Derived p-adic heights

Massimo Bertolini^{1} Henri Darmon^{2}

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Preliminary results 1.1 Notations and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The duality formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Derived p-adic heights 2.1 Derived heights for cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Comparison of pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Compatibility of the derived heights . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Derived p-adic heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Refined Birch-Swinnerton Dyer formulae . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Elliptic curves and anticyclotomic Z_{p}-extensions 3.1 The conjectures of Mazur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Applications of the derived p-adic heights . . . . . . . . . . . . . . . . . . . . . . . . 22 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Introduction

Let E be an elliptic curve defined over a number field K, and let K_{∞}/K be a ℤ_{p}-extension. Denote by Λ the Iwasawa algebra ℤ_{p}[[Γ]], with Γ = Gal(K_{∞}/K). Given any topological generator γ of Γ, write I for the ideal (γ − 1)Λ of Λ. Let E(K)_{p} = lim_{n^{⟵}} E(K)/p^{n}E(K) denote the p-adic completion of E(K). When E has

good ordinary reduction at the primes above p which are ramified in K_{∞}/K, there is a canonical symmetric pairing

⟨ , ⟩ : E(K)_{p} × E(K)_{p} → I/I^{2},

called the p-adic height pairing (for a definition, see for instance [MT1] or [Sc1]). Unlike the Néron-Tate canonical height, the p-adic height pairing can be degenerate in certain cases, and the phenomena associated to this degeneracy are poorly understood. This paper is devoted to the study of these questions. One of the main results of this paper, theorem 2.18, states the existence, for 1 ≤ k ≤ p − 1, of a sequence of canonical pairings, called derived p-adic heights,

⟨⟨ , ⟩⟩_{k} : S̄_{p}^{(k)} × S̄_{p}^{(k)} → I^{k}/I^{k+1} ⊗ ℚ,

where S̄_{p}^{(1)} = lim_{n^{⟵}} Sel_{p^{n}}(E/K) is the inverse limit of the p^{n}-Selmer groups with respect to the multiplication

by p maps, and for k ≥ 2 S̄_{p}^{(k)} denotes the null-space of ⟨⟨ , ⟩⟩_{k−1}. We show that these pairings are either symmetric or alternating, depending on whether k is odd or even, and the space of universal norms of S̄_{p}^{(1)} is contained in their null-space. We also show that the restriction of ⟨⟨ , ⟩⟩_{1} to E(K)_{p} is equal to ⟨ , ⟩. We give an alternate description of the null-spaces S̄_{p}^{(k)} in terms of the Λ-module structure of the Selmer group of E/K_{∞}. See thm.2.18 and thm.2.7.

1Partially supported by M.U.R.S.T. of Italy and Consiglio Nazionale delle Ricerche.

2Partially supported by a grant from the NSF, and NSERC.

The product of the discriminants of the above pairings (viewed as defined on (S̄_{p}^{(k)}/S̄_{p}^{(k+1)})) provides a generalization of the notion of p-adic regulator. This derived regulator is useful in all instances where the p-adic height ⟨⟨ , ⟩⟩_{1} is degenerate (i.e. the classical p-adic regulator vanishes) as a way of describing in a convenient manner the leading coefficient of the p-adic algebraic L-function associated to the data (E, K_{∞}/K). See thm.2.23 for the precise statement. When E is defined over ℚ, the anticyclotomic ℤ_{p}-extension of an imaginary quadratic field K provides a prototypical example of the above situation, since the Galois equivariance of ⟨⟨ , ⟩⟩_{1} forces degeneracy if the “plus” and “minus” part of S̄_{p}^{(1)} under the action of complex conjugation have different ranks. This case is analyzed in detail in the third part of the paper where, inspired by conjectures of Mazur ([Ma1], [Ma2]) we predict that the null-space of the second derived height consists exactly of the subspace of universal norms (cf. §3.2). We also explain how this fits into a (partly conjectural) picture describing the behaviour of Heegner points over the anticyclotomic tower.

1 Preliminary results

1.1 Notations and assumptions

We keep the notations of the introduction. For n ≥ 1, let K_{n}/K denote the subextension of K_{∞} of degree p^{n}. Given a place v of K, let K_{v} be the completion of K at v. If F is any finite extension of K, we write

F_{v} = ⨁_{w|v}F_{w},

where the sum is taken over all places of F above v. Functors on abelian categories will always be additive, e.g.,

H^{i}(F_{v}, M) := ⨁_{w|v}H^{i}(F_{w}, M),

if M is any Gal(K̄/K)-module. We make the following assumptions on (E, p, K_{∞}/K).

1. p ∤ 2#(E/E^{0}), where E/E^{0} denotes the group of connected components of the Néron model of E over Spec(O_{K}).

2. E has good reduction above p.

3. The image of the Galois representation ρ_{p} : Gal(K̄/K) → Aut(E_{p}) contains a Cartan subgroup of Aut(E_{p}) ≃ GL_{2}(𝔽_{p}).

4. The local norm mappings Norm_{v} : E((K_{n})_{v}) → E(K_{v}) are surjective for all primes v of K and for all finite subextensions K_{n} of K_{∞}/K.

Proposition 1.1 Assume that for all primes v ramified in K_{∞}/K, p ∤ #E(𝔽_{v}) and v is ordinary for E. Then assumption 4 is satisfied. Proof: This is proved in [Ma3], §4.

Using prop.1.1, it is easy to construct ℤ_{p}-extensions satisfying conditions 1-4, once E/K is fixed. First, almost all primes p of ℚ satisfy conditions 1-3, by Serre’s “open image theorem” [Se] and the theory of complex multiplication. Given a prime v of K above p where E has good reduction, let α_{v} and β_{v} denote the eigenvalues of the Frobenius at v. Assume for simplicity that E does not have complex multiplications, and that K_{v} has residue field 𝔽_{p} (the set of such v’s has density 1). The conditions that E be ordinary at v and p ∤ #E(𝔽_{v}) translate into a_{v} ≢ 0, 1 (mod p), where a_{v} is the rational integer a_{v} = α_{v} + β_{v}. If p ≥ 7, this is equivalent to a_{v} ≠ 0, 1 in ℤ by the Hasse bound a_{v} ≤ 2\sqrt{p}. Now choose a rational prime l ≠ p lying below primes of good reduction for E. The integer a_{v} is equal to the trace of F rob_{v} acting on the Tate module T_{l}E. By [Se], we may assume that Gal(K(E_{l^{n}})/K) is isomorphic to GL_{2}(ℤ/l^{n}ℤ) for all n ≥ 1. The Chebotarev density theorem applied to the extensions K(E_{l^{n}})/K for n → ∞ implies that the set of v as above such that a_{v} ≠ 0, 1 has has density 1. In conclusion, any ℤ_{p}-extension K_{∞}/K such that p lies below the primes v considered above satisfies conditions 1-4.

1.2 The duality formalism

In this section let L/K denote any finite subextension of K_{∞}/K. We review some duality theorems for the Galois cohomology of an elliptic curve, together with results of [BD] built on them. (In effect, these results hold more generally for finite abelian p-extensions.) 1.2.1 Duality (Reference: [Mi], chapter 1) Let v be a finite prime of K and m be a positive integer. Recall the local Tate pairing

⟨ , ⟩_{L_{v},m} : H^{1}(L_{v}, E_{m}) × H^{1}(L_{v}, E_{m}) → ℤ/mℤ,

defined by composing the cup product with the Weil pairing. (In view of our conventions, H^{1}(L_{v}, E_{m}) = ⨁_{w|v}H^{1}(L_{w}, E_{m}), thus ⟨ , ⟩_{L_{v},m} is actually a sum over the primes of L dividing v of the local Tate pairings.) The local Tate pairing is non-degenerate, symmetric and Galois-equivariant. The submodule of local points E(L_{v})/mE(L_{v}) is the orthogonal complement of itself under ⟨ , ⟩_{L_{v},m}. Hence there is also an induced non-degenerate pairing

[ , ]_{L_{v},m} : E(L_{v})/mE(L_{v}) × H^{1}(L_{v}, E)_{m} → ℤ/mℤ.

Later we shall need the following compatibility formulae for the local Tate pairings, which are consequence of standard properties of the cup product.

(1) ⟨cores_{L_{v}/K_{v}}(a), b⟩_{K_{v},m} = ⟨a, res_{L_{v}/K_{v}}(b)⟩_{L_{v},m}, ∀a ∈ H^{1}(L_{v}, E_{m}), ∀b ∈ H^{1}(K_{v}, E_{m}).

(2) [cores_{L_{v}/K_{v}}(a), b]_{K_{v},m} = [a, res_{L_{v}/K_{v}}(b)]_{L_{v},m}, ∀a ∈ E(L_{v})/mE(L_{v}), ∀b ∈ H^{1}(K_{v}, E)_{m}.

Let Σ be a finite, possibly empty, set of non-archimedean primes of K. The local Tate pairing gives rise in the obvious way to non-degenerate Galois-equivariant pairings

⟨ , ⟩_{L,m} : ⨁_{v∈Σ}H^{1}(L_{v}, E_{m}) × ⨁_{v∈Σ}H^{1}(L_{v}, E_{m}) → ℤ/mℤ.
[ , ]_{L,m} : ⨁_{v∈Σ}E(L_{v})/mE(L_{v}) × ⨁_{v∈Σ}H^{1}(L_{v}, E)_{m} → ℤ/mℤ.

Define the Σ-Selmer group of E/L to be

Sel_{m}^{Σ}(E/L) = {s ∈ H^{1}(L, E_{m}) : res_{w}(s) ∈ E(L_{w})/mE(L_{w}) ∀w | v, v ∉ Σ}.

When Σ is the empty set, one finds the usual Selmer group Sel_{m}(E/L). By definition, there is a map

Sel_{m}(E/L) → ⨁_{v∈S}E(L_{v})/mE(L_{v}).

By passing to the Pontryagin dual, we obtain a map

δ = δ_{Σ} : ⨁_{v∈S}H^{1}(L_{v}, E)_{m} → Sel_{m}(E/L)^{dual},

where H^{1}(L_{v}, E)_{m} is identified with (E(L_{v})/mE(L_{v}))^{dual} via the local Tate duality. The following, known as the Cassels dual exact sequence, plays a key role in our construction.

Proposition 1.2 There is an exact sequence

0 → Sel_{m}(E/L) → Sel_{m}^{Σ}(E/L) → ⨁_{v∈Σ}H^{1}(L_{v}, E)_{m} ⟶^{δ} Sel_{m}(E/L)^{dual},

where the first map is the canonical inclusion and the second one is induced by the natural map H^{1}(L, E_{m}) → H^{1}(L, E)_{m} followed by localization. Proof: [Mi], lemma 6.15, p.105.

The non-trivial point in the proof of 1.2 is to show the exactness at the third term of the sequence. Rephrased, this means that a #Σ-tuple of local classes of ⨁_{v∈Σ}H^{1}(L_{v}, E)_{m} pairs to zero with all the elements of the Selmer group Sel_{m}(E/L) under the local Tate pairing if and only if it comes from a global class of Sel_{m}^{Σ}(E/L). Half of this statement, namely, that the image of Sel_{m}^{Σ}(E/L) in Sel_{m}(E/L)^{dual} is zero, follows from the global reciprocity law of class field theory. (More precisely, given global classes α and β of H^{1}(L, E_{m}), their cup product composed with the Weil pairing gives an element α · β in the m-torsion Br(L)_{m} of the Brauer group of L. The local invariant at v of α · β is equal to the local Tate pairing ⟨res_{v}(α), res_{v}(β)⟩_{L_{v},m}. But the sum of the local invariants is zero by class field theory.) The other half follows from Tate’s global duality theorem [Mi], ch.I, thm.4.10. Write G for the Galois group of the extension L/K. The ℤ/mℤ[G]-valued pairing we introduce next will be an ingredient in the construction of the derived heights. Define

⟨ , ⟩ : ⨁_{v∈Σ}H^{1}(L_{v}, E_{m}) × ⨁_{v∈Σ}H^{1}(L_{v}, E_{m}) → ℤ/mℤ[G]

by the rule

⟨x, y⟩ := ∑_{g∈G}⟨x, y^{g}⟩_{L,m} · g^{−1} .

Write ϵ : ℤ/mℤ[G] → ℤ/mℤ for the augmentation map. Let λ ↦ λ^{∗} denote the involution of ℤ/mℤ[G] defined on group-like elements by g ↦ g^{−1}.

Proposition 1.3

1. The pairing ⟨ , ⟩ is non-degenerate. It is ℤ/mℤ[G]-linear in the first variable and ∗-linear in the second variable, i.e. for all λ ∈ ℤ/mℤ[G] we have ⟨λa, b⟩ = λ⟨a, b⟩, ⟨a, λb⟩ = λ^{∗}⟨a, b⟩.

2. ⟨a, b⟩ = ⟨b, a⟩^{∗}.

3. ϵ⟨a, b⟩ = −⟨cores_{L/K}(a), cores_{L/K}(b)⟩_{K,m}.

4. The image of ⨁_{v∈Σ}E(L_{v})/mE(L_{v}) in ⨁_{v∈Σ}H^{1}(L_{v}, E_{m}) is isotropic for ⟨ , ⟩.

5. The image of Sel_{m}^{Σ}(E/L) in ⨁_{v∈Σ}H^{1}(L_{v}, E_{m}) is isotropic for ⟨ , ⟩. Proof:

1. follows from the fact that ⟨ , ⟩_{L,m} is non-degenerate and G-equivariant.

2. follows from the symmetry and G-equivariance of ⟨ , ⟩_{L,m}.

3. is a consequence of formula (1) above.

4. follows from the isotropy of the local points with respect to the local Tate pairing.

5. follows from the global reciprocity law of class field theory.

1.2.2 Descent theory From now on, we shall work with p-groups (notationwise, replace the positive integer m with p^{m}).

Lemma 1.4 For any choice of Σ, the restriction map induces an isomorphism

res : Sel_{p^{m}}^{Σ}(E/K) → Sel_{p^{m}}^{Σ}(E/L)^{G}.

Proof: The Hochschild-Serre spectral sequence gives

H^{1}(G, H^{0}(L, E_{p^{m}})) → H^{1}(K, E_{p^{m}}) → H^{1}(L, E_{p^{m}})^{G} → H^{2}(G, H^{0}(L, E_{p^{m}})).

By assumption 3 of §1.1, E_{p}(L) = 0. Hence we get an isomorphism H^{1}(K, E_{p^{m}}) → H^{1}(L, E_{p^{m}})^{G}. It follows that Sel_{p^{m}}^{Σ}(E/K) injects into Sel_{p^{m}}^{Σ}(E/L)^{G}. Let s be an element of this last group. Then s is a class of H^{1}(K, E_{p^{m}}) whose image in H^{1}(L_{v}, E)^{G} is trivial for all v not in Σ. To conclude the proof, observe that restriction induces an isomorphism H^{1}(K_{v}, E) → H^{1}(L_{v}, E)^{G} for all v. For this, by the Hochschild-Serre spectral sequence, it suffices to check that Ĥ^{1}(G, E(L_{v})) and Ĥ^{2}(G, E(L_{v})) are both trivial. This is equivalent to Ĥ^{0}(G, E(L_{v})) = Ĥ^{1}(G, E(L_{v})) = 0 ([CF], thm.9, p.113). The first group is trivial by our assumption on the local norms. As for the second group, it is dual of the first by the compatibility formula (2) of §1.2.1, combined with the non-degeneracy of the local Tate pairing.

Definition 1.5 We call admissible set for (E, L/K, p^{m}) any finite set Σ of primes of K such that for all v in Σ:

1. res char(v) ≠ p and v is a prime of good reduction for E;

2. v is split in L;

3. E(K_{v})/p^{m}E(K_{v}) ≃ (ℤ/p^{m}ℤ)^{2};

4. Sel_{p^{m}}(E/K) injects into ⨁_{v∈Σ}E(K_{v})/p^{m}E(K_{v}) under the natural restriction map.

The existence of admissible set of primes for (E, L/K, p^{m}) (infinitely many, indeed) is proved in [BD], lemma 2.23. It follows from a standard argument based on the Chebotarev density theorem applied to M/K, M being the extension of L(E_{p^{m}}) cut out by the elements of the Selmer group Sel_{p^{m}}(E/K). The same argument shows that one can assume that the cardinality of Σ is equal to dim_{𝔽_{p}}(Sel_{p^{m}}(E/K) ⊗ 𝔽_{p}).

Proposition 1.6 Let Σ be an admissible set for (E, L/K, p^{m}). Then there is an exact sequence

0 → Sel_{p^{m}}(E/L) → Sel_{p^{m}}^{Σ}(E/L) → ⨁_{v∈Σ}H^{1}(L_{v}, E)_{p^{m}} ⟶^{δ} Sel_{p^{m}}(E/L)^{dual} → 0.

Proof: By prop.1.2 and duality, we only need to show that the map Sel_{p^{m}}(E/L) → ⨁_{v∈Σ}E(L_{v})/p^{m}E(L_{v}) is injective. Assume the contrary, and denote by s a non-zero element of the kernel. Then we can find a non-zero element s′ of the ℤ/p^{m}ℤ[G]-module ℤ/p^{m}ℤ[G]s fixed under the action of G. s′ is in the kernel of the above map. Since by lemma 1.5 s′ comes from Sel_{p^{m}}(E/K), this contradicts the hypothesis that Σ is admissible.

Lemma 1.7 Let Σ be an admissible set for (E, L/K, p^{m}).

1. The modules ⨁_{v∈Σ}E(L_{v})/p^{m}E(L_{v}), ⨁_{v∈Σ}H^{1}(L_{v}, E)_{p^{m}} and ⨁_{v∈Σ}H^{1}(L_{v}, E_{p^{m}}) are free ℤ/p^{m}ℤ[G]- modules of ranks 2#Σ, 2#Σ and 4#Σ, respectively.

2. We can identify ⨁_{v∈Σ}E(L_{v})/p^{m}E(L_{v}) and Sel_{p^{m}}^{Σ}(E/L) with submodules of ⨁_{v∈Σ}H^{1}(L_{v}, E_{p^{m}}) via the natural maps. Then, their intersection is equal to Sel_{p^{m}}(E/L). Proof: 1. follows immediately from the definition of admissible set and the local Tate duality. As for 2., the only non obvious thing is to show that the natural map Sel_{p^{m}}^{Σ}(E/L) → ⨁_{v∈Σ}H^{1}(L_{v}, E_{p^{m}}) is injective. Its kernel contains the elements of Sel_{p^{m}}(E/L) mapping to zero in ⨁_{v∈Σ}E(L_{v})/p^{m}E(L_{v}). Since Σ is admissible, the kernel is 0.

When Σ is an admissible set, the descent module Sel_{p^{m}}(E/L) has a simple and “predictable” Galois structure. This fact will play an important role in the construction of the derived heights.

Proposition 1.8 Let Σ be admissible for (E, L/K, p^{m}). Then Sel_{p^{m}}^{Σ}(E/L) is a free ℤ/p^{m}ℤ[G]-module of rank 2#Σ.

Proposition 1.8 is proved in [BD], thm.3.2. The proof consists in a counting argument, which exploits the exact sequence of prop.1.6 and the information on the ℤ/p^{m}ℤ[G]-module structure of ⨁_{v∈Σ}H^{1}(L_{v}, E)_{p^{m}} contained in lemma 1.7.

2 Derived p-adic heights

2.1 Derived heights for cyclic groups

In this section we define a sequence of canonical pairings associated with finite cyclic p-extensions. The first pairing turns out to be equal, when restricted to the points, to the pairing of [Sc1] (adapted to finite extensions as in [KST]) and of [MT1]: see §2.2. The successive pairings are each defined on the null-space of the previous. This generalizes results obtained in [BD] for cyclic extensions of prime degree. The notations are as follows. Let L = K_{n} for some n ≥ 1, and G = Gal(L/K) ≃ ℤ/p^{n}ℤ. Fix any generator γ of G. Write Λ for the group ring ℤ/p^{n}ℤ[G] and I = (γ − 1)Λ for its augmentation ideal. (Note: the order of G is the same as the order of the ring of coefficients ℤ/p^{n}ℤ.) Given a Λ-module M, let

N : M → M^{G}

denote the norm operator. We identify N with the element ∑_{g∈G} g of Λ.

Lemma 2.1 There exist operators D^{(0)}, ⋯, D^{(p−1)} ∈ Λ such that:

1. D^{(0)} = N;

2. (γ − 1)D^{(k)} = D^{(k−1)} for 1 ≤ k ≤ p − 1. In particular (γ − 1)^{k}D^{(k)} = N for 0 ≤ k ≤ p − 1.

Proof: Let

D^{(k)} := (−1)^{k}γ^{−k} ∑_{i=0}^{p^{n}−1} \binom{i}{k}γ^{i}.

The claim follows from a direct computation (cf. also [D], §3.1).

Lemma 2.2 For 0 ≤ k ≤ p − 1, the module I^{k}/I^{k+1} is isomorphic to ℤ/p^{n}ℤ equipped with the trivial G-action. Proof: By induction on k, the case k = 0 being trivial. Let 1 ≤ k ≤ p − 1. Consider the exact sequence

0 → C_{k} → I^{k−1}/I^{k} ⟶^{γ−1} I^{k}/I^{k+1} → 0.

Given α ∈ C_{k}, let x be any lift of α to I^{k−1}. Then there exists y ∈ Λ such that

(γ − 1)x = (γ − 1)^{k+1}y.

I.e., x = (γ − 1)^{k}y + z, where z ∈ Λ^{G}. Since Λ^{G} = NΛ, there exists w ∈ Λ such that Nw = z. By lemma 2.1, we may write N = (γ − 1)^{k}D^{(k)}. Thus

x = (γ − 1)^{k}(y + D^{(k)}w).

We have proved that C_{k} = 0 for 1 ≤ k ≤ p − 1. By the induction hypothesis, I^{k−1}/I^{k} ≃ ℤ/p^{n}ℤ. This proves lemma 2.2.

Lemma 2.3 Let M be a free Λ-module of finite rank. For 0 ≤ k ≤ p − 1, we have ker((γ − 1)^{k+1}) = D^{(k)}M, where we identify (γ − 1)^{k+1} with the operator on M defined by left multiplication by (γ − 1)^{k+1}. Proof: We may reduce to prove the lemma for M = Λ. We reason by induction on k. Note that by lemma 2.1

D^{(k)}Λ ⊂ ker((γ − 1)^{k+1}).

We have the exact sequence

0 → ker((γ − 1)^{k+1}) → Λ ⟶^{(γ−1)^{k+1}} I^{k+1} → 0.

By lemma 2.2, I^{k+1} ≃ (ℤ/p^{n}ℤ)^{p^{n}−k−1} as an abelian group. Thus

ker((γ − 1)^{k+1}) ≃ (ℤ/p^{n}ℤ)^{k+1}.

For k = 0, D^{(0)}Λ = NΛ = Λ^{G} is isomorphic to ℤ/p^{n}ℤ, hence the claim is true. In general, there is an exact sequence

0 → Ω_{k} → D^{(k)}Λ ⟶^{(γ−1)} D^{(k−1)}Λ → 0.

By the induction hypothesis, D^{(k−1)}Λ ≃ (ℤ/p^{n}ℤ)^{k}. Since Λ^{G} = NΛ ⊂ D^{(k)}Λ by lemma 2.1, we deduce Ω_{k} = Λ^{G} and D^{(k)}Λ ≃ (ℤ/p^{n}ℤ)^{k+1}. This concludes the proof of lemma 2.3.

Corollary 2.4 Let M be a free Λ-module of finite rank. Given x ∈ M^{G}, assume that there exists y ∈ M such that (γ − 1)^{k}y = x, with 0 ≤ k ≤ p − 1. Then there exists z ∈ M such that D^{(k)}z = y. In particular, Nz = x. Proof: y belongs to ker((γ − 1)^{k+1}).

Corollary 2.5 Let M be a free Λ-module of finite rank. For 0 ≤ k ≤ p − 1 we have ker D^{(k)} = I^{k+1}M, where D^{(k)} operates on M by left multiplication. Proof: We may assume M = Λ. By lemma 2.1, I^{k+1} ⊂ ker D^{(k)}. The exact sequence

0 → D^{(k)}Λ → Λ ⟶^{(γ−1)^{k+1}} I^{k+1} → 0,

which is a consequence of lemma 2.3, gives #(D^{(k)}Λ)#(I^{k+1}) = #(Λ). By combining this with the exact sequence

0 → ker D^{(k)} → Λ → D^{(k)}Λ → 0,

we get #(ker D^{(k)}) = #(I^{k+1}). The corollary follows.

Recall the involution ∗ : Λ → Λ defined on group-like elements by g^{∗} = g^{−1}.

Corollary 2.6 For 0 ≤ k ≤ p − 1, (D^{(k)})^{∗} = uD^{(k)}, where u is a unit of Λ. Proof: We have (D^{(k)})^{∗}Λ ⊂ ker(γ−1)^{k+1} = D^{(k)}Λ, since (γ−1)^{k+1}D^{(k)} = 0 implies ((γ−1)^{∗})^{k+1}(D^{(k)})^{∗} = 0, and (γ − 1)^{∗} = −γ^{−1}(γ − 1). But #((D^{(k)})^{∗}Λ) = #(D^{(k)}Λ), since ∗ is an automorphism of Λ. Hence (D^{(k)})^{∗}Λ = D^{(k)}Λ. The corollary follows.

Let Sel := Sel_{p^{n}}(E/K). By lemma 1.4 (applied to the empty set) the restriction map

Sel → Sel_{p^{n}}(E/L)^{G}

is an isomorphism. Hence we may abuse notation somewhat and identify Sel with its image in Sel_{p^{n}}(E/L) under restriction. We define a filtration on Sel

Sel = Sel^{(1)} ⊃ Sel^{(2)} ⊃ ⋯ ⊃ Sel^{(k)} ⊃ ⋯

by letting

Sel^{(k)} := {s ∈ Sel : ∃s̃ ∈ Sel_{p^{n}}(E/L) s.t. (γ − 1)^{k−1}s̃ = s}
:= Sel ∩ (γ − 1)^{k−1}Sel_{p^{n}}(E/L).

Since the operator γ − 1 is nilpotent, Sel^{(k)} = 0 for k sufficiently large.

Theorem 2.7 For 1 ≤ k ≤ p − 1, there exists a sequence of canonical pairings

⟨ , ⟩_{k} : Sel^{(k)} × Sel^{(k)} → I^{k}/I^{k+1}

such that:

1. ⟨s_{1}, s_{2}⟩_{k} = (−1)^{k+1}⟨s_{2}, s_{1}⟩_{k} for all s_{1}, s_{2} ∈ Sel^{(k)},

2. Sel^{(k+1)} is the null-space of ⟨ , ⟩_{k},

3. the norm space cores_{L/K}Sel_{p^{n}}(E/L) is contained in the null-space of all the pairings. Proof: Definition of ⟨ , ⟩_{k} Let Σ be an admissible set for (E, L/K, p^{n}). Let X = X_{Σ}, Y = Y_{Σ} denote the free Λ-modules of lemma 1.7 ⨁_{v∈Σ}E(L_{v})/p^{n}E(L_{v}) and Sel_{p^{n}}^{Σ}(E/L), respectively. Given s_{i} ∈ Sel^{(k)}, i = 1, 2, let s̃_{i} ∈ Sel_{p^{n}}(E/L) be such that (γ − 1)^{k−1}s̃_{i} = s_{i}. Lemma 1.7, 2. allows us to view s̃_{1}, resp. s̃_{2} as an element of X, resp. Y. By cor.2.4 we can find x_{1} ∈ X, y_{2} ∈ Y such that

D^{(k−1)}x_{1} = s̃_{1},
D^{(k−1)}y_{2} = s̃_{2}.

In particular, cores_{L/K}x_{1} = s_{1}, cores_{L/K}y_{2} = s_{2}. Let

⟨ , ⟩ : ⨁_{v∈Σ}H^{1}(L_{v}, E_{p^{n}}) × ⨁_{v∈Σ}H^{1}(L_{v}, E_{p^{n}}) → Λ

be the Λ-valued pairing defined in §1.2. Recall that ⟨ , ⟩ is Λ-linear in the first variable and ∗-linear in the second (prop.1.3, 1.). We have ⟨x_{1}, y_{2}⟩ ∈ I^{k}. This follows from cor.2.5 (applied to M = Λ) and cor.2.6, as

(D^{(k−1)})^{∗}⟨x_{1}, y_{2}⟩ = ⟨x_{1}, D^{(k−1)}y_{2}⟩
= ⟨x_{1}, s̃_{2}⟩ = 0,

where the last equality comes from prop.1.3, 4. We define

⟨s_{1}, s_{2}⟩_{k} := ⟨x_{1}, y_{2}⟩ (mod I^{k+1}).

We have to check that ⟨ , ⟩_{k} is well-defined. To this end, let x′_{1} ∈ X and y′_{2} ∈ Y be such that

D^{(k−1)}x′_{1} = s̃′_{1}, cores_{L/K}x′_{1} = s_{1},
D^{(k−1)}y′_{2} = s̃′_{2}, cores_{L/K}y′_{2} = s_{2},

with s̃′_{1} and s̃′_{2} elements of Sel_{p^{n}}(E/L). Then,

cores_{L/K}(x_{1} − x′_{1}) = 0, cores_{L/K}(y_{1} − y′_{1}) = 0.

Since X and Y are free Λ-modules, there exist ξ ∈ X and η ∈ Y such that x_{1} − x′_{1} = (γ − 1)^{∗}ξ and y_{1} − y′_{1} = (γ − 1)^{∗}η. It is enough to show that ⟨x_{1} − x′_{1}, y_{2}⟩ ∈ I^{k+1} and ⟨x_{1}, y_{2} − y′_{2}⟩ ∈ I^{k+1}. We have

(D^{(k)})^{∗}⟨x_{1} − x′_{1}, y_{2}⟩ = ⟨ξ, (γ − 1)D^{(k)}y_{2}⟩
= ⟨ξ, s̃_{2}⟩ = 0

by prop.1.3, 4., and

D^{(k)}⟨x_{1}, y_{2} − y′_{2}⟩ = ⟨(γ − 1)D^{(k)}x_{1}, η⟩
= ⟨s̃_{1}, η⟩ = 0,

the last equality being consequence of 1.3, 5. The pairing ⟨ , ⟩_{k} is also independent of the admissible set Σ. For, if Σ′ is another admissible set, Σ ∪ Σ′ is also admissible. The modules X_{Σ} and X_{Σ′} inject into X_{Σ∪Σ′}, and similarly for Y. Moreover, the restriction of the Λ-valued pairing on ⨁_{v∈Σ∪Σ′}H^{1}(L_{v}, E_{p^{n}}) to ⨁_{v∈Σ}H^{1}(L_{v}, E_{p^{n}}) coincides with the Λ-valued pairing on ⨁_{v∈Σ}H^{1}(L_{v}, E_{p^{n}}). Thus, the above calculations also show that ⟨ , ⟩_{k} is independent of Σ. Finally, ⟨ , ⟩_{k} is visibly independent of the choice of the generator γ of G.

Proof of 1. Let s_{1}, s_{2} ∈ Sel^{(k)}. There are s̃_{1}, s̃_{2} ∈ Sel_{p^{n}}(E/L) such that (γ − 1)^{k−1}s̃_{i} = s_{i}, i = 1, 2. As above, we can find x_{i} ∈ X, y_{i} ∈ Y for i = 1, 2 such that

D^{(k−1)}x_{i} = s̃_{i}, D^{(k−1)}y_{i} = s̃_{i}.

Hence

D^{(k−1)}(x_{i} − y_{i}) = 0, i = 1, 2.

By lemma 1.7, Z = ⨁_{v∈Σ}H^{1}(L_{v}, E_{p^{n}}) is a free Λ-module and we can view X and Y as submodules of Z. Then, cor.2.5 implies the existence of z_{i} ∈ Z such that

(γ − 1)^{k}z_{i} = x_{i} − y_{i}.

It follows from cor.2.5, applied to M = Λ, that ⟨x_{1} − y_{1}, x_{2} − y_{2}⟩ belongs to I^{k+1}. By the isotropy of X and Y with respect to ⟨ , ⟩ (prop.1.3) we find

⟨x_{1}, y_{2}⟩ = −⟨y_{1}, x_{2}⟩ (mod I^{k+1}).

Since the involution ∗ acts as (−1)^{k} on I^{k}/I^{k+1}, prop.1.3, 2. gives

⟨x_{1}, y_{2}⟩ = (−1)^{k+1}⟨x_{2}, y_{1}⟩ (mod I^{k+1}).

In view of the definition of our pairings, this concludes the proof of 1.

Proof of 2. By induction on k. By 1., it is enough to prove that the right null-space of ⟨ , ⟩_{k} is equal to Sel^{(k+1)}, for 1 ≤ k ≤ p − 1. Case k = 1. Let s_{2} be in the (right) null-space of ⟨ , ⟩_{1}. Let y_{2} ∈ Y such that cores_{L/K}y_{2} = s_{2}. Then ⟨x_{1}, y_{2}⟩ belongs to I^{2} for all x_{1} ∈ X mapping by corestriction to Sel. By cor.2.5 and 2.6, we get

0 = (D^{(1)})^{∗}⟨x_{1}, y_{2}⟩ = ⟨x_{1}, D^{(1)}y_{2}⟩.

Let β be the image of D^{(1)}y_{2} in ⨁_{v∈Σ}H^{1}(L_{v}, E)_{p^{n}} under the natural map. Since, by lemma 2.1, (γ − 1)D^{(1)}y_{2} = s_{2}, then β belongs to (⨁_{v∈Σ}H^{1}(L_{v}, E)_{p^{n}})^{G} = ⨁_{v∈Σ}H^{1}(K_{v}, E)_{p^{n}}. By the compatibility formula (2) of §1.2.1 we find immediately

⟨x_{1}, D^{(1)}y_{2}⟩ = −[s_{1}, β]_{K,p^{n}} · N

for all s_{1} in the Selmer group Sel, where [ , ]_{K,p^{n}} is the local pairing introduced in §1.2.1 and N = D^{(0)} denotes the norm operator. Hence [s_{1}, β]_{K,p^{n}} = 0 for all s_{1} ∈ Sel. By prop.1.2, there exists a global class α in Sel_{p^{n}}^{Σ}(E/K) mapping to β under the natural map. Thus, D^{(1)}y_{2} − α belongs to Sel_{p^{n}}(E/L), and

(γ − 1)(D^{(1)}y_{2} − α) = (γ − 1)D^{(1)}y_{2} = s_{2}.

In other words, s_{2} belongs to Sel^{(2)}, and this concludes the proof of the case k = 1. Claim

1. Let y_{2} ∈ Y be such that D^{(k−1)}y_{2} belongs to Sel_{p^{n}}(E/L). Then y_{2} induces a homomorphism ϕ in Hom(Sel, I^{k}/I^{k+1}) by the rule

ϕ(s_{1}) = ⟨x_{1}, y_{2}⟩ (mod I^{k+1})

where, given s_{1} ∈ Sel, x_{1} denotes any element of X such that cores_{L/K}x_{1} = s_{1}.

2. If the element s_{2} = cores_{L/K}y_{2} ∈ Sel^{(k)} belongs to the right null-space of ⟨ , ⟩_{k}, then we can view ϕ as a homomorphism in Hom(Sel/Sel^{(k)}, I^{k}/I^{k+1}). Proof of the Claim: ⟨x_{1}, y_{2}⟩ belongs to I^{k}. For, in view of cor.2.5,

(D^{(k−1)})^{∗}⟨x_{1}, y_{2}⟩ = ⟨x_{1}, D^{(k−1)}y_{2}⟩ = 0.

The last equality follows from prop.1.3, 5. Note that ϕ does not depend on the choice of x_{1}. For, given another x′_{1} ∈ X such that cores_{L/K}x′_{1} = s_{1}, by the freeness of X we can find ξ ∈ X such that x_{1} − x′_{1} = (γ − 1)^{∗}ξ. Then

(D^{(k)})^{∗}⟨x_{1} − x′_{1}, y_{2}⟩ = ⟨ξ, D^{(k−1)}y_{2}⟩ = 0.

Hence ⟨x_{1}, y_{2}⟩ ≡ ⟨x′_{1}, y_{2}⟩ (mod I^{k+1}). Finally, if s_{2} is in the null-space of ⟨ , ⟩_{k}, then ϕ(Sel^{(k)}) ⊂ I^{k+1}. This concludes the proof of the claim.

Now let k be at least 2. By the induction hypothesis, for 1 ≤ i ≤ k−1 the pairings ⟨ , ⟩_{i} induce identifications

Sel^{(i)}/Sel^{(i+1)} = Hom(Sel^{(i)}/Sel^{(i+1)}, I^{i}/I^{i+1}).

By lemma 2.2, multiplication by (γ − 1)^{k−i} identifies Hom(Sel^{(i)}/Sel^{(i+1)}, I^{i}/I^{i+1}) with Hom(Sel^{(i)}/Sel^{(i+1)}, I^{k}/I^{k+1}). Thus, we can find y^{(1)}, …, y^{(k−1)} in Y such that:

1. cores_{L/K}y^{(i)} = s^{(i)} ∈ Sel^{(i)},

2. D^{(i−1)}y^{(i)} = s̃^{(i)} ∈ Sel_{p^{n}}(E/L),

3. ⟨x_{1}, y_{2}⟩ = ⟨x_{1}, (γ − 1)^{k−1}y^{(1)}⟩ + ⋯ + ⟨x_{1}, (γ − 1)y^{(k−1)}⟩ (mod I^{k+1}) for all x_{1} as above.

Let y′_{2} = y_{2} − (γ − 1)^{k−1}y^{(1)} − ⋯ − (γ − 1)y^{(k−1)}. By lemma 2.1,

(i) cores_{L/K}y′_{2} = s_{2},

(ii) D^{(k−1)}y′_{2} = D^{(k−1)}y_{2} − s̃^{(1)} − ⋯ − s̃^{(k−1)} ∈ Sel_{p^{n}}(E/L).

By definition of y′_{2}, ⟨x_{1}, y′_{2}⟩ = 0 (mod I^{k+1}) for all x_{1} ∈ X such that cores_{L/K}x_{1} belongs to Sel. Hence, by cor.2.5 and 2.6,

0 = (D^{(k)})^{∗}⟨x_{1}, y′_{2}⟩ = ⟨x_{1}, D^{(k)}y′_{2}⟩.

Let β be the image of D^{(k)}y′_{2} in ⨁_{v∈Σ}H^{1}(L_{v}, E)_{p^{n}} under the natural map. Since (γ − 1)D^{(k)}y′_{2} = D^{(k−1)}y′_{2} belongs to Sel_{p^{n}}(E/L), then β belongs to (⨁_{v∈Σ}H^{1}(L_{v}, E)_{p^{n}})^{G} = ⨁_{v∈Σ}H^{1}(K_{v}, E)_{p^{n}}. By formula (2) of §1.2.1, we find ⟨x_{1}, D^{(k)}y′_{2}⟩ = −[s_{1}, β]_{K,p^{n}} · N for all s_{1} in the Selmer group Sel. It follows

[s_{1}, β]_{K,p^{n}} = 0

for all s_{1} in Sel. Then, by prop.1.2 there exists α ∈ Sel_{p^{n}}^{Σ}(E/K) mapping to β under the natural map. Hence D^{(k)}y′_{2} − α belongs to Sel_{p^{n}}(E/L). Moreover,

(γ − 1)^{k}(D^{(k)}y′_{2} − α) = s_{2},

i.e. s_{2} belongs to Sel^{(k+1)}. This concludes the proof of 2. Proof of 3. It follows from the factorization in Λ of cores_{L/K} as (γ − 1)^{k}D^{(k)} for 1 ≤ k ≤ p − 1 (lemma 2.1).

2.2 Comparison of pairings

We keep the notations of §2.1. The first pairing of thm.2.7 induces on points a pairing

E(K) × E(K) → I/I^{2},

which, by an abuse of notation, we still denote by ⟨ , ⟩_{1}. We show that it is equal to the “analytic” height pairings of Schneider [Sc1] (as formulated by K.-S. Tan [KST] for finite extensions) and of [MT1]. For the convenience of the reader, we recall the definition of ⟨ , ⟩_{1}. Fix an admissible set Σ for (E, L/K, p^{n}), and denote by X, resp. Y the free Λ-modules ⨁_{v∈Σ}E(L_{v})/p^{n}E(L_{v}), resp. Sel_{p^{n}}^{Σ}(E/L). Given P, resp. Q in E(K), we write ā, resp. b̄ for its image in X^{G}, resp. Y^{G}. Choose a ∈ X, b ∈ Y such that cores_{L/K}(a) = ā, cores_{L/K}(b) = b̄. Recall the Λ-valued pairing ⟨ , ⟩ of §1.2.1. The element ⟨a, b⟩ belongs to I. Equivalently by lemma 2.5, D^{(0)}⟨a, b⟩ = 0. But D^{(0)}⟨a, b⟩ = ⟨a, b̄⟩, and this is zero by prop.1.3, 4. The image of ⟨a, b⟩ in I/I^{2} depends only on P and Q : this is proved, in greater generality, in the course of constructing the pairings ⟨ , ⟩_{k} in the proof of thm.2.7. By definition

⟨P, Q⟩_{1} := ⟨a, b⟩ (mod I^{2}).

In [KST] an “algebraic” definition of height is introduced, and shown to be equal to the analytic pairings of Mazur-Tate and Schneider. Hence it suffices to show that ⟨ , ⟩_{1} is equal to the algebraic pairing of [KST], whose definition we now recall. Tan defines a homomorphism

Φ_{L/K} : E(K) ⊗ E(K) ⊗ Hom(G, ℤ/p^{n}ℤ) → ℤ/p^{n}ℤ

as follows. Let P, Q ∈ E(K) = H^{0}(G, E(L)), and ψ ∈ Hom(G, ℤ/p^{n}ℤ) = H^{2}(G, ℤ). The cup product of P and ψ gives an element P ∪ ψ ∈ H^{2}(G, E(L)). We have H^{2}(K, E) = 0 by [Mi], cor. I.6.24, p.111, since p is odd. Then the Hochschild-Serre spectral sequence defines a surjective homomorphism (transgression)

H^{1}(L, E)^{G} ⟶^{trg} H^{2}(G, E(L)) → 0.

Let γ ∈ H^{1}(L, E)^{G} be such that trg(γ) = P ∪ ψ. For all v, since restriction induces an isomorphism H^{1}(K_{v}, E) ≃ H^{1}(L_{v}, E)^{G} (see the proof of lemma 1.4), there exists α_{v} ∈ H^{1}(K_{v}, E) which maps in H^{1}(L_{v}, E)^{G} to the localization at v of γ. Let [ , ]_{K_{v},p^{n}} : E(K_{v})/p^{n}E(K_{v}) × H^{1}(K_{v}, E)_{p^{n}} → ℤ/p^{n}ℤ denote the local Tate duality of §1.2.1. Define

Φ_{L/K}(P ⊗ Q ⊗ ψ) = ∑_{v}[Q_{v}, α_{v}]_{K_{v},p^{n}}.

One can check that this is independent of the choices made. Identify I/I^{2} with G and G with its bi-dual, in the usual way. Then Φ_{L/K} defines a pairing

⟨ , ⟩_{KS} : E(K) × E(K) → I/I^{2}.

Theorem 2.8 We have ⟨P, Q⟩_{1} = ⟨P, Q⟩_{KS} for all P, Q ∈ E(K).

Proof: By definition

⟨P, Q⟩_{1} = ∑_{i=0}^{p^{n}−1}⟨a, b^{γ^{−i}}⟩_{L,p^{n}} · γ^{i} (mod I^{2})

= − ∑_{i=0}^{p^{n}−1}⟨a, b^{γ^{i}}⟩_{L,p^{n}} · i(γ − 1) (mod I^{2})

= ⟨a, D^{(1)}b⟩_{L,p^{n}} · (γ − 1) (mod I^{2}) ,

where ⟨ , ⟩_{L,p^{n}} denotes the local pairing of §1.2.1, a and b are chosen as above and D^{(1)} = − ∑_{i=0}^{p^{n}−1} iγ^{i} is the derivative operator of lemma 2.1. Let η be the image of D^{(1)}b in H^{1}(L, E). Since (γ − 1)D^{(1)} = D^{(0)} is the norm operator, η belongs to H^{1}(L, E)^{G}. Let ψ ∈ Hom(G, ℤ/p^{n}ℤ) be the homomorphism such that ψ(γ) = 1 (mod p^{n}ℤ).

Lemma 2.9 We have trg(η) = Q ∪ ψ.

Clearly ⟨a, D^{(1)}b⟩_{L,p^{n}} = ∑_{v}[P_{v}, η_{v}]_{K_{v},p^{n}}, where η_{v} ∈ H^{1}(K_{v}, E) corresponds to the localization of η under the isomorphism H^{1}(K_{v}, E) ≃ H^{1}(L_{v}, E)^{G}. Then, in view of the definition of the height pairings, lemma 2.9 implies that ⟨P, Q⟩_{1} = ⟨Q, P ⟩_{KS} for all P, Q ∈ E(K). Theorem 2.8 follows from the symmetry of ⟨ , ⟩_{1} (thm.2.7, 1.).

Proof of lemma 2.9: We make use of the explicit formula for computing the transgression homomorphism on cocycles given in [KST], proof of lemma 3.2. Let G_{K} = Gal(K̄/K), G_{L} = Gal(K̄/L). Let f : G_{L} → E(K̄) be a 1-cocycle representing η. Then

τf(τ^{−1}στ) − f(σ) = σl(τ) − l(τ) ∀σ ∈ G_{L}, ∀τ ∈ G_{K},

for some l(τ) ∈ E(K̄). The point l(τ) is determined modulo elements in E(L). Fix R ∈ E(K̄) such that p^{n}R = Q. It is easy to see that we can choose l(τ) = iR + e(τ), where e(τ) ∈ E_{p^{n}} and 0 ≤ i ≤ p^{n} − 1 is an integer such that τ maps to γ^{i} in G. With this choice of l(τ) we have

1. l(σ) = f(σ) for all σ ∈ G_{L},

2. l(τσ) = l(τ) + τl(σ) for all σ ∈ G_{L}, τ ∈ G_{K}. This follows from an explicit computation, observing that E_{p^{n}}(L) = 0 by assumption 3 of §1.1. Then a well-defined 2-cocycle g : G × G → E(L) representing tr(η) is given by the formula

g(γ^{i_{1}}, γ^{i_{2}}) = l(τ_{1}) + τ_{1}l(τ_{2}) − l(τ_{1}τ_{2}) 0 ≤ i_{1}, i_{2} ≤ p^{n} − 1,

where τ_{j}, j = 1, 2 is chosen so that it maps in G to γ^{i_{j}}. We find

g(γ^{i_{1}}, γ^{i_{2}}) = \{ \begin{matrix}0 & for i_{1} + i_{2} ≤ p^{n} − 1 \\ Q & for i_{1} + i_{2} > p^{n} − 1. \end{matrix}

But this is precisely a representative for Q ∪ ψ, as one can see by describing explicitly on cochains the cup product ([CF], pp.106-7) and the identification via coboundary Hom(G, ℤ/p^{n}ℤ) = H^{2}(G, ℤ). This concludes the proof of lemma 2.9.

Remark 2.10 The definition of the pairings ⟨ , ⟩_{1} and ⟨ , ⟩_{KS} works equally well when L/K is a finite abelian p-extension, not necessarily cyclic. The two pairings still coincide in this more general setting, as a consequence of their norm-compatibility (combined with thm.2.8). More precisely, given a tower K ⊂ L_{1} ⊂ L, write ν : I/I^{2} → I_{1}/I_{1}^{2} for the natural projection, I_{1} being the augmentation ideal of ℤ/p^{n}ℤ[Gal(L_{1}/K)]. Then a computation gives the identities (where we use the obvious notations)

1. ν(⟨P, Q⟩_{1,L/K}) = ⟨P, Q⟩_{1,L_{1}/K} for all P, Q ∈ E(K),

2. ν(⟨P, Q⟩_{KS,L/K}) = ⟨P, Q⟩_{KS,L_{1}/K} for all P, Q ∈ E(K).

2.3 Compatibility of the derived heights

We shall define the p-adic derived heights by compiling the derived heights corresponding to the finite layers of the ℤ_{p}-extension K_{∞}/K. To do this, we need to study the compatibility of the derived heights for finite cyclic groups under change of extension, and this is the goal of this section. Recall that K_{n} denotes the subextension of K_{∞}/K of degree p^{n}. Write G_{n} for the Galois group Gal(K_{n}/K), Λ_{n} for the group ring ℤ/p^{n}ℤ[G_{n}], and I_{n} for its augmentation ideal. Let Sel_{n} := Sel_{p^{n}}(E/K). Let

Sel_{n} = Sel_{n}^{(1)} ⊃ Sel_{n}^{(2)} ⊃ ⋯ ⊃ Sel_{n}^{(k)} ⊃ ⋯

denote the filtration of Sel_{n} defined by

Sel_{n}^{(k)} := {s ∈ Sel_{n} : ∃s̃ ∈ Sel_{p^{n}}(E/K_{n}) s.t. (γ_{n} − 1)^{k−1}s̃ = s}
:= Sel_{p^{n}}(E/K) ∩ (γ_{n} − 1)^{k−1}Sel_{p^{n}}(E/K_{n}).

Let

⟨ , ⟩_{k,n} : Sel_{n}^{(k)} × Sel_{n}^{(k)} → I_{n}^{k}/I_{n}^{k+1}

be the derived height pairings defined in §2.1. By abuse of notation we shall write m_{p} for any map induced in cohomology by E_{p^{n+1}} ⟶^{p} E_{p^{n}}. In particular, we have a map

m_{p} : Sel_{n+1} → Sel_{n}.

Since E_{p}(K) = 0 by our assumptions, Sel_{n} injects into Sel_{n+1} under the natural map, and m_{p} is induced by the multiplication by p on Sel_{n+1}. The next proposition contains the compatibility result we need. Let

ν_{n} : Λ_{n+1} → Λ_{n}

denote the natural projection of group rings, and also, by abusing notation, the induced map I_{n+1}^{k}/I_{n+1}^{k+1} → I_{n}^{k}/I_{n}^{k+1} for any k.

Proposition 2.11

1. For 1 ≤ k ≤ p, the map m_{p} respects the filtrations on Sel_{n+1} and Sel_{n}, i.e. m_{p}(Sel_{n+1}^{(k)}) ⊂ Sel_{n}^{(k)}.

2. For 1 ≤ k ≤ p − 1, we have

ν_{n}⟨s_{1}, s_{2}⟩_{k,n+1} = ⟨m_{p}s_{1}, m_{p}s_{2}⟩_{k,n}

for all s_{1}, s_{2} in Sel_{n+1}^{(k)}. Proof of Part 1. Fix any topological generator γ of Gal(K_{∞}/K), and let γ_{n} be the generator of Gal(K_{n}/K) corresponding to γ under the natural projection. Denote by

D_{n}^{(k)} ∈ Λ_{n}, 0 ≤ k ≤ p − 1

the operators defined in the proof of lemma 2.1 (with γ_{n} replacing γ). Write

q_{n} : ℤ/p^{n+1}[G_{n+1}] → ℤ/p^{n+1}ℤ[G_{n}],
π_{n} : ℤ/p^{n+1}ℤ[G_{n}] → ℤ/p^{n}ℤ[G_{n}]

for the natural projections. Thus their composite π_{n}q_{n} is equal to ν_{n}.

Lemma 2.12 For 0 ≤ k ≤ p − 1 there exists D_{n}^{(k)} ∈ ℤ/p^{n+1}ℤ[G_{n}] such that:

1. q_{n}D_{n+1}^{(k)} = pD_{n}^{(k)},

2. π_{n}D_{n}^{(k)} = D_{n}^{(k)}. Proof: By definition of D_{n}^{(k)}, we reduce to show that for all 0 ≤ i ≤ p^{n} − 1 the equality

∑_{j=0}^{p−1} \binom{i + jp^{n}}{k} = p\binom{i}{k}, 0 ≤ k ≤ p − 1

holds in ℤ/p^{n+1}ℤ. This follows from an easy induction argument.

We conclude the proof of part 1. Let s ∈ Sel_{n+1}^{(k)}. Let Σ be an admissible set for (E, K_{n+1}/K, p^{n+1}). Then by the results of §2.1 there exists y ∈ Sel_{p^{n+1}}^{Σ}(E/K_{n+1}) such that

(1)  D_{n+1}^{(k−1)}y := s̃ ∈ Sel_{p^{n+1}}(E/K_{n+1}), (γ_{n+1} − 1)^{k−1}s̃ = s.

Then

m_{p}s = m_{p}(cores_{K_{n+1}/K}y)
= cores_{K_{n}/K}(m_{p}cores_{K_{n+1}/K_{n}}y)
= (γ_{n} − 1)^{k−1}D_{n}^{(k−1)}(m_{p}cores_{K_{n+1}/K_{n}}y).

Let y′ := m_{p}cores_{K_{n+1}/K_{n}}y. We claim that

D_{n}^{(k−1)}y′ ∈ Sel_{p^{n}}(E/K_{n}).

For, D_{n}^{(k−1)}y′ ∈ Sel_{p^{n}}^{Σ}(E/K_{n}) since y′ ∈ Sel_{p^{n+1}}^{Σ}(E/K_{n})[p^{n}] = Sel_{p^{n}}^{Σ}(E/K_{n}), where the equality follows, for example, from prop.1.8 and E_{p}(K_{n}) = 0. Moreover, by lemma 2.12

D_{n}^{(k−1)}y′ = pD_{n}^{(k−1)}cores_{K_{n+1}/K_{n}}y
= D_{n+1}^{(k−1)}cores_{K_{n+1}/K_{n}}y
= cores_{K_{n+1}/K_{n}}D_{n+1}^{(k−1)}y
= cores_{K_{n+1}/K_{n}} s̃ ∈ Sel_{p^{n+1}}(E/K_{n}).

Thus, D_{n}^{(k−1)}y′ ∈ Sel_{p^{n+1}}(E/K_{n}) ∩ Sel_{p^{n}}^{Σ}(E/K_{n}) = Sel_{p^{n}}(E/K_{n}). We find

(2)  D_{n}^{(k−1)}y′ := s̃′ ∈ Sel_{p^{n}}(E/K_{n}), (γ_{n} − 1)^{k−1}s̃′ = m_{p}s.

In particular, m_{p}s belongs to Sel_{n}^{(k)}, as was to be shown.

Proof of Part 2. We begin with a couple of lemmas.

Lemma 2.13 Let F be a local field. Let p_{n} : ℤ/p^{n+1}ℤ → ℤ/p^{n}ℤ be the canonical projection. Then for all x, y in H^{1}(F, E_{p^{n+1}}), the local Tate pairing satifies

p_{n}(⟨x, y⟩_{F,p^{n+1}}) = ⟨m_{p}x, m_{p}y⟩_{F,p^{n}}.

Proof: Recall that the local Tate pairing is defined by composing the cup product with the Weil pairing w_{n} : E_{p^{n}} ⊗ E_{p^{n}} → μ_{p^{n}}. The lemma follows from the explicit definition of the cup product on cocycles ([CF], pp.106-7) combined with the relation w_{n+1}(ξ ⊗ η)^{p} = w_{n}(pξ ⊗ pη) for all ξ, η ∈ E_{p^{n+1}}.

Given an admissible set Σ for (E, K_{n}/K, p^{n}) we let

⟨ , ⟩_{(n)} : ⨁_{v∈Σ}H^{1}((K_{n})_{v}, E_{p^{n}}) × ⨁_{v∈Σ}H^{1}((K_{n})_{v}, E_{p^{n}}) → Λ_{n}

denote the non-degenerate pairing defined in §1.2.1.

Lemma 2.14 Let Σ be an admissible set for (E, K_{n+1}/K, p^{n+1}). For all x, y ∈ ⨁_{v∈Σ}H^{1}((K_{n+1})_{v}, E_{p^{n+1}}) we have

ν_{n}⟨x, y⟩_{(n+1)} = ⟨m_{p}(cores_{K_{n+1}/K_{n}}x), m_{p}(cores_{K_{n+1}/K_{n}}y)⟩_{(n)} .

Proof: Note that Σ is also admissible for (E, K_{n}/K, p^{n+1}) and for (E, K_{n}/K, p^{n}). The formulae (1) and (2) of §1.2.1 and lemma 2.13 give

ν_{n}⟨x, y⟩_{(n+1)} = ∑_{σ∈G_{n}} π_{n}⟨x, (cores_{K_{n+1}/K_{n}}y)^{σ}⟩_{K_{n+1},p^{n+1}} · σ^{−1}

= ∑_{σ∈G_{n}} π_{n}⟨cores_{K_{n+1}/K_{n}}x, (cores_{K_{n+1}/K_{n}}y)^{σ}⟩_{K_{n},p^{n+1}} · σ^{−1}

= ⟨m_{p}(cores_{K_{n+1}/K_{n}}x), m_{p}(cores_{K_{n+1}/K_{n}}y)⟩_{(n)}.

We conclude the proof of part 2. Let Σ be an admissible set for (E, K_{n+1}/K, p^{n+1}). Let s ∈ Sel_{n+1}^{(k)}. Then there exists x belonging to ⨁_{v∈Σ}E((K_{n+1})_{v})/p^{n+1}E((K_{n+1})_{v}) such that

(3)  D_{n+1}^{(k−1)}x := s̃ ∈ Sel_{p^{n+1}}(E/K_{n+1}), (γ_{n+1} − 1)^{k−1}s̃ = s.

Let x′ := m_{p}cores_{K_{n+1}/K_{n}}x. With the formal argument of the proof of part 1., we can show that

(4)  D_{n}^{(k−1)}x′ := s̃′ ∈ Sel_{p^{n}}(E/K_{n}), (γ_{n} − 1)^{k−1}s̃′ = m_{p}s.

In view of the definition of our pairings (thm.2.7), the claim follows from lemma 2.14, and the equations (1)-(4).

2.4 Derived p-adic heights

By prop.2.11, we may compile the pairings ⟨ , ⟩_{k,n} via the maps m_{p}, in order to define pairings on the inverse limit of the modules Sel_{n}^{(k)}. We may use thm.2.7 to obtain the properties of these new pairings, once we know that the modules Sel_{n}^{(k)} can be recovered from their inverse limit in the natural way. This is proved in prop.2.17, after a few preliminary definitions. The properties of the p-adic pairings we construct are summarized in thm.2.18.

Definition 2.15

1. We define the pro-p Selmer group of E/K_{n} S_{p}(E/K_{n}) := lim_{m^{⟵}} Sel_{p^{m}}(E/K_{n}), where the inverse limit is

with respect to the maps m_{p}.

2. In view of prop.2.11, we define a filtration on S_{p}(E/K)

S_{p}(E/K) = S_{p}^{(1)} ⊃ S_{p}^{(2)} ⊃ ⋯ ⊃ S_{p}^{(p−1)} ⊃ S_{p}^{(p)}

by letting S_{p}^{(k)} := lim_{n^{⟵}} Sel_{n}^{(k)}, the limit being taken by means of the maps m_{p}.

Since E_{p}(K_{n}) = 0 by our assumptions, we have that E(K_{n})_{p} is equal to E(K_{n}) ⊗ ℤ_{p}, and it injects into S_{p}(E/K_{n}). They coincide if and only if the p-primary part � (E/K_{n})_{p^{∞}} of the Shafarevich-Tate group of E/K_{n} is finite.

Given s = (s_{n})_{n≥1}, t = (t_{n})_{n≥1} ∈ S_{p}^{(k)} with 1 ≤ k ≤ p − 1, prop.2.11 allows us to define canonical pairings

⟨ , ⟩_{k} : S_{p}^{(k)} × S_{p}^{(k)} → I^{k}/I^{k+1}

by the rule

⟨s, t⟩_{k} = (⟨s_{n}, t_{n}⟩_{k,n})_{n≥1}.

Definition 2.16 Let N ⊂ M be an inclusion of free ℤ_{p}-modules of finite rank. Define the p-adic saturation Sat_{M}(N) of N in M to be the maximal submodule of M containing N with finite index.

The theory of elementary divisors guarantees the existence of Sat_{M}(N). To ease notations, we shall often write N̄ instead of Sat_{M}(N). In particular, we write S̄_{p}^{(k)} for Sat_{S_{p}^{(1)}}(S_{p}^{(k)}), 1 ≤ k ≤ p. Thus S̄_{p}^{(1)} = S_{p}^{(1)}. Let

US_{p}(E/K) := ⋂_{n≥1} cores_{K_{n}/K}(S_{p}(E/K_{n}))

denote the universal norm submodule of S_{p}(E/K). The next proposition is the key to relate the p-adic pairings ⟨ , ⟩_{k} to the pairings for the finite layers of K_{∞}/K. Proposition 2.17 For 1 ≤ k ≤ p the cokernel of the natural map S_{p}^{(k)} → Sel_{n}^{(k)} is bounded independently of n. Proof: By induction on k. For k = 1, the cokernel in Sel_{n} is (� (E/K)/Div(� (E/K)))[p^{n}], where Div is the functor which to every abelian group associates its divisible part. Assume that S_{p}^{(k)} → Sel_{n}^{(k)} has a cokernel whose order is bounded independently of n. Choosing an identification of I^{k}/I^{k+1} with ℤ_{p}, we may view the pairing ⟨ , ⟩_{k} as taking values in ℤ_{p}. By the structure theory of pairings over ℚ_{p}, there is a finite index submodule T_{p}^{(k)} of S_{p}^{(k)} such that the pairing ⟨ , ⟩_{k} restricted to T_{p}^{(k)} has the form

( \begin{matrix}p^{a_{1}}J \\ & p^{a_{s}}J \\ & & 0 \end{matrix} ) ,

relative to a basis e_{1}, …, e_{s}, e_{s+1}, ⋯, e_{r}, (resp. e_{1}, e1^{′}, …, e_{s}, es^{′}, e_{s+1}, …, e_{r}), where J = (1) (resp. J =

( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )) if ⟨ , ⟩_{k} is symmetric, i.e., k is odd (resp. ⟨ , ⟩_{k} is alternating, i.e., k is even). Let x be an

element of Sel_{n}^{(k+1)}. By the induction hypothesis, the cokernel of the natural map T_{p}^{(k)} → Sel_{n}^{(k)} is bounded independently of n, by p^{A}, say. Then y = p^{A}x belongs to the image of T_{p}^{(k)}. We have

⟨y, ξ⟩_{k,n} = 0,

for all ξ in Sel_{n}^{(k)}, hence for all ξ in the image of T_{p}^{(k)}. Let ỹ be any element of T_{p}^{(k)} mapping to y, so that, by definition of ⟨ , ⟩_{k},

⟨ỹ, ξ⟩_{k} ≡ 0 mod p^{n},

for all ξ in T_{p}^{(k)}. This implies that ỹ is of the form

ỹ = λ_{1}p^{n−a_{1}}e_{1} + ⋯ + λ_{s}p^{n−a_{s}}e_{s} + λ_{s+1}e_{s+1} + ⋯ + λ_{r}e_{r}

is k is odd, or

ỹ = λ_{1}p^{n−a_{1}}e_{1} + γ_{1}p^{n−a_{1}}e1^{′} + ⋯ + λ_{s}p^{n−a_{s}}e_{s} + γ_{s}p^{n−a_{s}}es^{′} + λ_{s+1}e_{s+1} + ⋯ + λ_{r}e_{r},

where λ_{i} and γ_{i} are scalars in ℤ_{p}. Let B denote the maximum of the a_{i}. Write

z̃ = p^{B}(λ_{s+1}e_{s+1} + ⋯ + λ_{r}e_{r}).

Since z̃ is in the null-space of ⟨ , ⟩_{k} resricted to T_{p}^{(k)}, it follows that p^{A}z̃ belongs to the null-space of ⟨ , ⟩_{k}. Hence p^{A+B}z̃ belongs to S_{p}^{(k+1)}. On the other hand, letting z be the image of z̃ in Sel_{n}^{(k+1)}, we have z = p^{A+B}x. Hence p^{2A+2B}x belongs to the image of S_{p}^{(k+1)}. Since A and B do not depend on n, this proves the claim.

Theorem 2.18 For 1 ≤ k ≤ p − 1, there exists a sequence of canonical pairings

⟨⟨ , ⟩⟩_{k} : S̄_{p}^{(k)} × S̄_{p}^{(k)} → I^{k}/I^{k+1} ⊗ ℚ

such that:

1. ⟨⟨s_{1}, s_{2}⟩⟩_{k} = (−1)^{k+1}⟨⟨s_{2}, s_{1}⟩⟩_{k} ∀s_{1}, s_{2} ∈ S̄_{p}^{(k)},

2. S̄_{p}^{(k+1)} is the null-space of ⟨⟨ , ⟩⟩_{k},

3. US_{p}(E/K) is contained in the null-space of all the pairings,

4. the restriction of ⟨⟨ , ⟩⟩_{1} to E(K)_{p} is equal to the p-adic height relative to (E, K_{∞}/K) (as defined in [MT1] or [Sc1]),

5. p^{A_{k}}⟨⟨ , ⟩⟩_{k} takes values in I^{k}/I^{k+1}, where p^{A_{k}} denotes the exponent of the finite group S̄_{p}^{(k)}/S_{p}^{(k)}.

Proof: Define ⟨⟨ , ⟩⟩_{k} by extending ⟨ , ⟩_{k} to S̄_{p}^{(k)} in the natural way.

1. It follows directly from thm. 2.7.

2. Observe that S̄_{p}^{(k+1)} is contained in the null-space of ⟨⟨ , ⟩⟩_{k}. As for the reverse inclusion, let y be in the null-space of ⟨⟨ , ⟩⟩_{k}. Then there exists A ≥ 0 such that p^{A}y belongs to the null-space of ⟨ , ⟩_{k}. By proposition 2.17, there exists B ≥ 0 such that p^{A+B}y maps to Sel_{n}^{(k+1)} for all n, and hence p^{A+B}y belongs to S_{p}^{(k+1)}. This proves 2.

3. By thm. 2.7, 3., it suffices to observe that US_{p}(E/K) is contained in lim_{n^{⟵}}(cores_{K_{n}/K}Sel_{p^{n}}(E/K_{n})), where

the limit is taken with respect to the maps m_{p}.

4. The homomorphisms Φ_{K_{n}/K} of §2.2 satisfy the compatibility relation [KST]

p_{m}^{n}Φ_{K_{n}/K}(P ⊗ Q ⊗ ψ) = Φ_{K_{m}/K}(P ⊗ Q ⊗ p_{m}^{n}ψ) ,

where, for n ≥ m, p_{m}^{n} : ℤ/p^{n}ℤ → ℤ/p^{m}ℤ denotes the canonical projection and, given any homomorphism ψ in Hom(Gal(K_{n}/K), ℤ/p^{n}ℤ), p_{m}^{n}ψ is viewed as a homomorphism in Hom(Gal(K_{m}/K), ℤ/p^{m}ℤ). Tan defines a p-adic height pairing by means of the homomorphism

Φ_{K_{∞}/K} = lim_{n^{⟵}} Φ_{K_{n}/K} : E(K)_{p} ⊗ E(K)_{p} ⊗ Hom(Γ, ℤ_{p}) → ℤ_{p},

where the tensor products and the homomorphisms are are of ℤ_{p}-modules. He shows that this coincides with the p-adic height of Mazur-Tate and Schneider. Statement 4. follows from thm.2.8 and the definition of ⟨⟨ , ⟩⟩_{1}.

5. We have ⟨⟨s, t⟩⟩_{k} = p^{−2A_{k}}⟨p^{A_{k}}s, p^{A_{k}}t⟩_{k} for all s, t ∈ S̄_{p}^{(k)}. Let s = (s_{n}), t = (t_{n}). It is enough to show that ⟨p^{A_{k}}s_{n}, p^{A_{k}}t_{n}⟩_{k,n} belongs to p^{A_{k}}(I_{n}^{k}/I_{n}^{k+1}) for all n. By the claim in the proof of thm.2.7, 2. there exists a homomorphism ϕ : Sel_{n} → I_{n}^{k}/I_{n}^{k+1} such that

⟨p^{A_{k}}s_{n}, p^{A_{k}}t_{n}⟩_{k,n} = ϕ(p^{A_{k}}s_{n}) = p^{A_{k}}ϕ(s_{n}).

This proves 5.

Definition 2.19 We call the canonical pairing ⟨⟨ , ⟩⟩_{k} of thm.2.18 the k-th derived p-adic height pairing.

It is possible to use the derived heights to generalize the notion of p-adic regulator. Choose a basis s_{1}, …, s_{r} for the free ℤ_{p}-module S_{p}(E/K) compatible with the filtration

S_{p}(E/K) = S̄_{p}^{(1)} ⊃ S̄_{p}^{(2)} ⊃ ⋯ ⊃ S̄_{p}^{(p)}.

This is possible because, by definition of p-adic saturation, the successive quotients S̄_{p}^{(k)}/S̄_{p}^{(k+1)} are free ℤ_{p}-modules. Say that the projection of s_{j_{k}+1}, …, s_{j_{k}+d_{k}} to S̄_{p}^{(k)}/S̄_{p}^{(k+1)} is a basis for S̄_{p}^{(k)}/S̄_{p}^{(k+1)}. Define the k-th partial regulator, 1 ≤ k ≤ p − 1,

R^{(k)} := det(⟨⟨s_{i}, s_{j}⟩⟩_{k})_{j_{k}+1≤i,j≤j_{k}+d_{k}}.

Notice that R^{(k)} depends on the choice of the basis, and is well defined up to multiplication by a p-adic unit. Given t ≥ 0, we let (I^{t}/I^{t+1} ⊗ ℚ)/ℤ_{p}^{×} denote the quotient of the multiplicative monoid I^{t}/I^{t+1} ⊗ ℚ by the action of the group of p-adic units ℤ_{p}^{×}. Let

ρ_{p}^{(k)} := rank_{ℤ_{p}}(S̄_{p}^{(k)}), ρ_{p} := ∑_{k=1}^{p} ρ_{p}^{(k)}.

Observe that if S̄_{p}^{(p)} = 0 (i.e., by thm.2.18, ⟨⟨ , ⟩⟩_{p−1} is non-degenerate) then ρ_{p} = ∑_{k=1}^{p−1} kd_{k}.

Definition 2.20 If ⟨⟨ , ⟩⟩_{p−1} is non-degenerate, we define the derived regulator R_{der} ∈ (I^{ρ_{p}}/I^{ρ_{p}+1} ⊗ℚ)/ℤ_{p}^{×} to be the product of the partial regulators R^{(1)} ⋯ R^{(p−1)}. Otherwise, we let R_{der} = 0.

By thm.2.18, R_{der} is non-zero when ⟨⟨ , ⟩⟩_{p−1} is non-degenerate. We say in this case that ρ_{p} is the order of vanishing of R_{der}. On the other hand, if US_{p}(E/K) is non-trivial, then thm.2.18 implies R_{der} = 0.

Remark 2.21 If � (E/K)_{p^{∞}} is finite, then S_{p}(E/K) ≃ E(K) ⊗ ℤ_{p} has a natural basis coming from an integral basis P_{1}, …, P_{r} for the Mordell-Weil group E(K). Let M ∈ M_{n}(ℤ_{p}) be an endomorphism sending P_{1}, …, P_{r} to a basis compatible with the filtration on S_{p}(E/K). Using this compatible basis, define the partial regulators R^{(k)} as above, and define the generalized regulator by the formula:

R_{der} = det(M)^{−2}R^{(1)} ⋯ R^{(p−1)}.

This is well-defined in I^{ρ_{p}}/I^{ρ_{p}+1} (not just up to a p-adic unit) and does not depend on the choice of M.

In the next section we shall relate R_{der} to the leading coefficient of the characteristic power series of the Pontryagin dual of Sel_{p^{∞}}(E/K_{∞}). Here we content ourselves with proving a parity statement for the order of vanishing.

Proposition 2.22 Assume that ⟨⟨ , ⟩⟩_{p−1} is non-degenerate. Then the order of vanishing of R_{der} has the same parity as the rank of the pro-p Selmer group S_{p}(E/K). In particular, if � (E/K)_{p^{∞}} is finite, then the order of vanishing of R_{der} has the same parity as the rank of the Mordell-Weil group E(K). Proof: With notations as above, we have

rank_{ℤ_{p}}(S_{p}(E/K)) = d_{1} + ⋯ + d_{p−1}.

For k even, by thm.2.18 ⟨⟨ , ⟩⟩_{k} is a non-degenerate alternating pairing on the free ℤ_{p}-module of rank d_{k} S̄_{p}^{(k)}/S̄_{p}^{(k+1)}. Hence d_{k} is even and

rank_{ℤ_{p}}(S_{p}(E/K)) ≡ d_{1} + d_{3} + ⋯ + d_{p−2} (mod 2).

On the other hand, the order of vanishing of R_{der} is equal to

d_{1} + 2d_{2} + 3d_{3} + ⋯ + (p − 1)d_{p−1} ≡ d_{1} + d_{3} + ⋯ + d_{p−2} (mod 2).

2.5 Refined Birch Swinnerton-Dyer formulae

We keep the notations of §2.4. Let

X_{∞} = Hom_{ℤ_{p}}(Sel_{p^{∞}}(E/K_{∞}), ℚ_{p}/ℤ_{p})

denote the Pontryagin dual of the Selmer group of E/K_{∞}. The main result of this section relates the order of vanishing and the leading coefficient of the characteristic ideal of X_{∞} to the derived regulator R_{der} defined before. Assume that X_{∞} is a torsion Λ-module. (Otherwise, both R_{der} and the characteristic ideal vanish, since US_{p}(E/K) ≠ 0.) Let L_{∞} denote a characteristic power series of X_{∞}. L_{∞} is determined up to a unit of Λ. Let σ_{p} ≥ 0 be the smallest exponent such that L_{∞} belongs to I^{σ_{p}}, and let L̄_{∞} denote the image of L_{∞} in (I^{σ_{p}}/I^{σ_{p}+1})/ℤ_{p}^{×}. Write char(X_{∞}) for the characteristic ideal of X_{∞}. Clearly σ_{p} and L̄_{∞} depend only on char(X_{∞}), and therefore we may call them the order of vanishing and the leading coefficient of char(X_{∞}).

Theorem 2.23 The order of vanishing σ_{p} is greater than ρ_{p}, and we have:

L̄_{∞} = #\frac{� (E/K)_{p^{∞}}}{Div(� (E/K)_{p^{∞}})} · R_{der}

in (I^{ρ_{p}}/I^{ρ_{p}+1} ⊗ ℚ)/ℤ_{p}^{×}, where Div denotes the divisible part.

Corollary 2.24

1. Assume that ⟨⟨ , ⟩⟩_{p−1} is non-degenerate. Then σ_{p} is equal to ρ_{p}.

2. Let p^{B} be the order of the finite group � (E/K)_{p^{∞}}/Div(� (E/K)_{p^{∞}}). Then p^{B}R_{der} belongs to (I^{ρ_{p}}/I^{ρ_{p}+1})/ℤ_{p}^{×}.

Remarks

1. When ⟨⟨ , ⟩⟩_{1} is non-degenerate, then R_{der} is equal to the usual p-adic regulator, and thm.2.23 contains as a particular case the Birch Swinnerton-Dyer formulae of Schneider [Sc2], [Sc3] (cf. also [PR2], [PR3]). On the other hand, when ⟨⟨ , ⟩⟩_{1} happens to be degenerate, the order of vanishing of char(X_{∞}) is strictly greater than the order of vanishing of the classical regulator, and one needs the refined Birch Swinnerton-Dyer formula of thm.2.23 to capture the order of vanishing and the leading coefficient of char(X_{∞}). See ch. 3 for an illustration of this.

2. If ⟨⟨ , ⟩⟩_{p−1} is degenerate, i.e. R_{der} is zero, the order of vanishing of char(X_{∞}) is stricly greater than ρ_{p} by thm.2.23. It is tempting to hope that the null-space of ⟨⟨ , ⟩⟩_{p−1} reduces to the p-adic saturation ŪS_{p}(E/K) of the universal norms for almost all primes p. It would follow that ⟨⟨ , ⟩⟩_{p−1} is non-degenerate if and only if X_{∞} is a torsion Λ-module. In this case, by cor.2.24, the order of vanishig is precisely ρ_{p}. See the next chapter for the formulation of stronger conjectures of this sort, when K_{∞}/K is the anticyclotomic ℤ_{p}-extension of an imaginary quadratic field.

Proof of Thm.2.23: Given n ≥ 0 fix an admissible set Σ = Σ_{n} for (E, K_{n}/K, p^{n}). Let t = 2#Σ. Fix any bases (x_{1}, …, x_{t}) and (y_{1}, …, y_{t}) for the free Λ_{n}-modules of rank t, X = ⨁_{v∈Σ}E((K_{n})_{v})/p^{n}E((K_{n})_{v}) and Y = Sel_{p^{n}}^{Σ}(E/K_{n}), respectively. By lemma 1.7, we can view X and Y as submodules of the free rank 2t Λ_{n}-module Z = ⨁_{v∈Σ}H^{1}((K_{n})_{v}, E_{p^{n}}). Thus, by restricting the pairing ⟨ , ⟩ = ⟨ , ⟩_{(n)} of §2.3 to X × Y we find a pairing

(∗) ⟨ , ⟩ : X × Y → Λ_{n}

(denoted in the same way by abusing notation). The next two lemmas, by relating char(X_{∞}) to ⟨ , ⟩, provide a the bridge beween char(X_{∞}) and the derived heights and regulator. The reader may find all the facts about the theory of Fitting ideals we need below in [MW], Appendix.

Lemma 2.25 The Fitting ideal Fitt_{Λ_{n}}(Sel_{p^{n}}(E/K_{n})^{dual}) of the Pontryagin dual of Sel_{p^{n}}(E/K_{n}) is a principal ideal, generated by the discriminant det(⟨x_{i}, y_{j}⟩_{1≤i,j≤t}) of the pairing (∗).

Proof: Let W = ⨁_{v∈Σ}H^{1}((K_{n})_{v}, E)_{p^{n}}. By the properties of the local Tate pairing, ⟨ , ⟩ : Z×Z → Λ_{n} induces a non-degenerate pairing [ , ] : X×W → Λ_{n}. Let y′_{i} be the image of y_{i} in W, and let x_{i}^{∨} be the basis of W which is dual to x_{i} with respect to the pairing [ , ]. By lemma 1.7, the exact sequence of prop.1.6 gives a presentation of the Λ_{n}-module Sel_{p^{n}}(E/K_{n})^{dual} with t generators and t relations. Hence Fitt_{Λ_{n}}(Sel_{p^{n}}(E/K_{n})^{dual}) is a principal ideal, generated by det(a_{ij}), where yj^{′} = ∑_{i} a_{ij}x_{i}^{∨}. Since a_{ij} = [x_{i}, y′_{j}] = ⟨x_{i}, y_{j}⟩, the result follows.

Lemma 2.26 Let μ_{n} : Λ → Λ_{n} be the canonical projection. Then μ_{n}(char(X_{∞})) = Fitt_{Λ_{n}}(Sel_{p^{n}}(E/K_{n})^{dual}).

Proof: Step 1 Let J_{n} be the ideal of Λ (p^{n}, (γ^{p^{n}} − 1)). Then there is a natural identification

X_{∞}/J_{n}X_{∞} = Sel_{p^{n}}(E/K_{n})^{dual}.

For, by combining the argument in the proof of lemma 1.4 with E_{p}(K_{∞}) = 0, one can prove that the restriction map composed with the inclusion E_{p^{n}} ⊂ E_{p^{∞}} induces a natural isomorphism Sel_{p^{n}}(E/K_{n}) = Sel_{p^{∞}}(E/K_{∞})^{Γ_{n}}[p^{n}], Γ_{n} being the Galois group Gal(K_{∞}/K_{n}). By taking duals, the identification follows. Step 2 Since Λ_{n} = Λ/J_{n}, step 1 and the theory of Fitting ideals give

μ_{n}(Fitt_{Λ}(X_{∞})) = Fitt_{Λ_{n}}(Sel_{p^{n}}(E/K_{n})^{dual}).

Lemma 2.25 implies that Fitt_{Λ}(X_{∞}) is a principal ideal. The next step shows that in this case Fitt_{Λ}(X_{∞}) is equal to the characteristic ideal char(X_{∞}), concluding the proof of lemma 2.26. Step 3 Sublemma Let T be a torsion Λ-module whose Fitting ideal Fitt_{Λ}(T) is principal. Then the characteristic ideal char(T) of T is equal to Fitt_{Λ}(T). Proof of the Sublemma: We use repeatedly without explicit mention the following fact: given an exact sequence of finitely generated Λ-modules

0 → M_{1} → M → M_{2} → 0,

then

Fitt_{Λ}(M_{1})Fitt_{Λ}(M_{2}) ⊂ Fitt_{Λ}(M) ⊂ Fitt_{Λ}(M_{2}).

By the classification theorem of Λ-modules there is an exact sequence

0 → C_{1} → T → ⨁_{i}Λ/Λf_{i} → C_{2} → 0,

with C_{1} and C_{2} finite. Since T is torsion, there is also an exact sequence

0 → D_{1} → ⨁_{i}Λ/Λf_{i} → T → D_{2} → 0,

with D_{1} and D_{2} finite. The first sequence implies easily

Fitt_{Λ}(T)Fitt_{Λ}(C_{2}) ⊂ Fitt_{Λ}(T/C_{1})Fitt_{Λ}(C_{2}) ⊂ Fitt_{Λ}(⨁_{i}Λ/Λf_{i}).

Similarly, from the second sequence we get

Fitt_{Λ}(⨁_{i}Λ/Λf_{i})Fitt_{Λ}(D_{2}) ⊂ Fitt_{Λ}((⨁_{i}Λ/Λf_{i})/D_{1})Fitt_{Λ}(D_{2}) ⊂ Fitt_{Λ}(T).

Since char(T) = Fitt_{Λ}(⨁_{i}Λ/Λf_{i}), by combining the two chains of inclusions we find

Fitt_{Λ}(T)Fitt_{Λ}(C_{2})Fitt_{Λ}(D_{2}) ⊂ char(T)Fitt_{Λ}(D_{2}) ⊂ Fitt_{Λ}(T).

Fitt_{Λ}(C_{2}) and Fitt_{Λ}(D_{2}) are finite index ideals in Λ. For, C_{2} and D_{2} are finite Λ-modules, and the Fitting ideal of a finitely generated Λ-module contains a positive power of the annihilator of that module. Hence there is a finite index ideal I (equal to Fitt_{Λ}(C_{2})Fitt_{Λ}(D_{2})) such that

Fitt_{Λ}(T) · I ⊂ char(T)

with finite index. By hypothesis, Fitt_{Λ}(T) = (θ) is a principal ideal. Let δ_{1} and δ_{2} be non-zero elements of I with no common irreducible factors. Since char(T) divides δ_{1}θ and δ_{2}θ, we conclude that char(T) divides θ and Fitt_{Λ}(T) is contained in char(T) with finite index. Let char(T) · (α) = Fitt_{Λ}(T). Then the finite module char(T)/Fitt_{Λ}(T) is isomorphic to Λ/(α). Thus α must be a unit of Λ. This finishes the proof of the sublemma, and also of lemma 2.26.

Recall the filtration S_{p}(E/K) = S̄_{p}^{(1)} ⊃ ⋯ ⊃ S̄_{p}^{(p−1)} ⊃ S̄_{p}^{(p)} defined in §2.4. It ends with 0, by thm.2.18, if and only if the pairing ⟨⟨ , ⟩⟩_{p−1} is non-degenerate. For 1 ≤ k ≤ p, we let S̄el_{n}^{(k)} denote the image of S̄_{p}^{(k)} in Sel_{p^{n}}(E/K) = Sel_{n}^{(1)} under the natural map. Since S̄_{p}^{(k)} is defined as the p-adic saturation in S_{p}(E/K) of the module S_{p}^{(k)}, the S̄el_{n}^{(k)} give rise to a filtration of Sel_{p^{n}}(E/K) such that

S̄el_{n}^{(k)}/S̄el_{n}^{(k+1)} ≃ (ℤ/p^{n}ℤ)^{d_{k}}, 1 ≤ k ≤ p − 1,
S̄el_{n}^{(p)} ≃ (ℤ/p^{n}ℤ)^{d_{p}},

where d_{k} = rank_{ℤ_{p}}(S̄_{p}^{(k)}/S̄_{p}^{(k+1)}) and d_{p} = rank_{ℤ_{p}} S̄_{p}^{(p)}. In view of the lemmas 1.7 and 1.4, we can identify Sel_{p^{n}}(E/K) with the intersection submodule of the free ℤ/p^{n}ℤ-modules X^{G_{n}} = ⨁_{v∈Σ}E(K_{v})/p^{n}E(K_{v}) and Y^{G_{n}} = Sel_{p^{n}}^{Σ}(E/K). Choose a basis (x̄_{1}, …, x̄_{t}), resp. (ȳ_{1}, …, ȳ_{t}) for X^{G_{n}}, resp. Y^{G_{n}}, compatible with the above filtration. Assume that for 1 ≤ k ≤ p − 1 the projection of (x̄_{j_{k}+1}, …, x̄_{j_{k}+d_{k}}) to S̄el_{n}^{(k)}/S̄el_{n}^{(k+1)} is a basis for S̄el_{n}^{(k)}/S̄el_{n}^{(k+1)}, and that (x̄_{j_{p}+1}, …, x̄_{j_{p}+d_{p}}) is a basis for S̄el_{n}^{(p)}. And similarly for (ȳ_{j_{k}+1}, …, ȳ_{j_{k}+d_{k}}). Let (x_{1}, …, x_{t}), resp. (y_{1}, …, y_{t}) be a basis for X, resp. Y such that cores_{K_{n}/K}x_{k} = x̄_{k} and cores_{K_{n}/K}y_{k} = ȳ_{k}. Write r for rank_{ℤ_{p}}S_{p}(E/K). Then r = ∑_{i=1}^{p} d_{i}.

Lemma 2.27 Let ϵ : Λ_{n} → ℤ/p^{n}ℤ be the augmentation map and let ⟨ , ⟩ : X × Y → Λ_{n} be the pairing (∗) introduced before. Then for some unit u ∈ ℤ/p^{n}ℤ we have the equality in ℤ/p^{n}ℤ

ϵ(det(⟨x_{i}, y_{j}⟩_{r+1≤i,j≤t})) = u · #\frac{� (E/K)_{p^{∞}}}{Div(� (E/K)_{p^{∞}})}.

Proof: Of course, ϵ(det(⟨x_{i}, y_{j}⟩_{r+1≤i,j≤t})) is equal to det(ϵ⟨x_{i}, y_{j}⟩_{r+1≤i,j≤t}). By prop.1.3, 3. ϵ⟨x_{i}, y_{j}⟩ = −⟨x̄_{i}, ȳ_{j}⟩_{K,p^{n}} for 1 ≤ i, j ≤ t. Since (x̄_{1}, …, x̄_{r}) is a basis for S̄el_{n}^{(1)} ⊂ Sel_{p^{n}}(E/K), the global reciprocity law (see prop.1.2) gives

(a) ⟨x̄_{i}, ȳ_{j}⟩_{K,p^{n}} = 0, 1 ≤ i ≤ r, 1 ≤ j ≤ t.

Similarly, since (ȳ_{1}, …, ȳ_{r}) is a basis for S̄el_{n}^{(1)}, the isotropy of the local points under the local Tate pairing (§1.2.1) gives

(b) ⟨x̄_{i}, ȳ_{j}⟩_{K,p^{n}} = 0, 1 ≤ i ≤ t, 1 ≤ j ≤ r.

Suppose that the group � (E/K)_{p^{∞}}/Div(� (E/K)_{p^{∞}}) is isomorphic to ℤ/p^{α_{1}}ℤ⨁ ⋯ ⨁ℤ/p^{α_{s}}ℤ. We have s ≤ t − r. Assume, without loss of generality, that α_{i} < n. The image of Sel_{p^{n}}^{Σ}(E/K) in ⨁_{v∈Σ}H^{1}(K_{v}, E)_{p^{n}} is generated by the images of ȳ_{r+1}, …, ȳ_{t}, and it is isomorphic to the direct sum of t − r summands ℤ/p^{n−α_{1}}ℤ⨁ ⋯ ⨁ℤ/p^{n−α_{s}}⨁ℤ/p^{n}ℤ⨁ ⋯ ⨁ℤ/p^{n}ℤ. Hence, by (a), (b) and the local Tate duality we find

det(⟨x̄_{i}, ȳ_{j}⟩_{r+1≤i,j≤t}) = u_{1} · #\frac{� (E/K)_{p^{∞}}}{Div(� (E/K)_{p^{∞}})}

for a unit u_{1} of ℤ/p^{n}ℤ. The lemma follows.

Let p^{A} be the maximum of the exponents of the finite groups S̄_{p}^{(k)}/S_{p}^{(k)}, 1 ≤ k ≤ p. Then for all n ≥ 1 we have p^{A} · S̄el_{n}^{(k)} ⊂ Sel_{n}^{(k)}. Consider the discriminant det(⟨p^{A}x_{i}, p^{A}y_{j}⟩_{1≤i,j≤t}). By lemma 2.25, it generates p^{2At}Fitt_{Λ_{n}}(Sel_{p^{n}}(E/K_{n})^{dual}). Although the admissible set Σ = Σ_{n} depends on n, we may assume that t = 2#Σ is independent of n (see the remark after definition 1.5: we may assume that t be equal to 2dim_{𝔽_{p}}(Sel_{p^{n}}(E/K) ⊗ 𝔽_{p})). In particular, p^{2tA}Fitt_{Λ_{n}}(Sel_{p^{n}}(E/K_{n})^{dual}) is non-zero for n sufficiently large. For 1 ≤ k ≤ p the elements (p^{A}x̄_{j_{k}+1}, …, p^{A}x̄_{j_{k}+d_{k}}), (p^{A}ȳ_{j_{k}+1}, …, p^{A}ȳ_{j_{k}+d_{k}}), belong to Sel_{n}^{(k)}. Choose (x′_{j_{k}+1}, …, x′_{j_{k}+d_{k}}) in X and (y′_{j_{k}+1}, …, y′_{j_{k}+d_{k}}) in Y such that:

1. cores_{K_{n}/K}x′_{j_{k}+i} = p^{A}x̄_{j_{k}+i}, cores_{K_{n}/K}y′_{j_{k}+i} = p^{A}ȳ_{j_{k}+i};

2. D_{n}^{(k−1)}x′_{j_{k}+i} belongs to Sel_{p^{n}}(E/K_{n}), D_{n}^{(k−1)}y′_{j_{k}+i} belongs to Sel_{p^{n}}(E/K_{n}).

For r+1 ≤ i ≤ t, let x′_{i} = p^{A}x_{i} and y′_{i} = p^{A}y_{i}. Let U and V in M_{t}(Λ_{n}) be t×t matrices with entries in Λ_{n} such

that (x_{1}, …, x_{t})U = (x1^{′}, …, xt^{′}), and (y_{1}, …, y_{t})V = (y1^{′}, …, yt^{′}). Since cores_{K_{n}/K}x′_{i} = cores_{K_{n}/K}p^{A}x_{i}

and cores_{K_{n}/K}y′_{i} = cores_{K_{n}/K}p^{A}y_{i}, we find ϵ(U) = ϵ(V) = p^{A}I_{t}, where I_{t} is the identity matrix in M_{t}(ℤ/p^{n}ℤ). Thus, we may replace det(⟨p^{A}x_{i}, p^{A}y_{j}⟩_{1≤i,j≤t}) by det(⟨x′_{i}, y′_{j}⟩_{1≤i,j≤t}) in the computation of the leading coefficient of p^{2At}Fitt_{Λ_{n}}(Sel_{p^{n}}(E/K_{n})^{dual}). Observe that for j_{k} + 1 ≤ i ≤ j_{k} + d_{k}, ⟨x′_{i}, y⟩ belongs to I_{n}^{k} for all y ∈ Y. For, D_{n}^{(k−1)}⟨x′_{i}, y⟩ = 0, by prop.1.3, 5. Let σ_{p} = d_{1} + 2d_{2} + ⋯ + (p − 1)d_{p−1} + pd_{p}. Then

det(⟨x′_{i}, y′_{j}⟩_{1≤i,j≤t}) ∈ I_{n}^{σ_{p}}.

Thus the order of vanishing of char(X_{∞}) is greater or equal than σ_{p}. This is enough to prove the theorem when ⟨⟨ , ⟩⟩_{p−1} is degenerate, i.e. d_{p} ≠ 0. For, σ_{p} > ρ_{p} = d_{1} + 2d_{2} + ⋯ + (p − 1)d_{p−1} and R_{der} = 0 by definition. From now on assume that ⟨⟨ , ⟩⟩_{p−1} is non-degenerate, hence ρ_{p} = σ_{p}. By lemma 2.27 and the above remarks, we find the equality in I^{ρ_{p}}/I^{ρ_{p}+1}

det(⟨x′_{i}, y′_{j}⟩_{1≤i,j≤t}) = u_{n} · p^{2A(t−r)} · #\frac{� (E/K)_{p^{∞}}}{Div(� (E/K)_{p^{∞}})} · ∏_{k=1}^{p−1} det(⟨x′_{i}, y′_{j}⟩_{j_{k}+1≤i,j≤j_{k}+d_{k}}),

for some unit u_{n} of ℤ/p^{n}ℤ. By the definition of the derived heights ⟨ , ⟩_{k,n} we have in I_{n}^{ρ_{p}}/I_{n}^{ρ_{p}+1}

∏_{k=1}^{p−1} det((⟨p^{A}x̄_{i}, p^{A}ȳ_{j}⟩_{k,n})_{j_{k}+1≤i,j≤j_{k}+d_{k}}) = ∏_{k=1}^{p−1} det(⟨x′_{i}, y′_{j}⟩_{j_{k}+1≤i,j≤j_{k}+d_{k}}).

But

p^{2Ar}μ_{n}(R_{der}) = ∏_{k=1}^{p−1} det((⟨p^{A}x̄_{i}, p^{A}ȳ_{j}⟩_{k,n})_{j_{k}+1≤i,j≤j_{k}+d_{k}}).

Theorem 2.23 follows.

3 Elliptic curves and anticyclotomic Z_{p}-extensions

In this chapter K_{∞} denotes the anticyclotomic ℤ_{p}-extension of an imaginary quadratic field K, defined by adjoining to K values of modular functions at complex multiplication points of p-power conductor. It is the only ℤ_{p}-extension of K that is dihedral over ℚ. We require that (E, p, K_{∞}/K) satisfy the assumptions of §1.1. We also assume that � (E/K)_{p^{∞}} is finite. Thus S_{p}(E/K) = E(K)_{p}, and we can readily translate standard analytic conjectures into statements on the rank of the pro-p Selmer group S_{p}(E/K). The goal of this chapter is to sketch the Iwasawa theory, still largely conjectural, for elliptic curves over ℚ with values in the intermediate extensions of K_{∞}/K. In this situation the degeneracy of the p-adic height pairing tends to be the rule, and the theory of derived heights allows a conceptual formulation of the theory.

3.1 The conjectures of Mazur

We begin by recalling some conjectures of Mazur (see [Ma1], and also [Ma2]) concerning the null-space of the p-adic height. Let τ denote a fixed complex conjugation. Thus, given g ∈ Gal(K_{∞}/K), we have g^{τ} = g^{−1}. Write S_{p}(E/K)^{±} for the submodule of S_{p}(E/K) on which τ acts as ±1, and r_{±} for the ℤ_{p}-rank of S_{p}(E/K)^{±}. Write r = r_{+} + r_{−} for rank_{ℤ_{p}}(S_{p}(E/K)). The following Galois equivariance property of the derived p-adic heights holds.

Lemma 3.1 For all s_{1}, s_{2} ∈ S̄_{p}^{(k)}, ⟨⟨τs_{1}, τs_{2}⟩⟩_{k} = (−1)^{k}⟨⟨s_{1}, s_{2}⟩⟩_{k}. Proof: The Galois equivariance of the local Tate pairing implies immediately the equality

⟨τx, τy⟩_{(n)} = τ⟨x, y⟩_{(n)}τ^{−1},

where ⟨ , ⟩_{(n)} denotes the Λ_{n}-valued pairing of §2.3. Hence by definition

⟨⟨τs_{1}, τs_{2}⟩⟩_{k} = ⟨⟨s_{1}, s_{2}⟩⟩_{k}^{∗},

where ∗ denotes the involution of Λ given on group-like elements by g^{∗} = g^{−1}. The lemma follows from the fact that ∗ acts on I^{k}/I^{k+1} ⊗ ℚ as (−1)^{k}.

In particular, we find that the p-adic height pairing ⟨⟨ , ⟩⟩_{1} is trivial when restricted to S_{p}(E/K)^{+}×S_{p}(E/K)^{+} and S_{p}(E/K)^{−} × S_{p}(E/K)^{−}. Hence, the null-space of ⟨⟨ , ⟩⟩_{1} has rank at least |r_{+} − r_{−}|. Following Mazur’s terminology we give:

Definition 3.2 We say that E is in the generic case if either E does not have complex multiplications or K is not the complex multiplication field of E.

Conjecture 3.3 (Mazur) Assume that E is in the generic case. Then the null-space of ⟨⟨ , ⟩⟩_{1} has rank |r_{+} − r_{−}|, i.e. rank_{ℤ_{p}} S̄_{p}^{(2)} = |r_{+} − r_{−}|.

In other words, the anticyclotomic p-adic height should be as non-degenerate as possible. We may reformulate conjecture 3.3 by saying that the pairing

⟨⟨ , ⟩⟩_{1} : S_{p}(E/K)^{+} × S_{p}(E/K)^{−} → I/I^{2},

obtained by restriction of ⟨⟨ , ⟩⟩_{1}, is either left or right non-degenerate. Since, by thm.2.18, the universal norm submodule US_{p}(E/K) is contained in the null-space of ⟨⟨ , ⟩⟩_{1}, conjecture 3.3 implies that US_{p}(E/K) is contained in either S_{p}(E/K)^{+} or S_{p}(E/K)^{−}, whichever has the larger rank. The following “growth number” conjecture gives precise information on the size of US_{p}(E/K).

Conjecture 3.4 (Mazur) Assume that E is in the generic case. If rank_{ℤ_{p}}(S_{p}(E/K)) is even, resp. odd then US_{p}(E/K) = 0, resp. US_{p}(E/K) ≃ ℤ_{p}.

Equivalently (see for example [B], §II.1) the Pontryagin dual X_{∞} of Sel_{p^{∞}}(E/K_{∞}) is a torsion Λ-module, resp. has rank 1 over Λ when rank_{ℤ_{p}}(S_{p}(E/K)) is even, resp. odd .

When K_{∞} is the cyclotomic ℤ_{p}-extension of a number field K, Schneider [Sc1] conjectures that the p-adic height is non-degenerate. In this case the generalized regulator R_{der} coincides with the usual p-adic regulator, and the characteristic ideal of the Λ-torsion module X_{∞} vanishes to order r = rank_{ℤ}E(K) (see thm.2.23). Moreover, the Mordell-Weil group E(K_{∞}) is a finitely generated ℤ-module, since E(K_{∞})_{tors} is finite under our assumptions (by [Ma3], §6).

3.2 Applications of the derived p-adic heights

Consider the second derived p-adic height

⟨⟨ , ⟩⟩_{2} : S̄_{p}^{(2)} × S̄_{p}^{(2)} → I^{2}/I^{3} ⊗ ℚ.

Assume conjecture 3.3. Then rank_{ℤ_{p}}(S̄_{p}^{(2)}) = |r_{+} − r_{−}| and S̄_{p}^{(2)} is contained in one of the eigenspaces of τ acting on S_{p}(E/K). Thus lemma 3.1 does not force any degeneracy of ⟨⟨ , ⟩⟩_{2}. On the other hand, we have the a priori information that the universal norms US_{p}(E/K) are contained in the null-space of ⟨⟨ , ⟩⟩_{2}. Conjecture 3.4 indicates that US_{p}(E/K) need not be trivial (see also §3.2.3 below). Inspired by the philosophy of conjecture 3.3, we formulate the following:

Conjecture 3.5 Assume that E is in the generic case. Then the null-space of the second derived p-adic height ⟨⟨ , ⟩⟩_{2} is equal to ŪS_{p}(E/K), the p-adic saturation of US_{p}(E/K) in S_{p}(E/K).

For the sake of clarity, let us make two separate discussions according to the parity of the rank of S_{p}(E/K).

3.2.1 The even rank case Assume that rank_{ℤ_{p}}S_{p}(E/K) is even. Assuming conjecture 3.4, we may reformulate 3.5 as follows.

Conjecture 3.6 Assume that E is in the generic case, and that rank_{ℤ_{p}}S_{p}(E/K) is even. Then the second derived p-adic height pairing is non-degenerate.

By theorem 2.18, this is equivalent to S̄_{p}^{(3)} = 0. Since we are assuming that US_{p}(E/K) is trivial, X_{∞} is a torsion Λ-module. By combining theorem 2.23 with the above conjecture, we obtain a conjectural statement for the order of vanishing of the characteristic ideal of X_{∞}.

Conjecture 3.7 Assume that E is in the generic case and that rank_{ℤ_{p}}(S_{p}(E/K)) is even. Then the characteristic ideal of X_{∞} vanishes to order r + |r_{+} − r_{−}| = 2max(r_{+}, r_{−}).

3.2.2 The odd rank case Assume that rank_{ℤ_{p}}S_{p}(E/K) is odd. Assuming conjecture 3.4, we reformulate 3.5 as follows.

Conjecture 3.8 Assume that E is in the generic case and that rank_{ℤ_{p}}S_{p}(E/K) is odd. Then the null-space of the second derived p-adic height pairing is a free rank one ℤ_{p}-module equal to ŪS_{p}(E/K).

Conjecture 3.8 provides a characterization of the space of universal norms US_{p}(E/K) (or, rather, its p-adic saturation) in terms of the second derived height. This brings up the challenge of finding a computational definition of the derived heights, in order to have a way of approximating numerically US_{p}(E/K).

If ŪS_{p}(E/K) is isomorphic to ℤ_{p}, then X_{∞} is a Λ-module of rank 1 and, by definition, the derived regulator R_{der} vanishes. The theory of derived heights may be used to define a modified derived regulator R′_{der}, which accounts for this situation. By theorem 2.18, conjecture 3.8 amounts to S̄_{p}^{(3)} = ŪS_{p}(E/K). Hence, with notations as in §2.3, the partial regulators R^{(k)} vanish for 3 ≤ k ≤ p−1. Let d_{i} = rank_{ℤ_{p}} S̄_{p}^{(i)}/S̄_{p}^{(i+1)}, i = 1, 2. Then we define

R′_{der} := R^{(1)}R^{(2)} ∈ (I^{d_{1}+2d_{2}}/I^{d_{1}+2d_{2}+1} ⊗ ℚ)/ℤ_{p}^{×}.

Note that the order of vanishing of R′_{der} is exactly d_{1} + 2d_{2}. By the above conjectures, this is equal to (r − 1) + (|r_{+} − r_{−}| − 1) = 2(max(r_{+}, r_{−}) − 1). Along the lines of theorem 2.23, one can prove a theorem connecting the modified regulator R′_{der} to the leading coefficient of the characteristic ideal of the Λ-torsion submodule (X_{∞})_{tors} of X_{∞}. (For reasons of brevity, we do not give details.)

Conjecture 3.9 Assume that E is in the generic case, and that rank_{ℤ_{p}}(S_{p}(E/K)) is odd. Then the characteristic ideal of (X_{∞})_{tors} vanishes to order (r − 1) + (|r_{+} − r_{−}| − 1).

3.2.3 Heegner points (Reference: [B]) Assume that E/ℚ is a modular elliptic curve and that K satisfies the Heegner hypothesis, i.e. all rational primes dividing the conductor of E are split in K. Then the analytic rank of E(K)_{p} is odd. Assuming standard conjectures, so is rank_{ℤ_{p}}(S_{p}(E/K)). There is a collection of complex multiplication points defined over K_{∞}, called Heegner points. Let E(E/K_{n})_{p} ⊂ S_{p}(E/K_{n}) denote the cyclic ℤ_{p}[Gal(K_{n}/K)]-module generated by any Heegner point of level K_{n}. Define the Λ-modules

Ê(E/K_{∞})_{p} := lim_{n^{⟵}} E(E/K_{n})_{p},

Ŝ_{p}(E/K_{∞}) := lim_{n^{⟵}} S_{p}(E/K_{n}),

where the projective limits are with respect to the natural corestriction mappings. Note that Ê(E/K_{∞})_{p} is contained in Ŝ_{p}(E/K_{∞}). One can show that Ŝ_{p}(E/K_{∞}) is a free Λ-module of finite rank (equal to the ℤ_{p}-rank of US_{p}(E/K)) and that Ê(E/K_{∞})_{p} is a cyclic Λ-module (cf. [B], ch. II). Thus Ê(E/K_{∞})_{p} is either isomorphic to Λ or 0. Analytic evidence combined with a theorem of Gross-Zagier lead to the natural expectation that some of the Heegner points of level K_{n} have infinite order, i.e., Ê(E/K_{∞})_{p} is isomorphic to Λ. Assuming this, it is proved in [B], §III.1 that the conclusion of conjecture 3.4, i.e. US_{p}(E/K) ≃ ℤ_{p}, holds. This is equivalent to Ŝ_{p}(E/K_{∞}) being a free Λ-module of rank 1. In this situation, Ŝ_{p}(E/K_{∞})/Ê(E/K_{∞})_{p} is a cyclic torsion Λ-module. The next conjecture blends a conjecture of B. Perrin-Riou [PR1] with conj. 3.9. Let char(Ŝ_{p}(E/K_{∞})/Ê(E/K_{∞})_{p}) denote the characteristic ideal of Ŝ_{p}(E/K_{∞})/Ê(E/K_{∞})_{p}. Conjecture 3.10 char(Ŝ_{p}(E/K_{∞})/Ê(E/K_{∞})_{p})^{2} vanishes to order (r − 1) + (|r_{+} − r_{−}| − 1). The results of [B], ch. III, based on the ideas of Kolyvagin [Ko], provide evidence for conjecture 3.10.

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