ANTICYCLOTOMIC IWASAWA THEORY OF CM ELLIPTIC CURVES II

ADEBISI AGBOOLA AND BENJAMIN HOWARD

Abstract. We study the Iwasawa theory of a CM elliptic curve E in the anticyclotomic Z_{p}- extension D_{∞} of the CM field K, where p is a prime of good, supersingular reduction for E. Our main result yields an asymptotic formula for the corank of the p-primary Selmer group of E along the extension D_{∞}/K.

1. Introduction

Let E be an elliptic curve over Q. Whilst much is known about the Iwasawa theory of E for primes of ordinary reduction, the same is unfortunately not true of Iwasawa theory at supersingular primes, for in this case the Iwasawa modules that one naturally considers are not torsion, and the obvious candidates for p-adic L-functions do not lie in the Iwasawa algebra. Nevertheless, there has recently been a great deal of progress in the study of the Iwasawa theory of elliptic curves at supersingular primes. In particular, S. Kobayashi has recently formulated a cyclotomic main conjecture for E within this framework (see [5]). His conjecture relates certain restricted ‘plus/minus’ Selmer groups of E to certain modified p-adic L-functions defined by R. Pollack (see [9]), and it is equivalent to a cyclotomic main conjecture that was proposed earlier by K. Kato and B. Perrin-Riou (see [4], [7] [6]). Kobayashi’s conjecture has recently been proved by Rubin and Pollack (see [15]) when E has complex multiplication, and Kobayashi himself, using methods of Kato, proves one divisibility of the main conjecture in the non-CM case. In both cases, the plus/minus Selmer groups are cotorsion modules over the cylotomic Iwasawa algebra, and so the corank of the p-Selmer group remains bounded as one ascends the cyclotomic Z_{p}-extension.

Suppose now that E has complex multiplication by the maximal order O of an imaginary quadratic field K. Let ψ denote the K-valued grossencharacter associated to E, and write 𝔣 for the conductor of ψ. Fix once and for all a rational prime p > 3 at which E has good reduction, and which is inert in K. Then E has supersingular reduction at p. We write 𝔭 for the unique prime of K above p. Let D_{∞} be the anticyclotomic Z_{p} extension of K, and let D_{n} ⊂ D_{∞} be the subfield such that [D_{n} : K] = p^{n}. The prime 𝔭 is totally ramified in D_{∞}, and we let 𝔭 also denote the unique place of D_{∞} above 𝔭.

In this paper, we study the Iwasawa theory of E over D_{∞}. We define anticyclotomic versions of Kobayashi’s restricted plus/minus Selmer groups, and we analyse their structure using the Euler system of twisted elliptic units (cf. [1]). In contrast to what happens in the cyclotomic case, it turns out that one of the restricted Selmer groups is a cotorsion Iwasawa module, while the other is not, and which module is cotorsion is determined by the sign in the functional equation of L(E/Q, s). Our main result, predicted by R. Greenberg [3, p. 247], is as follows (somewhat more information is contained in Theorem 5.4).

Date: Final version, March 23, 2005.

2000 Mathematics Subject Classification. 11G05, 11R23, 11G16.

The first author was partially supported by NSF grant DMS-0070449.

The second author was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship.

Theorem A. Let ϕ be Euler’s function and let ε = ±1 be the sign in the functional equation of L(E/Q, s). Write Sel_{𝔭^{∞}}(E/D_{n}) for the 𝔭-primary Selmer group of E/D_{n}, and O_{𝔭} for the local completion of O at 𝔭. Then there is an integer e, independent of n, such that

corank_{O_{𝔭}} Sel_{𝔭^{∞}}(E/D_{n}) = e + ∑_{1≤k≤n, (−1)^{k}=ε} ϕ(p^{k})

for all n ≫ 0.

The results in this paper may be viewed as a first step towards a supersingular main conjecture of the same type as that considered in [1]. However, in the present setting, we do not know how to define suitable anticyclotomic analogues of Pollack’s plus/minus p-adic L-functions. The essential missing ingredient is a construction of local elements along the lines of [5, §8.4].

2. Selmer groups

We write

T = T_{p}(E), W = E[p^{∞}]

for the p-adic Tate module and the goup of p-power torsion points in E(\overline{K}) respectively. Let F/K be any finite extension. For any place v of F, we define H_{f}^{1}(F_{v}, W) to be the image of E(F_{v})⊗(Q_{p}/Z_{p}) under the Kummer map

E(F_{v}) ⊗ (Q_{p}/Z_{p}) → H^{1}(F_{v}, W),

and we write H_{f}^{1}(F_{v}, T) for the orthogonal complement of H_{f}^{1}(F_{v}, W) with respect to the local Tate pairing. Note that H_{f}^{1}(F_{v}, W) = 0 if v ∤ p. If c ∈ H^{1}(F, W), then we write loc_{v}(c) for the image of c in H^{1}(F_{v}, W).

We define

• the relaxed Selmer group Sel_{rel}(F, W) by

Sel_{rel}(F, W) = {c ∈ H^{1}(F, W) | loc_{v}(c) ∈ H_{f}^{1}(F_{v}, W) for all v not dividing p};

• the true Selmer group Sel(F, W) by

Sel(F, W) = {c ∈ H^{1}(F, W) | loc_{v}(c) ∈ H_{f}^{1}(F_{v}, W) for all v};

• the strict Selmer group Sel_{str}(F, W) by

Sel_{str}(F, W) = {c ∈ Sel(F, W) | loc_{v}(c) = 0 for all v dividing p} .

We also define Sel_{rel}(F, T), Sel(F, T) and Sel_{str}(F, T) in a similar way. It follows from the definitions that there are inclusions

Sel_{str}(F, W) ⊂ Sel(F, W) ⊂ Sel_{rel}(F, W),

and similarly with W replaced by T. If F/K is an infinite extension, we define

Sel_{∗}(F, W) = \underrightarrow{lim} Sel_{∗}(F′, W) S_{∗}(F, T) = \underleftarrow{lim} Sel_{∗}(F′, T),

where the limits are taken with respect to restriction and corestriction, respectively, over all subfields F′ ⊂ F finite over K.

We now give the definition of a slightly modified (see Remark 3.2 below) form of Kobayashi’s restricted plus/minus Selmer groups. Let E denote the formal group of E over K_{𝔭}. Since E has supersingular reduction at p > 3, it is a standard fact that E is isomorphic to the unique (up to isomorphism) Lubin-Tate formal goup over K_{𝔭} with parameter −p. For n ≥ 0, let Ξ_{n}^{−} be the set of characters of Γ_{n} of exact order p^{k} with k odd, together with the trivial character. Let Ξ_{n}^{+} be the set of characters of Γ_{n} of exact order p^{k} with k even, excluding the trivial character. Define subspaces of E(D_{n,𝔭}) ⊗_{O_{𝔭}} K_{𝔭} by

E_{±}(D_{n,𝔭}) = {x ∈ E(D_{n,𝔭}) ⊗ K_{𝔭} | ∑_{σ∈Γ_{n}} χ(σ)x^{σ} = 0, ∀χ ∈ Ξ_{n}^{∓} } .

Let H_{±}^{1}(D_{n,𝔭}, W) be the image of E_{±}(D_{n,𝔭}) under the Kummer map

E(D_{n,𝔭}) ⊗ K_{𝔭} ⟶ E(D_{n,𝔭}) ⊗ (K_{𝔭}/O_{𝔭}) ⟶ H^{1}(D_{n,𝔭}, W)

and let H_{±}^{1}(D_{n,𝔭}, T) be the orthogonal complement of H_{±}^{1}(D_{n,𝔭}, W) with respect to the local Tate pairing. We define

Sel_{±}(D_{n}, W) = {c ∈ Sel_{rel}(D_{n}, W) | loc_{𝔭}(c) ∈ H_{±}^{1}(D_{n,𝔭}, W)};
Sel_{±}(D_{n}, T) = {c ∈ Sel_{rel}(D_{n}, T) | loc_{𝔭}(c) ∈ H_{±}^{1}(D_{n,𝔭}, T)} .

It follows from the definitions that we have inclusions

Sel_{str}(D_{n}, W) ⊂ Sel_{±}(D_{n}, W) ⊂ Sel(D_{n}, W),
Sel(D_{n}, T) ⊂ Sel_{±}(D_{n}, T) ⊂ Sel_{rel}(D_{n}, T).

In the limit, we define

Sel_{±}(D_{∞}, W) = \underrightarrow{lim} Sel_{±}(D_{n}, W) S_{±}(D_{∞}, T) = \underleftarrow{lim} Sel_{±}(D_{n}, T),

where the inverse limits are taken with respect to restriction and corestriction, respectively. In order to ease notation, we shall sometimes write

Sel_{∗}^{∞} = Sel_{∗}(D_{∞}, W) S_{∗} = S_{∗}(D_{∞}, T).

3. Ranks

Let 𝔞 be an integral ideal of O coprime to 6p𝔣, and write K_{𝔞} for the union of all ray class fields of K of conductor prime to 𝔞. Let c_{ell,𝔞} denote the Euler system of elliptic units for (Z_{p}(1), 𝔣p, K_{𝔞}) in the sense of [14]. Twisting c_{ell,𝔞} by the character ω_{cyc}^{−1}ψ_{𝔭}, where

ψ_{𝔭} : Gal(\overline{K}/K) ⟶ Aut_{O_{K}}(T) ≅ O_{𝔭}^{×},

and ω_{cyc} is the cyclotomic character, yields an Euler system c_{𝔞} for (T, 𝔣p, K_{𝔞}) (see [14, Chapter 6]). Then c_{𝔞}(F) ∈ Sel_{rel}(F, T) for every F ⊂ K_{𝔞} finite over K. For any L ⊂ K_{∞}, let

c_{𝔞}(L) = \underleftarrow{lim} c_{𝔞}(F′) ∈ S_{rel}(L, T),

where the inverse limit is taken over all subfields F′ of L that are finite over K. Let C_{𝔞}(F) be the O_{𝔭}[[Gal(F/K]]-submodule of S_{rel}(F, T) generated by c_{𝔞}(F), and write C(F) for the submodule generated by C_{𝔞}(F) as 𝔞 varies over all ideals that are coprime to 6p𝔣. We set C = C(D_{∞}).

Define

H_{±}^{1} = \underleftarrow{lim} H_{±}^{1}(D_{n,𝔭}, T) H^{1} = \underleftarrow{lim} H^{1}(D_{n,𝔭}, T)

and

H_{±}^{1}(D_{∞,𝔭}, W) = \underrightarrow{lim} H_{±}^{1}(D_{n,𝔭}, W).

Let W (ψ) denote the root number of ψ. In particular W (ψ) = ±1 and is equal to the sign in the functional equation of L(E/Q, s).

Proposition 3.1. The image of C in H^{1} is nontrivial, and lies in H_{ε}^{1} if and only if ε is equal to the sign of W (ψ). In particular C is nontrivial, and is contained in S_{ε} if and only if ε is the sign of W (ψ).

Proof. Let χ be any primitive character of Gal(D_{n}/K) for n > 0, and write W (χψ) for the root number of χψ. The following formula is proved by Greenberg in [3, page 247]:

W (χψ) = (−1)^{n+1}W (ψ).  (3.1)

If (−1)^{n} = W (ψ), the functional equation of L(E/Q, s) forces L(χψ, 1) = 0. On the other hand, the main result of [10] shows that if (−1)^{n} = −W (ψ), then L(χψ, 1) ≠ 0 for all but finitely many χ. The claim now follows from the reciprocity law of Coates-Wiles, which relates the localization of the elliptic units to the special value of twists of L(ψ, s) (see [15, Theorem 5.1] for example). □

Remark 3.2. The equality (3.1) (which is visibly incorrect when n = 0) is the reason for placing the trivial character in Ξ_{n}^{−}. In particular, our definitions differ from those of [15, Definition 3.1].

The reader should also note that in [12], the characters of Γ_{n} are indexed according to the parity of their conductors, while we have indexed them according to the parity of their orders. Hence our Ξ^{+} (respectively Ξ^{−}) is denoted by Ξ^{−} (respectively Ξ^{+}) in [12]. □

Set Λ = O_{𝔭}[[Gal(D_{∞}/K)]]. For any finitely generated Λ-module M, we write Char_{Λ}(M) for the characteristic ideal of M in Λ, and rk_{Λ}(M) for the Λ-rank of M. Define a Λ-module

X_{∗} = Hom_{Z_{p}}(Sel_{∗}^{∞}, Q_{p}/Z_{p}).

Let ι : Λ → Λ denote the canonical involution on Λ which is induced by inversion on group-like elements. We adopt the convention that Λ acts on X_{∗} via the rule (λ · f)(x) = f(λ^{ι}x) (cf. [1, Remark 1.18]).

The following two propositions are consequences of the work of Rubin.

Proposition 3.3. (i) The Λ-modules S_{rel} and X_{str} are torsion-free of rank one, and torsion, respectively. The Λ-module X_{rel} has rank one.

(ii) There is an equality of characteristic ideals

Char_{Λ}(X_{str}) = Char_{Λ}(S_{rel}/C).

Proof. The fact that S_{rel} is torsion-free of the same rank as X_{rel} may be proved exactly as in [1, Lemma 1.1.9] (the proof of which is essentially the same as that of [8, Proposition 4.2.3]). The remaining claims of (i) follow from the nontriviality of C using the theory of Euler systems as in [14]. Using (i), (ii) may be deduced from Rubin’s two variable main conjecture [13, Theorem 4.1(ii)] exactly as in [1, Proposition 2.4.16]. □

Proposition 3.4. The Λ-module H^{1} is torsion free of rank 2. The modules H_{±}^{1} have Λ-rank 1 and satisfy H_{+}^{1} ∩ H_{−}^{1} = 0. The modules H_{±}^{1}(D_{∞}, W) have Λ-corank one.

Proof. Write H_{f}^{1}(D_{∞,𝔭}, W) = \underrightarrow{lim} H_{f}^{1}(D_{n,𝔭}, W), where the injective limit is taken with respect to restriction maps. Let E[𝔭^{∞}] and E[𝔭^{∞}] denote the 𝔭-primary torsion subgroups of E and E respectively, and write K(E[𝔭^{∞}]) for the field obtained by adjoining the elements of E[𝔭^{∞}] to K. Set

V_{∞} = Hom_{O_{𝔭}} (H_{f}^{1}(D_{∞,𝔭}, W), E[𝔭^{∞}]) , V^{±} = Hom_{O_{𝔭}} (\frac{H_{f}^{1}(D_{∞,𝔭}, W)}{H_{±}^{1}(D_{∞,𝔭}, W)}, E[𝔭^{∞}]) .

We may view V_{∞} and V^{±} as being Λ-modules by identifying Gal(D_{∞}/K) with a subgroup of Gal(K(E[𝔭^{∞}])/K) in the obvious way. It is shown in [12, Propositions 1.1 and 8.1] that the Λmodule V_{∞} is torsion-free of rank 2, while the Λ-modules V^{±} are of rank one and satisfy V_{+}∩V_{−} = 0. The proposition now follows from the fact that fixing an identification of K_{𝔭}/O_{𝔭} with E[𝔭^{∞}] induces Λ-module isomorphisms

V ≃ H ⊗ Hom_{O_{𝔭}}(O_{𝔭}, T), V_{±} ≃ H_{±} ⊗ Hom_{O_{𝔭}}(O_{𝔭}, T).  (3.2)
□

Conjecture 3.5. (Rubin [12, Conjecture 2.2]) H^{1} = H_{+}^{1} ⊕ H_{−}^{1}.

Theorem 3.6. We have rk_{Λ}(S_{±}) = rk_{Λ}(X_{±}). If the sign of W (ψ) is ε, then X_{ε} has Λ-rank one, and X_{−ε} is Λ-torsion. In particular S_{str} = S_{−ε} = 0, as S_{rel} is torsion-free.

Proof. Global duality (see [14, Theorem 1.7.3]) gives the exact sequence

0 ⟶ S_{±} ⟶ S_{rel} ⟶ H^{1}/H_{±}^{1} ⟶ X_{±} ⟶ X_{str} ⟶ 0.  (3.3)

The first claim now follows from Propositions 3.3 and 3.4.

Since S_{±} ⊂ S_{rel}, the Λ-rank of S_{±} is at most one, and as C ⊂ S_{ε} is non-trivial, we see that the Λrank of S_{ε} is in fact equal to one. Next, we observe that if rk_{Λ}(S_{−ε}) = 1, then rk_{Λ}(S_{+}∩S_{−}) = 1, since both S_{+} and S_{−} are submodules of the rank one Λ-module S_{rel}. By Proposition 3.4, S_{+} ∩S_{−} ⊂ S_{str}, and so also rk_{Λ}(S_{str}) = 1. But then S_{rel}/S_{str} is a Λ-torsion module. This quotient injects into H^{1} which is torsion-free by Proposition 3.4. We conclude that S_{str} = S_{rel} and that the localization map S_{rel} ⟶ H^{1} is trivial, contradicting Proposition 3.1.

It now follows that rk_{Λ}(S_{−ε}) = rk_{Λ}(S_{str}) = 0, and since S_{rel} is torsion-free, this implies that both S_{−ε} and S_{str} are equal to zero. □

4. Characteristic ideals

Theorem 4.1. We have the equality of characteristic ideals

Char_{Λ}(X_{rel,Λ−tor}) = Char_{Λ}(X_{str}).

Proof. Let K_{∞}/K denote the unique Z_{p}^{2}-extension of K, and set Λ(K_{∞}) := O_{𝔭}[[Gal(K_{∞}/K)]]. Write X(K_{∞}) := Hom(Sel(K_{∞}), Q_{p}/Z_{p}). It follows from Rubin’s proof of the main conjecture that rk_{Λ(K_{∞})}(X(K_{∞})) = 1 (see e.g. [15, Remark 2.2] and[13, Theorem 5.3(iii)]). As X_{rel} = X (see [2, Remark 3.3], for instance), Proposition 3.3(i) implies that rk_{Λ}(X) = 1 also. Hence, if γ_{1} is any topological generator of Gal(K_{∞}/D_{∞}), then, since

\frac{X(K_{∞})}{(γ_{1} − 1)X(K_{∞})} ≃ X

(see [11, Proposition 1.2 and Theorem 2.1] or [2, p. 364–365]), we deduce that γ_{1} − 1 is coprime to Char_{Λ(K_{∞})}(X(K_{∞})_{tor}). The theorem now follows directly from [2, Theorem 3.24] and [1, Lemma 2.1.2] (see also [16, Corollary 6.5] for a more general result along these lines). □

Theorem 4.2. Suppose that the sign of W (ψ) is equal to ε. Then

Char_{Λ} (X_{ε,Λ−tor}) Char_{Λ} (\frac{H_{ε}^{1}}{S_{ε}} ) = Char_{Λ} (\frac{S_{rel}}{C} ) .  (4.1)

If we assume that Conjecture 3.5 is true then S_{ε} = S_{rel} and

Char_{Λ} (X_{−ε}) = Char_{Λ} (\frac{H_{ε}^{1}}{C} ) .  (4.2)

Proof. From Proposition 3.1, we see that C ⊂ S_{ε}, and so rk_{Λ}(C) = rk_{Λ}(S_{ε}) = 1. Via global duality (see [14, Theorem 1.7.3]), together with the fact that S_{str} = 0, we have the exact sequence

0 → S_{ε} → H_{ε}^{1} → X_{rel} → X_{ε} → 0.  (4.3)

As H^{1}/S_{ε} is Λ-torsion, it is not hard to check that this in turn yields the exact sequence

0 → H_{ε}^{1}/S_{ε} → X_{rel,Λ−tor} → X_{ε,Λ−tor} → 0.  (4.4)

The equality (4.1) now follows from (4.4) together with Theorem 4.1 and Proposition 3.3(ii).

Now assume Conjecture 3.5. Then (3.3) gives an injection S_{rel}/S_{ε} ↪ H^{1}/H_{ε}^{1} of a torsion module into a torsion-free module. Hence S_{ε} = S_{rel}. In order to show (4.2), we observe that, as S_{−ε} = 0, (3.3) yields

0 → H_{ε}^{1}/S_{ε} → X_{−ε} → X_{str} → 0.

Combining this with the exactness of

0 → S_{ε}/C → H_{ε}^{1}/C → H_{ε}^{1}/S_{ε} → 0

and with Proposition 3.3 proves the equality (4.2). □

Let ε be the sign of W (ψ), and write \overline{ψ} denote the complex conjugate of the grossencharacter ψ. Fix a generator c_{ε} of H_{ε}.

Theorem 4.3. Assume that Conjecture 3.5 holds. Then there exists a generator L_{−ε} of Char_{Λ}(H_{ε}^{1}/C) such that the following statement is true:

Let χ be any character of Γ of order p^{n}, where n > 0 and satisfies (−1)^{n+1} = W (ψ). Then

δ_{χ}(v_{ε}) · χ(L_{−ε}) = \frac{L(\overline{ψ}χ, 1)}{Ω_{E}}.

Here Ω_{E} ∈ R^{+} is the real period of a minimal model of E, v_{ε} ∈ V_{ε} is the image of c_{ε} under a fixed choice of the isomorphism (3.2), and δ_{χ} is the Coates-Wiles homomorphism defined in [12, §2]. Furthermore, δ_{χ}(v_{ε}) is always non-zero.

Proof. This is a direct consequence of [12, §10], once we fix a choice of isomorphism (3.2) above. (One must also bear in mind the last part of Remark 3.2.) □

Now Theorems 4.2 and 4.3 imply that if Conjecture 3.5 holds, then

L_{−ε}Λ = Char_{Λ}(H_{ε}^{1}/C) = Char_{Λ}(X_{−ε}).

Hence we see that Conjecture 3.5 implies that Char_{Λ}(X_{−ε}) is generated by an element which padically interpolates suitably normalised special values of twists of L(\overline{ψ}, s), and which may therefore be viewed as being a p-adic L-function attached to E.

5. Control theorems

Define

X_{n,∗} = Hom_{O_{𝔭}}(Sel_{∗}(D_{n}, W), K_{𝔭}/O_{𝔭}).

Our goal in this section is to explain how to recover the O_{𝔭}-rank of X_{n} from the Λ-modules X_{±}. Fix a topological generator γ ∈ Gal(D_{∞}/K) and define

ω_{n}^{+} = ∏_{1≤k≤n,k even} Φ_{p^{k}}(γ)  ω_{n}^{−} = (γ − 1) ∏_{1≤k≤n,k odd} Φ_{p^{k}}(γ)

where Φ_{p^{k}} is the p^{k}-th cyclotomic polynomial. Since χ(ω_{n}^{±}) = 0 for every χ ∈ Ξ_{n}^{±}, we have

ω_{n}^{∓} · E(D_{n,𝔭}) ⊂ E_{±}(D_{n,𝔭})  (5.1)
ω_{n}^{∓} · Sel(D_{n}, W) ⊂ Sel_{±}(D_{n}, W)

and similarly ω_{n}^{±} · E_{±}(D_{n,𝔭}) = 0.

Lemma 5.1. The natural map

f_{n} : H_{±}^{1}(D_{n,𝔭}, W) ⟶ H_{±}^{1}(D_{∞,𝔭}, W)[ω_{n}^{±}]

is injective, and the O_{𝔭}-corank of the cokernel of f_{n} is a bounded, non-decreasing function of n. If Conjecture 3.5 holds then the cokernel of f_{n} is finite for all n.

Proof. Let L denote the extension of K_{𝔭} obtained by adjoining E[𝔭] to K_{𝔭}. Then it follows from Lubin-Tate theory that L/K_{𝔭} is a totally ramified extension of degree p^{2} −1. Hence L∩D_{∞,𝔭} = K_{𝔭}, and we deduce that H^{0}(D_{∞,𝔭}, W) = 0. From the inflation-restriction sequence we deduce that

H^{1}(D_{n,𝔭}, W) ⟶ H^{1}(D_{∞,𝔭}, W),

and therefore also f_{n}, is injective. To prove the rest of the lemma, we compare the O_{𝔭}-coranks of H_{±}^{1}(D_{n,𝔭}, W) and H_{±}^{1}(D_{∞,𝔭}, W)[ω_{n}^{±}].

From Proposition 3.4 and the general structure theory of Λ-modules, we see that the O_{𝔭}-corank of H_{±}^{1}(D_{∞,𝔭}, W)[ω_{n}^{±}] is equal to rk_{O_{𝔭}} (Λ/ω_{n}^{±}Λ) + e(n), where e(n) is a non-decreasing, bounded function of n. If Conjecture 3.5 holds, then the Λ-module H_{±}^{1}(D_{∞}, W) is cotorsion-free, and so e(n) = 0 for all n. On the other hand, there is an isomorphism of K_{𝔭}[Gal(D_{n,𝔭}/K_{𝔭})]-modules

E(D_{n,𝔭}) ⊗_{O_{𝔭}} K_{𝔭} ≃ D_{n,𝔭} ≃ K_{𝔭}[Gal(D_{n,𝔭}/K_{𝔭})],  (5.2)

in which the first isomorphism is induced by the logarithm of the formal group E, and the second follows from the normal basis theorem of Galois theory. This implies that the O_{𝔭}-corank of H_{±}^{1}(D_{n,𝔭}, W) is equal to the O_{𝔭}-rank of Λ/ω_{n}^{±}. The result now follows immediately. □

The following result is an anticyclotomic analogue of Kobayashi’s control theorem (see [5, Theorem 9.3]).

Theorem 5.2. The natural map

X_{±}/ω_{n}^{±}X_{±} ⟶ X_{n,±}/ω_{n}^{±}X_{n,±}  (5.3)

is surjective. The O_{𝔭}-rank of the kernel is a bounded, nondecreasing function of n. If Conjecture 3.5 holds, then the kernel is finite for all n.

Proof. Write

L_{n,±} = H^{1}(D_{n,𝔭}, W)[ω_{n}^{±}]/H_{±}^{1}(D_{n,𝔭}, W),
L_{∞,±} = H^{1}(D_{∞,𝔭}, W)[ω_{n}^{±}]/H_{±}^{1}(D_{∞,𝔭}, W)[ω_{n}^{±}].

Let K denote the maximal extension of K unramified outside 𝔣𝔭 and set

H^{1}(D_{n,𝔣}, W) = ⊕_{v|𝔣}H^{1}(D_{n,v}, W), H^{1}(D_{∞,𝔣}, W) = ⊕_{v|𝔣}H^{1}(D_{∞,v}, W).

Consider the following commutative diagram with exact rows:

\xymatrix{0 \ar[r] & Sel_{±}(D_{n}, W)[ω_{n}^{±}] \ar[r] \ar[d] & H^{1}(K/D_{n}, W)[ω_{n}^{±}] \ar[r] \ar[d] & H^{1}(D_{n,𝔣}, W) ⊕ L_{n,±} \ar[d] \\ 0 \ar[r] & Sel_{±}(D_{∞}, W)[ω_{n}^{±}] \ar[r] & H^{1}(K/D_{∞}, W)[ω_{n}^{±}] \ar[r] & H^{1}(D_{∞,𝔣}, W) ⊕ L_{∞,±}}

The left-hand vertical arrow of this diagram is the dual of the map (5.3). Since H^{1}(K/D_{n}, W) ≅ H^{1}(K/D_{∞}, W)[γ^{p^{n}} − 1], the middle vertical arrow is an isomorphism. To prove the theorem, it therefore suffices (by the Snake Lemma) to show that the O_{𝔭}-corank of the kernel of the right-hand arrow is a bounded, non-decreasing function of n, and is finite for all n if Conjecture 3.5 holds.

For any place v of D_{∞} dividing 𝔣, the extension D_{∞,v}/D_{n,v} is either trivial (in which case there is nothing to check) or is the unique unramified Z_{p}-extension of D_{n,v}. Assume we are in the latter case. The kernel of

H^{1}(D_{n,v}, W) ⟶ H^{1}(D_{∞,v}, W)

is isomorphic to H^{1}(D_{∞,v}/D_{n,v}, E(D_{∞,v})[p^{∞}]), which is isomorphic to a quotient of E(D_{∞,v})[p^{∞}]. Since the Galois module W is ramified at all primes dividing 𝔣, it follows that E(D_{∞,v})[p^{∞}] is a proper O_{𝔭} submodule of W. This implies that E(D_{∞,v})[p^{∞}] is finite, because W is cofree of corank one over O_{𝔭}, and so any proper submodule of W is finite.

In order to control the kernel of L_{n,±} ⟶ L_{∞,±}, we apply the Snake Lemma to the diagram

\xymatrix{0 \ar[r] & H_{±}^{1}(D_{n,𝔭}, W) \ar[r] \ar[d] & H^{1}(D_{n,𝔭}, W)[ω_{n}^{±}] \ar[r] \ar[d] & L_{n,±} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & H_{±}^{1}(D_{∞,𝔭}, W)[ω_{n}^{±}] \ar[r] & H^{1}(D_{∞,𝔭}, W)[ω_{n}^{±}] \ar[r] & L_{∞,±} \ar[r] & 0.}

Just as in the proof of Lemma 5.1, we deduce from the inflation-restriction sequence that the middle vertical arrow of this diagram is injective; the same inflation-restriction sequnce also shows that that this arrow is surjective. We therefore deduce from Lemma 5.1 that the O_{𝔭}-corank of the kernel of the right-hand vertical arrow of this diagram is a bounded, non-decreasing function of n, and is finite for all n if Conjecture 3.5 holds. This completes the proof. □

Proposition 5.3. For any n, the natural map

X_{n} ⟶ (X_{n,+}/ω_{n}^{+}X_{n,+}) ⊕ (X_{n,−}/ω_{n}^{−}X_{n,−})

has finite kernel and cokernel.

Proof. Consider the dual map

Sel_{+}(D_{n}, W)[ω_{n}^{+}] ⊕ Sel_{−}(D_{n}, W)[ω_{n}^{−}] ⟶ Sel(D_{n}, W).  (5.4)

By (5.1) and the equality ω_{n}^{±}ω_{n}^{∓} = γ^{p^{k}} − 1, there is an inclusion

ω_{n}^{−} · Sel(D_{n}, W) + ω_{n}^{+} · Sel(D_{n}, W) ⊂ Sel_{+}(D_{n}, W)[ω_{n}^{+}] + Sel_{−}(D_{n}, W)[ω_{n}^{−}].

Since

Sel(D_{n}, W)/(ω_{n}^{−} · Sel(D_{n}, W) + ω_{n}^{+} · Sel(D_{n}, W))

is a module of cofinite type over the finite ring Λ/(ω_{n}^{+}, ω_{n}^{−}), it is finite, and therefore the same is true of the cokernel of (5.4). The kernel of (5.4) is isomorphic to

Sel_{+}(D_{n}, W)[ω_{n}^{+}] ∩ Sel_{−}(D_{n}, W)[ω_{n}^{−}]

which is again a cofinite type module over Λ/(ω_{n}^{+}, ω_{n}^{−}), and is therefore also finite. □

Combining Theorem 3.6, Theorem 5.2, and Proposition 5.3 we obtain the following result.

Theorem 5.4. Let ε be the sign of W (ψ). There is an integer e, independent of n, such that

corank_{O_{𝔭}}(Sel(D_{n}, W)) = rank_{O_{𝔭}}(Λ/ω_{n}^{ε}) + e

for n ≫ 0. If Conjecture 3.5 holds then the O_{𝔭}-corank of Sel(D_{n}, W) is equal to

rank_{O_{𝔭}}(Λ/ω_{n}^{ε}) + rank_{O_{𝔭}}(Y_{+}/ω_{n}^{+}Y_{+}) + rank_{O_{𝔭}}(Y_{−}/ω_{n}^{−}Y_{−})

for all n, where Y_{±} is the Λ-torsion submodule of X_{±}. □

Theorem A of the Introduction now follows from the first part of Theorem 5.4.

References

[1] A. Agboola, B. Howard. Anticyclotomic Iwasawa theory for CM elliptic curves. Preprint (2003).

[2] P. Billot. Quelques aspects de la descente sur une courbe elliptique dans le cas de réduction supersingulière. Comp. Math. 58 (1986) 341–369.

[3] R. Greenberg. On the Birch and Swinnerton-Dyer conjecture. Invent. Math. 72 (1983) 241–265.

[4] K. Kato. p-adic Hodge theory and values of zeta functions of modular forms. In: Cohomologies p-adiques et applications arithmétiques III. Asterisque 295 (2004), ix, 117–290.

[5] S. Kobayashi. Iwasawa theory for elliptic curves at supersingular primes. Invent. Math. 152 (2003) 1–36.

[6] B. Perrin-Riou. Arithmétique des courbes elliptiques à réduction supersingulière en p. Experiment. Math. 12 (2003), no. 2, 155–186.

[7] B. Perrin-Riou. Fonctions L p-adiques d’une courbe elliptique et points rationnelles. Ann. Inst. Fourier 43 (1993) 945–995.

[8] B. Perrin-Riou. Théorie d’Iwasawa et hauteurs p-adiques. Invent. Math. 109 (1992) 137–185.

[9] R. Pollack, On the p-adic L-function of a modular form at a supersingular prime. Duke Math. J. 118 (2003) 523–558.

[10] D. Rohrlich. On L-functions of elliptic curves and anticyclotomic towers. Invent. Math., 75 (1984) 383–408.

[11] K. Rubin. Elliptic curves and Z_{p}-extensions. Comp. Math. 56 (1985) 237–250.

[12] K. Rubin. Local units, elliptic units, Heegner points and elliptic curves. Invent. Math. 88 (1987) 405–422.

[13] K. Rubin. The “main conjectures” of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103 (1991) 25–68.

[14] K. Rubin. Euler Systems. Princeton University Press, (2000).

[15] K. Rubin, R. Pollack. The main conjecture for elliptic curves at supersingular primes. Ann. Math. 159 (2004) no. 1, 447–464.

[16] K. Wingberg. Duality theorems for abelian varieties over Z_{p}-extensions. Advanced Studies in Pure Mathematics 17 (1989) 471–492.

Department of Mathematics, University of California, Santa Barbara, CA, 93106

   E-mail address: agboola@math.ucsb.edu

Department of Mathematics, Harvard University, Cambridge, MA, 02138

Current address: Department of Mathematics, University of Chicago,, 5734 S. University Ave., Chicago, IL, 60637

   E-mail address: howard@math.uchicago.edu