ANTICYCLOTOMIC IWASAWA'S MAIN CONJECTURE FOR HILBERT MODULAR FORMS

ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS MATTEO LONGO Abstract. Let F/Q be a totally real extension and f an Hilbert modular cusp form of level n, with trivial central character and parallel weight 2, which is an eigenform for the action of the Hecke algebra. Fix a prime ℘ | n of F of residual characteristic p. Let K/F be a quadratic totally imaginary extension and K℘∞ be the ℘-anticyclotomic Zp -extension of K. The main result of this paper, generalizing the analogous result [5] of Bertolini and Darmon, states that, under suitable arithmetic assumptions, the characteristic power series of the Pontryagin dual of the Selmer group attached to (f, K℘∞ ) divides the p-adic L-function attached to (f, K℘∞ ), thus proving one direction of the Anticyclotomic Main Conjecture for Hilbert modular forms. Arithmetic applications are given. Contents 1. Introduction 2. Anticyclotomic Zp -extensions 3. CM points on quaternion algebras 3.1. Optimal embeddings and CM-points 3.2. The trace formula 3.3. Orientations and Gross points 4. p-adic L-functions 4.1. Modular forms on definite quaternion algebras 4.2. CM points on definite quaternion algebras 4.3. Anticyclotomic ℘-adic L functions 4.4. Interpolation properties 5. Selmer groups attached to Hilbert modular forms 5.1. Galois cohomology groups 5.2. Selmer groups 6. Iwasawa’s Main Conjecture 6.1. The main result 6.2. Applications to modular abelian varieties 7. The proof 7.1. The divisibility criterion 7.2. Admissible primes 7.3. Rigid pairs 7.4. Congruences between modular forms and the Euler system 7.5. Explicit reciprocity laws 7.6. The argument 2000 Mathematics Subject Classification. 11G10, 11G18, 11G40. Key words and phrases. Hilbert Modular Forms, Iwasawa Theory. 1 2 5 6 6 7 7 8 8 8 9 10 10 10 11 15 15 15 17 17 20 20 24 31 34 2 MATTEO LONGO References 38 1. Introduction Let F/Q be a totally real extension of degree d := [F : Q] and n a square-free integral ideal of the ring of integers OF of F . Let f ∈ S2 (n) be a Hilbert modular cusp form for the Γ0 (n) level structure with trivial central character and parallel weight 2. Let Tn be the Hecke algebra generated over Z by the Hecke operators acting on S2 (n). Assume that f is a normalized eigenform for the action of Tn and denote by φf : Tn → Q the morphism corresponding to f . Let aq (f ) := φf (Tq ) (respectively, aq (f ) := φf (Uq ))) be the eigenvalue of the Hecke operator at prime ideals q n (respectively, q | n). Define Kf := Q(aq (f ), q prime ideal of OF ) to be the field generated (over Q) by the eigenvalues of the Hecke algebra acting on f and denote by Of its ring of integers. Since the character of f is trivial, Kf is totally real by [43, Proposition 2.5]. Suppose finally that the morphism φf : Tn → Of is surjective. Fix p ≥ 5 a rational prime and assume for simplicity that p does not ramify in F/Q and Kf /Q. If p = 5, also assume that [F (ζ5 ) : F ] = 2, where ζ5 is a 5-th root of unity. Fix an embedding ιp : Q → Qp . Denote by π the prime ideal of Of corresponding to ιp and denote by Of,π the completion of Of at π. Say that f is ordinary at a prime ideal p | p if there exists a root αp of the Hecke polynomial at p such that ιp (αp ) is a unit. Suppose the following assumption is verified: Assumption 1.1. f is ordinary at all prime ideals p | p. Suppose that there exists a prime ideal ℘ | p such that ℘ | n. Suppose that either f is a newform or it comes from a newform of level n/℘ which is ordinary at all primes p dividing p via the procedure of p-stabilization. In the totally real case, see [35, Section 12.5.2] for this procedure; see also Nekov´ˇ [35, Chapter 12], [36], [37], Zhang [47], ar [48], [49], Cornut-Vatsal [9], [8], Howard [21] and Goren [16] for references on recent developments and results on the arithmetic theory of Hilbert modular forms. Let K/F be a totally imaginary quadratic extension. Assume that the discriminant of K/F and pn are prime to each other. Then K determines a factorization n = ℘n+ n− where a prime ideal q divides n+ if and only if q is split in K/F while divides n− if and only if it is inert in K/F . Suppose that p does not divide the class number hK of × × × × K and the index [OK : OF ] of OF in OK . The following assumption is crucial in this setting: Assumption 1.2. The number of prime ideals q ⊆ OF dividing n− and d = [F : Q] have the same parity. If d is even, assume n− = OF . Remark 1.3. The condition d even ⇒ n− = OF is assumed to obtain the isomorphism (9) in the text. See Remark 7.15. For the case of d even and n− = OF , see the discussions and the results of [29] and [30]. ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 3 As a consequence of Assumption 1.2, the special value at 1 of the complex L-function LK (f, χ, s) of f over K twisted by χ is non zero for infinitely many ramified ring class characters χ of conductor ℘m (see [8, Theorem 1.4]). Using the notion of Gross points it is possible to associate to f a p-adic L-function L℘,π (f /K) relative to ℘ and π. This is an element of the Iwasawa algebra Λ℘,π := Of,π [[G℘∞ ]], G℘∞ := Gal(K℘∞ /K) Z[F℘ :Qp ] p is the Galois group of the anticyclotomic Zp -extension K℘∞ /K associated to ℘. See Section 2 for the definition of K℘∞ and Section 4 for the construction of L℘,π (f /K). The p-adic L function L℘,π (f /K) is characterized by its interpolation properties of the complex L-functions LK (f, χ, s), where χ is as above: see Section 4.4 for details. On the other hand, there is a notion of Selmer group attached to f . Denote by ρf,π∞ : GF := Gal(F /F ) → GL2 (Of,π ) the π-adic Galois representation attached to f and define ρf,πn := ρf,π∞ (mod π n ). Suppose that the following condition is verified: Assumption 1.4. ρf,π is surjective. Denote by Tf,π∞ the GF -module associated to the representation ρf,π∞ . Let Vf,π∞ := Tf,π∞ ⊗Of,π Kf,π (where Kf,π := Frac(Of,π )). Define finally Af,π∞ := Vf,π∞ /Tf,π∞ and Af,πn := Af,π∞ [π n ] for all n ≥ 1. The Selmer group Selπ∞ (f /K℘∞ ) ⊆ H 1 (K℘∞ , Af,π∞ ) is defined in Section 5 by requiring suitable local conditions to global cohomology classes. Its Pontryagin dual Sel∨∞ (f /K℘∞ ) is a finitely generated Λ℘,π -module. Deπ note by Char℘,π (f /K) ∈ Λ℘,π the characteristic power series of Sel∨∞ (f /K℘∞ ). This element is well-defined only up π to units, while the ideal (Char℘,π (f /K)) of Λ℘,π generated by Char℘,π (f /K) depends only on Sel∨∞ (f /K℘∞ ). π The Anticyclotomic Iwasawa Main Conjecture relates the ideals of Λ℘,π generated by L℘,π (f /K) and Char℘,π (f /K); it can be stated as follows: Conjecture 1.5 (Anticyclotomic Iwasawa’s Main Conjecture). The ideals of Λ℘,π generated by L℘,π (f /K) and by Char℘,π (f /K) are equal. For any prime ideal q ⊆ OF , choose GFq = Gal(F q /Fq ) ⊆ Gal(F /F ) a decomposition group and denote by IFq its inertia subgroup. To state the main result, suppose that the following conditions on f are verified: Assumption 1.6. Define mf,π to be kernel of the morphism Tn → Of,π /π associated to f . The completion Tf of Tn at mf,π is isomorphic to Of,π . If this condition holds, say that f is π-isolated. Assumption 1.7. Let q | n and q p be a prime ideal. The maximal IFq -invariant submodule of Af,π∞ is free of rank one over Kf,π /Of,π . where 4 MATTEO LONGO Assumption 1.8. Ihara’s Lemma for Shimura curves over totally real fields, as stated in Assumption 7.18 in the text, holds. Remark 1.9. The technical condition in Assumption 1.8 is essential in the proof of Lemma 7.20 below. It consists in a version of Ihara’s Lemma for Shimura curves over totally real fields. If F = Q, the analogue of Assumption 1.8 holds thanks to [12, Theorem 2]. The results contained in [12] and successively refined in [13] are partially generalized to the totally real case in [25]. However, [25] do not cover the full generalization of [12, Theorem 2]. In this paper we follow [15], which assumes the generalization of Ihara’s Lemma as an hypothesis in [15, Hypothesis 5.9]. Similar results for Hilbert modular varieties hold thanks to [14]. For further discussions, see Remark 7.19. The main result, corresponding to Theorem 6.1 in the text, is the following: Theorem 1.10. Suppose that the assumptions listed above are satisfied. Then the characteristic power series Char℘,π (f /K) divides the p-adic L-function L℘,π (f /K). Under our arithmetic assumptions, the p-adic L-function does not vanish identically by [8, Theorem 1.4]: see Section 4.4. This shows that (see Corollary 6.2 in the text): Corollary 1.11. Assumptions as in Theorem 1.10. Then Sel∨∞ (f /K℘∞ ) is pseudoπ isomorphic to a torsion Λ℘,π -module. Remark 1.12. If Assumption 1.2 is not satisfied, then Sel∨∞ (f /K℘∞ ) is not pseudoπ isomorphic to a torsion Λ℘,π -module and the growth of Selπ∞ (f /K℘∞ ) is forced by the presence of Heegner points coming from a Shimura curve parametrization of the abelian variety Af associated to f (see Remark 6.4 for details on Af and its parametrization by the Jacobian variety of a suitable Shimura curve). For precise statements and results in this case, see [1] (over Q), [21] and [36] (over totally real number fields). Remark 1.13. Using the techniques announced by Skinner-Urban, it should be possible to prove the opposite divisibility L℘,π (f /K) | Char℘,π (f /K). Thus, combining with Theorem 1.10, it may be possible to obtain a proof of Conjecture 1.5. For references on Skinner-Urban methods, see [27, Section 2]. Remark 1.14. The proof of the main result is a generalization of [5], where the case of F = Q and Of,π = Zp is considered. In Section 7 the main steps of the proof are recalled and the necessary adaptations are performed. Probably, the assumptions that p does × × not divide hK [OK : OF ] and does not ramify in K could be relaxed; these conditions are assumed mainly to get a simpler description of the extension K℘∞ in Section 2 and, consequently, a simpler construction of L℘,π (f /K) in Section 4. Furthermore, the condition that φf : Tn → Of is surjective can be probably relaxed using the techniques of [38]. Theorem 1.10 can be used to study the arithmetic of abelian varieties of GL2 -type. The simplest case is that of an elliptic curve. Let A be an elliptic curve defined over F , of conductor n, without complex multiplication, which is ordinary at each prime ideal p | p. Suppose also that A is modular in the sense that there exists a Hilbert modular form f for the Γ0 (n)-structure, of parallel weight 2 and trivial central character, such that the -adic representation of A is isomorphic to the -adic representation associated to f , where is a rational prime. In this case, Of = Z, π = p and Of,π = Zp . Suppose ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 5 finally that f satisfies all the above assumptions. Note in particular that, since A does not have complex multiplication, there are only a finite number of primes such that the Galois representation on the -torsion points of A is not surjective. Theorem 1.10 can be used to study the characteristic power series of the Pontryagin dual Sel∨∞ (A/K℘∞ ) p of the p-primary Selmer group Selp∞ (A/K℘∞ ) of A over K℘∞ . Theorem 1.10 and the non-vanishing of L℘ (f /K) established in [8, Theorem 1.4] show that Sel∨∞ (A/K℘∞ ) is p always pseudo-isomorphic to a torsion Λ℘,p -module. The first application, corresponding to Corollary 6.11 in the text, is the following: Corollary 1.15. Assumptions as in Theorem 1.10. Furthermore, suppose [F℘ : Qp ] = 1 and A/F a modular elliptic curve as above. Then A(K℘∞ ) is finitely generated. For any Λ℘,p -module M and any finite order character χ : G℘∞ → O, where O is the ring of integers of a finite extension of Qp , extend χ to a homomorphism, denoted by the same symbol, χ : Λ℘,p → O and set M χ := M ⊗χ O, the tensor product being taken over Λ℘,p via χ. Let Xp∞ (A/K℘∞ ) be the p-primary part of the Shafarevich-Tate group of A over K℘∞ . The second application, corresponding to Corollary 6.9 in the text, is the following: Corollary 1.16. Assumptions as in Theorem 1.10. Furthermore, suppose [F℘ : Qp ] = 1 and A/F a modular elliptic curve as above. If LK (A, χ, 1) = 0, then A(K℘∞ )χ and Xp∞ (A/K℘∞ )χ are finite. 2. Anticyclotomic Zp -extensions Let the assumptions and notations be fixed as in Section 1. In particular, recall that × × p does not divide the class number of K and the index of OF in OK . For any integral ideal c ⊆ OF , let Oc := OF + cOK be the order of conductor c in K and define the ring class field Kc /K of K of conductor c to be the Galois extension of K such that Gal(Kc /K) Pic(Oc ) × K × /Oc K × . Denote by | ∗ | the norm on ideals of OF and set hc := #Pic(Oc ), so that hK := h(1) is the class number of K. By the Dedekind formula: (1) hc = hK |c| q|c 1− × [OK : K q × Oc ] |q|−1 , where q denotes a prime ideal of OF and K q = 1 (respectively, −1, 0) if q is split (respectively, inert, ramified) in K/F . The extension K℘m /K is unramified outside the places dividing ℘. Thanks to the fact that p does not ramify in K and does not divide × × hK [OK : OF ], it follows from (1), that [K℘m : K℘m−1 ] = |p| for all integers m ≥ 2 and that p [K℘ : K]. Define K℘∞ := lim K℘m . − → m Definition 2.1. The ℘-anticyclotomic Zp -extension K℘∞ /K is defined to be the unique subfield K℘∞ of K℘∞ such that: G℘∞ := Gal(K℘∞ /K) Z[F℘ :Qp ] . p 6 MATTEO LONGO The extension K℘∞ /F is Galois and non abelian. More precisely, the quotient Gal(K/F ) acts by conjugation on the normal subgroup Gal(K℘∞ /K) by the formula σ → τ στ = σ −1 , where τ is the choice of a complex conjugation raising the non trivial automorphism of Gal(K/F ). For any integer m ≥ 1, define the extension K℘m /K by requiring that G℘m := Gal(K℘m /K) (Z/pm Z)[F℘ :Qp ] . It follows from the above assumptions on p that K℘m is the maximal p-power subextension of K℘m /K. Denote by Λ℘,π the Iwasawa algebra of G℘∞ : Λ℘,π := Of,π [[G℘∞ ]] = lim Of,π [G℘m ] ← − m where the inverse limit is with respect to the canonical projection maps G℘m → G℘m−1 . Remark 2.2. There are other definitions of ring class fields of conductor c in the literature. Nekov´ˇ [36, Section 2.6] (see also Zhang [49]) defines the ring class field ar of conductor c to be the Galois extension Kc∗ corresponding via class field theory to × K × /K × Oc F × . On the other hand, [8] uses the definition given in this paper for the ring × class field K℘n , denoted K[P n ] therein. However, note that the quotient K × /K × Oc F × is isomorphic to Pic(Oc )/Pic(OF ), so, since p hK , the maximal Zp -extension contained ∗ in ∪∞ K℘n is exactly the extension K℘∞ in Definition 2.1. n=1 3. CM points on quaternion algebras This section is devoted to fix the notations for CM-points on quaternion algebras. Since we will need this notions both for totally definite quaternion algebras (in Section 4) and for quaternion algebras which are split in exactly one archimedean place (in Section 7.4), we will adopt a quite general view-point. 3.1. Optimal embeddings and CM-points. Let k denote a global or local field and D/k a quaternion algebra. Let O be an Eichler order of D. Let k /k be a quadratic extension and denote by r an order in k . Say that ψ is an optimal embedding of r into O if ψ : k → D is an injective homomorphism of k-algebras such that ψ(r) = ψ(k ) ∩ O. Two optimal embeddings ψ1 and ψ2 of r into O are said to be equivalent if there exists α ∈ O× so that, ∀x ∈ r, ψ1 (x) = α−1 ψ2 (x)α. The conductor of an optimal embedding ψ is the conductor of the order r. For more details, see [46, Chapitre II] when k is a local field and [46, Chapitre III] when k is a global field. Suppose now that k is a global field and, for any valuation v of k, let kv , kv , rv , Dv and Ov denote the completions of k, k , r, D and O, respectively, at v. In the following, by an abuse of notations, we will identify v with the integral prime ideal of k corresponding to it. Let d denote the discriminant of k /k, c the conductor of the quadratic order r, n the discriminant of the quaternion algebra D and m the level of the Eichler order O, and assume that m is square-free, c is prime to n and d is prime to cmn. Suppose that if v | n then v is inert in k . Suppose also that if v | m and v c (so rv is maximal) then v is split in k /k. This conditions ensure that the set of optimal embeddings of r into O is non-empty: see [46, page 94]. Following [17] and [3], define X(k ) := D× \D× × Hom(k , D)/O× ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 7 where the action of b ∈ D× and x ∈ O× on a pair (g, ψ) is b(g, ψ)x := (bgx, bψb−1 ). Say that a point (x, ψ) ∈ X(k ) is a CM-point of conductor c if ψ is an optimal embedding of r into Ox := xOx−1 ∩ D. Write CM(c) for the set of CM-points of conductor c in X(k ). Fix an embedding ψ : k → D which allows to see k as a subfield of B. Following [8], the set CM(c) can also be described as follows. Define: Y (k ) := ψ(k )× \D× /O× . Say that a point x ∈ Y (k ) has conductor c if k ∩ Ox = r. The set CM(c) can be identified with the set of points in Y (k ) of conductor c. To see this, note that there is a map from the set of points in Y (k ) of conductor c to X(k ) defined by x → (x, ψ). This map is a bijection. Injectivity: Suppose (x1 , ψ) = (x2 , ψ). Then there exists b ∈ D× and x ∈ O× such that x1 = bx2 x and ψ = bψb−1 . Since ψ(k ) is a maximal commutative subfield of D, it follows that b ∈ ψ(k ) and so x1 and x2 represent the same element in ψ(k × )\D× /O× . Surjectivity: Fix (x, ψ) ∈ X(k ) of conductor c. By the Skolem-Noether theorem, there exists b ∈ D× such that ψ(k ) = b−1 ψ(k )b. It follows that b−1 ψ(r)b = b−1 ψ(k )b ∩ xOx−1 , so ψ(r) = ψ(K) ∩ bxO(bx)−1 . Hence, (bx, ψ) belongs to the image of the set of points in CM of conductor c. Finally, note that (bx, ψ) = (x, b−1 ψb) = (x, ψ). The Galois group G(c) = Pic(r) = k /k r× acts on CM(c) by left translation: for every g ∈ G(c) and (x, ψ) ∈ CM(c), the action is given by x → (gx, ψ). Equivalently, if x ∈ Y (k ) has conductor c, the Galois is given by x → gx. 3.2. The trace formula. Fix representatives g1 = 1, . . . , gh of D× \D× /O× and define −1 Oj := gj Ogj ∩ D, so that O1 = O. Note that the number of CM-points of X(K) is equal to the number of non-equivalent optimal embedding of r into one of the Eichler orders Oj . Write Emb(r, Oj ) for the set of equivalence classes of optimal embeddings of r into Oj . For any place v of k, let mv be the number of non-equivalent local optimal embeddings of rv into Ov . Then mv is finite and mv = 1 for those v which do not divide mn. The following trace formula [46, Chapitre III, Th´or`me 5.11 and page 94] holds: e e h × × (2) |CM(c)| = j=1 |Emb(r, Oj )| = h(r) v|mn mv , where h(r) is the class number of r. 3.3. Orientations and Gross points. An orientation at v of a local optimal embedding ψ : kv → Dv of rv into Ov is the choice of an equivalence class of optimal embeddings. This can be made precise as follows. If v | nm and v c, then mv = 2. The choice of an orientation can be performed as follows. For v | nm and v c, define × Uv (r, O) := Hom(rv , Ov )/Ov . 8 MATTEO LONGO The choice of an orientation ov at the primes v | nm and v c is the choice of an element in Uv (r, O). Say that a point (x, ψ) ∈ CM(c) is oriented at a prime v | nm and v c (with respect to the chosen orientation ov ) if x−1 ψx and ov define the same element in Uv (r, O). For more details, see [47, Section 2.1.1]. Let now v | m and v | c, so rv is not maximal. In this case too, mv = 2 (see [46, page 94]). The choice of an orientation can be performed as follows. The set of maximal orders (respectively, Eichler orders of level v) of GL2 (kv ) can be identified with the set of vertexes Vv (respectively, unoriented edges Ev ) of the homogeneous tree Tv of degree |v|. Let v0 (respectively, e0 ) denote the vertex (respectively, the edge) corresponding to the maximal order GL2 (rv ) (respectively, the Eichler order Γ0 (v) ⊆ GL2 (rv ) of level v consisting of matrices which are upper triangular modulo v). Say that a vertex v is even (respectively, odd ) if its distance from v0 is even (respectively, odd) and define an orientation s, t : Ev → Vv by requiring that for any edge e, s(e) = veven and t(e) = vodd , where e is the edge joining veven and vodd and veven and vodd are even and odd, respectively. Let (x, ψ) ∈ CM(c), so that ψ : kv → Dv is an optimal embedding of rv into Ox . Fix an isomorphism ιv : Dv → M2 (kv ). Then Ox can be identified with an edge eOx = (s(eOx ), t(eOx )) is such a way that Ox is the intersection of the two maximal orders represented by s(eOx ) and t(eOx ). Finally, let r be the quadratic order containing r of conductor c/v. Say that (x, ψ) is oriented (with respect to the chosen orientations s, t) if the v-component ψv of ψ is an optimal embedding of rv into the maximal order corresponding to s(eOx ). Note that, in this case, ψ must be an optimal embedding of rv into the maximal order corresponding to t(eOx ). Fix orientations ov ∈ Uv (r, O) for v | mn and v c and orientations s, t : Ev → Vv for v | m and v | c. A Gross point of conductor c is a CM-point (x, ψ) ∈ CM(c) which is oriented at all v | mn. 4. p-adic L-functions 4.1. Modular forms on definite quaternion algebras. Let B/F be the quaternion algebra of discriminant n− which is ramified at all archimedean places. Fix an Eichler order R ⊆ B of level ℘n+ . Let f ∈ S2 (n) be a Hilbert modular cuspform of parallel weight 2 and trivial central character with respect to the Γ0 (n)-level structure. Let Tn be the Hecke algebra acting faithfully on S2 (n) (see [47, Section 3.1] for precise definitions). B Denote by S2 (℘n+ ) the C-vector space of functions: B × \B × /R× → C. B There is an action of the Hecke algebra Tn on S2 (℘n+ ) defined as usual via double cosets. The Jacquet-Langlands correspondence implies that there is a unique up to B scaling modular form f B ∈ S2 (℘n+ ) having the same eigenvalues as f under the action of the Hecke algebra. If the Hecke eigenvalues on a Hilbert modular form f are contained in a ring O, them f B can be normalized to take values in O. 4.2. CM points on definite quaternion algebras. Since all primes dividing the discriminant of B are inert in K, there exists an embedding K → B, so that K can be regarded as a subfield of B via this fixed embedding Ψ. Following the notations in ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 9 Section 3, define the set of CM-points by R to be: CMR := Ψ(K)× \B × /R× and say that a point x ∈ CMR has conductor c if Ψ(K) ∩ xRx−1 = Oc . Denote by CMR (c) the set of CM points of conductor c. Following Section 3, the set CMR (c) can also be described as the set of points in XR (K) := B × \(B × × Hom(K, B))/R× such that Ψ is an optimal embedding of Oc into the Eichler order B ∩xR× x−1 ; explicitly, Ψ(K) ∩ xR× x−1 = Ψ(Oc ). Since all primes dividing n+ are split in K, CMR (℘m ) is non empty for all m ≥ 1. × The group G℘m K × /K × O℘m acts on CMR (℘m ) by left translation, as described in Section 3. Fix a positive integer m. Choose orientations for the optimal embeddings of O℘m into R as in Section 3 for all primes q dividing n: this amounts to choose orientations oq ∈ Uq (O℘m , R) for all primes q | n+ n− and an orientation s, t : E℘ → V℘ at the prime ℘. Let Gr(℘m ) denote the set of Gross points of conductor ℘m with respect to these orientations and define: Gr(℘∞ ) := ∪∞ Gr(℘m ). m=1 If P = (x, Ψ) ∈ Gr(℘m ), then the local component Ψ℘ : K℘ → B℘ of Ψ is an optimal embedding of the completion O℘m ,℘ of O℘m at ℘ into xR℘ x−1 , where R℘ is the completion of R at ℘. Let eP = (s(eP ), t(eP )) ∈ E℘ be the edge corresponding to xR℘ x−1 as described in Section 3. Say that a sequence (Pm )m≥1 of points in Gr(℘∞ ), with Pm ∈ Gr(℘m ), is compatible if t(ePm ) = s(ePm+1 ) for all integers m ≥ 1. Remark 4.1. If P = (x, Ψ) ∈ Gr(℘m ) with m ≥ 1, then the pair (x, Ψ) also defines a CM-point of conductor ℘m−1 in XR0 (K), where R0 ⊃ R is an Eichler order of B of level n+ chosen is such a way that R0,℘ corresponds to s(eP ). 4.3. Anticyclotomic ℘-adic L functions. Let f B be the modular form on the quaternion algebra B associated to f via the Jacquet-Langlands correspondence and define the following map: η : ψ(K)× \B × /R× −→ B × \B × /R× −→ Of , where µ is the canonical projection. Choose points xm ∈ CMR (℘m ) in such a way that the sequence (xm )m is compatible. The orientation s, t : E℘ → Vp being fixed as above, the action of U℘ on an edge e ∈ E℘ can be described as U℘ (e) = e e , where the sum is over all edges e such that s(e ) = t(e). The choice of the compatible sequence of Gross points made before shows then that for m ≥ 2: (3) g∈Gal(K℘m /K℘m−1 ) µ fB µ(gxm ) = U℘ (µ(xm−1 )). Define the theta elements for m ≥ 1: −m ˜ θf,m := α℘ η(gxm )g ∈ Of,π [G℘m ]. g∈G℘m 10 MATTEO LONGO Denote by νm+1,m : Of,π [G℘m+1 ] → Of,π [G℘m ] the homomorphisms induced by the ˜ projection maps G℘m+1 → G℘m . By Equation (3), the elements θf,m verify the following relation: ˜ ˜ νm+1,m (θf,m ) = θf,m−1 . Taking the inverse limit with respect to the projection maps νm+1,m yields an element: ˜ ˜ θf := lim θf,m ∈ Of,π [[G℘∞ ]] := lim Of,π [G℘m ]. ← − ← − m m The group ring Of,π [[G℘∞ ]] is endowed with a canonical involution x → x∗ defined to be the extension by Of,π -linearity of the involution σ → σ −1 of G℘∞ . Define: ˜ ˜ L℘,π (f /K) := θf θ∗ ∈ Of,π [[G℘∞ ]]. f ˜ Since θf is well defined only up to multiplication by an element of G℘∞ , the definition of ˜ L℘,π (f /K) is independent on the choice of the Gross points xm . Set Λ℘,π := Of,π [[G℘∞ ]] ˜ and denote by λ : Λ℘,π → Λ℘,π the projection induced by the inclusion K℘∞ ⊆ K℘∞ . Definition 4.2. Define the anticyclotomic ℘-adic L-function attached to f and K to be the element ˜ L℘,π (f /K) := λ(L℘,π (f /K)) ∈ Λ℘,π . ∗ ˜ ˜ Furthermore, define θf,n := λ(θf,n ) and θf := λ(θf ), so that L℘,π (f /K) = θf θf and θf = lim θf,n . ← − n 4.4. Interpolation properties. Let χ : G℘∞ → O× be a ramified finite order character, where O is the ring of integers of a finite extension of Qp . Extend χ to an homomorphism, denoted by the same symbol, χ : Of,π [[G℘∞ ]] → O. Then χ(L℘,π (f /K)) = 0 if and only if LK (f, χ, 1) = 0. For a reference of this result in this form, see [45, Theorem 6.4]. For a more precise statement, see [48, Theorem 1.3.2], which is a generalization of [17]. The arithmetic assumptions we are working with imply that the sign of the functional equation of LK (f, χ, 1) is +1 and, by [8, Theorem 1.4], that LK (f, χ, 1) = 0 for infinitely many characters χ as above. Hence L℘,π (f /K) = 0. Since G℘∞ G℘∞ × ∆℘ and ∆℘ is finite, it follows that L℘,π (f /K) = 0. 5. Selmer groups attached to Hilbert modular forms 5.1. Galois cohomology groups. 5.1.1. Galois representations. Let Tf,π∞ be the GF = Gal(F /F )-module, free of rank 2 over Of,π , associated to the representation ρf,π∞ : Gal(F /F ) → GL2 (Of,π ); define Kf,π := Frac(Of,π ) and: Vf,π∞ := Tf,π∞ ⊗Of,π Kf,π ; Af,π∞ := Vf,π∞ /Tf,π∞ ; Tf,πn := Tf,π∞ /π n Tf,π∞ ; Af,πn = Af,π∞ [π n ]. As Of,π -modules, Af,π∞ (Kf,π /Of,π )2 while both Tf,πn and Af,πn are Of,π /π n -modules free of rank 2 and there is an isomorphism of GF -modules Tf,πn Af,πn . Furthermore, Af,π∞ lim Af,πn − → n and Tf,∞ lim Tf,πn ← − n ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 11 with respect to the canonical maps. 5.1.2. Global cohomology groups. Let ν denote a positive integer or ∞. Define the following groups: H 1 (K℘∞ , Af,πν ) := lim H 1 (K℘m , Af,πν ), − → m where the direct limit is with respect to the restriction maps, and ˆ H 1 (K℘∞ , Tf,πν ) := lim H 1 (K℘m , Tf,πν ), ← − m where the inverse limit is with respect to the corestriction maps. 5.1.3. Local cohomology groups. For each prime q ⊆ OF , let K℘m ,q := K℘m ⊗F Fq = ⊕q |q K℘m ,q where the sum is over the prime ideals q | q of the ring of integers OK℘m of K℘m and K℘m ,q is the completion of K℘m at q . For any Gal(K/K℘m )-module M , define H 1 (K℘m ,q , M ) := ⊕q |q H 1 (K℘m ,q , M ). Then define as above for ν a positive integer or ∞: H 1 (K℘∞ ,q , Af,πν ) := lim H 1 (K℘m ,q , Af,πν ), − → m where the direct limit is with respect to the restriction maps, and ˆ H 1 (K℘∞ ,q , Tf,πν ) := lim H 1 (K℘m ,q , Tf,πν ), ← − m where the inverse limit is with respect to the corestriction maps. 5.2. Selmer groups. The definitions of Selπn (f /K℘∞ ) and Selπ∞ (f /K℘∞ ) require the introduction of the following finite/singular and ordinary structures. For any prime ideal q of OF and any prime ideal q of OK℘m above q, choose a decomposition subgroup Gm,q ⊆ GK℘m at q and let Im,q ⊆ Gm,q denote the inertia subgroup. 5.2.1. Primes q np. Let M denote Af,πn or Tf,πn . Fix q ⊆ OF a prime ideal such that q np. The singular part of H 1 (K℘m ,q , M ) is 1 Hsing (K℘m ,q , M ) := ⊕q |q H 1 (Im,q , M ) unr Gal(K℘m ,q /K℘m ,q ) , where the sum is over all prime ideals q of OK℘m dividing q. The kernel of the residue 1 map ∂q : H 1 (K℘m ,q , M ) → Hsing (K℘m ,q , M ) is the finite part of H 1 (K℘m ,q , M ) and is 1 denoted by Hfin (K℘m ,q , M ). Define: 1 1 Hfin (K℘∞ ,q , Af,πn ) := lim Hfin (K℘m ,q , Af,πn ), − → m 1 Hsing (K℘∞ ,q , Af,πn ) 1 := lim Hsing (K℘m ,q , Af,πn ), − → m where the direct limits are with respect to restriction maps, and: 1 ˆ1 Hfin (K℘∞ ,q , Tf,πn ) := lim Hfin (K℘m ,q , Tf,πn ), ← − m 1 ˆ1 Hsing (K℘∞ ,q , Tf,πn ) := lim Hsing (K℘m ,q , Tf,πn ), ← − m where the inverse limits are with respect to the corestriction maps. The cohomology 1 ˆ1 groups Hfin (K℘∞ ,q , Af,πn ) and Hfin (K℘∞ ,q , Tf,πn ) are the exact annihilators of each other under the local Tate pairing , q (for a proof, see [34, Theorem 2.6]). If q = q1 q2 is split in K/F , the Frobenius element at qi topologically generates a finite index subgroup in 12 MATTEO LONGO G℘∞ . Hence there are only a finite number of prime ideals q of K℘∞ over q and for each of them, K℘m ,q is the unramified Zp -extension of Kq . It follows that any unramified class of H 1 (K℘m ,q , Af,πn ) becomes trivial after restriction to H 1 (K℘r ,q , Af,πn ) for r sufficiently large. Hence, if q is split in K/F : 1 ˆ1 Hfin (K℘∞ ,q , Af,πn ) = 0 and Hsing (K℘∞ ,q , Tf,πn ) = 0, where the second assertion follows from the non-degeneracy of the local Tate pairing. If q is inert in K/F , then it splits completely in K℘∞ (this observation is due to Iwasawa [23]). It follows that, if q is inert in K/F : ˆ1 Hsing (K℘∞ ,q , Tf,πn ) and 1 Hfin (K℘∞ ,q , Af,πn ) 1 Hom(Hsing (Kq , Tf,πn ) ⊗ Λ℘,π , Kf,π /Of,π ). 1 Hsing (Kq , Tf,πn ) ⊗ Λ℘,π Remark 5.1. To explain the above definitions, let be a prime number, K/Q a finite extension and A/K an abelian variety with good reduction. Let p = a prime and denote by GK and IK the absolute Galois group of K and its inertia subgroup, respectively. Finally, let κ : A(K) → H 1 (K, A[pn ]) denote the Kummer map, where n is a non-negative integer. Then Im(κ) = H 1 (GK /IK , A[pn ]) = Ker(H 1 (GK , A[pn ]) → H 1 (IK , A[pn ])GK /IK ). For a proof, see [34, Chapter 1, Proposition 3.8] or [18, Lemma 7]. 5.2.2. Primes q | n and q p. Fix a prime q p which divides n. By assumption 1.7, q Af,πn := Af,πn (q) IF Kf,π /Of,π . The ordinary part of the group H 1 (K℘m ,q , Af,πn ) is defined to be the unramified cohomology: (q) 1 Hord (K℘m ,q , Af,πn ) := H 1 (GK℘m ,q /IK℘m ,q , Af,πn ). Define: 1 1 Hord (K℘∞ ,q , Af,πn ) := lim Hord (K℘m ,q , Af,πn ), − → m where the direct limit is with respect to the restriction maps. Note that if q | n+ and 1 q p, then, by an argument similar to that of Section 5.2.1, Hord (K℘∞ ,q , Af,πn ) = 0. Remark 5.2. To explain the above definitions, let be a prime number, K/Q a finite extension and A/K an abelian variety with purely toric reduction. Suppose that there exists an extension E/Q such that [E : Q] = dim(A) and an embedding OE → End(A), where OE is the ring of integers of E. Let p = a prime and p a prime ideal of OE of residual characteristic p. Denote by GK and IK the absolute Galois group of K and its inertia subgroup, respectively. Suppose that the inertia invariants A[pn ]IK of A[pn ] are one-dimensional over the field OE /p. Finally, let κ : A(K) → H 1 (K, A[pn ]) denote the Kummer map, where n is a non-negative integer. Then Im(κ) = H 1 (GK /IK , A[pn ]IK ). ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 13 For a proof in the case n = 1, see [18, Lemma 4, Lemma 6 and Section 3.3]. The general case (n > 1) can be obtained by a direct generalization of the arguments used in the case n = 1. 5.2.3. Primes p | p. Let p | p be a prime ideal. Let IFp ⊆ GFp := Gal(Fp /Fp ) denote the inertia subgroup. By Assumption 1.1, f is ordinary at p. This condition implies that there is an exact sequence of IFp -modules: (4) 0 → Af,π∞ → Af,π∞ → Af,π∞ → 0 (p) (1) (p) (p) (1) such that the submodule Af,π∞ and the quotient Af,π∞ are both isomorphic to Kf,π /Of,π as groups and IFp acts on Af,π∞ via the cyclotomic character p : Gal(F /F ) → Aut(µp∞ ) describing the action of Gal(F /F ) on the group µp∞ of p-power roots of unity, and acts (1) trivially on Af,πn . Let λp,m : H 1 (K℘m ,p , Af,π∞ ) → H 1 (K℘m ,p , Af,π∞ ) be the map of cohomology groups induced by the inclusion Af,π∞ ⊆ Af,π∞ . Define 1 the ordinary part Hord (K℘m ,p , Af,π∞ ) of H 1 (K℘m ,p , Af,π∞ ) to be the maximal divisible subgroup of Image(λp,m ). Then define: 1 1 Hord (K℘∞ ,p , Af,π∞ ) := lim Hord (K℘m ,p , Af,π∞ ), − → m (p) (p) where the direct limit is with respect to the restriction maps. Remark 5.3. To justify the above definition, let A[p∞ ] be the maximal p-divisible group of A(K), where A/K is an ordinary abelian variety defined over a finite extension K of Qp . Let F be the formal group over OK attached to the N´ron model for A over e OK and define C := F(m)[p∞ ], where m is the maximal ideal of the algebraic closure of K. Finally, define the map: λ : H 1 (K, C) → H 1 (K, A[p∞ ]) induced by the inclusion C → A[p∞ ]. Then the image of the Kummer map κ : A(K) ⊗ Qp /Zp → H 1 (K, A[p∞ ]) is equal to the maximal divisible subgroup (Im(λ))div of Im(λ). For proofs, see [7, Proposition 4.5]. Moreover, if K∞ /K is a deeply ramified extension (see [7, Section 2] for definitions), then the image of the Kummer map A(K∞ )⊗Qp /Zp → H 1 (K∞ , A[p∞ ]) coincides with the image of λ : H 1 (K∞ , C) → H 1 (K∞ , A[p∞ ]) by [7, Proposition 4.3]. Note that for each prime p of K℘∞ over p, the extension K℘∞ ,p /Kp is deeply ramified. 1 The last lines of Remark 5.3 show that one could equivalently define Hord (K℘∞ ,p , Af,π∞ ) to be the image of λp,∞ : H 1 (K℘∞ ,p , Af,π∞ ) → H 1 (K℘∞ ,p , Af,π∞ ). Define 1 H(p, m, n) := Hord (K℘m ,p , Af,π∞ ) ∩ H 1 (K℘m ,p , Af,πn ). (p) For any subgroup H ⊆ H 1 (K℘m ,p , Af,πn ), use the isomorphism Af,πn Tf,πn to define 1 a subgroup H∗ ⊆ H 1 (K℘m ,p , Tf,πn ) such that H H∗ . Then define Hord (K℘m ,p , Af,πn ) to be the maximal subgroup of H 1 (K℘m ,p , Af,πn ) containing H(p, m, n) and such that 14 MATTEO LONGO 1 1 Hord (K℘m ,p , Af,πn ) and Hord (K℘m ,p , Af,πn )∗ are the exact annihilators of each other under the local Tate pairing at p. Finally, set: 1 1 Hord (K℘∞ ,p , Af,πn ) := lim Hord (K℘m ,p , Af,πn ), − → m where the direct limit is with respect to the restriction maps. Remark 5.4. Let A[pn ] be the pn -torsion of an abelian variety A/K as in Remark 5.3. The image of the Kummer map κ : A(K)/pn → H 1 (K, A[pn ]) contains the subgroup H := Im(λ)div ∩ H 1 (K, A[pn ]), where λ is the map defined in Remark 5.3. Since Im(κ) is maximal isotropic for the local Tate pairing, then it coincides with the maximal isotropic subgroup of H 1 (K, A[pn ]) containing H. 5.2.4. Selmer groups. Let Mf,πn denote Af,πn or Tf,πn . For any prime q, let resq : H 1 (K℘∞ , Mf,πn ) → H 1 (K℘∞ ,q , Mf,πn ) denote the restriction map. For a prime q ⊆ OF not dividing np, let ∂q denote the residue map 1 ∂q : H 1 (K℘∞ ,q , Mf,πn ) → Hsing (K℘∞ ,q , Mf,πn ) and, by an abuse of notations, denote also by ∂q the map obtained by composing resq with ∂q . If s ∈ H 1 (K℘∞ , Mf,πn ) satisfies ∂q (s) = 0, write vq (s) for the image of s in 1 Hfin (K℘∞ , Mf,πn ). Definition 5.5. The Selmer group Selπn (f /K℘∞ ) attached to f, n and K℘∞ is the group of elements s ∈ H 1 (K℘∞ , Af,πn ) satisfying: (1) (2) (3) (4) Primes Primes Primes Primes 1 q np: resq (s) ∈ Hfin (K℘∞ ,q , Af,πn ). 1 q | n− and q p: resq (s) ∈ Hord (K℘∞ ,p , Af,πn ). + q | n and q p: resq (s) = 0. 1 p | p: resp (s) ∈ Hord (K℘∞ ,p , Af,πn ). The Selmer group Selπ∞ (f /K℘∞ ) is defined to be the direct limit Selπ∞ (f /K℘∞ ) := lim Selπn (f /K℘∞ ) − → n with respect to the inclusion maps. Let s ⊆ OF be a square free ideal prime to n. The compactified Selmer group ˆ ˆ1 Hs (K℘∞ , Tf,πn ) attached to f, n and K℘∞ is the groups of elements κ ∈ H 1 (K℘∞ , Tf,πn ) such that resq (κ), resq (s) q =0 for all s ∈ Selπn (f /K℘∞ ) and all q s, where , q is the local Tate pairing. The global reciprocity law of class field theory implies that for any s ∈ Selπn (f /K℘∞ ) and any ˆ1 κ ∈ Hs (K℘∞ , Tf,πn ): (5) q|s ∂q (κ), vq (s) q = 0. ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 15 6. Iwasawa’s Main Conjecture 6.1. The main result. Let Sel∨∞ (f /K℘∞ ) := Hom(Selπ∞ (f /K℘∞ ), Kf,π /Of,π ) π be the Pontryagin dual of Selπ∞ (f /K℘∞ ). Since Sel∨∞ (f /K℘∞ ) has a structure of finitely π generated Λ℘,π -module, there is an exact sequence: (6) 0 → M → Sel∨∞ (f /K℘∞ ) → Λr ⊕s Λ℘,π /(fi ) → N → 0, i=1 ℘,π π where fi = 0 and M and N are pseudo-null Λ℘,π -modules (for definitions of pseudo-null Λ℘,π -modules, as well as for the notion of pseudo-isomorphism of Λ℘,π -modules, we refer to Section 7.1). Define the characteristic power series of Sel∨∞ (f /K℘∞ ) to be: π Char℘ (f /K) := 0, if r = 0 s i=1 fi , if r = 0. The main result which will by proved in Section 7 is the following: Theorem 6.1. Suppose that the assumptions listed in the Introduction are satisfied. The characteristic power series Char℘,π (f /K) of the Pontryagin dual Sel∨∞ (f /K℘∞ ) of π Selπ∞ (f /K℘∞ ) divides the p-adic L-function L℘,π (f /K). Corollary 6.2. Suppose that the assumptions listed in the Introduction are satisfied. Then Sel∨∞ (f /K℘∞ ) is pseudo-isomorphic to a torsion Λ℘,π -module. π Proof. By [8, Theorem 1.4], L℘,π (A/K) is not identically zero, so Char℘,π (f /K) = 0. The proof of this result is based on a generalization of the argument in [5]. In Section 7 a sketch of the argument with the necessary adaptations to the totally real case will be presented. 6.2. Applications to modular abelian varieties. 6.2.1. Modular abelian varieties. Let A/F be an abelian variety. Denote by End(A) its endomorphism ring and define E := EndQ (A) = End(A) ⊗Z Q. Say that A is of GL2 -type if [EndQ (A) : Q] = dim(A) and End(A) is the ring of integers OE of E. For any ideal I ⊆ OE , denote by A[I] the I-torsion in A, by A[I ∞ ] the I-primary subgroup of A and by TI (A) the I-adic Tate module of A. Finally, let ρA,I : Gal(F /F ) → Aut(TI (A)) be the representation of Gal(F /F ) on TI (A). Definition 6.3. Say that an abelian variety of GL2 -type A/F as above is modular if there exists a cuspidal Hilbert modular form f of Γ0 (n)-level for some ideal n ⊆ OF , parallel weight 2, trivial central character, which is an eigenform for the Hecke algebra Tn , such that E = Kf and the -adic representation ρA, of Gal(F /F ) on the -adic Tate module T (A) of A is equivalent to the -adic representation ρf, attached to f , where is a prime number. 16 MATTEO LONGO Remark 6.4. Since n− = OF when d is even, Shimura’s construction generalized to this context (see [47, Theorem B and Section 3]) shows that for f as above there is a modular abelian variety A/F whose associated eigenform is f . Note that Definition 6.3 applies also to the case of n− = OF and d even, which however is not considered in this paper. For results in this important case, see [29] and [30]. Assume that the abelian variety A/F satisfies the following: Assumption 6.5. (1) A/F is a modular abelian variety in the sense of Definition 6.3. (2) The modular form f associated to A by Definition 6.3 satisfies the assumptions listed in the Introduction. (3) A/F has good reduction at all primes q n. (4) A/F has purely toric reduction at all primes q | n and q p. (5) A/F has ordinary reduction at all prime ideals p | p. Remark 6.6. If A is ordinary at p | p, then the associated Hilbert modular form is also ordinary at p (see [16, Chapter 3, Section 6.2]), so Assumption 1.1 is automatically satisfied if A satisfies Condition 5 in Assumptions 6.5. Let A/F satisfy Assumption 6.5 above. Define the Selmer groups: Selπn (A/K℘m ) := Ker(H 1 (K℘m , A[π n ]) → q H 1 (K℘m ,q , A(K℘m ,q ))), where the product is over all prime ideals q of K℘m , Selπn (A/K℘∞ ) := lim Selπn (A/K℘m ) − → m where the direct limit is with respect to the restriction maps, and Selπ∞ (A/K℘∞ ) := lim Selπn (A/K℘∞ ) − → n where the direct limit is with respect to the maps induced by A[π n ] ⊆ A[π n+1 ]. Lemma 6.7. There are isomorphisms: Selπn (f /K℘∞ ) Selπn (A/K℘∞ ) and Selπ∞ (A/K℘∞ ) Selπ∞ (f /K℘∞ ). In particular, the characteristic power series of their Pontryagin duals are the same. Proof. To show the first isomorphism it is necessary to compare the local conditions used in the definition of Selπn (f /K℘∞ ) with the image of the local Kummer map κq : A(K℘∞ ,q )/π n → H 1 (K℘∞ ,q , A[π n ]) for all prime ideals q in the ring of integers of K℘∞ . The equality of the local conditions follows from Remark 5.1 for primes q np, from Remark 5.2 for primes q | n, q p and from Remark 5.4 for primes p | p. The second isomorphism follows by taking the direct limit over n. ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 17 6.2.2. Arithmetic applications of the main result. Let A/F satisfy Assumption 6.5 above. Define the ℘-adic L-function associated to A/K to be L℘,π (A/K) := L℘,π (f /K). Then Theorem 6.1 and Corollary 6.2 can be restated as follows: Theorem 6.8. The characteristic power series Char℘,π (A/K) of the Pontryagin dual Sel∨∞ (A/K℘∞ ) of the π-primary Selmer group Selπ∞ (A/K℘∞ ) of A over K℘∞ divides π the ℘-adic L-function L℘,π (A/K) of A over K. In particular, Sel∨∞ (A/K℘∞ ) is pseudoπ isomorphic to a torsion Λ℘,π -module. This result on the abelian variety A/F can be used to deduce the following corollaries. Denote by LK (A, s) and LK (A, χ, s) the complex L-function of A over K and its twist by finite order characters χ : G℘∞ → C× . For any character χ : G℘∞ → O× , where O is the valuation ring of a finite extension of Qp , denote by the same symbol χ : Λ℘,π → O ¯ its extension. Choose an embedding Qp → C such that χ can also be considered as a complex-valued character. For any Λ℘,π -module M , let M χ := M ⊗χ Of,π . Finally, let Xπ∞ (A/K℘∞ ) denote the π-primary Tate-Shafarevich group of A/K℘∞ which is defined by the exactness of the following sequence: 0 → A(K℘∞ ) ⊗ (Eπ /OE,π ) → Selπ∞ (A/K℘∞ ) → Xπ∞ (A/K℘∞ ) → 0, where Eπ and OE,π are the completions of E = EndQ (A) and OE = End(A) at π. Corollary 6.9. Suppose that [F℘ : Qp ] = 1. If LK (A, χ, 1) = 0, then A(K℘∞ )χ and Xπ∞ (A/K℘∞ )χ are finite. Proof. In this case the Iwasawa algebra Λ℘,π is isomorphic to a power series ring over Of,π in one variable and all pseudo-null Λ℘,π -modules are finite. By the interpolation formula, χ(L℘ (f /K)) = 0. By Theorem 6.8, χ(Char℘,π (A, K)) = 0. Hence Selπ∞ (A/K℘∞ )χ is finite and the result follows. Corollary 6.10. Suppose [F℘ : Qp ] = 1 and the torsion subgroup A(K℘∞ )tors of A(K℘∞ ) finite. Then A(K℘∞ ) is finitely generated. Proof. As in the proof of Corollary 6.9, note that all pseudo-null Λ℘,π -modules are finite. By Theorem 6.8, Sel∨∞ (A/K℘∞ ) is a torsion Λ℘,π -module. The result follows from the π classification of torsion Λ℘,π -modules because A(K℘∞ )tors is finite. Corollary 6.11. Suppose [F℘ : Qp ] = 1 and A an elliptic curve. Then A(K℘∞ ) is finitely generated. Proof. By definition, A does not have complex multiplication, hence by [32, Proposition 6.12, (ii)], A(K℘∞ )tors is finite and Corollary 6.10 applies. Remark 6.12. The finiteness of A(K℘∞ )tors for more general abelian varieties of GL2 type is proved for example in [32, Proposition 6.12, (i)] under the condition that the Zp -extension is the cyclotomic one. This explains the finiteness assumption added in Corollary 6.10. 7. The proof 7.1. The divisibility criterion. The argument of the proof of Theorem 6.1 is based on the generalization of [5, Proposition 3.1], which will be obtained in the next Proposition 7.4. For its proof, we need two preliminary results which, for lacking of precise references, are stated in the following as Lemma 7.1, Lemma 7.2 and Lemma 7.3. 18 MATTEO LONGO Let Λ := R[[T1 , . . . , Tm ]] be a ring of formal power series in m ≥ 1 variables, where R is the ring of integers of a finite extension of Qp and p is a prime number. Choose an uniformizer of R. Recall that the Noetherian integral domain Λ is an UFD (see for example [41]), so every height one prime ideal of Λ is principal (see for example [31, Theorem 20.1]). A finitely generated Λ-module X is said to be pseudo-null if its support SuppΛ (X) contains only prime ideals of height greatest or equal to 2. Two Λ-modules X and Y are said to be pseudo-isomorphic if there exist two pseudo-null Λ-modules A and B and an exact sequence of Λ-modules: 0 → A → X → Y → B → 0. Let X be a finitely generated Λ-module. By [6, Section 4, 4, Th´or`mes 4, 5], X is e e pseudo-isomorphic to a Λ-module of the form Λr ⊕s Λ/(gi ), that is, there exists an i=1 exact sequence of Λ-modules: (7) 0 → A → X → Λr ⊕s Λ/(gi ) → B → 0, i=1 where r, s are non-negative integers, A, B are pseudo-null Λ-modules and gi ∈ Λ. By definition the characteristic power series CharΛ (X) attached to the Λ-module X is CharΛ (X) := s gi if r = 0 and 0 otherwise. The characteristic power series i=1 CharΛ (X) is well-defined only up to units in Λ; the characteristic ideal (CharΛ (X)) of Λ that it generates is then well defined. Lemma 7.1. Let F , G be elements of Λ. Then F divides G if and only if for all morphisms ϕ : Λ → O, where O is the ring of integers of a finite extension of Qp , ϕ(F ) divides ϕ(G). Proof. One direction is obvious. For the other direction, prove the following equivalent statement: If F does not divide G, then there exists a homomorphism ϕ : Λ → O, where O is the ring of integers of a finite extension of Qp , such that ϕ(F ) does not divide ϕ(G). The proof is by induction. The case m = 1 is an easy consequence of the Weierstrass preparation theorem, so suppose the statement true for m − 1 and prove it for m. For T := T1 and W := ∞ ∞ n n (T2 , . . . , Tm ), write F = and G = where an , bn ∈ R[[W ]] for n=0 an T n=0 bn T n = 0, . . . , ∞. If a0 b0 , then, by the inductive hypothesis, there exists an homomorphism ϕ : R[[W ]] → O for some O as above such that ϕ(a0 ) ϕ(b0 ). Extend ϕ to a morphism, denoted by the same letter ϕ : Λ → O, by setting ϕ(T ) := 0. Then ϕ(F ) does not divide ϕ(G). Hence, in the following suppose that a0 | b0 . If a0 | b0 , since F does not divide G, there are elements cn ∈ R[[W ]], n = 0, . . . , N − 1 and N ≥ 1, such that bn = n ai cn−i for n = 0, . . . , N − 1 and a0 does not divide i=0 bN − N ai cN −i . Hence by the inductive hypothesis, there exists a morphism ϕ : i=1 R[[W ]] → O for some O as above such that ϕ(a0 ) ϕ(bN − N ai cN −i ). Extend i=1 ϕ to a morphism, denoted by the same letter ϕ : Λ → O, by setting ϕ(T ) := T . Hence, ϕ(F ) does not divide ϕ(G) in O[[T ]]. By the inductive hypothesis, there exists a morphism ϕ : O[[T ]] → O such that ϕ (ϕ(F )) does not divide ϕ (ϕ(G)). Defining ϕ := ϕ ◦ ϕ : Λ → O , yields ϕ (F ) ϕ (G). Lemma 7.2. Let I = (x1 , . . . , xn ) with n ≥ 2 be an ideal of Λ such that I ⊆ P for all prime ideals P of Λ of height one. Then I contains at least two elements without common irreducible factors. ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 19 Proof. The proof is by induction on n. The case n = 2 is immediate, so suppose the result true for n − 1 and prove it for n. Denote by f the common greatest divisor of the xi for i = 1, . . . , n−1 and write xi := xi /f for i = 1, . . . , n−1. Then J := (x1 , . . . , xn−1 ) is not contained in any prime ideal of Λ of height one, so, by the inductive hypothesis, there are two elements, say g and h, without common irreducible factors. Then f g and f h are in I and g, h do not have common irreducible factors. Furthermore, any irreducible factor z of xn may divide g or h (but not both of them) and does not divide f (if it does, then I ⊆ (z), which contradicts our assumption). Write xn = ks where an irreducible factor z of xn divides k if and only if z divides gh. Then any irreducible factor of s is prime to gh. If s is invertible, then xn and f (g + h) ∈ I do not have irreducible common factors: any irreducible factor of xn does not divide f and divides m exactly one between g and h. Suppose s is not invertible and write s = t sj j , where j=1 sj are irreducible and mj are non negative integers. If sj | g +nh for some integer n = 0, then sj g + (m + n)h for all integers m = 0, except possibly those m such that p | m in the case when (sj ) = ( ): indeed, if sj | g + (m + n)h, then sj | mh and, since sj h, sj | m, and this is possible only if sj is a constant, hence (sj ) = ( ), so that p | m. It follows that if s, then s and g + mh do not have common irreducible factors for all integers m except possibly a finite number of them, while if | s, then s and g + mh do not have common irreducible factors for infinitely many integers m. Choose an integer m = 0 such that s and g + mh do not have common irreducible factors, with the additional condition that p m if | k. Note that there are infinitely many integers m verifying these conditions, even if R = Z2 : indeed, the condition p m is required only if | k, but in this case s and there are only a finite number of integers m such that s and g + mh do not have common irreducible factors. We claim that xn and f (g + mh) ∈ I do not have common irreducible factors. Indeed, let z | xn be an irreducible factor, so that z | k or z | s. If z | k then z f and z divides exactly one between g and h. If z | h, then, since z g, z g + mh. If z | g, then z mh: indeed, z h and, since m = 0, the only case when z | m is that of (z) = ( ) and p | m, but our additional condition on m stipulates that p m when | k. Since z | g and z mh, then z g + mh. Hence in any case if z | k then z f (g + mh). If z | s, then z = sj for some j, hence z g + mh and since sj f then sj f (g + mh). The claim follows, thus completing the proof. If X is a finitely generated Λ-module, denote by FittΛ (X) (respectively, AnnΛ (X)) its Fitting ideal (respectively, its annihilator ideal) over Λ. Lemma 7.3. Let X be a finitely generated pseudo-null Λ-module. Then FittΛ (X) contains at least two elements with no common irreducible factors. Proof. Recall that a prime ideal P of Λ belongs to the support SuppΛ (X) of X in Λ if and only if the annihilator AnnΛ (X) of X in Λ is contained in P (see for example [31, page 26]). Fix a prime ideal P of Λ of height 1. By definition of pseudo-null Λ-module, P ∈ SuppΛ (X), so AnnΛ (X) ⊆ P . Suppose that X is generated over Λ by h elements. Then by [33, Appendix, 8], AnnΛ (X)h ⊆ FittΛ (X), hence, since P is a prime ideal, FittΛ (X) ⊆ P for all prime ideals P of height 1. The result follows from Lemma 7.2. Proposition 7.4. Let X be a finitely generated Λ-module and L ∈ Λ. Suppose that ϕ(L) belongs to FittO (X ⊗ϕ O) for all homomorphisms ϕ : Λ → O, where O is the ring of integers of a finite extension of Qp . Then L belongs to (CharΛ (X)). 20 MATTEO LONGO Proof. If X is not Λ-torsion, then FittΛ (X) = 0. Since FittO (X ⊗ϕ O) is equal to ϕ(FittΛ (X)), it follows that ϕ(L) = 0 for all ϕ as above and hence, by Lemma 7.1, L = 0. Assume in the following that X is a Λ-torsion module. Since B in the exact sequence (7) is pseudo-null, by Lemma 7.3 there are at least two elements x1 and x2 in FittΛ (B) without common irreducible factors. Tensoring the exact sequence (7) with O yields ϕ(xi )FittO (X ⊗ϕ O) ⊆ (ϕ(CharΛ (X))) for i = 1, 2. By assumption, ϕ(CharΛ (X)) divides ϕ(xi L) for i = 1, 2 and hence, by Lemma 7.1, CharΛ (X) divides xi L for i = 1, 2. Since x1 and x2 do not have common irreducible factors, CharΛ (X) divides L and the result follows. 7.2. Admissible primes. A prime ideal ⊆ OF is said to be n-admissible if: (1) does not divide np; (2) is inert in K/F ; (3) π does not divide | |2 − 1; (4) π n divides | | + 1 + a (f ) or | | + 1 − a (f ). Let be an n-admissible prime. Then 1 Hsing (K , Tf,πn ) 1 Of,π /π n and Hfin (K , Tf,πn ) Of,π /π n . To show this, note that, since Tf,πn is unramified at , 1 Hsing (K , Tf,πn ) = HomGK (IK , Tf,πn ). Since p, all homomorphisms above factors through the tame inertia subgroup. The Frobenius Frob (K) of K at (where, by an abuse of notations, denotes the unique prime of K above ) acts on IK by | |2 and on Tf,πn it acts with eigenvalues | |2 and 1 (which are distinct in Of,π /π n ). Hence, 1 Hsing (K , Tf,πn ) Of,π /π n . For the finite cohomology, since Tf,πn is unramified at , 1 Hfin (K , Tt,πn ) 1 Hfin (K Tf,πn /(Frob (K) − 1). Hence, as above, , Tf,πn ) Of,π /π n . Since is inert in K, it splits completely in K℘∞ . It follows that: ˆ1 ˆ1 Hsing (K℘∞ , , Tf,πn ) Λ℘,π /π n Λ℘,π and Hfin (K℘∞ , , Tf,πn ) Λ℘,π /π n Λ℘,π . Proposition 7.5. Let s ∈ H 1 (K, Af,π ) be a non-zero element. Then there exist infinitely many admissible primes such that ∂ (s) = 0 and v (s) = 0. Proof. This is a direct generalization of [5, Theorem 3.2]. A similar argument will be given in Proposition 7.13. 7.3. Rigid pairs. Let ρ = ρf,π denote the representation of GF = Gal(F /F ) on the k := Of,π /π-vector space Af,π . The k-vector space adρ := Hom(Af,π , Af,π ) is endowed with an action of GF by conjugation of endomorphisms. The GF -module adρ is called the adjoint representation of ρ. Denote by ad0 ρ the k-subspace of trace-zero endomorphisms in adρ with the induced action of GF . Define the following local structures for the cohomology of ad0 ρ: 1 Primes q np: Define Hfin (Fq , ad0 ρ) := H 1 (GFq /IFq , ad0 ρ) to be the unramified cohomology. ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 21 Primes q | n, q p: As in the previous case, define 1 Hord (Fq , ad0 ρ) := H 1 (GFq /IFq , (ad0 ρ)IFq ) to be the unramified cohomology. (1) (p) Primes p | p: Let ad(1) ρ denote the subspace Hom(Af,π , Af,π ) of ad0 ρ. Define: 1 Hord (Fp , ad0 ρ) := ker(H 1 (Fp , ad0 ρ) → H 1 (IFp , ad0 ρ/ad(1) ρ)). 1-admissible primes : If is a 1-admissible prime, denote by ad( ) ρ the unique one dimensional k-vector subspace of ad0 ρ on which the absolute Frobenius Frob (F ) of F at acts with eigenvalue | | (the existence of this subspace follows because the Frobenius at acts on Af,π with eigenvalues ±| | and ±1, so that the eigenvalues of its action on ad0 ρ are | |, | |−1 and 1, while its uniqueness follows because | |2 = 1 in k). Define 1 Hord (F , ad0 ρ) := H 1 (F , ad( ) ρ) 1 1 and Hfin (F , ad0 ρ) to be the kernel of the canonical map H 1 (F , ad0 ρ) → Hord (F , ad0 ρ). 0 The group H 1 (F , ad ρ) is two dimensional over k and there is a decomposition in one-dimensional k-vector spaces: 1 1 H 1 (F , ad0 ρ) = Hfin (F , ad0 ρ) ⊕ Hord (F , ad0 ρ). See for example [40, Section 3, Lemma 1] for details. Let s be a square-free product of 1-admissible primes. Define the s-Selmer group Sels (F, ad0 ρ) attached to ad0 ρ to be the k-vector space consisting of those classes ξ ∈ H 1 (F, ad0 ρ) such that: 1 (1) Primes q np: resq (ξ) ∈ Hfin (F , ad0 ρ); 1 (2) Primes | s: res (ξ) ∈ Hord (Fq , ad0 ρ); 1 (3) Primes q | n and q p: resq (ξ) ∈ Hord (Fq , ad0 ρ); 1 (4) Primes p | p: resp (ξ) ∈ Hord (Fp , ad0 ρ); Denote by R the minimal nearly ordinary universal deformation ring attached to ρ with determinant the cyclotomic character. See [15, Section 3.8] for detailed definitions. Let mf,π := Ker(Tn → k) and denote by Tf the completion of Tn at mf,π . Then R is isomorphic to Tf by [15, Theorem 11.1]. Remark 7.6. The condition [F (ζ5 ) : F ] = 2 when p = 5 in the Introduction is required to apply [15]. Lemma 7.7. The modular form f is π-isolated if and only if SelOF (F, ad0 ρ) is trivial. Proof. f is π-isolated if and only if Tf Of,π , and this condition is equivalent to the isomorphism R Of,π . Now R Of,π if and only if m/(π, m2 ) = 0, and this condition is equivalent to SelOF (F, ad0 ρ) = 0 by [15, Proposition 3.35]. Assume from now on that Assumption 1.6 is verified, so that f is π-isolated. If s is a (possibly empty) square free product of 1-admissible primes, let Sel(s) (F, ad0 ρ) be the group defined in the same way as Sels (F, ad0 ρ) but with no conditions imposed at the prime dividing s. Let Sel[s] (F, ad0 ρ) denote the subgroup of Sels (F, ad0 ρ) consisting of classes which are trivial at the primes dividing s. These notations can be combined: if s1 , s2 , s3 are pairwise coprime square-free product of 1-admissible primes, define the group Sels1 (s2 )[s3 ] (F, ad0 ρ) := Sels1 (F, ad0 ρ) ∩ Sel(s2 ) (F, ad0 ρ) ∩ Sel[s3 ] (F, ad0 ρ). 22 MATTEO LONGO Let ad0 ρ∗ := Hom(ad0 ρ, k) be the dual representation of ad0 ρ. Then define the dual Selmer group of Sels (F, ad0 ρ) to be the subgroup Sels (F, ad0 ρ∗ ) of H 1 (F, ad0 ρ∗ ) consisting of those elements t ∈ H 1 (F, ad0 ρ∗ ) such that resq (s), resq (t) 0 q =0 for all s ∈ Sels (F, ad ρ) and for all prime ideals q, where , q is the local Tate pairing at q. Define as above the Selmer groups Sels (F, ad0 ρ∗ ), Sel(s) (F, ad0 ρ∗ ), Sel[s] (F, ad0 ρ∗ ) and Sels1 (s2 )[s3 ] (F, ad0 ρ∗ ). The groups Sel(s) (F, ad0 ρ) and Sel[s] (F, ad0 ρ∗ ) are dual to each other, and the same is true for Sels (F, ad0 ρ) and Sels (F, ad0 ρ∗ ). Lemma 7.8. If is an admissible prime for f , then Sel( ) (F, ad0 ρ) and Sel( ) (F, ad0 ρ∗ ) are one dimensional over k. Proof. The groups SelOF (F, ad0 ρ) and SelOF (F, ad0 ρ∗ ) have the same cardinality by [10, Theorem 2.19]. Furthermore, SelOF (F, ad0 ρ) = 0 by Lemma 7.7 because f is πisolated. Hence SelOF (F, ad0 ρ∗ ) = 0. Since #Sel( ) (F, ad0 ρ)/#Sel[ ] (F, ad0 ρ∗ ) = #k by [10, Theorem 2.19], it follows that Sel( ) (F, ad0 ρ) is one dimensional over k. Replacing ad0 ρ by ad0 ρ∗ and repeating the same argument shows that Sel( ) (F, ad0 ρ∗ ) is one dimensional too. Lemma 7.9. Let be an admissible prime for f and suppose that Sel (F, ad0 ρ) = 0. Then Sel (F, ad0 ρ) k. Proof. Thanks to the inclusion Sel 2 (F, ad0 ρ) ⊆ Sel( 2 ) (F, ad0 ρ), this is immediate from Lemma 7.8. Fix a pair of admissible primes 1 = 1 v 2 : Sel( 1 ) (F, ad ρ) → Hfin (F 2 , ad0 ρ) 1 v ∗2 : Sel( 1 ) (F, ad0 ρ∗ ) → Hfin (F 2 , ad0 ρ∗ ) for the restriction maps at 2 . 2. 0 Write Lemma 7.10. Suppose that Sel 1 (F, ad0 ρ) = 0 and v 2 , v ∗2 are both non trivial. Then Sel 1 2 (F, ad0 ρ) = 0. Proof. By Lemma 7.8, choose generators ξ and ξ ∗ of the one dimensional k-vector spaces Sel( 1 ) (F, ad0 ρ) and Sel( 1 ) (F, ad0 ρ∗ ). Note that Sel 1 (F, ad0 ρ) ⊆ Sel( 1 ) (F, ad0 ρ) k 0 0 0 and Sel 1 (F, ad ρ) = 0 by assumption. Hence ξ ∈ Sel 1 (F, ad ρ) and Sel 1 (F, ad ρ) k. Since Sel 1 (F, ad0 ρ) and Sel 1 (F, ad0 ρ∗ ) have the same cardinality by [10, Theorem 2.19], ξ ∗ ∈ Sel 1 (F, ad0 ρ∗ ). By [10, Theorem 2.19]: (8) #Sel 1 ( 2 ) (F, ad0 ρ)/#Sel 1 [ 2 ] (F, ad0 ρ∗ ) = #k. Note that Sel 1 [ 2 ] (F, ad0 ρ∗ ) ⊆ Sel 1 (F, ad0 ρ∗ ) k. Hence, either Sel 1 [ 2 ] (F, ad0 ρ∗ ) = 0 or Sel 1 [ 2 ] (F, ad0 ρ∗ ) k, generated by ξ ∗ . In the second case, ξ ∗ ∈ Sel 1 [ 2 ] (F, ad0 ρ∗ ) 1 implies that res 2 (ξ ∗ ) = 0 in Hfin (F 2 , ad0 ρ∗ ). The assumption v ∗2 (ξ ∗ ) = 0 excludes this possibility, so Sel 1 [ 2 ] (F, ad0 ρ∗ ) = 0. By (8), Sel 1 ( 2 ) (F, ad0 ρ) k. The inclusion Sel 1 (F, ad0 ρ) ⊆ Sel 1 ( 2 ) (F, ad0 ρ) implies Sel 1 ( 2 ) (F, ad0 ρ) = Sel 1 (F, ad0 ρ) and both of them are generated by ξ. Finally, note that Sel 1 2 (F, ad0 ρ) ⊆ Sel 1 ( 2 ) (F, ad0 ρ), so, as above, either Sel 1 2 (F, ad0 ρ) is trivial or is one dimensional. In the second case, it is ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 23 isomorphic to Sel 1 ( 2 ) (F, ad0 ρ) and hence also to Sel 1 (F, ad0 ρ). So the reduction of ξ at 2 should be both ordinary (it belongs to Sel 1 2 (F, ad0 ρ)) and finite (it belongs to Sel 1 2 (F, ad0 ρ)), hence trivial. The assumption v 2 (ξ) = 0 excludes this possibility, so Sel 1 2 (F, ad0 ρ) is trivial. Lemma 7.11. If Sel 1 (F, ad0 ρ) = 0, Sel 2 (F, ad0 ρ) = 0 and v 2 is the trivial map, then Sel 1 2 (F, ad0 ρ) = 0. Proof. Since Sel 2 (F, ad0 ρ) = 0, by [10, Theorem 2.19], Sel( 1 ) 2 (F, ad0 ρ) is one dimensional. By Lemma 7.8, choose a generator ξ of Sel( 1 ) (F, ad0 ρ). Since v 2 (ξ) = 0, the restriction to F 2 of this class must be ordinary, and so belongs to Sel( 1 ) 2 (F, ad0 ρ). Hence, Sel( 1 ) 2 (F, ad0 ρ) is generated by ξ and Sel( 1 ) 2 (F, ad0 ρ) Sel( 1 ) (F, ad0 ρ). Note that if ξ ∈ Sel 1 ( 2 ) (F, ad0 ρ), then also ξ ∈ Sel 1 (F, ad0 ρ). By assumption Sel 1 (F, ad0 ρ) = 0, so ξ ∈ Sel 1 ( 2 ) (F, ad0 ρ). Since ξ ∈ Sel( 1 ) 2 (F, ad0 ρ) and ξ ∈ Sel 1 ( 2 ) (F, ad0 ρ), then Sel 1 2 (F, ad0 ρ) is trivial because it is the intersection of Sel 1 ( 2 ) (F, ad0 ρ) and Sel( 1 ) 2 (F, ad0 ρ)). Definition 7.12. A pair ( 1 , 2 ) of admissible primes is said to be a rigid pair if Sel 1 2 (F, ad0 ρ) is trivial. Choose s ∈ H 1 (K, Af,π ), s = 0. Assume that s belongs to a specific eigenspace for the complex conjugation τ , so that τ (s) = δs with δ = ±1. Write M = K(Af,πn ) and let Ms /M be the extension cut out by s, so that Gal(Ms /M ) Af,π via s. Let GM := Gal(M /M ). Since f is π-isolated, Sel( 1 ) (F, ad0 ρ) and Sel( 1 ) (F, ad0 ρ∗ ) are one dimensional over ¯ ¯ k. Let ξ and ξ ∗ be generators. The images ξ and ξ ∗ of ξ and ξ ∗ in H 1 (M, ad0 ρ∗ ) = Hom(GM , ad0 ρ∗ ) ¯ ¯ cut out extensions Mξ and Mξ∗ of M whose Galois groups are identified via ξ and ξ ∗ with 0 0 ∗ 0 ad ρ and ad ρ respectively (that is, Gal(Mξ /M ) ad ρ and Gal(Mξ∗ /M ) ad0 ρ∗ ). Denote by Ms,ξ,ξ∗ the compositum of Ms , Mξ and Mξ∗ . Since the representations Af,π , ad0 ρ and ad0 ρ∗ are pairwise non isomorphic and absolutely irreducible, then and Gal(Ms,ξ,ξ∗ /F ) (Af,π × ad0 ρ × ad0 ρ∗ ) Gal(M/F ) where the action of Gal(M/F ) on the normal subgroup (Af,π , ad0 ρ, ad0 ρ∗ ) is given by ¯ ¯ ¯ ¯ ¯ (v, w, w∗ )(τ j , T ) = (δ j T v, T wT −1 , T w∗ T −1 det(T )). Proposition 7.13. Let 1 be admissible such that Sel 1 (F, ad0 ρ) = 0. Fix a non trivial element s ∈ H 1 (K, Af,π ). For any n there exists infinitely many n-admissible primes 2 such that ∂ 2 (s) = 0, v 2 (s) = 0 and ( 1 , 2 ) is a rigid pair. Proof. By Lemma 7.9, ξ ∈ Sel 1 (F, ad0 ρ), so that ξ ∗ ∈ Sel 1 (F, ad0 ρ∗ ) too. The Galois group Gal(Ms,ξ,ξ∗ /F ) contains an element (v, w, w∗ , τ, T ) such that: (1) T acts on Af,π with eigenvalues δ and λ where λ is an element of (Of,π /π m )× of order prime to p and = ±1; (2) v belongs to the unique line in Af,π where T acts by δ; (3) w belongs to the unique line in ad0 ρ fixed by T ; (4) w∗ belongs to the unique line in ad0 ρ∗ fixed by T . H 1 (M, ad0 ρ) = Hom(GM , ad0 ρ) 24 MATTEO LONGO Choose 2 pn and unramified in Ms,ξ,ξ∗ so that the Frobenius element Frob 2 (Ms,ξ,ξ∗ /F ) of Gal(Ms,ξ,ξ∗ /F ) at 2 verifies the relation: Frob 2 (Ms,ξ,ξ∗ /F ) = (v, w, w∗ , τ, T ). By the Chebotarev density theorem, there are infinitely many such primes. Then 2 has the desired properties. To show that 2 is n-admissible, note that the Frobenius element Frob 2 (K/F ) of Gal(K/F ) at 2 verifies the relation Frob 2 (K/F ) = τ , which implies that is inert in K. The congruences a 2 (f ) ≡ δ + λ (mod π n ) and | 2 | ≡ δλ (mod π n ) enjoyed by the characteristic polynomial of Frobenius show a 2 (f ) ≡ δ(| 2 |+1) (mod π n ). Finally, since λ = ±1, it follows that | 2 | ≡ ±1 (mod π n ). Hence 2 is an n-admissible prime. Moreover, 2 has the properties stated in the theorem. First, note that ∂ 2 (s) = 0. Indeed, if l is a prime ideal of Ms,ξ,ξ∗ dividing 2 , then res 2 (s) ∈ Ker H 1 (K 2 , Af,π ) → H 1 (Ms,ξ,ξ∗ ,l , Af,π ) . 1 Since H 1 (Ms,ξ,ξ∗ ,l , Af,π ) ⊇ H 1 (K unr , Af,π ) ⊇ Hsing (K 2 , Af,π ), it follows that ∂ 2 (s) = 0 2 (here Ms,ξ,ξ∗ ,l is the completion of Ms,ξ,ξ∗ at l). For the proof that v 2 (s) = 0: Let l be a prime ideal in M dividing and set c := [M : F ]. Denote by Frobl (Ms,ξ,ξ∗ /M ) a Frobenius element of Gal(Ms,ξ,ξ∗ /M ) at l. Note that: Frobl (Ms,ξ,ξ∗ /M ) = (v, w, w∗ , τ, T )c = (cv, cw, cw∗ , 1, 1). Let s be the image of s in Gal(Ms /M ). Since c is even and prime to p by property 1 of ¯ T, s(Frobl (Ms,ξ,ξ∗ /M )) = s(cv) = cs(v) = 0 and res 2 (s) = 0. So, v 2 (s) = 0. Since ¯ ¯ ¯ ξ(Frobl (Ms,ξ,ξ∗ /M )) = ξ(cw) = cξ(w) = 0, ¯ ¯ ¯ ξ ∗ (Frobl (Ms,ξ,ξ∗ /M )) = ξ ∗ (cw∗ ) = cξ ∗ (w) = 0, Lemma 7.10 implies Sel 1 2 (F, ad0 ρ∗ ) = 0, so ( 1 , 2 ) is a rigid pair. Proposition 7.14. Let 1 be admissible such that Sel 1 (F, ad0 ρ) = 0. Fix a non trivial element s ∈ H 1 (K, Af,π ). For any n there exists infinitely many n-admissible primes 2 such that ∂ 2 (s) = 0, v 2 (s) = 0 and either Sel 2 (F, ad0 ρ) k or Sel 2 (F, ad0 ρ) = 0 and ( 1 , 2 ) is a rigid pair. Proof. Choose a prime 2 so that Frob 2 (Ms,ξ,ξ∗ /F ) = (v, 0, 0, τ, T ). The same computations as in Proposition 7.13 show that 2 is admissible and that v 2 (s) = 0. Note that ξ(w) = 0 and ξ ∗ (w∗ ) = 0. If Sel 2 (F, ad0 ρ) = 0, by Lemma 7.11 Sel 1 2 (F, ad0 ρ) is trivial, so ( 1 , 2 ) is a rigid pair. If Sel 2 (F, ad0 ρ) = 0, then it is one-dimensional by Lemma 7.9. 7.4. Congruences between modular forms and the Euler system. ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 25 7.4.1. Raising the level in one prime. Fix an n-admissible prime . Let Tn+ ,n− be the Hecke algebra acting on the space of modular forms which are new at n− . It is known that there exists a morphism f : Tn+ ,n− → Of,π /π n such that: (1) Primes q n : f (Tq ) ≡ aq (f ) (mod π n ); (2) Primes q | n: f (Uq ) ≡ aq (f ) (mod π n ); (3) f (U ) ≡ (mod π n ), where π n divides | | + 1 − aq (f ). This result follows from a generalization to the case n > 1 of [39]. For details, see [30, Theorem 3.3]. 7.4.2. The Euler system. Denote by X ( ) the Shimura curve (defined over F ) whose complex points are given by X ( ) (C) = B × \H± × B × /R× , where H± := C − R, B/F is a quaternion algebra of discriminant n− which is ramified in exactly one of the archimedean places and R ⊆ B is an Eichler order of level ℘n+ . Let J ( ) be the jacobian variety (defined over F ) associated to X ( ) . Denote by Tp (J ( ) ) the p-adic Tate module of J ( ) and by Φ the group of connected components of the fiber at of the N´ron model of J ( ) over OK . Denote by If the kernel of the map e f . By [30], which generalizes the result of [29] to the present situation, there exists an Hecke equivariant isomorphism of Gal(F /F )-modules: (9) ν : Tp (J ( ) )/If → Tf,πn . Remark 7.15. It is not known at the present if (9) is an isomorphism when the degree d of F over Q is even and n− = OF . This explains the second part in Assumption 1.2. Following Section 3, an Heegner point Pm of conductor ℘m is a CM-point of conductor ℘ in ( ) XR (K) := B × \Hom(K, B) × B × /R× . Let σ be the archimedean place where B is split and fix an isomorphism ι∞ of B ⊗σ R in M2 (R). Then B acts on H± by fractional linear transformations via ι∞ and the set Hom(K, B) can be embedded in H± by sending Ψ ∈ Hom(K, B) to the fixed point of Ψ(K × ) acting on H± whose imaginary part is positive. Hence, a CM-point P ∈ ( ) XR (K) of conductor ℘m can be viewed as a point in X ( ) (C) and the theory of complex multiplication shows that, in fact, P ∈ X ( ) (K℘m ). Furthermore, the Galois action on CM-points of conductor ℘m described in Section 3 translates into the usual Galois action of G℘m on X ( ) (K℘m ). For more details, see [44, Chapter 9]. Recall the choice of orientations made in Section 4.2 and fix an orientation as ex( ) plained in Section 3 at the prime . Define the set of Gross points Gr( ) (℘m ) in XR with respect to these orientations. Write Pm = (xm , Ψm ). Let ePm = (s(ePm ), t(ePm )) ∈ E℘ be the edge corresponding to xm R℘ x−1 as described in Section 3. Say that a sequence m (Pm )m≥1 of points in Gr( ) (℘∞ ), with Pm ∈ Gr( ) (℘m ), is compatible if t(ePm ) = s(ePm+1 ) for all integers m ≥ 1. Choose a sequence of compatible Heegner points (Pm )m≥1 with Pm ∈ Gr( ) (℘m ). For the modular interpretation of Heegner points, which will not be recalled here, we refer to [47, Section 2]. Since If is not Eisenstein, there is an isomorphism m J ( ) (K℘m )/If → Pic(X ( ) )(K℘m )/If . 26 MATTEO LONGO + Denote by Pm the image of Pm in J ( ) (K℘m )/If . Define ∗ −m + Pm := αp Pm . ∗ Since (Pm )m≥1 is compatible, it is easily seen that the points Pm are norm-compatible. Their images under the Kummer map followed by the map induced by ν J ( ) (K℘m )/If → H 1 (K℘m , Tp (J ( ) )/If ) → H 1 (K℘m , Tf,πn ) yield a sequence of cohomology classes, κm ( ), which are compatible under corestriction. ˜ ˆ 1 (K℘∞ , Tf,πn ). Define finally the class Taking limit defines a class κ( ) ∈ H ˜ ˆ κ( ) ∈ H 1 (K℘∞ , Tf,πn ) to be the corestriction of κ( ) from K℘∞ to K℘∞ . ˜ ˆ Lemma 7.16. κ( ) ∈ H 1 (K℘∞ , Tf,πn ). Proof. It is enough to observe, as in [5, beginning of Section 8], that κ( ) is constructed from a sequence of global points of X ( ) , so it belongs to the usual Selmer group of J ( ) relative to the Galois module Tp (J ( ) )/If . For completeness, let us provide some ˆ details on this proof. From the definition of H 1 (K℘∞ , Tf,πn ), we see that it is enough to show that: 1 κ (1) resq (˜ m ( )) ∈ Hfin (K℘m ,q , Tf,πn ) for primes q of K℘m which do not divide np ; 1 κ (2) resq (˜ m ( )) ∈ Hord (K℘m ,q , Tf,πn ) for primes q dividing n− but not p; 1 κ (3) resp (˜ m ( )) ∈ Hord (K℘m ,p , Tf,πn ) for primes p of K℘m which divide p. For (1), Remark 5.1 shows that the image of the Kummer map J ( ) (K℘m ,q ) → H 1 (K℘m ,q , J ( ) [pn ]) is unramified; the result follows then taking quotient by If . For (3), note that the Kummer map J ( ) (K℘m ,p ) → H 1 (K℘m ,p , Tf,n ) factors through the maximal ordinary abelian subvariety J ( ),ord of J ( ) ; the result follows then by Remark 5.4, again taking quotients by If . For (2), the analogue of [5, Corollary 5.18] (see (21) with the prime q replacing m ) shows that if the quotient Φq /If of the group of connected components Φq at q of J ( ) by If is trivial, then resq (˜ m ( )) is unramified; on the other hand, the κ vanishing of Φq /If follows because f is ramified at q. Indeed, if Φq /If = 0, then there is an Of,π /π n -valued modular form of level n /q which is congruent to f , and hence to f , modulo π n ; so the π-adic representation associated to f should be unramified at q, which is not the case. 7.4.3. Raising the level in two primes. Choose distinct n-admissible primes 1 and 2 so that π n divides both | 1 | + 1 − 1 a 1 (f ) and | 2 | + 1 − 2 a 2 (f ), with 1 , 2 equal to ±1. Let T 1 be the Hecke algebra acting on the Shimura curve X ( 1 ) . Assume that f is π-isolated. The map arising from Kummer theory composed with (9) yields a map: J ( 1 ) (K 2 )/If 1 → H 1 (K 2 , Tp (J ( 1 ) )/If 1 ) → H 1 (K 2 , Tf,πn ) 1 whose image is equal to Hfin (K 2 , Tf,πn ) because both Tp (J ( 1 ) ) and Tf,πn are unramified at 2 . For the same reason and the fact that 2 p, the map induced by reduction modulo 2 : J ( 1 ) (K 2 )/If 1 → J ( 1 ) (F 2 )/If 1 2 ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 27 is an isomorphism, where F 2 is the residue field of the ring of integers of K 2 . The iden2 1 tification Hfin (K 2 , Tf,πn ) Of,π /π n and the inverse of the above map yield a surjective map: (10) J ( 1 ) (F 2 )/If 1 → Of,π /π n . 2 Let S 2 ⊆ X ( 1 ) (F 2 ) be the set of supersingular points of X ( 1 ) in characteristic 2 2 and let Div(S 2 ) and Div0 (S 2 ) be the set of formal divisors and the set of formal degree zero divisors with Z-coefficients supported on S 2 . Let the Hecke algebra T 1 act on Div(S 2 ) and Div0 (S 2 ) via Albanese functoriality (it makes no difference if the Picard functoriality were chosen: see the discussion in [5, Section 9]). Since If 1 is not Eisenstein, there is an identification Div(S 2 )/If 1 Div0 (S 2 )/If 1 , so there is a map: γ : Div(S 2 ) → Of,π /π n . ¯ ¯ Write T for the image of T ∈ T 1 into T 1 /If 1 , so that for primes q n 1 : Tq ≡ aq (f ) ¯ ¯ (mod π n ), for primes q | n: Uq ≡ aq (f ) (mod π n ) and U 1 ≡ 1 (mod π n ). Lemma 7.17. For x ∈ Div(S 2 ) the following relations hold: ¯ (1) For q n 1 : γ(Tq x) = Tq γ(x); ¯ (2) For q | n 1 : γ(Uq x) = Uq γ(x); ¯ (3) γ(T 2 x) = T 2 γ(x); (4) γ(Frob 2 (F )(x)) = 2 γ(x), where, as above, Frob (F ) is the absolute Frobenius of F at . Proof. The first two relations can be obtained from the identification between the groups 1 Hfin (K 2 , Tf,πn ) and Tf,πn /(Frob22 (F ) − 1). The last two relations follow from EichlerShimura. For more details, see [5, Lemma 9.1]. Before going on with the raising the level result, we state an analogue of Ihara’s Lemma in the context of Shimura curves over totally real fields. First recall the setting of [22]: Define G∞ := SL2 (R)/{±1} and, for any prime q of F , Gq := {g ∈ GL2 (Fq ) : × valq (det(g)) ≡ 0 (mod 2)}/Fq , where valq is the normalized valuation of Fq . Let × × i∞ : B → G∞ and iq : B → Gq be the injections. Let OF [1/q] be the ring of q-integers of F and U ⊆ B any OF [1/q]-order. Define ΓU := {γ ∈ U : NB/F (γ) = 1}/{±1}, ˜ where NB/F : B → F is the norm map. Let ΓU be the pull-back of the group × GL2 (Oq )/Oq under the map iq : ΓU → Gq . Denote by XU the Shimura curve defined over a suitable abelian extension of F whose complex points are: ˜ XU (C) = i∞ (ΓU )\H, where H is the upper complex plane. Suppose that ΓU is torsion-free. Denote by JU the Jacobian variety of XU . Let Fq2n be the field with q 2n elements, where q is the residue ss characteristic of and q and |q| = q n for a positive integer n. Let JU (Fq2n ) be the set of supersingular points in JU (Fq2n ). Then by [22, Section 3, (G)] there is a canonical isomorphism: (11) ss JU (Fq2n )/JU (Fq2n ) Γ(U)ab , where, if G is a group, Gab is the abelianization of G. 28 MATTEO LONGO Let U ⊆ B × be a compact open subgroup and define XU → Spec(F ), where F is a suitable abelian extension of F , to be the Shimura curve whose complex points are: t (12) XU (C) = B \B × H /U i=1 × × ± Xi (C), Xj (C) = Γj \H where Γi ⊆ B × are suitable arithmetic subgroups. Write JU for the jacobian variety of XU . Fix a prime q such that the q-component Uq of U is isomorphic to GL2 (OF,q ). For × ˜ any i = 1, . . . , t, let Γi denote the subgroup of norm-one elements in Γj [1/q]/OF [1/q]. Assume that (13) ˜ all the groups Γi are torsion free. t i=1 ˜ Let Ji denote the jacobian variety of Xj and set Γ := set of supersingular points in J(Fq2n ), then from (11): (14) ss JU (Fq2n )/JU (Fq2n ) ˜ Γi . If J ss (Fq2n ) denotes the ˜ Γab . ˜ By fixing an embedding of B into M2 (F 2 ), one obtains an action of Γi on the Bruhat˜ Tits tree Tq of PGL2 (Fq ). Let v0 be the vertex of Tq such that the stabilizer Γvi,0 of vi,0 ˜ ˜ in Γi is the image of Γi in Γi . Let ei,0 be the edge originating from vi,0 and such that the ˜ ˜ stabilizer Γei,0 of ei,0 in Γi the image of the subgroup Γi of Γi obtained as in (12) but with U ∩ U0 (q) replacing U , where U0 (q)q is the standard upper triangular subgroup Γ0 (q) of GL2 (Fq ) and U0 (q)q = GL2 (OF,q ) for q = q. More explicitly, t XU ∩U0 (q) = i=1 Xi , with Xi = Γi \H. ˜ Write vi,1 for the target of ei,0 . The group Γi acts on the tree Tq with the closed edge ˜ ˜ ˜ ˜ attached to ei,0 as a fundamental region. Set Γv0 := t Γvi,0 , Γv1 := t Γvi,1 and i=1 i=1 ˜ ˜ Γe0 := t Γei,0 . Hence, taking the product over all i = 1, . . . , t of the exact sequence i=1 ˜ in [42, Proposition 13, Section II, 2.8] for i = 1, M = Fp , G = Γi yields: d ˜ ˜ ˜ ˜ 0 → Hom(Γ, Fp ) → Hom(Γv0 , Fp ) ⊕ Hom(Γv1 , Fp ) −→ Hom(Γe0 , Fp ). For i = 1, . . . , t there are natural injective maps as in [28, Section 1, Equation (3)]: (15) ψi : Ji (C) → Hom(Γi , S) and ψi : Ji (C) → Hom(Γi , S), where S := {z ∈ C : |z| = 1}. Hence in the above exact sequence the modules appearing in the source and in the target of d correspond to the p-torsion of Ji and Ji respectively, where Ji is the jacobian variety of Xi . Suppose now that U is contained in some Eichler order of B of level r and let g be a modular form with coefficients in a finite field F, of weight 2, level K and trivial central character, which is an eigenform for the quotient T of the Hecke algebra of level rn− acting faithfully on JU (recall that the discriminant of B is n− ). Let mg be the kernel of the homomorphism T → F associated to g. Assumption 7.18. Let U be an open compact subgroup of B × such that (13) is ˜ verified. If g is irreducible then Hom(Γ, Fp )[mg ] = 0. ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 29 Remark 7.19. The technical condition in Assumption 7.18 is essential in the proof of Lemma 7.20 below. It consists in a version of Ihara’s Lemma for Shimura curves over totally real fields. Indeed, if F = Q, Assumption 1.8 holds thanks to [12, Theorem 2] because under the above identifications the map d corresponds to the map αp in [12, Theorem 2]. The result of [12, Theorem 2] can be understood as an analogue of Ihara’s Lemma in the context of Shimura curves over Q. The results contained in [12] and successively refined in [13] are partially generalized to the totally real case in [25]. However, [25] do not cover the full generalization of [12, Theorem 2]. It might be possible that the techniques in [25] and [26] can be used to prove some results in the direction of an analogue of [12, Theorem 2]. In this paper we follow [15], which assumes a suitable generalization to totally real fields of Ihara’s Lemma as an hypothesis, although Assumption 7.18 is stated in a different form with respect to [15, Hypothesis 5.9]. Similar results for Hilbert modular varieties hold: see [14]. ˜ The main consequence of Assumption 7.18 is that, under this condition, Γab /mg = 0. Let now R be a finite extension of Q and fix a maximal ideal v of R such that Rv /v F, where Rv is the completion of R at v. Suppose that g is a modular form with coefficients in Rv /v m for some integer m ≥ 1 of weight 2, level U and trivial central character, which is an eigenform for the Hecke algebra T; let Ig denote the kernel of the associated homomorphism T → Rv /v m and note that mg is the maximal ideal containing Ig . If ˜ the above conditions on g are satisfied, Γab /mg = 0 and hence, by Nakayama’s Lemma, ˜ Γab /Ig = 0. By (14), (16) ss the canonical map JU (Fq2n ) → JU (Fq2n )/Ig is surjective. Suppose from now on that Assumption 7.18 is verified. Lemma 7.20. The map γ is surjective. Proof. Write X = X ( 1 ) and J = J ( 1 ) . Let J ss (F 2 ) be the set of supersingular points 2 in J(F 2 ), where F 2 is the quadratic extension of the residue field F of OF at 2 . Since 2 2 the map (10) is surjective, it is enough to show that (17) the canonical map J ss (F 2 ) → J(F 2 )/If 1 is surjective. 2 2 X(C) = B × \B × × H± /R× . Define X to be the Shimura curve defined over F whose complex points are X (C) = B × \B × × H± /R × , where R ⊆ R is defined by requiring that, for a fixed isomorphism ι℘ : R ⊗OF OF,℘ { a b c d Recall that X is the Shimura curve defined over F whose complex points are ∈ GL2 (OF,℘ )|c ≡ 0 mod ℘}, b R ⊗OF OF,℘ correspond to the elements which are congruent to 1 1 mod ℘, while 0 R ⊗OF OF,q = R ⊗OF OF,q if q = ℘. Since R ⊆ R, there is a canonical projection map u : X → X and also, by Picard (respectively, Albanese) functoriality, maps u∗ : J → J (respectively, u∗ : J → J), where J and J are the jacobian varieties of X and X respectively. Write as above s t X(C) = i=1 Xi (C) and X (C) = j=1 Xj (C) 30 MATTEO LONGO where Xi = Γi \H and Xj (C) = Γj \H for suitable arithmetic subgroups Γi and Γj ; here s and t are suitable integers such that t ≥ s. The canonical projection u : X → X can be decomposed as t projections Xj → Xi(j) and if i(j1 ) = i(j2 ) (that is, two projections have the same target), then Γj1 = Γj2 . For details, see [20, Section 3]. Write finally Ji and Jj for the jacobian varieties of Xi and Xj , respectively. ˜ The subgroups Γj of norm one elements in Γj [1/ 2 ]/OF [1/ 2 ] are torsion free (see for example [19, Lemma 7.1], after noticing that p is not ramified in the extension K/Q). Now view f 1 as a mod π n eigenform on X and write If for its associated ideal in the 1 Hecke algebra T 1 acting faithfully on J . Write mf 1 for the maximal ideal containing If . Since mf 1 corresponds to an irreducible representation, it follows from (16) that 1 (18) the canonical map J ss (F 2 ) → J (F 2 )/If 2 2 1 is surjective. We need the generalization to this context of [28], which can be obtained as follows. For any j = 1, . . . , t, let i(j) such that i(j(i)) = i, that is, u∗ maps Ji(j) into Jj . An element x belongs to Σj := Ker(Ji(j) (C) → Jj (C)) if and only if the kernel of the map ψi(j) (x) associated to x as in (15) contains Γj . Set Σ := Ker(J(C) → J (C)). Using the fact that Γj1 = Γj2 if i(j1 ) = i(j2 ), we get an injection: 0 → Σ → ⊕s Hom(Γi /Γj(i) , S). i=1 The order of the group R× /R × is prime to p, hence the same is true for the order of (g −1 R× g)/(g −1 R × g) for any g ∈ B × . Since the groups Γi /Γj(i) are contained in (g −1 R× g)/(g −1 R × g) for suitable elements g ∈ B × , it follows the order of any Γi /Γj(i) is prime to p, so the same is true for Σ. Dualizing shows that the cokernel of the map u∗ : J (F 2 ) → J(F 2 ) has order prime to p. It follows that 2 2 (19) the canonical map J (F 2 ) → J(F 2 )/If 1 is surjective. 2 2 Finally, combining (18) and (19) shows (17). Let B /F be the totally definite quaternion algebra of discriminant n− 1 2 and R an B Eichler order of B of level ℘n+ . For any ring C, denote by S2 (℘n+ , C) the C-module of functions: × × × B \B /R → C. This module is endowed with an action on the Hecke algebra Tn 1 2 . B Proposition 7.21. There exists g ∈ S2 (℘n+ , Of,π /π n ) such that: (1) Prime ideals q n 1 2 : Tq (g) ≡ aq (f )g (mod π n ); (2) Prime ideals q | n: Uq (g) ≡ aq (f )g (mod π n ); (3) U 1 g ≡ 1 g (mod π n ) and U 2 g ≡ 2 g (mod π n ). Furthermore, if ( 1 , 2 ) is a rigid pair, then g can be lifted to a π-isolated form in B S2 (℘n+ ) taking values in Of,π . Proof. Write T 1 (respectively, T 2 , 1 ) for the quotient of the Hecke algebra Tn 1 (respectively, Tn 1 2 ) acting on cusp forms of weight 2, trivial central character, Γ0 (n 1 ) (respectively, Γ0 (n 1 2 )) level structure and new at n− 1 . Write f 1 : T 1 → Of,π /π n ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 31 for the modular form satisfying f 1 ≡ f (mod π n ). This form has the properties that Tq (f 1 ) ≡ aq (f )f 1 (mod π n ) for all q n 1 , Uq (f 1 ) ≡ aq (f )f 1 (mod π n ) for all q | n and U 2 (f 1 ) ≡ 1 f 1 (mod π n ). Let R1 ⊆ R be an Eichler order of level ℘n+ 2 and denote by X ( 2 , 1 ) the Shimura curve (over F ) whose complex points are given by: X ( 2 , 1 ) (C) = B × \B × × H± /R× . 1 Recall from above the set S 2 ⊆ X ( 1 ) (F 2 ) of supersingular points of X ( 1 ) in charac2 teristic 2 . By [48, Section 5.4]: (20) S2 B × \B × /R × . It follows that the character group X 2 of X ( 2 , 1 ) at 2 is identified with the module Div0 (S 2 ). Furthermore, the action of T 2 , 1 on X 2 induced from the action on Pic(X ( 2 , 1 ) ) by Picard functoriality is compatible with the standard Albanese action of T 2 , 1 via correspondences in the set of supersingular points. Therefore, γ can also be viewed as a Of,π /π n -valued modular form on B × \B × /R × . Denote by g this modular form. Since γ is surjective by Lemma 7.20, the image of g is not contained in any proper subgroup of Of,π /π n . ∗ ∗ To show that g has the desired properties, write Tq (with q n 1 2 ) and Uq (with q | n 1 2 ) for the Hecke operators in T 2 , 1 and Tq and Uq for the Hecke operators in ∗ ∗ T 1 . By Lemma 7.17, Tq g = aq (f )g (mod π n ) and Uq g = aq (f )g (mod π n ). By [25, ∗ Lemma 7.2], U 2 x = Frob 2 (F )x for x ∈ S 2 . Hence Lemma 7.17 yields (U ∗2 g)(x) = γ(Frob 2 (F )x) = 2 g(x). For the final part of the statement: The modular form g yields a surjective morphism φg : T 2 , 1 → Of,π /π n ; if ( 1 , 2 ) is a rigid pair, then T 2 , 1 Of,π and therefore φg lifts to characteristic zero. 7.5. Explicit reciprocity laws. The two following theorems explore the relations between the classes κ( ) constructed in Section 7.4 and the ℘-adic L-functions of Section 4. Their proofs are similar to the proofs of the corresponding results [5, Theorem 4.1, Theorem 4.2]. We will present a sketch of the arguments: for more details, the reader is referred to [5]. See also [29, Section 5.3] and [30, Section 3.5] where a result similar to that of Theorem 7.22 is proved. Recall the notations for ∂ and v before Definition 5.5. Theorem 7.22 (First Explicit Reciprocity Law). v (κ( )) = 0 and the equality ∂ (κ( )) ≡ θf ˆ1 holds in Hsing (K℘∞ , , Tf,πn ) G℘∞ . (mod π n ) × Λ℘,π /π n Λ℘,π up to multiplication by elements in Of,π and ˜ Proof. Denote by ∂ the residue map ˆ ˆ1 H 1 (K℘∞ , Tf,πn ) → Hsing (K℘∞ , , Tf,πn ) ˆ ˆ1 (these cohomology groups are defined for H 1 (K℘∞ , Tf,πn ) and Hsing (K℘∞ , , Tf,πn ) by ∗ ˜ ˜ replacing K℘∞ by K℘∞ ). In is enough to show that ∂ ({Pm }m ) ≡ θf mod π n (note the ∗ ˆ abuse of notations for the image of {Pm }m in H 1 (K℘∞ , Tf,πn )). 32 MATTEO LONGO Recall the notations of Section 6.2: Let B/F be the quaternion algebra which is ramified at all archimedean places and whose discriminant is Disc(B) = n− . Denote by R ⊆ B an Eichler order of level ℘n+ . Recall that End(Pm ) O℘m , where End(Pm ) is defined in [47, Section 2.1.1]. The Heegner point Pm is described in [47, Section 2.1.2] in terms of a certain abelian variety Am with additional structures. Let k denote as in [47, Section 2.2] the residue field of ¯ the maximal unramified extension of OK, . Denote by Am the reduced abelian variety ¯ ¯ over k and by End(Pm ) the endomorphism ring of Am as defined in [47, Section 2.3.3]. ¯m ) ⊗Z Q B. Tensoring by Q the map Then, by [47, Section 2.3.3], End(P ¯ End(Pm ) → End(Pm ) induced by reduction of endomorphism yields an embedding ψ : K → B. Let H := C − F be the -adic upper half plane, where C is the completion of an ( ) algebraic closure of F . The C -points of the special fiber X at of the Shimura curve X ( ) can be described by using the Cerednik-Drinfeld theorem: X (C ) ( ) B × \(B × × H )/R[1/ ]× , where R[1/ ] is the Eichler OF [1/ ]-order of B of level ℘n+ and OF [1/ ] is the ring of ( ) -integers of F . Then the point Pm reduces to the point Pm = (1, z) ∈ X (K ), where z is one of the two fixed points of ψ(K × ) acting on H . The integrality property of Pm follows because, since is inert in K/F , then it splits completely in K℘∞ . Let V and E are, respectively, the set of geometrically irreducible components and ( ) the set of singular points, respectively, of X . By [48, Lemma 5.4.4], the set V can be × × identified with Be \B × /R× , where Be is the set of elements of B with even order at ℘. ( ) ( ) The reduction of Pm in the special fiber X of X belongs to a single geometrically irreducible component: this is because, since is inert in K and O℘m ⊗ Z is maximal, ψ(O℘m ⊗ Z ) is contained in a unique maximal order, hence the action of ψ(K × ) on V ∪ E has a unique fixed point which is a vertex. Denote by r(Pm ) the corresponding × element in Be \B × /R× . Fix a prime ∞ of K℘∞ dividing and set m := ∞ ∩ K℘m . Note that the different choices of ∞ are permuted by the multiplication by an element of G℘∞ , and the same ˜ dependence holds for the definition of θf . Let Φ m be the group of connected components of the fiber at m of the N´ron model of J ( ) over OK℘m . There is a specialization map e δ m : J ( ) (K℘m ) → Φ m which fits into the following commutative diagram: (21) J ( ) (K℘m )/If ↓δm Φ m /If −→ H 1 (K℘m , Tf,πn ) ˜ ↓∂ 1 −→ Hsing (K℘m , m , Tf,πn ). 1 where the map Φ m /If → Hsing (K℘m , m , Tf,πn ) is an isomorphism. The Heegner point Pm satisfies, by [4, Appendix, Section 2], the following relation: δ m (Pm ) = ω (r(Pm )), where ω : Z0 [V ] → Φ is the map arising from the exact sequence 0 → X → X∨ → Φ → 0 ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 33 connecting Φ with the character group X of the maximal torus of the special fiber of ( ) × J and its Z-dual X ∨ . Recall the identification of V with Be \B × /R× and note that the last double coset space can be identified with two copies of B × \B × /R× by sending × a class [b] in Be \B × /R× to the class [b] in the first copy of B × \B × /R× if the ℘-adic valuation of b is even and to the class of [b] of the second copy otherwise. It follows that evaluation on Heegner points gives rise to an Hecke equivariant map: 1 B × \B × /R× → Φ m /If → Hsing (K℘m , m , Tf,πn ) Of,π /π n which, by multiplicity one, is equal to the modular form f B up to multiplication by an element in (Of,π /π n )× . ˜ It follows from above that ∂ (Pm ) = f B (r (Pm )) mod π n . The result follows now ∗ ˜f because the action of G℘∞ on Gr( ) (℘m ) is compatible from the definition of Pm and θ with the action of G℘∞ on Gr(℘m ) and, by our choice of the orientation at ℘, the compatibility of the sequence {Pm } translates into the compatibility of Gross points. Theorem 7.23 (Second Explicit Reciprocity Law). Let primes. Let g be as in Proposition 7.21. The equality v 2 (κ( 1 )) = θg ˆ1 holds in Hfin (K℘∞ , 2 , Tf,πn ) G℘∞ . × Λ℘,π /π n Λ℘,π up to multiplication by elements in Of,π and 1 and 2 be two n-admissible Proof. Consider the sequence {Pm }m of Heegner points. Fix (as in the proof of the above theorem) a prime 2,∞ of K℘∞ above 2 and let 2,m := 2,∞ ∩ K℘m . Since 2 is ¯ inert in K, the points Pm reduce modulo 2,∞ to supersingular points Pm ∈ X ( 1 ) (F 2,m ), where F 2,m is the residue field of K℘m at 2,m . Identify F 2,m with F 2 for all m. Then 2 ¯ ¯ Pm can be viewed as a point in S 2 , and hence, by Equation (20), Pm can be identified with an element in B × \B × /R × . Reduction modulo 2,m of endomorphism as in the proof of Theorem 7.22 yields by extension of scalars an embedding ϕ : K → B , which is independent of m. The Galois ¯ action of G℘∞ on Pm is compatible with the action of G℘∞ on Pm via ϕ. Write: −m ˜ θg,m = α℘ σ∈G℘m ¯ g(σ Pm ) · σ ∈ Of,π /π n [G℘m ] together with the isomorphism ˜ ˜ so that θg = lim θg,m ∈ Of,π /π n [[G∞ ]]. The choice of ← − m 1 Hfin (K 2 , Tf,πn ) Of,π /π n yields identifications: 2,∞ 1 Hfin (K℘m , 2 , Tf,πn ) = Of,π /π n [G℘m ], ˆ1 Hfin (K℘∞ , 2 , Tf,πn ) = Of,π /π n [[G℘∞ ]], where these cohomology groups are defined as in Section 5.2.1. By the definition of γ, 1 ∗ ˜ the image of Pm in Hfin (K℘m , 2 , Tf,πn ) corresponds to θg,m (mod π n ) and so the image ∗ ˜g . Define the class κ( 1 ) to be the ˜ of the compatible sequence {Pm } corresponds to θ 1 1 ∗ ˆ ˆ κ image of {Pm } in H (K℘∞ , Tf,πn ). It follows that v 2 (˜ ( 1 )) ∈ Hfin (K∞, 2 , Tf,πn ) is equal ˜g (mod π n ). Since κ( 1 ) is the corestriction of κ( 1 ) from K℘∞ to K℘∞ , the result to θ ˜ follows. 34 MATTEO LONGO Corollary 7.24. The equality v 1 (κ( 2 )) ≡ v 2 (κ( 1 )) (mod π n ) × holds in Λ℘,π /π n Λ℘,π up to multiplication by elements in Of,π and G℘∞ . Proof. Since the definition of g is symmetric in 1 and 2, this is obvious. 7.6. The argument. The remaining part of the section is devoted to the proof of Theorem 6.1. Being ℘ fixed, denote Selπ∞ (f /K℘∞ ) (respectively, Selπn (f /K℘∞ )) simply by Self,∞ (respectively, Self,n ). By Proposition 7.4, it is enough to show that ϕ(θf )2 belongs to FittO (Sel∨ ⊗ϕ O) for all ϕ ∈ Hom(Λ, O) where O is the ring of integer of f,∞ a finite extension of Qp . For this, by [33, Appendix, 10], it is enough to show that (22) ϕ(θf )2 belongs to FittO (Sel∨ ⊗ϕ O) for all integrs n ≥ 1. f,n tf := ordν (ϕ(θf )). If ϕ(θf ) = 0, then ϕ(θf )2 belongs trivially to FittO (Sel∨ ⊗ϕ O) for all n ≥ 1, so f,n assume ϕ(θf ) = 0. If Sel∨ ⊗ϕ O is trivial, then its Fitting ideal is equal to O and, f,∞ again, ϕ(θf )2 belongs trivially to FittO (Sel∨ ⊗ϕ O) for all n ≥ 1, so assume that f,n FittO (Sel∨ ⊗ϕ O) = 0. The theorem is proved now by induction on tf . f,n 7.6.1. Construction of κϕ ( ). Let be any (n + tf )-admissible prime and enlarge { } to a (n + tf )-admissible set S: such a set consists of s distinct (n + tf )-admissible primes such that the map 1 Self,n+tf (K) → ⊕ ∈S Hfin (K , Af,πn+tf ) is injective (Proposition 7.5 shows that such a set exists). Denote by s the square-free product of the primes in S and let ˆ ˆ1 κ( ) ∈ H 1 (K℘∞ , Tf,πn+tf ) ⊆ Hs (K℘∞ , Tf,πn+tf ) be the cohomology class attached to . 1 Proposition 7.25. The group Hs (K℘∞ , Tf,πn ) is free of rank s over Λ℘,π /π n . Fix O and ϕ as above. Write ν for an uniformizer of O. Set Proof. This statement can be proved by a direct generalization of [2, Theorem 3.2] as suggested in [5, Proposition 3.3]. Let κϕ ( ) be the image of κ( ) in ˆ1 M := Hs (K℘∞ , Tf,πn+tf ) ⊗ϕ O. Note that, by Proposition 7.25, M is free of rank s over Of,π /π n+tf . By Theorem 7.22, (23) t := ordν (κϕ ( )) ≤ ordν (∂ (κϕ ( ))) = ordν (ϕ(θf )). Choose an element κϕ ( ) ∈ M so that ν t κϕ ( ) = κϕ ( ). This element is well defined ˜ ˜ modulo the ν t -torsion subgroup of M; to remove this ambiguity, denote by κϕ ( ) the 1 image of κϕ ( ) in Hs (K℘∞ , Tf,πn ) ⊗ϕ O. The following properties of κϕ ( ) holds: ˜ (1) ordν (κϕ ( )) = 0 (because ordν (κϕ ( )) = t ≤ tf ); ˆ1 (2) ∂q (κϕ ( )) = 0 for all q n− (because κ( ) ∈ Hs (K℘∞ , Tf,n+tf )); (3) v (κϕ ( )) = 0 (by Theorem 7.22); (4) ∂ (κϕ ( )) = tf − t (by Theorem 7.22 and formula (23)); ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 35 (5) The element ∂ (κϕ ( )) belongs to the kernel of the homomorphism: (24) ˆ1 η : Hsing (K℘∞ , , Tf,πn ) ⊗ϕ O → Sel∨ ⊗ϕ O. f,n To prove the last statement use the global reciprocity law of class field theory (5) as follows (see more details in [5, Lemma 4.6]). Denote by Iϕ the kernel of ϕ. First note that it is enough to show that η (∂ κϕ ( ))(s) = 0 for all s ∈ Self,n [Iϕ ]. Note that, by the global reciprocity law of class field theory: ∂q (˜ ϕ ( )), sq κ q|S q =0 for all s ∈ Self,n+tf [Iϕ ]. On the other hand, ν t κϕ ( ) = κϕ ( ) has trivial residue at ˜ all the primes q = (it is finite at that primes) so the element ∂q (˜ ϕ ( )) annihilates κ 1 t 1 ν Hfin (K∞,q , Af,πn+tf ])[Iϕ ], which contains Hfin (K∞,q , Af,πn )[Iϕ ]). Hence, if s belongs to Self,n [Iϕ ], then the terms in the above sum corresponding to primes q = are all zero. 1 It follows that ∂ (κϕ ( )) annihilates the image of Self,n [Iϕ ] in Hfin (K∞, , Af,πn ), so it belongs to the kernel of η . 7.6.2. Case of tf = 0. This is the basis for the induction argument. First, recall the following result. Proposition 7.26. The natural map H 1 (K, Af,π ) → H 1 (K℘∞ , Af,πn )[m] induced by restriction is an isomorphism. Proof. This result can be obtained as in [5, Theorem 3.4] by analyzing the inflationrestriction exact sequence ℘ 0 → H 1 (Gal(K℘m /K), Af,πn ) → H 1 (K, Af,πn ) → ℘ → H 1 (K℘m , Af,πn )Gal(K℘m /K) → H 2 (Gal(K℘m /K), Af,πn ) m where GK℘ is the absolute Galois group of K℘m , and the exact sequence GK m GK m AGKn−1 → H 1 (K, Af,π ) → H 1 (K, Af,πn ) → H 1 (K, Af,πn ) → H 2 (K, AGK ) f,π f,π induced by 0 → Af,π → Af,πn → Af,πn−1 → 0 and noticing that, since ρf,π is surjective, ℘ Af,πn = AGK = 0. For details, see [5, Theorem 3.4]. f,π m π GK Then we can state the basis of the inductive argument. Proposition 7.27. If tf = 0 then Sel∨ = 0. f,n Proof. To prove this, note that, for all n-admissible primes , Theorem 7.22 implies that ˆ1 Hsing (K℘∞ , , Tf,πn )⊗ϕ O is generated by ∂ (κϕ ( )) (as O-module) and that the map η in (24) is trivial. Assume now that Sel∨ is not trivial. Then Nakayama’s lemma implies f,n that the group Sel∨ /m = (Self,n [m])∨ is not trivial, where m is the maximal ideal of f,n Λ℘,π . Let now s ∈ Self,n [m] be a non trivial element. Proposition 7.26 allows to consider s as an element of H 1 (K, Af,π ). Invoke Proposition 7.5 to choose an n-admissible prime such that ∂ (s) = 0 and v (s) = 0. Then the non degeneracy of the local Tate pairing implies that η is trivial, which is a contradiction. 36 MATTEO LONGO 7.6.3. The minimality property. As a corollary of Proposition 7.25, note that ˆ (25) the corestriction map H 1 (K℘∞ , Tf,πn )/m → H 1 (K, Tf,π ) is injective. s Let now Π be the set of primes of OF so that: (1) is n + tf -admissible; (2) The number t = ordν (κϕ ( )) is minimal among the set of n + tf -admissible primes. By Proposition 7.5, Π = ∅. Proposition 7.28. t < tf . Proof. To prove this assertion, assume on the contrary that t ≥ tf . Since by definition t ≤ tf , then t = tf for all n + tf admissible primes . Use Proposition 7.26 to choose a non trivial element in H 1 (K, Af,π )∩Self,n (recall that by assumption, Sel∨ ⊗ϕ O = 0, so f,n Self,n [m] = 0). By Proposition 7.5, choose a n + tf admissible prime such that v (s) = 0. Now by the property 5 enjoyed by the class κϕ ( ), it follows that ordν (∂ κϕ ( )) = 0, ˆ1 so that ∂ κϕ ( ) is a generator of Hsing (K∞, , Tf,πn ) ⊗ϕ O. By Nakayama’s lemma again, ˆ1 the image of ∂ (κϕ ( )) in Hsing (K∞, , Tf,πn )/m ⊗ϕ O is not trivial. Use (25) to identify this last module with H 1 (K, Tf,π )⊗O; then it follows that the natural image of ∂ (κϕ ( )) in H 1 (K, Tf,π ) ⊗ O is not trivial. By Property 5 enjoyed by the class κϕ ( ) again, it follows that ∂ (κϕ ( )) is orthogonal to v (s) with respect to the local Tate pairing, contradicting the fact that ∂ (κϕ ( )) and v (s) are both supposed to be non trivial and the fact that the Tate pairing is a perfect duality between one-dimensional O/ν-vector spaces. 7.6.4. Rigid pairs with the minimality property. This step is devoted to the proof that there exist primes 1 , 2 ∈ Π so that ( 1 , 2 ) is a rigid pair. To prove this, start by choosing any prime 1 ∈ Π and denote by s the image of κϕ ( 1 ) in ˆ1 (Hs (K∞ , Tf,πn )/m) ⊗ϕ O/(ν), where m is the maximal ideal of Λ℘,π . By (25), view s as a non-zero element in H 1 (K, Tf,π ) ⊗ O/(ν). Note that ∂q (s) = 0 for all q 1 n. By Propositions 7.13 and 7.14, choose a n + tf admissible prime 2 so that ∂ 2 (s) = 0, v 2 (s) = 0 and either ( 1 , 2 ) is a rigid pair or Sel 2 (F, ad0 ρ) is one-dimensional. The following relation holds: (26) t = ordν (κϕ ( 1 )) ≤ ordν (κϕ ( 2 )) ≤ ordν (v 1 (κϕ ( 2 ))). The first inequality follows from the minimality property of t using that 1 ∈ Π and that 2 is a n + tf -admissible prime. By the choice of 2 and Corollary 7.24, ordν (v 1 (κϕ ( 2 ))) = ordν (v 2 (κϕ ( 1 ))). Now note that ordν (v 2 (κϕ ( 1 ))) ≥ ordν (κϕ ( 1 )) and that the strict inequality holds if and only if v 2 (s) = 0, so, since v 2 (s) = 0, ordν (v 1 (κϕ ( 2 ))) = ordν (κϕ ( 1 )). Combining this with the inequalities in formula (26) shows that: (27) t = ordν (κϕ ( 1 )) = ordν (κϕ ( 2 )). It follows that 2 ∈ Π. If ( 1 , 2 ) is not a rigid pair, then Sel 2 (F, ad0 ρ) is one dimensional (this is the case only if Sel 1 (F, ad0 ρ) = 0). In this case, by Proposition 7.13, choose a n + tf admissible prime 3 so that ∂ 3 (s) = 0, v 3 (s) = 0 and ( 2 , 3 ) is a rigid pair. Repeat the argument above with 2 replacing 1 and 3 replacing 2 to show that 3 ∈ Π. ANTICYCLOTOMIC IWASAWA’S MAIN CONJECTURE FOR HILBERT MODULAR FORMS 37 In any case then, either ( 1 , 2 ) or ( 2 , 3 ) is a rigid pair and the claim at the beginning of follows. 7.6.5. The congruence argument. Choose by the result explained in Subsection 7.6.4 a rigid pair ( 1 , 2 ) with 1 , 2 ∈ Π. Note that, by Theorem 7.23, (28) t = tg = ordν (θg ) (here g is the congruent modular form attached to ( 1 , 2 ) by Proposition 7.21). There is an exact sequence of Λ-modules: (29) 0 → Self1 2 → Sel∨ → Sel∨1 , 2 ] → 0, f,n [ where Sel[ 1 , 2 ] ⊆ Self,n is defined by the condition that the restriction at the primes 1 and 2 must be trivial and Self1 2 is the kernel of the surjection of duals. There is an inclusion: 1 1 (Self1 2 )∨ ⊆ Hfin (K℘∞ , 1 , Af,πn ) ⊕ Hfin (K℘∞ , 2 , Af,πn ). 1 1 The dual of Hfin (K℘∞ , 1 , Af,πn ) ⊕ Hfin (K℘∞ , 2 , Af,πn ), by the non-degeneracy of the local ˆ1 ˆ1 Tate pairing, is Hsing (K℘∞ , 1 , Af,πn ) ⊕ Hsing (K℘∞ , 2 , Af,πn ), so the above inclusion leads to a surjection: ˆ1 ˆ1 ηf : Hsing (K℘∞ , 1 , Af,πn ) ⊕ Hsing (K℘∞ , 1 , Af,πn ) → Self1 2 . ϕ ˆ1 Recall that, since 1 is n-admissible, Hsing (K℘∞ , 1 , Af,πn ) Λ℘,π /π n . Let ηf be the ϕ map induced by ηf after tensoring by O via ϕ. Then the domain of ηf is isomorphic to (O/ϕ(π n ))2 . By property 5 above enjoyed by the classes κϕ ( 1 ) and κϕ ( 2 ), the ϕ kernel of ηf contains (∂ 1 κϕ ( 1 ), 0) and (0, ∂ 2 κϕ ( 2 )). The same property combined with equations (27) and (28) yields: tf − tg = ordν (∂ 1 (κϕ ( 1 ))) = ordν (∂ 2 (κϕ ( 2 ))). It follows that: (30) ν 2(tf −tg ) belongs to the Fitting ideal of Self1 2 ⊗ϕ O. 0 → Selg1 2 → Sel∨ → Sel∨1 , 2 ] → 0, g,n [ and a surjection: ˆ1 ˆ1 ηg : Hfin (K℘∞ , 1 , Af,πn ) ⊕ Hfin (K℘∞ , 1 , Af,πn ) → Selg1 2 . ϕ Let ηg be the map induced by ηg after tensoring by O via ϕ. By the global reciprocity ϕ law of class field theory, the kernel of ηg contains the elements Repeat now the argument with the modular form g: there is an exact sequence: (v 1 (κϕ ( 1 )), v 2 (κϕ ( 1 ))) = (v 1 (κϕ ( 1 )), 0), (v 1 (κϕ ( 2 )), v 2 (κϕ ( 2 ))) = (0, v 2 (κϕ ( 2 ))), where the equalities follow from property 3 above enjoyed by the classes κϕ ( 1 ) and κϕ ( 2 ). Note that ordν (v 2 κϕ ( 1 )) = ordν (v 1 κϕ ( 2 )) = tg − t = 0. From this it follows that the module Selg1 2 is trivial. As a consequence, there is an isomorphism: (31) Sel∨ ⊗ϕ O → Sel∨1 g,n [ ∼ 2] ⊗ϕ O. 38 MATTEO LONGO 7.6.6. The inductive argument. Now assume that the theorem is true for all t < tf and prove that it is true for tf . Recall that t = tg < tf . Since ( 1 , 2 ) is a rigid pair, the modular form g satisfies the assumptions in the theorem, so, by the inductive hypothesis, (32) ϕ(θg ) belongs to the Fitting ideal of Sel∨ ⊗ϕ O. g,n ν 2tf = ν 2(tf −tg ) ν 2tg ∈ FittO (Self1 2 ⊗ϕ O) · FittO (Sel∨ ⊗ϕ O), by (30) and (32) g,n = FittO (Self1 2 ⊗ϕ O) · FittO (Sel∨1 [ ⊆ FittO (Sel∨ f,n ⊗ϕ O), by (29). 2] Now use the theory of Fitting ideals: ⊗ϕ O), by (31) Since by definition ord(θf ) = tf , it follows that ϕ(θf )2 ∈ FittO (Sel∨ ⊗ϕ O), thus f,n proving (22) and therefore Theorem 6.1. References [1] M. Bertolini. Selmer group and Heegner points in anticyclotomic Zp -extensions. Compositio Math. 99 (1995), no 2, 153–182. [2] M. Bertolini, H. 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