ mrq for all prime ideals q | m

}.

(6)Fix an ideal n ⊆ r, a rational prime p prime to 2n and an open compactsubgroup S ⊆ GL2(r) such that U0(n) ⊇ S ⊇ U1(n) and the p-component Spof S is GL2(rp). For any positive integer α, set

S0(pα) := S ∩ U0(pα), S1(pα) := S ∩ U1(pα), S(pα) := S ∩ U(pα).

Denote by Γλ(pα) and Γλ(pα) (respectively, Γλ0(pα) and Γ

λ

0(pα); Γλ1(pα) and

Γλ

1(pα)) the arithmetic groups associated as in (3) to S(pα) (respectively,S0(pα); S1(pα)).

When B = M2(F ) we set:

Sk,w(U,C) := Sk,w,I(U,M2(F ),C)

Sk,w(Γλ(U),C) := Sk,w,I(Γλ(U),M2(F ),C).

Any modular form fλ ∈ Sk,w(Γλ(pα),C) has a Fourier expansion of the form

fλ(z) =∑

ξ∈gλd,ξ�0

aλ(ξ)e2πi(ξ·z),

where the notations are as follows: gλ is an ideal represented by the normof gλ, d is the different ideal of F/Q, ξ � 0 if and only if, by definition, ξ istotally positive, and (ξ · z) :=

∑σ∈I σ(ξ)zσ is the scalar product. For details,

see [6, Corollary 4.3].

2.2 Hecke operators

The right action on Sk,w(S(pα),C) of the Hecke algebra R(S(pα),∆0(npα)),which is by definition the free Z-module generated by double cosets

T (x) := S(pα)xS(pα)

for x ∈ ∆0(npα) with multiplication defined by∑i

aiT (x) ·∑j

bjT (y) :=∑i,j

aibjT (xiyj),

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can be described as follows. Fix x ∈ ∆0(npα) and 1 ≤ λ ≤ h. Let µ ∈{1, . . . , h} such that det(x)tλt

−1µ is trivial in the strict class group of F . Set

xλ :=(

1 00 tλ

). Let αλ such that S(pα)xS(pα) = S(pα)x−1

λ αλxµS(pα) andform the finite disjoint coset decomposition Γλ(p

α)αλΓµ(pα) =∑

j Γλ(pα)αλ,j

where αλ,j ∈ GL2(F ) ∩ xλ∆0(npα)x−1µ . Define

gµ :=∑j

fλ|αλ,j and f |T (x) := (g1, . . . , gh).

Denote by F the composite field of all the images σ(F ) of F under theelements σ ∈ I and by r its ring of integers. Fix an r-algebra A ⊆ Csuch that for every x ∈ F× and every σ ∈ I, the A-ideal xσA is generatedby a single element of A. For any prime ideal q ⊆ r, choose a generator{qσ} ∈ A of the principal ideal qσA. Define {q}v :=

∏σ∈I{qσ}vσ . Write a

fractional ideal m of r as a product of prime ideals m =∏

q qm(q) and define

{m}v :=∏

q({q}v)m(q). For any element x ∈ F×, denote by mx the fractionalr-ideal corresponding to x and define {x}v := {mx}v. Modify the Heckeoperators T (x) ∈ R(S(pα),∆0(npα)) by setting:

T0(x) := ({det(x)}v)−1T (x).

Denote by hk,w(S(pα), A) the A-subalgebra of End(Sk,w(S(pα),C)) gener-ated over A by operators T0(x) for x ∈ ∆0(npα). By [7, Proposition 1.1],hk,w(S(pα), A) is commutative.

For any x ∈ F and m = (mσ)σ ∈ Z[I], set xm :=∏

σ∈I σ(x)mσ . Forany integral ideal m, choose λ = λ(m) such that m is equivalent to tλd inthe strict ideal class group of F and let ξm ∈ tλd be such that ξm � 0 andm = ξm(tλd)−1. Define the modified Fourier coefficients as in [6, Corollary3.4]:

C(m, f) :=aλ(ξm)ξvmbv,λ

, where bv,λ :=N(tλ)

{(tλd)v}. (7)

Suppose that f ∈ Sk,w(S(pα),C) is an eigenform for h(S(pα), A) such thatC(r, fλ) = 1 for all λ = 1, . . . , h (call such a form normalized). Then by [6,Corollary 4.2]:

f |T (m) = C(m, f)f.

The group Gα := S0(pα)r×/S(pα)r× acts on Sk,w(S(pα),C) via the op-erator ω(an+2v

p )−1T (x) for x = ( a bc d ) ∈ S0(pα), where ω is the Teichmullercharacter.

For any prime ideal q ⊆ r, choose an element q ∈ r× such that qr = qand the l-component ql of q is equal to 1 for all prime ideals l ⊆ r, l 6= q.

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Define T (q) := T(

1 00 q

)for all prime ideals q ⊆ r and T (q, q) := T

(q 00 q

)for

prime ideals q ⊆ r such that q - n. By [7, Proposition 1.1], if S ⊇ U1(n), thenhk,w(S(pα), A) is generated over A by the operators induced from the actionof Gα and T0(q) for all prime ideals q.

If f is an eigenform for the Hecke algebra hk,w(S(pα), A), then its eigen-values are algebraic numbers. If k ∼ t, then they are algebraic integers. De-fine Sk,w(S(pα), A) ⊆ Sk,w(S(pα),C) to be the A-module consisting of formswhose Fourier expansion has coefficients in A. The A-module Sk,w(S(pα), A)is stable under the action of hk,w(S(pα), A).

2.3 Nearly ordinary Hecke algebras

Choose an embedding ι : Q ↪→ Qp, so that any algebraic number is equippedwith a p-adic valuation. Fix a ring of integers O of a finite extension of thecompletion of ι(F ). After choosing an embedding i : Qp ↪→ C, the r-algebraO satisfies the conditions of §2.2.

By [7, Lemma 2.2], hk,w(S(pα),O) is free of finite rank over O and hencecan be decomposed as a direct sum:

hk,w(S(pα),O) = hn,ordk,w (S(pα),O)⊕ hss

k,w(S(pα),O) (8)

such that the image of T0(p) in the first factor, the nearly ordinary part, is aunit while its image in the other factor is topologically nilpotent. For any pairof non negative integers β ≥ α the map T0(x) 7→ T0(x) for x ∈ ∆0(npα) in-duces a surjective ring homomorphism ρβα : hk,w(S(pβ),O)→ hk,w(S(pα),O).Define:

hk,w(S(p∞),O) := lim←−hk,w(S(pα),O)

andhn,ordk,w (S(p∞),O) := lim←−h

n,ordk,w (S(pα),O)

where the inverse limits are with respect to the maps ρβα. By [7, Theo-rem 2.3], for any weight (k, w) there is an isomorphism hk,w(S(p∞),O) 'h2t,t(S(p∞),O) which takes T (q) to T (q) and T (q, q) to T (q, q) for all primeideals q - p. This isomorphism induces an isomorphism between the nearlyordinary parts

hn,ordk,w (S(p∞),O) ' R := hn,ord

2t,t (S(p∞),O). (9)

Set SF := S ∩ F×, SF (pα) := S(pα) ∩ F×, Zα := SF r×/SF (pα)r× and

Z∞ := lim←−Zα. By [7, Lemma 2.1] the map ( a bc d ) 7→ (a−1p dp, a) induces an

isomorphism Gα ' (r/pα)× × Zα and hence an isomorphism

G := lim←−Gα ' r×p × Z∞.

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Since hk,w(S(pα),O) is a O[Gα]-algebra and this algebra structure is com-

patible with the maps ρβα, it follows that hn,ordk,w (S(p∞),O) is a Λ := O[[G]]-

algebra. Write G ' W ×Gtors where Gtors is the torsion subgroup of G andW is torsion-free. Note that W is well determined only up to isomorphism;fix from now on a choice of W . The ring Λ := O[[W ]] is isomorphic to thepower series ring O[[X1, . . . , Xs]] in d < s < 2d variables. If Leopoldt’s con-jecture holds, then s = d+1. By [7, Theorem 2.4], the nearly ordinary Heckealgebra R = hn,ord

2t,t (S(p∞),O) is a torsion-free Λ-module of finite type.

For any finite order character ε : S0(pα)/S1(pα) → Q×, define char-acters ελ : Γλ0(pα)/Γλ1(pα) → C× by setting ελ(γ) := ε(t−1

λ γtλ). Denoteby Sk,w(S0(pα), ε,C) the C-subspace of Sk,w(S(pα),C) consisting of formsf = (f1, . . . , fh) such that for any λ = 1, . . . , h and any γ ∈ Γλ0(pα),(fλ|γ)(z) = ε−1

λ (γ)f(z). Suppose that O contains the values of ε and setSk,w(S0(pα), ε,O) := Sk,w(S0(pα), ε,C) ∩ Sk,w(S(pα),O). Denote by

hk,w(S0(pα), ε,O)

the O-subalgebra of End(Sk,w(S0(pα), ε,C)) generated over O by operators

T0(x) for x ∈ ∆0(npα). Define hn,ordk,w (S0(pα), ε,O) to be the maximal factor

of hk,w(S0(pα), ε,O) such that the image of T0(p) is a unit in that factor. (Inthe following the ελ’s will often be simply denoted by ε).

Definition 2.1. An eigenform f ∈ Sk,w(S0(pα), ε,C) for the Hecke algebrahk,w(S(pα), A) is said to be p-nearly ordinary if the eigenvalue of T0(p) is ap-adic unit.

Denote by r×+ the group of totally positive units of r. Define

Zα := SF r×+/SF (pα)r×+ (10)

and Z∞ := lim←−Zα. The kernel of the natural surjection map Z∞ → Z∞is finite and annihilated by a power of 2. Denote by χcyc : Z∞ → Z×p thecyclotomic character defined by χcyc(x) = xt =

∏σ∈I σ(x) = N(x). Let

ε : Z∞ → O× be a character factoring through Zα. Suppose that εχn+2vcyc

factors through Z∞, where if n + 2v = mt with m ∈ Z, then χn+2vcyc is by

definition χmcyc. Let

Pn,v,ε : G ' r×p × Z∞ −→ O×

be the character defined by Pn,v,ε(a, z) := εχn+2vcyc (z)av. Denote by the same

symbol the homomorphism Λ → O deduced from Pn,v,ε by extension of

scalars. The kernel of this homomorphism is a prime ideal of Λ, denoted

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be the same symbol Pn,v,ε. To simplify notations, set P := Pn,v,ε. Let

ΛP (respectively, RP ) denote the localization of Λ (respectively, R) in P .

Then RP is free of finite rank over ΛP and the natural surjective morphismR → hk,w(S0(pα), ε,O) induces by [7, Theorem 2.4] an isomorphism:

RP/PRP ' hk,w(S0(pα), ε,K), (11)

where K := Frac(O) is the fraction field of O. In particular, the dimensionof the K-vector space hk,w(S0(pα), ε,K) does not depend on ε and (k, w) andis equal to the ΛP -rank of RP .

Let L := Frac(Λ) be the fraction field of Λ and fix an algebraic closureL of L. Let θ : R → L be a Λ-algebra homomorphism. The image Im(θ) ofθ is finite over Λ. Denote by I the integral closure of Im(θ) in the fractionfield K := Frac(Im(θ)). Define

X (I) := HomO−alg(I,Qp)

and denote byA(I) the subset of κ ∈ X (I) consisting of points whose restric-tion to Λ coincide with the restriction to Λ of some character Pn(κ),v(κ),ε(κ) asabove. Points inA(I) are called arithmetic. In this case, set k(κ) := n(κ)−2tand w(κ) := k(κ) + v(κ) − t. Let C(κ) denote the conductor of ε restrictedto the torsion free part W of Z∞ and εW the restriction of ε to W . Let

Ztors

∞ denote the maximal torsion subgroup of Z∞ and let ψ : Ztors

∞ → L be

the composite of θ with the natural map Ztors

∞ → R× induced by the actionof G on R. Denote by r×tors

p the maximal torsion subgroup of r×p . For anyκ ∈ X (I), define

θκ := κ ◦ θ : R → Qp.

By [7, Corollary 2.5], if κ ∈ A(I) and θ restricted to r×torsp is the character

x 7→ xv(κ), then θκ(T (q)) are algebraic numbers for all prime ideals q andthere exists a p-nearly ordinary eigenform

fκ ∈ Sk(κ),w(κ)

(U0(nC(κ)), εWψω

−(n(κ)+2v(κ)),C)

(unique up to multiplication by constants) such that fκ|T (q) = θκ(T (q))fκ,where ω is the Teichmuller character and if n(κ) + 2v(κ) = mt with m ∈ Z,then ω−(n(κ)+2v(κ)) := ω−m. Conversely, if α > 0 and f ∈ Sk,w(U1(npα),C)is a p-nearly ordinary eigenform, then there exists κ ∈ A(I) and θ as abovesuch that f is a constant multiple of fκ.

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3 Λ-adic modular symbols

3.1 Classical modular symbols

Fix B/F a quaternion algebra which is unramified at all finite places anddenote by r ∈ {0, . . . , d} the number of archimedan places where B is split.Define the modular variety associated to an open compact subgroup U of

to be the complex variety:

X(U) := UC∞+\GA/GQ.

By strong approximation,

X(U) 'h(U)∐λ=1

Γλ(U)\Hr.

Suppose that E is a (right or left) O[Γλ(U)]-module for all λ. Denote by Ethe coefficient system on X(U) associated to E. Then

Hr(X(U), E) 'h(U)⊕λ=1

Hr(Γλ(U)\Hr, E) 'h(U)⊕λ=1

Hr(Γλ(U), E).

For any ω ∈ Hr(X(U), E), write ωλ for its projection to Hr(Γλ(U)\Hr, E).

Definition 3.1. The group of modular symbols on X(U) associated to E isthe group Hr

cpt(X(U), E) of cohomology with compact support.

Suppose that E is a right tλ∆0(npα)t−1λ ∩GQ for all λ. Define an action

of the Hecke algebra R(S(pα),∆0(npα)) by the formula:

ωλ(z)|T (x) :=∑j

ωλ(αλ,jz)|αλ,j ∈ Hr(Γµ(pα)\Hr, E)

(same notations as in §2.2). Equivalently, identifying ωλ with a d-cocycle bythe above isomorphism, T (x) can be defined as in [2] by the formula:

(ωλ)|T (x)(γ0, . . . , γd) :=∑j

ωλ(tj(γ0), . . . , tj(γd))|αλ,j,

where tj : Γµ(pα) → Γλ(pα) are defined for γ ∈ Γµ(pα) by the equationsΓλαλ,jγ = Γλαλ,l and αλ,jγ = tj(γ)αλ,l. The group of modular symbolsHr

cpt(X(S(pα)), E) is an R(S(pα),∆0(npα))-module if E is.

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If r ≥ 1, the modular symbol ω(f) associated to f ∈ Sk,w,IB(S(pα),B,C)can be described as follows. For any ring R, let L(n,R) be the R-moduleof homogeneous polynomials in 2r variables X = (Xσ)σ∈I , and Y = (Yσ)σ∈Iof degree nσ in Xσ, Yσ. Recall that γ 7→ γ∗ denotes the main involution ofM2(R) and define a right action of GL2(F )r on L(n,C) by

(P |γ)(X, Y ) := det(γ)vP((X, Y )γ∗

)where (X, Y )γ∗ is matrix multiplication and the usual multi-index notationsare used. Denote by L(n, v,C) the right representation of B obtained com-posing the above action with the injection B× ↪→ GL2(F )r. The differentialform

ω(fλ)(z) := f(z)(zX + Y )k−2dz

(usual multi-index notations) satisfies the transformation formula for anyγ ∈ Γλ(pα):

(ω(fλ))(γ(z))|γ = ω(fλ)(z)

hence, by [11], ω(fλ) ∈ Hr(Γλ(pα)\Hd,L(n, v,C)), where L(n, v,C) is thecoefficient system on Γλ(pα)\Hr associated to L(n, v,C). If B = M2(F ),since fλ is a cusp form, it can be proved that ω(fλ) has compact support.Hence, we always have:

ω(f) := (ω(f1), . . . , ω(fh)) ∈ Hrcpt

(X(S(pα),L(n, v,C)

).

For any character ε as above, write L(n, v, ε, R) for the ∆0(n)-moduleL(n, v, R) with the action of ∆0(n) twisted by ε, that is, denoting by |ε thisnew action: P |εγ := ε(γ)P |γ for γ ∈ ∆0(n). If fλ ∈ Sk,w(S0(pα), ε,C), thenω(fλ) ∈ Hr(Γλ0(pα),L(n, v, ε,C)), where L(n, v, ε,C) is the coefficient systemassociated to L(n, v, ε,C). Hence

ω(f) ∈ Hrcpt

(X(S0(pα)),L(n, v, ε,C)

).

A straightforward calculation shows that the map f 7→ ω(f) is equivariantfor the action of R(S(pα),∆0(npα)).

3.2 Λ-adic modular symbols

We assume in this section that r ≤ 1, so that B is a division algebra. Recallalso that B is assumed to be unramified at all finite places of F .

Define e = eS := S ∩ F×. Then Zα = (rp/pαrp)

×/e, so G = r×p × r×p /e,where e is the closure of e in r×p . It follow that G ' (r×p × r×p )/e via the map

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(x, y) 7→ (xy, y). Embed diagonally e in GL2(rp) and call C the image. LetN be the standard lower unipotent subgroup of GL2(rp). Define the sets:

Xα := S(pα)r×\Sr×, X := lim←−Xα. (12)

To describe these sets, let Nα = ( a bc d ) ∈ GL2(rp) such that a, d = 1 mod pα

and c = 0 mod pα. We have a surjective map S(pα)\S → Xα. Note thatS(pα)\S = Nα\GL2(rp), and that the kernel of the above map is Nα\Nαr

×,so we have a bijection of sets Nαr

×\GL2(rp)→Xα. Finally, since NC isthe closure of ∩Nαr

× in rp, we see that X is identified canonically withNC\GL2(rp).

Let Fp = F ⊗QQp =∏

p|p Fp and Y := N(Fp)C\GL2(Fp), where N(Fp) isthe group of lower triangular matrices with entries in Fp and diagonal entries1. Write rp =

∏p|p rp, where rp is the completion of r at p. Define (r2

p)′ to be

the set of primitive vectors of r2p, that is, the pair of elements (a, b) ∈ r2

p suchthat at least one of a and b does not belong to p. Set (r2

p)′ :=

∏p|p(r

2p)′. The

map

g =

(a bc d

)7→((a, b), det(g)

)defines a bijection between X and e\((r2

p)′× r×p ), where the action of e ∈ e is

e · ((x, y), z) = ((ex, ey), e2z).

Define πp to be the element in GL2(F ) whose p-component is(

1 00 p

)and

is 1 outside p. Note that πp normalizes N(Fp) =∏

p′|pN(Fp′) because(

1 00 p

)normalizes N(Fp). Hence, it is possible to define an action of πp on Y byletting πp act on its p-component as:

N(Fp)g ∗ πp = N(Fp)π−1p gπp.

Identify Y with e\((F 2p )′ × F×p ), where (F 2

p )′ := {(x, y) ∈ F 2p : xy 6= 0} via

the map γ = ( a bc d ) 7→ ((a, b), det(γ)). Then

((x, y), z) ∗ πp = ((x, py), z),

where for any y =∏

p′|p yp, write py :=∏

p′|p,p′ 6=p yp′ × pyp. In particular, πpdoes not affect the determinant of the matrix.

Let Gπ be the semigroup generated by GL2(rp) and πp for all divisors pof p. Using that any element s ∈ Gπ can be expressed as a word in termsof GL2(rp) and πp, and that the actions of the πp’s commute, extend the ∗action to Gπ by letting any πp act through ∗ and elements of GL2(rp) throughright multiplication, so that

N(Fp)g ∗ s = N(Fp)∏p|p

π−c(p)p gs

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for any N(Fp)g ∈ Y and s ∈ Gπ, where c(p) is the number of times πpappears in the expression of s. Since this number does not depend on thespecific expression we chose, the action of Gπ on Y is well defined.

Let Y ′ denote the smallest subset of Y containing X and stable underGπ. Define DX (respectively, DY ′) to be the O-module of O-values measureson X (respectively, on Y ′). For γ ∈ Gπ and µ ∈ DY ′ , define µ ∗ γ by theintegration formula:∫

Y ′ϕ(η)d(µ ∗ γ)(η) :=

∫Y ′ϕ(η ∗ γ)dµ(η),

where ϕ is any Cp-valued step function on Y ′. Denote by DXi−→ DY ′ the

canonical inclusion defined by extending measures by zero and by DY ′p−→ DX

the canonical projection map. If µ ∈ D and s ∈ Gπ, define

µ ∗ s = p(i(µ) ∗ s).

Since by [2, Lemma 3.1] the kernel of p is stable under Gπ, the action is welldefined.

By the choice of tλ in §2, Γλ(pα) ⊆ GL2(rp) for all λ. Hence, DX is a rightO[Γλ(pα)]-module for all λ. Denote by DX the coefficient system associatedto DX and set:

W := Hr(X(S),DX).

Remark 3.2. Since SO2(R) is compact and isomorphic to the unit circle C1

and the Γλ are discrete, the stabilizer (Γλ)z0 of any element z0 ∈ Hr is a finite

cyclic group. Since the groups Γλ

are torsion-free, it follows that if γ ∈ Γλ

stabilizes an element z0 ∈ Hr, then γ belongs to the center of Γλ and henceacts trivially on DX . It follows that the sheaf DX is well-defined.

Lemma 3.3. Let Z be a topological space that is an inverse limit of finitediscrete topological spaces Zα for α in some indexing set. Then the space ofO-valued measures on Z is isomorphic to lim←−Fns(Zα,O) where Fns(Zα,O) =

OZα is the space of continuous O-valued functions on Zα.

Proof. If φ ∈ lim←−Fns(Zα,O), then µ can be written as a compatible sequenceof the form {

∑x∈Zα ax ·x}α. Let pα : Z → Zα be the natural projection map

and for each x ∈ Zα set Uα,x = p−1α (x). Then this is isomorphic to the space

of O-valued measures on Z by the map

φ 7−→ µ such that µ(Uα,x) = ag. (13)

This defines a measure due to the compatibility of the sequence.

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For each α, let pα : X(S(pα)) → X(S). Let Fα = pα∗O be the directimage of the constant sheaf O on X(S). Fix a point x ∈ X(S) and defineYα = Yα,x to be the fiber p−1

α (x) of x under pα. By Lemma 3.3, lim←−OYα is the

space of O-valued measures on the space lim←−Yα. Now, for the double cosetGL2(F )xSS∞ in X(S), there is a natural map

S(pα)\S = S(pα)p\GL2(rp) −→ Yα

given byS(pα)pz 7−→ S(pα)S∞xzGL2(F ).

This map induces a bijection from S(pα)(r× ∩ SS∞)\GL2(rp) to Yα. Hencethe inverse limit lim←−Yα can be identified with NC\GL2(rp) which is finallyidentified with X.

Lemma 3.4. The sheaves DX and lim←−α

Fα on X(S) are isomorphic.

Proof. Let U ∈ X(S) be an open set. For each α and w ∈ Yα, let Xα,w ⊂ Xbe the inverse image of w under the natural projection map X → Yα. Letu ∈ U , then choose x ∈ GL2(FA) such that u = SS∞xGL2(F ). If s ∈ DX(U)is a section in DX , we can express s(u) = SS∞(x, µ(x))GL2(F ) for someµ ∈ DX , depending on s. In fact, since s is a locally constant section, thisexpression is valid in a neighborhood Uu of u. Then define a map:

DX(U)φα(U)−−−→ Fα(U) = O(p−1

α (U))s 7−→ u 7→

∑w∈Yα µ(x)(Xα,w).

This map is independent of the choice of x since a different representativex′ of the double coset u would yield the same measure µ(x) because s is asection on X(S).

The compatible maps φα then give rise to a map

φ(U) : DX(U)→ (lim←−α

Fα)(U).

At the level of the stalks this map is the isomorphism in (13). It follows thatφ is an isomorphism of sheaves.

Proposition 3.5. We have a canonical isomorphism:

W ∼= lim←−Hr(X(S(pα)),O).

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Proof. If r = 0 the result is obvious so we can assume that r = 1. We knowfrom lemma 3.4 and [5, Theorem 4.5] that there is a short exact sequence

0 −→ lim←−(1){H0(X(S(pα)),O)} →W→ lim←−H

1(X(S(pα)),O)→ 0.

So we need to show that the kernel is zero. Let Hα = H0(X(S(pα)),O) and

we know that Hα ≡ ⊕H0(Γλ(pα),O). The action of Γ

λ(pα) on O is trivial so

the zeroth cohomology group is just O. Hence if α ≥ β, the transition mappαβ : Hα → Hβ in the inverse system {Hα} is just the inflation map and is

given by multiplication by [Γλ(pβ) : Γ

λ(pα)] in the λth component. If we fix β

then the filtration given by images of Hα in Hβ, for α ≥ β, stabilizes if p doesnot divide the indices above and is a sub-filtration of O ⊃ πO ⊃ π2O . . .otherwise. In the first case {Hα} satisfies the Mittag-Leffler condition and inthe second case O is complete under the filtration. Hence from [5, Corollary4.3] in the first case and [5, Proposition 4.2] in the second case, the firstderived functor of inverse limit is 0 in either case.

Since tλ∆0(n)t−1λ ∩ GQ ⊆ Gπ for all λ = 1, . . . , h, it follows that W

is an RO(S,∆0(n)) := O ⊗Z R(S,∆0(n))-module. Let G act on X by leftmultiplication. Define G′ to be the multiplicative subset of (r × r)/r× con-sisting of pair of elements (x, y) such that x and y are prime to p. The map(a, d) 7→ ω(an+2v

p )−1T ( a 00 d ) for (a, d) ∈ G′ considered in §2.2 is multiplica-

tive, hence extends to a O-algebra homomorphism O[G′] → RO(S,∆0(n)).

On the other hand, G′ ⊆ G, hence O[G′] embeds naturally on Λ = O[[G]].

Form the Λ-algebra

H := RO(S,∆0(n))⊗O[G′] Λ.

Since the action of G′ on W extends to a continuous action of G, it followsthat W is an H-module.

From the fact that h2t,t(S(pα),O) is generated overO by T (q) for all primeideals q and those operators coming from the action of Gα, it follows thatthere is are surjective homomorphisms of Λ-algebras H → h2t,t(S(p∞),O)and H → R.

Define a subset X ′ of X as follows:

X ′ =

{x = NC

(a bc d

)∈ X

∣∣∣∣ a ∈ r×p

}. (14)

It is easy to check that the definition does not depend on the choice ofthe representative matrix used to define it and that X ′ can be identifiedwith the set e\(r×p × rp × r×p ) under the above identification between X and

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e\((r2p)′× r×p ). From now on denote elements of X by ((x, y), z), where (x, y)

is the first arrow on the matrix and z is its determinant.Let Pn,v,ε be an arithmetic point of weight (n, v) and character ε factoring

through Zα. Define the specialization map ρn,v,ε : DX → L(n, v, ε,O) at Pby:

µ 7−→ ρn,v,ε(µ) :=

∫X′zvε(x)(xY − yX)ndµ(x, y, z).

Suppose that the conductor of ε is pα for some non negative integer α. Asimple computation shows that

ρn,v,ε(µ ∗ γ) = ρn,v,ε(µ)|εγ

for γ ∈ GL2(rp) ∩ ∆0(pα). It follows that the specialization map ρn,v,ε isGL2(rp) ∩ ∆0(pα)-equivariant. Letting K := Frac(O), there are GL2(rp) ∩∆0(pα)-equivariant maps:

ρn,v,ε : W −→Wn,v,ε := Hr(X(S(pα)),L(n, v, ε,O)).

Proposition 3.6. Let Φ ∈W.

1. For any prime ideal q of r prime to p: ρn,v,ε(Φ∗T (q)) = (ρn,v,ε(Φ))|T (q).

2. ρn,v,ε(Φ ∗ T (p)) = (ρn,v,ε(Φ))|T0(p).

Proof. The equivariance for the action of Hecke operators T (q) in the firststatement is immediate because of the GL2(rp)∩∆0(pα)-equivariance of ρn,v,ε.It remains to check the action of T (p). To see this, write ΓλπΓλ =

∐t Γλαλ,t

and note that

ρn,v,ε(Φ ∗ T0(p)) =

∫X′

∑t

zvε(x)(xY − yX)n d(Φ ∗ αλ,t)

=

∫X′

∑t

zvε(x)(xY − yX)n d(π−1Φαλ,t)

=∑t

∫X′{pv}−1zvε(x)(xY − yX)n|αλ,t d(Φ)

= ρn,v,ε(Φ)|T0(p).

This proves the second formula.

Note that

W =h∏λ=1

Wλ, where Wλ = Hr(Γλ\Hr,DX)

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Wn,v,ε =h∏λ=1

Wλn,v,ε, where Wλ

n,v,ε := Hr(Γλ

0(pα)\Hr,L(n, v, ε,K)).

Any element Φ ∈ W will be written as (Φλ)λ=1,...,h while any element ofω ∈Wn,v,ε will be denoted as (ωλ)λ=1,...,h. Define:

Φλ,θn,v,ε := ρn,v,ε(Φλ) ∈Wλn,v,ε. (15)

3.3 The Control Theorem

In this section we will state and prove a control theorem for W. We willassume throughout this section that r ≤ 1, so that B is a division algebra;also recall that B is assumed to be unramified at all finite places.

Fix θ : R → L (where L = Frac(Λ)) and denote as in §2.3 by I theintegral closure of Λ in Im(θ). Recall the specialization map

θκ := κ ◦ θ : R −→ Qp

which corresponds to an eigenform fκ. Let Pκ be the kernel of θκ. Set

Wκ :=h∏λ=1

Wλκ, with Wλ

κ := Hr(Γλ

0(pα)\Hr,L(n, v, ε,Kκ))

whereKκ := fraction field of the image of θκ.

The map θκ extends to the localizationRPκ ofR at Pκ and one can intertwineθκ with the map ρκ defining

ρκ : W×RPκ −→Wκ (16)

by ∑λ

Φλ × rλ 7−→∑λ

ρκ(Φλ) · θκ(rλ).

This is well defined because, if g belongs to the free part W of G and isrepresented by the matrix ( a 0

0 d ) ∈ GL2(rp), then we have for µ ∈ DX ,

ρκ(gµ) = ρκ(g)ρκ(µ).

Since ρκ(φg, r) = ρκ(φ, gr) for g ∈ W and by continuity the same is true forany element in Λ, the map (16) induces a homomorphism

ρκ : W⊗Λ RPκ −→Wκ

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which is Hecke equivariant.For any H-module M , let Mord denote its ordinary part, that is, the

maximal subspace of M on which the T (p) operator acts as a unit. Leth : H → R be the natural map obtained by the action of Hecke operators onΛ-adic cusp forms. For any arithmetic point κ, let hκ be the composition ofh with the localization morphism R → RPκ . For any H⊗Λ RPκ-module M ,let

Mhκ = {m ∈M | (T (q)⊗1)m = (1⊗hκ(T (q)))·m for all prime ideals q in r}

denote the hκ-eigenspace of M . If fκ is a classical eigenform for an arithmeticpoint κ, let

Wfκκ = {φ ∈Wκ| T0(q)φ = aq(g)φ}

denote the fκ-eigenspace of Wκ, where aq(g) is the eigenvalue of the T0(q)operator on fκ. Hence, there is a map:

ρκ : (W⊗Λ RPκ)hκ −→Wfκκ .

In the case r = 1 we also have an action of complex conjugation τ onX(S(pα)). The action of τ corresponds to the action on X(S(pα)) deducedfrom the function z 7→ −z on the complex upper half plane H by the canonicalprojection H → X(S(pα)) (here z is the complex conjugate of z). For anymodule M over which τ acts and for each sgn ∈ {±1}, let M sgn denote thesgn-eigenmodule of M for the action of τ .

We are now ready to state our main theorem. We need to make theassumption that the generators of Pκ are regular in RPκ or equivalently, sinceRPκ is a local ring, that RPκ is regular. This condition is a generalizationof the notion of primitive eigneforms to the nearly ordinary setting. SeeRemark 3.8 for a more precise discussion of this issue.

Theorem 3.7. Let κ ∈ A(I) be an arithmetic point of weight (nκ, vκ) andcharacter εκ. Assume that RPκ is a regular local ring. Then the map:

ρκ : (W⊗Λ RPκ)hκ/Pκ(W⊗Λ RPκ)hκ −→Wfκκ

is an isomorphism. If r = 1, then for each sgn ∈ {±1} we also have anisomorphism:

ρκ : (W⊗Λ RPκ)hκ,sgn/Pκ(W⊗Λ RPκ)hκ,sgn −→Wfκ,sgnκ .

Remark 3.8. As mentioned before Theorem 3.7, the condition that RPκ isa regular local ring is a suitable generalization of the concept of primitiveeigenforms in the nearly ordinary setting. To be more precise, note that,

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since Pn,v is generated by a regular sequence in Λ, then Pκ is generatedby a regular sequence in R if the extension of local rings RPκ/ΛPn,v,ε isunramified. In the case F = Q, the assumption that f is a primitive formimplies that the corresponding prime Pκ is unramified over Pn,0,ε (recall thatin the rational case one always has v = 0). Similar conditions are also true inthe case of families of Hilbert modular forms of parallel weight thanks to [13,§12.7.7]. As an alternative choice of Pκ, one could consider the reduced Heckealgebra Rred generated by operators T (q) and T (q, q) for primes q - pn. ThenRred/Pn,v,εRred is a local factor of a product of fields by the semisemplicityof Rred and Hida’s control theorem. So it is a field and Rred

Pκis a regular ring.

The primes of R containing the Hecke operators dividing the tame level thusverify the property required in Theorem 3.7.

The proof of this Theorem will be given in §3.7. Before explaining theproof, we need some preliminary results, stated in §§3.4, 3.5 and 3.6.

3.4 Freeness of WRecall that we are assuming in this section that r ≤ 1, so that B is a divisionalgebra. We start by providing a different description of W. Define

Vα := Hr(X(S(pα)), K/O), V∞ := lim−→α

Vα,

V∗α := HomO(Vα, K/O), V∗∞ := HomO(V∞, K/O)

where the direct limit is computed with respect to the projection mapsX(S(pβ))→ X(S(pα)) for β ≥ α. The Hecke algebras hk,w(S0(p∞), ε,O) de-fined in §2.3 can be equivalently introduced as lim←−

α

h′k,w(S0(pα), ε,O), where

h′k,w(S0(pα), ε,O) is the image in EndO(Hr(X(S(pα)),L(n, v, ε,K/O)) of thealgebra generated over O by the Hecke operators; the same observation holdsfor the ordinary parts: see [8, §3] for details.

By Lemma 3.4,

W = Hr(X(S),DX) ' Hr(X(S), lim←−Fα).

SinceHr(X(S),Fα) = Hr(X(S), pα∗O) ' Hr(X(S(pα)),O),

there is an isomorphism

W ∼−→ lim←−α

Hr(X(S(pα)),O). (17)

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By Poincare duality:

Hr(X(S(pα)),O) ' Hr(X(S(pα)),O). (18)

By Pontryagin duality there is a canonical isomorphism:

HomO(Hr(X(S(pα)), K/O), K/O) ' Hr(X(S(pα)),O), (19)

where K := Frac(O). Since

V∞ = lim−→α

Hr(X(S(pα)), K/O)

there is an isomorphism

HomO(

lim−→α

Hd(X(S(pα)), K/O), K/O) ∼−→ V∗∞. (20)

By composing the maps (17), (18), (19), (20), we get an isomorphism

W '−→ V∗∞. (21)

The isomorphism (21) is equivariant for the action of the Hecke operatorsand, if r = 1, also for the action of the complex conjugation τ .

Proposition 3.9. The group Word of ordinary Λ-adic modular symbols is aΛ-module free of finite rank.

Proof. This follows from Proposition 3.5 and [7, Th 3.8] in light of (21).

3.5 Description of Ker(ρκ)

Proposition 3.10. The kernel of ρn,v,ε is equal to Pn,v,εWord.

Proof. This result is [2, Theorem 5.1], so only a sketch of the proof will begiven.

Recall that a cocycle in Hq(Γλ,DX) is represented by the class of anelement in HomΓλ(Fq,DX), where F• is a projective resolution of Z overZ[Γλ] by right Γλ-modules. Let ϕ(m) denote the characteristic functions ofπmY , for m ≥ 0 an integer.

We first prove that Ker(ρn,v,ε) ⊇ Pn,v,εWord. Let Φ ∈ Pn,v,εWord andwrite Φ = (Φ1, . . . ,Φh). Fix λ and represent Φλ by a cocycle z as above. Itfollows from [2, Lemma 6.3] that

∫Xϕ(m)dz(f) = 0 for all f ∈ F λ

d and all

characteristic functions ϕ(m). Since the function((x, y), z

)7−→ ε(x)zv(xY − yX)k

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appearing in the specialization map ρn,v,ε can be written as an uniform limitof functions ϕ(m), the inclusion follows.

Now we show that Ker(ρn,v,ε) ⊆ Pn,v,εWord. Let c ∈ HomΓλ(F λk ,DX) and

choose b such that c = T (pm)b: this is possible because T (p) induces anisomorphism on Word and, since p is a principal ideal of r, the T (p) operator

preserves each of the cohomology groups Hr(Γλ,DX). Set π :=

(1 00 p

). Write

ΓλπmΓλ =∐

t Γλαλ,t and γλ,t := π−mαλ,t. By [2, Lemma 6.1],∫X

ϕ(m)(y) dc(f)(y) =∑t

∫X

ϕ(m)(y ∗ αλ,t)db(fγ−1λ,t )(y).

Since Xm ∩ X ∗ αλ,t = ∅ for αλ,t 6= 1 by [2, Lemma 6.6], the above sum isequal to ∫

Xm

db(fγ−1λ,1)(y) = ρ∗n,v,ε(Φλ)(fγ

−1λ,1) = 0.

From [2, Lemma 6.3] if follows that b takes values in Pn,v,εDX . Hence, b

belongs to the image of Hr(Γλ, Pn,v,εDX) in Hr(Γ

λ, DX) which, by [2, Lemma

1.2], is equal to Pn,v,εHr(Γ

λ,DX).

3.6 Dimension bounds

We retain the notations and assumptions from previous sections, so r ≤ 1and B is a division algebra. Let Pκ be an arithmetic point such that RPκ

is a regular local ring. Let RPκ denote the localization of R at Pκ. SetKPκ := Frac(RPκ). Let V∗,ord

∞ denote the ordinary submodule of V∗∞. Forany arithmetic character Pn,v,ε which factors through RPκ , set V∗,ord

∞,Pn,v,ε :=

V∗,ord∞ ⊗Λ RPn,v,ε .

Proposition 3.11. The RPn,v,ε-module V∗,ord∞,Pn,v,ε is free of rank 2r. If r = 1

then for each sgn ∈ {±1}, its sgn-eigenmodule is free of rank one.

Proof. Since RPn,v,ε is a flat extension of ΛPn,v,ε and V∗,ord∞ is free of finite

rank, say s, over Λ, it follows that V∗,ord∞,Pn,v,ε is free of rank s over RPn,v,ε .

If r = 0 define V := V∗,ord∞,Pn,v,ε . If r = 1 then V∗,ord

∞,Pn,v,ε splits into the directsum of two submodules

V∗,ord∞,Pn,v,ε = V+ ⊕ V− (22)

such that the complex conjugation τ acts on V± as ±; in this case, let Vdenote one of the two modules V±. By [7, §4, p. 187] there is an isomorphism:

V/Pn,v,εV'−→ hord

k,w(S0(pα), ε,K).

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By (11), hordk,w(S0(pα), ε,K) is free of rank one over RPn,v,ε/Pn,v,εRPn,v,ε . Thus

V/Pn,v,εV ' RPn,v,ε/Pn,v,εRPn,v,ε .

So the Rn,v,ε-rank of V is one and the result follows.

Corollary 3.12. The hκ-eigenspace of WL⊗LKPκ has dimension at least 2r

over KPκ.

Proof. As in the proof of Proposition 3.11, let V := V∗,ord∞,Pn,v,ε if r = 0 and

V := Vsgn for sgn ∈ {±1} if r = 1, where Vsgn is defined in (22). Set

Vκ := V ⊗RPn,v,ε RPκ .

We have an Hecke-equivariant map of finite dimensional KPκ-vector spaces

WL ⊗L KPκ −→ Vκ ⊗RPκ KPκ .

If r = 1, this map is also equivariant for the action of complex conjugationτ . Since H if r = 0 (respectively, H[τ ] if r = 1) is commutative and thehκ-eigenspace of Vκ ⊗RPκ KPκ in non-trivial, being of dimension 1 over KPκ ,it follows that the hκ-eigenspace (respectively, the (hκ, sgn)-eigenspace) inWL⊗LKPκ is non-zero too (for this linear algebra argument, see [15, Lemma5.10]). This concludes the proof in both cases.

3.7 Proof of the Control Theorem

Before proving Theorem 3.7, recall the following Lemma. Let Ih := Ker(h)be the kernel of the canonical map h : H → R.

Lemma 3.13. Let M be an H-module and P be an ideal in Λ. Suppose thatP is generated by an M-regular sequence (x1, . . . , xr). Then the image of themap

i∗ : Ext∗H(H/Ih, PM) −→ Ext∗H(H/Ih,M)

induced by the inclusion i : PM →M is equal to PExt∗H(H/Ih,M).

Proof. The proof of this lemma is a slight modification of that of [2, Lemma1.2] and will be omitted.

Remark 3.14. In order to apply this lemma in the proof of the main theo-rem, we first note that for any RPκ-module M with a H-action, the Heckeeigenspace Mhκ is equal to HomH(RPκ ,M) and hence to Ext0

H(R,M). Sincewe assume that the generators of Pκ form a regular sequence in RPκ , it fol-lows that they form a regular sequence for M = W⊗RPκ since it is free overRPκ .

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Proof of the Control theorem. The proof of the Control Theorem follows [4]and can be described as follows. Recall that we have to show that for eachprimitive arithmetic point κ of weight (n(κ), v(κ)) and character ε(κ) themap

ρκ : (W⊗Λ RPκ)hκ/Pκ(W⊗Λ RPκ)hκ −→Wκ

is an isomorphism.Since hκ(T (p)) is a unit in R, the module (W⊗Λ RPκ)hκ is contained in

the ordinary part Word ⊗Λ RPκ . Since, by Proposition 3.10, Word ⊗Λ ΛPκ

is a free ΛPκ-module of finite rank, it follows that Word ⊗Λ RPκ is a freeRPκ-module of finite rank. By Proposition 3.10 and the fact that RPκ isfree over ΛPκ , it follows that the kernel of the map Word ⊗Λ RPκ → Word

0 isPκ(W⊗ΛRPκ)ord. This is the same as Pκ(W⊗ΛRPκ)hκ by Lemma 3.13 andthe remark following it. Hence, we get an injective map

(W⊗Λ RPκ)hκ/Pκ(W⊗Λ RPκ)hκ ↪−→Wfκκ .

Since Wfκκ is 2r-dimensional, to prove the surjectivity of the map it suf-

fices to show that (W ⊗Λ RPκ)hκ has RPκ-rank at least 2r. Recall that byCorollary 3.12 the hκ-eigenspace of WL ⊗L KPκ has dimension 2r over KPκ .The intersection of this eigenspace with W ⊗Λ RPκ is a RPκ-submodule of(W⊗Λ RPκ)hκ of rank 2r. The surjectivity of the above map follows.

Finally, in the case r = 1, since the specialization map commutes with theaction of the complex conjugations, we deduce the statement on eigenspaces.

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