The Tamagawa Number Conjecture of Adjoint Motives of Modular Forms

Ann. Scient. Éc. Norm. Sup., 4e série, t. 37, 2004, p. 663 à 727.

THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES OF MODULAR FORMS

BY FRED DIAMOND, MATTHIAS FLACH AND LI GUO

ABSTRACT.–Letf be a newform of weight k 2,levelN with coefﬁcients in a number ﬁeld K, and A the adjoint motive of the motive M associated to f. We carefully discuss the construction of the realisations of M and A, as well as natural integral structures in these realisations. We then use the method of Taylor and Wiles to verify the λ-part of the Tamagawa number conjecture of Bloch and Kato for L(A, 0) and L(A, 1).Hereλ is any prime of K not dividing Nk!, and so that the mod λ representation associated to f is absolutely irreducible when restricted to the Galois group over Q( (−1)(−1)/2) where λ | . The method also establishes modularity of all lifts of the mod λ representation which are crystalline of Hodge–Tate type (0,k− 1).  2004 Elsevier SAS

RÉSUMÉ.–Soientf une forme nouvelle de poids k, de conducteur N, à coefﬁcients dans un corps de nombres K,etA le motif adjoint du motif M associé à f. Nous présentons en détail les réalisations des motifs M et A avec leurs réseaux entiers naturels. En utilisant les méthodes de Taylor–Wiles nous prouvons la partie λ-primaire de la conjecture de Bloch–Kato pour L(A, 0) et L(A, 1).Iciλ est une place de K ne divisant pas Nk! et telle que la représentation modulo λ associée à f, restreinte au groupe de Galois du corps Q( (−1)(−1)/2) avec λ | , est irréductible. Notre méthode démontre aussi la modularité de toutes les représentations λ-adiques cristallines de type de Hodge–Tate (0,k − 1) congrues à la représentation associée à f modulo λ.  2004 Elsevier SAS

0. Introduction

This paper concerns the Tamagawa number conjecture of Bloch and Kato [4] for adjoint motives of modular forms of weight k 2. The conjecture relates the value at 0 of the associated L-function to arithmetic invariants of the motive. We prove that it holds up to powers of certain “bad primes”. The strategy for achieving this is essentially due to Wiles [88], as completed with Taylor in [86]. The Taylor–Wiles construction yields a formula relating the size of a certain module measuring congruences between modular forms to that of a certain Galois cohomology group. This was carried out in [88] and [86] in the context of modular forms of weight 2,where it was used to prove results in the direction of the Fontaine–Mazur conjecture [40]. While it was no surprise that the method could be generalized to higher weight modular forms and that the resulting formula would be related to the Bloch–Kato conjecture, there remained many technical details to verify in order to accomplish this. In particular, the very formulation of the conjecture relies on a comparison isomorphism between the -adic and de Rham realizations of the motive provided by theorems of Faltings [31] or Tsuji [87], and veriﬁcation of the conjecture requires

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 0012-9593/05/ 2004 Elsevier SAS. All rights reserved 664 F. DIAMOND, M. FLACH AND L. GUO the careful application of such a theorem. We also need to generalize results on congruences between modular forms to higher weight, and to compute certain local Tamagawa numbers.

0.1. Some history

Special values of L-functions have long played an important role in number theory. The underlying principle is that the values of L-functions at integers reflect arithmetic properties of the object used to define them. A prime example of this is Dirichlet’s class number formula; another is the Birch and Swinnerton–Dyer conjecture. The Tamagawa number conjecture of Bloch and Kato [4], refined by Fontaine, Kato and Perrin-Riou [55,41,38], is a vast generalization of these. Roughly speaking, they predict the precise value of the first non-vanishing derivative of the L-function at zero (hence any integer) for every motive over Q. This was already done up to a rational multiple by conjectures of Deligne and Beilinson; the additional precision of the Bloch–Kato conjecture can be thought of as a generalized class number formula, where ideal class groups are replaced by groups defined using Galois cohomology. Dirichlet’s class number formula amounts to the conjecture for the Dedekind zeta function for a number field at s =0or 1. The conjecture is also known for Dirichlet L-functions (including the Riemann zeta function) at any integer [62,4,52,8,35]. It is known up to an explicit set of bad primes for the L-function of a CM elliptic curve at s =1if the order of vanishing is 1 [11,69,58]. There are also partial results for L-functions of other modular forms at the central critical value [45,56,59,64,89] and for values of certain Hecke L-functions [49,48,57]. For a more detailed survey of known results we refer to [35]. Here we consider the adjoint L-function of a modular form of weight k 2 at s =0and 1. Special values of the L-function associated to the adjoint of a modular form, and more generally, twists of its symmetric square, have been studied by many mathematicians. A method of Rankin relates the values to Petersson inner products, and this was used by Ogg [65], Shimura [81], Sturm [84,85], Coates and Schmidt [10,73] to obtain nonvanishing results and rationality results along the lines of Deligne’s conjecture. Hida [51] related the precise value to a number measuring congruences between modular forms. In the case of forms corresponding to (modular) elliptic curves, results relating the value to certain Galois cohomology groups (Selmer groups) were obtained by Coates and Schmidt in the context of Iwasawa theory, and by one of the authors, who in [34] obtained results in the direction of the Bloch–Kato conjecture. A key point of Wiles’ paper [88] is that for many elliptic curves, modularity could be deduced from a formula relating congruences and Galois cohomology [88]. This formula could be regarded as a primitive form of the Bloch–Kato conjecture for the adjoint motive of a modular form. His attempt to prove it using the Euler system method introduced in [34] was not successful except in the CM case using generalizations of results in [47] and [70]. Wiles, in work completed with Taylor [86], eventually proved his formula using a new construction which could be viewed as a kind of “horizontal Iwasawa theory”. In this paper, we refine the method of [88] and [86], generalize it to higher weight modular forms and relate the result to the Bloch–Kato conjecture. Ultimately, we prove the conjecture for the adjoint of an arbitrary newform of weight k 2 up to an explicit finite set of bad primes. We should stress the importance of making this set as small and explicit as possible; indeed the refinements in [22,13,5] which completed the proof of the Shimura–Taniyama–Weil conjecture can be viewed as work in this direction for weight two modular forms. In this paper, we make use of some of the techniques introduced in [22] and [13], as well as the modification of Taylor– Wiles construction in [24] and [42]. One should be able to improve our results using current technology in the weight two case (using [13,13,72]), and in the ordinary case (using [28,82]); one just has to relate the results in those papers to the Bloch–Kato conjecture. Finally we remark

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 665 that Wiles’ method has been generalized to the setting of Hilbert modular forms by Fujiwara [42], Skinner–Wiles [83] (using Shimura curves) and Dimitrov [29] (using Hilbert modular varieties). Dimitrov’s work allows to relate the Selmer group with the special value of the adjoint L-function of the Hilbert modular form but to verify the Bloch–Kato conjecture it remains to relate the appearing period with a motivic period.

0.2. The framework

The Bloch–Kato conjecture is formulated in terms of “motivic structures,” a term referring to the usual collection of cohomological data associated to a motive. This data consists of: • vector spaces M?, called realizations, for ?=B, dR and for each rational prime , each with extra structure (involution, filtration or Galois action); • comparison isomorphisms relating the realizations; • a weight filtration. Suppose that f is a newform of weight k 2 and level N. Much of the paper is devoted to the construction of the motivic structure Af for which we prove the conjecture. This construction is not new; it is due for the most part to Eichler, Shimura, Deligne, Jannsen, Scholl and Faltings [79,15,53,75,31]. We review it however in order to collect the facts we need and set things up in a way suited to the formulation of the Bloch–Kato conjecture. For proofs of results not readily found in the literature, we direct the reader to [25]. Let us briefly recall here how the construction works. We start with the modular curve XN parametrizing elliptic curves with level N structure. Then one takes the Betti, de Rham and -adic cohomology of XN with coefficients in a sheaf defined as the (k − 2)nd symmetric power of the relative cohomology of the universal elliptic curve over XN . These come with the additional structure and comparison isomorphisms needed to define a motivic structure MN,k,the comparison between -adic and de Rham cohomology being provided by a theorem of Faltings [31]. The structures MN,k can also be defined as in [75] using Kuga-Sato varieties; this has the advantage of showing they arise from “motives” and provides the option of applying Tsuji’s comparison theorem [87]. However the construction using “coefficient sheaves” is better suited to defining and comparing lattices in the realizations which play a key role in the proof. The structures MN,k also come with an action of the Hecke operators and a perfect pairing. The Hecke action is used to “cut out” a piece Mf , which corresponds to the newform f and has rank two over the field generated by the coefficients of f. The pairing comes from Poincaré duality, is related to the Petersson inner product and restricts to a perfect pairing on Mf .We 0 finally take the trace zero endomorphisms of Mf to obtain the motivic structure Af =ad Mf . The construction also yields integral structures Mf and Af , consisting of lattices in the various realizations and integral comparison isomorphisms outside a set of bad primes. Our presentation of the Bloch–Kato conjecture is much influenced by its reformulation and generalization due to Fontaine and Perrin-Riou [41]. Their version assumes the existence of a category of motives with conjectural properties. Without assuming conjectures however, they define a category SPMQ(Q) of premotivic structures whose objects consist of realizations with additional structure and comparison isomorphisms. The category of mixed motives is supposed to admit a fully faithful functor to it, and a motivic structure is an object of the essential image. Their version of the Bloch–Kato conjecture is then stated in terms of Ext groups of motivic structures, but whenever there is an explicit “motivic” construction of (conjecturally) all the relevant extensions, the conjecture can be formulated entirely in terms of premotivic structures. This happens in our case, for all the relevant Ext’s conjecturally vanish. There will therefore be no further mention of motives in this paper. We make several other slight modifications to the framework of [41]:

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 666 F. DIAMOND, M. FLACH AND L. GUO

• We use premotivic structures with coefficients in a number field K, as in [38]. • We forget about the -adic realization and comparison isomorphisms at a finite set of “bad” primes S. • We work with S-integral premotivic structures. This yields a version of the conjecture which predicts the value of L(Af , 0) up to an S-unit in K. The conjecture is independent of the choice of integral structures, but the formalism is convenient and certain lattices arise naturally in the proof. We make our set S explicit: Let Sf be the set of finite primes λ in K such that either: • λ | Nk!,or • M M the two-dimensional residual Galois representation f,λ/λ f,λ is not absolutely irre- (−1)/2 ducible when restricted to GF ,whereF = Q( (−1) ) and λ | . Note that since Sf includes the set of primes dividing Nk!, we will only be applying Faltings’ comparison theorem in the “easy” case of crystalline representations whose associated Dieudonné module has short filtration length.

0.3. The main theorems

Our main result can be stated as follows.

THEOREM 0.1 (= Theorem 2.15). – Let f be a newform of weight k 2 and level N with coefﬁcients in K.Ifλ is not in Sf , then the λ-part of the Bloch–Kato conjecture holds for Af and Af (1). The main tool in the proof is the construction of Taylor and Wiles, which we axiomatize (Theorem 3.1), and apply to higher weight forms to obtain the following generalization of their class number formula.

THEOREM 0.2 (= Theorem 3.7). – Let f be a newform of weight k 2 and level N with coefficients in K. Suppose Σ is a finite set of rational primes containing those dividing N. Suppose that λ is a prime of K which is not in Sf and does not divide any prime in Σ.Then the OK,λ-module 1 A HΣ(GQ,Af,λ/ f,λ) Σ has length vλ(ηf ). Σ Here ηf , defined in Section 1.7.3, is a generalization of the congruence ideal of Hida and Wiles; it can also be viewed as measuring the failure of the pairing on Mf to be perfect on Mf . Another consequence of Theorem 0.2, is the following result in the direction of Fontaine- Mazur conjecture [40].

THEOREM 0.3 (= Theorem 3.6). – Suppose ρ : GQ → GL2(Kλ) is a continuous geometric representation whose restriction to G is ramiﬁed and crystalline and its associated Dieudonné module has ﬁltration length less than − 1. If its residual representation is modular and absolutely irreducible restricted to Q( (−1)(−1)/2) where λ | ,thenρ is modular.

Contents

0 Introduction ...... 663 0.1 Somehistory...... 664 0.2 Theframework...... 665 0.3 Themaintheorems...... 666

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 667

1 The adjoint motive of a modular form ...... 668 1.1 Generalities and examples of premotivic structures ...... 668 1.1.1 Premotivicstructures...... 668 1.1.2 Integralpremotivicstructures...... 669 1.1.3 Basicexamples...... 671 1.2 Premotivic structures for level N modular forms ...... 673 1.2.1 Level N modular curves ...... 673 1.2.2 Bettirealization...... 674 1.2.3 -adicrealizations...... 674 1.2.4 deRhamrealization...... 674 1.2.5 Crystalline realization ...... 676 1.2.6 Weightﬁltration...... 676 1.3 The action of GL2(Af ) ...... 677 1.3.1 Action on modular forms ...... 677 1.3.2 Actiononpremotivicstructures...... 677 1.4 The premotivic structure for forms of level N and character ψ ...... 678 1.4.1 σ-constructions...... 678 1.4.2 Premotivic structure of level N and character ψ ...... 679 1.5 Duality...... 679 1.5.1 Duality at level N ...... 679 1.5.2 Duality for σ-constructions...... 681 1.5.3 Duality for level N, character ψ ...... 682 1.6 Premotivicstructureofanewform...... 682 1.6.1 Heckeaction...... 682 1.6.2 Premotivicstructureforaneigenform...... 683 1.6.3 The L-function...... 684 1.7 Theadjointpremotivicstructure...... 685 1.7.1 Realizations of the adjoint premotivic structure ...... 685 1.7.2 Eulerfactorsandfunctionalequation...... 686 1.7.3 Variationofintegralstructures...... 686 1.8 Reﬁnedintegralstructures...... 687 1.8.1 Σ-levelstructure...... 688 1.8.2 Heckeactionandlocalization...... 689 1.8.3 Ihara’sLemma...... 691 1.8.4 Comparisonofintegralstructures...... 693

2 The Bloch–Kato conjecture for Af and Af (1) ...... 694 2.1 Galois cohomology ...... 695 2.2 Orderofvanishing...... 699 2.3 Deligne’speriod...... 700 2.4 Bloch–Katoconjecture...... 707

3 TheTaylor–Wilesconstruction...... 715 3.1 Anaxiomaticformulation...... 715 3.2 Consequences...... 721 Acknowledgements...... 724

References...... 724

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 668 F. DIAMOND, M. FLACH AND L. GUO

1. The adjoint motive of a modular form

1.1. Generalities and examples of premotivic structures

1.1.1. Premotivic structures For a field F , F¯ will denote an algebraic closure, and GF =Gal(F/F¯ ). We fix an embedding Q¯ → Q¯ p for each prime p, and an embedding Q¯ → C.IfF is a number field, we let IF denote the set of embeddings F → Q¯ , which we identify with the set of embeddings F → C via our fixed one of Q¯ in C.

We write Gp for GQp , Ip for the inertia subgroup of Gp,andFrobp for the geometric Frobenius ∼ ⊂ element in Gp/Ip = GFp . We identify Ip Gp with their images in GQ. If K is a number field, then Sf (K) denotes the set of finite places of K. Suppose that λ ∈ Sf (K) divides ∈ Sf (Q).LetBdR = BdR, and Bcrys = Bcrys, be the rings defined by Fontaine [37, §2], [41, I.2.1]. Suppose that V is a finite-dimensional vector space over Kλ with a ⊗ G continuous action of G (i.e., a λ-adic representation of G). Then DdR(V )=(BdR Q V ) is ⊗ G a filtered finite-dimensional vector space over Kλ,andDcrys(V )=(Bcrys Q V ) is a filtered finite-dimensional vector space over Kλ equipped with a Kλ-linear endomorphism φ.The representation V is called de Rham if dimKλ DdR(V )=dimKλ V ,andV is called crystalline if dimKλ Dcrys(V )=dimKλ V . We recall that if V is crystalline, then V is de Rham. A λ-adic representation V of GQ is pseudo-geometric [41, II.2] if it is unramified outside of a finite number of places of Q and its restriction to G is de Rham. The representation V is said to have good reduction at p if its restriction to Gp is crystalline (resp. unramified) if p = (resp. p = ). We work with categories of premotivic structures based on notions from [41,38,4]. For a number field K,weletPMK denote the category of premotivic structures over Q with coefficients in K. In the notation of [41, III.2.1], this is the category SPMQ(Q) ⊗ K of K-modules in SPMQ(Q). Thus an object M of PMK consists of the following data: • a finite-dimensional K-vector space MB with an action of GR; i • a finite-dimensional K-vector space MdR with a finite decreasing filtration Fil , called the Hodge filtration; • for each λ ∈ Sf (K), a finite-dimensional Kλ vector space Mλ with a continuous pseudo- geometric action of GQ; • a C ⊗ K-linear isomorphism

∞ I : C ⊗ MdR → C ⊗ MB

respecting the action of GR (where GR acts on C ⊗ MB diagonally and acts on C ⊗ MdR via the ﬁrst factor); • for each λ ∈ Sf (K),aKλ-linear isomorphism

λ ⊗ → IB : Kλ K MB Mλ

respecting the action of GR (where the action on Mλ is via the restriction GR → GQ determined by our choice of embedding Q¯ → C); • ∈ ⊗ for each λ Sf (K),aBdR, Q Kλ-linear isomorphism

λ ⊗ ⊗ → ⊗ I : BdR, Q Kλ K MdR BdR, Q Mλ

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 669

respecting ﬁltrations and the action of GQ (where is the prime which λ divides, Kλ and

Mλ are given the degree-0 filtration, Kλ and MdR are given the trivial GQ -action and the action on Mλ is determined by our choice of embedding Q¯ → Q¯ ); i • increasing weight filtrations W on MB, MdR and each Mλ respecting all of the above data, and such that R ⊗ MB with its Galois action and weight filtration, together with the Hodge filtration on C ⊗ MB defined via IB, defines a mixed Hodge structure over R (see [41, III.1]). S If S Sf (K) is a set of primes of K,weletPMK denote the category defined in exactly the S same way, but with Sf (K) replaced by the complement of S.IfS ⊆ S ,weuse· to denote the S S forgetful functor from PMK to PMK . S ⊗ The category PMK is equipped with a tensor product, which we denote K , and an internal hom, which we denote HomK . There is also a unit object, which we denote simply by K.These are defined in the obvious way; for example, (M ⊗K N)B is the K[GR]-module MB ⊗K NB. K If K ⊆ K ,weletS denote the set of primes in Sf (K ) lying over those in S,thenK ⊗K · S SK defines a functor from PMK to PMK . S ∈ If M is an object of PMK , then for each prime p and each λ/S, we can associate a representation of the Deligne–Weil group of Qp (see [16, §8]), which we denote by WDp(Mλ). For λ not dividing p, the representation is over Kλ;forλ dividing p,wehavethatMλ|Gp is potentially semistable [3, Theorem 0.7], so the construction in [41, I.2.2] yields a representation Qur ⊗ | over p Qp Kλ. We recall that Mλ Gp is crystalline if and only if WDp(Mλ) is unramified (in the sense that the monodromy operator and the inertia group act trivially), in which case Qur ⊗ ⊗ −1 S WDp(Mλ)= p Qp Dcrys(Mλ) with Frobp acting via 1 φ . An object M of PMK • has good reduction at p if WDp(Mλ) is unramified for all λ/∈ S; • is L-admissible at p if the Frobenius semisimplifications of WDp(Mλ),forλ/∈ S,form a compatible system of K-rational representations of the Deligne–Weil group of Qp (see [16, §8]); • is L-admissible everywhere if it is L-admissible at p for all primes p. If M is L-admissible at p, then the local factor associated to WDp(Mλ) is of the form P (p−s)−1 for some polynomial P (u) ∈ K[u] independent of λ not in S. For an embedding −s τ : K → C we put Lp(M,τ,s)=τP(p ) and we regard the collection {Lp(M,τ,s)}τ∈I as a ∼ K meromorphic function on C with values in CIK = K ⊗ C.IfS is finite and M is L-admissible everywhere, then its L-function L(M,s):= Lp(M,s) p is a holomorphic K ⊗ C-valued function in some right half plane Re(s) >r(with components L(M,τ,s)= p Lp(M,τ,s)). 1.1.2. Integral premotivic structures Before introducing integral premotivic structures, we recall some of the theory of Fontaine and Laffaille [39]. We let MF denote the category whose objects are finitely generated Z-modules equipped with • a decreasing filtration such that Fila A = A and Filb A =0for some a, b ∈ Z, and for each i i ∈ Z, Fil A is a direct summand of A; • Z i i → ∈ Z i| i+1 i -linear maps φ :Fil A A for i satisfying φ Fili+1 A = φ and A = Im φ . It follows from [39, 1.8] that MF is an abelian category. Let MFa denote the full subcategory of objects A satisfying Fila A = A and Fila+ A =0and having no non-trivial quotients A such a+−1 MF a MF a that Fil A = A ,andlet tor denote the full subcategory of consisting of objects

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 670 F. DIAMOND, M. FLACH AND L. GUO

MF 0 f, of finite length. So tor is the category denoted MFtor in [39], and it follows from [39, 6.1] MF a MF a that and tor are abelian categories, stable under taking subobjects, quotients, direct products and extensions in MF. MF 0 Fontaine and Laffaille define a contravariant functor U S from tor to the category of finite continuous Z[G]-modules and they prove it is fully faithful [39, 6.1]. We let V denote the V Q Z functor defined by (A)=Hom(U S(A), / ) and we extend it to a fully faithful functor on MF0 by setting V(A) = projlim V(A/nA).ThenV defines an equivalence between 0 MF and the full subcategory of Z[G]-modules whose objects are isomorphic to quotients of the form L1/L2,whereL2 ⊂ L1 are finitely generated submodules of short crystalline representations. Here we define a crystalline representation V to be short if the following hold • 0 ⊗ G Fil D = D and Fil D =0,whereD =(Bcrys Q V ) ; • ⊗ Q − if V is a nonzero quotient of V ,thenV Q ( 1) is ramified. In particular, the essential image of V is closed under taking subobjects, quotients and finite direct sums. Furthermore, one sees from [39, 8.4] that the natural transformations Q ⊗ → ⊗ V G Z A Bcrys Z (A) , ⊗ → ⊗ V (1) Bcrys Z A Bcrys Z (A) and 0 ⊗ φ=1 → Q ⊗ V Fil (Bcrys Z A) Z (A) are isomorphisms. a If K is a number field and λ ∈ Sf f(K) is a prime over ,weletOλ = OK,λ and let Oλ-MF a 0 denote the category of Oλ-modules in MF . We can regard V as a functor from Oλ-MF to the category of Oλ[G]-modules. O MF0 ⊗ O MF0 If A and A are objects of λ- such that A Oλ A defines an object of λ- ,then there is a canonical isomorphism

V ⊗ ∼ V ⊗ V (A Oλ A ) = (A) Oλ (A ).

Analogous assertions hold for HomOλ (A, A ). PMS We now define a category K of S-integral premotivic structures as follows. We let O O ∈ ∈ M PMS S = K,S denote the set of x K with vλ(x) 0 for all λ/S. An object of K consists of the following data: • a finitely generated OK -module MB with an action of GR; i • a finitely generated OS -module MdR with a finite decreasing filtration Fil , called the Hodge filtration; • for each λ ∈ Sf (K), a finitely generated Oλ-module Mλ with continuous action of GQ M ⊗ inducing a pseudo-geometric action on λ Oλ Kλ; 0 • for each λ/∈ S, an object Mλ-crys of Oλ-MF ; • an R ⊗OK -linear isomorphism

∞ I : C ⊗MdR → C ⊗MB

respecting the action of GR; • for each λ in Sf (K), an isomorphism

λ M ⊗ O ∼ M IB : B OK λ = λ

respecting the action of GR;

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 671

• for each λ/∈ S,anOλ-linear isomorphism

λ M ⊗ O ∼ M IdR : dR O λ = λ-crys

respecting ﬁltrations; • for each λ/∈ S,anOλ-linear isomorphism

λ I : V(Mλ-crys) →Mλ

respecting the action of GQ ,where is the prime which λ divides; i • increasing weight ﬁltrations W on Q ⊗MB, Q ⊗MdR and each Q ⊗Mλ respecting all of the above data and giving rise to a mixed Hodge structure. With the evident notion of morphism this becomes an OK -linear abelian category. Note also Q ⊗· PMS S Q ⊗M Q ⊗M that there is a natural functor from K to PMK ,whereweset( )? = ? for ?=B, dR and λ for λ/∈ S, with induced additional structure and comparison isomorphisms. (The comparison Iλ for Q ⊗Mis deﬁned as the composite

⊗ O ⊗ M ∼ ⊗ M → ⊗ V M → ⊗ M BdR, Z λ O dR = BdR, Z λ-crys BdR, Z ( λ-crys) BdR, Z λ,

λ λ where the maps are respectively, IdR, the canonical map (1) and I , each with scalars extended to BdR,.) ⊂ ·S PMS PMS M If S S , we define a functor from K to K in the obvious way. We say that is MS M ⊗ O O M S -flat if dR = dR OK,S K,S is flat over K,S . Note that if is S -flat,thensoisany M O ⊗ · subobject of .LetK be a finite extension of K. We also have a natural functor K OK PMS PMSK from K to K . We say that M has good reduction at p,isL-admissible at p or is L-admissible everywhere according to whether the same is true for Q ⊗M. Note that if M is L-admissible at p and p is not invertible in O,thenM necessarily has good reduction at p. M M PMS M⊗ M PMS M ⊗ For objects and of K , we can form OK in K provided λ-crys Oλ M MF 0 ∈ λ-crys defines an object of λ for all λ/S. In particular this holds if there exist positive a M a M integers a, a such that Fil dR =0, Fil dR =0and a + a <for all primes not O N N PMS N⊗ N invertible in .If and are objects of K such that OK is as well, and if M→N M →N PMS α : and α : are morphisms in K , then there is a well-defined morphism ⊗ M⊗ N→M ⊗ N PMS α α : OK OK in K . Analogous assertions hold for the formation and M M properties of HomOK ( , ). M PMS M O Note that if is an object of K ,thenEnd is a finitely generated K -module. If I is O M M PMS an K -submodule of End , then we define an object [I] of K as the kernel of

r (x1,...,xr):M→M where x1,...,xr generate I. This is independent of the choice of generators. This applies in particular when I is the image in EndM of an ideal in a commutative OK -algebra R mapping to EndM, or the augmentation ideal in OK [G] where G is a group acting on M. In the latter case, we write MG instead of M[I].

1.1.3. Basic examples 1 The object Q(−1) in PMQ is the weight two premotivic structure defined by H (Gm). To give an explicit description, let ε denote the generator of Z (1) = lim µ n (Q¯ ) defined by ←− 2πi/n (e )n via our fixed embedding Q¯ → C.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 672 F. DIAMOND, M. FLACH AND L. GUO

• T 1 G C Z ∼ −1Z ⊂ C − Let B = HB( m( ), ) = (2πi) with complex conjugation in GR acting by 1, and let Q(−1)B = Q ⊗TB . • T 1 G Z Z − Let dR = HdR( m/ ), which with its Hodge filtration is isomorphic to [ 1] (where [n] denotes a shift by n in the filtration, so Fili V [n]=Fili+n V ). Write ι for the canonical dx T ∼ Z − Q − Q ⊗T 1 Q − Q basis x of dR = [ 1] and let ( 1)dR = dR,soFil ( 1)dR = ι and 2 Fil Q(−1)dR =0. • T 1 G Z ∼ Z Z Z Q − Let = Het( m,Q¯ , ) = HomZ ( (1), )= δ where δ(ε)=1,andlet ( 1) = Q ⊗T. 1 • Let T-crys denote the object of MF defined by Z ⊗TdR = Z ι with φ (ι)=ι. ∞ −1 • I : C ⊗Z TdR → C ⊗Z TB is defined by 1 ⊗ ι → 2πi ⊗ (2πi) . • Z ⊗T →T ⊗ −1 → IB : B is defined by 1 (2πi) δ. • T ⊗ Z →T → IdR : dR Z -crys is given by ι ι . • ⊗ Q − → ⊗ Q − ⊗ → ⊗ I : BdR, [ 1] BdR, Q ( 1) is defined by 1 ι t δ where t = log[ε] (see e.g., [41, I.2.1.3]). 0 ∼ • For >2, T-crys is an object of MF and I is induced by an isomorphism V(T-crys) = T. {2} The above data defines objects in PMQ and PMQ which we denote by Q(−1) and T . These could be described equivalently by H2(P1), or indeed H2(X) for any smooth, proper, geometrically connected curve X over Q. The Tate premotivic structure Q(1) is the object of PMQ defined by HomQ(Q(−1), Q) and Q(n) is defined by Q(1)⊗n for integer n.WehaveL(Q(n),s)=ζ(s + n) where ζ is the Riemann ζ-function. More generally, for any object M in PMK and integer n, M(n) is defined ⊗n S,⊗n S as M ⊗Q Q(1) . For any integer n 0, T defines an object of PMQ where S is any set ⊗ ∼ of primes containing those dividing (n +1)!; note also that Q ⊗TS, n = Q(−n)S. For any number field F ⊂ Q¯ ,letMF denote the premotivic structure MF of weight zero defined by H0(Spec F ), called the Dedekind premotivic structure of F . To give an explicit description, let S denote the set of primes dividing D =Disc(F/Q).Welet IF IF •MF,B = Z with the natural action of GR,andMF,B = Q ⊗MF,B = Q . (Recall that we identified IF with the set of embeddings F → C via the chosen embedding Q¯ → C,so −1 for α : IF → Z and σ ∈ GR,wedefineσα by τ → α(σ ◦ τ)); 0 1 •MF,dR = OF [1/D] with Fil MF,dR = MF,dR and Fil MF,dR =0,andMF,dR = Q ⊗MF,dR = F ; •M Z ⊗M ZIF Q ⊗M QIF F, = F,B = with the natural action of GQ,andMF, = F, = −1 (so for α : IF → Z, σ ∈ GQ and τ : F → Q¯ ,wehave(σα)(τ)=α(σ ◦ τ)); •MF,-crys = Z ⊗MF,dR = Z ⊗OF for /∈ S, with the same filtration as MF,dR and 0 with φ = φ. ∞ ∞ ⊗ The comparisons IB and IdR are identity maps, I is defined by I (1 x)(τ)=τ(x) after IF identifying C ⊗ MF,B with C ,andI is defined similarly. We thus obtain objects MF of S PMQ and MF of PMQ,andL(MF ,s) is the ζ-function of F .IfF is Galois over Q, then there is a natural action of G =Gal(F/Q) on MF , where for α ∈MF,B, g ∈ G and τ ∈ IF ,wehave gα by τ → α(τ ◦ g). × × × Let ψ : Zˆ → K be a character, regarded also as a character of A and GQ via the Zˆ× ∼ A× R× Q× ∼ ab isomorphisms = / >0 = GQ , where the first isomorphism is induced by the natural inclusion Zˆ× → A× and the second is given by class field theory. (Our convention is that a Q× Q uniformizer in p maps to Frobp in the Galois group of any abelian extension of unramified at p.) If F is a Galois extension of Q such that ψ is trivial on the image of GF in GQ, then we can regard ψ as a character of G =Gal(F/Q) and define the Dirichlet premotivic structure Mψ as G (V ⊗ MF ) where V = K with G acting by ψ. The construction is independent of the choice of 2πi/N F and embedding of F in Q¯ . To describe Mψ explicitly, we choose F = Q(e ) ⊂ C where

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ψ has conductor N.Weletτ0 : F → Q¯ denote the embedding compatible with our fixed Q¯ → C, × ∼ and we regard ψ as a Dirichlet character via the canonical isomorphism (Z/N Z) = G.LetS M PMS denote the set of primes in K lying over those dividing N and define an object ψ of K V⊗M G V OS by ( F ) where = K with G acting by ψ.Wethenhave: •M O OIF ◦ → −1 ψ,B is the K -submodule of K spanned by the map bB defined by τ0 g ψ (g), where τ0 is the inclusion of F in Q¯ ; •Mψ,dR is the O = OK [1/N ]-submodule of OK ⊗OF [1/N ] spanned by 2πia/N bdR = ψ(a) ⊗ e , a

× 1 0 where a runs over (Z/N Z) with Fil Mψ,dR =0and Fil Mψ,dR = Mψ,dR; •M O ⊗ M ψ,λ = λ OK ψ,B with GQ acting via ψ; • M O ⊗ M M 0 for λ N, ψ,λ-crys = λ OK ψ,dR with the same ﬁltration as ψ,dR and φ = ψ−1(). The comparison isomorphisms are induced from those of MF . Similarly, we get the object Mψ of PMK by setting Mψ,? = Q ⊗Mψ,? with comparison isomorphisms induced from ∞ ⊗ ⊗ those of MF . In particular, we have I (1 bdR)=Gψ(1 bB) where Gψ is the Gauss sum 2πia/N ⊗ C ⊗ a e ψ(a) in K. Q ⊗M ∼ S We have that ψ = Mψ , Mψ has good reduction at all primes not dividing the conductor −1 of ψ and is L-admissible everywhere, and L(Mψ,s) is the Dirichlet L-function L(ψ ,s).

1.2. Premotivic structures for level N modular forms

In this section we review the construction of premotivic structures associated to the space of modular forms of weight k and level N. More precisely, if k 2 and N 3, we construct objects S of PMQ whose de Rham realization contains the space of such forms, where S = SN is the set of primes dividing Nk!.

1.2.1. Level N modular curves These premotivic structures are obtained from the cohomology of modular curves. Let k and N be integers with k 2 and N 3.LetT =SpecZ[1/N k!], and consider the functor which associates to a T -scheme T the set of isomorphism classes of generalized elliptic curves over T with level N structure [18, IV.6.6].By [18, IV.6.7],the functor is represented by a smooth, proper curve over T . We denote this curve by X¯,andwelets¯: E¯ → X¯ denote the universal generalized elliptic curve with level N structure. We let X denote the open subscheme of X¯ over which E¯ is smooth. Then X is the complement of a reduced divisor, called the cuspidal divisor,whichwe ∞ −1 | ∞ ¯ × ∞ denote by X .WeletE =¯s X, s =¯s E and E = E X¯ X . Using the arguments of [18, VII.2.4], one can check that E¯ is smooth over T and E∞ is a reduced divisor with strict normal crossings (in the sense of [46, 1.8] as well as [1, XIII.2.1]). Let us also recall the standard description of

s¯an : E¯an → X¯ an, where we use an to denote the associated complex analytic space. We let XN = XN,t, t∈(Z/N Z)×

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 674 F. DIAMOND, M. FLACH AND L. GUO where for each t, XN,t denotes a copy of Γ(N)\H, the quotient of the complex upper half-plane H, by the principal congruence subgroup Γ(N) of SL2(Z). Similarly we let X¯N = X¯N,t, t∈(Z/N Z)×

¯ ¯ alg where XN,t is the compactification of XN,t obtained by adjoining the cusps. We write XN for the corresponding algebraic curve over C. × For each t ∈ (Z/N Z) , we define the complex analytic surface EN,t to be a copy of the quotient Γ(N)\ (H × C)/(Z × Z) , (m, n) ∈ Z × Z H × C (τ,z) → (τ,z + mτ + n) γ = ab ∈ Γ(N) where acts on via ,and cd acts −1 by sending the class of (τ,z) to that of (γ(τ), (cτ + d) z). We can regard EN = EN,t as a complex analytic family of elliptic curves over XN with level N-structure defined by the pair of sections (τ,τ/N) and (τ,t/N) on XN,t. We can then extend EN to a family E¯N of generalized elliptic curves with level N-structure over X¯N using analytic Tate curves, as in [18, VII.4]. One checks that E¯N is algebraic, so E¯N → X¯N is the analytification of a generalized elliptic ¯alg → ¯ alg ¯ alg → ¯ → an curve EN XN . The resulting morphism XN XC induces an isomorphism XN X . The analytification of the universal generalized elliptic curve with level N-structure is therefore isomorphic to E¯N → X¯N with the level N-structure defined above.

1.2.2. Betti realization 1 an an To construct the Betti realization, define FB as the locally constant sheaf R s∗ Z on X .Let F k k−2 F B =SymZ B, where our convention for defining symmetric powers is to take coinvariants M 1 an F k M 1 an F k under the symmetric group. We then define B = H (X , B),and c,B = Hc (X , B). an F k Identifying X with XN as above, we find that B is identified with the locally constant sheaf defined by Γ(N)\(H × Symk−2 Z2) where Γ(N) acts on Z2 by left-multiplication. It follows that i an F k ∼ i k−2 Z2 (2) H (X , B) = H Γ(N), Sym . t∈(Z/N Z)×

In particular, it follows easily that MB has no -torsion if k =2or does not divide N(k − 2)!. an The actions of complex conjugation in GR on MB and Mc,B are induced by its action on E an and X .WeletMB = Q ⊗MB and Mc,B = Q ⊗Mc,B.

1.2.3. -adic realizations 1 k k−2 For any ﬁnite prime ,weletF denote the -adic sheaf R s∗Z on X.LetF =Sym F , Z M 1 F k M 1 F k M M = H (XQ¯ , ) and c, = Hc (XQ¯ , ).ThenGQ acts on and c, by transport of structure. We let M = Q ⊗M and Mc, = Q ⊗Mc,. A standard construction using the comparison between classical and étale cohomology [2, XI.4.4, XVII.5.3] yields the ∼ ∼ isomorphisms Z ⊗MB = M and Z ⊗Mc,B = Mc,.

1.2.4. de Rham realization The construction of the de Rham realization is similar to the one given in [74] except we use the language of log schemes [54]. We let NE (resp. NX ) denote the log structure on E¯ (resp.

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X¯ ) associated to E∞ (resp. X∞) [54, 1.5]. By [54, 3.5, 3.12], we have an exact sequence of O coherent locally free E¯ -modules → ∗ 1 → 1 → 1 → (3) 0 s¯ ωX/T¯ ωE/T¯ ωE/¯ X¯ 0, where ω1 denotes the sheaf of logarithmic relative differentials defined in [54, 1.7]. The sheaves 1 1 1 ∞ ωX/T¯ and ωE/¯ X¯ are invertible, and can be identified, respectively, with ΩX/T¯ (X ) and the F sheaf of regular differentials for s¯ (denoted ωE/¯ X¯ in [18, I.2.1]). Define dR as the locally free 1 • • 1 ∗ O ¯ O → sheaf R s¯ ωE/¯ X¯ of X¯ -modules on X,whereωE/¯ X¯ is the complex d : E¯ ωE/¯ X¯ .Thishas 2 1 1 0 F F ∗ F F a decreasing filtration with Fil dR =0, Fil dR =¯s ωE/¯ X¯ ,andFil dR = dR. We denote 1 k k−2 Fil FdR simply as ω.WedefineF as the filtered sheaf of O ¯ -modules SymO FdR,andwe dR X X¯ F k F k − ∞ let c,dR = dR( X ). The (logarithmic) Gauss–Manin connection

1 ∇ F →F ⊗O : dR dR X¯ ωX/T¯

F k F k induces logarithmic connections on dR and c,dR satisfying Griffiths transversality. We set M 1 ¯ • F k M 1 ¯ • F k • G dR = H (X,ω ( dR)) and c,dR = H (X,ω ( c,dR)), where we write ω ( ) for the complex associated to the module G with its connection. The filtrations on MdR and Mc,dR F k F k Q ⊗M Q ⊗M are defined by those on dR and c,dR.WeletMdR = dR and Mc,dR = c,dR. 1 0 ∼ −1 2 ∼ 1 ∗ F Letting ω =¯s ωE/¯ X¯ ,wehavegr dR = ω by Grothendieck–Serre duality and ω = ωX/T¯ 2 ∼ 1 by the Kodaira–Spencer isomorphism [18, VI.4.5.2]. It follows that ω = ωX/T¯ , and one deduces that   0 ¯ k−2 ⊗ 1 − H (X,ω ωX/T¯ ), if i = k 1; i M ∼ − gr dR =  H1(X,ω¯ 2 k), if i =0; 0, otherwise. Similarly one finds   0 ¯ k−2 ⊗ 1 − H (X,ω ΩX/T¯ ), if i = k 1; i M ∼ − ∞ gr c,dR =  H1(X,ω¯ 2 k(−X )), if i =0; 0, otherwise. k−1 ⊗(k−2) Pulling back to t H and trivializing by (2πi) (dz) dτ yields an isomorphism k−1 (4) α : C ⊗ Fil MdR → Mk Γ(N) t∈(Z/N Z)× where Mk(Γ(N)) is the space of modular forms of weight k with respect to Γ(N).Bythe q-expansion principle [18, VII], the map 1/N Mk Γ(N) → C[[q ]] t∈(Z/N Z)× t∈(Z/N Z)×

2πiτ/N 1/N k−1 that sends f(τ)=g(e ) to g(q ) identiﬁes Fil MdR as the subset of Mk Γ(N) t∈(Z/N Z)×

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 676 F. DIAMOND, M. FLACH AND L. GUO ∞ Z C whose q-expansion at has coefficients in [1/N k!,µN ], which we view as a subring of t 2πit/N via the embedding defined by (e )t. The same assertions hold with MdR replaced by Mc,dR and Mk(Γ(N)) by Sk(Γ(N)), the subspace of cusp forms. To construct of the comparison isomorphisms I∞, apply GAGA [76] and the Poincaré Lemma • an to conclude that the pull-back of ω (FdR) to X defines a resolution of C ⊗FB .Taking C⊗Fk C⊗ F k an → ¯ an symmetric powers then provides resolutions of B and j! B where j : X X ,and taking cohomology yields the desired comparisons. We refer the reader to [25] for more details.

1.2.5. Crystalline realization We define the crystalline realization using the language of logarithmic crystals as in [31]. Suppose that Y¯ is a smooth, proper scheme over Spec Z with a relative divisor D with strict normal crossings, and let Y = Y¯ − D. For each integer a with 0 a − 2, Faltings [31, MF ∇ ¯ Z §2i)] defines a category [0,a](Y ). In the case of Y = Y =Spec , Faltings’ category can be MF0 a+1 identified with the full subcategory of tor whose objects A satisfy Fil A =0. Assuming F MF ∇ does not divide 2N,let -crys denote the inverse system in [0,1](XZ ) defined by reduction n mod of FdR with its filtration, logarithmic Gauss–Manin connection and locally defined − F k Frobenius maps. Assuming further that >k 1,welet -crys denote the inverse system in ∇ k−2 MF − (XZ ) defined by SymO F crys.If>k, we obtain an object [0,k 2] Y¯ -

1 k M Z F -crys = Hcrys(X , -crys)

0 of MF whose underlying filtered module can be identified with Z ⊗MdR (see [31, §4c)]). Similarly, taking cohomology with compact support (in the sense of [31]) yields an object Mc,-crys whose underlying filtered module is Z ⊗Mc,dR. The above identifications provide the comparison isomorphisms IdR. The construction of the comparisons I relies on Faltings’ comparison theorem between -adic and crystalline D MF∇ cohomology. Faltings [31, Theorem 2.6] defines a functor from [0,a](Y ) to the category of finite locally constant étale sheaves on YQ so that V =Hom(D(·), Q /Z ) coincides with ∼ that of [39] for Y =SpecZ.If Nk!,thenwehaveV(F-crys) = F by [31, Theorem 6.2], V F k ∼ F k V M ∼ M V M ∼ M so ( -crys) = by [31, IIh], giving ( -crys) = and ( c,-crys) = c, by [31, Theorem 5.3].

1.2.6. Weight ﬁltration There is a natural map Mc,? →M? for each realization respecting all of the data and comparison isomorphisms. Setting   0, if i0 is chosen to annihilate the torsion in MB and M[r] denotes the kernel of multiplication by r on

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S M.) We then let M! denote the premotivic structure im(Mc →Mtf ) in PMQ, pure of weight k − 1.WeletM! = Q ⊗M!.

1.3. The action of GL2(Af )

In this section we deﬁne the adelic action on premotivic structures associated to modular forms.

1.3.1. Action on modular forms We first recall the adelic definition of modular curves and forms. Suppose that U is an open compact subgroup of GL2(Af ) where Af denotes the finite adeles. Let U∞ denote the stabilizer × of i in GL2(R),soU∞ = R SO2(R). The analytic modular curve XU of level U is defined as the quotient

GL2(Q)\ GL2(A)/UU∞.

The analytic structure is characterized by requiring that if g is in GL2(Af ), then the map H → XU → Q ∈ + R deﬁned by γ(i) GL2( )gγUU∞,γ GL2 ( ), is holomorphic. −k If φ :GL2(A) → C is such that φ(δxuv)=detv (ci + d) φ(x) for all δ ∈ GL2(Q), ab ∈ A ∈ ∈ ∞ → C x GL2( ), u U and v = cd U , then we deﬁne φg : H −1 k ab ∈ + R φg(γ(i)) = (det γ) (ci + d) φ(gγ) for γ = cd GL2 ( ).

We say that such a function φ is a modular form of level U if for all g ∈ GL2(Af ), φg is a −1 ∩ + Q modular form of weight k with respect to gUg GL2 ( ). We denote this space by Mk(U), and similarly deﬁne Sk(U), the space of cusp forms of level U. Suppose now that U and U are open compact subgroups of GL2(Af ),andg is an element of −1 GL2(Af ) such that g U g ⊂ U. Note that right multiplication by g induces a holomorphic map XU → XU , and inclusions Mk(U) → Mk(U ) and Sk(U) → Sk(U ). We thus obtain an action of GL2(Af ) on A = lim M (U) and A0 = lim S (U). k → k k → k U U

Suppose now that U = UN for some N 3,whereUN ⊂ GL2(Af ) is the kernel of the Zˆ → Z Z ∈ Z Z × reduction map GL2( ) GL2( /N ). For each class t ( /N ) , we choose an element ∈ Zˆ Z Z 10 gt GL2( ) whose image in GL2( /N ) is 0 t−1 . We identify XN with XU via the maps ηt : XN,t → XU deﬁned by

Γ(N) · γ(i) → GL2(Q) · gtγ · UU∞ ∈ + R for γ GL2 ( ). We identify Mk(U) with t Mk(Γ(N)) via the isomorphism β deﬁned by

β(φ)t = φgt . (Note that η and β are independent of the choices of the gt.) We thus obtain isomorphisms

− − (5) β−1 ◦ α : C ⊗ Filk 1 M ∼ M (U) and A ∼ C ⊗ lim Filk 1 M(N) , dR = k k = → dR N where α was deﬁned in (4).

1.3.2. Action on premotivic structures For h ∈ GL2(Af ) and integers N,N 3, we call (h, N, N ) an admissible triple if both h and −1 −1 −1 N N h ∈ M2(Zˆ).If(h, N, N ) is an admissible triple, then N|N and h UN h ⊂ UN .Let

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 678 F. DIAMOND, M. FLACH AND L. GUO

E/¯ X¯ (resp. E¯/X¯ ) denote the universal generalized elliptic curve with level N (respectively −1 N ) structure. Note that right multiplication by N h ∈ M2(Zˆ) defines an endomorphism of ¯ Z Z 2 E [N ]=( /N )/X .WedefineG to be its image which is a finite flat subgroup of E . Right multiplication by N −1N h−1 defines an injective map

(Z/N Z)2 → (Z/N Z)2/((Z/N Z)2(N h−1)

which gives rise to a level N-structure on E /G extending to (E¯ /G)cont, the contraction of ¯ ¯ E /G whose cuspidal ﬁbers are N-gons [18, IV.1].By the universal property of E/X¯ , this deﬁnes ¯ → ¯ ¯ → ¯ × ¯ amapX X such that (E /G)cont E X¯ X as generalized elliptic curves with level N-structures. Composing with the natural map E¯ → (E¯ /G)cont,wegetacommutativediagram

E¯ E¯ (6)

X¯ X.¯

−1 Suppose that g ∈ M2(Zˆ) ∩ GL2(Af ) with g UN g ⊆ UN . One can then factor g = rh so that r ∈ Z and (h, N, N ) is admissible. Suppose that S is a set of primes containing those ∅ M M S M M S dividing N k!.For = , c and !, we write = (N) and = (N ) for the objects S of PMQ defined in Section 1.2.6. For ?=B, dR, and -crys with N k!,weusethetop F F row of (6) to define compatible maps from ? to the pullback of ? along the bottom row, take M →M symmetric products and then take cohomology, yielding morphisms [h] : .Wethen M →M k−2 obtain morphisms [g] : by defining [g] = r [h], and this is independent of the factorization g = rh.Furthermore,ifN 3 is an integer, g ∈ M2(Zˆ) ∩ GL2(Af ) is such that − , 1 ⊆ ◦ g UN g UN and S contains the set of primes dividing N k!,then[g ] [g] =[g g] in S PMQ for = ∅,cand !,whereS is the set of primes dividing N k!. In particular, we obtain ∼ an action of GL2(Zˆ)/UN = GL2(Z/N Z) on M. We note the following: −1 LEMMA 1.1. – If g ∈ M2(Zˆ) ∩ GL2(Af ) and UN ⊂ gUN g ⊆ GL2(Zˆ), then the injective −1 − M → M gUN g 1 morphism [g]c : c ( c) has cokernel killed by detg . −1 Suppose now that g ∈ GL2(Af ) with g UN g ⊆ UN . We can then write g = rh for some S r ∈ Q so that (h, N, N ) is admissible and obtain morphisms in PMQ which we also denote [g]. These behave naturally under composition and the resulting action of GL2(Af ) is compatible with the isomorphisms in (5).

1.4. The premotivic structure for forms of level N and character ψ

1.4.1. σ-constructions Suppose U is any open compact subgroup of GL2(Zˆ).LetK be a number field with ring of integers OK .LetV be a finite dimensional vector space over K and let σ : U → AutK (V ) be a continuous representation of U.Define Sσ = ∈ Sf (Q)| |k! or GL2(Z) ⊂ ker σ ,

⊂ K ⊂ and suppose that S Sf (K) with Sσ S. Choose N 3 such that UN ⊂ ker σ and N is divisible only by primes in S,andlet S M = M(N) .SinceUN is normal in U, by Section 1.3.2, we have a group action of U on

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M.LetV be an OK -lattice in V that is stable under the action of U. We then have an object M⊗V PMS M⊗V M ⊗V V of K deﬁned by ( )? = ? where all additional structures on are trivial. Letting U act diagonally on M⊗V, we obtain an object

M(σ)=(M⊗V)U

PMS M M ⊗V U of K as in Section 1.1.1. We also define objects (σ) =( ) for =tf or c and define M(σ)! =im M(σ)c →M(σ)tf . U We remark that M(σ)! may lie properly in (M! ⊗V) and that M(σ) and M(σ)tf may rely on | M the choice of N as above. However using the fact that if N is another choice with N N and c M S M → M UN denotes (N )c , then the natural map c ( c) is an isomorphism (by Lemma 1.1), we conclude that M(σ)c and M(σ)! are independent of N.WealsodefineM(σ) = M(σ) ⊗ Q S ∅ in PMK for = , c and !; these are also independent of N. Let U and U be two open compact subgroups contained in GL2(Zˆ).Letσ : U → AutK (V ) and σ : U → AutK (V ) be two representations with stable OK -lattices V and V . Suppose we are given a g ∈ GL2(Af ) ∩ M2(Zˆ) and a K-linear homomorphism τ : V → V such that −1 ∈ ∈ ∩ −1 τ(σ(g ug)v)=σ (u)τ(v) for all v V and u U1 = U gUg .LetS be a subset of K ∪ K ∪ K ∈ Z Sf (K) containing Sσ Sσ Sg where Sg is the set of such that g / GL2( ). Choosing suitable N and N and a coset decomposition U = i giU1, the formula x ⊗ v → [gig]x ⊗ σ (gi)τ(v) i

deﬁnes maps [U gU] : M(σ) →M(σ ). The map is independent of the choices for = c and !,andfor = ∅ after tensoring with Q,

1.4.2. Premotivic structure of level N and character ψ × × Suppose that k 2 and N 1.Letψ be a character Zˆ → K of conductor dividing N. Let U = U (N) denote the set of matrices ab ∈ GL (Zˆ) with c ∈ NZˆ.Defineσ = σ(N,ψ) 0 cd 2 → × ab −1 by the character ψ : U0(N) K sending to ψ (aN ),whereaN denotes the image of Z cd a in p|N p.DefineV = V (N,ψ) to be the vector space K with an action of U by σ.Let V O ⊂ K M M = K V . Note that Sσ = SN .Welet (N,ψ) denote the premotivic structure (σ) for = c or !,andletM(N,ψ) = M(σ) for = ∅, c or !. k−1 ∼ Recall that the isomorphism C ⊗ Fil M(M)dR = Mk(UM ) in (5) respects the action of GL2(Zˆ). It follows that for any embedding K → C, the isomorphism identifies C ⊗K k−1 Fil M(N,ψ)dR with Mk(N,ψ), the space of classical modular forms of weight k,level k−1 N and character ψ. Under this isomorphism, Fil M(N,ψ)dR corresponds to the set of forms whose q-expansion at ∞ has coefficients in τ(OS ). Replacing M(N,ψ) by M(N,ψ)c or M(N,ψ)! gives the same identifications, but for the space of cusp forms Sk(N,ψ).

1.5. Duality

We now deﬁne duality morphisms arising from pairings on the realizations of the premotivic structures associated to modular forms.

1.5.1. Duality at level N H 2 2 ¯ For N 3,welet = HN denote the premotivic structure Hc (XN )=H (XN ). H 2 ¯ an Z H 2 ¯ Z H 1 ¯ 1 More precisely, we let B = H (X , ), = H (XQ¯ , ), dR = H (X,ΩX/T¯ ) and

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 680 F. DIAMOND, M. FLACH AND L. GUO

H 2 ¯ O ∈ -crys = H (XZ , ¯Z ) for /SN . These come equipped with additional structure crys X ,crys SN and comparison isomorphisms making H an object of PMQ ,andweletH = Q ⊗H. Q Z Z 2 ∼ Let F = (µN ). The Weil pairing on ( /N )X = E[N] deﬁnes an isomorphism between (Z/N Z)X and µN,X, hence a morphism X → Spec OF [1/N ]. The ﬁbers being geometrically connected, this induces an isomorphism

(7) MF (−1) →H

SN in PMQ whose realizations are given by Poincaré and Serre dualities. Furthermore, the action of u ∈ GL2(Zˆ) on H corresponds to that of det u on MF , where the action on H arises from × the action on X¯ (see Section 1.3.2) and that of Zˆ on MF is via the isomorphism of class field theory (see Section 1.1.3). 1 an an Recall from Section 1.2.2 that FB denotes the sheaf R s∗ Z on X . The cup product defines −1 an a morphism FB ⊗FB → (2πi) Z of locally constant sheaves on X , inducing a morphism F k ⊗Fk → 2−kZ B B (2πi) defined on sections by

k−2 (8) x1 ⊗···⊗xk−2 ⊗ y1 ⊗···⊗yk−2 → xi ∪ yσ(i).

σ∈Σk−2 i=1

Taking cohomology and composing this with the cup product yields a morphism ∼ ( , )B : Mc,B ⊗MB →HB(2 − k) = MF (1 − k)B, which induces Mc,B → HomZ(MB, MF (1 − k)B). Defining pairings ( , )dR and ( , ), one finds that they respect the comparison isomorphisms and weight filtrations yielding morphisms (9) δ : Mc → Hom M, MF (1 − k) and δ! : M! → Hom M!, MF (1 − k)

S in PMQ if SN ⊂ S. The pairing ( , )dR is compatible with the Petersson inner product −1 k−2 (g,h)Γ(N) =(−2i) g(τ)h(τ)(Imτ) dτ ∧ dτ¯ Γ(N)\H for g ∈ Sk Γ(N) ,h∈ Mk Γ(N)

k−1 k−1 as follows. For g ∈ C ⊗ Fil Mc,dR and h ∈ C ⊗ Fil MdR, write α(g)=(gt)t ∈ Sk Γ(N) and α(h)=(ht)t ∈ Mk Γ(N) . t∈(Z/N Z)× t∈(Z/N Z)×

After extending scalars to C for the pairing ( , )dR,wehave ∞ −1 ∞ k−1 k−1 ∞ ⊗ − ⊗ (10) πt g,(I ) (F 1)I h dR =(k 2)!(4π) (gt,ht)Γ(N) ι t∈(Z/N Z)×

2πit/N where πt : C ⊗ MF,dR = C ⊗ F → C is deﬁned by e ∈ µN (C).

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1.5.2. Duality for σ-constructions For the rest of the section, we assume U is an open compact subgroup of GL2(Zˆ) satisfying detU = Zˆ×,andletψ : Zˆ× → K× be a continuous character. For a continuous representation → V V O σ : U AutOK ,letσˆ denote the representation deﬁned by HomOK ( , K ). Suppose N 3 ⊂ K ⊂ is such that UN ker σ, the conductor of ψ divides N and SN S. Restricting the pairings ( , )? to U-invariants, we get a morphism M ⊗ −1 ◦ → M M − (11) δN : c σˆ (ψ det) HomOK (σ), ψ(1 k) which depends on the choice of N. Tensoring with Q and normalizing by dividing by [U : UN ], we get an isomorphism −1 δ¯: Mc σˆ ⊗ (ψ ◦ det) → HomK M(σ),Mψ(1 − k) which is independent of N, and we similarly deﬁne −1 (12) δ¯! : M! σˆ ⊗ (ψ ◦ det) → HomK M!(σ),Mψ(1 − k) .

We say that U is sufficiently small if U acts freely on GL2(Q)\ GL2(A)/U∞. In particular ⊂ U is sufficiently small if U U1(d) for some d 4,whereU1(d) denotes the preimage in Zˆ Z Z 1 ∗ GL2( ) of the subgroup of GL2( /d ) consisting of matrices of the form 0 ∗ . One then has a description of M(σ)!,B in terms of the cohomology of the curve XU with coefficients in a sheaf depending on σ. Poincaré duality on XU then shows that the isomorphism δ¯ arises from an injective morphism −1 M ⊗ ◦ → O M M − σˆ (ψ det) ! Hom K (σ)!, ψ(1 k) whose cokernel C satisfies C =0for N(k − 2)!. We deduce the following:

LEMMA 1.2. – Suppose that U has a sufﬁciently small open compact normal subgroup U × such that det U = Zˆ and [U : U ].If>k− 2 and ker σ ⊂ UN for some N not divisible by ,thenδ¯λ arises from an isomorphism −1 M ⊗ ◦ → O M M − σˆ (ψ det) !,λ Hom K,λ (σ)!,λ, ψ(1 k)λ for every λ dividing .

Suppose now that σ, σ, g, τ and S areasinSection1.4,sowehavemorphisms

[U gU]τ, : M(σ) → M(σ ) for = ∅, c and !. We then also have morphisms −1 ⊗ −1 ◦ → ⊗ −1 ◦ U detg g U τ t⊗ψ(det(g)), : M σ (ψ det) M σ (ψ det)

T T T which we denote by [U gU]τ,. One ﬁnds then that [U gU]τ,c (respectively, [U gU]τ,!)isthe adjoint of [U gU]τ (respectively, [U gU]τ,!) with respect to the pairing δ¯, (respectively, δ¯!).

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 682 F. DIAMOND, M. FLACH AND L. GUO

1.5.3. Duality for level N, character ψ We now define duality morphisms for premotivic structures defined in Section 1.4.2. Suppose K ⊂ now that N 1, ψ has conductor dividing N and SN S.LetU = U0(N) and σ = σ(N,ψ) be the representation on V = V (N,ψ) as in Section 1.4.2. Let V be the one dimensional representation HomK (V,K) ⊗ Kψ−1◦det of U with natural lattice V . We denote by σ = −1 σˆ ⊗ (ψ ◦ det) the representation on V .Defineω : V→V by sending v0 to vˆ0 ⊗ 1,where ∈ V O vˆ0 Hom( , )is such thatvˆ0(v0)=1. 0 −1 Z Let w denote N 0 N in p|N GL2( p). The operator [UwU]ω, defines an isomorphism

(13) M(N,ψ) → M(σ )

PMS ∅ in K for = , c and !, restricting to an isomorphism on integral structures for −1 −1 = c and !. One ﬁnds that [UwU]ω =[Uw U]ωt⊗ψ(det w) and [UwU]ω is adjoint to k−2 −1 k−2 −1 N [Uw U]ωt⊗ψ(det w)),c and coincides with ψ(−1∞)N [Uw U]ω. Composing the operator [UwU]ω with the duality morphism δ¯, we obtain a duality isomorphism (14) δˆ: M(N,ψ) → HomK M(N,ψ)c,Mψ(1 − k) .

Similar assertions hold for M(N,ψ)! yielding an isomorphism δˆ!.SinceM(N,ψ)=0unless k−2 ψ(−1∞)=(−1) , we ﬁnd that the corresponding perfect pairing is alternating.

1.6. Premotivic structure of a newform

We keep the notation of Section 1.4.2. In particular, we assume N 1, ψ is a K-valued K M Dirichlet character of conductor dividing N, S is a set of primes containing SN and (N,ψ) and M(N,ψ) are premotivic structures associated t modular forms of weight k,levelN and character ψ. We describe the premotivic structures associated to Hecke eigenforms.

1.6.1. Hecke action We now deﬁne the action of Hecke operators on the premotivic structures deﬁned in Section 1.4.2. For each rational prime p, we have an action of the classical Hecke operator Tp on the spaces Mk(N,ψ) and Sk(N,ψ).LetT˜ denote the polynomial algebra over OK generated by the variables tp for all primes p. The operators Tp commute on Mk(N,ψ) and Sk(N,ψ) making them T˜-modules with tp acting as Tp. Denote their annihilators a ⊂ a,letT = T˜/a and T = T˜/a. PROPOSITION 1.3. – There is a natural action of T on M(N,ψ)! and of T on M(N,ψ)c and M(N,ψ) compatible with the isomorphisms

k−1 ∼ k−1 ∼ Fil M(N,ψ)!,dR ⊗K C = Sk(N,ψ), Fil M(N,ψ)dR ⊗K C = Mk(N,ψ), the natural morphisms

M(N,ψ)c → M(N,ψ)! → M(N,ψ) and the duality morphisms δˆ and δˆ! of (14). ∪ K Proof. – For S = S S{p}, the double coset operator p 0 U0(N) U0(N) −1 01 p ψ(pp)

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 683

M S M S S deﬁnes endomorphisms of (N,ψ)c , (N,ψ)! and M(N,ψ) . It is straightforward to check the compatibility with Tp on Sk(N,ψ) and Mk(N,ψ) and with the indicated morphisms of S T˜ objects of PMK . These double coset operators commute, yielding an action of on M(N,ψ),? for = c, ! and ∅ and ?=B, dR and λ, restricting to an action on M(N,ψ),? for = c and ! and ?=B and λ. k−1 If T ∈ a,thenT annihilates Fil M(N,ψ)!,dR and is compatible with I∞,soitalso annihilates

− k 1 ∞ −1 ∞ k−1 Fil M(N,ψ)!,dR =(I ) ◦ (id ⊗c) ◦ I ) Fil M(N,ψ)!,dR .

From the opposition of ﬁltrations in the Hodge structure, we deduce that the action of T˜ on T λ M(N,ψ)!,dR and M(N,ψ)!,B factors through . From the compatibility with IB , we deduce the same for M(N,ψ)!,λ. The same argument shows that the action of T˜ on M(N,ψ)? factors through T for ?=B, dR and λ. It follows then from the compatibility with δˆ that the action on M(N,ψ)c,? factors through T for these realizations. Suppose now that λ is not in S. There is then a unique action of T on M(N,ψ)c,λ-crys λ compatible with its action on M(N,ψ)λ and the comparison isomorphism I .Forp not divisible by λ, the action of Tp is given by the above double coset operator, hence is compatible with λ T ⊗ Q T IdR as well. Since such Tp generate the K-algebra , it follows that the action of on M λ M(N,ψ)c,dR preserves the localization of (N,ψ)dR at λ and is compatible with IdR. We thus T M PMS T obtain the desired action of on the object (N,ψ)c of K . Similarly we conclude that acts on M(N,ψ) and T acts on M(N,ψ)! as desired. 2

1.6.2. Premotivic structure for an eigenform k−1 Now suppose that f is an eigenform in Fil M(N,ψ)!,dR for the action of T.Sofor ∈ T O T →O T we have T (f)=θf (T )f for some K -linear homomorphism K . We assume n f is normalized so that its q-expansion an(f)q at ∞ has leading term q.Wethenhave ap(f)=θf (Tp) ∈OK for all primes p.LetIf =kerθf and Mf = M(N,ψ)![If ] in the notation M PMS M ⊗ S of Section 1.1.2; thus f is an object of K and Mf = f OK K is in PMK .Then

k−1 (15) Fil Mf,dR = OK,Sf and Mf is a premotivic structure of rank 2 over K.TheGQ-module Mf,λ is irreducible. We write M¯ f,λ for the residual representation Mf,λ/λMf,λ. For each embedding τ : K → C, we obtain a (classical) normalized eigenform τ(f)= n τ(an(f))q in Sk(N,ψ).Conversely,iff is a normalized eigenform in Sk(N,ψ), its q-expansion coefficients an(f) generate a number field Kf ⊂ C, and taking K ⊃ Kf ,we k−1 can regard f as an eigenform in Fil M(N,ψ)!,dR and consider the associated premotivic structures Mf and Mf . We say that f is a newform of level N if each (equivalently, some) τ(f) is a classical newform k−1 of level N.Ifg is a normalized eigenform in Fil M(N,ψ)!,dR, then there is a unique newform f of some level Nf dividing N such that ap(f)=ap(g) for all p not dividing N/Nf . In that case, a straightforward construction using double-coset operators defines an isomorphism ∼ S Mf = Mg in PMK (see Proposition 1.4 below for the cases we need). If f is a newform of level N, then the pairing δˆ! on M(N,ψ)! restricts to a perfect alternating pairing on Mf , i.e., an isomorphism

∧2 ∼ − (16) K Mf = Mψ(1 k).

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 684 F. DIAMOND, M. FLACH AND L. GUO

With our normalization of Tp, the Eichler–Shimura relation on M(N,ψ)!,λ due to Deligne [15] takes the form 2 − −1 −1 k−1 (17) Frobp ψ(p) Tp Frobp +ψ(p) p =0 −1 for all p not dividing N,whereψ(p)=ψ(pN )=ψ(pp) . It follows that Frobp on Mf,λ has characteristic polynomial

2 −1 −1 k−1 (18) X − ψ(p) ap(f)X + ψ(p) p .

1.6.3. The L-function n Suppose that f = anq ∈ Sk(N,ψ) is a newform of weight k, conductor Nf and character ψf . Associated to f is the L-function with Euler product factorization: −s −s k−1−2s −1 −s −1 L(f,s)= ann = 1 − app + ψf (p)p (1 − app ) . n 1 pNf p|Nf

A A0 There is also an irreducible GL2( f )-subrepresentation π(f) of k with central character 2−k U (N ) ψf such that f spans the image of π(f) 1 f under the isomorphism A0 U1(Nf ) ∼ ( k) = Sk Γ1(Nf ) ,

A× ⊂ A× where we view ψf as a character on f . (Recall from Section 1.3.1 that

− A0 = lim S (U ) ∼ lim Filk 1 M(N) ⊗ C.) k → k N = → !,dR N N A0 Moreover, we have the decomposition k = f π(f) where f runs over newforms of weight k ∼ ⊗ of any conductor and character. For each f we have a factorization π(f) = pπp(f) where πp(f) is an irreducible admissible representation of GL2(Qp) and ⊗ is a restricted tensor product. Suppose now that f is as above with Kf ⊂ K ⊂ C.Foreveryprimep of Q and λ/∈ S,the ss representation Dpst(Mf,λ|Gp) of the Weil–Deligne group of Qp is K-rational and corresponds via local Langlands to πp(f) (where we extend scalars to C via τ and normalize the local Langlands correspondence as in [9]). For λ not dividing p and p not dividing N, this is the Eichler–Shimura relation (18); for λ dividing p and p dividing N, this is due to Deligne, Langlands and Carayol [9]; for λ|p, p/∈ S, this is due to Scholl [75]. It follows that Mf is L-admissible everywhere 1 and that its L-function is related to that of f by the formula

(19) L(Mf ⊗K M −1 ,s)=L(f,s), ψf where the Euler factors defining the first L-function are viewed as C-valued via the inclusion K ⊂ C. More generally, for a newform f with coefficients in a number field K,wehave L(Mg ⊗K M −1 ,τ,s)=L τ(g),s ψg for each embedding τ : K → C, so (19) holds as an identity of K ⊗ C-valued functions.

1 S In fact, the main theorem of [71] shows that Mf = M for an object M of PMK which is L-admissible everywhere.

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1.7. The adjoint premotivic structure

1.7.1. Realizations of the adjoint premotivic structure Suppose now that f is a newform of weight k, character ψ and level N, with coefﬁcients in 0 K.WedeﬁneAf =ad Mf to be the kernel of the trace morphism

HomK (Mf ,Mf ) → K.

S ⊇ K It is a premotivic structure in PMK for S SN . For ?=B,dR or λ, Af,? has an integral structure given by Af,? = a ∈ End(Mf,?) | tr(a)=0 .

The extra structures on the realizations of Af are obtained by restrictions from those of End(Mf ). For example the ﬁltration on Af,dR is given by n i n+i Fil Af,dR = a ∈AdR ⊆ End(Mf,dR) | a(Fil Mf,dR) ⊆ Fil Mf,dR, ∀j   Af,dR,n 1 − k,  0 0 {a ∈Af,dR | a(Fil Mf,dR) ⊆ Fil Mf,dR}, 1 − kk− 1. 0 Note that deﬁning Af,λ-crys as above does not yield an object of MF since the non-trivial graded pieces are in degree 1 − k, 0 and k − 1 (though one can obtain such an object by suitably twisting if − 1 > 2(k − 1)). ∼ S There is a canonical isomorphism detK Af = K in PMK which restricts to an isomorphism

A ∼ O (20) detOK f,? = K,? for ? ∈{B, dR,λ},whereOK,B = OK and OK,dR = OK,S . (To see this, note that 00 ∧ 10 ∧ 01 10 0 −1 00 is independent of the choice of basis used to represent an endomorphism.) We note also that Af and HomK (Af ,K) are canonically isomorphic, the isomorphism being deﬁned by the pairing

(21) α ⊗ β → tr(α ◦ β) on each realization of Af . A× → × D Suppose that ψ is a character K of conductor D and that S contains SK as well. Let 2 2 f ⊗ ψ denote the newform (of weight k, level dividing ND and character ψ(ψ ) ) associated n S to the normalized eigenform g = (n,D)=1 ψ (nD)anq .ThenMf⊗ψ is an object of PMK and one checks that the double coset operator U (ND2) 11/D U (N) 0 01 0 1 ∼ ∼ induces an isomorphism Mf ⊗K Mψ = Mg. It follows that Mf ⊗K Mψ = Mf⊗ψ ,so ∼ (22) Af⊗ψ = Af

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 686 F. DIAMOND, M. FLACH AND L. GUO

S in PMK . We may therefore assume f has minimal conductor among its twists when considering Af .WealsonotethatifwereplaceK by K ⊃ K and S by a subset S of the primes over those S in S,thenAf is replaced by (Af ⊗K K ) .

1.7.2. Euler factors and functional equation Ip For each prime p,weletcp = vp(N) and let δp denote the dimension of Mf,λ for any λ not dividing p,so   2, if p N, | δp =  1, if p N and ap =0, 0, if p|N and ap =0. nv nv We set Lp (Af ,s)=Lp(Af ,s) if δp > 0,andLp (Af ,s)=1if δp =0.WeletΣe =Σe(f) denote the set of primes p such that δp =0and Lp(Af ,s) =1 ,andset nv nv L (Af ,s)= Lp (Af ,s)= Lp(Af ,s).

p p/∈Σe(f)

We call the primes in Σe exceptional for f. Recall that if δp =2, then writing −s k−1−2s −1 −s −1 −s −1 Lp(f,s)= 1 − app + ψ(p)p =(1− αpp ) (1 − βpp ) , we have − −1 −s −1 − −s −1 − −1 −s −1 Lp(Af ,s)=(1 αpβp p ) (1 p ) (1 αp βpp ) .

If δp =1,then −1−s −1 (1 − p ) if πp(f) is special; (23) Lp(Af ,s)= −s −1 (1 − p ) if πp(f) is principal series.

Shimura [80] proved that L(Af ,s) extends to an entire function on the complex plane. Recall that we regard L(M,s) as taking values in K ⊗ C. Each embedding τ : K → C gives a map K ⊗ C → C and we write L(M,τ,s) for the composite with L(M,s). Moreover, the work of Gelbart and Jacquet [43] and others (see [73]) shows that

Λ(A ,s)=L(A ,s)ΓR(s +1)ΓC(s + k − 1) f f s +1 =22−k−sπ(1−2k−3s)/2L(A ,s)Γ Γ(s + k − 1) f 2 satisﬁes the functional equation

(24) Λ(Af ,s)=(Af ,s)Λ(Af , 1 − s), where (Af ,s) is as deﬁned by Deligne [16]. Here we have used that Af and HomK (Af ,K) are isomorphic (using (21)).

1.7.3. Variation of integral structures K We maintain the above notation, but now we fix a prime λ of K not in SN and let S = Sf (K) \{λ}. For each finite set of primes Σ ⊂ S, we shall define integral structures on Mf and ∧2 K Mf . We then compare these as Σ varies, showing that under certain hypotheses, the integral

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∧2 structure on Mf is invariant, but the variation on K Mf is controlled by Euler factors of the adjoint L-function. Σ δp Let N = N p∈Σ p . Setting an =0 if n is divisible by a prime in Σ and an = an Σ n Σ otherwise, we have that f = anq is an eigenform of level N with associated newform M PMS f. The construction of Section 1.6.2 thus yields a premotivic structure f Σ in K contained Σ in M(N ,ψ)!. The pairing δˆ! defined in Section 1.5.3 restricts to an alternating pairing on Mf Σ , which Proposition 1.4 below shows is non-degenerate, hence induces an isomorphism 2 2 ∧ M Σ → M (1 − k).Theimageof∧ M Σ therefore defines an integral structure for K f ψ OK f − ΣM − Σ ⊂ Σ Mψ(1 k), necessarily of the form ηf ψ(1 k) for some fractional ideal ηf K. We call ηf O Σ the (naive, Σ-finite) congruence K -ideal of f. Note that ηf is well-behaved under extension O ⊗O · ⊂ of scalars K K if K K . Σ δp For positive integers m dividing N /N = p∈Σ p ,weletm denote the morphism Σ M(N,ψ)! → M(N ,ψ)! defined by the operator −1 m−1 U (N Σ) m 0 U (N) = m1−k U (N Σ) 10 U (N) . 0 01 0 1 0 0 m 0 1

We also define the endomorphism φm of M(N,ψ)! by k−1 • φ1 =1, φp = −Tp, φp2 = ψ(p)p ; • φm1m2 = φm1 φm2 if (m1,m2)=1. We also define Σ γ = mφm : M(N,ψ)! → M(N ,ψ)! m and let γt denote its adjoint with respect to the pairings defined in Section 1.5.3.

PROPOSITION 1.4. – S Σ (a) The morphism γ restricts to an isomorphism Mf → Mf Σ in PM with γdR(f)=f . (b) We have t nv −1 ◦ Σ γ γ = φN /Nf Lp (Af , 1) p∈Σ ˆ on Mf ,soδ! is non-degenerate on Mf Σ . (c) If M¯ f,λ is an irreducible (OK /λ)[GQ]-module, then γ induces an isomorphism M →MΣ PMS f,λ f,λ in and Σ ∅ nv −1 ηf,λ = ηf,λ Lp (Af , 1) . p∈Σ

Proof. – Part (a) and the formula in (b) follow from straightforward double-coset computations similar to those in Chapter 2 of [88] (see also p. 121 of [14]). The non-degeneracy of the

Σ M M¯ pairing follows from φN /Nf being non-zero on Mf ; in fact it is invertible on f,λ.If f,λ n is irreducible, then the image of Mf,λ must be of the form λ Mf Σ,λ for some n 0;since Σ Σ 2 γdR(f)=f , we see that n =0. The formula for ηf,λ in part (c) then follows from part (b). 1.8. Reﬁned integral structures

We now modify some of the constructions of the preceding sections in order to obtain perfect pairings on integral structures and to account for the congruences corresponding to Euler factors nv missing from L (Af ,s). We also prove some technical results needed for Section 3.2.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 688 F. DIAMOND, M. FLACH AND L. GUO

\{ } ∈ K We maintain the notation of Section 1.7.3. In particular S = Sf (K) λ for some λ/SN . We assume also that f has minimal conductor among its twists. We further assume that the representation of GQ on M¯ f,λ is irreducible; moreover if 3 ∈ λ, then we require its restriction to

GQ(µ3) to be absolutely irreducible.

1.8.1. Σ-level structure Recall that Σe denotes the set of exceptional primes deﬁned in Section 1.7.2. Since we assume f has minimal conductor among its twists, we have p ∈ Σe if and only if Lp(Af ,s)= −s −1 (1 + p ) , which is equivalent to Mf,λ|Gp being an absolutely irreducible representation induced from a character of GF where F is the unramiﬁed quadratic extension of Qp.In particular, its conductor exponent cp = vp(N) is even and δp =0.Welet

Σ1 = { p ∈ Σe | p ≡−1modλ }.

The irreducibility hypotheses allow us to choose an auxiliary prime r>3 not dividing N −1 ∈O× such that Lr(Af , 1) K,S by Lemma 3 of [27]. We then deﬁne Σ cp cp+δp N1 = p p .

p/∈Σ1∪Σ p∈Σ∪{r} Σ Σ Σ Σ Σ Z We then define U = U0(N1 ). Note that U = p Up , where the subgroup Up of GL2( p) is determined by whether p ∈ Σ. Next we shall define a representation of U Σ as a tensor product of Σ | certain representations of the Up for p N. If p ∈ Σ or p/∈ Σ ,weletV = O with ab ∈ U Σ acting via ψ(a) (which is trivial if 1 p K cd p 10 ∈ cp ∩ Z cp =0). For p Σ1,weletUp = U0(p ) GL2( p) and gp = 0 pcp/2 . We then define a V Z representation p of GL2( p) by the following lemma: LEMMA 1.5. – There is a finite extension K of K,aprimeλ of O = OK over λ and a O V Z finite flat -module p with a continuous action of GL2( p) such that the following hold: Z V ⊗ GL2( p) → C (a) ( p O ,τ πp(τ(f))) is one-dimensional for any embedding τ : K ; V V O Z (b) p/λ p is an absolutely irreducible ( /λ )[GL2( p)]-module; (c) there is a homomorphism of O [GL2(Zp)]-modules V ∼ V O −1 ◦ ωp : p = HomO p, (ψ det)

such that ωp,λ is an isomorphism; (d) there is a homomorphism of O [Up]-modules

Up V →V τp :res −1 gp p p gp GL2(Zp)gp

such that τp,λ is surjective. O× → ¯ × × Proof. – Let ε : F Kλ be the restriction of a character of F corresponding via class field theory to one from which Mf,λ|Gp is induced. The minimality of the conductor of f among cp/2 twists implies that ε/(ε ◦ Frobp) has conductor p OF .WeletK be a finite extension of K over which the K-rational representation denoted Θ(ε) in Section 3 [13] is defined. Then part (a) follows from compatibility with the local Langlands correspondence and its explicit description in Section 3 of [44]. Part (b) is contained in Lemma 3.2.1 of [13]. Part (c), after tensoring with K, follows from the first paragraph of Section 3.3 of [13]. Rescaling and applying (b) gives

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 689 the desired homomorphism ωp. Part (d), after tensoring with C via τ, follows from the fact Up that (Vp ⊗O,τ πp(τ(f))) is one-dimensional. It therefore holds after tensoring with K ,and rescaling again gives the desired homomorphism. 2 V Replacing K by a larger K (and λ by λ ) if necessary to define the representations p for ∈ VΣ V V ∈ \ VΣ VΣ p Σ1,welet p = p or p according to whether p Σ1 Σ.Wethenlet = p p , Σ Σ → VΣ Σ M Σ σ : U AutOK ( ) and consider the Σ-level premotivic structures M(σ ) and (σ ) ∪{ }⊂ Σ Σ Σ Σ for = c and !. Note that if Σ1 r Σ,thenN1 = N , σ = σ(N ,ψ) as in Section 1.7.3. Σ Σ We now define a perfect pairing on M(σ )! giving rise to a perfect pairing on M(σ )!,λ. − Σ 0 1 ∈ A Σ V VΣ Define w = w1 = Σ Σ GL2( f ).Letσ = σ , = and define N1 0 N1 V→ V O −1 ◦ ∼ ⊗ VΣ O −1 ◦ ω : HomOK , K(ψ det) = HomOK p , K (ψ det) as the tensor product of the maps ωp,whereωp is defined in Lemma 1.5 if p ∈ Σ1 − Σ, and by sending a generator v0 to the map v0 → 1 otherwise. We then have that ω σ(w−1uw)v = σ(u)ω(v)

Σ for all u ∈ U and v ∈V, so the operator [UwU]ω is well-deﬁned and induces a morphism M →M ⊗ −1 ◦ (σ)! σˆ (ψ det) !.

Composing with the isomorphism of (12), we obtain an isomorphism ˆΣ → − (25) δ! : M(σ)! HomK M(σ)!,Mψ(1 k)

Σ arising from a perfect alternating pairing on M(σ)!. Moreover Lemma 1.2 with U = U ∩U1(r) yields the following:

COROLLARY 1.6. – The pairing δˆ!,λ of (25) restricts to an isomorphism M → M M − (σ)!,λ HomOK,λ (σ)!,λ, ψ(1 k)λ .

1.8.2. Hecke action and localization We now define an action of Hecke operators on the Σ-level premotivic structures. For a finite Ψ set of primes Ψ, we write T˜ for the OK -subalgebra of T˜ generated by the variables tp for p/∈ Ψ. Let ΨΣ denote the finite set of primes p/∈ Σ such that δp =1or p ∈ Σ1. For primes p/∈ ΨΣ,we write Tp for the double coset operator Σ p 0 Σ Tp = U U −1 . 01 p ψ (pp)

ΨΣ Σ Σ Σ As in Section 1.6.1, we obtain an action of T˜ on M(σ ), M(σ )c and M(σ )! factoring − ΨΣ k 1 Σ through the quotient of T˜ by the annihilator of Fil M(σ )dR. Moreover the Hecke operators are self-adjoint with respect to the pairing of (25). Recall that we are assuming irreducibility of the representation of GQ on M¯ f,λ.Oneway we use this hypothesis is to relate cohomology groups with different supports after localizing at maximal ideals of Hecke algebras. For the rest of the section, Σ and Ψ denote ﬁnite subsets of Ψ S with ΨΣ ⊂ Ψ and m is the maximal ideal of the maximal ideal T˜ generated by λ and the Σ∪{r} elements tp − ap(f ) for all primes p/∈ Ψ.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 690 F. DIAMOND, M. FLACH AND L. GUO

Σ We shall need to consider a slightly more general setting than that of the M(σ ), but can then Σ restrict attention to Betti and λ-adic realizations. Suppose that N = N1 D for some positive integer D not divisible by any primes in Σ1 ∪{}\Σ,andthatU is an open compact subgroup Σ Ψ of GL2(Af ) satisfying U1(N ) ⊂ U ⊂ U0(N ). Setting σ = σ |U, we deﬁne an action of T˜ on the M(σ),? for ? ∈{B,λ} and Ψ ⊃ ΨΣ by letting tp act as k−2 10 − Tp = p U 0 p 1 U 1.

λ M One checks that the action respects the comparison isomorphism IB ,that (σ),? is stable for ∈{c, !}, and that the resulting action coincides with the ones defined above if U = U Σ. Σ Σ Let Σ =Σ∪ Σ1, U = U ∩ U and σ = σ |U .Defineg ∈ GL2(Af ) with gp as in Σ Σ Lemma 1.5 for p ∈ Σ1 \ Σ and gp =1otherwise. Define α : V →V by ⊗pαp with αp = τpgp as in Lemma 1.5 for p ∈ Σ1 \Σ and the identity otherwise. The operator [U gU]α,c,B then defines Ψ a T˜ -linear homomorphism M(σ)c,B →M(σ )c,?.

LEMMA 1.7. – The map [U gU]α,c,B is injective. Proof. – Let d = pcp/2, M = M (N d), M = M (N d2), V = VΣ and V = p∈Σ1\Σ c c c c VΣ −1 V→ U −1V→ −1V . Writing g α as a composite Indg−1U g g g , we can write [U gU]α,c as

M ⊗V U → M ⊗ U −1V U → M ⊗ −1V g−1U g → M ⊗V U ( c,B ) ( c,B Indg−1U g g ) ( c,B g ) ( c,B ) where the last map is deﬁned by [g]c,B ⊗ g. The ﬁrst map is injective since V is irreducible, the second is an isomorphism by Shapiro’s Lemma, and the last is injective by Lemma 1.1. 2

Suppose for the moment that we also have U ⊂ U1(r). Letting Γ=SL2(Z) ∩ U,wehavethat \ F k Γ acts freely on H and XU can be identified with Γ H. We write B for the locally constant sheaf k−2 1 SymZ R s∗Z,wheres is the natural projection EU → XU with EU =Γ\(H × C)/(Z × Z) defined as in Section 1.2.1. The representation σ defines an action of Γ on V,andwelet Fσ denote the locally constant sheaf on XU defined by Γ\(H ×V). We can then identify M 1 F k ⊗F ⊗ 1 F k ⊗F (σ)c,B with Hc (XU , B σ) and M(σ)B with K OK H (XU , B σ).Ifσ is the trivial representation of U on OK , then the usual action of the Hecke operator Tp on 1 F k ⊗O 1 F k ⊗O M Hc (XU , B K ) and H (XU , B K ) is compatible with the ones we defined on (σ)c and M(σ).

LEMMA 1.8. – If U ⊂ U1(r) and σ is trivial, then the natural map

1 F k ⊗O → 1 F k ⊗O Hc (XU , B K )m H (XU , B K )m is an isomorphism.

We recall the idea of the proof, which is standard. The kernel of the map is torsion-free, and the k−1 cokernel has no λ-torsion since N (k − 2)! ∈/ λ. After tensoring with K, one has Tp = p +1 on the kernel and cokernel for all p ≡ 1modN . Thus if m is in the support of the kernel or cokernel, then Tp − 2 ∈ m for all p ≡ 1modN , p/∈ Ψ. Arguing as in Proposition 2 of [26], one obtains a contradiction to the hypothesis that M¯ f,λ is absolutely irreducible.

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 691

Let us now return to the case of arbitrary U with U1(N ) ⊂ U ⊂ U0(N ).WeletU = Ψ U ∩ U1(r) and consider the commutative diagram of T˜ -modules

M M 1 F k ⊗O (σ)c,B (σ )c,B Hc (XU , B K )

M M 1 F k ⊗O (σ)!,B (σ )!,B H (XU , B K )tf .

The horizontal maps in the top row are injective (the ﬁrst by Lemma 1.7) and the right most vertical map is an isomorphism after localizing at m by Lemma 1.8. We thus have:

COROLLARY 1.9. – The localization at m of the natural map M(σ)c,? →M(σ)!,? is an isomorphism for ? ∈{B,λ}.

Note that the case of ?=λ follows from that of ?=B. In fact, M(σ),λ is isomorphic as a Ψ T˜ -module to the completion of M(σ),B at λ. Note also that M(σ),λ is isomorphic to the direct sum of its localizations at the maximal ideals over λ in its support as a T˜Ψ-module. Finally we shall need the following generalization of a lemma of De Shalit in Section 3.2. LEMMA 1.10. – Suppose that U has -power index in U0(N ) and let ∆=U0(N )/U.Then M − O (σ)!,λ,m is a free K,λ[∆]-module. × Proof. – Let U = U ∩ U1(r) and σ = σ|U . Note that (Z/rZ) has order not divisible ∼ × ∼ by ,andthatU0(N )/U = ∆ × (Z/rZ) acts on M(σ )c,λ,m. It follows that M(σ)c,λ,m = × M (Z/rZ) O M M − O (σ )c,λ,m is an K,λ[∆]-module summand of (σ )c,λ,m and (σ)c,λ,m is an K,λ[∆]- M − O module summand of (σ )c,λ,m. Note also that the ring K,λ[∆] is local. Suppose ﬁrst that k =2, ψ is trivial and Σ1 ⊂ Σ. The argument of Proposition 1 of [86] shows − 1 F k ⊗O O that H (XU , B K,λ) is free over K,λ[∆], hence so is its summand

− − − M ∼ 1 F k ⊗O ∼ 1 F k ⊗O (σ )c,λ,m = Hc (XU , B K,λ)m = H (XU , B K,λ)m,

λ where the ﬁrst isomorphism is gotten from IB and the second from Lemma 1.8. It follows that M − M − its summand (σ)c,λ,m is free, hence so is (σ)!,λ,m by Corollary 1.9. ⊂ O ⊗ F F Suppose next that k>2, ψ is non-trivial or Σ1 Σ. Denoting K,λ OK σ by σ ,λ,we have 1 k ∼ M(σ )c,λ, if i =1, H (X , F ⊗F ) = c U B σ ,λ 0, otherwise

0 F k ⊗F (the case i =2following from the vanishing of H (XU , B σ /λ)). Note that this holds in the case U = U0(N ) as well, and the Serre–Hochschild spectral sequence with respect to → i M the cover XU XU0(N )∩U1(r) gives H (∆, (σ )c,λ)=0for all i>0. By [6, VI.8.10], it M M − ∼ M − 2 follows that (σ )c,λ is free, hence so is its summand (σ)!,λ,m = (σ)c,λ,m. 1.8.3. Ihara’s Lemma For ﬁnite subsets Σ ⊂ Σ of S = Sf (K) \{λ}, we shall deﬁne a morphism

Σ Σ → Σ γΣ : M(σ )! M(σ )! generalizing the one in Section 1.7.3. We shall prove a result needed in Section 3.2—that this map is injective with torsion-free cokernel on certain localizations of the integral Betti realization. The

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 692 F. DIAMOND, M. FLACH AND L. GUO result stems from a lemma of Ihara which has been generalized in various ways for applications to congruences between modular forms (see [68,20] and Chapter 2 of [88]). For positive integers m dividing N Σ ∪{r}/N Σ∪{r},wedeﬁne Σ 1−k Σ 10 Σ Σ → Σ Σ,m = m U 0 md U α,! : M(σ )! M(σ )!, where d and α are as in Lemma 1.7. We then deﬁne Σ Σ γΣ = Σ,mφm m

Σ ∪{r} Σ∪{r} where the sum is over positive divisors of N /N and φm is defined in Section 1.7.3. Σ T˜Ψ Σ Σ Note that γΣ is -linear where Ψ is the union of ΨΣ and the set of primes dividing N1 /N1 . ⊂ ⊂ Σ Σ ◦ Σ One checks also that if Σ Σ Σ ,thenγΣ = γΣ γΣ . We let m denote the maximal ideal of T˜ΨΣ defined as in Section 1.8.2, and similarly define m using Σ. Note that m might not lie over m, but that they lie over the same maximal ideal m of T˜Ψ. The argument in the first part of the proof of the lemma on p. 491 of [88] shows that T˜Ψ ΨΣ Σ Σ and T˜ havethesameimageinEndK M(σ )!.Sincem is in the support of M(σ )!,B ,it Σ Σ follows that the localization map M(σ )!,B,m →M(σ )!,B,m is an isomorphism. Composing its inverse with the map

Σ Σ Σ M(σ )!,B,m →M(σ )!,B,m →M(σ )!,B,m

Σ induced by γΣ , we obtain a morphism

Σ Σ (26) M(σ )!,B,m →M(σ )!,B,m

Σ which we denote γm = γΣ,m. Similarly, we have a morphism

Σ Σ Σ M →M (27) γˆm =ˆγΣ,m : (σ )!,λ,m (σ )!,λ,m .

LEMMA 1.11. – The OK,S -linear (respectively, OK,λ-linear) map γm (respectively, γˆm) is injective with torsion-free cokernel.

Proof. – First note the lemma is equivalent to the injectivity of γm mod λ, and by the formula Σ Σ ◦ Σ ∪{ } ∈ γΣ,m = γΣ,m γΣ,m, we can assume Σ =Σ p for some p/Σ. Note that the case p = r is Σ Σ ◦ clear, and that if p divides N1 /N1 ,thenTp γ =0and m =(m ,tp). Thus by Corollary 1.9, it sufﬁces to prove that if p = r,then

Σ Σ (28) M(σ )c,B,m /λ →M(σ )c,B,m /λ is injective. First we consider the case δp =0.Ifp/∈ Σ1 then γ is the identity, so we assume p ∈ Σ1.Using part (b) of Lemma 1.5, the argument in the proof of Lemma 1.7 carries over mod λ,givingthe injectivity of M ⊗V U → M ⊗V U ( c /λ) ( c /λ) , hence that of (28).

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 693

∈ ∪{ } Having proved the lemma for p Σ1 r, we may assume for the remaining cases (δp =1 10 ∪{ }⊂ V V Σ Σ δp or 2, p = r), that Σ1 r Σ. Setting g = 0 p , 0 = (N,ψ)/λ, N = N , N = N p , U = U0(N ), U = U0(N ), A = M(N )c,B ⊗V0 and A = M(N )c,B ⊗V0, it sufﬁces to show that the map

δp U → U (29) Am (A )m i=0

δp defined by ([1]c, [g]c,...,[g]c ) is injective. U g−1U g Suppose now that δp =1. Then Lemma 1.1 yields an isomorphism (A ) → (A ) in- Σ 2 duced by [g]c (each module being identified with the U -invariants in M(N p )c,B ⊗V0). U g−1 Ug Furthermore one finds that [1]c (respectively, [g]c)mapsA isomorphically (A ) (respectively, (A)U ). Therefore it suffices to prove these have trivial intersection. Suppose then − that x ⊗ v ∈ A with v =0 is invariant under U and g 1Ug. Since the p-part of the conductor − cp ∈ cp 1Z a 0 of ψ is p , we may choose a 1+p p so that ψ(a) =1. One checks that h = 01 is in −1 the subgroup of GL2(Zˆ) generated by U and g Ug. Therefore x ⊗ v = h(x ⊗ v)=x ⊗ ψ(a)v implies that x =0. Finally consider the case δp =2.LetA = M(N p)c,B ⊗V0 and U = U0(N p).SinceU is −1 generated by U and g U g, the argument in the case δp =1applied to U instead of U now yields an exact sequence

(30) AU → (A)U × (A)U → (A)U −[g]c where the maps are given by and ([1]c, [g]c). We combine this with Lemma 3.2 of [20], [1]c whose proof shows that the map

1 F k 2 → 1 F k Hp (X1(N ), B/λ) Hp (X1(N ,p), B/λ)

induced by ([1], [g]) is injective, where X1(N ,p) is the modular curve associated to U1(N ) ∩ U0(p). Lemma 1.8 then gives the injectivity of 1 F k 2 → 1 F k Hc X1(N ), B/λ m Hc X1(N ,p), B/λ m ,

U 2 → U whence the injectivity of ([1]c, [g]c):(A )m (A )m . Combining this with the exactness of the localization at m of (30) we deduce the injectivity of (29). 2

1.8.4. Comparison of integral structures We now generalize Proposition 1.4 to the setting of the refined integral structures defined MΣ M Σ ˜Σ ˜Σ T˜ΨΣ in Section 1.8.1. Define f,1 = (σ )![If ] where If is the preimage of If Σ∪{r} in . Σ∪{r} Σ (Recall that f is the eigenform of level N defined in Section 1.7.3 and Ig was defined in Section 1.6.2.) Using strong multiplicity one and Lemma 1.5(a), one sees that k−1 Σ Σ dimK Fil Mf,1,dR =1and therefore that Mf,1,dR has rank two over K,whereMf,1 = ⊗ MΣ ∪{ }⊂ MΣ M K OK f,1. Note that if Σ1 r Σ,then f,1 = f Σ . ⊂ Σ Σ → If Σ Σ , then the restriction of γΣ (defined in Section 1.8.3) defines a morphism Mf,1 Σ T˜Ψ Σ Mf,1. (This follows from -linearity with Ψ as in Section 1.8.3 and the fact that TpγΣ =0for | Σ Σ T˜ΨΣ ˜Σ p N1 /N1 .) Note that the maximal ideal of defined in Section 1.8.2 is simply (If ,λ),so MΣ →M Σ the natural map f,1,B (σ )!,B,m is injective. It therefore follows from Lemma 1.11 that

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 694 F. DIAMOND, M. FLACH AND L. GUO

Σ Σ Σ →∼ Σ γΣ is injective on Mf,1, hence induces an isomorphism Mf,1 Mf,1. Moreover it restricts to MΣ →M∼ Σ isomorphisms f,1,? f,1,? for ?=λ and dR. Σ Σ ˆΣ ˆΣ Now let γΣ denote the transpose of γΣ with respect to the pairings δ! and δ! deﬁned in ˜ Σ Σ Σ ∪ Section 1.8.1. Using Ψ-linearity again, we see that γΣ maps Mf,1 to Mf,1. Recall that if Σ1 { }⊂ Σ r Σ , then the pairing on Mf,1 = Mf Σ is alternating and non-degenerate (Proposition 1.4). ⊂ Σ It follows that the same is true for arbitrary Σ and that for any Σ Σ , the restriction of γΣ is an isomorphism. We thus obtain an isomorphism

∧2 Σ → − K Mf,1 Mψ(1 k).

We let ηΣ be the ideal in O such that ∧2 MΣ maps isomorphically to ηΣ M (1 − k) . f,1 K,λ OK,λ f,1,λ f,1 ψ λ ∪{ }⊂ Σ Σ Note that if Σ1 r Σ,thenηf,1 = ηf,λ. We now state the generalization of Proposition 1.4.

PROPOSITION 1.12. – Suppose that Σ ⊂ Σ are ﬁnite subsets of S = Sf (K) \{λ}. Σ Σ → Σ S MΣ →∼ (a) The morphism γΣ restricts to an isomorphism Mf,1 Mf,1 in PM with f,1,? MΣ f,1,? for ?=λ and dR. (b) We have Σ ◦ Σ Σ −1 γΣ γΣ = βf,Σ Lp(Af , 1) p∈Σ

Σ Σ O Σ ∩ \ ∅ on Mf,1 for some non-zero βf,Σ in K,S . Moreover βf,Σ = ϕN Σ /N Σ if Σ1 Σ Σ= (cf. Proposition 1.4). ˆΣ Σ (c) The pairing δ! is non-degenerate and S-integral on Mf,1, and Σ ⊂ ∅ −1 ηf,1 ηf,1 Lp(Af , 1) . p∈Σ

Proof. – Part (a) and the ﬁrst part of (c) have already been shown, and the formula in (c) follows from the one in (b). Part (b) reduces to the case Σ =Σ∪{p} for some p/∈ Σ.Ifp = r, ∈O× ∈ ∪{ } the result is clear since Lr(Af , 1) K,S .Ifp/Σ1 r , the computation is the same as in ∈ Σ Σ Σ ◦ Proposition 1.4. Finally, for p Σ1, we factor γΣ =[U gU ]α = γ2 γ1 where

Σ Σ Σ Σ γ1 =[U1U ]1,! : M(σ )! → M(σ)! and γ2 =[U gU]α,! : M(σ)! → M(σ )!,

Σ Σ| Σ where U = U0(N1 p) and σ = σ U. Deﬁning a pairing on M(σ)! exactly as for M(σ )!, t t O using the same w and ω,weﬁndthatγ1γ1 = p +1 and γ2γ2 = β for some K [U]-linear endomorphism β of VΣ, necessarily a scalar by Lemma 3.2.1 of [13]. The desired formula Σ 2 follows with βf,Σ = pβ.

2. The BlochÐKato conjecture for Af and Af (1)

In this section we shall deduce the Bloch–Kato conjecture from the main result, Theorem 3.7, of Section 3 below. More precisely, we prove the λ-part of the Bloch–Kato conjecture [4] for Af and Bf := Af (1),wheref is a newform of weight k 2, conductor N 1, with coefﬁcients in

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 695 the number ﬁeld K,andλ is a prime of K not contained in the set     λ | Nk!, or the (OK /λ)[GF ]-module M¯ f,λ is not absolutely irreducible, (31) Sf =   . where F = Q( (−1)(−1)/2) and λ |

By (22) we can assume that f has minimal conductor among its twists and we shall do so in this section. Our formulation of the conjecture follows Fontaine and Perrin-Riou [41], generalized to motives with coefficients in K. For a more systematic discussion of the Bloch–Kato conjecture for motives with coefficients we refer to [7]. If Af is the scalar extension of a premotivic structure with coefficients in a subfield K ⊆ K then Theorem 4.1 and Lemma 11b) of [7] show that the conjecture over K implies the one over K (in the context of Deligne’s conjecture this was already noted in [17, Rem 2.10]). So we need not be concerned with finding the smallest coefficient field for Af .

2.1. Galois cohomology

i i For any field F and continuous GF -module M we write H (F,M) for Hcont(GF ,M).Let V be a continuous finite-dimensional representation of GQ over Q, unramified at all but finitely many primes, and let T ⊆ V be a GQ-stable Z-lattice. We set W := V/T. For each place p of Q 1 Q ⊆ 1 Q , Bloch and Kato (see [4] or [41]) define a subspace Hf ( p,V) H ( p,V) by   1 Q ∞ Hur( p,V) p = , , 1 Q 1 Q → 1 Q ⊗ Hf ( p,V):= ker(H ( p,V) H ( p,Bcrys V )) p = , 0 p = ∞, where 1 Q 1 F Ip 1 Q → 1 Hur( p,M):=H ( p,M )=ker H ( p,M) H (Ip,M) for any Gp-module M. They then define groups 1 Q 1 Q → 1 Q Hf ( p,W):=im Hf ( p,V) H ( p,W) and a Selmer group H1(Q ,M) H1(Q,M):=ker H1(Q,M) → p , f H1(Q ,M) p f p where M is either V or W and the sum is over all places p of Q.Forp/∈{, ∞}, note that 1 Q 1 Q Hf ( p,W) is the maximal divisible subgroup of Hur( p,W) and that the two groups coincide if W Ip is divisible, e.g. when W is unramified. For any finite set Σ of prime numbers not containing we define a larger Selmer group 1 Q 1 Q 1 Q 1 Q → H ( p,W) ⊕ H ( ,W) HΣ( ,W):=ker H ( ,W) 1 1 H (Qp,W) H (Q,W) p/∈Σ∪{,∞} ur f without local conditions at p ∈ Σ ∪{∞} and slightly relaxed local conditions at primes p 1 Q 1 Q where Hur( p,W) is not divisible. The group Hf ( ,W) appears in the Bloch–Kato conjecture 1 Q whereas HΣ( ,W) can be analyzed using the Taylor–Wiles method in our situation.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 696 F. DIAMOND, M. FLACH AND L. GUO

D Z D D ⊗ Q D D D LEMMA 2.1. – Put T =HomZ (T, (1)), V = T Z and W = V /T , and denote by M ∗ the Pontryagin dual of a locally compact abelian group M.Set

1 Q D −1 1 Q D Hf ( p,T ):=ι Hf ( p,V )

1 D 1 D where ι : H (Qp,T ) → H (Qp,V ) is the natural map. If Σ is nonempty and contains all 1 Q 1 Q 0 Q D 1 Q D primes where Hf ( p,W) = Hur( p,W), and if moreover H ( ,V )=Hf ( ,V )=0then there is an exact sequence → 1 Q → 1 Q → 1 Q D ∗ → 0 Q D ∗ → 0 Hf ( ,W) HΣ( ,W) Hf ( p,T ) H ( ,W ) 0. p∈Σ∪{∞}

Proof. – By [32, Proposition 1.4] there is a long exact sequence

1 Q D,∗ → 1 Q → 1 →ρ H ( p,W) ρ−→ 1 Q D ∗ 0 Hf ( ,W) H (GS ,W) 1 Hf ( ,T ) H (Qp,W) p∈S f 2 2 0 D ∗ → H (GS ,W) → H (Qp,W) → H (Q,T ) → 0 p∈S where GS is the Galois group of the maximal extension of Q unramiﬁed outside S := {, ∞}∪Σ 1 Q D −1 1 Q D and Hf ( ,T )=ι Hf ( ,V ). By our assumption

0 Q D 1 Q D H ( ,V )=Hf ( ,V )=0,

0 Q D → 1 Q D D,∗ the natural (boundary) map H ( ,W ) Hf ( ,T ) is an isomorphism. The map ρ is Pontryagin dual to the restriction map

D 0 Q D 1 Q D ρ→ 1 Q D H ( ,W )=Hf ( ,T ) Hf ( p,T ). p∈S

D 0 Q D 0 Q D ∼ 1 Q D Clearly, ρ is injective as H ( ,W ) injects into H ( p,W ) = Hf ( p,T )tor for any p ∈ S \{∞}= ∅. This argument also shows that ρD,∗ restricted to

1 Q H ( p,W) ∼ 1 Q D ∗ L := 1 = Hf ( p,T ) H (Qp,W) p∈Σ∪{∞} f p∈Σ∪{∞} is still surjective since the dual map is still injective. On the other hand we have ρ−1(L)= 1 Q 2 HΣ( ,W) which yields the lemma.

Suppose now that Kλ is a ﬁnite extension of Q with ring of integers Oλ and uniformizer λ. For i =1, 2,letVi be representations of GQ over Kλ which are pseudo-geometric and short as deﬁned in Sections 1.1.1 and 1.1.2 respectively. Suppose that Li is a GQ-stable Oλ-lattice in Vi and set

V =HomKλ (V1,V2),T=HomOλ (L1,L2),W= V/T. For n 1, put

n ∼ n Wn = {x ∈ W | λ x =0} = T/λ T

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 697 and note that we have a natural isomorphism

1 1 n −n H (F,W )=Ext n (L /λ L ,λ L /L ) n Oλ/λ [GF ] 1 1 2 2

n n since L1/λ L1 is free over Oλ/λ (here F = Q or F = Qp). Since Vi is short the G-modules n Li/λ Li are in the essential image of the functor

V MF0 →O − : tor λ[G] Mod of Section 1.1.2. Let 1 Q ⊆ 1 Q Hf ( ,Wn) H ( ,Wn) n be the subset of extensions of Oλ/λ [G]-modules

−n n (32) 0 → λ L2/L2 →E→L1/λ L1 → 0 so that E is in the essential image of V. Using the stability of this essential image under 1 Q O direct sums, subobjects and quotients, one checks that Hf ( ,Wn) is a λ-submodule, and 1 Q 1 Q 1 Q → that Hf ( ,Wn) is the preimage of Hf ( ,Wn+1) under the natural map H ( ,Wn) 1 H (Q,Wn+1). 1 Q O PROPOSITION 2.2. – The group Hf ( ,W) is divisible of λ-corank 0 Q − 0 d =dimKλ H ( ,V)+dimKλ V dimKλ Fil Dcrys(V ).

Moreover, the natural isomorphism

1 Q ∼ 1 Q lim−→ H ( p,Wn) = H ( p,W) n induces isomorphisms

1 Q ∼ 1 Q 1 Q ∼ 1 Q lim−→ Hur( p,Wn) = Hur( p,W), lim−→ Hf ( ,Wn) = Hf ( ,W). n n

1 Q Proof. – The divisibility of Hf ( ,W) follows from its deﬁnition, as does the fact that its 1 Q corank coincides with dimKλ Hf ( ,V)=d (see [4] for this last identity). The statement 1 concerning Hur follows from the fact that continuous group cohomology commutes with direct 1 limits. For Hf we ﬁrst note that −1 1 Q ⊆ 1 Q ιn Hf ( ,W) Hf ( ,Wn)

1 1 where ιn : H (Q,Wn) → H (Q,W) is the natural map. Indeed, on the level of extensions the −n n map ιn is given by pushout via λ L2/L2 → V2/L2, pullback via L1 → L1/λ L1,andthe H1(Q ,W) ∼ Ext1 (L ,V /L ) isomorphism = Oλ[G] 1 2 2 . Similarly, the map

1 1 π : H (Q,V) → H (Q,W)

→ → ∈ 1 Q is given by pushout via V2 V2/L2 and pullback via L1 V1.SoforE Hf ( ,V) all ﬁnite subquotients of the locally compact continuous Oλ[G]-module π(E) are in the essential image

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 698 F. DIAMOND, M. FLACH AND L. GUO

V E E ∈ 1 Q E of . Hence, if is as in (32) and ιn( )=π(E) for E Hf ( ,V) then lies in the essential image of V. So we obtain an inclusion of torsion groups

1 Q ←∼ −1 1 Q ⊆ 1 Q Hf ( ,W) lim−→ ιn Hf ( ,W) lim−→ Hf ( ,Wn) n n which is an isomorphism if and only if the induced inclusion on the λ-torsion submodules is an 1 Q isomorphism (as Hf ( ,W) is divisible). There is an exact sequence → 0 Q → 1 Q → 1 Q → (33) 0 H ( ,W)/λ Hf ( ,W1) lim−→ Hf ( ,Wn) [λ] 0, n and we need to prove that the right hand term has κ := Oλ/λ-dimension d. Pick objects Di 0 ∼ of MF so that V(D ) = L .ThenD := HomO (D , D ) is also an object of MF and ∼ i i λ 1 2 D⊗O K = D (V ) (see Eq. (1) in Section 1.1.2). Put D¯ = D /λ, D¯ = D/λ so that λ∼ λ crys ∼ i i V(D¯i) = Li/λLi and V(D¯) = W1.Forallj ∈ Z we have

j j D j D¯ (34) dimKλ Fil Dcrys(V )=dimOλ Fil =dimκ Fil .

Denote by κ-MF the category of κ-modules in MF.Then

1 Q 1 D¯ D¯ (35) dimκ Hf ( ,W1)=dimκ Extκ-MF ( 1, 2) and 0 Q 0 Q − 0 Q (36) dimκ H ( ,W)/λ =dimκ H ( ,W1) dimKλ H ( ,V) D¯ D¯ − 0 Q =dimκ Homκ-MF ( 1, 2) dimKλ H ( ,V). There is an exact sequence ¯ ¯ ¯ ¯ 0 ¯ (37) 0 → Homκ-MF (D1, D2) → Homκ,Fil(D1, D2)=Fil D 0 1−→φ D¯ D¯ D→¯ 1 D¯ D¯ → Homκ( 1, 2)= Extκ-MF ( 1, 2) 0 (see diagram (61) below for a similar computation) and the combination of (33)–(38) then shows that indeed 1 Q 2 dimκ lim−→ Hf ( ,Wn) [λ]=d. n

COROLLARY 2.3. – Suppose that L¯ is a two-dimensional GQ-representation over the ﬁnite | ∼ V D¯ D¯ MF 0 ﬁeld κ of characteristic >2 so that L G = ( ) for some object of κ- with 1 D¯ 0 ¯ ⊂ ¯ ¯ ¯ dimκ Fil =1.Letadκ L adκ L := Homκ(L, L) be the endomorphisms of trace zero. Then 1 Q 0 ¯ 0 Q 0 ¯ dimκ Hf ( , adκ L)=1+dimκ H ( , adκ L).

Proof. – From (38) applied to D¯1 = D¯2 = D¯ we have

1 Q ¯ · − 0 Q ¯ dimκ Hf ( , adκ L)=2 2 3+dimκ H ( , adκ L)

D¯ D¯ MF 1 Q and (38) applied to 1 = 2 = κ[0] (the unit object of κ- )showsthatdimκ Hf ( ,κ)= 0 Q ¯ ⊕ 0 ¯ 2 dimκ H ( ,κ).Since>2 we have adκ L = κ adκ L which gives the lemma.

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 699

Finally we record the following fact in this subsection.

LEMMA 2.4. – The set Sf deﬁned in (31) is ﬁnite.

Proof. – Suppose that λ does not divide Nk! and M¯ f,λ is reducible. Its semisimplification is of the form ψ1 ⊕ ψ2 where ψ1 and ψ2 are characters of Gal(Q(µN)/Q). The representation is necessarily ordinary at (see [30]), so one of the characters is unramified at and the other 1−k → has restriction χ on I where χ : GQ Aut(µ) is the cyclotomic character. It follows that k−1 ap ≡ p +1modλ for all p ≡ 1modN. If this holds for infinitely many λ,thenweget k−1 ap = p +1for all such p, violating the Ramanujan conjecture (a theorem of Deligne [15]). Having established irreducibility of M¯ f,λ for all but finitely many λ, the proof is then finished by the following lemma. 2

LEMMA 2.5. – Suppose that λ does not divide N(2k − 1)(2k − 3)k!.IfM¯ f,λ is irreducible, then its restriction to GF is absolutely irreducible.

Proof. – Consider the restriction of M¯ f,λ to I. By results of Deligne and Fontaine (see [30]), 1−k ⊕ 1−k ⊕ (1−k) this restriction has semisimpliﬁcation of the form χ 1 or ψ ψ (after extending scalars if necessary), where ψ is a fundamental character of level 2, according to whether or not a is a unit mod λ. Suppose that M¯ f,λ is irreducible but its restriction to GF is not absolutely irreducible. Then (after extending scalars) M¯ f,λ is induced from a character of GF , and its restriction to I is induced from a character of its subgroup of index 2. It follows that the ratio of the characters 2 into which this restriction decomposes is quadratic. Since ψ has order − 1, this forces either ( − 1)|2(k − 1) or ( +1)|2(k − 1) and we arrive at a contradiction. 2

2.2. Order of vanishing

D Suppose that M is an L-admissible object of PMK and let M =HomK (M,K(1)).We recall the conjectured order of vanishing of L(M,s) at s =0[41, III. 4.2.2].

CONJECTURE 2.6. – Let τ : K → C be an embedding and λ any ﬁnite prime of K.Then

1 Q D − 0 Q D ords=0 L(M,τ,s)=dimKλ Hf ( ,Mλ ) dimKλ H ( ,Mλ ).

THEOREM 2.7. – Conjecture 2.6 holds for both M = Af and M = Bf if λ is not in Sf .More precisely, we have ords=0 L(Af ,τ,s)=ords=0 L(Bf ,τ,s)=0and

0 Q ∼ 1 Q ∼ 1 Q ∼ 0 Q ∼ H ( ,Af,λ) = Hf ( ,Af,λ) = Hf ( ,Bf,λ) = H ( ,Bf,λ) = 0 if λ/∈ Sf . Proof. – Lemma 2.12 below shows that nv L(Af ,τ,1) = L (Af ,τ,1) Lp(Af ,τ,1)

p∈Σe(f) is a nonzero multiple of the Petersson inner product of f with itself and hence it follows that L(Bf ,τ,0) = L(Af ,τ,1) =0 for each τ. It follows from the functional equation (24) that

(k − 1)(A ) (38) L(A ,τ,0) = f L(A ,τ,1) =0 f 2π2 f

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 700 F. DIAMOND, M. FLACH AND L. GUO for each τ as well. The absolute irreducibility of Mf,λ for each λ implies that

EndKλ[GQ](Mf,λ)=Kλ,

0 so H (Q,Af,λ)=0, and since Mf,λ is not isomorphic to Mf,λ(1),wealsohave 0 H (Q,Bf,λ)=0. It follows then from [41, II.2.2.2] (see also [32, Corollary 1.5]) that

1 Q 1 Q dimKλ Hf ( ,Af,λ)=dimKλ Hf ( ,Bf,λ)

1 Q for all λ and hence that Theorem 2.7 is implied by the vanishing of Hf ( ,Af,λ). Theorem 3.7 shows that H1(Q,A / ad0 M ) ⊂ H1 (Q,A / ad0 M ) f f,λ Oλ f,λ Σ f,λ Oλ f,λ is ﬁnite for λ in Sf . Since the kernel of

H1(Q,A ) → H1(Q,A / ad0 M ) f f,λ f f,λ Oλ f,λ

O 1 Q 2 is ﬁnitely generated over λ we deduce Hf ( ,Af,λ)=0and Theorem 2.7 follows.

2.3. Deligne’s period

We now recall the formulation in [41] of Deligne’s conjecture [17] for the “transcendental part” of L(M,0) for M = Af or Bf . The authors there actually discuss the more general conjecture of Beilinson concerning the leading coefﬁcient L∗(M,0) for premotivic structures arising from motives, but their formulation relies on the conjectural existence of a category of mixed motives with certain properties. We restrict our attention to those M,suchasAf and Bf ,forwhich L(M,0) =0 and which are critical in the sense of Deligne. In that case Beilinson’s conjecture reduces (conjecturally) to Deligne’s, which can be stated without reference to the category of mixed motives. Under these hypotheses, the fundamental line for M is the K-line deﬁned by

+ ∆f (M)=HomK (detK MB , detK tM )

+ 0 where indicates the subspace ﬁxed by F∞ and tM = MdR/ Fil MdR. Furthermore the composite

∞ −1 R ⊗ + → C ⊗ + (I−→) R ⊗ → R ⊗ MB ( MB) MdR tM is an R ⊗ K-linear isomorphism. Its determinant over R ⊗ K deﬁnes a basis for R ⊗ ∆f (M) called the Deligne period, denoted c+(M).

CONJECTURE 2.8. – There exists a basis b(M) for ∆f (M) such that L(M,0) 1 ⊗ b(M) = c+(M).

There are various rationality results for L(Af , 0) and L(Bf , 0) in the literature (see for example [73, Theorem 2.3]) although the precise relationship with Conjecture 2.8 for M = Af or Bf is not always clear. In this section we recall the proof of Conjecture 2.8 for M = Af and Bf and give convenient natural descriptions for b(Af ) and b(Bf ).

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 701

+ We begin by observing that Af,B and tAf are one-dimensional over K. Furthermore, complex conjugation

F∞ : Mf,B → Mf,B + has trace zero and commutes with F∞,soitisabasisforAf,B. Note also that the natural map

k−1 k−1 Af,dR → HomK (Fil Mf,dR,Mf,dR/ Fil Mf,dR) factors through an isomorphism

→ k−1 k−1 (39) tAf HomK (Fil Mf,dR,Mf,dR/ Fil Mf,dR).

The fundamental line ∆f (Af ) can therefore be identiﬁed with

k−1 k−1 HomK (Fil Mf,dR ⊗ Q · F∞,Mf,dR/ Fil Mf,dR).

We shall describe b(Af ) by specifying the image of the canonical basis f ⊗ F∞ for k−1 Fil Mf,dR ⊗ K · F∞ where we view f as an element of Mf,dR by (15). Recall that we deﬁned in (16) a perfect alternating pairing

· , · : Mf ⊗K Mf → Mψ(1 − k), and this induces an isomorphism k−1 k−1 Mf,dR/ Fil Mf,dR → HomK Fil Mf,dR,Mψ(1 − k)dR .

We shall eventually define b(Af ) by specifying the element f,b(Af )(f ⊗F∞) of Mψ(1 − k)dR. + We can make a similar analysis of the fundamental line ∆f (Bf ). One finds that Bf,B and tBf + − ⊗ Q are two-dimensional over K. Note that Bf,B can be identified with Af,B (1)B and that the natural map − → + − ⊕ − + Af,B HomK (Mf,B,Mf,B) HomK (Mf,B,Mf,B) defined by restrictions is an isomorphism. We therefore have an isomorphism

+ → detK Bf,B K(2)B

∧ −1 − which is canonical up to sign. To ﬁx the choice of sign, we use α α as a basis for detK Af,B + → − where α : Mf,B Mf,B is any K-linear isomorphism. Next note that the natural map k−1 Bf,dR → HomK Mf,dR,Mf (1)dR → HomK Fil Mf,dR,Mf (1)dR factors through an isomorphism

→ k−1 tBf HomK (Fil Mf,dR,Mf (1)dR).

Using the isomorphism

k−1 k−1 detK Mf,dR → Fil Mf,dR ⊗K (Mf,dR/ Fil Mf,dR)

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 702 F. DIAMOND, M. FLACH AND L. GUO

(with choice of sign again indicated by the ordering), we ﬁnd that detK tBf is naturally isomorphic to k−1 k−1 (40) HomK (Fil Mf,dR,Mf,dR/ Fil Mf,dR) ⊗ Q(2)dR.

We can therefore identify ∆f (Bf ) with k−1 k−1 (41) HomK Fil Mf,dR ⊗ Q(2)B, (Mf,dR/ Fil Mf,dR) ⊗ Q(2)dR , and we arrive at a canonical isomorphism ⊗ Q ⊗ + →∼ ∆f (Af ) ∆f (2) Af,B ∆f (Bf ).

+ Q 2 −2 Fixing the basis F∞ of Af,B and the basis β of ∆f ( (2)) which sends (2πi) to ι , this deﬁnes an isomorphism of K-lines

(42) tw : ∆f (Af ) → ∆f (Bf ) so that tw(φ)(x ⊗ y)=φ(x ⊗ F∞) ⊗ β(y).

LEMMA 2.9. – We have 1 (R ⊗ tw) c+(A ) = − c+(B ). f 2π2 f ∞ C ⊗ ∼ C ⊗ Proof. – Let IM : Mf,dR = Mf,B be the comparison isomorphism for Mf . Via the C ⊗ ∼ C ⊗ ∞ natural isomorphism EndK (Mf )? = EndC⊗K ( Mf,?) where ?=B or ?=dR, IM ∞ ∞ ∞ ◦ ◦ ∞ −1 induces the comparison isomorphism I for both End(Mf ) and Af : I (φ)=IM φ (IM ) . + A similar formula holds for c (Af ). R ⊗ R ⊗ k−1 ∞ + − Suppose now that x is a K-basis of Fil Mf,dR and write IM (x)=y + y with ± ∈ C ⊗ ± y Mf,B.Then

+ ⊗ ∞ −1 ⊗ ∞ ∞ −1 + − − R ⊗ k−1 c (Af )(x F∞)=(IM ) (1 F∞)IM (x)=(IM ) (y y )mod Fil Mf,dR.

On the other hand we have α(y+)=λy− for some λ ∈ (C ⊗ K)× and therefore α−1(y−)= λ−1y+. Hence ∞ −1 ∧ ∞ −1 −1 ∞ −1 + ∧ ∞ −1 −1 − (I ) (α)(x) (I ) (α )(x)=(IM ) α(y ) (IM ) α (y ) ∞ −1 − ∧ ∞ −1 −1 + =(IM ) λy (IM ) λ y 1 = (I∞)−1(y+ + y−) ∧ (I∞)−1(y+ − y−) 2 M M 1 = x ∧ (I∞)−1(y+ − y−) 2 M + and in the description (41) of R ⊗ ∆f (Bf ) the element c (Bf ) is given by 1 − c+(B ) x ⊗ (2πi)2 ⊗ ι2 =(2πi)2 (I∞)−1(y+ − y−) mod R ⊗ Filk 1 M f 2 M f,dR 2 + = −2π c (Af )(x ⊗ F∞). In view of the deﬁnition of tw in (42) this gives the lemma. 2 nv −s −1 Recall that Σe(f) is the set of primes p such that Lp (Af ,s)=1but Lp(Af ,s)=(1+p ) . We write bdR for the basis of Mψ,dR deﬁned in Section 1.1.3, and pick η ∈{0, 1} so that

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 703

× η ≡ k mod 2. Note that by Proposition 5.5 of [17], we have (Mf ⊗ Mψ−1 )/(Mψ−1 ) ∈ K . × The same proposition together with (20) gives (Af ) ∈ K .

THEOREM 2.10. – Let b(Af ) ∈ ∆f (Af ) be deﬁned by the formula

− k η − 2 ⊗ − i ((k 2)!) (Mf Mψ 1 ) −1 k−1 f,b(Af )(f ⊗ F∞) = (1 + p ) · (bdR ⊗ ι ), 2(Mψ−1 )(Af ) p∈Σe(f) and b(Bf ) ∈ ∆f (Bf ) by the formula (43) b(Bf )=(1− k)(Af )tw b(Af ) .

+ + Then L(Af , 0)(1 ⊗ b(Af )) = c (Af ) and L(Bf , 0)(1 ⊗ b(Bf )) = c (Bf ). Proof. – If we show

− k η − − ⊗ − nv + i (k 1)!(k 2)!(Mf Mψ 1 )L (Af , 1) k−1 ⊗ ∞ · ⊗ f,c (Af )(f F ) = 2 (bdR ι ) 4π (Mψ−1 ) in C ⊗ Mψ(1 − k)dR, then the statement concerning b(Af ) is an immediate consequence of the + functional equation (38). The identity L(Bf , 0)(1 ⊗ b(Bf )) = c (Bf ) then follows by applying + (R ⊗ tw) to the identity L(Af , 0)(1 ⊗ b(Af )) = c (Af ) and using (38) and Lemma 2.9. × ab −1 As in Section 1.4.2 put U = U0(N),letσ : U → K be the representation → ψ (aN ) cd and set M(N,ψ)=M(σ)=M(N )(σ) for some N 3 so that UN ⊆ U.Putw = 0 −1 ∈ A → ⊗ −1 ◦ N 0 GL2( f ) and denote by W =[UwU]ω : M(N )(σ) M(N )(ˆσ (ψ det)) the −1 ∈ isomorphism in (13). Note that wwN U so that we can work with w instead of wN .Forany one-dimensional K-representation σ of U whose kernel contains UN we shall view M(N )(σ) ∞ as a sub-PMK -structure of K ⊗ M(N ). With I denoting the comparison isomorphism for both Mf and K ⊗ M(N ) we have + ∞ −1 ∞ (44) f,c (A )(f ⊗ F∞) = f,(I ) (1 ⊗ F∞)I f f −1 ∞ −1 ∞ ⊗ ∞ =[U : UN ] f,(I ) (1 F )I Wf N where this last pairing is the one defined in (12). k−1 −1 We proceed with the computation of Wf ∈ Fil M(N )(ˆσ ⊗ (ψ ◦ det))!,dR. Note that the field Kf generated by the Fourier coefficients of the newform f is either totally real or a CM field and hence has a well defined automorphism ρ induced by complex conjugation. It is known ρ ∞ ρ 2πizn that the Fourier expansion f (z)= n=1 ane is a newform of conductor N and character −1 k−1 ψ [63, 4.6.15(2)], hence represents an element of Fil M(N )(ˆσ)!,dR. Let P1,P2 be the canonical N -torsion sections on the moduli scheme X of level N introduced in Section 1.2.1, denote by ζ = P1,P2∈Γ(X,OX) their Weil pairing and consider the resulting morphism X → Spec(OF ) where F = Q(ζ). This induces isomorphisms ∼ 0 MF = H (X) and ∼ 0 U (45) Mψ = H (X) ⊗ Kψ−1◦det

2πiN where in the deﬁnition of Mψ in Section 1.1.3 we have to replace e by ζ.Then −1 Mψ ⊗K M(N )(ˆσ)! has a natural map into M(N )(ˆσ ⊗ (ψ ◦ det))! via the isomorphism (45) followed by cup product on X.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 704 F. DIAMOND, M. FLACH AND L. GUO

LEMMA 2.11. – We have

− k η ⊗ − i (Mf Mψ 1 ) ρ (46) Wf = ψ(−1) bdR ∪ f N(Mψ−1 ) where bdR is the basis of Mψ,dR deﬁned in Section 1.1.3.

Proof. – We ﬁx an embedding τ : K → C and compute the images of both sides in Sk(UN ) (we shall suppress τ in the notation and view all elements of K as complex numbers via τ). U Let φ ∈ (Sk(UN ) ⊗C Cσ) denote the element corresponding to f under the isomorphism (5). Recall that the isomorphism ∼ β : Sk(UN ) = Sk Γ(N ) t∈(Z/N Z)× was deﬁned before (5) by −1 k β(F )t γ(i) := (det γ) j(γ,i) F (gtγ) + ab 10 ∈ R ≡ − for γ GL2( ) , j cd ,z = cz + d and gt 0 t 1 mod N .Wehaveφ(xu)= −1 × σ (u)φ(x) for all u ∈ U and β(φ)t(z)=f(z) for all t ∈ (Z/N Z) since gt ∈ U and σ(gt)=1. Recall the analytic description XN = t∈(Z/N Z)× XN ,t of X and of P1,P2 from τ t −2πit/N Section 1.2.1. One checks that (τ, N ), (τ, N ) = e . Hence N /N bdR = ψ(a) ⊗ ζ ∈ C ⊗ F, a∈(Z/N Z)× 0 C when viewed as an element of HdR(XN )= t is given by t → ψ(a)e−2πiat/N = ψ(−t)−1 ψ(a)e2πia/N a∈(Z/N Z)× a∈(Z/N Z)× −1 −1 η = ψ(−t) Gψ = ψ(−t) i (Mψ−1 ,τ). ρ ρ ρ ρ If now φ ∈ Sk(UN ) corresponds to f then β(φ )t(z)=f (z) is again independent of t and the right hand side of (46) is given by

k i (M ⊗ M −1 ,τ) (47) t → f ψ ψ(t)−1f ρ. N

The perfect pairing Mf ⊗K Mf → Mψ(1 − k) and the identity of Hecke eigenvalues ∗ ∼ ∼ ⊗ − − − [63, (4.6.17)] induce an isomorphism Mf = Mf K Mψ 1 (k 1) = Mf ρ (k 1) so that the functional equation for Λ(Mf ⊗ Mψ−1 ,τ,s) can be written

−s (48) Λ(Mf ⊗ Mψ−1 ,τ,s)=(Mf ⊗ Mψ−1 ,τ)N Λ(Mf ρ ⊗ Mψ,τ,k− s).

k/2 −k + Recall that the deﬁnition (g|kγ)(z)=det(γ) j(γ,z) g(γ(z))for γ ∈ GL2(R) deﬁnes a R + → C 0 −1 right action of GL2( ) on functions g : H .PutWN = N 0 . By [63, Theorem 4.3.6] we have k −s+k/2 Λ(Mf ⊗ Mψ−1 ,τ,s)=Λ(f,s)=i N Λ(f|kWN ,k− s)

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 705 which together with (48) yields

ρ −1 k k/2 f = (Mf ⊗ Mψ−1 ,τ) i N f|kWN .

Hence (47) becomes k k/2−1 −1 (49) t → (−1) N ψ(t) f|kWN . Turning to the left hand side of (46) we have

−1 −1 (Wφ)(x):=φ(xw) and φ(wh)=φ(WQWN h)=φ(WN h)

+ where h ∈ GL2(R) and WQ ∈ GL2(Q) is the matrix with image w (resp. WN )inGL2(Af ) + (resp. GL2(R)). For γ ∈ GL2(R) we have −k det(h)−1j(h, i)kφ(γh)=det(γ)det(γh)−1j γ,h(i) j(γh,i)kφ(γh) −k =det(γ)j γ,h(i) f γh(i) 1−k/2 =det(γ) (f|kγ) h(i) . Combining these equations we ﬁnd that Wφcorresponds to

−1 k −1 k −1 t → det(h) j(h, i) (Wφ)(gth)=det(h) j(h, i) φ(ww gtwh) −1 k −1 −1 =det(h) j(h, i) σ (w gtw)φ(wh) =det(h)−1j(h, i)kσ−1(w−1g w)φ(W −1h) t N −1 −1 −1 1−k/2 | −1 = σ (w gtw)det(WN ) (f kWN ) h(i) . | 2 − k Since f kWN =( 1) f this last expression equals

−1 −1 k k/2−1 (50) σ (w gtw)(−1) N (f|kWN )(h(i)). 10 −1 t−1 ∗ −1 −1 −1 For g ≡ −1 mod N, we have w g w ≡ mod N and σ (w g w)=ψ(t )= t 0 t t 01 t ψ(t)−1. So (49) and (50) agree which ﬁnishes the proof of the lemma. 2

The deﬁnition (11) of the pairing on σ-constructions shows that (x, α ∪ y)N =(x, y) ⊗K α where α ∈ Mψ and (x, y) is the K-linear extension of the Q(1 − k)-valued pairing on M(N )L in (9). Combining this with Lemma 2.11 the last term in (44) equals

−1 ∞ −1 ∞ ρ (51) [U : UN ] (f,(I ) (1 ⊗ F∞)I f ) ⊗K αdR in C ⊗ K(1 − r)dR ⊗K Mψ,dR where − k η ⊗ − i (Mf Mψ 1 ) ∞ −1 ∞ αdR = ψ(−1) (I ) (1 ⊗ F∞)I bdR N(Mψ−1 ) − k η ⊗ − i (Mf Mψ 1 ) −1 = ψ(−1) ψ(−1) bdR N(Mψ−1 ) ∞ (with I also denoting the comparison isomorphism for Mψ). For any premotivic structure ∞ ∞ ρ ρ ρ we have (F∞ ⊗ F∞)I = I (F∞ ⊗ 1) and we have (F∞ ⊗ 1)(f )=f since f ∈ K ⊗ M(N )dR ⊂ C ⊗ K ⊗ M(N )dR. Hence

∞ −1 ∞ ρ ∞ −1 ∞ ρ (I ) (1 ⊗ F∞)I f =(I ) (F∞ ⊗ 1)I f .

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 706 F. DIAMOND, M. FLACH AND L. GUO

∼ IK Under the natural isomorphism C⊗K ⊗M(N )B = (C⊗M(N )B) the action of F∞ ⊗1⊗1 on the left hand side gets transformed into the action sending (xτ ) to τ → (F∞,τ ⊗ 1)(xτ ) where F∞,τ is complex conjugation acting on C in the factor indexed by τ. Hence the τ-component of (51) equals −1 ∞ −1 ∞ ρ [U : UN ] τ(f), (I ) (F∞,τ ⊗ 1)I τ(f ) ⊗C τ(αdR) − − − 1 − k 1 ⊗ k 1 =[U : UN ] (k 2)!(4π) φ(N ) τ(f),τ(f) Γ(N ) τ(αdR) ι where φ is Euler’s function and we have used (10). Therefore (51) equals

−1 k−1 k−1 (52) [U : UN ] (k − 2)!(4π) φ(N )(f,f)Γ(N ) · αdR ⊗ ι ¯ ¯ [Γ1(N):Γ(N )] − k−1 · ⊗ k−1 = φ(N )(k 2)!(4π) (f,f)Γ1(N) αdR ι [U : UN ] in C ⊗ Mψ(1 − k)dR where [Γ¯1(N):Γ(¯ N )] is the degree of the covering Γ(N )\H → × Γ1(N)\H. Since the maps det: U → (Z/N Z) and SL2(Z) → SL2(Z/N Z) are surjective one ﬁnds [U : UN ]=φ(N ) SL2(Z) ∩ U :SL2(Z) ∩ UN = φ(N ) Γ0(N):Γ(N ) (53) = φ(N )φ(N) Γ1(N):Γ(N ) = φ(N )φ(N)δ(N) Γ¯1(N):Γ(¯ N ) where δ(N)=1if N>2 and δ(N)=2if N 2 (note that −1 ∈ Γ1(N) iff N 2 whereas −1 ∈/ Γ(N )). Combining this with Lemma 2.12 below we ﬁnd that (52) equals

k−1 nv k−η − − i (M ⊗ M −1 ) (k 2)!(4π) · (k 1)!δ(N)Nφ(N)L (Af , 1) · f ψ · ⊗ k−1 k k+1 bdR ι φ(N)δ(N) 4 π N(Mψ−1 ) k−η nv i (k − 2)!(k − 1)!L (A , 1)(M ⊗ M −1 ) f f ψ · ⊗ k−1 = 2 bdR ι . 4π (Mψ−1 ) This ﬁnishes the proof of Theorem 2.10. 2

LEMMA 2.12. – If f is a newform of conductor N, weight k and with coefﬁcients in the number ﬁeld K, we have

(k − 1)!δ(N)Nφ(N)Lnv(A ,τ,1) τ(f),τ(f) = f Γ1(N) 4kπk+1 for any embedding τ : K → C and δ(N) as in (53). Proof. – We ﬁx τ and write f for τ(f) to ease notation. By Theorem 5.1 of [51] (essentially a reformulation of a theorem of Rankin and Shimura), we have

4kπk+1(f,f) L(k,f,ψ¯)= Γ1(N) (k − 1)!δ(N)NNψφ(N/Nψ) ¯ ¯ where L(s, f, ψ)= p Lp(s, f, ψ), ¯ −1 − ¯ 2 −s − ¯ −s − ¯ 2 −s Lp(s, f, ψ) = 1 ψ(p)αpp 1 ψ(p)αpβpp 1 ψ(p)βp p

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 707 and αp,βp are deﬁned as in Section 1.7.2 for p N and αp + βp = ap,αpβp =0for p|N. Denote by Mp the exact power of p dividing an integer M. To show the lemma it sufﬁces to show that

¯ φ(Np/Nψ,p) nv φ(Np) (54) Lp(k,f,ψ) = Lp (Af ,τ,1) Np/Nψ,p Np for all primes p.Ifp N,thisisimmediatefromSection1.7.2.IfNp = p and Nψ,p =1, 2 k−2 we have ap = ψ(p)p by [63, Theorem 4.6.17(2)] and πp(f) is special so that (54) holds true by (23). The only other case in which ap =0 is when Np = Nψ,p [63, Theorem 4.6.17]. In this case ψ¯(p)=0and hence Lp(k,f,ψ¯)=1whereas πp(f) is principal series so that nv − −1 −1 Lp (Af ,τ,1) = (1 p ) = Np/φ(Np) by (23). Finally, if Np > 1 and ap =0,then ¯ nv − −1 2 Lp(k,f,ψ)=Lp (Af ,τ,1) = 1, Np/Nψ,p > 1 and both sides in (54) equal (1 p ). Remark. – In the following, we shall not need the full precision of Theorem 2.10 but only the k−η 2 −1 fact that i ((k − 2)!) (Mf ⊗ Mψ−1 )/2(Mψ−1 )(Af ) is a unit in O = OK [(Nk!) ].This in turn is a consequence of Lemma 2.13 below.

LEMMA 2.13. – Let M be an object of PMK which is L-admissible everywhere and let τ : K → C be an embedding. Then (M,τ)=(M,τ,0) is a unit in Z[c(M)−1] where Z is the ring of algebraic integers. | ⊗ C Proof. – By deﬁnition (M,τ)= p (Dpst(Mλ Gp) Kλ,τ ,ψp,dxp) is a product over all places p of Q where the additive characters ψp and the Haar measures dxp are chosen as in [17, 5.3] and τ : Kλ → C is any extension of τ. The assumption that M is L-admissible at | ⊗ C p implies that the isomorphism class of Dpst(Mλ Gp) Kλ,τ is independent of τ .The deﬁnition of in [16, (8.12)] and [16, Theorem 6.5 (a),(b)] show that | ⊗ C | ∈ ∞ Dpst(Mλ Gp) Kλ,τ ,ψp,dxp = τ Dpst(Mλ Gp),ψp,dxp τ Kλ(µp ) .

Replacing τ by γτ , γ ∈ Aut(C/K(µp∞ )), and using the L-admissibility again, we deduce | ⊗ C ∈ ∞ from this formula that (Dpst(Mλ Gp) Kλ,τ ,ψp,dxp) K(µp ). The remark after [16, (8.12.4)] shows that can be directly expressed in terms of the λ-adic representation Mλ for λ p. Namely | | ss − | Ip −1 Dpst(Mλ Gp),ψp,dxp = 0 (Mλ Wp) ,ψp,dxp det( Frob Mλ )

ss where 0 is introduced in [16, §5] and (Mλ|Wp) is the semisimpliﬁcation of Mλ as a representation of Wp.Nowforanyλ p the Wp-representation Mλ is the restriction of a continuous Gp-representation, hence carries a Wp-stable Oλ-lattice. This implies, on the one − | Ip ∈O× hand, that det( Frob Mλ ) λ and on the other hand, via [16, Theorem 6.5(c)], that ss × 0((Mλ|Wp) ,ψp,dxp) ∈Oλ[µp∞ ] . Noting that with our choice of ψp,dxp the epsilon factor equals 1 (resp. a power of i)forp c(M) (resp. p = ∞) the lemma follows. 2

2.4. BlochÐKato conjecture

We now recall the formulation of the λ-part of the Bloch–Kato conjecture. We assume that M is a premotivic structure in PMK such that M is critical, L(M,0) =0 and Conjecture 2.8 holds. We assume that λ is a prime of K such that

0 Q ∼ 1 Q ∼ 1 Q D ∼ 0 Q D ∼ (55) H ( ,Mλ) = Hf ( ,Mλ) = Hf ( ,Mλ ) = H ( ,Mλ ) = 0.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 708 F. DIAMOND, M. FLACH AND L. GUO

This is conjectured to hold for all λ under our hypotheses on M and it implies Conjecture 2.6. If M = Af or Af (1) and λ/∈ Sf then (55) holds by Theorem 2.7. Fontaine and Perrin-Riou [41, II.4] define an Oλ-lattice δf,λ(M) in Kλ ⊗K ∆f (M).They assume K = Q, denote their lattice ∆S(T ) (where S is a finite set of primes and T is a Galois-stable lattice in Mλ) and then prove it is independent of the choice of S and T .One checks that the definition and independence argument carry over to arbitrary K by taking determinants relative to Oλ and Kλ instead of Z and Q. The arguments of [41, II.5] carry over as well, giving another description of δf,λ(M) for which we need more notation. Choose a Galois stable lattice Mλ ⊂ Mλ and a free rank one Oλ-module ω ⊂ Kλ ⊗K detK tM .Welet + + M O M ⊗ θ( λ)=det λ λ , regarded as a lattice in Kλ K detK MB via the comparison isomorphism B MD M O ⊂ D M Iλ .Welet λ =HomOλ ( λ, λ(1)) Mλ . The Tate–Shafarevich group of

H1(Q,M /M ) X(M ):= f λ λ λ 1 Q ⊗ O Hf ( ,Mλ) (Kλ/ λ)

1 Q M is always ﬁnite and can be identiﬁed with Hf ( ,Mλ/ λ) under our hypothesis 1 Q MD Hf ( ,Mλ)=0. The same holds for λ . Furthermore, by the main result of [33] (also X M X MD O [41, II.5.4.2]), ( λ) and ( λ ) have the same length. In fact, there is an λ-linear isomorphism X MD ∼ X M Q Z (56) ( λ ) = HomZ ( λ), / .

Finally, the Tamagawa ideal of Mλ relative to ω is deﬁned as 0 M 0 M · 0 M · 0 M Tamω( λ)=Tam,ω( λ) Tam∞( λ) Tamp( λ), p=

0 M M where the factors are deﬁned as in I.4.1 (and II.5.3.3) of [41]. Recall that Tamp( λ)=1if λ is unramiﬁed at p = and that

0 M 1 R M O Tam∞( λ)=FittOλ H ( , λ)= λ if is odd. The argument of [41, I.4.2.2] shows that if p = ,then

GQ 0 M 1 M p Tamp( λ)=FittOλ H (Ip, λ)tor from which it is not hard to deduce that

0 M 0 MD (57) Tamp( λ)=Tamp( λ ). M ⊗ Viewing HomOλ (θ( λ),ω) as a lattice in Kλ K ∆f (M), we have by [41, Theorem II.5.3.6]

0 Q M · 0 Q D MD FittOλH ( ,Mλ/ λ) FittOλH ( ,Mλ / λ ) δ (M)= HomO θ(M ),ω . (58) f,λ D 0 λ λ O X M · M Fitt λ ( λ ) Tamω( λ) The λ-part of the Bloch–Kato conjecture can then be formulated as follows:

CONJECTURE 2.14. – Let M in PMK be critical, b(M) as in Conjecture 2.8, λ aplaceof K such that (55) holds and δf,λ(M) as in (58).Then δf,λ(M)= 1 ⊗ b(M) Oλ.

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 709

THEOREM 2.15. – Let f be a newform and Sf the set of places deﬁned in (31).Then Conjecture 2.14 holds for both M = Af and M = Af (1) and any λ/∈ Sf . Proof. – Suppose M, b(M) and λ are as in Conjecture 2.14 and S is a set of places of Q ∞ \{ ∞} containing , and those where Mλ is ramiﬁed. Assume Σ:=S , is nonempty and −1 ∈ Σ Lp(M,0) =0for all p Σ.Putb (M)= p∈Σ Lp(M,0)b(M). By [41, Proof of I.4.2.2] we have for p ∈ Σ, 1 Q M −1 0 M FittOλ Hf ( p, λ)=Lp(M,0) Tamp( λ) −1 D MD since Lp(M,0) =0. The exact sequence of Lemma 2.1 applied to W = Mλ / λ then implies that Conjecture 2.14 is equivalent to

0 Q D MD FittOλ H ( ,Mλ / λ ) Σ (59) HomO θ(M ),ω = 1 ⊗ b (M) O D D 0 λ λ λ O 1 Q M M Fitt λ HΣ( ,Mλ / λ )Tam,ω( λ)

1 Q D MD where HΣ( ,Mλ / λ ) was defined in Section 2.1. We shall first prove Theorem 2.15 for −1 M = Bf = Af (1) in which case the condition Lp(M,0) =0 for the reformulation (59) of Conjecture 2.14 is satisfied. A 0 M B A Recall that f,λ =adOλ f,λ and put f,λ = f,λ(1). Using the identification (39) of tAf =detK tAf we let k−1 k−1 ωA = Oλ ⊗O HomO(Fil Mf,dR, Mf,dR/ Fil Mf,dR) ∼ k−1 M ⊗ O M ⊗ O k−1 M ⊗ O = HomOλ (Fil f,dR O λ, f,dR O λ/ Fil f,dR O λ) ∼ k−1 M M k−1 M = HomOλ (Fil f,λ-crys, f,λ-crys/ Fil f,λ-crys). ⊗ Q ⊗ −2 Similarly, identifying detK tBf with detK tAf (2)dR we define ωB as ωA ι . Fix a prime λ/∈ Sf and let Σ be the set of primes dividing N if N>1 or put Σ={p} for → Σ some prime λ p if N =1. The isomorphism γ : Mf Mf of Proposition 1.4 satisfies t −1 nv −1 γ = γ φ Lp (Bf , 0) p∈Σ

× by Proposition 1.4 where φ = (−a ) ψ(p)pk−1 ∈O (it is well known that δp=1 p δp=2 λ 2 k−1 2 k−2 ap = ψ(p)p or ap = ψ(p)p if δp =1[63, 4.6.17]). Moreover, γ induces an isomorphism → Σ Σ Σ Bf =HomK Mf ,Mf (1) Bf := HomK Mf ,Mf (1)

→ Σ and an isomorphism γ :∆f (Bf ) ∆f (Bf ) so that ⊗ 2 ⊗ 2 −1 ⊗ 2 ⊗ 2 γ(b) x (2πi) ι = γdRb γdR (x) (2πi) ι

∈ ∈ Σ for b ∆f (Bf ) and x Mf,dR.Forsuchb and x we have ⊗ 2 ⊗ 2 Σ −1 t ⊗ 2 ⊗ 2 x, γ(b) x (2πi) ι = γ (x),γdRγ(b) x (2πi) ι dR −1 −1 ⊗ 2 ⊗ 2 nv −1 = γdR (x),γdR γ(b) x (2πi) ι φ Lp (Bf , 0) p∈Σ −1 −1 ⊗ 2 ⊗ 2 nv −1 (60) = γdR (x),b γdR (x) (2πi) ι φ Lp (Bf , 0) . p∈Σ

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 710 F. DIAMOND, M. FLACH AND L. GUO

− Recall that Filk 1 MΣ = O·f Σ = O·γ(f) by Propositions 1.4 where O = K ∩O . f,dR λ∈Sf λ Note that if b is an Oλ-basis for Σ HomO θ(B ),ωB λ f,λ ∼ k−1 MΣ ⊗ B MΣ k−1 MΣ ⊗ O ⊗ −2 = HomOλ Fil f,dR O θ( f,λ), ( f,dR/ Fil f,dR) O λ ι , then O · Σ ⊗ 2 ⊗ 2 MΣ k−1 MΣ ⊗ O λ b f (2πi) ι =( f,dR/ Fil f,dR) O λ and hence O · Σ Σ ⊗ 2 ⊗ 2 Σ O ΣM − λ f ,b f (2πi) ι = ληf ψ(1 k)dR Σ where ηf was deﬁned before Proposition 1.4. On the other hand by (60), Theorem 2.10 and the remark after the proof of Theorem 2.10, we have for λ/∈ Sf , Σ Σ Σ 2 2 Σ Oλ · f ,γb (Bf ) f ⊗ (2πi) ⊗ ι O · Σ ⊗ 2 ⊗ 2 nv −1 = λ f,b (Bf ) f (2πi) ι Lp (Bf , 0) p∈Σ 2 2 = Oλ · f,b(Bf ) f ⊗ (2πi) ⊗ ι Lp(Bf , 0)

p∈Σe(f) k−1 = Oλ(bdR ⊗ ι )=OλMψ(1 − k)dR.

Eq. (59) for M = Bf therefore reduces to

0 D D FittO H (Q,B /B ) λ f,λ f,λ Σ O 0 ηf = λ. O 1 Q D BD B Fitt λ HΣ( ,Bf,λ/ f,λ)Tam,ωB ( f,λ)

A BD Using Proposition 2.16 below, the fact that f = f and the vanishing of the group 0 H (Q,Af,λ/Af,λ) for λ ∈ Sf , this identity reduces to

1 Q A O Σ FittOλ HΣ( ,Af,λ/ f,λ)= ληf which is Theorem 3.7. By (56), (57) and Proposition 2.16, the factor in front of Hom(θ(Mλ),ω) in (58) is the same A for M = Af and M = Bf . The isomorphism tw defined in (42) maps HomOλ (θ( f,λ),ωA) to B HomOλ (θ( f,λ),ωB), hence δf,λ(Af ) to δf,λ(Bf ). Theorem 2.15 for M = Af therefore follows from Theorem 2.15 for M = Bf , together with Theorem 2.10 and the fact that (1 − k)(Af ) is a unit in Oλ. 2 0 0 ROPOSITION A B O ∈ P 2.16. – We have Tam,ωA ( f,λ)=Tam,ωB ( f,λ)= λ for λ/Sf . Proof. – With the notation in Section 1.1.1, we further denote by MF the additive category of filtered φ-modules as defined in [36, 1.2.1], by Kλ-MF the category of Kλ-modules in MF and a by Kλ-MF the full subcategory of Kλ-MF with filtration restrictions as in Section 1.1.1. Scalar −⊗ Q O MF → extension Z induces an exact functor λ- Kλ-MF where the notion of exactness in MF is defined in [36, 1.2.3]. a Now assume that D1, D2 are torsion free objects of Oλ-MF for some a and put Di = D ⊗ Q D D D MF i Z .Set =HomOλ ( 1, 2) and D =HomKλ (D1,D2) which are objects of and

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 711

∼ D⊗ Q MF respectively. We also have D = Z . An elementary computation shows that the ﬁrst two rows in the following commutative diagram are exact

− 0 1 φ π 1 HomO MF (D1, D2) 0 D D ExtO MF (D1, D2) 0 λ- Fil λ- 0

0 1

1−φ π 0 1 0 HomK (D1,D2) D Ext (D1,D2) 0 λ-MF Fil D Kλ-MF

θ0 ι ι θ1

1−φ 0 0 e 1 Q 0 H (Q,V) Fil D(V ) D(V ) Hf ( ,V) 0

(id,0)

0 ⊕ 1 Q 0 H (Q,V) D(V ) D(V ) tV Hf ( ,V) 0 (61) where π is defined as follows. For η ∈D, define an extension Eη of D1 by D2 in Oλ-MF with underlying Oλ-module D2 ⊕D1, filtration

i i i Fil Eη := Fil D2 ⊕ Fil D1

i i and Frobenius maps φ :Fil Eη →Eη (62) φi(x, y)= φi(x)+ηφi(y),φi(y) .

Thesamedefinitionsforη ∈ D lead to an extension in Kλ-MF. Then π(η) is the class of the 1 1 Yoneda extension Eη in Ext (we shall identify Ext with the group of Yoneda extensions throughout). To explain the remaining part of diagram (61), we first recall the notion of admissibility from ⊗ ∼ [36, 3.6.4]. A filtered φ-module D in MF is called admissible if the natural map Bcrys Q D = ⊗ Bcrys Q V (D ) is an isomorphism where V (D ) is the G-representation

0 φ⊗φ=1 V (D )=Fil (D ⊗ Bcrys) .

The functor D → V (D) is fully faithful and exact on the category of admissible ﬁltered φ-modules, and induces an equivalence of this category with the category Repcris(G) of D ⊗ Q D crystalline Kλ[G]-representations (see [36, 3.6.5]). If D = Zl l for some object of MF 0,thenD is admissible by [39, Theorem 8.4], and for such D we have a natural ∼ V D ⊗ Q D ⊗ Q D MFa isomorphism V (D ) = ( ) Z by (1). If D = Zl l for some object of then we can extend the deﬁnition of V by V(D)=V(D[−a])(a) (Tate twist) and we deduce again that D is admissible. In particular, D1 and D2 are admissible, and then D is admissible by [36, Proposition 3.4.3]. Putting V := V (D) and Vi := V (Di) we have an isomorphism of

G-representations V =HomKλ (V1,V2) by [36, 3.6]. Coming back to diagram (61), the map ι is just the natural map induced by

1−→⊗− ⊗ ∼ ⊗ ← 0 Q ⊗ D Bcrys Q D = Bcrys Q V H ( ,Bcrys Q V )=:D(V )

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 712 F. DIAMOND, M. FLACH AND L. GUO and e is the boundary map in Galois cohomology induced from the short exact sequence of

GQ -modules

→ → 0 ⊗ 1−−→φ⊗φ ⊗ → (63) 0 V (D) Fil (Bcrys Q D) Bcrys Q D 0

a as in the proof of [4, Lemma4.5(b)]. It is clear that Kλ-MF is closed under extensions inside Kλ-MF, hence we obtain a chain of isomorphisms

i i i θ :Ext (D1,D2) ← Ext a (D1,D2) Kλ-MF Kλ-MF i i →v Exti (V ,V ) →∆ Exti K , Hom (V ,V ) → Hi (Q ,V) Repcris(G) 1 2 Repcris(G) λ Kλ 1 2 f for i =0, 1.Here∆1 sends a Yoneda extension

0 → V2 → V3 → V1 → 0

· ⊆ to the pull back to Kλ 1V1 HomKλ (V1,V1) of the induced extension → → → → 0 HomKλ (V1,V2) HomKλ (V1,V3) HomKλ (V1,V1) 0.

The maps vi (deﬁned by applying V to a Yoneda extension) are isomorphisms because V is fully faithful and exact. The three lower rows in (61) with the indicated maps form a commutative diagram, and all these rows are exact (see [4, Lemma 4.5(b)] for the two lower rows). We shall verify the identity θ1π = eι, all the others being straightforward. Consider the commutative diagram

1−φ⊗φ 0 ⊗ B ⊗Q D 0 V2 Fil (Bcrys Q D2) crys 2 0

1⊗ψ

0 ⊗ ⊕ φ=1 0 ⊗ φ=1 0 V2 Fil (Bcrys Q (D2 D1)) Fil (Bcrys Q D1) 0 (64) where all unnamed arrows are natural projection or inclusion maps, the top row is (63) with D replaced by D2, and the action of φ on D2 ⊕ D1 is given by (62). For ψ ∈ D, the extension eι(ψ) ⊗ ⊂ ⊗ 1 −1 is the pullback of (63) under Kλ(1 ψ) Bcrys Q D. To compute (∆ ) eι(ψ) apply the − exact functor HomKλ (V1, ) to diagram (64). Via the isomorphisms ⊗ ∼ ⊗ ⊗ HomKλ (V1,Bcrys Q D2) = HomBcrys ⊗Kλ (Bcrys Q V1,Bcrys Q D2) ∼ ⊗ ⊗ = HomBcrys ⊗Kλ (Bcrys Q D1,Bcrys Q D2) ∼ = B ⊗Q Hom (D ,D ), crys Kλ 1 2 0 ⊗ ∼ 0 ⊗ HomKλ V1, Fil (Bcrys Q D2) = Fil Bcrys Q HomKλ (D1,D2) , the ﬁrst row becomes isomorphic to (63) and the image of ∈ 0 ⊗ φ=1 1V1 HomKλ (V1,V1)=HomKλ V1, Fil (Bcrys Q D1)

⊗ ⊗ 1 −1 in Bcrys Q D is 1 ψ. Hence (∆ ) eι(ψ) is represented by the lower row in (64). But from the deﬁnition of π it is immediate that the lower row in (64) is the image of π(ψ) under the functor V . This gives the identity θ1π = eι.

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 713

V D Put Ti = ( i) for i =1, 2 and T =HomOλ (T1,T2) with its natural G-action. Then, since 1 T1 is torsion free, H (Q,T) naturally identiﬁes with the set of equivalence classes of extensions of Oλ[G]-modules

(65) 0 → T2 → T3 → T1 → 0. a a Since the functor V is exact on Oλ-MF , and since Oλ-MF is closed under extensions in Oλ-MF, we obtain maps

i i ∼ i i ∼ i Θ :Ext (D , D ) ← Ext a (D , D ) → Ext (T ,T ) H (Q ,T) Oλ-MF 1 2 Oλ-MF 1 2 Oλ[G] 1 2 = analogous to the maps θi. The faithfulness of V implies that Θ0 is injective and fullness of V implies that Θ0 is surjective and that Θ1 is injective. The image of Θ1 lies in the subgroup 1 Q ∈ 1 Q | ⊗ Q ∈ 1 Q Hf ( ,T):= [T3] H ( ,T) [V3]:=[T3 Z ] Hf ( ,V) V D ⊗ Q ∼ D ⊗ Q since ( 3) Z = V ( 3 Z ) is a crystalline representation. Conversely, if (65) lies 1 Q in Hf ( ,T),theG-module T3 is a submodule of a crystalline representation V3 so that a D(V3) lies in Kλ-MF and hence T3 lies in the essential image of the Fontaine–Laffaille functor V, T3 = V(D3), say. Since V is full the extension (65) is the image of a sequence a 0 →D2 →D3 →D1 → 0 in Oλ-MF and since V is fully faithful and exact, this sequence is exact, hence represents an element of Ext1 (D , D ). We conclude that Oλ-MF 1 2

1 1 D D ∼ 1 Q (66) Θ :ExtOλ-MF ( 1, 2) = Hf ( ,T)

i i i i i i i is an isomorphism. It is clear that θ =˜ Θ where ˜ : H (Q,T) → H (Q,V) are the natural maps. The last row in (61) induces an isomorphism − − − det H0(Q ,V) ⊗ det 1 H1(Q ,V) ∼ det D ⊗ det 1 D ⊗ det 1 t Kλ Kλ Kλ f = Kλ Kλ Kλ Kλ Kλ V − ∼ det 1 t = Kλ V and the Tamagawa ideal is deﬁned in [41, I.4.1.1] so that

0 −1 1 ∼ 0 −1 (67) detO H (Q ,T) ⊗O det H (Q ,T) Tam (T )ω . λ λ Oλ f = ,ω

Using the fact that Θ0 is an isomorphism together with (66) and (61) one computes that the − left hand side in (67) equals det 1 D/ Fil0 D so that Tam0 (T )=O if ω is a basis of Oλ ,ω λ D 0 D detOλ / Fil . 0 These arguments apply to D1 = D2 = Mf,λ-crys which is an object of Oλ-MF if λ/∈ Sf , more speciﬁcally if N and >k.WehaveT1 = T2 = Mf,λ and T = Af,λ ⊕Oλ. Our choice 0 A O of ωA then ensures that Tam,ωA ( f,λ)= λ.ForBf = Af (1) we can use the same argument 1 as long as both D1 = Mf,λ-crys and D2 = Mf,λ-crys[1] are objects of Oλ-MF .Thisisthe case if >k+1or if = k +1and Mf,λ-crys has no nonzero quotient A in Oλ-MF with − Filk 1 A = A.

LEMMA 2.17. – If a ≡ 0modλ,thenMf,λ-crys has no nonzero quotient A in Oλ-MF with − Filk 1 A = A. Proof. – By [75] we know that the characteristic polynomial of φ on

2 −1 −1 k−1 M := Mf,λ-crys isX − ψ ()aX + ψ () ,

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 714 F. DIAMOND, M. FLACH AND L. GUO

2 hence φ¯ has characteristic polynomial X on M¯ := M⊗O (Oλ/λ).Since

− (68) M¯ = φ¯(M¯ )+φ¯k−1(Filk 1 M¯ ) k−1 01 dimO Fil M¯ =1 φ¯ φ¯ = and λ/λ ,themap is nonzero, hence conjugate to 00 .Since − − k−1φ¯k−1 =0on Filk 1 M¯ and because of (68), we have Filk 1 M¯ =ker(φ¯)=φ¯(M¯ ).Itis now easy to see that M¯ is a simple object in Oλ-MF: Any proper subobject N ⊂ M¯ is φ¯- − − stable, hence contained in ker(φ¯)=Filk 1 M¯ and we have Filk 1 N = N. But again by (68) we − − ﬁnd φ¯k−1(Filk 1 M¯ ) ⊆ ker(φ¯)=φ¯(M¯ ) so that N =Filk 1 N =0.IfA is a nonzero quotient − of M then A¯ is a nonzero quotient of M¯ hence equal to M¯ and we ﬁnd Filk 1 A¯ = A¯ and − Filk 1 A = A. 2

It remains to prove Proposition 2.16 for Bf,λ in the ordinary case a ≡ 0modλ (and ∼ ∗ = k +1). We use the fact that Bf = Af (1) and appeal to the following conjecture, a slight generalization (from Z to Oλ) of conjecture CEP (V ) of [67] (we also use a similar generalization of [67, Proposition C.2.6]). Let V be a crystalline representation of G over Kλ and T ⊆ V a G-stable Oλ-lattice. Let ω (resp. ω∗) be a lattice of 0 ∗ 0 ∗ detKλ D(V )/ Fil D(V ) resp. detKλ D V (1) / Fil D V (1)

⊗ ∗,−1 so than we obtain a lattice ω ω of detKλ D(V ) via the exact sequence ∗ (69) 0 → D V ∗(1) / Fil0 D V ∗(1) → D(V ) → D(V )/ Fil0 D(V ) → 0.

∗ ∈ ⊗ ∗ ⊗ ∗,−1 Let η(T,ω,ω ) Bcrys Q Kλ be such that detOλ T = η(T,ω,ω )ω ω under the ⊗ ∼ ⊗ comparison isomorphism Bcrys Q detKλ V = Bcrys Q detKλ D(V ). One shows that ∗ ∈ Qur ⊗ ∗ ∈ ⊗ η(T,ω,ω ) Q Kλ [67, Lemme C.2.8] and that in fact η(T,ω,ω ) 1 Kλ up to an Zur ⊗ O × element in ( Z λ) . ∈ Z j j+1 CONJECTURE 2.18. – For j , put hj(V )=dimKλ Fil D(V )/ Fil D(V ), and put Γ∗(j)=(j − 1)! if j>0 and Γ∗(j)=(−1)j((−j)!)−1 if j 0.Then

0 Tam,ω(T ) ∗ − ∗ O O − hj (V ) λ 0 ∗ = λ Γ ( j) η(T,ω,ω ). ∗ Tam,ω (T (1)) j

Remark. – One can show that upon taking the norm from Kλ to Q all quantities in this formula transform into the corresponding quantities obtained by viewing V as a representation Q × O× → Q× Z× over rather than Kλ. Since the norm map Kλ / λ / is injective it sufﬁces to prove the conjecture for Kλ = Q.

We make Conjecture 2.18 more explicit for V = Af,λ.Inthiscasewehavehj(V )=1for ∗ − − − hj (V ) − ∈ i = 1, 0, 1 and hj(V )=0otherwise so that j Γ ( j) = 1.Forλ/Sf , equation ⊗ ∼ ⊗ (20) shows that the isomorphism Bcrys Q detKλ V = Bcrys Q detKλ D(V ) is induced by V PMS A A ⊗ O the functor for the unit object in K , hence sends detOλ f,λ to detOλ f,dR O λ. M ⊗ O The computation of tBf in (40) works with Mf,dR replaced by f,dR O λ and the pairing (21) on Af gives a perfect pairing (Af,dR ⊗O Oλ) ⊗ (Af,dR ⊗O Oλ) →Oλ. Hence we ﬁnd ⊗ −1 A ⊗ O that ωA ωB is a basis of detOλ f,dR O λ via the exact sequence (69). We conclude that η(Af,λ,ωA,ωB)=1and that Conjecture 2.18 reduces to the assertion O 0 A O 0 A λ Tam,ωA ( f,λ)= λ Tam,ωB f,λ(1) .

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 715

Moreover, we know from the ﬁrst part of the proof that the left hand side equals Oλ if λ/∈ Sf . Now Conjecture 2.18 is shown in [66] for Kλ = Q and V an ordinary representation of

GQ (combine Proposition 4.2.5, Theorem 3.5.4 of loc. cit.) under the assumption of another conjecture Rec(V) which has meanwhile been proved in [12]. Ordinarity of Af,λ is implied by ordinarity of Mf,λ which in turn is implied by a ≡ 0modλ. This ﬁnishes the proof of Proposition 2.16. 2

3. The TaylorÐWiles construction

Our method for computing the Selmer group of Af,λ is based on that of Wiles [88] and his work with Taylor [86]. We ﬁrst give an axiomatic formulation of the method of [88] and [86], made possible by the simpliﬁcations due to Faltings ([86], appendix), Lenstra [60], Fujiwara [42] and one of the authors [24]. This formulation makes no reference to deformation rings and group rings that appear in other axiomatizations of the method. We then verify these axioms in the context of modular forms of higher weight.

3.1. An axiomatic formulation

In this section, we fix a prime λ of a number field K and let κ = OK /λ.Welet denote the rational prime in λ and F the quadratic subfield of Q(µ). We also fix a continuous, odd, irreducible representation

ρ0 : GQ → Autκ(V0) where V0 is two-dimensional over κ. We define the Serre weight k of ρ0 as in [78], but using geometric normalizations. (Thus k is the integer associated in [78] to the representation on Homκ(V0,κ).) We impose the following three conditions on the representations ρ0 we consider: • ρ0 has minimal conductor among its twists. • The restriction of ρ0 to GF is absolutely irreducible. • The Serre weight k of ρ0 satisfies 2 k − 1. The last condition is equivalent to ρ0|I being equivalent over κ¯ to a representation of the form • 1−k ⊕ (1−k) ψ ψ where ψ is a fundamental character of level two, 1 ∗ • or 1−k ,peuramifiéifk =2. 0 χ →O× 1−k −1 We let ψ : GQ λ denote the Teichmüller lift of χ (det ρ0 ); thus ψ is unramified at and −1 1−k −1 1−k has order prime to ,andψ χ is a lift of detρ0.Weletδ denote ψ χ . We consider continuous geometric -adic representations

→ ρ : GQ AutKρ (Vρ) where Vρ is two-dimensional over a finite extension Kρ of Kλ contained in K¯λ, ρ has determinant δ and reduction isomorphic to ρ0 over κ¯.WeletOρ denote the ring of integers of Kρ. We say such a representation ρ is an allowable lift of ρ0 if its restriction to G is short and crystalline. For a prime p = ,wesayρ is minimally ramified at p if the following hold: ∼ • If #ρ0(Ip) = ,thenρ(Ip) = ρ0(Ip). • Ip If #ρ0(Ip)=,thendimKρ Vρ =1. Suppose we are given a set N of allowable lifts. We assume the K¯λ-isomorphism classes of the elements of N are distinct. For each ρ,weletΣρ denote the set of primes at which ρ is not minimally ramified. For each set of primes

Σ ⊆ Σ0 := {p | p = }

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 716 F. DIAMOND, M. FLACH AND L. GUO

Σ Σ Σ we let N denote the set of ρ in N such that Σρ ⊂ Σ and write V for the direct sum over N ¯ ¯ ⊗ N Σ of Vρ = Kλ Kρ Vρ. We assume that is finite if Σ is finite. Σ0 A trellis for N is an OλGQ-submodule L of V such that for each finite set Σ ⊂ Σ0,theOλ- Σ ∩ Σ ¯ ⊗ Σ → Σ module L := L V is finitely generated and the map Kλ Oλ L V is an isomorphism. N ∩ ¯ ¯ ⊗ →∼ One checks that if ρ in is such that Kρ = Kλ,thenLρ := L Vρ satisfies Kλ Oλ Lρ Vρ. ∈NΣ (To see this, for each σ = ρ , choose gσ such that tr ρ(gσ) =trσ(gσ). Then the map V Σ → V Σ defined by (g2 − tr ρ(g )g − det ρ(g )) has kernel V¯ . It follows that its σ= ρ σ σ σ σ σ ρ Σ Σ ∼ → ¯ ⊗O → restriction to a map L σ= ρ L has kernel Lρ, and therefore that Kλ λ Lρ Vρ.) For such ρ,welet A =(ad0 V )/(ad0 L ). ρ Kλ ρ Oλ ρ

Ip One checks that if ρ is minimally ramiﬁed at p,thenAρ is divisible. A system of perfect pairings ϕ for L is an Oλ[GQ]-isomorphism Σ Σ Σ −1 1−k → O O ϕ : L Hom λ L , λ(ψ χ ) for each ﬁnite Σ ⊂ Σ0. Since the V¯ρ are irreducible, non-isomorphic and have determinant δ,we Σ Σ see that for each ρ in N , ϕ induces a K¯λGQ-isomorphism

∧2 V¯ → K¯ (δ) K¯ λ ρ λ

Σ Σ which we denote by ϕρ . Moreover if Kρ = Kλ,thenϕρ arises from an injection

∧2 →O Oλ Lρ λ(δ).

We say that a prime q is horizontal if the following hold • q ≡ 1mod; • ρ0 is unramified at q; • ρ0(Frobq) has distinct eigenvalues. Z Z × If Q is a finite set of horizontal primes, we let ∆Q denote the maximal quotient of q∈Q( /q ) of -power order. For each q ∈ Q, we choose an eigenvalue αq ∈ κ¯ of ρ0(Frobq) and let µq,0 × denote the unramified character Gq → κ¯ sending Frobq to αq. Suppose that ξ is a character → ¯ × ∈NQ ∈ ∆Q Kλ . We say that ρ is a ξ-lift of ρ0 if for each q Q,wehave ∼ V¯ρ = K¯λ(µq,ρ) ⊕ K¯λ(δ/µq,ρ)

¯ → ¯ × | as KλGq-modules for some lift µq,ρ : Gq Kλ of µq,0 with µq Iq corresponding via local class ﬁeld theory to ξ|∆{q}.

THEOREM 3.1. – Let N be a set of allowable lifts of ρ0 (with distinct K¯λ-isomorphism Σ classes and finite N for each finite Σ ⊂ Σ0), L a trellis for N and ϕ a system of perfect pairings for L. Suppose that •N∅ = ∅; • if Σ ⊂ Σ0 is a finite set of primes and ρ ∈N∅,then Σ ∅ Σ 0 −1 ϕρ = ϕρβρ Lp(adKρ Vρ, 1) p∈Σ

Σ O for some βρ in ρ;

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 717

• if Q is a ﬁnite set of horizontal primes, then Q ∈O ∈N∅ (i) βρ λ is independent of ρ , Q ∅ (ii) #N #N · #∆Q, and N Q → ¯ × (iii) there is a ξ-lift of ρ0 in for each ξ :∆Q Kλ . Then every allowable lift of ρ0 is isomorphic over K¯λ to some ρ in N . Furthermore if Kρ = Kλ, then the lengths of H1 (Q,A ) O (δ)/ϕΣ(∧2 L ) Σ ρ and λ ρ Oλ ρ coincide for any ﬁnite subset Σ of Σ0 containing Σρ.

Proof. – One checks that to prove the theorem, we can replace Kλ by any ﬁnite extension and so assume that κ contains the eigenvalues of the elements of the image of ρ0.Notealso ∅ that the hypotheses ensure the existence of an element ρmin of N . We may assume also that

Kλ = Kρmin , and we write simply Vmin, Lmin and Amin for Vρmin , Lρmin and Aρmin . We first recall the results we need from the deformation theory of Galois representations. See [19,61] and Appendix A of [13] for more details. We let C denote the category of complete local Noetherian Oλ-algebras. Recall that if A is an object of C with maximal ideal m,thenan A-deformation of V0 is an isomorphism class of free A-modules M endowed with continuous AGQ-action ρM : GQ → AutA M such that M/mM is (A/m)GQ-isomorphic to (A/m) ⊗κ V0. For a prime p = , we say that an A-deformation of V0 is minimally ramified at p if the following hold: ∼ • If #ρ0(Ip) = ,thenρM (Ip) = ρ0(Ip). Ip • If #ρ0(Ip)=,thenM/M is free of rank one over A. Suppose that Σ is a finite subset of Σ0. We say that M is of type Σ if the following hold: • the AGQ-module M is minimally ramified outside Σ; • ∧2 ⊗ O the AGQ-module AM is isomorphic to A Oλ λ(δ); • there exists an object A0 of C with maximal ideal m0 and finite residue field so that ∼ ⊗ Z n M = A A0 M0 and for every n>0,the G-module M0/m0 M0 is an object of the MF0 category tor. Consider the functor on C which associates to A the set of A-deformations of ρ0 of type Σ. By the results of Mazur and Ramakrishna, this functor is representable by an object of C. 2 We denote this object RΣ and let M Σ denote the universal deformation. We recall also that RΣ is O Σ Σ topologically generated over λ by the elements tg for g in GQ,wheretg denotes the trace of the endomorphism g of the free RΣ-module M Σ. In particular, RΣ has residue field κ. Σ1 Σ1 If Σ1 ⊂ Σ2,thenM is an R -deformation of V0 of type Σ2 and hence gives rise to a natural surjection RΣ2 → RΣ1 . Suppose now that ρ is in N and Σρ ⊂ Σ.ThenOρ is an object of C and there is an Oρ- ⊗ deformation M of ρ0 of type Σ so that Vρ is KρGQ-isomorphic to Kρ Oρ M. We thus obtain a continuous Oλ-algebra homomorphism Σ Σ → θρ : R Kρ

⊗ Σ Σ ⊃ so that Kρ RΣ M is isomorphic to Vρ. The maps θρ for varying Σ Σρ are compatible with Σ Σ the natural surjections R 2 → R 1 deﬁned above. Note also that if Kρ = Kλ,thenA = Oλ and Σ Σ →O θρ deﬁnes a surjection R λ. In that case we have a natural isomorphism Σ Σ 2 O ∼ 1 Q (70) HomOλ pρ /(pρ ) ,Kλ/ λ = HΣ( ,Aρ)

2 Following [19] and of [13], we note that it is not necessary to assume A has residue ﬁeld κ or to use strict equivalence classes of deformations.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 718 F. DIAMOND, M. FLACH AND L. GUO

O Σ Σ of λ-modules where pρ is the kernel of θρ . (We omit the proof, which is now standard; see for example Proposition 1.2 of [88] or Section 2 of [14], and use Proposition 2.2 above to identify the local condition on .) In particular this is the case for ρ = ρmin and any ﬁnite Σ ⊂ Σ0. We regard V Σ as a module for RΣ via Σ (71) R → Kρ ρ∈N Σ

Σ Σ Σ deﬁned by the maps θρ . Note that if g is in GQ,thentg acts on V via the endomorphism tr ρ(g) = g + δ(g)g−1

Σ Σ which is given by an element of OλGQ. It follows that L is stable under the action of R and Σ Σ Σ1 Σ2 that φ is R -linear. If Σ1 ⊂ Σ2, then regarding L as an R -module via the natural surjection to RΣ, we see that the inclusion LΣ1 → LΣ2 is RΣ2 -linear, as is its adjoint with respect to ϕΣ1 and ϕΣ2 . O Σ Σ Σ Σ We define the finite flat λ-algebra T to be the image of R in EndOλ L . The maps θρ induce an isomorphism of finite K¯λ-algebras ¯ ⊗ Σ → ¯ Kλ Oλ T Kλ ρ∈N Σ

Σ → such that tg (tr ρ(g))ρ∈N Σ for g in GQ. (The injectivity follows from that of

¯ ⊗ Σ → ¯ ⊗ Σ Kλ Oλ T Kλ Oλ EndOλ L ,

Σ Σ and the surjectivity from the distinctness of the θρ .) In particular T is reduced and

Σ · N Σ · Σ (72) rankOλ L =2 # =2 rankOλ T .

N Σ PΣ Σ Σ Suppose that ρ is an element of such that Kρ = Kλ. Write ρ for the image of pρ in T Σ PΣ Σ PΣ Σ Σ and Iρ for the annihilator of ρ in T . Note that ρ (resp., Iρ )isthesetofelementsinT whose image in K¯λ has trivial component at ρ (resp., at each ρ = ρ). Now consider the Oλ-module Σ Σ Σ PΣ Σ Σ Ωρ = L / L [ ρ ]+L [Iρ ] .

Σ O We deﬁne ηρ as the annihilator of the ﬁnite torsion λ-module

O (δ)/ϕΣ(∧2 L ). λ ρ Oλ ρ

Σ Σ Σ Σ Σ Σ We shall write pmin, Ωmin and ηmin for pρmin , Ωρmin and ηρmin . O Σ O Σ 2 LEMMA 3.2. – The λ-module Ωρ is isomorphic to ( λ/ηρ ) . Σ Proof. – Note that the kernel of the projection L → Vρ coincides with that of the surjective composite Σ →∼ Σ O → O L HomOλ L , λ(δ) HomOλ Lρ, λ(δ) ,

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 719 where the ﬁrst map is ϕΣ and the second is the natural surjection. Denoting this kernel by ⊥ ⊂ Σ PΣ ⊥ ⊂ Σ Σ Lρ ,wehaveLρ L [ ρ ] and Lρ L [Iρ ]. Furthermore both inclusions are equalities since ¯ Σ Σ ⊥ they become so after tensoring with Kλ and L /Lρ and L /Lρ are torsion-free. Therefore the O Σ λ-module Ωρ is isomorphic to the cokernel of the map → Σ ⊥ →∼ O Lρ L /Lρ HomOλ Lρ, λ(δ) induced by ϕΣ, which in turn is isomorphic to O ⊗ O Σ HomOλ Lρ, λ(δ) Oλ λ/ηρ

(in fact, canonically so as an OλGQ-module). 2 Suppose now that Q is a ﬁnite set of horizontal primes. For each q ∈ Q, we have chosen an eigenvalue αq of ρ0(Frobq). As in Lemma 2.44 of [14],

Q ∼ Q Q ⊕ Q Q M = R (µq ) R (δ/µq )

Q Q → Q × as an R Gq-module for some lift µq : Gq (R ) of µq,0. (Recall that the character µq,0 × was deﬁned before the statement of the theorem and is κ -valued since we enlarged Kλ.) The Q restriction of µq to the inertia group Iq factors through

→ Z× → Iq q ∆{q} where the first map is gotten from local class field theory and the second is the natural projection. Q × Q We thus obtain a homomorphism ∆{q} → (R ) for each q ∈ Q. We can thus regard R as an Q Q Oλ[∆Q]-algebra, and so regard L as an Oλ[∆Q]-module. Note that every ρ ∈N is a ξ-lift → ¯ × for a unique ξ = ξρ :∆Q Kλ ,andthen∆Q acts on Vρ via ξρ. Q Now let P denote the augmentation ideal of Oλ[∆Q], i.e., the kernel of the map O →O → Q PQ O λ[∆Q] λ defined by g 1 for g in ∆Q.LetI denote the annihilator of in λ[∆Q], i.e., the principal ideal generated by t = g. Now consider the O -module ∆Q g∈∆Q λ ΩQ = LQ/ LQ[PQ]+LQ[IQ] .

Q Q Q Q Q ∅ LEMMA 3.3. – L is free over Oλ[∆Q], and L /P L is isomorphic over R to L . ¯ ∼ ¯ → Proof. – Note that Kλ[∆Q] = ξ Kλ via g (ξ(g))ξ , the product being over all characters → ¯ × Q ξ :∆Q Kλ . Hypothesis (c) of the theorem ensures that this algebra acts faithfully on V , Q Q and hence that Oλ[∆Q] acts faithfully on L .Furthermore,ifρ is in N ,thenρ is in ∅ Q Q ∅ N if and only if ∆Q acts trivially on Vρ,sowehaveL [P ]=L . It follows also that ∅ ⊂ Q ⊂ ∅ Q Q ∅ ⊥ Q #∆QL t∆Q L L . Thus L [I ]=(L ) ,soΩ is isomorphic to the cokernel of the ∅ ∅ Q endomorphism νQ of L obtained by composing the inclusion L → L with its adjoint with respect to ϕQ and ϕ∅. Our hypotheses on the pairings (including (a)) ensure that Q −3 2 −1 − 2 ∈ Q νQ =#∆Qβ q qtFrobq δ (Frobq) (q +1) #∆QR . q∈Q

Q ∅ ∅ Q Q 2 ∼ Therefore Ω has length at least that of L /#∆QL .SinceP /(P ) = Oλ ⊗ ∆Q,weget

Q PQ PQ 2 lengthOλ Ω d lengthOλ /( )

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 720 F. DIAMOND, M. FLACH AND L. GUO

∅ where d is the Oλ-rank of L . On the other hand, hypothesis (b) gives

Q · N Q · · N ∅ O rankOλ L =2 # 2 #∆Q # = d rankOλ λ[∆Q].

Q Theorem 2.4 of [24] therefore implies that L is free over O[∆Q] of rank d. It follows that Q Q Q ∅ Q L /P L is free of rank d over Oλ. Since the adjoint of L → L is surjective with kernel ∼ ∅ containing PQLQ, it follows that LQ/PQLQ = L . 2 The following is proved exactly as in Chapter 3 of [88] (see Theorem 2.49 of [14]), except that we use Corollary 2.3 above instead of Proposition 1.9 of [88] (or Proposition 2.27 of [14]).

LEMMA 3.4. – There exists an integer r 0 and sets of horizontal primes Qn for each n 1 such that the following hold: • #Qn = r; n • q ≡ 1mod for each q ∈ Qn; • RQn is generated by r elements as an O-algebra. We are now ready to prove that R∅ is a complete intersection over which L∅ is free of rank two. We let r and Q = Qn for n 1 be as in Lemma 3.4. Setting A = κ[[S1,...,Sr]], ∅ ∅ B = κ[[X1,...,Xr]], R = κ ⊗O R and H = κ ⊗O L , we shall define B-modules Hn and maps φn : A → B, ψn : B → R and πn : Hn → H satisfying the hypotheses of Theorem 1.3 of → → [24]. We first choose surjective κ-algebra homomorphisms A κ[∆Qn ] and B Rn where n ⊗ Qn → ⊂ n Rn = κ O R . Note that the kernel of A κ[∆Qn ] is contained in mA mA.Defineψn as the composite B → Rn → R and define φn : A → B so the diagram

A B

κ[∆Qn ] Rn

Qn commutes. We consider Ln = κ ⊗O L as a B-module via B → Rn, and define Hn as n Qn ∅ n Ln/m Ln and πn as the map induced by L → L .ThenHn is free over A/m ,andπn A ∼ A induces Hn/mAHn → H. We can therefore apply Theorem 1.3 of [24] to conclude that R is ∅ a complete intersection over which H is a free module. Since T is finite and flat over Oλ,it follows that R∅ → T ∅ is an isomorphism since it is so after tensoring with κ. Moreover these rings are complete intersections over which L∅ is a free module of rank 2. We now apply the implication (c) ⇒ (b) of Theorem 2.4 of [24] to the R∅-module L∅ and ∅ prime ideal pmin. We thus obtain the formula 2 · length H1(Q,A )=2· length p∅ /(p∅ )2 =length Ω∅ =2· v (η∅ ) Oλ ∅ min Oλ min min Oλ min λ min where the first equality follows from (70) and the last from Lemma 3.2. Suppose now that Σ is a finite subset of Σ0. Applying (70) and Lemma 3.2 again, together with the inequality 1 Q 1 Q − 0 lengthOλ HΣ( ,Amin) lengthOλ H∅ ( ,Amin) vλ Lp(adK Vmin, 1) p∈Σ obtained from the exact sequence of Lemma 2.1, we find that · Σ Σ 2 · 1 Q · Σ Σ 2 lengthOλ pmin/(pmin) =2 lengthOλ HΣ( ,Amin) 2 v(ηmin)=lengthOλ Ωmin.

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 721

We can then apply the implication (a) ⇒ (c) of Theorem 2.4 of [24] to conclude RΣ is a complete intersection over which LΣ is a free module of rank 2. The second assertion of Theorem 3.1 follows from another application of Theorem 2.4 of [24], (70) and Lemma 3.2. To deduce the ﬁrst assertion of Theorem 3.1, note that the map ¯ ⊗ Σ → ¯ Kλ Oλ R Kλ ρ∈N Σ induced by (71) is an isomorphism. Every allowable lift of ρ0 arises, up to K¯λ-isomorphism, ¯ ¯ ⊗ Σ Σ N 2 from a Kλ-linear map Kλ Oλ R for some Σ. It therefore arises from θρ for some ρ in .

3.2. Consequences

Let us now return to the setting of Theorem 2.15, namely that f is a newform of weight k 2, character ψ and conductor N with coefficients in a number field K,andλ is a prime of K not in the set Sf defined in (31). We let denote the rational prime in λ;letκ = OK /λ M¯ ⊗ M M and f,λ = κ OK,λ f,λ where f,λ is defined in Section 1.6.2. We then consider the representation

ρ0 : GQ → Autκ M¯ f,λ. Enlarging K and replacing f by a twist if necessary, we can assume that κ contains the eigenvalues of all elements of ρ0(GQ) and ρ0 has minimal conductor among its twists. We shall now construct a set of lifts of ρ0 from modular forms satisfying the hypotheses of Theorem 3.1. Suppose that g is a newform of the same weight k and character ψ, but any conductor Ng not divisible by . We suppose that g has coefficients in a subfield Kg of K¯λ generated over K by the coefficients of g. The inclusion of Kg in K¯λ determines a prime λg of Kg over and identifies ¯ Kg,λg with a finite extension of Kλ in Kλ. The representation

Q → ρg : G AutKg,λg Mg,λg is an allowable lift of ρ0 if and only if

(73) ap(g) ≡ ap(f)modλg for all but finitely many p.WeletN denote the set of ρg such that (73) holds. Note that the ρg ∈N are inequivalent for distinct g. From the work of Ribet and others, one knows that N∅ is non-empty. (See the discussions ∅ following Theorem 1 of [27] and Corollary 1.2 of [21].) Choose fmin ∈N and enlarge K if necessary so that Lemma 1.5 holds for f = fmin and all primes p ∈ Σ1. For each finite subset Σ Σ of Σ0 = Sf (Q) \{}, we consider the Oλ[GQ]-module M(σ )!,λ defined in Section 1.8.1, and endowed with an action of T˜ΨΣ in Section 1.8.2. Let mΣ denote the maximal ideal of Σ∪{r} T˜ΨΣ T˜ΨΣ → → defined there, i.e., the kernel of the map κ defined by tp ap(fmin )modλ ∈ Σ M Σ for p/ΨΣ.WeletL = (σ )!,λ,mΣ . Q If Q is a finite set of horizontal primes for ρ0,thenweletDQ = q∈Q q and define U1 as the kernel of the homomorphism ∅ → Z Z × → U0(N1 DQ) ( /q ) ∆Q q∈Q

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 722 F. DIAMOND, M. FLACH AND L. GUO ab where the ﬁrst map sends cd to (aq)q and the second is the natural projection. We let Q ∅ Q O M Q σ1 denote the restriction of σ to U1 . Then the λ[GQ]-module (σ1 )!,λ is endowed T˜Ψ∅ Q T˜Ψ∅ → with an action of and we let m1 denote the kernel of the map κ deﬁned by → {r} ∈ ∪ → ∈ tp ap(fmin)modλ for p/Ψ∅ Q and tq αq for q Q,whereαq is a chosen eigenvalue of Q M Q ∈NQ ρ0(Frobq).WeletL1 = (σ1 ) Q . Recall that for each ρ , there is a unique character !,λ,m1 → ¯ × ξρ :∆Q Kλ such that ρ is a ξρ-lift of ρ0.Wealsouseξρ to denote the corresponding character Q Q ∼ Z Z × → of GQ factoring through Gal( (µDQ )/ ) = ( /DQ ) ∆Q.

LEMMA 3.5. – There is a K¯λ[GQ]-linear isomorphism

Σ ¯ ⊗ Σ →∼ Σ ι : Kλ Oλ L V for each ﬁnite subset Σ of Σ0, and Q ¯ ⊗ Q →∼ ¯ −1 ι1 : Kλ Oλ L1 Vρ(ξρ ) ρ∈NQ for each ﬁnite set Q of horizontal primes for ρ0.

TΣ O ⊗ T˜ΨΣ M Σ ⊗ Proof. – Let denote the image of λ OK in EndOλ (σ )!,λ. Identifying Kλ Oλ Σ Σ Σ Σ Σ T T T ⊗O mΣ with the product of p over minimal primes p contained in m , we see that Kλ λ L Σ is isomorphic to the direct sum of M(σ )!,λ,p for such p. Σ Suppose that g is a newform with coefficients in Kg and ρg ∈N (where K ⊂ Kg ⊂ K¯λ as in the definition of N ). The Σ-level structures σΣ defined in Section 1.8.1 are then the same min for f and g (in fact, if p/∈ Σ then cp(fmin)=cp(g) and δp(fmin)=δp(g),ifp ∈ Σ then ∈ \ V cp(fmin)+δp(fmin)=cp(g)+δp(g),andifp Σ1 Σ then the representation p defined in Lemma 1.5 for f min works for g as well). We thus have

Σ ⊂ ⊗ Σ Mg,1,λg Kg,λg Kλ M(σ )!,λ

TΣ → → Σ∪{r} ∈ Σ giving rise a homomorphism Kg,λg defined by Tp ap(g ) for p/ΨΣ. Letting pg Σ ⊂ ΣTΣ Σ Σ denote its kernel, we have pg m . Moreover pg = pg if and only if g and g are conjugate under GKλ . Next we check that every such minimal p ⊂ mΣTΣ arises this way. Indeed every minimal Σ Ψ prime p of T is the kernel of a homomorphism T˜ Σ → Kh arising from an eigenform h of ∪{ } Z Σ r V ⊗ GL2( p) level N . Moreover the newform g associated to h satisfies ( p OK ,τ πp(g)) =0 ∈ \ → C Σ∪{r} V ⊂ ΣTΣ for all p Σ1 Σ, τ : K (where N and p are defined using fmin). If p m , then (73) holds, so ρg is an allowable lift of ρ0. In particular, we have that ρg is unramified at r,socr(g)=0. Combining the inequality cp(g) cp(fmin) for p/∈ Σ with the condition on πp(g) for p ∈ Σ1 \ Σ, we conclude that ρg is minimally ramified outside Σ. Finally, the condition Σ that Tp ∈ m for p ∈ Σ ∪{r} implies that ap(h) ∈ λh for such p, from which we deduce that Σ∪{r} ap(h)=0and therefore that h = g . ⊂ ΣTΣ Σ We have now shown that the set of minimal p m is precisely the set of pg where g runs ∈NΣ TΣ over GKλ -orbits of newforms g such that ρg .Forsuchp,wehavethat p is reduced, so ⊗ M Σ ∼ M Σ ¯ that Kλ Oλ (σ )!,λ[p] = (σ )!,λ,p. Extending scalars to Kλ gives ¯ ⊗ M Σ ∼ ¯ ⊗ Σ Kλ Kλ (σ )!,λ,p = Kλ Kg,λg Mg,1,λg , { | Σ } g pg =p

◦ 4e SÉRIE – TOME 37 – 2004 – N 5 THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES 723 and summing over p gives the desired isomorphism. Q The construction of ι1 is similar, so we omit the details and note the only signiﬁcant difference Q in the proof. Starting with a newform g such that ρg is a ξ-lift of ρ0 in N , we consider ⊗ Q Σ 2 the newform associated to g ξ. This in turn gives an eigenform g1 of level N DQr with Q Q ∈ TQ ar(g1 )=0 and aq(g1 ) reducing to αq for all q Q. Working with a Hecke algebra 1 TΣ TQ QTQ deﬁned analogously to above, one checks that the minimal primes of 1 contained in m1 1 Q 2 correspond to the GKλ -orbits of the eigenforms g1 .

O Σ Σ → Σ Recall that in Section 1.8.3 we defined an λ[GQ]-linear homomorphism γˆΣ : L L for Σ ⊂ Σ and proved in Lemma 1.11 that it is injective with torsion-free cokernel as an Σ Σ Σ Σ O -module. Recall also that γˆ =ˆγ ◦ γˆ if Σ ⊂ Σ ⊂ Σ , so we can consider L := lim L λ Σ Σ Σ −→ ⊂ Σ ⊂ over all finite Σ Σ0 with respect to the inclusions γˆΣ for Σ Σ . Note that the isomorphisms Σ Σ Σ ⊂ Σ ι in Lemma 3.5 can be chosen so that the γˆΣ are compatible with the inclusions V V . Taking their direct limit, we get an isomorphism ¯ ⊗ ∼ ι : Kλ Oλ L0 = Vρ ρ∈N so that ι(L) is a trellis with ι(L)Σ = ιΣ(LΣ) and with a system of perfect pairings provided by Corollary 1.6. We now verify the remaining hypotheses of Theorem 3.1. The second bullet and part (a) of the third follow from Proposition 1.12(b). To establish parts (b) and (c), we appeal to Lemma 1.10 ∪ −1 − with Ψ=Σ1 Q. Combined with Lemma 3.5, this implies that ρ∈N Q Vρ(ξρ ) is a free ¯ −1 2 Kλ[∆Q]-module with ∆Q acting on each Vρ(ξρ ) via ξρ. We conclude that the number of ξ- Q lifts of ρ0 in N is independent of ξ, giving (b) and (c). → THEOREM 3.6. – Suppose ρ : GQ AutKλ V is a continuous geometric representation whose restriction to G is ramified, crystalline and short. If ρ0 is modular and its restriction to GF is absolutely irreducible, where F is the quadratic subfield of Q(µ),thenρ is modular.

Proof. – Note that we may enlarge Kλ in order to prove the theorem. We can then apply Theorem 3.1 to the set N just constructed for the twist ρ0 ⊗κ ψ of minimal conductor, where ⊗ ˜ ψ is unramiﬁed at . Writing˜for Teichmüller liftings, we conclude that ρ Kλ ψ ψ is modular, 1−k 2 2 where ψ is a character of -power order such that χ ψ detρ has order not divisible by .

THEOREM 3.7. – Let f be a newform of weight k 2 and level N with coefficients in the number field K. Suppose that λ is a prime of K not in the set Sf defined in (31), and let Oλ be the ring of integers in Kλ. Suppose that Σ is a finite set of primes not containing such that Mf,λ is minimally ramified outside Σ. Then the Oλ-module

1 Q A HΣ( ,Af,λ/ f,λ)

Σ Σ has length vλ(ηf ) where ηf was deﬁned before Proposition 1.4.

Proof. – Enlarging K and applying Theorem 3.1 to the set N just constructed for the twist ρ0 ⊗κ ψ of minimal conductor, we conclude that the theorem holds for a twist of f, hence for f itself. 2

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 724 F. DIAMOND, M. FLACH AND L. GUO

Acknowledgements

Research on this project was carried out while the ﬁrst author worked at Cambridge, MIT and Rutgers, visited the IAS, IHP and Paris VII, and received support from the EPSRC, NSF and an AMS Centennial Fellowship. The second author would like to thank the IAS for its hospitality and acknowledge support from the NSF and the Sloan foundation. The third author was supported in part by an NSF grant and a research grant from University of Georgia in the early stages of this project, and thanks the IAS for its hospitality.

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(Manuscrit reçu le 19 avril 2002 ; accepté le 21 septembre 2004.)

Fred DIAMOND Department of Mathematics, Brandeis University, Waltham, MA 02454, USA E-mail: [email protected]

Matthias FLACH Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA E-mail: ﬂ[email protected]

Li GUO Department of Mathematics and Computer Science, University of Rutgers at Newark, Newark, NJ 07102, USA E-mail: [email protected]

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