Kolyvagin’s result on the vanishing of � (E/K)[p^{∞}] and its consequences for anticyclotomic Iwasawa theory

Ahmed Matar, Jan Nekovář

Abstract: We discuss improvements of Kolyvagin’s classical result about the vanishing of the p-primary part of the Tate–Šafarevič group of an elliptic curve E (defined over Q) over an imaginary quadratic field K satisfying the Heegner hypothesis for which the basic Heegner point y_{K} ∈ E(K) is not divisible by an odd prime p. Combining Kolyvagin’s theorem with a new abstract Iwasawa-theoretical result, we deduce, under suitable assumptions, that similar vanishing holds for all layers in the anticyclotomic Z_{p}-extension of K.

0. Introduction

(0.1) Let E be an elliptic curve over Q of conductor N and K an imaginary quadratic field of discriminant D_{K} in which all primes dividing N split. Fix a modular parameterisation φ : X_{0}(N) ⟶ E and an ideal N ⊂ O_{K} such that O_{K}/N ≃ Z/NZ. The basic Heegner point y_{K} ∈ E(K) attached to these data is, by definition, the trace y_{K} := Tr_{H_{1}/K}(y_{1}) of the Heegner point of conductor one y_{1} := φ([C/O_{K} ⟶ C/N^{−1}]) ∈ E(H_{1}) defined over the Hilbert class field H_{1} of K. (0.2) If y_{K} ∉ E(K)_{tors} and D_{K} ≠ −3, −4, Kolyvagin [K1, Thm. A] proved that the groups E(K)/Zy_{K} and � (E/K) are finite, and that #� (E/K) divides [E(K) : Zy_{K}]^{2} multiplied by a product of several error terms. The p-primary parts of these error terms vanish in the following situation (each of the respective assumptions (a), (b) and (c) implies that the corresponding error term a, b, c in [K1, Cor. 11, Cor. 12, Cor. 13] is relatively prime to p; the error term d is equal to 1, since p ≠ 2).

(0.3) Theorem (Kolyvagin, special case of [K1, Cor. 13]). Assume that D_{K} ≠ −3, −4 and that p ≠ 2 is a prime number satisfying the following conditions.

(a) ∀n_{1}, n_{2} ≥ 0 H^{1}(K(E[p^{n_{1}+n_{2}}])/K, E[p^{n_{1}}]) = 0.

(b) Neither of the (±1)-eigenspaces E[p]^{±} for the action of complex conjugation is stable under the action of G_{Q} := Gal(\overline{Q}/Q). Equivalently, the (mod p) Galois representation \overline{ρ}_{E,p} : G_{Q} ⟶ Aut_{F_{p}}(E[p]) ≃ GL_{2}(F_{p}) is irreducible.

(c) E(K)[p] = 0. If y_{K} ∉ E(K)_{tors}, then E(K)/Zy_{K} is finite and

p^{m_{0}}� (E/K)[p^{∞}] = 0, #� (E/K)[p^{∞}] divides p^{2m_{0}},

where m_{0} := sup{m ≥ 0 | y_{K} ∈ p^{m}E(K)} (thus E(K) ⊗ Z_{p} ≃ Z_{p} and p^{m_{0}} = [E(K) ⊗ Z_{p} : Z_{p}(y_{K} ⊗ 1)]).

(0.4) For p ≠ 2, the assumption (b) in Theorem 0.3 implies (c). Moreover, the assumptions (a), (b) and (c) are satisfied if \overline{ρ}_{E,p} has “big image” (e.g., if it is surjective).

Gross [G] gave a self-contained account of Kolyvagin’s proof of Theorem 0.3 in the simplest case when \overline{ρ}_{E,p} is surjective and m_{0} = 0. One step in the argument ([G, beginning of §9]) required an additional assumption p ∤ D_{K}.

(0.5) Theorem ([G, Prop. 2.1, Prop. 2.3]). Assume that D_{K} ≠ −3, −4 and that p ∤ 2D_{K} is a prime number for which \overline{ρ}_{E,p} : G_{Q} ⟶ GL_{2}(F_{p}) is surjective. If y_{K} ∉ pE(K), then E(K) ⊗ Z_{p} = Z_{p}y_{K} ≃ Z_{p} and � (E/K)[p^{∞}] = 0.

(0.6) In [K2], Kolyvagin proved the following structure theorem for � (E/K)[p^{∞}], which refines Theorem 0.3 (under the “big image” assumption for the p-adic Galois representation ρ_{E,p} : G_{Q} ⟶ Aut_{Z_{p}}(T_{p}(E)) ≃ GL_{2}(Z_{p})).

(0.7) Theorem (Kolyvagin, [K2, Thm. C, Thm. D]). Assume that D_{K} ≠ −3, −4 and that p ≠ 2 is a prime number for which ρ_{E,p} : G_{Q} ⟶ GL_{2}(Z_{p}) has “big image” (e.g., that ρ_{E,p} is surjective). If y_{K} ∉ E(K)_{tors}, then

� (E/K)[p^{∞}] ≃ X ⊕ X,  X ≃ ⨁_{i≥0} Z/p^{m_{i}−m_{i+1}}Z,  m_{0} ≥ m_{1} ≥ ⋯ ≥ m_{∞} := inf m_{i},

where m_{0} is as in Theorem 0.3 and m_{i} for i > 0 is defined in a similar way in terms of certain linear combinations of Heegner points of higher conductors. In particular, #� (E/K)[p^{∞}] = p^{2(m_{0}−m_{∞})}.

(0.8) The divisibility #X | p^{m_{0}} was reproved by Howard [H1, Thm. A] using the formalism of anticyclotomic Kolyvagin systems, under the assumptions that ρ_{E,p} is surjective, D_{K} ≠ −3, −4 and p ∤ 2ND_{K}. (0.9) For p ≠ 2, the condition (a) in Theorem 0.3 was studied in detail by Cha [Ch, Thm. 2], who showed

that it is satisfied if p ∤ D_{K}, p^{2} ∤ N and E(K)[p] = 0, except when p = 3 and \overline{ρ}_{E,3}(G_{K}) = ( \begin{matrix}F_{3}^{×} & F_{3} \\ 0 & 1 \end{matrix} ).

Therefore the conclusions of Theorems 0.3 and 0.5 hold (for D_{K} ≠ −3, −4) whenever p ∤ 2D_{K}, p^{2} ∤ N and \overline{ρ}_{E,p} is irreducible. He also showed [Ch, Thm. 21, Rmk. 25] that the statement of Theorem 0.7 holds under the same assumptions. (0.10) The authors of a collective article [GJPST] had made an attempt at generalising Cha’s results. However, the cohomological calculations in [GJPST, Lemma 5.7, Lemma 5.9] and [GJPST, proof of Proposition 5.4] are incorrect (see [LW, Lemma 8]), the statement of [GJPST, Proposition 5.8] is correct but the proof is not, and the discussion of Kolyvagin’s method (in the form presented in [G]) in [GJPST, §5] is seriously flawed. In particular, the assertion to the effect that the surjectivity of \overline{ρ}_{E,p} in Theorem 0.5 can be replaced by the vanishing of the groups H^{i}(K(E[p])/K, E[p]) for i = 1, 2 and of E′(K)[p] for all Q-isogenies E ⟶ E′, is incorrect, for the following reason: the current state of the art requires an irreducibility assumption for \overline{ρ}_{E,p} (or its restriction to G_{K}) in order to obtain, by Kolyvagin’s method, an upper bound on the size of � (E/K)[p^{∞}] without any error terms. As a result, [GJPST, Thm. 3.7] remains unproved. (0.11) Lawson and Wuthrich [LW] extended and simplified the cohomological calculations of [Ch], and corrected various mistakes from [GJPST]. In [LW, Thm 1, Thm 2], they gave a complete classification of pairs (E, p) consisting of an elliptic curve E over Q and a prime number p ≠ 2 for which H^{1}(Q(E[p])/Q, E[p]) ≠ 0 (and similarly for H^{1}(Q(E[p^{n}])/Q, E[p^{n}]) ≠ 0, where n > 1 and p > 3). They also classified pairs (E, p) for which H^{2}(Q(E[p])/Q, E[p]) ≠ 0.

Their results imply that the condition (a) in Theorem 0.3 (for p ≠ 2) is always satisfied if \overline{ρ}_{E,p} is irreducible. Consequently, the conclusions of Theorems 0.3 and 0.7 hold (for D_{K} ≠ −3, −4) if \overline{ρ}_{E,p} is irreducible and p ≠ 2.

However, the claims made in [LW, Thm. 14] about the validity of Theorem 0.3 in situations when (a) holds but \overline{ρ}_{E,p} is reducible are unjustified, for reasons explained in 0.10.

We recall the methods of [Ch] and [LW] and prove a mild generalisation of some of their results in §5. We also prove the following variant of Theorem 0.5.

(0.12) Theorem (= Theorem 6.7). Assume that p ≠ 2 and that E[p] is an irreducible F_{p}[G_{Q}]-module (which implies that E(K)[p] = 0).

(1) Assume that (K, p) ≠ (Q(\sqrt{−3}), 3). If y_{K} ∉ pE(K), then

E(K) ⊗ Z_{p} = Z_{p}(y_{K} ⊗ 1) ≃ Z_{p}, � (E/K)[p^{∞}] = 0.

(2) Assume that (K, p) = (Q(\sqrt{−3}), 3); then y_{K} ∈ 3E(K). If 3 ∤ a_{3}, assume, in addition, that ρ(G_{K}) is not a cyclic group of order four. If y_{K} ∉ 3^{2}E(K), then

Z_{3} ≃ E(K) ⊗ Z_{3} ⊃ 3E(K) ⊗ Z_{3} = Z_{3}(y_{K} ⊗ 1), � (E/K)[3^{∞}] = 0.

(0.13) We now turn to Iwasawa-theoretical results. Fix a prime number p and denote by K_{∞} = ⋃_{n≥1} K_{n} the anticyclotomic Z_{p}-extension of K. In this case Γ := Gal(K_{∞}/K) ≃ Z_{p}, K_{n} = K_{∞}^{Γ_{n}}, where Γ_{n} = Γ^{p^{n}} ≃ p^{n}Z_{p}, and Gal(K_{∞}/Q) = Γ ⋊ {1, c}, with complex conjugation c acting on Γ by g ↦ g^{−1}. Denote by Λ := Z_{p}[[Γ]] the Iwasawa algebra of Γ. (0.14) From now on until the end of Introduction we assume that p ≠ 2 and that E has good ordinary reduction at p. The Selmer module Sel_{p^{∞}}(E/K_{∞}) := \underrightarrow{lim}_{n} Sel_{p^{∞}}(E/K_{n}) (resp. S_{p}(E/K_{∞}) := \underleftarrow{lim}_{n} S_{p}(E/K_{n}))

(see §1.4 for the notation) is a Λ-module of cofinite (resp. finite) type, of corank (resp. rank) equal to one, as predicted by one of Mazur’s conjectures formulated in [Mz, §18]. This conjecture is a consequence of another conjecture of Mazur [Mz, §19] (proved independently by Cornut [Co1, Co2] and Vatsal [Va]) combined with an Euler system argument along the tower K_{∞}/K ([B, Thm. A] under some additional assumptions; the general case is proved in [N1, §2] together with [N3, Thm. 3.2]; see also [H1, Thm. B]). Another proof of [B, Thm. A], which had applications to the study of the anticyclotomic μ-invariant, was given in [M, Thm. A].

In [B, Thm. B], Bertolini also proved a Λ-adic variant of Kolyvagin’s annihilation result [K1, Cor. 12] for the torsion submodule of the Pontryagin dual of Sel_{p^{∞}}(E/K_{∞}) (assuming the validity of Mazur’s conjecture [Mz, §19]). This result was subsequently generalised by Howard [H1, Thm. B], who proved one half of a conjecture of Perrin-Riou [PR, Conj. B] for Heegner points along K_{∞}/K, namely, a Λ-adic variant of Kolyvagin’s result #X | p^{m_{0}} (in the notation of Theorem 0.7). (0.15) The proofs of [B, Thm. B] and [H1, Thm. B] relied on fairly detailed arguments involving the Euler system and the Kolyvagin system of Heegner points along K_{∞}/K, respectively. The main insight of the present work is that in the simplest case when y_{K} ∉ pE(K), one can obtain — under certain assumptions — precise information about the structure of the Z_{p}[Γ/Γ_{n}]-modules E(K_{n}) ⊗ Q_{p}/Z_{p} and � (E/K_{n})[p^{∞}] from Theorem 0.5 (and its variant Theorem 0.12) by purely Iwasawa-theoretical methods, combined with the norm relations for the Heegner points of p-power conductor, without applying any Euler system arguments along the tower K_{∞}/K. The following results are proved in §4.

(0.16) Theorem (= Theorem 4.8). If p ≠ 2 is a prime number such that

(a) E(K)[p] = 0,

(b) p ∤ N · a_{p} · (a_{p} − 1) · c_{Tam}(E/Q),

(c) y_{K} ∉ E(K)_{tors},

(d) rk_{Z} E(K) = 1 and � (E/K)[p^{∞}] = 0, then � (E/K_{∞})[p^{∞}] = 0 and the Pontryagin dual of E(K_{∞}) ⊗ Q_{p}/Z_{p} = Sel_{p^{∞}}(E/K_{∞}) is a free module of rank one over Z_{p}[[Gal(K_{∞}/K)]].

(0.17) Theorem (= Theorem 4.9). If p ≠ 2 is a prime number such that

(a) E(K)[p] = 0, (b’) p ∤ N · a_{p} · (a_{p} − 1) · (a_{p} − η_{K}(p)) · c_{Tam}(E/Q), (c’) y_{K} ∉ pE(K),

(d) rk_{Z} E(K) = 1 and � (E/K)[p^{∞}] = 0, then, for every intermediate field K ⊂ L ⊂ K_{∞}, � (E/L)[p^{∞}] = 0 and the Pontryagin dual of E(L) ⊗ Q_{p}/Z_{p} = Sel_{p^{∞}}(E/L) is a free module of rank one over Z_{p}[[Gal(L/K)]]. For every integer n ≥ 0, rk_{Z} E(K_{n}) = p^{n}, � (E/K_{n})[p^{∞}] = 0 and E(K_{n}) ⊗ Z_{p} is generated over Z_{p}[Gal(K_{n}/K)] by the traces to K_{n} of the Heegner points of p-power conductor.

(0.18) Above, a_{p} denotes the p-th coefficient of the L-function L(E/Q, s) = ∑_{n≥1} a_{n}n^{−s}, the value η_{K}(p) is equal to 1, −1, 0, respectively, if p splits, is inert, or is ramified in K/Q, and c_{Tam}(E/Q) = ∏_{ℓ|N} c_{Tam,ℓ}(E/Q) is the product of the local Tamagawa factors of E at all primes of bad reduction. (0.19) If K = Q(\sqrt{−3}) and p = 3, the conditions (a) and (c’) in Theorem 0.17 cannot be satisfied simultaneously, by Proposition 4.11 below. In general, (a) and (c’) should imply both (d) and p ∤ c_{Tam}(E/Q) (see (6.2.1)). (0.20) What is the role of the individual assumptions in Theorem 0.16 and Theorem 0.17? The condition (a) implies that E(K_{∞})[p] = 0. The assumption p ∤ N · a_{p} is equivalent to E having good ordinary reduction at p, and the remaining part p ∤ (a_{p} − 1) · c_{Tam}(E/Q) of (b) ensures (when combined with (a)) that Mazur’s control theorem holds along the tower K_{∞}/K without any error terms: Sel_{p^{k}}(E/K_{n}) ⟶^{∼} Sel_{p^{k}}(E/K_{∞})^{Γ_{n}} for all k, n ≥ 0. The condition (d) implies that Sel_{p^{∞}}(E/K) ≃ Q_{p}/Z_{p}. Finally, the norm relations for the Heegner points of p-power conductor imply that, for a suitable non-zero element m ∈ Z_{p}, the multiple y_{K} ⊗ m ∈ E(K) ⊗ Z_{p} is a universal norm from the projective system {E(K_{n}) ⊗ Z_{p}}, and the condition p ∤ (a_{p} − 1) · (a_{p} − η_{K}(p)) ensures that m ∈ Z_{p}^{×}. (0.21) One can combine Theorem 0.17 with the Euler system results over K (but not over K_{∞}) discussed in 0.1–0.11. Kolyvagin’s result alluded to in 0.2 tells us that the condition rk_{Z} E(K) = 1 in Theorem 0.16(d) follows from (c), and therefore can be dropped. Likewise, the condition (d) in Theorem 0.17 follows from (c’), whenever the conclusions of Theorem 0.5 hold. Combining Theorem 0.5 (with weaker assumptions, supplied by [Ch], [LW] and Theorem 6.7(1)) with Theorem 0.17, we obtain the following result.

(0.22) Theorem (= Theorem 6.9). If p ≠ 2 is a prime number such that E[p] is an irreducible F_{p}[G_{Q}]- module, p ∤ N · a_{p} · (a_{p} − 1) · (a_{p} − η_{K}(p)) · c_{Tam}(E/Q) and y_{K} ∉ pE(K), then the conclusions of Theorem 0.17 hold.

(0.23) The case K = Q(\sqrt{−3}), p = 3 is different, as already mentioned in Theorem 0.12 and in 0.19. The point is that, if E(K)[3] = 0, then y_{K} = 3z_{K}, where z_{K} ∈ E(K) is a linear combination of the traces to K of the Heegner points of conductors 1 and q, for any prime q ∤ 3N satisfying a_{q} ≢ 1 + η_{K}(q) (mod 3) (there are infinitely many such primes q).

(0.24) Theorem (= Theorem 6.10). Assume that K = Q(\sqrt{−3}), p = 3, E[3] is an irreducible F_{3}[G_{Q}]- module, ρ(G_{K}) is not a cyclic group of order four and 3 ∤ a_{3} · (a_{3} − 1) · c_{Tam}(E/Q). If y_{K} ∉ 3^{2}E(K), then the conclusions of Theorem 0.17 hold, with the following modification: each E(K_{n}) ⊗ Z_{3} is generated over Z_{3}[Gal(K_{n}/K)] by the traces to K_{∞} of the Heegner points of conductors dividing 3^{∞}q, for any prime q as in 0.23.

(0.25) Analogous results hold for anticyclotomic Z_{p}^{m}-extensions and basic CM points on abelian varieties of GL(2)-type with real multiplication occurring as simple quotients of Jacobians of Shimura curves over totally real number fields. This will be discussed in a separate publication. (0.26) Let us describe the contents of this article in more detail. The goal of §1-§3 is to prove two abstract results (Theorems 3.4 and 3.5) on Selmer groups of 𝔭-ordinary abelian varieties in dihedral Iwasawa theory. The framework is general enough to apply in the context of 0.25, not just in the situation involving classical Heegner points on elliptic curves. In §4 we recall the norm relations for Heegner points and combine them with Theorems 3.4 and 3.5 in order to deduce Theorems 0.16 and 0.17. In §5-§6 we give a proof of Kolyvagin’s result on vanishing of � (E/K)[p^{∞}] in the form of Theorem 0.12. When combined with Theorems 0.16 and 0.17, this implies Theorems 0.22 and 0.24. Again, the general theory developed in §5 is applicable in the context of 0.25. Acknowledgements. Some of the work on this article was carried out by the second named author when he was visiting Centre Interfacultaire Bernoulli (CIB) at Ecole Polytechnique Fédérale de Lausanne during the semester “Euler systems and special values of L-functions” in fall 2017, and when he was staying at Imperial College London as an ICL–CNRS fellow in spring 2018. He is grateful to both institutions for their generous support.

1. Generalities

(1.1) Throughout §1-§3,

• for any perfect field k, denote by G_{k} = Gal(\overline{k}/k) its absolute Galois group.

• For an integer n ≥ 1 invertible in k, denote by χ_{n,k} : G_{k} ⟶ (Z/nZ)^{×} the cyclotomic character given by the action of G_{k} on μ_{n}(\overline{k}).

• K is a number field.

• F r(v) will always denote the arithmetic Frobenius element.

• p is a prime number; if K is not totally imaginary, we assume that p ≠ 2.

• B is an abelian variety over K with good reduction at all primes of K above p; let B^{t} be the dual abelian variety.

• If v is a finite prime of a finite extension L of K, denote by B_{v} the Néron model of B ⊗_{K} L_{v} over O_{L_{v}}, by B̃_{v} its special fibre (over the residue field k(v) of v), and by π_{0}(B̃_{v}) = B̃_{v}/B̃_{v}^{∘} the G_{k(v)}-module of its connected components.

• M is a totally real number field with ring of integers O_{M}.

• We are given a ring morphism i : O_{M} ⟶ End(B) and an O_{M}-linear isogeny λ : B ⟶ B^{t} which is symmetric in the sense that λ = λ^{t}. Above, we use a scheme-theoretic notation: the ring of endomorphisms of B defined over a field L containing K is denoted by End(B ⊗_{K} L) (not by End_{L}(E)).

Throughout, one can replace O_{M} by any order in M whose index in O_{M} is prime to p, but the current setting is sufficient for the arithmetic applications we have in mind. (1.2) The decomposition

O_{M} ⊗ Z_{p} = ∏_{𝔭|p} O_{M_{𝔭}}

(where 𝔭 runs through all primes of M above p) induces decompositions

B[p^{∞}] = ⨁_{𝔭} B[𝔭^{∞}], T_{p}(B) = ⨁_{𝔭} T_{𝔭}(B).

Fix, once for all, a prime 𝔭 | p in M and set

O := O_{M_{𝔭}},  K := M_{𝔭},  A := B[𝔭^{∞}],  T := T_{𝔭}(B).

Throughout §1-§3, we assume that

• B has good 𝔭-ordinary reduction at each prime v of K above p in the sense that

rk_{O} T_{𝔭}(B̃_{v}) = \frac{1}{2} rk_{O} T_{𝔭}(B) (= dim(B)/[M : Q]).

This condition is weaker than requiring B to have good ordinary reduction at v (which is equivalent to B having good 𝔭′-ordinary reduction at v for all 𝔭′ | p in M).

(1.3) Pontryagin duality. For any discrete or compact topological Z_{p}-module X, let us denote by

D(X) := Hom_{cont,Z_{p}}(X, Q_{p}/Z_{p})

the Pontryagin dual of X. In the special case when X is a topological O-module, so is D(X), and there are canonical isomorphisms of O-modules

D(X) ⟶^{∼} Hom_{cont,O}(X, Hom_{Z_{p}}(O, Q_{p}/Z_{p})),

Hom_{Z_{p}}(O, Q_{p}/Z_{p}) ⟶^{∼} Hom_{Z_{p}}(O, Z_{p}) ⊗_{Z_{p}} Q_{p}/Z_{p} = Hom_{Z_{p}}(O, Z_{p}) ⊗_{O} K/O,

where Hom_{Z_{p}}(O, Z_{p}) is a free O-module of rank one. A choice of an isomorphism of O-modules

O ⟶^{∼} Hom_{Z_{p}}(O, Z_{p})  (1.3.1)

is equivalent to choosing a generator a ∈ 𝒟_{O/Z_{p}}^{−1} of the inverse different, via the pairing

O × O ⟶ Z_{p},  (x, y) ↦ Tr_{K/Q_{p}}(axy).  (1.3.2)

As in [N2, (0.4.1)], we let

T^{∗} := D(A), A^{∗} := D(T).

The Weil pairing

( , ) : T_{p}(B) × T_{p}(B^{t}) ⟶ Z_{p}(1)

is Z_{p}-bilinear and G_{K}-equivariant. It satisfies (αx, y) = (x, α^{t}y), for all α ∈ End(B) (where α^{t} denotes the dual isogeny to α). In particular, it induces an eponymous pairing

( , ) : T_{𝔭}(B) × T_{𝔭}(B^{t}) ⟶ Z_{p}(1)  (1.3.3)

giving rise to isomorphisms of O[G_{K}]-modules

D(A)(1) = T^{∗}(1) = Hom_{Z_{p}}(T_{𝔭}(B), Z_{p})(1) ⟶^{∼} T_{𝔭}(B^{t}), A^{∗}(1) = D(T)(1) ⟶^{∼} B^{t}[𝔭^{∞}].

Once we fix an isomorphism (1.3.1) via (1.3.2), we can pass from the Weil pairing (1.3.3) to its O-bilinear version, namely

( , )_{O} : T_{𝔭}(B) × T_{𝔭}(B^{t}) = T × T^{∗}(1) ⟶ O(1), (x, y) = Tr_{K/Q_{p}}(a(x, y)_{O}),

which induces an isomorphism of O[G_{K}]-modules

T^{∗}(1) = Hom_{O}(T_{𝔭}(B), O)(1) ⟶^{∼} T_{𝔭}(B^{t}).

The symmetric isogeny λ from (1.1) defines morphisms of O[G_{K}]-modules

λ_{∗} : T_{𝔭}(B) ↪ T_{𝔭}(B^{t}), B[𝔭^{∞}] ↠ B^{t}[𝔭^{∞}]

with finite cokernel and kernel, respectively. The Weil pairing attached to λ

( , )_{O,λ} : T_{𝔭}(B) × T_{𝔭}(B) = T × T ⟶ O(1),  (x, y)_{O,λ} := (x, λ_{∗}(y))_{O}  (1.3.4)

is skew-symmetric; in other words, λ_{∗} : T ⟶ T^{∗}(1) satisfies (λ_{∗})^{∗}(1) = −λ_{∗}. (1.4) Classical Selmer groups. For every finite extension L/K, p-power descent on B over L gives rise to the classical Selmer groups Sel_{p^{k}}(B/L) ⊂ H^{1}(L, B[p^{k}]) sitting in the standard exact sequences

0 ⟶ B(L) ⊗ Z/p^{k}Z ⟶ Sel_{p^{k}}(B/L) ⟶ � (B/L)[p^{k}] ⟶ 0.  (1.4.1)

Their respective inductive and projective limits

Sel_{p^{∞}}(B/L) := \underrightarrow{lim}_{k} Sel_{p^{k}}(B/L) ⊂ H^{1}(L, B[p^{∞}]), S_{p}(B/L) := \underleftarrow{lim}_{k} Sel_{p^{k}}(B/L) ⊂ H^{1}(L, T_{p}(B))

coincide with the corresponding Bloch–Kato Selmer groups

H_{f}^{1}(L, B[p^{∞}]) ⊂ H^{1}(L, B[p^{∞}]), H_{f}^{1}(L, T_{p}(B)) ⊂ H^{1}(L, T_{p}(B)),

by [BK, (3.11.1), (3.11.2)]. All groups in (1.4.1) and in the limit exact sequences

0 ⟶ B(L) ⊗ Q_{p}/Z_{p} ⟶ Sel_{p^{∞}}(B/L) ⟶ � (B/L)[p^{∞}] ⟶ 0,
0 ⟶ B(L) ⊗ Z_{p} ⟶ S_{p}(B/L) ⟶ T_{p}� (B/L)[p^{∞}] ⟶ 0

are O_{M} ⊗ Z_{p}-modules. After tensoring with O over O_{M} ⊗ Z_{p}, we obtain exact sequences

0 ⟶ B(L) ⊗_{O_{M}} O_{M}/𝔭^{ke} ⟶ Sel_{𝔭^{ke}}(B/L) ⟶ � (B/L)[𝔭^{ke}] ⟶ 0  (1.4.2)

(where e = e_{𝔭} is the ramification index of 𝔭 above p) and

0 ⟶ B(L) ⊗_{O_{M}} K/O ⟶ Sel_{𝔭^{∞}}(B/L) ⟶ � (B/L)[𝔭^{∞}] ⟶ 0,
0 ⟶ B(L) ⊗_{O_{M}} O ⟶ S_{𝔭}(B/L) ⟶ T_{𝔭}� (B/L)[p^{∞}] ⟶ 0.

Again,

Sel_{𝔭^{∞}}(B/L) = H_{f}^{1}(L, B[𝔭^{∞}]) = H_{f}^{1}(L, A), S_{𝔭}(B/L) = H_{f}^{1}(L, T_{𝔭}(B)) = H_{f}^{1}(L, T).

The same discussion applies to B^{t}; one obtains

Sel_{𝔭^{∞}}(B^{t}/L) = H_{f}^{1}(L, B^{t}[𝔭^{∞}]) = H_{f}^{1}(L, A^{∗}(1)), S_{𝔭}(B^{t}/L) = H_{f}^{1}(L, T_{𝔭}(B^{t})) = H_{f}^{1}(L, T^{∗}(1)).

If L ⊂ \overline{K} is an arbitrary algebraic extension of K, we let

Sel_{𝔭^{ke}}(B/L) := \underrightarrow{lim}_{F,res} Sel_{𝔭^{ke}}(B/F)  (k ∈ N ∪ \{∞\}),  S_{𝔭}(B/L) := \underleftarrow{lim}_{F,cor} S_{𝔭}(B/F),

where F runs through all intermediate fields K ⊂ F ⊂ L such that [F : K] < ∞. (1.5) Greenberg’s Selmer groups. Let v | p be a prime of K above p. As B has good 𝔭-ordinary reduction at v, there are exact sequences of O[G_{K_{v}}]-modules

0 ⟶ T_{v}^{+} ⟶ T ⟶ T_{v}^{−} ⟶ 0,  0 ⟶ A_{v}^{+} ⟶ A ⟶ A_{v}^{−} ⟶ 0  (1.5.1)

in which

T_{v}^{−} = T_{𝔭}(B̃_{v}), A_{v}^{−} = B̃_{v}[𝔭^{∞}],

and the Pontryagin dual of (1.5.1) is isomorphic to

0 ⟶ A^{∗}(1)_{v}^{+} ⟶ A^{∗}(1) ⟶ A^{∗}(1)_{v}^{−} ⟶ 0, 0 ⟶ T^{∗}(1)_{v}^{+} ⟶ T^{∗}(1) ⟶ T^{∗}(1)_{v}^{−} ⟶ 0,

where

T^{∗}(1)_{v}^{−} = T_{𝔭}(B̃_{v}^{t}), A^{∗}(1)_{v}^{−} = B̃_{v}^{t}[𝔭^{∞}].

In addition, λ : B ⟶ B^{t} induces maps

T ↪ T^{∗}(1),  T_{v}^{±} ↪ T^{∗}(1)_{v}^{±},  A ↠ A^{∗}(1),  A_{v}^{±} ↠ A^{∗}(1)_{v}^{±}

with finite cokernel (for T, T_{v}^{±}) and kernel (for A, A_{v}^{±}), respectively.

Fix a finite set S of primes of K containing all archimedean primes, all primes above p and all primes at which B has bad reduction. If L is a finite extension of K, let L_{S} be the maximal algebraic extension of L unramified outside primes above S; set G_{L,S} := Gal(L_{S}/L). Denote by Σ_{L} (resp. Σ′_{L}) the set of all primes of L above p (resp. the set of all nonarchimedean primes of L above S ∖ Σ_{K}). For each X = T, A, T^{∗}(1), T^{∗}(1), the Greenberg Selmer group over L and its strict counterpart are defined, respectively, by

S_{X}(L) := Ker (H^{1}(G_{L,S}, X) ⟶ ⨁_{v∈Σ_{L}} H^{1}(I_{v}, X_{v}^{−}) ⊕ ⨁_{v∈Σ′_{L}} H^{1}(I_{v}, X))

S_{X}^{str}(L) := Ker (H^{1}(G_{L,S}, X) ⟶ ⨁_{v∈Σ} H^{1}(G_{L_{v}}, X_{v}^{−}) ⊕ ⨁_{v∈Σ′_{L}} H^{1}(I_{v}, X)) ,

where I_{v} ⊂ G_{L_{v}} = Gal(\overline{L}_{v}/L_{v}) denotes the inertia group at v. These groups do not depend on S, and the morphisms

S_{T} (L) ⊗_{O} K/O ↪ S_{A}(L), S_{T^{∗}(1)}(L) ⊗_{O} K/O ↪ S_{A^{∗}(1)}(L)

(as well as their strict counterparts) have finite cokernels. (1.6) Selmer complexes and extended Selmer groups. In the notation of 1.5, the Selmer complex attached to X = T, A, T^{∗}(1), T^{∗}(1) over L is defined as

C̃_{f}^{•}(L, X) = Cone (C_{cont}^{•}(G_{L,S}, X) ⊕ ⨁_{v∈Σ_{L}∪Σ′_{L}} U_{v}^{+}(X) ⟶ ⨁_{v∈Σ_{L}∪Σ′_{L}} C_{cont}^{•}(G_{L_{v}}, X)) [−1],

where

U_{v}(X)^{+} = \{ \begin{matrix}C_{cont}^{•}(G_{L_{v}}, X_{v}^{+}), & v ∈ Σ_{L} \\ C_{cont}^{•}(G_{L_{v}}/I_{v}, X^{I_{v}}), & v ∈ Σ′_{L}. \end{matrix}

Up to a canonical quasi-isomorphism, C̃_{f}^{•}(L, X) does not depend on S; its cohomology groups are denoted by H̃_{f}^{i}(L, X). (1.7) Comparison of Selmer groups. For each X = T, A, T^{∗}(1), T^{∗}(1), there is an exact sequence

0 ⟶ H̃_{f}^{0}(L, X) ⟶ H^{0}(L, X) ⟶ ⨁_{v∈Σ_{L}} H^{0}(L_{v}, X_{v}^{−}) ⟶ H̃_{f}^{1}(L, X) ⟶ S_{X}^{str}(L) ⟶ 0,

by [N2, Lemma 9.6.3]. In addition, [N2, Lemma 9.6.7.3] implies that there are exact sequences

0 ⟶ S_{T}^{str}(L) ⟶ S_{𝔭}(B/L) ⟶ ⨁_{v∈Σ_{L}} H^{1}(L_{v}, T_{v}^{−})_{tors} ⊕ ⨁_{v∈Σ′_{L}} H^{1}(L_{v}, T)/H_{ur}^{1}(L_{v}, T)

0 ⟶ Sel_{𝔭^{∞}}(B/L) ⟶ S_{A}^{str}(L) ⟶

⟶ ⨁_{v∈Σ_{L}} Im (H^{1}(L_{v}, A_{v}^{+}) ⟶ H^{1}(L_{v}, A)) /div ⊕ ⨁_{v∈Σ′_{L}} H_{ur}^{1}(L_{v}, A),

in which

H^{1}(L_{v}, T_{v}^{−})_{tors} ⟶^{∼} H^{0}(L_{v}, A_{v}^{−})/div = B̃_{v}(k(v))[𝔭^{∞}]

D (Im (H^{1}(L_{v}, A_{v}^{+}) ⟶ H^{1}(L_{v}, A)) /div) ⊆ H^{0}(L_{v}, A^{∗}(1)_{v}^{−})/div = B̃_{v}^{t}(k(v))[𝔭^{∞}].

The same lemma implies that, for each v ∈ Σ′_{L}, the O-modules H^{1}(L_{v}, T)/H_{ur}^{1}(L_{v}, T) and H_{ur}^{1}(L_{v}, A) have the same finite length, equal to the local Tamagawa factor Tam_{v}(T, 𝔭) defined in 1.8 below.

Of course, one can replace B by B^{t}, T by T^{∗}(1) and A by A^{∗}(1) in the above discussion. (1.8) Local Tamagawa factors. In the notation of 1.6, if v ∤ p is a finite prime of L, the local Tamagawa factor Tam_{v}(T, 𝔭) is defined as in [N2, 7.6.10] (following [FoPR, Prop. 4.2.2(ii)]), namely

Tam_{v}(T, 𝔭) := ℓ_{O} (H^{1}(I_{v}, T)_{tors}^{F r(v)=1})

(where F r(v) is the arithmetic Frobenius at v and ℓ_{O}(Z) denotes the length of any O-module Z). This is a non-negative integer (since the group H^{1}(I_{v}, T)_{tors} ≃ H^{0}(I_{v}, A)/div is finite), equal to zero if v ∉ Σ′_{L}.

It will be more convenient to use geometric notation; let us write

Tam_{v}(B/L, 𝔭) := Tam_{v}(T, 𝔭), Tam(B/L, 𝔭) := ∑_{v∈Σ′_{L}} Tam_{v}(B/L, 𝔭).

The equality

Tam_{v}(T^{∗}(1), 𝔭) = Tam_{v}(T, 𝔭)  (1.8.1)

proved in [N2, 10.2.8] then implies that

Tam_{v}(B^{t}/L, 𝔭) := Tam_{v}(T^{∗}(1), 𝔭) = Tam_{v}(B/L, 𝔭), Tam(B^{t}/L, 𝔭) = Tam(B/L, 𝔭).

This cohomological definition agrees with the geometric one, namely, that

Tam_{v}(B/L, 𝔭) := ℓ_{O} (π_{0}(B̃_{v})^{G_{k(v)}} ⊗_{O_{M}} O) .  (1.8.2)

In particular, if M = Q, then 𝔭 = p and Tam_{v}(B/L, 𝔭) is equal to the p-adic valuation of the usual local Tamagawa factor c_{Tam,v}(B/L) = #H^{0}(k(v), π_{0}(B̃_{v})).

Note that (1.8.2) also implies (1.8.1), by the G_{k(v)}-equivariance and nondegeneracy of Grothendieck’s monodromy pairing π_{0}(B̃_{v}) × π_{0}(B̃_{v}^{t}) ⟶ Q/Z.

2. Comparison of Selmer groups, duality, control theorems

(2.1) Conditions on B. Given a finite extension L/K, consider the following conditions.

(A1)_{B,L,𝔭} There is an isomorphism of O[G_{L}]-modules j : T ⟶^{∼} T^{∗}(1) (where T = T_{𝔭}(B)) such that j^{∗}(1) = −j. (A2)_{B,L,𝔭} Tam(B/L, 𝔭) = 0 and ⨁_{v∈Σ_{L}} B̃_{v}(k(v))[𝔭] = 0. (A3)_{B,L,𝔭} B(L)[𝔭] = 0. (A4)_{B,L,𝔭} Sel_{𝔭^{∞}}(B/L) ⟶^{∼} K/O (which is equivalent to D(Sel_{𝔭^{∞}}(B/L)) ⟶^{∼} O).

(2.2) Proposition. (1) The conditions (A2)_{B,L,𝔭} and (A2)_{B^{t},L,𝔭} are equivalent.

(2) If k ∈ {3, 4} and if the conditions (A1)_{B,L,𝔭} and (Ak)_{B,L,𝔭} hold, so does (Ak)_{B^{t},L,𝔭}.

(3) If (A3)_{B,L,𝔭} holds, then Sel_{𝔭^{ke}}(B/L) = Sel_{𝔭^{∞}}(B/L)[𝔭^{ke}] holds, for all k ≥ 1.

(4) If L′/L is a finite extension of p-power degree which is unramified at all primes of L at which B has bad reduction, then the conditions (A2)_{B,L,𝔭} and (A2)_{B,L′,𝔭} are equivalent.

(5) If L′/L is a finite Galois extension of p-power degree, then the conditions (A3)_{B,L,𝔭} and (A3)_{B,L′,𝔭} are equivalent.

(6) If dim(B) = [M : Q], then (A1)_{B,L,𝔭} holds, and the isomorphism j : T ⟶^{∼} T^{∗}(1) induces isomorphisms of O[G_{L_{v}}]-modules X_{v}^{±} ⟶^{∼} X^{∗}(1)_{v}^{±}, for X = T, A and all v ∈ Σ_{L}.

Proof. (1) Combine (1.8.1) with the fact that B̃_{v}(k(v))[𝔭^{∞}] and B̃_{v}^{t}(k(v))[𝔭^{∞}] have the same cardinality. The statement (2) is immediate, while (3) follows from (1.4.2). The statements (4) and (5) are consequences of the fact that, if a p-group G acts on a finite set X, then #(X^{G}) ≡ #(X) (mod p).

In the situation of (6), the given O_{M}-linear symmetric isogeny λ = λ^{t} : B ⟶ B^{t} induces an isomorphism of K[G_{K}]-modules λ_{∗} : T_{𝔭}(B) ⊗_{O} K ⟶^{∼} T_{𝔭}(B^{t}) ⊗_{O} K and a G_{K}-equivariant, O-bilinear, skew-symmetric pairing ⟨ , ⟩_{O,λ} from (1.3.4), which is non-degenerate when tensored with K and which satisfies T_{𝔭}(B^{t}) = λ_{∗}{y ∈ T_{𝔭}(B) ⊗_{O} K | ∀x ∈ T_{𝔭}(B) ⟨x, y⟩_{O,λ} ∈ O}.

As T = T_{𝔭}(B) is a free O-module of rank two, the matrix of the pairing ⟨ , ⟩_{O,λ} in any basis of T

over O is of the form ( \begin{matrix}0 & b \\ −b & 0 \end{matrix} ), for some b ∈ O ∖ \{0\}. This implies that T_{𝔭}(B^{t}) = b^{−1}λ_{∗}(T_{𝔭}(B)), hence

j := b^{−1} ∘ λ_{∗} : T_{𝔭}(B) ⟶^{∼} T_{𝔭}(B^{t}) has the required property.

The maps X_{v}^{±} ⟶ X^{∗}(1)_{v}^{±} are isomorphisms, since T_{v}^{−} is the unique quotient of T which is free of rank one over O on which I_{v} acts trivially.

(2.3) Proposition. Let L be a finite extension of K.

(1) If (A2)_{B,L,𝔭} holds, then

Sel_{𝔭^{∞}}(B/L) = H_{f}^{1}(L, A) = S_{A}^{str}(L) = H̃_{f}^{1}(L, A),  A = B[𝔭^{∞}]
Sel_{𝔭^{∞}}(B^{t}/L) = H_{f}^{1}(L, A^{∗}(1)) = S_{A^{∗}(1)}^{str}(L) = H̃_{f}^{1}(L, A^{∗}(1)),  A^{∗}(1) = B^{t}[𝔭^{∞}]
S_{𝔭}(B/L) = H_{f}^{1}(L, T) = S_{T}^{str}(L) = H̃_{f}^{1}(L, T),  T = T_{𝔭}(B)
S_{𝔭}(B^{t}/L) = H_{f}^{1}(L, T^{∗}(1)) = S_{T^{∗}(1)}^{str}(L) = H̃_{f}^{1}(L, T^{∗}(1)),  T^{∗}(1) = T_{𝔭}(B^{t}).

(2) In general (without assuming any (Ak)_{B,L,𝔭}), there are isomorphisms of O-modules

\begin{aligned}D(H̃_{f}^{i}(L, T)) & ≃ H̃_{f}^{3−i}(L, A^{∗}(1))  (= 0 if i ≠ 1, 2, 3), \\ D(H̃_{f}^{i}(L, A)) & ≃ H̃_{f}^{3−i}(L, T^{∗}(1))  (= 0 if i ≠ 0, 1, 2). \end{aligned}

(3) If (A2)_{B,L,𝔭} holds, then

H̃_{f}^{i}(L, A) = \{ \begin{matrix}a submodule of B(L)[𝔭^{∞}], & i = 0 \\ Sel_{𝔭^{∞}}(B/L), & i = 1 \\ D(S_{𝔭}(B^{t}/L)), & i = 2 \\ 0, & i ≠ 0, 1, 2 \end{matrix}

H̃_{f}^{i}(L, T) = \{ \begin{matrix}S_{𝔭}(B/L), & i = 1 \\ D(Sel_{𝔭^{∞}}(B^{t}/L)), & i = 2 \\ a quotient of D(B^{t}(L)[𝔭^{∞}]), & i = 3 \\ 0, & i ≠ 1, 2, 3 \end{matrix}

(4) If (A3)_{B,L,𝔭} holds, then H̃_{f}^{0}(L, A) = 0 = H̃_{f}^{3}(L, T^{∗}(1)). Dually, if (A3)_{B^{t},L,𝔭} holds, then H̃_{f}^{0}(L, A^{∗}(1)) = 0 = H̃_{f}^{3}(L, T).

Proof. The equalities of the various Selmer groups in (1) follow from the discussion in 1.7. The statement (2) is a consequence of [N2, Thm. 6.3.4, Prop. 6.7.7], while (3) is a combination of (1) and (2). Finally, (4) follows from (2) and the fact that H̃_{f}^{0}(L, A) ⊂ H^{0}(L, A).

(2.4) Iwasawa theory. Fix a Galois extension K_{∞}/K such that Γ := Gal(K_{∞}/K) ≃ Z_{p}^{d} (d ≥ 1) and let Λ := O[[Γ]] = \underleftarrow{lim}_{F} O[Γ_{F}], where F runs through all fields K ⊂ F ⊂ K_{∞} such that [F : K] < ∞, and

Γ_{F} := Gal(F/K).

For every intermediate field K ⊂ L ⊂ K_{∞} (not necessarily of finite degree over K), let

Γ^{L} := Gal(K_{∞}/L),  Γ_{L} := Gal(L/K) = Γ/Γ^{L},  Λ_{L} := O[[Γ_{L}]].

The corresponding Iwasawa-theoretical Selmer modules

H̃_{f}^{i}(L, A) := \underrightarrow{lim}_{F,res} H̃_{f}^{i}(F, A),  H̃_{f,Iw}^{i}(L/K, T) := \underleftarrow{lim}_{F,cor} H̃_{f}^{i}(F, T)

(K ⊂ F ⊂ L, [F : K] < ∞) are Λ_{L}-modules of cofinite and finite type, respectively.

The standard involution ι : Λ_{L} ⟶ Λ_{L} is induced by the inverse map Γ_{L} ⟶ Γ_{L}, γ ↦ γ^{−1}. For any Λ_{L}-module N we denote by N^{ι} the Λ_{L}-module equal to N as an O-module, but on which every r ∈ Λ_{L} acts as ι(r) does on N. Note that ι induces an isomorphism of Λ_{L}-modules ι : Λ_{L} ⟶^{∼} Λ_{L}^{ι}.

This involution appears naturally when one compares Pontryagin duality between Λ_{L}-modules of finite type (compact) and cofinite type (discrete), defined by

D_{Λ_{L}}(N) := Hom_{cont,Z_{p}}(N, Q_{p}/Z_{p}), (rf)(n) := f(rn) (r ∈ Λ_{L}, f ∈ D_{Λ_{L}}(N), n ∈ N),

with Pontryagin duality for O-modules with a continuous linear action of Γ_{L}: in this case

D(N) := Hom_{cont,Z_{p}}(N, Q_{p}/Z_{p}), (γ · f)(n) := f(γ^{−1}(n)) (γ ∈ Γ_{L}, f ∈ D(N), n ∈ N).

In other words,

D_{Λ_{L}}(N) = D(N)^{ι}.

(2.5) Proposition. Assume that K ⊂ L ⊂ K_{∞} is an arbitrary intermediate field.

(1) In general (without assuming any (Ak)_{B,L,𝔭}), there are isomorphisms of Λ_{L}-modules

D_{Λ_{L}}(H̃_{f,Iw}^{i}(L/K, T)) ≃ H̃_{f}^{3−i}(L, A^{∗}(1))^{ι}  (= 0 if i ≠ 1, 2, 3),
D_{Λ_{L}}(H̃_{f}^{i}(L, A)) ≃ H̃_{f,Iw}^{3−i}(L/K, T^{∗}(1))^{ι}  (= 0 if i ≠ 0, 1, 2).

(2) If (A2)_{B,L,𝔭} holds, then

H̃_{f}^{i}(L, A) = \{ \begin{matrix}a submodule of B(L)[𝔭^{∞}], & i = 0 \\ Sel_{𝔭^{∞}}(B/L), & i = 1 \\ D_{Λ_{L}}(S_{𝔭}(B^{t}/L))^{ι}, & i = 2 \\ 0, & i ≠ 0, 1, 2 \end{matrix}

H̃_{f,Iw}^{i}(L/K, T) = \{ \begin{matrix}S_{𝔭}(B/L), & i = 1 \\ D_{Λ_{L}}(Sel_{𝔭^{∞}}(B^{t}/L))^{ι}, & i = 2 \\ a quotient of D_{Λ_{L}}(B^{t}(L)[𝔭^{∞}])^{ι}, & i = 3 \\ 0, & i ≠ 1, 2, 3 \end{matrix}

(3) (Exact control theorem) If (A2)_{B,L,𝔭} and (A3)_{B,L,𝔭} hold, then the canonical map

Sel_{𝔭^{∞}}(B/L) ⟶^{∼} Sel_{𝔭^{∞}}(B/K_{∞})^{Γ^{L}}

is an isomorphism (idem if we replace everywhere B by B^{t}).

(4) If (A2)_{B,L,𝔭} and (A3)_{B,L,𝔭} are satisfied, then there is an exact sequence of Λ_{L}-modules of cofinite type

0 ⟶ H^{1}(Γ^{L}, Sel_{𝔭^{∞}}(B/K_{∞})) ⟶ D_{Λ_{L}}(S_{𝔭}(B^{t}/L))^{ι} ⟶ D_{Λ_{L}}(S_{𝔭}(B^{t}/K_{∞})_{Γ^{L}})^{ι}
⟶ H^{2}(Γ^{L}, Sel_{𝔭^{∞}}(B/K_{∞})) ⟶ 0

(again, we can interchange everywhere B with B^{t}).

(5) If (A2)_{B,L,𝔭} and (A3)_{B,L,𝔭} are satisfied, then there is an isomorphism of Λ_{L}-modules of finite type

S_{𝔭}(B/L) ⟶^{∼} Hom_{Λ_{L}}(D_{Λ_{L}}(Sel_{𝔭^{∞}}(B/L)), Λ_{L})

(as before, we can replace everywhere B by B^{t}).

(6) If (A2)_{B,L,𝔭} and (A3)_{B,L,𝔭} are satisfied and if Γ_{L} ≃ Z_{p}^{r} (0 ≤ r ≤ d), then

rk_{Λ_{L}} S_{𝔭}(B/L) = rk_{Λ_{L}} S_{𝔭}(B^{t}/L) = cork_{Λ_{L}} Sel_{𝔭^{∞}}(B/L) = cork_{Λ_{L}} Sel_{𝔭^{∞}}(B^{t}/L).

Proof. (1), (2) Apply Proposition 2.3(2)-(3) over all intermediate fields K ⊂ F ⊂ L such that [F : K] < ∞ (which is legitimate, thanks to Proposition 2.2(4)-(5)) and take the inductive (resp. the projective) limit. (3),(4) In the spectral sequence from [N2, Prop. 8.10.12]

′\overline{E}_{2}^{i,j} = H^{i}(Γ^{L}, H̃_{f}^{j}(K_{∞}, A)) ⟹ H̃_{f}^{i+j}(L, A)

(which is a consequence of the “exact control theorem for Selmer complexes” [N2, Prop. 8.10.1]) we have ′\overline{E}_{2}^{i,j} = 0 if j ≠ 1, 2, by (1) applied to K_{∞} and the fact that B(K_{∞})[𝔭] = 0 (which follows from (A3)_{B,K,𝔭}, by Proposition 2.2(5)).

(5) The duality theorem [N2, Thm. 8.9.12] applies in this case, giving rise to a spectral sequence

E_{2}^{i,j} = Ext_{Λ_{L}}^{i}(H̃_{f,Iw}^{3−j}(L/K, T^{∗}(1)), Λ_{L})^{ι} ⟹ H̃_{f,Iw}^{i+j}(L/K, T)

satisfying E_{2}^{i,j} = 0 for j ≠ 1, 2 (as in the proof of (3) and (4)). Therefore

S_{𝔭}(B/L) = H̃_{f,Iw}^{1}(L/K, T) ≃ E_{2}^{0,1} ≃ Hom_{Λ_{L}}(D_{Λ_{L}}(Sel_{𝔭^{∞}}(B/L)), Λ_{L}).

(6) This follows from (5) and the fact that there exists a constant C ≥ 0 such that, for every k ≥ 1 and every finite extension F/K, the kernel and the cokernel of the map

Sel_{𝔭^{k}}(B/F) ⟶ Sel_{𝔭^{k}}(B^{t}/F)

induced by λ : B ⟶ B^{t} is killed by 𝔭^{C}. (2.6) Notation. For every field K ⊂ L ⊂ K_{∞} we are going to abbreviate as

0 ⟶ Z(B/L) ⟶ X(B/L) ⟶ Y (B/L) ⟶ 0  (2.6.1)

the terms in the exact sequence

0 ⟶ D_{Λ_{L}}(� (B/L)[𝔭^{∞}]) ⟶ D_{Λ_{L}}(Sel_{𝔭^{∞}}(B/L)) ⟶ D_{Λ_{L}}(B(L) ⊗_{O_{M}} K/O) ⟶ 0.

Proposition 2.5(3) tells us that, under the conditions (A2)_{B,K,𝔭} and (A3)_{B,K,𝔭}, there are canonical isomorphisms of Λ_{L}-modules

X(B/K_{∞})_{Γ^{L}} ⟶^{∼} X(B/L),  (2.6.2)

hence also

X(B/L′)_{Gal(L′/L)} ⟶^{∼} X(B/L) (K ⊂ L ⊂ L′ ⊂ K_{∞}).

3. Freeness of compact Selmer groups and the vanishing of � [𝔭^{∞}]

(3.1) Consider another condition on B and K.

(A5)_{B,K} There exists a subfield K^{+} ⊂ K such that [K : K^{+}] = 2, K_{∞}/K^{+} is a Galois extension with Galois group Γ^{+} = Γ ⋊ {1, c}, where c^{2} = 1 and ∀γ ∈ Γ cγc^{−1} = γ^{−1}, and there exists an abelian variety B^{+} over K^{+} with good reduction at all primes of K^{+} above p, equipped with a ring morphism i^{+} : O_{M} ⟶ End(B^{+}) and a symmetric O_{M}-linear isogeny λ^{+} = (λ^{+})^{t} : B^{+} ⟶ (B^{+})^{t}, such that B^{+} has good 𝔭-ordinary reduction at all primes of K^{+} above p, and that the base change of (B^{+}, i^{+}, λ^{+}) from K^{+} to K is isomorphic to (B, i, λ).

(3.2) Proposition. (1) If (A5)_{B,K} holds, then

cork_{O} Sel_{𝔭^{∞}}(B/K) ≡ cork_{Λ_{L}} Sel_{𝔭^{∞}}(B/K_{∞}) (mod 2),
cork_{O} Sel_{𝔭^{∞}}(B/K) ≥ cork_{Λ_{L}} Sel_{𝔭^{∞}}(B/K_{∞}).

(2) If the conditions (A2)_{B,K,𝔭}, (A3)_{B,K,𝔭} and (A4)_{B,K,𝔭} are satisfied, then, for every intermediate field K ⊂ L ⊂ K_{∞} such that Γ_{L} = Gal(L/K) ≃ Z_{p}^{r} (0 ≤ r ≤ d), X(B/L) = D_{Λ_{L}}(Sel_{𝔭^{∞}}(B/L)) is a cyclic Λ_{L}-module.

(3) If the conditions (A2)_{B,K,𝔭}, (A3)_{B,K,𝔭}, (A4)_{B,K,𝔭} and (A5)_{B,K} are satisfied, then, for every intermediate field K ⊂ L ⊂ K_{∞}, the Λ_{L}-modules X(B/L) and S_{𝔭}(B/L) are free of rank one, the Λ_{L}/𝔭^{ke}-module D_{Λ_{L}}(Sel_{𝔭^{ke}}(B/L)) is free of rank one (for every k ≥ 1), and the canonical maps

X(B/L′)_{Gal(L′/L)} ⟶^{∼} X(B/L), S_{𝔭}(B^{t}/L′)_{Gal(L′/L)} ⟶^{∼} S_{𝔭}(B^{t}/L)

are isomorphisms of Λ_{L}-modules, if K ⊂ L ⊂ L′ ⊂ K_{∞}. In particular, if [L : K] < ∞, then

rk_{O_{M}} B(L) + cork_{O} � (B/L)[𝔭^{∞}] = [L : K].

Proof. (1) If B = E is an elliptic curve, this is [N2, Prop. 10.7.19]. The general case follows from [N2, Thm. 10.7.17(iv)] (for χ = χ′ = 1).

(2) X(B/L) is a Λ_{L}-module of finite type satisfying X(B/L)_{Γ_{L}} ⟶^{∼} X(B/K) ⟶^{∼} O, by (2.6.2) (for L/K replacing K_{∞}/L) and (A4)_{B,K,𝔭}, respectively. If the image of x ∈ X(B/L) generates X(B/K) as an Omodule, then (X(B/L)/Λ_{L}x)_{Γ_{L}} = 0, hence X(B/L) = Λ_{L}x by Nakayama’s Lemma.

(3) It is enough to treat the case L′ = K_{∞}. According to (2) applied to L = K_{∞}, we have X(B/K_{∞}) ⟶^{∼} Λ/J for some ideal J ⊂ Λ. On the other hand, (1) together with (A4)_{B,K,𝔭} imply that rk_{Λ} X(B/K_{∞}) = 1, hence J = 0 and X(B/K_{∞}) is free of rank one over Λ. The control theorem (2.6.2) then yields X(B/L) ⟶^{∼} Λ_{L} as a Λ_{L}-module. The statement about Sel_{𝔭^{ke}}(B/L) then follows from Proposition 2.2(3).

It remains to show that S_{𝔭}(B^{t}/K_{∞})_{Γ^{L}} ⟶^{∼} S_{𝔭}(B^{t}/L) is an isomorphism, which is equivalent, by Proposition 2.5(4), to the vanishing of H^{i}(Γ^{L}, Sel_{𝔭^{∞}}(B/K_{∞})) for i = 1, 2. We claim that the latter group vanishes for all i > 0. Indeed, its Pontryagin dual H_{i}(Γ^{L}, X(B/K_{∞})) ≃ H_{i}(Γ^{L}, Λ) is the i-th homology group of the Koszul complex of Λ with respect to the sequence (γ′_{1} −1, …, γ′_{t} −1), where γ′_{1}, …, γ′_{t} is any basis of Γ^{L} ≃ Z_{p}^{t} over Z_{p}. We can take γ′_{i} = γ_{i}^{p^{n_{i}}} (1 ≤ i ≤ t, n_{i} ≥ 0), for a suitable basis γ_{1}, …, γ_{d} of Γ over Z_{p}. Therefore Λ = O[[X_{1}, …, X_{d}]] (X_{i} = γ_{i} − 1) and γ′_{i} − 1 = (1 + X_{i})^{p^{n_{i}}} − 1, which implies that (γ′_{1} − 1, …, γ′_{t} − 1) is a regular sequence in Λ, hence H_{i}(Γ^{L}, Λ) = 0 for i > 0. (3.3) Notation. For an arbitrary algebraic extension K′/K, let us write

N_{K′/K}(B ⊗ O) := \underleftarrow{lim}_{F,Tr}(B(F) ⊗_{O_{M}} O) ⊂ S_{𝔭}(B/K′),

where F runs through all intermediate fields K ⊂ F ⊂ K′ such that [F : K] < ∞.

We are going to consider the following conditions.

(A6)_{B,K′/K,𝔭} Im(N_{K′/K}(B ⊗ O) ⟶ B(K) ⊗_{O_{M}} O) ≠ 0. (A7)_{B,K′/K,𝔭} Im(N_{K′/K}(B ⊗ O) ⟶ B(K) ⊗_{O_{M}} O/𝔭) ≠ 0.

(3.4) Theorem. Assume that the conditions (A2)_{B,K,𝔭}, (A3)_{B,K,𝔭}, (A4)_{B,K,𝔭}, (A5)_{B,K} and (A6)_{B,K_{∞}/K,𝔭} are satisfied. Then, for every intermediate field K ⊂ L ⊂ K_{∞} such that Γ_{L} = Gal(L/K) ≃ Z_{p}^{r} (0 ≤ r ≤ d),

� (B/L)[𝔭^{∞}] = 0, B(L) ⊗_{O_{M}} K/O = Sel_{𝔭^{∞}}(B/L)

(the Pontryagin dual of the latter group being a free module of rank one over Λ_{L}).

Proof. Induction on r. If r = 0, then L = K. In this case B(K)⊗_{O_{M}} O ≠ 0 and B(K)[𝔭] = 0 by (A6)_{B,K_{∞}/K,𝔭} and (A3)_{B,K,𝔭}, respectively. This means that B(K) ⊗_{O_{M}} O ≃ O^{m} and B(K) ⊗_{O_{M}} K/O ≃ (K/O)^{m} for some m ≥ 1. On the other hand, B(K) ⊗_{O_{M}} K/O ⊂ Sel_{𝔭^{∞}}(B/K) ≃ K/O (by (A4)_{B,K,𝔭}); thus m = 1 and � (B/K)[𝔭^{∞}] = 0.

Assume that r > 0. In the notation of 2.6, we need to show that Z(B/L) = 0, which is equivalent to X(B/L)/Z(B/L) = Y (B/L) = D_{Λ_{L}}(B(L) ⊗_{O_{M}} K/O) not being Λ_{L}-torsion, since X(B/L) ≃ Λ_{L}, by Proposition 3.2(3). Note that the canonical map

B(F′) ⊗_{O_{M}} K/O ⟶ (B(F) ⊗_{O_{M}} K/O)^{Gal(F/F′)}

is injective, whenever K ⊂ F′ ⊂ F ⊂ K_{∞} and [F : K] < ∞, since B(K_{∞})[𝔭] = 0 (by Proposition 2.2(5) and (A3)_{B,K,𝔭}).

If r = 1, write L = ⋃_{n≥1} K_{n}, where Gal(K_{n}/K) ≃ Z/p^{n}Z. If Y (B/L) were Λ_{L}-torsion, it would be a free O-module of finite type (since Y (B/L)[𝔭] = 0), hence B(L) ⊗_{O_{M}} O = B(K_{m}) ⊗_{O_{M}} O for some m ≥ 1. This would imply that

∀k ≥ 0 ∀n ≥ m N_{K_{n+k}/K_{n}}(B(K_{n+k}) ⊗_{O_{M}} O) ⊂ p^{k}(B(K_{n}) ⊗_{O_{M}} O),

hence N_{L/K}(B ⊗ O) = 0, which contradicts (A6)_{B,K_{∞}/K,𝔭}.

Assume that r > 1. If Y (B/L) were Λ_{L}-torsion, there would be γ ∈ Γ_{L} ∖ Γ_{L}^{p} such that (γ − 1) ∉ Supp_{Λ_{L}}(Y (B/L)). The fixed field L′ := L^{γ=1} satisfies Γ_{L′} ≃ Z_{p}^{r−1}. By construction, Y (B/L)/(γ − 1) is a torsion module over Λ_{L}/(γ − 1) = Λ_{L′}, hence so is its quotient Y (B/L′); but this is false by the induction hypothesis.

(3.5) Theorem. Assume that the conditions (A2)_{B,K,𝔭}, (A3)_{B,K,𝔭}, (A4)_{B,K,𝔭}, (A5)_{B,K} and (A7)_{B,K_{∞}/K,𝔭} are satisfied.

(a) For every intermediate field K ⊂ L ⊂ K_{∞} the following statements hold.

� (B/L)[𝔭^{∞}] = 0,  B(L) ⊗_{O_{M}} K/O = Sel_{𝔭^{∞}}(B/L),  N_{L/K}(B ⊗ O) = S_{𝔭}(B/L)

and both S_{𝔭}(B/L) and D_{Λ_{L}}(Sel_{𝔭^{∞}}(B/L)) are free modules of rank one over Λ_{L} = O[[Gal(L/K)]]. In the special case when [L : K] < ∞, then B(L)⊗_{O_{M}} O = S_{𝔭}(B/L) is a free O-module of rank rk_{O_{M}} B(L) = [L : K].

(b) If, in addition, (A1)_{B,K,𝔭} is satisfied, then the canonical maps

N_{L′/K}(B ⊗ O) = S_{𝔭}(B/L′) ⟶ S_{𝔭}(B/L) = N_{L/K}(B ⊗ O)

are surjective, for arbitrary intermediate fields K ⊂ L ⊂ L′ ⊂ K_{∞}. Furthermore, S_{𝔭}(B/L) is generated as a Λ_{L}-module by the image x_{L} of any element x ∈ N_{K_{∞}/K}(B ⊗ O) whose image \overline{x}_{K} in B(K) ⊗_{O_{M}} O/𝔭 is non-zero.

Proof. (a) Fix intermediate fields K ⊂ L ⊂ L′ ⊂ K_{∞}. For every finite extension F/K,

Cone(C̃_{f}^{•}(F, X) ⟶^{λ_{∗}} C̃_{f}^{•}(F, X^{∗}(1))) (X = T, A)

is quasi-isomorphic to a complex of O/𝔭^{C}-modules, where 𝔭^{C} Ker(λ)(\overline{K})[𝔭^{∞}] = 0. Therefore the kernel and cokernel of

H̃_{f}^{i}(F, X) ⟶ H̃_{f}^{i}(F, X^{∗}(1)) (X = T, A)

is killed by 𝔭^{C}, for every i. Thus the same is true for the kernels and cokernels of the maps

S_{𝔭}(B/L) ⟶ S_{𝔭}(B^{t}/L), Sel_{𝔭^{∞}}(B/L) ⟶ Sel_{𝔭^{∞}}(B^{t}/L).

Combined with the freeness results and the isomorphisms in Proposition 3.2(3), this implies that the canonical map

j_{L′/L} : S_{𝔭}(B/L′)_{Gal(L′/L)} ⟶ S_{𝔭}(B/L)

is a morphism between two free Λ_{L}-modules of rank one, whose kernel and cokernel is killed by 𝔭^{C}. Therefore Ker(j_{L′/L}) = 0 and the maps in the commutative diagram

N_{L′/K}(B ⊗ O)_{Gal(L′/L)} ⟶^{k_{L′/L}} S_{𝔭}(B/L′)_{Gal(L′/L)}
↓^{i_{L′/L}}  ↓^{j_{L′/L}}
N_{L/K}(B ⊗ O)  ⟶^{k_{L}}  S_{𝔭}(B/L)

satisfy

Ker(j_{L′/L}) = 0 = Ker(k_{L}), 𝔭^{C} Coker(j_{L′/L}) = 0.

Moreover, S_{𝔭}(B/K) ⟶^{∼} O (by (A4)_{B,K,𝔭}) and Coker(k_{K} ∘ i_{L′/K}) = 0, by (A7)_{B,K_{∞}/K,𝔭}. As a result, j_{L′/K} is an isomorphism and

0 = Coker(k_{L′/K}) = (S_{𝔭}(B/L′)/N_{L′/K}(B ⊗ O))_{Gal(L′/K)},

which implies that

N_{L′/K}(B ⊗ O) = S_{𝔭}(B/L′) ≃ Λ_{L′}  (3.5.1)

(for arbitrary K ⊂ L′ ⊂ K_{∞}), by Nakayama’s Lemma.

In the special case when [L′ : K] < ∞, it follows from (3.5.1) that

B(L′) ⊗_{O_{M}} O = S_{𝔭}(B/L′), rk_{O_{M}} B(L′) = rk_{O} S_{𝔭}(B/L′) = [L′ : K].

As a result, both X(B/L′) and Y (B/L′) in the exact sequence (2.6.2) are free modules over O of the same rank [L′ : K], hence Z(B/L′) = 0 and � (B/L′)[𝔭^{∞}] = 0. This proves the Theorem in the special case when [L : K] < ∞. The general case follows by taking the inductive limit over all subfields of L of finite degree over K.

(b) In this case the arguments in the proof of (a) go through if one replaces the map λ_{∗} by the map j_{∗} induced by the isomorphism j : T_{𝔭}(B) ⟶^{∼} T_{𝔭}(B^{t}) from (A1)_{B,K,𝔭}. The constant C is then replaced by zero, which means that the map j_{L′/L} : S_{𝔭}(B/L′)_{Gal(L′/L)} ⟶ S_{𝔭}(B/L) is an isomorphism between two free Λ_{L}-modules of rank one. If \overline{x}_{K} ≠ 0, then S_{𝔭}(B/K_{∞})/Λx = 0, by Nakayama’s Lemma. It follows that S_{𝔭}(B/K_{∞}) = Λx and, after applying j_{K_{∞}/L}, that S_{𝔭}(B/L) = Λ_{L}x_{L}.

4. An application to Heegner points

(4.1) Ring class fields. Let K be an imaginary quadratic field of discriminant D_{K}. Denote by η_{K} : (Z/|D_{K}|Z)^{×} ⟶ {±1} the primitive quadratic character attached to K. For each prime p ∤ 2D_{K}, we have η_{K}(p) = (\frac{D_{K}}{p} ); if p | D_{K}, then η_{K}(p) = 0.

For any integer m ≥ 1, denote by O_{m} := Z + mO_{K} ⊂ O_{K} the order of conductor m in K and by H_{m} the ring class field of K of conductor m (H_{1} is the Hilbert class field of K). The Galois groups of the intermediate extensions in the diagram

Q = K^{+} ↪ K ↪ H_{1} ↪ H_{m}

are as follows.

G_{m} := Gal(H_{m}/K) ≃ Pic(O_{m}),  Gal(H_{m}/Q) = G_{m} ⋊ {1, c},  ∀g ∈ G_{m} cgc^{−1} = g^{−1}

(where c is complex conjugation) and there is an exact sequence

\frac{O_{K}^{×}}{Z^{×}} ⟶ \frac{(O_{K} ⊗ Z/mZ)^{×}}{(Z/mZ)^{×}} ⟶ G_{m} ⟶ G_{1} ⟶ 0.

The first group in this sequence is cyclic of order

u_{K} := #(O_{K}^{×}/Z^{×}) = \{ \begin{matrix}3, & D_{K} = −3 \\ 2, & D_{K} = −4 \\ 1, & D_{K} ≠ −3, −4. \end{matrix}

(4.2) The anticyclotomic Z_{p}-extension K_{∞}/K. For a fixed prime number p, the tower of fields

Q = K^{+} ↪ K ↪ H_{1} ↪ H_{p} ↪ H_{p^{2}} ↪ ⋯ ↪ H_{p^{∞}} := ⋃_{n≥0} H_{p^{n}}

has the following properties.

• ∀n ≥ 1 Gal(H_{p^{n+1}}/H_{p^{n}}) ≃ Z/pZ.

• If p ≠ 2, then Gal(H_{p^{∞}}/H_{p}) ≃ Z_{p}.

• Gal(H_{1}/K) ≃ Pic(O_{K}) = Cl_{K}.

• Gal(H_{p}/H_{1}) is a cyclic group of order u_{K}^{−1}(p − η_{K}(p)).

• The torsion subgroup Δ := Gal(H_{p^{∞}}/K)_{tors} is finite. Its fixed field K_{∞} := (H_{p^{∞}})^{Δ} satisfies Gal(K_{∞}/K) ≃ Z_{p} and Gal(K_{∞}/Q) = Gal(K_{∞}/K) ⋊ {1, c}, as in (A5). Write K_{∞} = ⋃_{n≥0} K_{n}, where Gal(K_{n}/K) ≃ Z/p^{n}Z.

(4.3) Heegner points. Assume that

• E is an elliptic curve over Q of conductor N.

• φ : X_{0}(N) ⟶ E is a modular parameterisation of E (sending i∞ to the origin) of the smallest degree.

• K is an imaginary quadratic field satisfying the Heegner condition

(Heeg)  all primes dividing N split in K/Q.

Fix an ideal N ⊂ O_{K} such that O_{K}/N ≃ Z/NZ. If m ≥ 1 is an integer such that (m, N) = 1, then N_{m} := N ∩ O_{m} is an invertible ideal of O_{m} satisfying O_{m}/N_{m} ≃ N_{m}^{−1}/O_{m} ≃ Z/NZ.

The Heegner points of conductor m on X_{0}(N) and E, respectively, are defined as

x_{m} := [C/O_{m} ⟶ C/N_{m}^{−1}] ∈ X_{0}(N)(H_{m}), y_{m} := φ(x_{m}) ∈ E(H_{m})

(up to a sign, y_{m} does not depend on the choice of N). The basic Heegner point on E is defined as

y_{K} := Tr_{H_{1}/K}(y_{1}) ∈ E(K).

A general modular parameterisation φ′ : X_{0}(N) ⟶ E of E (sending i∞ to the origin) is obtained by composing φ with a non-trivial element a ∈ End(E) = Z. The Heegner points y′_{m} := φ′(x_{m}) corresponding to φ′ are therefore equal to y′_{m} = ay_{m}.

(4.4) Norm relations. Fix a prime number p ∤ N and let a_{p} := p + 1 − #Ẽ_{p}(F_{p}). For any integer m ≥ 1 relatively prime to pN, the Heegner points of conductors mp^{n} on E are related as follows [PR, 3.1, Prop. 1].

∀n ≥ 1  Tr_{H_{mp^{n+1}} /H_{mp^{n}}} (y_{mp^{n+1}}) = a_{p}y_{mp^{n}} − y_{mp^{n−1}},

∃ σ ∈ Gal(H_{m}/K)  u_{K,m} · Tr_{H_{mp}/H_{m}}(y_{mp}) = \{ \begin{matrix}a_{p}y_{m}, & if η_{K}(p) = −1 \\ (a_{p} − σ)y_{m}, & if η_{K}(p) = 0 \\ (a_{p} − σ − σ^{−1})y_{m}, & if η_{K}(p) = 1, \end{matrix}

u_{K,m} = \{ \begin{matrix}u_{K}, & if m = 1, \\ 1, & if m > 1. \end{matrix}
u_{K} · Tr_{H_{p}/K}(y_{p}) = (a_{p} − 1 − η_{K}(p))y_{K}.

(4.5) Universal norms in the p-ordinary case. Assume that E has good ordinary reduction at a prime number p (which is equivalent to p ∤ N · a_{p}). In this case the polynomial defining the Euler factor of E at p factors in Z_{p}[X] as

X^{2} − a_{p}X + p = (X − α_{p})(X − β_{p}), α_{p} ∈ Z_{p}^{×}, β_{p} ∈ pZ_{p}^{×},  (4.5.1)
α_{p} + β_{p} = a_{p}, α_{p}β_{p} = p.  (4.5.2)

In addition, |ι(α_{p})| = |ι(β_{p})| = \sqrt{p}, for every embedding ι : Q(α_{p}) ↪ C.

Define, for every integer n ≥ 0,

z_{n} := α_{p}^{−n}y_{p^{n+1}} − α_{p}^{−n−1}y_{p^{n}} ∈ E(H_{p^{n+1}}) ⊗ Z_{p}.

These elements are norm compatible, namely

∀n ≥ 1 Tr_{H_{p^{n+1}} /H_{p^{n}}} (z_{n}) = z_{n−1}.  (4.5.3)

In addition, the bottom element z_{0} = y_{p} − α_{p}^{−1}y_{0} satisfies

u_{K} · Tr_{H_{p}/K}(z_{0}) = (a_{p} − 1 − η_{K}(p))y_{K} − α_{p}^{−1}(p − η_{K}(p))y_{K} = (α_{p} − 1)(1 − α_{p}^{−1}η_{K}(p))y_{K}.  (4.5.4)

(4.6) Proposition. Assume that p ∤ N · a_{p}.

(1) The element (α_{p} − 1)(1 − α_{p}^{−1}η_{K}(p)) (y_{K} ⊗ 1) ∈ E(K) ⊗ Z_{p} is contained in

u_{K} · Im(N_{H_{p^{∞}} /K}(E ⊗ Z_{p}) ⟶ E(K) ⊗ Z_{p}) ⊂ u_{K} · Im(N_{K_{∞}/K}(E ⊗ Z_{p}) ⟶ E(K) ⊗ Z_{p}).

(2) If v runs through all primes of K above p, then

∏_{v|p} #Ẽ_{p}(k(v)) ≡ (1 − α_{p})(1 − α_{p}η_{K}(p)) (mod p).

(3) If a_{p} ≢ 1, η_{K}(p) (mod p), then p ∤ ∏_{v|p} #Ẽ_{p}(k(v)) and

y_{K} ⊗ 1 ∈ u_{K} · Im(N_{H_{p^{∞}} /K}(E ⊗ Z_{p}) ⟶ E(K) ⊗ Z_{p}) ⊂ u_{K} · Im(N_{K_{∞}/K}(E ⊗ Z_{p}) ⟶ E(K) ⊗ Z_{p}).

Proof. (1) This is a consequence of the norm relations (4.5.3) and (4.5.4).

(2) The term on the left hand side is equal to (p + 1 − a_{p})(p + 1 − η_{K}(p)a_{p}) if η_{K}(p) ≠ 0, resp. to p + 1 − a_{p} if η_{K}(p) = 0. The claim follows from the fact that a_{p} ≡ α_{p} (mod p).

(3) This is an immediate consequence of (1) and (2). (4.7) We are now ready to combine the abstract Iwasawa-theoretical results of §1-§3 with the norm relations summarised in Proposition 4.6.

(4.8) Theorem. If p ≠ 2 is a prime number such that

(a) E(K)[p] = 0,

(b) p ∤ N · a_{p} · (a_{p} − 1) · c_{Tam}(E/Q),

(c) y_{K} ∉ E(K)_{tors},

(d) rk_{Z} E(K) = 1 and � (E/K)[p^{∞}] = 0, then � (E/K_{∞})[p^{∞}] = 0 and the Pontryagin dual of E(K_{∞}) ⊗ Q_{p}/Z_{p} = Sel_{p^{∞}}(E/K_{∞}) is a free module of rank one over Z_{p}[[Gal(K_{∞}/K)]].

(4.9) Theorem. If p ≠ 2 is a prime number such that

(a) E(K)[p] = 0, (b’) p ∤ N · a_{p} · (a_{p} − 1) · (a_{p} − η_{K}(p)) · c_{Tam}(E/Q), (c’) y_{K} ∉ pE(K),

(d) rk_{Z} E(K) = 1 and � (E/K)[p^{∞}] = 0, then, for every intermediate field K ⊂ L ⊂ K_{∞}, � (E/L)[p^{∞}] = 0 and the Pontryagin dual of E(L) ⊗ Q_{p}/Z_{p} = Sel_{p^{∞}}(E/L) is a free module of rank one over Z_{p}[[Gal(L/K)]]. For every integer n ≥ 0, rk_{Z} E(K_{n}) = p^{n}, � (E/K_{n})[p^{∞}] = 0 and E(K_{n}) ⊗ Z_{p} is generated over Z_{p}[Gal(K_{n}/K)] by the traces of Heegner points of p-power conductor.

Proof. Theorem 4.8 and Theorem 4.9 follow from Theorem 3.4 and Theorem 3.5, respectively, applied to B = E, M = Q and 𝔭 = p. Indeed, the conditions (A1)_{E,K,p} and (A5)_{E,K} are immediate, (A2)_{E,K,p}, (A3)_{E,K,p} and (A4)_{E,K,p} follow from (b), (a) and (d), respectively. Finally, (A6)_{E,K_{∞}/K,p} (resp. (A7)_{E,K_{∞}/K,p}) is a consequence of (c) and Proposition 4.6(1) (resp. of (b’), (c’), (d) and Proposition 4.6(3)). (4.10) If K = Q(\sqrt{−3}) and p = 3, then the conditions (a) and (c’) in Theorem 4.9 can never be satisfied simultaneously. This is a special case of the following divisibility result, which is probably well known, but for which we have not found any reference.

(4.11) Proposition. If a prime number p divides u_{K} (i.e., if (K, p) = (Q(i), 2) or (Q(\sqrt{−3}), 3), then E(K)[p] ≠ 0 or y_{K} ∈ pE(K). In particural, if y_{K} ∉ E(K)_{tors}, then the index [E(K) : Zy_{K}] is divisible by p.

Proof. Assume that E(K)[p] = 0. According to Proposition 5.25, there are infinitely many prime numbers q ∤ pN such that p ∤ Ẽ_{q}(F_{q}) = q + 1 − a_{q}. Any such q satisfies q ≡ η_{K}(q) (mod 2u_{K}), and therefore p ∤ (η_{K}(q) + 1 − a_{q}), since p | u_{K}. The last of the norm relations in 4.4

(a_{q} − 1 − η_{K}(q))y_{K} = u_{K} Tr_{H_{q}/K}(y_{q}) ∈ u_{K} E(K) ⊂ pE(K)

then implies that y_{K} ⊗ 1 ∈ p(E(K) ⊗ Z_{(p)}), hence y_{K} ∈ pE(K). (4.12) It may be worthwhile to reformulate the phenomenon encountered in Proposition 4.11 in more abstract terms, in the general situation of 4.3. Define the group of Heegner points

E(K)_{HP} ⊂ E(K)

to be the subgroup of E(K) generated by the points

y_{K,m} := Tr_{H_{m}/K}(y_{m}),  (4.12.1)

for all integers m ≥ 1 relatively prime to N.

The norm relations in 4.4 imply, firstly, that E(K)_{HP} is generated by y_{K,1} = y_{K} and by the points y_{K,q} (where q runs through all primes not dividing N), and, secondly, that u_{K}y_{K,q} ∈ Zy_{K} for all such q. It follows that

u_{K}E(K)_{HP} ⊂ Zy_{K} ⊂ E(K)_{HP};

in particular,

E(K)_{HP} = Zy_{K} if u_{K} = 1.

Let us now consider the more interesting case u_{K} ≠ 1, when (K, u_{K}) = (Q(i), 2) or (Q(\sqrt{−3}, 3). In either case u_{K} = p is a prime dividing D_{K}, which implies that p ∤ N, and therefore E has good reduction at p. In addition, χ_{p,K} = 1 if p = 3 (and χ_{p,Q} = 1 if p = 2), where χ_{p,K} is the cyclotomic character defined in §1.1.

(4.13) Proposition. Assume that u_{K} = p > 1 and E(K)[p] ≠ 0.

(1) E has good ordinary reduction at p, \overline{ρ}_{E,p} = ( \begin{matrix}χ_{p,Q} & ∗ \\ 0 & 1 \end{matrix} ) or ( \begin{matrix}1 & ∗ \\ 0 & χ_{p,Q} \end{matrix} ) in some basis of E[p], and

a_{p} ≡ 1 (mod p).

(2) For every prime q not dividing N we have a_{q} − 1 − η_{K}(q) ≡ 0 (mod p).

(3) Zy_{K} ⊂ E(K)_{HP} ⊂ Zy_{K} + E(K)[p].

Proof. (1) The assumption E(K)[p] ≠ 0 together with χ_{p,K} = 1 imply that, in a suitable basis of E[p],

\overline{ρ}_{E,p}|_{G_{K}} = ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ). Therefore \overline{ρ}_{E,p} = ( \begin{matrix}αχ_{p,Q} & ∗ \\ 0 & α \end{matrix} ) for some character α : G_{Q} ⟶ Gal(K/Q) ⟶ \{±1\},

which rules out the case of supersingular reduction at p, by [S1, Prop. 12].

If p = 2, then α = 1 and 2 ∤ a_{2}, for trivial reasons. If p = 3, then α = 1 or α = χ_{3,Q}. In either case, the semisimplification \overline{ρ}_{E,3}^{ss} is isomorphic to 1 ⊕ χ_{3,Q}. On the other hand, (\overline{ρ}_{E,3}|_{G_{Q_{3}}} )^{ss} ≃ β ⊕ βχ_{3,Q_{3}} for an unramified character G_{Q_{3}}/I_{3} ⟶ {±1} such that a_{3} ≡ β(F r(3)) (mod 3); but β = 1 by the previous discussion.

(2) The case q = p is treated in (1). If q ∤ pN, then a_{q} = Tr(ρ_{E,p}(F r(q)) ≡ q + 1 (mod p), by (1). However, q ≡ η_{K}(q) (mod p).

(3) For each prime q ∤ N, the point y_{K,q} − ((a_{q} − 1 − η_{K}(q))/p)y_{K} lies in E(K)[p], thanks to (2) and the norm relations in 4.4. In particular, y_{K,q} ∈ Zy_{K} + E(K)[p].

(4.14) Proposition. Assume that u_{K} = p > 1 and E(K)[p] = 0.

(1) There are infinitely many primes q ∤ pN satisfying a_{q} − 1 − η_{K}(q) ≢ 0 (mod p).

(2) If q is as in (1), then y_{K} ∈ p(Zy_{K} + Zy_{K,q}) and E(K)_{HP} = Zy_{K} + Zy_{K,q} = Zz_{K}, where z_{K} ∈ E(K)_{HP} does not depend on q and satisfies pz_{K} = y_{K}.

(3) If y_{K} ∈ E(K)_{tors} is of order m, then p ∤ m and E(K)_{HP} = Zy_{K} ≃ Z/mZ.

(4) If y_{K} ∉ E(K)_{tors}, then E(K)_{HP} ≃ Z and Zy_{K} = pE(K)_{HP}.

Proof. (1) If a_{q} −1−η_{K}(q) ≡ 0 (mod p) for all but finitely many primes q ∤ pN, then #Ẽ_{q}(F_{q}) = q +1−a_{q} ≡ q − η_{K}(q) ≡ 0 (mod p) for all such q, hence E(Q(μ_{p}))[p] ≠ 0, by Proposition 5.25. This contradicts our assumption E(K)[p] = 0, since K ⊃ Q(μ_{p}).

(2) The norm relation py_{K,q} = (a_{q} − 1 − η_{K}(q))y_{K} together with p ∤ (a_{q} − 1 − η_{K}(q)) imply that y_{K} ∈ p(Zy_{K} + Zy_{K,q}). Fix a prime q′ ∤ qN; then py_{K,q′} = (a_{q′} − 1 − η_{K}(q′))y_{K}. There exists n ∈ Z such that (a_{q} − 1 − η_{K}(q))n ≡ a_{q′} − 1 − η_{K}(q′) (mod p); then p(y_{K,q′} − ny_{K,q}) ∈ Zpy_{K}, hence y_{K,q′} − ny_{K,q} ∈ Zy_{K} + E(K)[p] = Zy_{K}. Therefore E(K)_{HP} = Zy_{K} + Zy_{K,q}. Finally, there is a unique z_{K} ∈ Zy_{K} + Zy_{K,q} such that pz_{K} = y_{K}; then y_{K,q} = (a_{q} − 1 − η_{K}(q))z_{K}, hence Zy_{K} + Zy_{K,q} = Zz_{K}. (3), (4) This follows from (2) and E(K)[p] = 0. (4.15) Let us now specialise to the case K = Q(\sqrt{−3}) and p = 3. Assume, in addition, that E(K)[3] = 0. As we saw in the proofs of Propositions 4.11 and 4.14, there are infinitely many primes q ∤ 3N such that

#Ẽ_{q}(F_{q}) = q + 1 − a_{q} ≢ 0 (mod 3),

thanks to Proposition 5.25 below. The point

y_{K,q} = Tr_{H_{q}/K}(y_{q}) ∈ E(K)

satisfies

3y_{K,q} = (a_{q} − 1 − η_{K}(q))y_{K}

with a_{q} − 1 − η_{K}(q) ≡ a_{q} − 1 − q ≢ 0 (mod 3), and therefore y_{K} ∈ 3E(K).

Fix such a prime q. The discussion in §4.5-§4.9 needs to be modified as follows. Assume that 3 ∤ a_{3} and let, in the notation of (4.5.1) and (4.5.2), for every integer n ≥ 0,

z_{n,q} := α_{3}^{−n}y_{3^{n+1}q} − α_{3}^{−n−1}y_{3^{n}q} ∈ E(H_{3^{n+1}q}) ⊗ Z_{3}.

These elements are again norm compatible

∀n ≥ 1 Tr_{H_{3^{n+1}q}/H_{3^{n}q}}(z_{n,q}) = z_{n−1,q}

and the bottom element z_{0,q} = y_{3q} − α_{3}^{−1}y_{q} ∈ H_{3q} ⊗ Z_{3} satisfies

Tr_{H_{3q}/H_{q}}(z_{0,q}) = (a_{3} − σ)y_{q} − 3α_{3}^{−1}y_{q} (σ ∈ Gal(H_{q}/K)),
Tr_{H_{3q}/K}(z_{0,q}) = (a_{3} − σ − β_{3})Tr_{H_{q}/K}(y_{q}) = (α_{3} − 1)y_{K,q}.

(4.16) Proposition. Assume that K = Q(\sqrt{−3}), p = 3, E(K)[3] = 0 and 3 ∤ a_{3}. As in 4.15, fix a prime number q ∤ 3N such that 3 ∤ (a_{q} − 1 − q) and define y_{K,q} ∈ E(K) by (4.12.1).

(1) The element (α_{3} − 1)(y_{K,q} ⊗ 1) ∈ E(K) ⊗ Z_{3} is contained in

Im(N_{H_{3^{∞}q}/K}(E ⊗ Z_{3}) ⟶ E(K) ⊗ Z_{3}) ⊂ Im(N_{K_{∞}/K}(E ⊗ Z_{3}) ⟶ E(K) ⊗ Z_{3}).

(2) The only prime v_{3} = (\sqrt{−3}) of K above 3 satisfies

#Ẽ_{3}(k(v_{3})) = #Ẽ_{3}(F_{3}) ≡ 1 − α_{3} (mod 3).

(3) If a_{3} ≢ 1 (mod 3), then 3 ∤ #Ẽ_{3}(k(v_{3})) and

(y_{K,q} ⊗ 1) ∈ Im(N_{H_{3^{∞}q}/K}(E ⊗ Z_{3}) ⟶ E(K) ⊗ Z_{3}) ⊂ Im(N_{K_{∞}/K}(E ⊗ Z_{3}) ⟶ E(K) ⊗ Z_{3}).

Proof. The statements (1) and (2) follow, respectively, from the norm relations in 4.15 and from the fact that #Ẽ_{3}(F_{3}) = 3 + 1 − a_{3}. The statement (3) is a consequence of (1) and (2).

(4.17) Theorem. If K = Q(\sqrt{−3}), p = 3 and if

(a) E(K)[3] = 0, (b’) 3 ∤ a_{3} · (a_{3} − 1) · c_{Tam}(E/Q), (c’) y_{K,q} ∉ 3E(K) (for a fixed prime q ∤ 3N satisfying 3 ∤ (a_{q} − 1 − q)),

(d) rk_{Z} E(K) = 1 and � (E/K)[3^{∞}] = 0, then, for every intermediate field K ⊂ L ⊂ K_{∞}, � (E/L)[3^{∞}] = 0 and the Pontryagin dual of E(L) ⊗ Q_{3}/Z_{3} = Sel_{3^{∞}}(E/L) is a free module of rank one over Z_{3}[[Gal(L/K)]]. For every integer n ≥ 0, rk_{Z} E(K_{n}) = 3^{n}, � (E/K_{n})[3^{∞}] = 0 and E(K_{n}) ⊗ Z_{3} is generated over Z_{3}[Gal(K_{n}/K)] by the traces to K_{n} of the Heegner points of conductors dividing 3^{∞}q.

Proof. The proof of Theorem 4.9 applies, except that we use Proposition 4.16 instead of Proposition 4.6.

5. Vanishing of certain Galois cohomology groups (after [Ch] and [LW])

(5.1) One of the ingredients of Kolyvagin’s method for obtaining upper bounds on the size of Selmer groups Sel_{p^{n}}(E/K) ⊂ H^{1}(K, E[p^{n}]) in the situation of 4.3 is a passage to the extension L_{n} := K(E[p^{n}]) of K over which the Galois action on E[p^{n}] becomes trivial. The inflation-restriction sequence

0 ⟶ H^{1}(L_{n}/K, E[p^{n}]) ⟶ H^{1}(K, E[p^{n}]) ⟶ H^{1}(L_{n}, E[p^{n}])^{Gal(L_{n}/K)} ⟶ H^{2}(L_{n}/K, E[p^{n}])

implies that such a passage entails no loss of information, provided that H^{1}(L_{n}/K, E[p^{n}]) = 0. Sufficient criteria for the vanishing of H^{i}(L_{n}/K, E[p^{n}]) were given in [Ch, Thm. 2] (for i = 1); a complete answer in the case K = Q was obtained in [LW, Thm. 1, Thm. 2] (for i = 1, 2). These questions were also considered, from a slightly different point of view, in [CS1, §5] and [CS2, §3].

In §5.2-§5.21 we recall the approach adopted in [Ch] and [LW], first in an abstract setting, then for 𝔭-power torsion in an abelian variety B of GL(2)-type with real multiplication (which includes the case of elliptic curves). Unlike [Ch] and [LW], we are only interested in the “easy case” when B[𝔭] is an irreducible Galois module. (5.2) Assume that we are given the following data:

• a prime number p,

• a finite extension K/Q_{p}, with ring of integers O, uniformiser π and residue field k = O/π,

• a free O-module T of finite rank r ≥ 1; set \overline{T} := T/π,

• a closed subgroup G ⊂ Aut_{O}(T) ≃ GL_{r}(O).

The π-adic filtration on T induces a filtration G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ ⋯ by open normal subgroups

G_{n} := Ker(G ↪ Aut_{O}(T) ⟶ Aut_{O}(T/π^{n})),

which have the following properties:

• G_{0}/G_{1} ↪ Aut_{k}(\overline{T}) ≃ GL_{r}(k),

• ∀n ≥ m ≥ 1 G_{n}/G_{m+n} ↪ End_{O}(π^{n}T/π^{m+n}) ≃ M_{r}(O/π^{m}) (1 + π^{n}A ↦ A (mod π^{m})),

• ∀m, n ≥ 1 [G_{m}, G_{n}] ⊂ G_{m+n}, which implies that the adjoint action of g ∈ G/G_{m+n} on G_{n}/G_{m+n}

(given by ad(g)h := ghg^{−1}) factors through G/G_{m}. (5.3) We are interested in establishing sufficient criteria for the vanishing of the cohomology groups H^{1}(G/G_{n}, T/π^{m}) (where n ≥ m ≥ 1). Firstly, dévissage implies that

if H^{1}(G/G_{n}, \overline{T}) = 0, then ∀m ∈ \{1, …, n\} H^{1}(G/G_{n}, T/π^{m}) = 0.  (5.3.1)

Secondly, the inflation-restriction sequence for G_{n}/G_{n+1} ◁ G/G_{n+1} (where n ≥ 1)

0 ⟶ H^{1}(G/G_{n}, \overline{T}) ⟶ H^{1}(G/G_{n+1}, \overline{T}) ⟶ H^{1}(G_{n}/G_{n+1}, \overline{T})^{G/G_{n}}  (5.3.2)

has the following properties: G_{n}/G_{n+1} ↪ End_{k}(\overline{T}) acts trivially on \overline{T}, the action of G/G_{n} on \overline{T} factors through G/G_{1} ↪ Aut_{k}(\overline{T}), and so does the adjoint action of G/G_{n} on End_{k}(\overline{T}) and its F_{p}[G/G_{n}]-submodule G_{n}/G_{n+1}. As a result,

H^{1}(G_{n}/G_{n+1}, \overline{T})^{G/G_{n}} = Hom_{F_{p}}(G_{n}/G_{n+1}, \overline{T})^{G/G_{1}}.  (5.3.3)

Putting together (5.3.1)–(5.3.3), we obtain the following statement.

(5.4) Proposition. Assume that n ≥ 1 and that ∀n′ ∈ \{1, …, n − 1\} Hom_{F_{p}}(G_{n′}/G_{n′+1}, \overline{T})^{G/G_{1}} = 0. If H^{1}(G/G_{1}, \overline{T}) = 0, then ∀m ∈ \{1, …, n\} H^{1}(G/G_{n}, T/π^{m}) = 0.

(5.5) It will be convenient to investigate the conditions in Proposition 5.4 in the following axiomatic setting. Throughout §5.5-§5.18,

• p is a prime number,

• k is a finite extension of F_{p}, of degree f = [k : F_{p}],

• V is a finite-dimensional k-vector space, of dimension r ≥ 1,

• H ⊂ GL(V) ≃ GL_{r}(k) is a subgroup,

• W ⊂ End_{k}(V) ≃ M_{r}(k) is an F_{p}[H]-submodule (with respect to the adjoint action of H).

• Denote by P H the image of H under the projection GL(V) ⟶ P GL(V).

In order to verify the assumptions of Proposition 5.4, we must be able to answer the following two questions (for V = \overline{T}, H = G/G_{1} and W = G_{n}/G_{n+1}, where n ≥ 1). Question (Q1): When is H^{1}(H, V) = 0? Question (Q2): When is Hom_{F_{p}}(W, V)^{H} = 0?

There is an extensive literature devoted to (Q1); see [Gu, Thm. A] for fairly general results (valid when k is an arbitrary field of characteristic p and H is a finite subgroup of GL(V)).

As noted in [Ch], [LW] and [CS1, CS2], one can often deduce the vanishing statements in (Q1) and (Q2) by applying the following elementary observations.

(5.5.1) If p ∤ #H, then ∀i > 0 H^{i}(H, V) = 0. (5.5.2) (Sah’s Lemma [Sa, Prop. 2.7(b)]) If M is a k[H]-module for which there exists a central element

z ∈ Z(H) acting on M by a scalar λ ∈ k^{×} ∖ {1}, then ∀i ≥ 0 H^{i}(H, M) = 0. (5.6) Following [Ch], [LW] and [CS1, CS2], we say that H contains a non-trivial homothety if H ∩ Z(GL(V)) = H ∩ k^{×} · id_{V} ≠ {1} (or, which is equivalent, that the projection H ⟶ P H is not an isomorphism).

If H contains a non-trivial homothety, Sah’s Lemma implies that

∀i ≥ 0 H^{i}(H, V) = 0 = H^{i}(H, Hom_{F_{p}}(W, V)).

In particular, the vanishing property in both questions (Q1) and (Q2) always holds.

(5.7) Proposition. Assume that at least one of the following two conditions is satisfied.

(a) p ∤ #H;

(b) V = ⨁ V_{i} is a direct sum of simple k[H]-modules of dimensions dim_{k}(V_{i}) ≤ (p + 1)/2. Then:

(1) End_{k}(V) is a semisimple k[H]-module.

(2) End_{k}(V) is a semisimple F_{p}[H]-module.

(3) Every F_{p}[H]-submodule W ⊂ End_{k}(V) is a direct summand.

(4) If Hom_{F_{p}}(End_{k}(V), V)^{H} = 0, then Hom_{F_{p}}(W, V)^{H} = 0, for every F_{p}[H]-submodule W ⊂ End_{k}(V).

Proof. The implications (2) ⟹ (3) ⟹ (4) and (a) ⟹ (1),(2) are automatic, and (2) follows from (1), since the Jacobson radical of F_{p}[H] is contained in the Jacobson radical of k[H] ([FD, ch. 2, ex. 6, 50, 53(c)]). If V = ⨁ V_{i} is as in (b), so is its dual V^{∗} = ⨁ V_{i}^{∗}. Semisimplicity of the k[H]-module End_{k}(V) = ⨁_{i,j} V_{i}^{∗} ⊗V_{j} then follows from [S2, Cor. 1]. (5.8) In view of Proposition 5.7, it is natural to investigate (Q2) for W = End_{k}(V). In this case there is a non-degenerate F_{p}-bilinear symmetric pairing

( , ) : W × W ⟶ F_{p}, (A, B) := Tr_{k/F_{p}}(Tr(AB)),

which is invariant under the adjoint action of GL(V) and satisfies (λA, B) = (A, λB), for all λ ∈ k. It induces, therefore, an isomorphism of (k ⊗_{F_{p}} k)[H]-modules

W ⊗_{F_{p}} V ⟶^{∼} Hom_{F_{p}}(W, V).

One can rewrite the tensor product on the left hand side in terms of the Galois group

Δ := Gal(k/F_{p}) = {φ^{i} | i ∈ Z/fZ}, φ(a) = a^{p},

as follows. The ring isomorphism

k ⊗_{F_{p}} k ⟶^{∼} ∏_{σ∈Δ} k, a ⊗ b ↦ (σ ↦ aσ(b))

induces an isomorphism of (k ⊗_{F_{p}} k)[H]-modules

W ⊗_{F_{p}} V ⟶^{∼} ⨁_{σ∈Δ} W ⊗_{k} V^{(σ)}, V^{(σ)} := V ⊗_{k,σ} k.

In concrete terms, if we fix a basis of V over k, the (faithful) action ρ : H ↪ GL(V) ≃ GL_{r}(k) of H on V gives rise to a twisted action ρ^{(σ)} : H ↪ GL(V^{(σ)}) ≃ GL_{r}(k) given by ρ^{(σ)} = σ ∘ ρ.

Using this language, the k[H]-module W = End_{k}(V) corresponds to the adjoint action ad(ρ) = Hom_{k}(ρ, ρ) = ρ^{∗} ⊗_{k} ρ : H ⟶ GL(W) ≃ GL_{r^{2}}(k), and

Hom_{F_{p}}(End_{k}(V), V) ⟶^{∼} ⨁_{σ∈Δ} (ad(ρ) ⊗_{k} ρ^{(σ)}) .

If p ∤ dim_{k}(V), then there is a decomposition ad(ρ) = ad^{∘}(ρ) ⊕ k, where ad^{∘}(ρ) = End_{k}^{∘}(V) := End_{k}(V)^{Tr=0} and the trivial representation corresponds to the scalar endomorphisms k · id_{V}. Therefore

Hom_{F_{p}}(End_{k}(V), V) ⟶^{∼} (⨁_{σ∈Δ} ρ^{(σ)}) ⊕ ⨁_{σ∈Δ} (ad^{∘}(ρ) ⊗_{k} ρ^{(σ)})

if p ∤ dim_{k}(V). The previous discussion can be summed up as follows.

(5.9) Proposition. If ρ : H ↪ GL(V) denotes the (faithful) action of H on V, then the condition Hom_{F_{p}}(End_{k}(V), V)^{H} = 0 is equivalent to ∀σ ∈ Δ (ad(ρ) ⊗_{k} ρ^{(σ)})^{H} = 0. If p ∤ dim_{k}(V), the latter condition is equivalent to the conjunction of ρ^{H} = 0 and ∀σ ∈ Δ (ad^{∘}(ρ) ⊗_{k} ρ^{(σ)})^{H} = 0.

(5.10) A split dihedral example. Assume that p ≠ 2 and that n > 1 is an odd integer dividing #k^{×} = p^{f} − 1. Denote by D_{2n} the dihedral group of order 2n and by C_{n} ◁ D_{2n} its unique cyclic subgroup of order n. Fix an element s ∈ D_{2n} ∖ C_{n}; then s^{2} = 1 and sgs^{−1} = g^{−1}, for all g ∈ C_{n}.

For any character ψ : C_{n} ⟶ k^{×}, the induced representation

I(ψ) := Ind_{C_{n}}^{D_{2n}}(ψ) : D_{2n} ⟶ GL(V) ≃ GL_{2}(k)

has the following properties.

• In a suitable basis, I(ψ)|_{C_{n}} = ( \begin{matrix}ψ & 0 \\ 0 & ψ^{−1} \end{matrix} ), I(ψ)(s) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ).

• The image of I(ψ) is contained in the normaliser N(C) of a split Cartan subgroup C ⊂ GL(V).

• det(I(ψ)) = {±1} ⊂ k^{×}, det(I(ψ)|_{C_{n}}) = {1}.

• I(ψ) ≃ I(ψ) ⊗ sgn ≃ I(ψ^{−1}) ≃ I(ψ)^{∗}, where sgn : D_{2n} ⟶ D_{2n}/C_{n} ⟶^{∼} {±1}.

• I(ψ) is irreducible if and only if ψ ≠ 1.

• I(1) ≃ 1 ⊕ sgn.

• I(ψ_{1}) ⊗ I(ψ_{2}) ≃ I(ψ_{1}ψ_{2}) ⊕ I(ψ_{1}ψ_{2}^{−1}).

• ad(I(ψ)) = I(ψ)^{∗} ⊗ I(ψ) ≃ I(ψ) ⊗ I(ψ), ad^{∘}(I(ψ)) ≃ I(ψ^{2}) ⊕ sgn.

• ∀i ∈ Z/fZ I(ψ)^{(φ^{i})} = I(ψ^{p^{i}}).

• ad^{∘}(I(ψ)) ⊗ I(ψ)^{(φ^{i})} ≃ I(ψ^{p^{i}}) ⊕ I(ψ^{p^{i}+2}) ⊕ I(ψ^{p^{i}−2}).

• dim_{k} I(ψ)^{C_{n}} = 2 dim_{k} I(ψ)^{D_{2n}} is equal to 2 (resp. to 0) if ψ = 1 (resp. if ψ ≠ 1).

(5.11) A nonsplit dihedral example. Let k_{2} ≃ F_{p^{2f}} be a quadratic extension of k. Assume that p ≠ 2 and that n > 1 is an odd integer dividing #(k_{2}^{×}/k^{×}) = p^{f} + 1.

For any character ψ′ : C_{n} ⟶ Ker(N_{k_{2}/k} : k_{2}^{×} ⟶ k^{×}) ⊂ k_{2}^{×} we define

J(ψ′) : D_{2n} ⟶ GL(V) ≃ GL_{2}(k)

as follows. Let V = k_{2}; the regular representation j : k_{2} = End_{k_{2}}(V) ⊂ End_{k}(V) identifies k_{2}^{×} with a nonsplit Cartan subgroup C = j(k_{2}^{×}) ⊂ GL(V) and Ker(N_{k_{2}/k} : k_{2}^{×} ⟶ k^{×}) with C ∩ SL(V). We let

J(ψ′)|_{C_{n}} := j ∘ ψ′, J(ψ′)(s) = s′,

for any element s′ ∈ N(C) of the normaliser of C with eigenvalues ±1. Explicitly, fix α ∈ k_{2}^{×} such that d := α^{2} ∈ k^{×} and write j in terms of the basis 1, α of k_{2} over k:

j(a + bα) = ( \begin{matrix}a & bd \\ b & a \end{matrix} ) , (a, b ∈ k).

We can then take s′ = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). The representation J(ψ′) has the following properties.

• J(τ ∘ ψ′) ≃ J(ψ′), for any τ ∈ Gal(k_{2}/k).

• Up to isomorphism, J(ψ′) does not depend on any choices.

• The image of J(ψ′) is contained in the normaliser N(C) of a nonsplit Cartan subgroup C ⊂ GL(V).

• det(J(ψ′)) = {±1} ⊂ k^{×}, det(J(ψ′)|_{C_{n}}) = {1}.

• J(ψ′) ⊗_{k} k_{2} ≃ I(ψ′) (where we consider ψ′ on the right hand side as a character ψ′ : C_{n} ⟶ k_{2}^{×}, and I(ψ′) : D_{2n} ⟶ GL_{2}(k_{2})).

(5.12) Proposition. Assume that p ≠ 2 and that n > 1 is an odd integer.

(1) If n | (p^{f} − 1) and if the character ψ : C_{n} ⟶ k^{×} in 5.10 is injective, then the following properties of the representation ρ := I(ψ) : D_{2n} ↪ GL(V) ≃ GL_{2}(k) are equivalent.

Hom_{F_{p}}(End_{k}(V), V)^{C_{n}} ≠ 0 ⟺ Hom_{F_{p}}(End_{k}(V), V)^{D_{2n}} ≠ 0 ⟺

∃ε ∈ {±1} ∃i ∈ Z/fZ p^{i} ≡ 2ε (mod n) ⟺ ∃ε ∈ {±1} ∃i ∈ Z/fZ p^{i} ≡ 2ε (mod n) and n | ((2ε)^{f} − 1).

(2) If n | (p^{f} +1) and if the character ψ′ : C_{n} ⟶ Ker(N_{k_{2}/k}) in 5.11 is injective, then the following properties of the representation ρ := J(ψ′) : D_{2n} ↪ GL(V) ≃ GL_{2}(k) are equivalent.

Hom_{F_{p}}(End_{k}(V), V)^{C_{n}} ≠ 0 ⟺ Hom_{F_{p}}(End_{k}(V), V)^{D_{2n}} ≠ 0 ⟺
∃i ∈ Z/2fZ p^{i} ≡ 2 (mod n) ⟺ ∃i ∈ Z/2fZ p^{i} ≡ 2 (mod n) and n | (2^{f} − (−1)^{i}).

Proof. The first two equivalences in (1) follow from Proposition 5.9 combined with the discussion in 5.10; the third one from the fact that the congruences p^{i} ≡ 2ε (mod n) and p^{f} ≡ 1 (mod n) imply (2ε)^{f} ≡ 1 (mod n). The statement (2) follows from the isomorphism J(ψ′) ⊗_{k} k_{2} ≃ I(ψ′) combined with (1) for the pair (ψ′, k_{2}). (5.13) Definition Given a finite field k ≃ F_{p^{f}} of characteristic p ≠ 2, we say that an odd integer n > 1 is k-exceptional if either n | (p^{f} − 1) and the equivalent conditions in Proposition 5.12(1) are satisfied, or n | (p^{f} + 1) and the equivalent conditions in Proposition 5.12(2) are satisfied. Such a k-exceptional integer must divide 2^{f} − 1 or 2^{f} + 1. Examples. (1) If k = F_{p}, then n is k-exceptional if and only if n = 3 and p ≠ 3.

(2) If k = F_{p^{2}}, then n is k-exceptional if and only if n ∈ {3, 5} and p ≡ ±2 (mod n).

(3) If k = F_{p^{3}}, then n is k-exceptional if and only if n ∈ {3, 7, 9} and p ≡ ±2, ±4 (mod n). (5.14) From now on, we focus our attention on the case dim_{k}(V) = 2. Recall Dickson’s classification of subgroups H ⊂ GL(V) ≃ GL_{2}(k) [Di, §260].

• If p | #H, then either H acts reducibly on V, or H contains SL(V′), for some F_{p}-vector subspace V′ ⊂ V such that V′ ⊗_{F_{p}} k = V.

• If p ∤ #H, then either H is contained in the normaliser N(C) of a Cartan subgroup C ⊂ GL(V) (which implies that P H ⊂ P GL(V) is cyclic or dihedral), or P H is isomorphic to A_{4}, S_{4} or A_{5}.

The following Proposition gives a complete list of subgroups H ⊂ GL_{2}(k) (for p ≠ 2) acting irreducibly on k^{2} and not containing a non-trivial homothety (cf. [Ch, Thm. 8], [LW, Lemma 4]).

(5.15) Proposition. Assume that dim_{k}(V) = 2 ≠ p. If H ⊂ GL(V) acts irreducibly on V and does not contain a non-trivial homothety, then:

(1) There exists a Cartan subgroup C ⊂ GL(V) such that H ⊂ N(C); in particular, p ∤ #H.

(2) The subgroup H ∩ C is contained in C ∩ SL(V); it is cyclic of order n > 2, where 2 ∤ n and n divides #C/#k^{×} = #k ∓ 1.

(3) If H ⊄ C (which is automatic if C is split), then H is isomorphic to the dihedral group D_{2n} of order 2n, and det(H) = {±1} ⊂ k^{×}.

[In concrete terms, H is isomorphic either to D_{2n} or to C_{n}, and its action on V is given by I(ψ) (if H ≃ D_{2n}) or J(ψ′) (if H ≃ D_{2n} or C_{n}), for an injective character ψ resp. ψ′.]

(4) Conversely, if H ⊂ GL(V) is a subgroup satisfying (1)–(2) for some Cartan subgroup C ⊂ GL(V) and if H ⊄ C if C is split, then H acts irreducibly on V and does not contain a non-trivial homothety.

(5) If k = F_{3}, then no such H exists.

Proof. The irreducibility assumption together with the absence of non-trivial homothety in H imply, by Dickson’s classification, that p ∤ #H and that H ≃ P H is cyclic, dihedral or isomorphic to A_{4}, S_{4} or A_{5}. However, the representation theory of H over \overline{F}_{p} is the same as over C, since p ∤ #H. The groups A_{4}, S_{4}, A_{5} do not admit a faithful representation into GL_{2}(C), therefore there is no such a representation into GL_{2}(\overline{F}_{p}), which leaves us only with the cases H ≃ P H ≃ C_{n} or D_{2n}, for some integer n ≥ 1. In particular, H ⊂ N(C) for some Cartan subgroup C ⊂ GL(V) and H ∩ k^{×} · id_{V} = {id_{V} }, which implies that H ∩ C ≃ P (H ∩ C) ⊂ C/k^{×} · id_{V} is cyclic of order n > 2 (by irreducibility), where n | #(C/k^{×} · id_{V}).

If C ≃ k_{2}^{×} is nonsplit, then n | (p^{f} + 1) and, for each a ∈ H ∩ C, det(a) = N_{k_{2}/k}(a) = a^{p^{f} +1} = 1.

If C is split, then n | (p^{f} − 1) and H ⊄ C. For fixed s ∈ H ∖ (H ∩ C) and any a ∈ H ∩ C, (as)^{2} = a(sas^{−1}) = det(a) id_{V} ∈ H ∩ C ∩ k^{×} · id_{V} = {id_{V} }, hence det(a) = 1.

In either case, the cyclic group H ∩ C is contained in C ∩ SL(V). Its order n > 1 is odd, since the only element of order two in C ∩ SL(V) is −id_{V} ∉ H.

The above discussion implies that the pair (H, ρ : H ↪ GL(V)) is of the form (C_{n}, J(ψ′)|_{C_{n}}), (D_{2n}, I(ψ)) or (D_{2n}, J(ψ′)), where I(ψ) (resp. J(ψ′)) is as in 5.10 (resp. 5.11), with ψ (resp. ψ′) injective. In each of these three cases H acts irreducibly on V and does not contain a non-trivial homothety. This proves parts (1)–(4) of the proposition. Finally, (5) follows from the fact that there is no odd n > 2 dividing 3 ± 1.

(5.16) Theorem ([CS1, Thm. 9] if k = F_{p}). Assume that dim_{k}(V) = 2 ≠ p and that H acts semisimply on V.

(1) ∀i > 0 H^{i}(H, V) = 0.

(2) If H acts irreducibly on V, then the following conditions are equivalent.

(a) For every F_{p}-submodule W ⊂ End_{k}(V), Hom_{F_{p}}(W, V)^{H} = 0.

(b) Hom_{F_{p}}(End_{k}^{∘}(V), V)^{H} = 0.

(c) The pair (H, ρ : H ↪ GL(V)) is not of the form

(C_{n}, J(ψ′)|_{C_{n}}),  (D_{2n}, I(ψ)),  (D_{2n}, J(ψ′)),

for any k-exceptional n > 1.

Proof. (1) It is enough to assume that p | #H, which rules out the reducible semisimple case, when H is contained in a split Cartan subgroup. By Dickson’s classification, H contains SL(V′), which in turn contains the homothety −1 ∈ k^{×} ∖ {1}; we conclude by Sah’s Lemma.

(2) If H contains a non-trivial homothety, (a), (b) and (c) are satisfied. If H does not contain a non-trivial homothety, then p ∤ #H, by Proposition 5.15(1). The irreducibility assumption implies that (V^{(σ)})^{H} = 0, for all σ ∈ Gal(k/F_{p}). Therefore (b) is equivalent to the same statement with End_{k}^{∘}(V) replaced by End_{k}(V). The equivalence (a) ⟺ (b) then follows from the case (a) of Proposition 5.7, and the equivalence (b) ⟺ (c) from Proposition 5.12 combined with Proposition 5.15.

(5.17) Theorem. In the situation of 5.2, assume that p ≠ 2 = r, that the group G_{0}/G_{1} acts irreducibly on the k-vector space \overline{T}, and that G_{0}/G_{1} is not isomorphic to C_{n} or D_{2n}, for any k-exceptional odd integer n > 1. Then

∀m_{1} ≥ m_{2} ≥ 1 H^{1}(G/G_{m_{1}}, T/π^{m_{2}}T) = 0.

Proof. Combine Theorem 5.16 with Proposition 5.4.

(5.18) Corollary. Assume that p ≠ 2 = r and that G_{0}/G_{1} acts irreducibly on the k-vector space \overline{T}. If at least one of the conditions (a)–(g) below holds, then

∀m_{1} ≥ m_{2} ≥ 1 H^{1}(G/G_{m_{1}}, T/π^{m_{2}}T) = 0.

(a) det(G_{0}/G_{1}) ⊄ {±1} ⊂ k^{×}.

(b) det(G_{0}/G_{1}) = {±1} ⊂ k^{×} and G_{0}/G_{1} is not isomorphic to D_{2n}, for any k-exceptional odd integer n > 1.

(c) det(G_{0}/G_{1}) = {1} ⊂ k^{×} and G_{0}/G_{1} is not isomorphic to C_{n}, for any k-exceptional odd integer n > 1.

(d) det(G_{0}/G_{1}) = F_{p}^{×} and p > 3.

(e) k = F_{p} and p = 3.

(f) k = F_{p}, p > 3 and G_{0}/G_{1} ≄ A_{3}, S_{3}.

(g) k = F_{p} and det(G_{0}/G_{1}) = F_{p}^{×}.

(5.19) Consider the following geometric situation:

• p ≠ 2 is a prime number,

• K is a field of characteristic different from p,

• M is a totally real number field,

• 𝔭 | p is a prime of M above p; let K := M_{𝔭}, O := O_{K}, k := O/𝔭;

• B is an abelian variety over K of dimension dim(B) = [M : Q], equipped with a ring morphism i : O_{M} ⟶ End(B) and a symmetric O_{M}-linear isogeny λ = λ^{t} : B ⟶ B^{t}; let T := T_{𝔭}(B) be as in (1.2).

In this case T is a free O-module of rank r = 2. Denote by G ⊂ Aut_{O}(T) ≃ GL_{2}(O) the image of the Galois representation ρ_{B,𝔭} : G_{K} ⟶ Aut_{O}(T). In the notation of 5.2, we have T/𝔭^{n} = B[𝔭^{n}] (in particular, \overline{T} = B[𝔭]), G/G_{n} = Gal(K(B[𝔭^{n}])/K) ⊂ Aut_{O}(T/𝔭^{n}) ≃ GL_{2}(O/𝔭^{n}) and G_{0}/G_{1} is the image of the residual Galois representation \overline{ρ}_{B,𝔭} : G_{K} ⟶ Aut_{k}(B[𝔭]) ≃ GL_{2}(k). The Weil pairing attached to λ implies that det(ρ_{B,𝔭}) : G_{K} ⟶ O^{×} is given by the p-adic cyclotomic character, hence det(\overline{ρ}_{B,𝔭}) = χ_{p,K} : G_{K} ⟶ F_{p}^{×} ⊂ k^{×} is the (mod p) cyclotomic character. Applying Theorem 5.17 and Corollary 5.18 in this situation, we obtain the following results.

(5.20) Theorem. In the situation of 5.19, assume that B[𝔭] is an irreducible k[G_{K}]-module, and that \overline{ρ}_{B,𝔭}(G_{K}) ⊂ GL_{2}(k) is not isomorphic to C_{n} or D_{2n}, for any k-exceptional odd integer n > 1. Then

∀m_{1} ≥ m_{2} ≥ 1 H^{1}(K(B[𝔭^{m_{1}}])/K, B[𝔭^{m_{2}}]) = 0.

(5.21) Corollary. Assume that B[𝔭] is an irreducible k[G_{K}]-module. If at least one of the conditions (a)–(g’) below holds, then

∀m_{1} ≥ m_{2} ≥ 1 H^{1}(K(B[𝔭^{m_{1}}])/K, B[𝔭^{m_{2}}]) = 0.

(a) χ_{p,K}(G_{K}) ⊄ {±1} ⊂ F_{p}^{×}. (a’) K ⊃ Q and Q(μ_{p})^{+} ⊄ K. (a”) K is an imaginary quadratic field and p > 3.

(b) χ_{p,K}(G_{K}) = \{±1\} ⊂ F_{p}^{×} and \overline{ρ}_{B,𝔭}(G_{K}) is not isomorphic to D_{2n}, for any k-exceptional odd integer n > 1. (b’) Q(μ_{p})^{+} ⊂ K, Q(μ_{p}) ⊄ K and \overline{ρ}_{B,𝔭}(G_{K}) is not isomorphic to D_{2n}, for any k-exceptional odd integer n > 1.

(c) χ_{p,K}(G_{K}) = \{1\} ⊂ F_{p}^{×} and \overline{ρ}_{B,𝔭}(G_{K}) is not isomorphic to C_{n}, for any k-exceptional odd integer n > 1. (c’) Q(μ_{p}) ⊂ K and \overline{ρ}_{B,𝔭}(G_{K}) is not isomorphic to C_{n}, for any k-exceptional odd integer n > 1.

(d) χ_{p,K}(G_{K}) = F_{p}^{×} and p > 3. (d’) K ⊃ Q, K ∩ Q(μ_{p}) = Q and p > 3.

(e) k = F_{p} and p = 3. (e’) K is an imaginary quadratic field and k = F_{p}.

(f) k = F_{p}, p > 3 and \overline{ρ}_{B,𝔭}(G_{K}) ≄ A_{3}, S_{3}.

(g) k = F_{p} and χ_{p,K}(G_{K}) = F_{p}^{×}. (g’) K ⊃ Q, K ∩ Q(μ_{p}) = Q and k = F_{p}.

(5.22) If M = Q, then 𝔭 = p, K = Q_{p}, O = Z_{p} and B = E is an elliptic curve. In this case much more precise results were proved in [Ch, Thm. 2] and [LW, Thm. 11], under suitable assumptions on K.

We now prove several auxiliary results that will be needed in §6 (Proposition 5.25 was already used in the proofs of Propositions 4.11 and 4.14).

(5.23) Proposition. Let V be a two-dimensional vector space over a field k.

(1) If G ⊂ GL(V) is a subgroup satisfying ∀g ∈ G det(1 − g | V) = 0, then there exists a one-dimensional subspace W ⊂ V such that W^{G} ≠ 0 or (V/W)^{G} ≠ 0. Equivalently, there exists a basis of V in which

G ⊂ H_{1} := ( \begin{matrix}1 & ∗ \\ 0 & ∗ \end{matrix} ) or G ⊂ H_{2} := ( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} ).

(2) If G ⊂ GL(V) is a subgroup satisfying ∀g ∈ G Tr(g − 1 | V) = 0 and if the characteristic of k is not equal to 2, then there exists a one-dimensional subspace W ⊂ V such that W^{G} = W and (V/W)^{G} = V/W.

Equivalently, there exists a basis of V in which G ⊂ ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ).

Proof. (1) The eigenvalues of any g ∈ G are equal to 1 and det(g). In particular, if g ∈ G ∩ SL(V), then g is unipotent and Tr(g) = 2, det(g) = 1.

If #(G ∩ SL(V)) > 1, then there exists a basis of V in which g_{0} := ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ∈ G. If g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈

G ∩ SL(V), then ad − bc = 1 and a + d = Tr(g) = 2 = Tr(g_{0}g) = a + c + d, which implies that c = 0, and both eigenvalues of g are equal to a = d = 1; thus G ∩ SL(V) ⊂ H_{1} ∩ H_{2} and G ⊂ {g ∈ GL(V) | gg_{0}g^{−1} ⊂

H_{1} ∩ H_{2}\} = ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ). This means that G is contained in the union of the subgroups H_{1} and H_{2} of GL(V),

and therefore is contained in one of them.

If the group G ∩ SL(V) is trivial, then det : G ⟶^{∼} det(G) ⊂ k^{×} is an isomorphism and G is abelian. As a result, for each g ∈ G ∖ {id_{V} }, the direct sum decomposition V = V^{g=1} ⊕ V^{g=det(g)} is G-stable, hence

G ⊂ H′_{1} ∪ H′_{2}, where H′_{1} := ( \begin{matrix}1 & 0 \\ 0 & ∗ \end{matrix} ) and H′_{2} := ( \begin{matrix}∗ & 0 \\ 0 & 1 \end{matrix} ). Again, this implies that G ⊂ H′_{1} or G ⊂ H′_{2}.

(2) For each g ∈ G we have 2 det(1 − g | V) = Tr(g − 1)Tr(g) − Tr(g^{2} − 1) = 0. Part (1) then implies that there exists i ∈ {1, 2} such that G ⊂ H_{i}^{Tr=2} = H_{1} ∩ H_{2}.

(5.24) Corollary. Assume that, in the situation of 5.2, r = 2 and Y ⊂ G is a subset that maps surjectively on G_{0}/G_{1}.

(1) If ∀g ∈ Y det(1 − g | T) ≡ 0 (mod π), then there is a basis of \overline{T} in which G_{0}/G_{1} ⊂ ( \begin{matrix}1 & ∗ \\ 0 & ∗ \end{matrix} ) or

G_{0}/G_{1} ⊂ ( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} ).

(2) If p ≠ 2 and ∀g ∈ Y Tr(g − 1 | T) ≡ 0 (mod π), then there is a basis of \overline{T} in which G_{0}/G_{1} ⊂ ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ).

(5.25) Proposition. If, in the situation of 5.19, K is a number field and there exists a finite set S of finite primes of K (containing all primes above p and all primes at which B has bad reduction) such that

∀v ∉ S #B̃_{v}(k(v)) ≡ 0 (mod p),

then \overline{ρ}_{B,𝔭} is isomorphic to ( \begin{matrix}1 & ∗ \\ 0 & χ_{p,K} \end{matrix} ) or ( \begin{matrix}χ_{p,K} & ∗ \\ 0 & 1 \end{matrix} ). In particular, B(K(μ_{p}))[𝔭] ≠ 0.

Proof. For each v ∉ S,

det_{O}(1 − F r(v) | T_{𝔭}(B)) = #B̃_{v}(k(v)) ≡ 0 (mod 𝔭).

The statement of the proposition follows from Corollary 5.24(1) applied to T = T_{𝔭}(B), G = Im(G_{K} ⟶ Aut_{O}(T)) and Y = {F r(v) | v ∉ S} (which maps surjectively on G_{0}/G_{1} = Im(G_{K} ⟶ Aut_{k}(B[𝔭])), by the Čebotarev density theorem for K(B[𝔭])/K).

(5.26) Proposition. Let E be an elliptic curve over Q of conductor N and K a quadratic field of discriminant D_{K} relatively prime to N. Let ρ := \overline{ρ}_{E,p} : G_{Q} ⟶ Aut_{F_{p}}(E[p]) ≃ GL_{2}(F_{p}), for a prime number p ≠ 2.

(1) The field L := Q(E[p]) has the following property:

ρ(G_{Q}) ≠ ρ(G_{K}) ⟺ L ∩ K = K ⟺ D_{K} = p^{∗} := (−1)^{(p−1)/2}p.

(2) If ρ is irreducible, so is ρ|_{G_{K}}.

(3) If ρ|_{G_{K}} is irreducible, but not absolutely irreducible, then p = 3, K = Q(\sqrt{−3}), E has good ordinary reduction at 3, ρ(G_{K}) is a cyclic group of order 4 and ρ(G_{Q}) is a dihedral group of order 8.

Proof. (1) We have ρ(G_{Q}) ⟶^{∼} Gal(L/Q) and Gal(L/L ∩ K) ≃ Gal(KL/K) ⟶^{∼} ρ(G_{K}), which yields the first equivalence in (1). A prime number ℓ is unramified in K/Q if and only if ℓ ∤ D_{K}; it is unramified in L/Q if ℓ ∤ pN. As (N, D_{K}) = 1, the equality L ∩ K = K implies that {ℓ | D_{K}} ⊂ {ℓ | D_{K}} ∩ {ℓ | pN} ⊂ {p}, hence D_{K} = p^{∗}. Conversely, Q(\sqrt{p^{∗}}) ⊂ Q(μ_{p}) ⊂ L.

(2) If ρ is irreducible but ρ|_{G_{K}} is not, then ρ|_{G_{K}} is semisimple (since G_{K} is a normal subgroup of G_{Q}) and its image is contained in a split Cartan subgroup C_{s} of GL_{2}(F_{p}). Moreover, ρ(G_{Q}) ≠ ρ(G_{K}), hence D_{K} = p^{∗} and p ∤ N, which means that E has good reduction at p.

If the reduction at p is supersingular, then ρ(G_{Q_{p}}) = N(C_{ns}) is the normaliser of a nonsplit Cartan subgroup C_{ns} of GL_{2}(F_{p}), by [S1, Prop. 12]. In particular, #ρ(G_{K}) is a multiple of #N(C_{ns})/2 = p^{2} − 1 > (p − 1)^{2} = #C_{s} ≥ #ρ(G_{K}), which is impossible.

If the reduction at p is ordinary, then the restriction of ρ to the inertia group I_{p} ⊂ G_{Q_{p}} is given

by ( \begin{matrix}χ_{p,Q_{p}} & ∗ \\ 0 & 1 \end{matrix} ), by [S1, Prop. 11]. On the other hand, ρ|_{G_{K}} = α ⊕ α^{c}, where α : G_{K} ⟶ F_{p}^{×} is a

character and α^{c}(g) := α(c̃gc̃^{−1}), for any c̃ ∈ G_{Q} ∖ G_{K}. Consequently, the restrictions to the inertia group I_{𝔭} ⊂ G_{K_{𝔭}} (where 𝔭 | p is the only prime of K above p) satisfy {α|_{I_{𝔭}}, α^{c}|_{I_{𝔭}}} = {χ_{p,K_{𝔭}}|_{I_{𝔭}}, 1}. As a result, χ_{p,K_{𝔭}}|_{I_{𝔭}} = 1, which implies that χ_{p,Q_{p}}^{2}(I_{p}) = 1, p = 3 and K = Q(\sqrt{−3}). In this case χ_{3,K} = 1, hence

ρ(G_{K}) ⊂ C_{s} ∩ SL_{2}(F_{3}) = \{±I\}. As ρ(G_{Q}) contains ρ(c̃) ∼ ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), we have ρ(G_{Q}) ≃ (Z/2Z)^{a} for some

a ≤ 2, which contradicts the irreducibility of ρ.

(3) Firstly, ρ(G_{K}) is contained in C_{ns} but not in C_{ns} ∩ C_{s} = F_{p}^{×} · I. Secondly, ρ(G_{Q}) contains ρ(c̃) ∉ C_{ns}, hence ρ(G_{Q}) ≠ ρ(G_{K}); thus D_{K} = p^{∗} and E has good reduction at p.

If the reduction at p is supersingular, then

#ρ(G_{K}) = #ρ(G_{Q})/2 ≥ #ρ(G_{Q_{p}})/2 = #C_{ns} ≥ #ρ(G_{K}).

It follows that ρ(G_{K}) = C_{ns} and det ρ(G_{K}) = N_{F_{p^{2}} /F_{p}}(F_{p^{2}}^{×}) = F_{p}^{×}, which is equivalent to Q(μ_{p}) ∩ K = Q, but this is not true.

If the reduction at p is ordinary, then the restriction of ρ to I_{p} is of the form ( \begin{matrix}χ_{p,Q_{p}} & ∗ \\ 0 & 1 \end{matrix} ) ⊂ C_{ns},

which implies again that χ_{p,K_{𝔭}}|_{I_{𝔭}} = 1, p = 3, K = Q(\sqrt{−3}) and χ_{3,K} = 1, hence ρ(G_{K}) is contained in C_{ns} ∩ SL_{2}(F_{3}), which is a cyclic group of order 4. On the other hand, #ρ(G_{K}) > 2, by the irreducibility of ρ|_{G_{K}}, which implies the statements about the structure of ρ(G_{K}) and ρ(G_{Q}). (5.27) Genus theory of quadratic fields. Let K be a quadratic field, R = {q | D_{K}} the set of prime numbers ramified in K/Q, C the strict ideal class group of K, H the strict Hilbert class field of K (the maximal abelian extension of K unramified over K at all finite primes) and K_{gen} := H ∩ Q^{ab} the genus field of K. The Galois groups in the tower Q ↪ K ↪ K_{gen} ↪ H are as follows.

G := Gal(H/K) ≃ C, G_{+} := Gal(H/Q) satisfies ∀g_{+} ∈ G_{+} ∖ G ∀g ∈ G g_{+}^{2} ∈ G, g_{+}gg_{+}^{−1} = g^{−1} Gal(H/K_{gen}) = [G_{+}, G_{+}] = G^{2} ≃ C^{2}, Gal(K_{gen}/K) ≃ C/C^{2}.

There is a unique factorisation

D_{K} = ∏_{q∈R} D_{q},  D_{q} ≡ 0, 1 (mod 4),  |D_{q}| = a power of q

(if q ≠ 2, then D_{q} = q^{∗} := (−1)^{(q−1)/2}q). In terms of this factorisation,

K_{gen} = Q(\{\sqrt{D_{q}}\}_{q∈R})

is the compositum of the quadratic fields K(q) := Q(\sqrt{D_{q}}), for all q ∈ R.

(5.28) Proposition. For each q ∈ R, the compositum H(q) of all subfields of H unramified over Q outside q∞ is equal to

H(q) = \{ \begin{matrix}H, & if R = \{q\} \\ K(q), & if R ≠ \{q\}. \end{matrix}

Proof. The case R = {q} is immediate. Assume that R ≠ {q}. For each q′ ∈ R ∖ {q} and each prime v in H above q′, the inertia subgroup I_{v} ⊂ Gal(H/Q) = G_{+} is of the form I_{v} = {1, h_{v}}, where h_{v}^{2} = 1 and h_{v} ≠ G. By definition, H(q) is the fixed field of the subgroup G(q) ⊂ G generated by the I_{v}, for all q′ ∈ R ∖ {q} and v | q′. If g ∈ G, then gh_{v}g^{−1} ∈ I_{g(v)} and g^{2} = gh_{v}g^{−1}h_{v}^{−1} ∈ G(q); thus G^{2} ⊂ G(q) and H(q) ⊂ K_{gen}, but the only subfields of K_{gen} unramified over Q outside q∞ are Q and K(q). (5.29) For an arbitrary quadratic field K, its ring class field H_{n} of conductor n ≥ 1 is an abelian extension of K characterised by the fact that the reciprocity map of class field theory induces an isomorphism

K_{+}^{×}\K̂^{×}/Ô_{n}^{×} ⟶^{∼} Gal(H_{n}/K),

where K̂ = K ⊗ Ẑ, Ô_{n} = (Z + nO_{K}) ⊗ Ẑ and K_{+}^{×} ⊂ K^{×} is the subgroup of elements that are positive under all real embeddings K ↪ R. For n = 1, H_{1} is the strict Hilbert class field of K. In general, H_{n} is a Galois extension of Q and

∀g ∈ Gal(H_{n}/K) ∀g_{+} ∈ Gal(H_{n}/Q) ∖ Gal(H_{n}/K) g_{+}^{2} ∈ Gal(H_{n}/K), g_{+}gg_{+}^{−1} = g^{−1}.

In particular, Gal(H_{n}/Q)^{ab} ⟶^{∼} (Z/2Z)^{a} for some a ≥ 0. (5.30) In the situation of 5.2, assume that we are given surjective morphisms G_{Q} ⟶^{ρ} G ⟶^{χ} Z_{p}^{×} whose composition is the cyclotomic character, and a surjective O-bilinear pairing ⟨ , ⟩ : T × T ⟶ O satisfying

∀g ∈ G ∀x, y ∈ T ⟨gx, gy⟩ = χ(g)⟨x, y⟩.

For each m ≥ 1, let ρ_{m} be the composition ρ_{m} : G_{Q} ⟶ G ⟶ G/G_{m} ↪ Aut_{O/π^{m}}(T/π^{m}) and define

L_{m} := Q(T/π^{m}T) = \overline{Q}^{Ker(ρ_{m})}.

By definition, if g ∈ G_{m} (m ≥ 1), then (g − 1)T ⊂ π^{m}T and

∀x, y ∈ T (χ(g) − 1)⟨x, y⟩ = ⟨gx, gy⟩ − ⟨x, y⟩ = ⟨(g − 1)x, gy⟩ + ⟨x, (g − 1)y⟩ ∈ π^{m}O,

hence χ(g) ∈ 1 + π^{m}O, by the surjectivity of ⟨ , ⟩. This implies that

∀m ≥ 1 L_{m} = Q(T/π^{m}T) ⊃ Q(μ_{p^{t}}),

where t is the smallest integer such that t ≥ m/e and e := ord_{π}(p) is the ramification index of K/Q_{p}. In particular,

L_{∞} := ⋃_{m≥1} L_{m} ⊃ Q(μ_{p^{∞}}).

(5.31) Proposition. Assume that we are in the situation of 5.30 with p ≠ 2, that K is a quadratic field of discriminant D_{K} and that ρ : G_{Q} ⟶ Aut_{O}(T) is unramified outside pN∞ (i.e., that L_{∞}/K is unramified outside pN∞). Fix m, n ≥ 1.

(1) For every algebraic extension F/Q we have (T/π^{m}T)^{G_{F}} = (T/π^{m}T)^{G_{F ∩Lm}}.

(2) If L_{m} ⊂ H_{n}, then p = 3, 1 ≤ m ≤ e and 3 | nD_{K}.

(3) If (n, pN) = 1, then KL_{∞} ∩ H_{n} = KL_{∞} ∩ H_{1}.

(4) If (N, D_{K}) = 1, then the extension (L_{∞} ∩ H_{1})/Q is unramified outside p∞.

(5) If (pN, D_{K}) = 1, then L_{∞} ∩ H_{1} = Q.

(6) If (N, D_{K}) = 1 and D_{K} = p^{∗} := (−1)^{(p−1)/2}, then K ⊂ L_{1}.

(7) If (N, D_{K}) = 1, p | D_{K} and D_{K} ≠ p^{∗}, then L_{∞} ∩ H_{1} = Q(\sqrt{p^{∗}}) = L_{1} ∩ H_{1} and L_{∞} ∩ K = Q.

(8) If (N, D_{K}) = 1, D_{K} = p^{∗} and r = rk_{O}(T) = 2, then \overline{T}^{G_{H_{1}}} = \overline{T}^{G_{K}}.

Proof. (1) This is true by the definition of L_{m}.

(2) If t is the smallest integer such that t ≥ m/e, then L_{m} ⊂ H_{n} implies that Q(μ_{p^{t}}) ⊂ L_{m}∩Q^{ab} ⊂ H_{n}∩Q^{ab} = a compositum of quadratic fields unramified outside nD_{K}∞. Therefore φ(p^{t}) ≤ 2, p = 3, t = 1 and 3 | nD_{K}.

(3) The extension KL_{∞}/K (resp. H_{n}/K) is unramified outside {v | pN∞} (resp. {v | n∞}); thus (KL_{∞} ∩ H_{n})/K is an abelian extension unramified at all finite places, so it must be contained in H_{1}.

(4) The extension L_{∞}/Q (resp. H_{1}/Q) is unramified outside {ℓ | pN∞} (resp. {ℓ | D_{K}∞}); thus (L_{∞} ∩ H_{1})/Q is unramified outside p∞.

(5) In this case (L_{∞} ∩ H_{1})/Q is unramified outside ∞, so L_{∞} ∩ H_{1} = Q.

(6) K = K(p) := Q(\sqrt{p^{∗}}) ⊂ Q(μ_{p}) ⊂ L_{1}.

(7) The quadratic field K(p) = Q(\sqrt{p^{∗}}) is contained in both Q(μ_{p}) ⊂ L_{1} and in H_{1} (by genus theory); thus K(p) ⊂ L_{1} ∩ H_{1} ⊂ L_{∞} ∩ H_{1}. On the other hand, (4) tells us that L_{∞} ∩ H_{1} is contained in H(p), but H(p) = K(p) in our case, by Proposition 5.28.

(8) If p = 3, then K = Q(\sqrt{−3}) = H_{1}. If p > 3, then L_{1} ⊄ H_{1} by (2), which means that d := dim_{k} \overline{T}^{G_{H_{1}}} ≤ 1. There is nothing to prove if d = 0. If d = 1, then Gal(L_{1} ∩ H_{1}/Q) acts on the line \overline{T}^{G_{H_{1}}} = \overline{T}^{G_{L_{1}∩H_{1}}} by a character α : Gal(L_{1} ∩ H_{1}/Q) ⟶ Gal(L_{1} ∩ H_{1}/Q)^{ab} ⟶ k^{×}. However, Gal(L_{1} ∩ H_{1}/Q)^{ab} is a quotient of

Gal(H_{1}/Q)^{ab} = Gal(K_{gen}/Q) = Gal(K/Q), which means that G_{K} acts on \overline{T} by ( \begin{matrix}1 & ∗ \\ 0 & χ_{p,K} \end{matrix} ). As χ_{p,K} ≠ 1

for p > 3, it follows that \overline{T}^{G_{K}} = \overline{T}^{G_{H_{1}}} , as claimed.

6. Kolyvagin’s result on the vanishing of � (E/K)[p^{∞}]

(6.1) Throughout §6, let:

• E be an elliptic curve over Q of conductor N,

• φ : X_{0}(N) ⟶ E a modular parameterisation of E sending i∞ to the origin,

• K an imaginary quadratic field in which all primes dividing N split,

• N an ideal of O_{K} such that O_{K}/N ≃ Z/NZ.

As in 4.3, these data determine the Heegner points y_{m} ∈ E(H_{m}) on E, defined over the ring class fields H_{m} of conductors m ≥ 1 relatively prime to N, and the basic Heegner point y_{K} = Tr_{H_{1}/K}(y_{1}).

(6.2) If y_{K} ∉ E(K)_{tors} (and D_{K} ≠ −3, −4), then the groups E(K)/Zy_{K} and � (E/K) are finite ([K1, Thm. A]) and the Néron–Tate height of y_{K} is given by the formula of Gross and Zagier [GZ, Thm. V.2.1] (Gross and Zagier considered only the case when D_{K} is odd; for even D_{K} the corresponding formula is a special case of [Z, Thm. 1.2.1]). Combining their formula with the conjecture of Birch and Swinnerton-Dyer, Gross and Zagier observed [GZ, Conj. V.2.2] that, if y_{K} ∉ E(K)_{tors}, then the conjecture of Birch and Swinnerton-Dyer for E over K holds if and only if

[E(K) : Zy_{K}] =^{?} (#� (E/K))^{1/2} u_{K} c_{Tam}(E/Q) c_{Manin}(φ),  (6.2.1)

where c_{Tam}(E/Q) = ∏_{ℓ|N} c_{Tam,ℓ}(E/Q) is the product of all non-archimedean local Tamagawa factors of E over Q, u_{K} = #(O_{K}^{×}/Z^{×}) and c_{Manin}(φ) ∈ Z_{>0} is the Manin constant for φ.

Recall that, for any elliptic curve E′ defined over any number field K′, the Cassels–Tate pairing on the finite abelian group � (E′/K′)/div with values in Q/Z is alternating and non-degenerate, which implies that � (E′/K′)/div is of the form X ⊕ X, for some maximal isotropic subspace X. In particular, #(� (E′/K′)/div) = (#X)^{2} is a square.

In §0.8-§0.9 we discussed Kolyvagin’s results on a conjectural divisibility

if y_{K} ∉ E(K)_{tors}, then [E(K) : Zy_{K}]/(#� (E/K))^{1/2} ∈ Z_{(p)},  (6.2.2)

for a fixed prime p ≠ 2. Jetchev [J, Thm. 1.1] proved, under suitable assumptions, a sharpening of (6.2.2) in the following form:

if y_{K} ∉ E(K)_{tors}, then ∀ℓ | N [E(K) : Zy_{K}]/((#� (E/K))^{1/2}c_{Tam,ℓ}(E/Q)) ∈ Z_{(p)},  (6.2.3)

in line with (6.2.1). (6.3) The simplest case of the expected divisibility (6.2.2) is the following statement:

if y_{K} ∉ pE(K) + E(K)_{tors}, then E(K) ⊗ Z_{p} = Z_{p}(y_{K} ⊗ 1) ≃ Z_{p} and � (E/K)[p^{∞}] = 0  (6.3.1)

(if E(K)[p] = 0, then pE(K) + E(K)_{tors} = pE(K)). As recalled in §0.3-§0.4, (6.3.1) was deduced by Kolyvagin [K1] from his more general annihilation result [K1, Cor. 13] under the assumption that p ≠ 2, u_{K} = 1 and ρ := \overline{ρ}_{E,p} : G_{Q} ⟶ Aut_{F_{p}}(E[p]) ≃ GL_{2}(F_{p}) has “large image”.

A more direct exposition of Kolyvagin’s proof of (6.3.1) in the case when p ∤ 2D_{K} and ρ is surjective was given by Gross [G, Proposition 2.1, Proposition 2.3]. It turns out that the arguments in [G] are valid under weaker assumptions, as we are now going to explain. We begin by extracting from [G] the conditions on E, K and p used in the proof. After that we show that only one of them (an irreducibility assumption) really matters.

(6.4) Proposition ([G, Prop. 2.1, Prop. 2.3 and its proof]. If p ≠ 2 is a prime number and if the conditions (C1)–(C6) below are satisfied, then the implication (6.3.1) holds. (C1) u_{K} = 1 (i.e., D_{K} ≠ −3, −4). (C2) For each n ≥ 1 relatively prime to pND_{K}, E(H_{n})[p] = 0. (C3) E(Q)[p] = 0. (C4) For i = 1, 2, H^{i}(K(E[p])/K, E[p]) = 0. (C5) The restriction of ρ = \overline{ρ}_{E,p} to G_{K} is absolutely irreducible. (C6) Neither of the two subgroups E[p]^{±} ⊂ E[p] (:= the (±1)-eigenspaces for the action of complex conjugation) contains a non-zero G_{K}-stable subgroup (equivalently, ρ is irreducible).

Proof. The conditions (C1) and (C2) are used in [G, §3-§5] in order to construct Kolyvagin’s derivative classes and establish their basic properties, and (C3) is needed in the proof of [G, Prop. 6.2(1)] for v | N. In the general discussion in [G, §7-§8], no additional conditions are needed. Things begin to get more interesting in [G, §9]. The condition (C4) implies the statement of [G, Prop. 9.1] (the proof of which relied on the assumption that p ∤ D_{K}; this was not stated explicitly in [G, Prop. 2.1, Prop. 2.3]). The irreducibility conditions (C5) and (C6) are used, respectively, in the proofs of [G, Prop. 9.3] and [G, Prop. 9.5(2)] (note that the proof of [G, Prop. 9.3] relies on the fact that End_{F_{p}[G_{K}]}(E[p]) = F_{p}). The rest of the proof in [G, §9-§10] goes through unchanged.

(6.5) Proposition. For any prime number p ≠ 2, the conditions (C2), (C3), (C4) and (C6) in Proposition 6.4 follow from (C5). Therefore the implication (6.3.1) holds if p ≠ 2, D_{K} ≠ −3, −4 and E[p] is an absolutely irreducible F_{p}[G_{K}]-module (the latter condition implies that E(K)[p] = 0).

Proof. The implication (C5) ⟹ (C6) is straightforward, and (C3) follows from (C6) and the fact that dim_{F_{p}} E(Q)[p] ≤ 1. The implication (C5) ⟹ (C2) is a special case of Proposition 5.31(8). Finally, (C4) follows from Sah’s Lemma (5.5.2) and the fact that ρ(G_{K}) ⊂ GL_{2}(F_{p}) contains a non-trivial homothety (by Proposition 5.15, since # det(ρ(G_{K})) = #χ_{p,K}(G_{K}) > 2 for p > 3). (6.6) We are now ready to reprove (and slightly extend) the refinement of Kolyvagin’s result on (6.3.1) established by Cha [Ch, the case m = 0 of Thm. 21].

(6.7) Theorem. Assume that p ≠ 2 and that E[p] is an irreducible F_{p}[G_{Q}]-module (which implies that E(K)[p] = 0).

(1) Assume that (K, p) ≠ (Q(\sqrt{−3}), 3). If y_{K} ∉ pE(K), then

E(K) ⊗ Z_{p} = Z_{p}(y_{K} ⊗ 1) ≃ Z_{p}, � (E/K)[p^{∞}] = 0.

(2) Assume that (K, p) = (Q(\sqrt{−3}), 3); then y_{K} ∈ 3E(K). If 3 ∤ a_{3}, assume, in addition, that ρ(G_{K}) is not a cyclic group of order four. If y_{K} ∉ 3^{2}E(K), then

Z_{3} ≃ E(K) ⊗ Z_{3} ⊃ 3E(K) ⊗ Z_{3} = Z_{3}(y_{K} ⊗ 1), � (E/K)[3^{∞}] = 0.

Proof. According to Proposition 5.26, the assumptions imply that E[p] is an absolutely irreducible F_{p}[G_{K}]- module. If u_{K} = 1, the statement follows from Proposition 6.5. It remains to consider the two fields K = Q(i) and K = Q(\sqrt{−3}), when u_{K} = 2 and u_{K} = 3, respectively. We distinguish two separate cases. Case 1: p ∤ u_{K} (equivalently, either K = Q(i) and p > 2, or K = Q(\sqrt{−3}) and p > 3). We modify the constructions in [G] as follows. For any square-free integer n we let H′_{n} to be the compositum inside H_{n} of the ring class fields H_{ℓ}, where ℓ runs through all prime numbers dividing n. The Galois group G_{n} := Gal(H′_{n}/H_{1}) is then canonically isomorphic to ∏_{ℓ|n} G_{ℓ}, where G_{ℓ} = Gal(H_{ℓ}/H_{1}) is a cyclic group of order #(G_{ℓ}) = (ℓ − η_{K}(ℓ))/u_{K}. If, in addition, (n, N) = 1, we define y_{n} := Tr_{H_{n}/H′_{n}}(φ(x_{n})) ∈ E(H′_{n}).

One considers only square-free products n of Kolyvagin primes ℓ satisfying [G, (3.1)-(3.2)]. For each such an ℓ fix a generator σ_{ℓ} ∈ G_{ℓ} and define D_{n} := ∏_{ℓ|n} D_{ℓ} ∈ Z[G_{n}], where each D_{ℓ} is defined as in [G, §3], except that ℓ + 1 is replaced by #(G_{ℓ}) = (ℓ + 1)/u_{K}. The norm relation [G, 3.7(1)] is replaced by u_{K}Tr_{ℓ}(y_{ℓm}) = a_{ℓ} · y_{m} (which implies that [G, 3.6] still holds, since p ∤ u_{K}); the congruence relation [G, 3.7(2)] does not change.

The points P_{n} ∈ E(H′_{n}) are defined as in [G, (4.1)], except that we replace H_{n} (denoted by K_{n} in [G]) by H′_{n}. The vanishing statement E(H_{n})[p] = 0 of [G, 4.3] (i.e., (C2) in Proposition 6.4) still holds, by Proposition 6.5.

Kolyvagin’s classes c(n) ∈ H^{1}(K, E[p]) are then defined by res_{H′_{n}/K}(c(n)) = δ_{n}[P_{n}] ∈ H^{1}(H′_{n}, E[p]) (hence c(1) = δy_{K}). These classes (and their images d(n) ∈ H^{1}(K, E)[p]) have all the properties listed in [G, §6-§7] (except that H_{n} needs to be replaced by H′_{n}). In the formula [G, p. 246, l. 2] one needs to replace Q_{n} by u_{K}Q_{n}, but this is harmless for the argument proving the key statement [G, 6.2(2)], since p ∤ u_{K}.

The rest of the proof goes through as in the situation considered in Proposition 6.4. Case 2: p | u_{K} (equivalently, K = Q(\sqrt{−3}) and p = u_{K} = 3). According to Proposition 4.14(1), there exist infinitely many primes q ∤ 3N satisfying 3 ∤ (q + 1 − a_{q}) (which is equivalent to 3 ∤ (η_{K}(q) + 1 − a_{q})); fix once for all such a prime q.

Consider square-free products n of primes ℓ ∤ 3Nq satisfying Kolyvagin’s condition [G, (3.2)] (which implies that η_{K}(ℓ) = −1, by [G, (3.3)]). For each such n we consider the point y_{n} := φ(x_{qn}) ∈ E(H_{qn}). The Galois group G_{n} := Gal(H_{qn}/H_{q}) is canonically isomorphic to ∏_{ℓ|n} G_{ℓ}, and each G_{ℓ} is cyclic of order ℓ + 1. We define D_{n} and Tr_{ℓ} as in [G, §3]. The statements of [G, 3.6-3.7] and the definition of P_{n} in [G, (4.1)] are unchanged, except that each H_{n} (denoted by K_{n} in [G]) needs to be replaced by H_{qn} (so that P_{n} ∈ E(H_{qn})). One obtains again classes c(n) ∈ H^{1}(K, E[p]) and d(n) ∈ H^{1}(K, E)[p]), with c(1) = δy_{K,q}. They have all the properties listed in [G, §6-§7], except that H_{n} needs to be replaced by H_{qn}.

The rest of the proof goes through as in the situation considered in Proposition 6.4, except that y_{K} in [G, §9-§10] needs to be replaced by y_{K,q}, and P_{ℓ} ∈ H_{ℓ} in [G, §10] by P_{ℓ} ∈ H_{qℓ}. The conclusion is that Sel_{3}(E/K) = (Z/3Z) · δy_{K,q}, which is equivalent to � (E/K)[3^{∞}] = 0 and E(K) ⊗ Z_{3} = Z_{3}(y_{K,q} ⊗ 1) ≃ Z_{3}, since E(K)[3] = 0. In particular, E(K) ⊗ Z_{3} = E(K)_{HP} ⊗ Z_{3} ≃ Z_{3}, which implies that Z_{3}(y_{K} ⊗ 1) = 3E(K) ⊗ Z_{3}, by Proposition 4.14(4). (6.8) Combining Theorem 6.7 with Theorems 4.9 and 4.17, respectively, we obtain the following results.

(6.9) Theorem. Assume that p ≠ 2, E[p] is an irreducible F_{p}[G_{Q}]-module and p ∤ N · a_{p} · (a_{p} − 1) · (a_{p} − η_{K}(p)) · c_{Tam}(E/Q). If y_{K} ∉ pE(K), then the conclusions of Theorem 4.9 hold.

(6.10) Theorem. Assume that K = Q(\sqrt{−3}), p = 3, E[3] is an irreducible F_{3}[G_{Q}]-module, ρ(G_{K}) is not a cyclic group of order four and 3 ∤ a_{3} · (a_{3} − 1) · c_{Tam}(E/Q). If y_{K} ∉ 3^{2}E(K), then the conclusions of Theorem 4.17 hold, with the following modification: each E(K_{n}) ⊗ Z_{3} is generated over Z_{3}[Gal(K_{n}/K)] by the traces to K_{∞} of the Heegner points of conductors dividing 3^{∞}q, for any prime q as in 0.23.

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Ahmed Matar, Department of Mathematics, University of Bahrain, P.O. Box 32038, Sukhair, Bahrain

Jan Nekovář, Sorbonne Université, Campus Pierre et Marie Curie, Institut de Mathématiques de Jussieu, Théorie des Nombres, Case 247, 4 place Jussieu, 75252 Paris cedex 05, France