ANTICYCLOTOMIC IWASAWA THEORY OF CM ELLIPTIC CURVES

ADEBISI AGBOOLA AND BENJAMIN HOWARD

WITH AN APPENDIX BY KARL RUBIN

Abstract. We study the Iwasawa theory of a CM elliptic curve E in the anticyclotomic Z_{p}-extension of the CM field, where p is a prime of good, ordinary reduction for E. When the complex L-function of E vanishes to even order, Rubin’s proof of the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the p-power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg shows that it is not a torsion module. In this paper we show that in the case of odd order of vanishing the dual of the Selmer group has rank exactly one, and we prove a form of the Iwasawa main conjecture for the torsion submodule.

0. Introduction and statement of results

Let K be an imaginary quadratic field of class number one, and let E_{/Q} be an elliptic curve with complex multiplication by the maximal order O_{K} of K. Let ψ denote the K-valued grossencharacter associated to E, and fix a rational prime p > 3 at which E has good, ordinary reduction.

Write Q_{p}^{unr} ⊂ C_{p} for the maximal unramified extension of Q_{p}, and let R_{0} denote the completion of its ring of integers. If F/K is any Galois extension, then we write Λ(F) = Z_{p}[[Gal(F/K)]] for the generalised Iwasawa algebra, and we set Λ(F)_{R_{0}} = R_{0}[[Gal(F/K)]]. Let C_{∞} and D_{∞} be the cyclotomic and anticyclotomic Z_{p}-extensions of K, respectively, and let K_{∞} = C_{∞}D_{∞} be the unique Z_{p}^{2}-extension of K.

As p is a prime of ordinary reduction for E, it follows that p splits into two distinct primes pO_{K} = 𝔭𝔭^{∗} over K. A construction of Katz gives a canonical measure

L ∈ Λ(K_{∞})_{R_{0}},

the two-variable p-adic L-function, denoted μ_{𝔭^{∗}}(K_{∞}, ψ_{𝔭^{∗}}) in the text, which interpolates the value at s = 0 of twists of L(ψ^{−1}, s) by characters of Gal(K_{∞}/K). It is a theorem of Coates [3] that the Pontryagin dual of the Selmer group Sel_{𝔭^{∗}}(E/K_{∞}) ⊂ H^{1}(K_{∞}, E[𝔭^{∗∞}]) is a torsion Λ(K_{∞})-module, and a fundamental theorem of Rubin, the two-variable Iwasawa main conjecture, asserts that the characteristic ideal of this torsion module is generated by L. In many cases this allows one to deduce properties of the 𝔭^{∗}-power Selmer group of E over subfields of K_{∞}. For example, if we identify

Date: Final version. January 11, 2006.

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

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

The second author is supported by a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship.

The author of the appendix is partially supported by NSF grant DMS-0140378.

Λ(K_{∞}) ≅ Λ(D_{∞})[[Gal(C_{∞}/K)]]

and choose a topological generator γ ∈ Gal(C_{∞}/K), then we may expand L as a power series in (γ − 1)

L = L_{0} + L_{1}(γ − 1) + L_{2}(γ − 1)^{2} + ⋯

with L_{i} ∈ Λ(D_{∞})_{R_{0}}. Standard “control theorems” imply that the characteristic ideal of

X^{∗}(D_{∞}) =^{def} Hom(Sel_{𝔭^{∗}}(E/D_{∞}), Q_{p}/Z_{p})

(the reader should note Remark 1.1.8) is then generated by the constant term L_{0}. If the sign in the functional equation of L(E/Q, s) is equal to 1, then theorems of Greenberg imply that L_{0} ≠ 0, and so X^{∗}(D_{∞}) is a torsion Λ(D_{∞})-module. In sharp contrast to this, when the sign of the functional equation is −1, the constant term vanishes and X^{∗}(D_{∞}) is not torsion, as was proved by Greenberg even before Rubin’s proof of the main conjecture (see [6]).

The following is the main result of this paper (which appears in the text as Theorems 2.4.17 and 3.1.5) and was inspired by conjectures of Mazur [12], Perrin-Riou [18] and Mazur-Rubin [15] concerning Heegner points.

Theorem A. Suppose the sign of the functional equation of L(E/Q, s) is −1. Then X^{∗}(D_{∞}) is a rank one Λ(D_{∞})-module. If X ⊂ Λ(D_{∞}) is the characteristic ideal of the torsion submodule of X^{∗}(D_{∞}), then

X · R = (L_{1})

as ideals of Λ(D_{∞})_{R_{0}} ⊗_{Z_{p}} Q_{p}, where R is the regulator of the Λ(D_{∞})-adic height pairing (defined in Section 3).

A different statement of the Iwasawa main conjecture over D_{∞}, involving elliptic units and including the case where the sign in the functional equation is equal to 1, is also contained in Theorem 2.4.17. Similar results in the Heegner point case alluded to above can be found in [8]. T. Arnold [1] has recently generalized Theorem A from the case of elliptic curves with complex multiplication to CM modular forms of higher weight.

The following result is due to K. Rubin. It establishes a conjecture made in an earlier version of this paper, and a proof is given in the Appendix.

Theorem B. Under the assumptions and notation of the Theorem A, the linear term L_{1} is nonzero.

While we have stated our results in terms of the 𝔭^{∗}-adic Selmer group, they may equally well be stated in terms of the 𝔭-adic Selmer group. If one replaces 𝔭^{∗} by 𝔭 in the above theorem, then X, L_{1}, and R are replaced by X^{ι}, L_{1}^{ι}, and R^{ι}, respectively, where ι is the involution of Λ(D_{∞}) induced by inversion on Gal(D_{∞}/K). The decomposition E[p^{∞}] ≅ E[𝔭^{∞}] ⊕ E[𝔭^{∗∞}] of Gal(K^{al}/K)-modules induces a decomposition of Λ(D_{∞})-modules

Sel_{p}(E/D_{∞}) ≅ Sel_{𝔭}(E/D_{∞}) ⊕ Sel_{𝔭^{∗}}(E/D_{∞})

which shows that, when the sign in the functional equation is −1, the full p-power Selmer group Sel_{p}(E/D_{∞}) has Λ(D_{∞})-corank 2, as was conjectured by Mazur [12].

An outline of this paper is as follows. The first section gives definitions and fundamental properties of various Selmer groups associated to E, with special attention to the anticyclotomic tower. In the second section, we recall the definition of Katz’s p-adic L-function and the Euler system of elliptic units, and we state a theorem of Yager which relates the two. Our discussion of these topics closely follows the excellent book of de Shalit [4]. Work of Rubin allows one to “twist” the elliptic unit Euler system into an Euler system for the 𝔭-adic Tate module T_{𝔭}(E), and we show, using nonvanishing results of Greenberg, that the restriction of the resulting Euler system to the anticyclotomic extension is nontrivial. Applying the main results of [25] shows that a certain “restricted” Selmer group, contained in Sel_{𝔭^{∗}}(E/D_{∞}), is a cotorsion module; using this we show that X^{∗}(D_{∞}) has rank one. Using a form of Mazur’s control theorem, we then deduce that the characteristic ideal of a restricted Selmer group over K_{∞} does not have an anticyclotomic zero. This restricted two-variable Selmer group is related to elliptic units by Rubin’s proof of the two-variable main conjecture, and the nonvanishing of its characteristic ideal along the anticyclotomic line allows us to descend to the anticyclotomic extension and relate the restricted Selmer group over D_{∞} to the elliptic units. In the third section we use results of Perrin-Riou and Rubin on the p-adic height pairing to relate the twisted elliptic units to the linear term L_{1} of Katz’s L-function. Acknowledgments. We are very grateful to Karl Rubin for writing an appendix to this paper.

0.1. Notation and conventions. We write ψ for the K-valued grossencharacter associated to E, and we let 𝔣 denote its conductor. Note that since p is a prime of good reduction for E, it follows that p is coprime to 𝔣.

Let Q^{al} ⊂ C be the algebraic closure of Q in C, and let τ be complex conjugation, also denoted by z ↦ z̄. Fix an embedding i_{𝔭} : Q^{al} ↪ C_{p} lying above the prime 𝔭, and let i_{𝔭^{∗}} = i_{𝔭} ∘ τ be the conjugate embedding.

We write R for the field of fractions of R_{0}. If M is any Z_{p}-module, we define

M_{R_{0}} = M ˆ⊗_{Z_{p}}R_{0}, M_{R} = M_{R_{0}} ⊗_{R_{0}} R.

The Pontryagin dual of M is denoted

M^{∨} =^{def} Hom_{Z_{p}}(M, Q_{p}/Z_{p}).

If M is any Z_{p}-module of finite or cofinite type equipped with a continuous action of G_{K} = Gal(Q^{al}/K) and F is a (possibly infinite) Galois extension of K, we let

H^{i}(F, M) = lim_{←} H^{i}(F′, M)

where the inverse limit is over all subfields F′ ⊂ F finite over K and is taken with respect to the natural corestriction maps. If 𝔮 is 𝔭 or 𝔭^{∗}, we define

H^{i}(F_{𝔮}, M) = lim_{←} ⨁_{w|𝔮} H^{i}(F′_{w}, M), H^{i}(F_{𝔮}, M) = lim_{→} ⨁_{w|𝔮} H^{i}(F′_{w}, M).

(Here the inverse (respectively, direct) limit is taken with respect to the corestriction (respectively, restriction) maps.) These groups have natural Λ(F) := Z_{p}[[Gal(F/K)]]-module structures.

For a positive integer n, C_{n} is the unique subfield of C_{∞} with [C_{n} : K] = p^{n}; the field D_{n} is defined similarly. If 𝔪 is an ideal of K, we denote by K(𝔪) the ray class field of conductor 𝔪. If 𝔫 is another ideal of K, we let K(𝔪𝔫^{∞}) = ∪_{k}K(𝔪𝔫^{k}). We write N(𝔪) for the absolute norm of the ideal 𝔪.

1. A little cohomology

We define canonical generators π and π^{∗} of the ideals 𝔭 and 𝔭^{∗} by π = ψ(𝔭) and π^{∗} = ψ(𝔭^{∗}), so that π^{∗} = π^{τ}. Define G_{K}-modules

W_{𝔭} = E[𝔭^{∞}] W_{𝔭^{∗}} = E[𝔭^{∗∞}],

and let T_{𝔭} and T_{𝔭^{∗}} be the π and π^{∗}-adic Tate modules, respectively. Note that the action of τ on E[p^{∞}] interchanges W_{𝔭} and W_{𝔭^{∗}}, and so induces a group isomorphism T_{𝔭} ≅ T_{𝔭^{∗}}. If we set V_{𝔭} = T_{𝔭} ⊗ Q_{p} and V_{𝔭^{∗}} = T_{𝔭^{∗}} ⊗ Q_{p}, then there is an exact sequence

(1.1)  0 ⟶ T_{𝔮} ⟶ V_{𝔮} ⟶ W_{𝔮} ⟶ 0

where 𝔮 = 𝔭 or 𝔭^{∗}. For every place v of K and any finite extension F of K or K_{v}, the G_{F} -cohomology of this sequence implies that

H^{0}(F, W_{𝔭})_{/div} ≅ H^{1}(F, T_{𝔭})_{tor},

where the subscript /div indicates the quotient by the maximal divisible submodule, and the subscript tor indicates the Z_{p}-torsion submodule. The Weil pairing restricts to a perfect pairing T_{𝔭} × T_{𝔭^{∗}} ⟶ Z_{p}(1).

1.1. Selmer modules. Let 𝔮 be either 𝔭 or 𝔭^{∗} and set 𝔮^{∗} = τ(𝔮).

Lemma 1.1.1. The primes of K above p are finitely decomposed in K_{∞}.

Proof. This follows from Proposition II.1.9 of [4]. □

Lemma 1.1.2. The degree of K_{𝔮}(E[𝔮]) over K_{𝔮} is p − 1.

Proof. This follows from the theory of Lubin-Tate groups. See for example Chapter 1 of [4]. □

Lemma 1.1.3. For any intermediate field K ⊂ F ⊂ K_{∞} the Λ(F)-module

A_{𝔣}(F) = ⨁_{v|𝔣} H^{0}(F_{v}, W_{𝔭})

has finite exponent. If all primes dividing 𝔣 are finitely decomposed in F, then A_{𝔣}(F) is finite.

Proof. Fix a place v|𝔣 of F. The extension F_{v}/K_{v} is unramified, while W_{𝔭} is a ramified Gal(K_{v}^{alg}/K_{v})-module (by the criterion of Néron-Ogg-Shafarevich). Since W_{𝔭} ≅ Q_{p}/Z_{p} has no proper infinite submodules we conclude that E(F_{v})[𝔭^{∞}] is finite, and so ⨁_{v|𝔣} H^{0}(F_{v}, W_{𝔭^{∗}}) has finite exponent. This group is finite if all primes above 𝔣 are finitely decomposed in F. □

Lemma 1.1.4. Let F/K be a finite extension, let v be a prime of F not dividing p, and let V be a finite dimensional Q_{p}-vector space with a linear action of G_{F_{v}}. If V and Hom(V, Q_{p}(1)) both have no G_{F_{v}}-invariants then H^{1}(F_{v}, V) = 0.

Proof. This follows from Corollary 1.3.5 of [25] and local duality, or from standard properties of local Euler characteristics. □

In particular, Lemma 1.1.4 implies that H^{1}(F_{v}, V_{𝔮}) = 0 for v not dividing p. If v is any place of F, we define the finite or Bloch-Kato local conditions

H_{f}^{1}(F_{v}, T_{𝔮}) = \{ \begin{matrix}H^{1}(F_{v}, T_{𝔮})_{tor} & if v | 𝔮^{∗} \\ H^{1}(F_{v}, T_{𝔮}) & else \end{matrix}

H_{f}^{1}(F_{v}, V_{𝔮}) = \{ \begin{matrix}H^{1}(F_{v}, V_{𝔮}) & if v | 𝔮 \\ 0 & else \end{matrix}

H_{f}^{1}(F_{v}, W_{𝔮}) = \{ \begin{matrix}H^{1}(F_{v}, W_{𝔮})_{div} & if v | 𝔮 \\ 0 & else \end{matrix}

The submodules H_{f}^{1}(F_{v}, T_{𝔮}) and H_{f}^{1}(F_{v}, W_{𝔮}) are the preimage and image, respectively, of H_{f}^{1}(F_{v}, V_{𝔮}) under the maps on cohomology induced by the exact sequence (1.1). If M is any object for which we have defined H_{f}^{1}(F_{v}, M), we define the relaxed Selmer group Sel_{rel}(F, M) to be the set of all c ∈ H^{1}(F, M) such that loc_{v}(c) ∈ H_{f}^{1}(F_{v}, M) for every place v not dividing p. We define the true Selmer group Sel(F, M) to be the subgroup consisting of all c ∈ Sel_{rel}(F, M) such that loc_{v}(c) ∈ H_{f}^{1}(F_{v}, M) for all v, including those above p. Finally, we define the strict Selmer group to be all those c ∈ Sel(F, M) such that loc_{v}(c) = 0 at the v lying above p. By definition there are inclusions

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

and all are Λ(F)-modules. Our definitions of Sel(F, T_{𝔮}) and Sel(F, W_{𝔮}) agree with the usual definitions of the Selmer groups defined by the local images of the Kummer maps; see Section 6.5 of [25].

Lemma 1.1.5. Let S denote the set of places of K dividing p𝔣 and let K_{S}/K be the maximal extension of K unramified outside S. For any K ⊂ F ⊂ K_{S} finite over K,

H^{1}(K_{S}/F, T_{𝔭}) = Sel_{rel}(F, T_{𝔭}).

Proof. For any v ∉ S, the local condition H_{f}^{1}(F_{v}, T_{𝔭}) is exactly the subgroup of unramified classes by [25] Lemma 1.3.5, while for v ∈ S the local condition defining the relaxed Selmer group is all of H^{1}(F_{v}, T_{𝔭}). □

If F/K is a (possibly infinite) extension we define Λ(F)-modules

S(F, T_{𝔮}) = lim_{←} Sel(F′, T_{𝔮}) Sel(F, W_{𝔮}) = lim_{→} Sel(F′, W_{𝔮})

where the limits are with respect to corestriction and restriction respectively, and are taken over all subfields F′ ⊂ F finite over K. We also define strict and relaxed Selmer groups over F in the obvious way, e.g. S_{str}(F, T_{𝔮}) = lim_{←} Sel_{str}(F′, T_{𝔮}), and so on. Recall the notation

H^{i}(F_{𝔮}, M) = lim_{←} ⨁_{v|𝔮} H^{i}(F′_{v}, M),

and set

H_{f}^{1}(F_{𝔮}, M) = lim_{←} ⨁_{v|𝔮} H_{f}^{1}(F′_{v}, M), H_{f}^{1}(F_{𝔮}, M) = lim_{→} ⨁_{v|𝔮} H_{f}^{1}(F′_{v}, M).

The canonical involution of Λ(F) which is inversion on group-like elements is denoted ι : Λ(F) ⟶ Λ(F). This involution induces a functor from the category of Λ(F)-modules to itself, which on objects is written as M ↦ M^{ι}. If 𝔮 = 𝔭 or 𝔭^{∗}, there is a perfect local Tate pairing

(1.2)  H^{1}(F_{𝔮}, T_{𝔭}) × H^{1}(F_{𝔮}, W_{𝔭^{∗}}) ⟶ Q_{p}/Z_{p}

which satisfies (λx, y) = (x, λ^{ι}y) for λ ∈ Λ(F). Under this pairing, the submodules H_{f}^{1}(F_{𝔮}, T_{𝔭}) and H_{f}^{1}(F_{𝔮}, W_{𝔭^{∗}}) are exact orthogonal complements.

Proposition 1.1.6. Suppose F/K is a Z_{p}-extension in which 𝔭^{∗} ramifies. Then H^{1}(F_{𝔭}, T_{𝔭}) and H^{1}(F_{𝔭^{∗}}, T_{𝔭}) are rank one, torsion-free Λ(F)-modules.

Proof. Let 𝔮 = 𝔭 or 𝔭^{∗}. The claim that H^{1}(F_{𝔮}, T_{𝔭}) has rank 1 is Proposition 2.1.3 of [19]. By Proposition 2.1.6 of the same, the Λ(F)-torsion submodule of H^{1}(F_{𝔮}, T_{𝔭}) is isomorphic to H^{0}(F_{𝔮}, T_{𝔭}), so it suffices to show that E(F_{v})[𝔭^{∞}] is finite for every place v of F above 𝔮. If 𝔮 = 𝔭 this is immediate from Lemma 1.1.2. If 𝔮 = 𝔭^{∗} then E[𝔭^{∞}] generates an unramified extension of K_{𝔮}, and by hypothesis the intersection of this extension with F_{v} is of finite degree over K_{𝔮}. Hence E(F_{v})[𝔭^{∞}] is finite. □

Proposition 1.1.7. If F/K is an abelian extension such that H_{f}^{1}(F_{𝔭^{∗}}, T_{𝔭}) = 0, then there are exact sequences

(1.3)  0 ⟶ S(F, T_{𝔭}) ⟶ S_{rel}(F, T_{𝔭}) ⟶^{loc_{𝔭^{∗}}} H^{1}(F_{𝔭^{∗}}, T_{𝔭})
(1.4)  0 ⟶ Sel_{str}(F, W_{𝔭^{∗}}) ⟶ Sel(F, W_{𝔭^{∗}}) ⟶^{loc_{𝔭^{∗}}} H^{1}(F_{𝔭^{∗}}, W_{𝔭^{∗}}),

and the images of the rightmost arrows are exact orthogonal complements under the local Tate pairing. Under the same hypotheses there are exact sequences

(1.5)  0 ⟶ S_{str}(F, T_{𝔭}) ⟶ S(F, T_{𝔭}) ⟶^{loc_{𝔭}} H^{1}(F_{𝔭}, T_{𝔭})
(1.6)  0 ⟶ Sel(F, W_{𝔭^{∗}}) ⟶ Sel_{rel}(F, W_{𝔭^{∗}}) ⟶^{loc_{𝔭}} H^{1}(F_{𝔭}, W_{𝔭^{∗}}),

and again the images of the rightmost arrows are exact orthogonal complements under the sum of the local pairings. The hypotheses hold if F contains a Z_{p}-extension in which 𝔭^{∗} ramifies.

Proof. If H_{f}^{1}(F_{𝔭^{∗}}, T_{𝔭}) = 0 then H_{f}^{1}(F_{𝔭^{∗}}, W_{𝔭^{∗}}) = H^{1}(F_{𝔭^{∗}}, W_{𝔭^{∗}}) by local duality. The exactness of the sequences is now just a restatement of the definitions. The claims concerning orthogonal complements are consequences of Poitou-Tate global duality, c.f. Theorem 1.7.3 of [25].

Let v be a place of F above 𝔭^{∗} and let F′ ⊂ F_{v} be finite over K_{𝔭^{∗}}. From the cohomology of (1.1) and the fact that H^{0}(F′, V_{𝔭}) = 0 we have

H_{f}^{1}(F′, T_{𝔭}) ≅ H^{0}(F′, W_{𝔭}) ≅ E(F′)[𝔭^{∞}].

Taking the inverse limit over F′ ⊂ F_{v}, we see that

lim_{←} H_{f}^{1}(F′, T_{𝔭}) = 0

whenever F_{v} contains an infinite pro-p extension of K_{𝔭^{∗}} whose intersection with K_{𝔭^{∗}}(E[𝔭^{∞}]) is of finite degree over K_{𝔭^{∗}}. The extension of K_{𝔭^{∗}} generated by E[𝔭^{∞}] is unramified, so this will be the case whenever F contains a Z_{p}-extension in which 𝔭^{∗} ramifies. □

If F/K is an abelian extension we define Λ(F)-modules

X(F) = Sel(F, W_{𝔭})^{∨}
X_{rel}(F) = Sel_{rel}(F, W_{𝔭})^{∨}
X_{str}(F) = Sel_{str}(F, W_{𝔭})^{∨}.

Define X^{∗}(F), X_{rel}^{∗}(F), and X_{str}^{∗}(F) similarly, replacing 𝔭 by 𝔭^{∗}.

Remark 1.1.8. Because of the behavior of the local pairing (1.2) under the action of Λ(F), we adopt, for the entirety of the paper, the convention that Λ(F) acts on X(F) via (λ · f)(x) = f(λ^{ι}x). Thus the map

H^{1}(F_{𝔭^{∗}}, T_{𝔭}) ⟶ X^{∗}(F)

induced by localization at 𝔭^{∗} and the local pairing is a map of Λ(D_{∞})-modules. The same convention is adopted for X^{∗}(F), X_{rel}(F), etc.

Lemma 1.1.9. Let F/K be a Z_{p} or Z_{p}^{2} extension of K. There is a canonical isomorphism of Λ(F)-modules

S_{rel}(F, T_{𝔭}) ≅ Hom_{Λ(F)}(X_{rel}(F), Λ(F)).

In particular S_{rel}(F, T_{𝔭}) and X_{rel}(F) have the same Λ(F)-rank, and S_{rel}(F, T_{𝔭}) is torsion-free.

Proof. The proof is essentially the same as that of [19] Proposition 4.2.3. Suppose that L ⊂ F is finite over K. Let S denote the set of places of K consisting of the infinite place and the prime divisors of p𝔣, and let K_{S}/K be the maximal extension of K unramified outside S. By Lemma 1.1.5

S_{rel}(L, T_{𝔭}) ≅ H^{1}(K_{S}/L, T_{𝔭}) ≅ lim_{←} H^{1}(K_{S}/L, E[𝔭^{k}]).

On the other hand, Lemma 1.1.2 shows that E(L)[𝔭] = 0 (since [L : K] is a power of p), and so the Gal(K_{S}/L)-cohomology of

0 ⟶ E[𝔭^{k}] ⟶ W_{𝔭} ⟶^{π^{k}} W_{𝔭} ⟶ 0

shows that H^{1}(K_{S}/L, E[𝔭^{k}]) ≅ H^{1}(K_{S}/L, W_{𝔭})[𝔭^{k}].

If we define

X_{S}(L) = H^{1}(K_{S}/L, W_{𝔭})^{∨}

then

Hom_{Λ(L)}(X_{S}(L), Λ(L)) ≅ Hom_{Z_{p}}(X_{S}(L), Z_{p})

via the augmentation map Λ(L) ⟶ Z_{p}. The right hand side is isomorphic to the p-adic Tate module of H^{1}(K_{S}/L, W_{𝔭}), so by the above

(1.7)  S_{rel}(L, T_{𝔭}) ≅ Hom_{Λ(L)}(X_{S}(L), Λ(L)).

As in the proof of Lemma 1.1.5, for any v ∉ S the unramified classes in H^{1}(L_{v}, W_{𝔭}) agree with the local condition H_{f}^{1}(L_{v}, W_{𝔭}), and so we have the exact sequence

(1.8)  0 ⟶ Sel_{rel}(L, W_{𝔭}) ⟶ H^{1}(K_{S}/L, W_{𝔭}) ⟶ ⨁_{v|𝔣} H^{1}(L_{v}, W_{𝔭}).

The Pontryagin dual of the final term is isomorphic to

⨁_{v|𝔣} H^{1}(L_{v}, T_{𝔭^{∗}}) ≅ ⨁_{v|𝔣} H^{0}(L_{v}, W_{𝔭^{∗}}),

where we have used Lemma 1.1.4 and the cohomology of the short exact sequence relating T_{𝔭^{∗}}, V_{𝔭^{∗}}, and W_{𝔭^{∗}}.

If A_{𝔣}^{∗}(F) denotes the module of Lemma 1.1.3, with 𝔭 replaced by 𝔭^{∗}, we may take the limit as L varies, and the dual sequence to (1.8) reads

A_{𝔣}^{∗}(F)^{∨} ⟶ X_{S}(F) ⟶ X_{rel}(F) ⟶ 0

where the first term is a torsion Λ(F)-module (even a torsion Z_{p}-module) by Lemma 1.1.3. Applying the functor Hom_{Λ(F)}(·, Λ(F)) and combining this with (1.7) gives the result. □

Remark 1.1.10. A similar argument can be used to show that S(F, T_{𝔭}) and X(F) have the same Λ(F)-rank, and similarly for the strict Selmer groups. See [19] Proposition 4.2.3, for example. When F = D_{∞}, these facts will fall out during the more detailed analysis of the relationship between X_{str} and and X_{rel} given in Theorem 1.2.2.

1.2. Anticyclotomic Iwasawa modules.

Lemma 1.2.1. There are isomorphisms of Λ(D_{∞})-modules

S(D_{∞}, T_{𝔭})^{ι} ≅ S(D_{∞}, T_{𝔭^{∗}}) X(D_{∞})^{ι} ≅ X^{∗}(D_{∞}),

and similarly for the relaxed and restricted Selmer groups.

Proof. The action of G_{K} on the full Selmer group

Sel(D_{∞}, E[p^{∞}]) ≅ Sel(D_{∞}, W_{𝔭}) ⊕ Sel(D_{∞}, W_{𝔭^{∗}})

extends to an action of G_{Q}, and complex conjugation interchanges the 𝔭 and 𝔭^{∗}- primary components. Since Gal(D_{∞}/Q) is of dihedral type, we may view complex conjugation as an isomorphism

Sel(D_{∞}, W_{𝔭})^{ι} ≅ Sel(D_{∞}, W_{𝔭^{∗}}),

and so X(D_{∞})^{ι} ≅ X^{∗}(D_{∞}). The other claims are proved similarly. □

The remainder of this subsection is devoted to a proof of the following result.

Theorem 1.2.2. If r(·) denotes Λ(D_{∞})-rank, then r(X(D_{∞})) = r(S(D_{∞})) and the same holds for the strict and relaxed Selmer groups. Furthermore,

r(X_{rel}(D_{∞})) = 1 + r(X_{str}(D_{∞}))

and the Λ(D_{∞})-torsion submodules of X_{rel}(D_{∞}) and X_{str}(D_{∞}) have the same characteristic ideals, up to powers of pΛ(D_{∞}).

Let O be the ring of integers of some finite extension Φ/Q_{p}, and let

χ : Gal(D_{∞}/K) ⟶ O^{×}

be a continuous character of Gal(D_{∞}/K). If M is any Z_{p}-module, define M(χ) = M ⊗ O(χ). From Lemma 1.1.2 it follows that ψ_{𝔮} : G_{K} ⟶ Z_{p}^{×} ⟶ (Z_{p}/pZ_{p})^{×} is surjective, and from this it is easy to see that the residual representation of T_{𝔮}(χ) is nontrivial and absolutely irreducible. Combining this with Lemma 1.1.4 and the duality

V_{𝔭}(χ) × V_{𝔭^{∗}}(χ^{−1}) ⟶ Q_{p}(1),

we have that H^{1}(K_{v}, V_{𝔮}(χ)) = 0 for every v not dividing p. We define generalized Selmer groups

H_{rel}^{1}(K, W_{𝔮}(χ)) H_{str}^{1}(K, W_{𝔮}(χ)),

where the first group consists of classes which are everywhere trivial at primes not dividing p, and which lie in the maximal divisible subgroup of H^{1}(K_{v}, W_{𝔮}(χ)) at primes v above p, and the second group consists of classes which are everywhere locally trivial.

If χ is the trivial character, then H_{str}^{1}(K, W_{𝔮}) = Sel_{str}(K, W_{𝔮}), but H_{rel}^{1}(K, W_{𝔮}) may be slightly smaller than Sel_{rel}(K, W_{𝔮}). If we define

Sel_{rel}(K, W_{𝔮}(χ)) ⊂ H^{1}(K, W_{𝔮}(χ))

to be the subgroup of classes which are locally trivial at all primes not dividing p, and impose no conditions at all above p (so that this agrees with our previous definition when χ is trivial), then we can bound the index of

(1.9)  H_{rel}^{1}(K, W_{𝔮}(χ)) ⊂ Sel_{rel}(K, W_{𝔮}(χ))

as follows. The quotient injects into

H^{1}(K_{𝔭}, W_{𝔮}(χ)) ⊕ H^{1}(K_{𝔭^{∗}}, W_{𝔮}(χ))

modulo its maximal divisible subgroup. Thus, using the exact sequence (1.1) and local duality, the order of the quotient is bounded by the order of

H^{0}(K_{𝔭}, W_{𝔮^{∗}}(χ^{−1})) ⊕ H^{0}(K_{𝔭^{∗}}, W_{𝔮^{∗}}(χ^{−1})).

It is easy to see that this group is finite, and bounded by some constant which does not depend on χ, provided O remains fixed.

Our reason for working with the slightly smaller group H_{rel}^{1} is the following

Proposition 1.2.3. (Mazur-Rubin) For every character χ there is a non-canonical isomorphism of O-modules

H_{rel}^{1}(K, W_{𝔭}(χ)) ≅ (Φ/O) ⊕ H_{str}^{1}(K, W_{𝔭^{∗}}(χ^{−1})).

Proof. All references in this proof are to [14]. It follows from Theorem 4.1.13 and Lemma 3.5.3 that

H_{rel}^{1}(K, W_{𝔭}(χ))[p^{i}] ≅ (Φ/O)^{r}[p^{i}] ⊕ H_{str}^{1}(K, W_{𝔭^{∗}}(χ^{−1}))[p^{i}]

for every i, where r is the core rank (Definition 4.1.11) of the local conditions defining the Selmer group H_{rel}^{1}(K, W_{𝔭}(χ)). A formula of Wiles, Proposition 2.3.5, shows that the core rank is equal to

corank H^{1}(K_{𝔭}, W_{𝔭}(χ)) + corank H^{1}(K_{𝔭^{∗}}, W_{𝔭}(χ)) − corank H^{0}(K_{v}, W_{𝔭}(χ)),

in which v denotes the unique archimedean place of K and corank means corank as an O-module. The first two terms are each equal to 1 by the local Euler characteristic formula, and the third is visibly 1. Hence the core rank is 1. Letting i → ∞ proves the claim. □

Restriction gives a map

H^{1}(K, W_{𝔮}(χ)) ⟶ H^{1}(D_{∞}, W_{𝔮})(χ)^{Gal(D_{∞}/K)},

and since H^{0}(D_{∞}, W_{𝔮}(χ)) = 0, the Hochschild-Serre sequence (see Proposition B.2.5 of [25]) implies that this map is an isomorphism.

Lemma 1.2.4. The above restriction isomorphism induces injective maps

H_{str}^{1}(K, W_{𝔮}(χ)) ⟶ Sel_{str}(D_{∞}, W_{𝔮})(χ)^{Gal(D_{∞}/K)}
H_{rel}^{1}(K, W_{𝔮}(χ)) ⟶ Sel_{rel}(D_{∞}, W_{𝔮})(χ)^{Gal(D_{∞}/K)}

whose cokernels are finite and bounded as χ varies (provided O remains fixed).

Proof. A class d ∈ Sel_{str}(D_{∞}, W_{𝔮})(χ)^{Gal(D_{∞}/K)} is the restriction of some class c ∈ H^{1}(K, W_{𝔮}(χ)) which is in the kernel of

(1.10)  H^{1}(K_{v}, W_{𝔮}(χ)) ⟶ H^{1}(D_{∞,v}, W_{𝔮}(χ))

for every place v of D_{∞}. Let Γ_{v} = Gal(D_{∞,v}/K_{v}), so that Γ is either trivial or isomorphic to Z_{p}. If Γ_{v} = 0 then (1.10) is an isomorphism. If Γ_{v} ≅ Z_{p} with generator γ, then the cokernel is trivial and the kernel is isomorphic to M/(γ −1)M where M = H^{0}(D_{∞,v}W_{𝔮}) ⊗ O. If v is a prime of good reduction not dividing p, then D_{∞,v} is the unique unramified Z_{p}-extension of K_{v}. It follows that M = 0 if E[𝔮] ⊄ K_{v}, while M = W_{𝔮} if E[𝔮] ⊂ K_{v}. In either case, since (γ − 1) acts as a nontrivial scalar on W_{𝔮}, we must have M/(γ − 1)M = 0. If v is a prime of bad reduction, or if v lies above p, then M is finite. Thus the kernel of (1.10) is trivial for almost all v, and finite and bounded by a constant independent of χ.

A class d ∈ Sel_{rel}(D_{∞}, W_{𝔮})(χ)^{Gal(D_{∞}/K)} is the restriction of some class c ∈ H^{1}(K, W_{𝔮}(χ)) which is in the kernel of (1.10) at every prime not dividing p. The above argument shows that the cokernel of

Sel_{rel}(K, W_{𝔮}(χ)) ⟶ Sel_{rel}(D_{∞}, W_{𝔮})(χ)^{Gal(D_{∞}/K)}

is finite with a bound of the desired sort, and so the claim follows from our bound on the index of (1.9). □

Corollary 1.2.5. For any χ : Gal(D_{∞}/K) ⟶ O^{×}, the O-coranks of

Sel_{rel}(D_{∞}, W_{𝔭})(χ)^{Gal(D_{∞}/K)} Sel_{str}(D_{∞}, W_{𝔭})(χ)^{Gal(D_{∞}/K)}

differ by 1, and the quotients by the maximal O-divisible submodules have the same order, up to O(1) as χ varies.

Proof. By Lemma 1.2.1,

Sel_{str}(D_{∞}, W_{𝔭^{∗}})(χ^{−1})^{Gal(D_{∞}/K)} ≅ Sel_{str}(D_{∞}, W_{𝔭})(χ)^{Gal(D_{∞}/K)}.

Combining this with Proposition 1.2.3 and Lemma 1.2.4 gives the stated result. □

Lemma 1.2.6. We have the equality r(X_{rel}(D_{∞})) = 1 + r(X_{str}(D_{∞})), and the Λ(D_{∞})-torsion submodules of X_{rel}(D_{∞}) and X_{str}(D_{∞}) have the same characteristic ideals, up to powers of pΛ(D_{∞}).

Proof. Choose a generator γ ∈ Gal(D_{∞}/K) and identify Λ(D_{∞}) with Z_{p}[[S]] via γ − 1 ↦ S. Assume O is chosen large enough that the characteristic ideals of the torsion submodules of X_{str}^{∗}(D_{∞}) and X_{rel}^{∗}(D_{∞}) split into linear factors. Let 𝔪 ⊂ O be the maximal ideal, and fix pseudo-isomorphisms

X_{rel}^{∗}(D_{∞}) ⊗_{Z_{p}} O ∼ A ⊕ A_{p} X_{str}^{∗}(D_{∞}) ⊗_{Z_{p}} O ∼ B ⊕ B_{p}

where A_{p} and B_{p} are torsion modules with chracteristic ideals generated by powers of p, and A and B are of the form

A ≅ O[[S]]^{a} ⊕ ⨁_{ξ∈𝔪} A_{ξ} B ≅ O[[S]]^{b} ⊕ ⨁_{ξ∈𝔪} B_{ξ}

where each A_{ξ} is isomorphic to ⨁_{i} O[[S]]/(S−ξ)^{e_{i}} for some exponents e_{i} = e_{i}(A, ξ), and similarly for B. Define

P = {ξ ∈ 𝔪 | A_{ξ} ≠ 0 or B_{ξ} ≠ 0},

and to any ξ ∉ P we define a character χ_{ξ} by χ_{ξ}(γ) = (ξ + 1)^{−1}. Then for any ξ ∉ P we have

a = rank_{O} A/(S − ξ)A = corank_{O} Sel_{rel}(D_{∞}, W_{𝔭})(χ_{ξ})^{Gal(D_{∞}/K)}

and similarly for B. The corollary above immediately implies that a = b + 1, hence r(X_{rel}(D_{∞})) = r(X_{str}(D_{∞})) + 1.

Now fix ξ ∈ P and choose a sequence x_{k} → ξ with x_{k} ∈ 𝔪 − P for all k. As k varies, the O-length of the torsion submodule of A/(S − x_{k})A is given by

v(x_{k} − ξ) · ∑_{i} e_{i}(A, ξ) + O(1)

where v is the valuation on O, and similarly for B. Applying the corollary, we have

v(x_{k} − ξ) · ∑_{i} e_{i}(A, ξ) = v(x_{k} − ξ) · ∑_{i} e_{i}(B, ξ),

up to O(1) as k varies. Letting k → ∞ shows that ∑_{i} e_{i}(A, ξ) = ∑_{i} e_{i}(B, ξ), proving that the torsion submodules of X_{str}(D_{∞}) and X_{rel}(D_{∞}) have the same characteristic ideals, up to powers of pΛ(D_{∞}). □

The following corollary completes the proof of Theorem 1.2.2.

Corollary 1.2.7. We have the equality of ranks r(X(D_{∞})) = r(S(D_{∞})), and the same holds for the strict and relaxed Selmer groups.

Proof. For the relaxed Selmer groups, this equality of ranks was proved in Lemma 1.1.9. Let A and B be the cokernels of

S_{rel}(D_{∞}, T_{𝔭}) ⟶ H^{1}(D_{∞,𝔭^{∗}}, T_{𝔭}), S(D_{∞}, T_{𝔭}) ⟶ H^{1}(D_{∞,𝔭}, T_{𝔭}),

respectively. Then Propositions 1.1.7 and 1.2.1 give

r(A) + r(X_{str}(D_{∞})) = r(X(D_{∞}))
r(B) + r(X(D_{∞})) = r(X_{rel}(D_{∞}))
r(A) + r(S_{rel}(D_{∞}, T_{𝔭})) = 1 + r(S(D_{∞}, T_{𝔭}))
r(B) + r(S(D_{∞}, T_{𝔭})) = 1 + r(S_{str}(D_{∞}, T_{𝔭})).

By Lemma 1.2.6, the first two equalities imply that r(A) + r(B) = 1. The second two equalities then imply that r(S_{rel}(D_{∞}, T_{𝔭})) = 1 + r(S_{str}(D_{∞}, T_{𝔭})). We deduce, using Lemma 1.1.9, that r(X_{str}(D_{∞})) = r(S_{str}(D_{∞}, T_{𝔭})). Similarly, the equality r(X(D_{∞})) = r(S(D_{∞}, T_{𝔭})) is deduced from Lemma 1.1.9 by adding the middle two equalities. □

2. L-functions and Euler systems

In this section we recall the definition of Katz’s L-function, the construction of the elliptic units, and state Yager’s theorem relating the two. Our presentation follows [4], to which the reader is referred for more details on these topics. Using results of Rubin, we then twist the elliptic units into an Euler system more suitable for our purposes and use the twisted Euler system to compute the corank of the 𝔭^{∗}-power Selmer group over D_{∞}.

2.1. The p-adic L-function. For any integers k, j, we define a grossencharacter (of type A_{0}, although we shall never consider any other type) of type (k, j) to be a Q^{al}-valued function, ϵ, defined on integral ideals prime to some ideal 𝔪, such that if 𝔞 = αO_{K} with α ≡ 1 (mod 𝔪) then ϵ(𝔞) = α^{k}ᾱ^{j}. We have the usual notion of the conductor of a grossencharacter, and the usual L-function defined to be (the analytic continuation of)

L(ϵ, s) = ∏_{𝔩} \frac{1}{1 − ϵ(𝔩)N(𝔩)^{−s}}

where the product is over all primes 𝔩 of K, with the convention that ϵ(𝔩) = 0 for 𝔩 dividing the conductor of ϵ. For any ideal 𝔪, the notation L_{𝔪}(ϵ, s) means the L-function without Euler factors at primes dividing 𝔪, and

L_{∞,𝔪}(ϵ, s) = \frac{Γ(s − min(k, j))}{(2π)^{s−min(k,j)}}L_{𝔪}(ϵ, s).

Finally, if ϵ has conductor 𝔣_{ϵ} and type (k, j), set R(ϵ, s) = (d_{K}N(𝔣_{ϵ}))^{s/2}L_{∞,𝔣_{ϵ}}(ϵ, s), where d_{K} denotes the discriminant of K. Then we have the functional equation R(ϵ, s) = W_{ϵ} · R(ϵ̄, 1 + k + j − s) for some constant W_{ϵ} of absolute value one (the “root number” associated to ϵ). If we take ϵ = ψ to be the grossencharacter of our elliptic curve, then the functional equation reads R(ψ, s) = W_{ψ} · R(ψ, 2 − s), since ψ̄(𝔩) = ψ(𝔩̄) implies that L(ψ, s) = L(ψ̄, s). In particular W_{ψ} must be ±1.

To any grossencharacter of conductor dividing 𝔪, we associate p-adic Galois characters

ϵ_{𝔮} : Gal(K(𝔪p^{∞})/K) ⟶ C_{p}^{×}

by the rule ϵ_{𝔮}(σ_{𝔞}) = i_{𝔮}(ϵ(𝔞)), where 𝔮 is 𝔭 or 𝔭^{∗}, and σ_{𝔞} is the Frobenius of 𝔞. The character ψ_{𝔮} agrees with the character

Gal(K(𝔣𝔮^{∞})/K) ⟶ Aut(T_{𝔮}) ≅ Z_{p}^{×},

and the formalism of the Weil pairing implies that ψ_{𝔭}ψ_{𝔭^{∗}} is the cyclotomic character.

Theorem 2.1.1. (Katz) There are measures

(2.1)  μ_{𝔭} ∈ Λ(K(𝔣p^{∞}))_{R_{0}}  μ_{𝔭^{∗}} ∈ Λ(K(𝔣p^{∞}))_{R_{0}}

such that if ϵ is a grossencharacter of conductor dividing 𝔣p^{∞} of type (k, j) with 0 ≤ −j < k, one has the interpolation formula

(2.2)  α_{𝔭}(ϵ) ∫ ϵ_{𝔭} dμ_{𝔭} = (1 − \frac{ϵ(𝔭)}{p}) · L_{∞,𝔣𝔭^{∗}}(ϵ^{−1}, 0)

where α_{𝔭}(ϵ) ∈ C_{p} is a nonzero constant, the integral is over Gal(K(𝔣p^{∞})/K), and the right hand side is interpreted as an element of C_{p} via the embedding i_{𝔭}. As usual, ϵ(𝔭) = 0 if 𝔭 divides the conductor of ϵ. Similarly, if ϵ has infinity type (k, j) with 0 ≤ −j < k and conductor dividing 𝔣p^{∞}, then

α_{𝔭^{∗}}(ϵ) ∫ ϵ_{𝔭^{∗}} dμ_{𝔭^{∗}} = (1 − \frac{ϵ(𝔭^{∗})}{p}) · L_{∞,𝔣𝔭}(ϵ^{−1}, 0)

for some nonzero α_{𝔭^{∗}}(ϵ) where the right hand side is embedded in C_{p} via i_{𝔭^{∗}}.

Proof. This is Theorem II.4.14 of [4]. □

Remark 2.1.2. Our measure μ_{𝔭} is de Shalit’s μ_{p}(𝔣𝔭^{∗∞}). The measure μ_{𝔭} is canonically associated to the field K, the ideal 𝔣 and the embedding i_{𝔭}. In particular it does not depend on the elliptic curve E. The constants α_{𝔭}(ϵ) and α_{𝔭^{∗}}(ϵ) can be made explicit.

Remark 2.1.3. It can be deduced either from the interpolation formulae (2.1), (2.2) or from a result of Yager (see Theorem 2.2.1 below) that the involution of Λ(K(𝔣p^{∞}))_{R_{0}} induced by complex conjugation interchanges μ_{𝔭} and μ_{𝔭^{∗}}.

If ϵ is a grossencharacter of conductor dividing 𝔣p^{∞}, we define

L_{𝔭,𝔣}(ϵ) = ∫_{Gal(K(𝔣p^{∞})/K)} ϵ_{𝔭}^{−1} dμ_{𝔭},

and similarly with 𝔭 replaced by 𝔭^{∗}, so that the interpolation formula reads

(2.3)  L_{𝔭,𝔣}(ϵ) = (1 − \frac{ϵ(𝔭)^{−1}}{p} ) · L_{∞,𝔣𝔭^{∗}}(ϵ, 0)

up to a nonzero constant, provided that ϵ has infinity type (k, j) with 0 ≥ −j > k.

If χ is a Z_{p}^{×}-valued character of Gal(F/K) for some abelian extension F/K, we let

Tw_{χ} : Λ(F)_{R_{0}} ⟶ Λ(F)_{R_{0}}

be the ring automorphism induced by γ ↦ χ(γ)γ on group-like elements. Suppose 𝔮 = 𝔭 or 𝔭^{∗}, F is an extension of K contained in K(𝔣p^{∞}), and χ is a Z_{p}^{×}-valued character of Gal(K(𝔣p^{∞})/K). Define μ_{𝔮}(F; χ) to be the image of Tw_{χ}(μ_{𝔮}) under the natural projection

Λ(K(𝔣p^{∞}))_{R_{0}} ⟶ Λ(F)_{R_{0}},

and for any integral O_{K}-ideal 𝔞 prime to 𝔣p, define λ(F; χ, 𝔞) to be the image of Tw_{χ}(σ_{𝔞} − N𝔞). Let

μ_{𝔮}(F; χ, 𝔞) = μ_{𝔮}(F; χ)λ(F; χ, 𝔞).

In particular, the measure μ_{𝔮}(D_{∞}; ψ_{𝔭^{∗}}, 𝔞) will be of crucial interest.

If F/K is any subextension of K(𝔣p^{∞}) and ϵ is a grossencharacter such that ϵ_{𝔮} factors through Gal(F/K), then

∫ ϵ_{𝔮}^{−1} dμ_{𝔮}(F; ψ_{𝔭^{∗}}, 𝔞) = (ϵ_{𝔮}^{−1}ψ_{𝔭^{∗}}(σ_{𝔞}) − N𝔞) ∫ ϵ_{𝔮}^{−1}ψ_{𝔭^{∗}} dμ_{𝔮}

(2.4)  = \{ \begin{matrix}i_{𝔭}(ϵ^{−1}ψ̄(𝔞) − N𝔞) · L_{𝔭,𝔣}(ϵψ̄^{−1}) & if 𝔮 = 𝔭 \\ i_{𝔭^{∗}}(ϵ^{−1}ψ(𝔞) − N𝔞) · L_{𝔭^{∗},𝔣}(ϵψ^{−1}) & if 𝔮 = 𝔭^{∗} \end{matrix}

where the integral on the left is over Gal(F/K) and the integral on the right is over Gal(K(𝔣p^{∞})/K).

Fix a generator σ of Gal(K_{∞}/D_{∞}). The cyclotomic character defines a canonical isomorphism

<>: Gal(K_{∞}/D_{∞}) ≅ Gal(C_{∞}/K) ≅ 1 + pZ_{p}.

Following Greenberg, we define the critical divisor

Θ = σ− < σ > σ^{−1} ∈ Λ(K_{∞})_{R_{0}}.

If 𝔮 is 𝔭 or 𝔭^{∗}, we have a canonical factorization ψ_{𝔮} = χ_{𝔮}η_{q} where χ_{𝔮} takes values in μ_{p−1} and η_{𝔮} takes values in 1 + pZ_{p}. It is trivial to verify, using the fact that η_{𝔭}η_{𝔭^{∗}} =<>, that Tw_{η_{q}}(Θ) generates the kernel of the natural projection Λ(K_{∞})_{R_{0}} ⟶ Λ(D_{∞})_{R_{0}}.

Let K = K(E[p^{∞}]) so that K ⊂ K(𝔣p^{∞}) by the theory of complex multiplication. We have a natural isomorphism

Gal(K/K) ⟶^{ψ_{𝔭}×ψ_{𝔭^{∗}}} Z_{p}^{×} × Z_{p}^{×},

and hence if we define Δ = Gal(K/K_{∞}), every character of Δ is of the form χ_{𝔭}^{a}χ_{𝔭^{∗}}^{b} for some unique 0 ≤ a, b < p − 1. If χ is any character of Δ, we let e(χ) ∈ Λ(K)_{R_{0}} be the associated idempotent, satisfying γe(χ) = χ(γ)e(χ) We may also view e(χ) as a map Λ(K)_{R_{0}} ⟶ Λ(K_{∞})_{R_{0}}, hopefully without confusion. If χ is the trivial character, this map is the natural projection.

Theorem 2.1.4. (Greenberg) Let 𝔮 = 𝔭 or 𝔭^{∗}, 𝔮^{∗} = 𝔮̄, and denote by W the sign in the functional equation of ψ. If 1 denotes the trivial character, so that μ_{𝔮}(K, 1) is the image of μ_{𝔮} in Λ(K)_{R_{0}}, then

(1) the critical divisor divides e(χ_{𝔮})μ_{𝔮}(K, 1) if and only if W = −1,

(2) the critical divisor divides e(χ_{𝔮^{∗}})μ_{𝔮}(K, 1) if and only if W = 1.

Proof. The first claim is exactly the case k_{0} = 0 of [6], Proposition 6. For the second claim, the case k_{0} = p − 2 of the same proposition shows that the critical divisor divides e(χ_{𝔮^{∗}})μ_{𝔮}(K, 1) if and only if W_{p−2} = −1, where W_{p−2} is the sign in the functional equation of ψ^{2(p−2)+1}. Let m denote the number of roots of unity in K. Proposition 1 of [6], together with the fact that p ≡ 1 (mod m), implies that W_{p−2} = −W. □

Corollary 2.1.5. The measure μ_{𝔭}(D_{∞}, ψ_{𝔭^{∗}}) is nonzero if and only if W = −1. The measure μ_{𝔭^{∗}}(D_{∞}, ψ_{𝔭^{∗}}) is nonzero if and only if W = 1.

Proof. Let 𝔮 = 𝔭 or 𝔭^{∗}. We have μ_{𝔮}(D_{∞}, ψ_{𝔭^{∗}}) = 0 if and only if Tw_{η_{𝔭^{∗}}} (Θ) divides μ_{𝔮}(K_{∞}, ψ_{𝔭^{∗}}), which occurs if and only if Θ divides

Tw_{η_{𝔭^{∗}}^{−1}}(μ_{𝔮}(K_{∞}, ψ_{𝔭^{∗}})) = Tw_{ψ_{𝔭^{∗}}^{−1}}(e(1)μ_{𝔮}(K, ψ_{𝔭^{∗}}))
= e(χ_{𝔭^{∗}})Tw_{ψ_{𝔭^{∗}}^{−1}}(μ_{𝔮}(K, ψ_{𝔭^{∗}}))
= e(χ_{𝔭^{∗}})μ_{𝔮}(K, 1).

The claim now follows from Theorem 2.1.4. □

Corollary 2.1.5 shows that one of the measures μ_{𝔭}(D_{∞}, ψ_{𝔭^{∗}}), μ_{𝔭^{∗}}(D_{∞}, ψ_{𝔭^{∗}}) (depending upon the value of W) is non-zero. We conclude this subsection by describing an alternative approach to showing this fact, using root number calculations and non-vanishing theorems for complex-valued L-functions.

Suppose that θ is a C^{×}-valued idele class character of K, of conductor f_{θ}. Then we may write θ = ∏_{v} θ_{v}, where the product is over all places of K. We define the integer n(θ) by the equality θ_{∞}(z) = z^{n(θ)}|z|^{s}. Hence, if θ is associated to a grossencharacter of K of type (k, j), then

θ_{∞}(z) = z^{k}z̄^{j} = z^{k−j}|z|^{2j},

and so n(θ) = k − j.

Let W_{θ} denote the root number associated to θ. It follows easily from standard properties of root numbers (see, for example [10], especially Proposition 2.2 on page 30 and the definition on page 32) that W_{θ} = W_{θ/|θ|}.

Proposition 2.1.6. (Weil) Suppose that θ_{1} and θ_{2} are C^{×}-valued idele class characters of absolute value 1. Assume also that f_{θ_{1}} and f_{θ_{2}} are relatively prime. Then

W_{θ_{1}}W_{θ_{2}}θ_{1}(f_{θ_{2}})θ_{2}(f_{θ_{1}}) = \begin{cases}W_{θ_{1}θ_{2}} & if n(θ_{1})n(θ_{2}) ≥ 0; \\ (−1)^{ν}W_{θ_{1}θ_{2}} & if n(θ_{1})n(θ_{2}) < 0, \end{cases}

where ν = inf{|n(θ_{1})|, |n(θ_{1})|} mod 2.

Proof. See [26], pages 151-161 (especially Section 79). □

Let K[p^{n}] denote the ring class field of K of conductor p^{n}, and set K[p^{∞}] = ∪_{n≥1}K[p^{n}].

Proposition 2.1.7. Let ξ be a grossencharacter of K whose associated Galois character factors through Gal(K[p^{∞}]/K) and is of finite order.

(a)(Greenberg) The following equality holds

W_{ξψ} = W_{ψ}.

(b) Let e be a positive integer, and let ϵ be a grossencharacter of type (−e, e) associated to K, of trivial conductor (such a grossencharacter always exists for a suitable choice of e). Then

W_{ϵξψ} = −W_{ψ}.

Proof. (a) This is proved on page 247 of [6]. (Note that the proof given in [6] assumes that the Galois character associated to ξ factors through Gal(D_{∞}/K). It is easy to see that the same proof holds if we instead assume that the Galois character associated to ξ factors through Gal(K[p^{∞}]/K).)

(b) From part (a), we see that it suffices to show that

W_{ϵξψ} = −W_{ξψ}.

The proof of this equality proceeds by applying Proposition 2.1.6 to the idele class characters θ_{1} = ξψ/|ξψ| and θ_{2} = ϵ/|ϵ|.

We first note that since ϵ has trivial conductor, the same is true of θ_{2}, and so θ_{1}(f_{θ_{2}}) = 1. Next, we observe that since E is defined over Q, it follows that f̄_{θ_{1}} = f_{θ_{1}}. Since ϵ is of type (e, −e) and has trivial conductor, this implies that θ_{2}(f_{θ_{1}}) = 1.

Let δ_{K} denote the different of K/Q. It follows from standard formulae for global root numbers (see [9], Chapter XIV, §8, Corollary 1, for example) that

(2.5)  W_{ϵ} = i^{−2e}ϵ(δ_{K}^{−1}) = (−1)^{e}ϵ(δ_{K}^{−1}).

It is not hard to check that the different of any imaginary quadratic field of class number one has a generator δ satisfying δ/|δ| = i. This implies (since ϵ has trivial conductor and is of type (e, −e)) that ϵ(δ_{K}^{−1}) = (−1)^{e}. It now follows from (2.5) that

W_{ϵ} = 1 = W_{θ_{2}}.

Finally, we note that n(θ_{1}) = 1 and n(θ_{2}) = −2e, whence it follows that ν = 1. Putting all of the above together gives

W_{θ_{1}θ_{2}} = −W_{θ_{1}}W_{θ_{2}},

from which we deduce that

W_{ϵξψ} = −W_{ξψ} = −W_{ψ},

as claimed. □

Now suppose that W_{ψ} = −1, and let κ = ϵξ be a grossencharacter as in Proposition 2.1.7(b) whose associated Galois character factors through Gal(D_{∞}/K). (Note that once a choice of ϵ is fixed, then there are infinitely many choices of ξ such that the Galois character associated to ϵξ factors through Gal(D_{∞}/K).) Then κψ̄^{−1} is of type (−e, e − 1), and so it lies within the range of interpolation of (2.3). Hence we have (from (2.3) and (2.4))

∫ κ_{𝔭}^{−1}dμ_{𝔭}(D_{∞}; ψ_{𝔭^{∗}}, 𝔞) = i_{𝔭}(κ^{−1}ψ̄(𝔞) − N𝔞)L_{𝔭,𝔣}(κψ̄^{−1})

= i_{𝔭}(κ^{−1}ψ̄(𝔞) − N𝔞) (1 − \frac{κ(𝔭)^{−1}}{p} ) L_{∞,𝔣𝔭^{∗}}(κψ̄^{−1}, 0)

= i_{𝔭}(κ^{−1}ψ̄(𝔞) − N𝔞) (1 − \frac{κ(𝔭)^{−1}}{p} ) L_{∞,𝔣𝔭^{∗}}(κψ, 1),

where for the last equality we have used the fact that ψψ̄ = N.

Next we note that Proposition 2.1.7(b) implies that W_{κψ} = 1. It now follows from a theorem of Rohrlich (see page 384 of [20]) that, for all but finitely many choices of κ, we have L_{∞,𝔣𝔭^{∗}}(κψ, 1) ≠ 0. Hence the measure μ_{𝔭}(D_{∞}; ψ_{𝔭^{∗}}, 𝔞) is non-zero, and so the same is true of μ_{𝔭}(D_{∞}; ψ_{𝔭^{∗}}).

We now turn to the measure μ_{𝔭^{∗}}(D_{∞}; ψ_{𝔭^{∗}}). Suppose that W_{ψ} = 1, and let ξ be any character of Gal(D_{∞}/K) of finite order. Then the grossencharacter ξψ^{−1} is of type (−1, 0), and so lies within the range of interpolation of (2.3). Hence, just as above, we have

∫ ξ_{𝔭^{∗}}^{−1}dμ_{𝔭^{∗}}(D_{∞}; ψ_{𝔭^{∗}}, 𝔞) = i_{𝔭^{∗}}(ξ^{−1}ψ(𝔞) − N𝔞)L_{𝔭^{∗},𝔣}(ξψ^{−1})

= i_{𝔭^{∗}}(ξ^{−1}ψ(𝔞) − N𝔞) (1 − \frac{ξ(𝔭^{∗})^{−1}}{p} ) L_{∞,𝔣𝔭}(ξψ^{−1}, 0)

= i_{𝔭^{∗}}(ξ^{−1}ψ(𝔞) − N𝔞) (1 − \frac{ξ(𝔭^{∗})^{−1}}{p} ) L_{∞,𝔣𝔭}(ξψ̄, 1).

Now Proposition 2.1.7(a) implies that that W (ξψ̄) = W (ψ) = 1 . It therefore follows from the theorem of Rohrlich quoted above that, for all but finitely many choices of ξ, we have L_{∞,𝔣𝔭}(ξψ̄, 1) ≠ 0. This in turn implies that μ_{𝔭^{∗}}(D_{∞}; ψ_{𝔭^{∗}}, 𝔞) is non-zero, whence it follows that μ_{𝔭^{∗}}(D_{∞}; ψ_{𝔭^{∗}}) is non-zero also.

2.2. Elliptic units. If F/K is any finite extension and 𝔮 = 𝔭 or 𝔭^{∗}, we define U_{𝔮}(F) to be the direct sum over all places w dividing 𝔮 of the principal units of F_{w}. If F/K is any (possibly infinite) extension, we define U_{𝔮}(F) to be the inverse limit with respect to the norm maps of the groups U_{𝔮}(F′) as F′ ⊂ F ranges over the finite extensions of K. Let 𝔞 be an O_{K}-ideal with (𝔞, 𝔣p) = 1, and let I(𝔞) denote the set of all ideals of O_{K} prime to 𝔞.

If L ⊂ C is a lattice with CM by O_{K}, we define an elliptic function

Θ(z; L, 𝔞) = \frac{Δ(L)}{Δ(𝔞^{−1}L)} ∏ \frac{Δ(L)}{(℘(z, L) − ℘(u, L))^{6}}

where the product is over the nontrivial u ∈ 𝔞^{−1}L/L, and Δ is the modular discriminant. For any 𝔪 ∈ I(𝔞), let w_{𝔪} be the number of roots of unity congruent to 1 modulo 𝔪. If 𝔪 is not a prime power then Θ(1; 𝔪, 𝔞) is a unit of K(𝔪), and if 𝔩 ∈ I(𝔞) is prime we have the distribution relation

Norm_{K(𝔪𝔩)/K(𝔪)}Θ(1; 𝔪𝔩, 𝔞)^{e} = \{ \begin{matrix}Θ(1; 𝔪, 𝔞) & if 𝔩 | 𝔪 \\ Θ(1; 𝔪, 𝔞)^{1−σ_{𝔩}^{−1}} & else \end{matrix}

where e = w_{𝔪}/w_{𝔪𝔩}. In particular, the sequence Θ(1; 𝔣p^{k}, 𝔞) is norm compatible for k > 0. We denote by β(𝔞) and β^{∗}(𝔞) the images of this sequence in U_{𝔭}(K(𝔣p^{∞})) and U_{𝔭^{∗}}(K(𝔣p^{∞})), respectively.

Define J ⊂ Λ(K(𝔣p^{∞}))_{R_{0}} to be the annihilator of μ_{p^{∞}}. Then J is the ideal generated by σ_{𝔟} − N(𝔟) as 𝔟 ranges over integral ideals prime to 𝔣p.

Theorem 2.2.1. (Yager) There are isomorphisms of Λ(K(𝔣p^{∞}))_{R_{0}}-modules

U_{𝔭}(K(𝔣p^{∞}))_{R_{0}} ≅ J, U_{𝔭^{∗}}(K(𝔣p^{∞}))_{R_{0}} ≅ J,

which take

β(𝔞) ↦ (σ_{𝔞} − N(𝔞)) · μ_{𝔭}, β^{∗}(𝔞) ↦ (σ_{𝔞} − N(𝔞)) · μ_{𝔭^{∗}}.

Proof. This is Proposition III.1.4 of [4]. □

2.3. The twisted Euler system. We continue to denote by 𝔞 a nontrivial O_{K}- ideal prime to 𝔣p. If 𝔪 ∈ I(𝔞), define a unit

ϑ_{𝔞}(𝔪) = Norm_{K(𝔪𝔣p)/K(𝔪)}Θ(1; 𝔪𝔣p, 𝔞).

If we let K_{𝔞} = ∪_{𝔪∈I(𝔞)}K(𝔪), the elements

ϑ_{𝔞}(𝔪) ∈ H^{1}(K(𝔪), Z_{p}(1)),

with 𝔞 fixed, form an Euler system for (Z_{p}(1), 𝔣p, K_{𝔞}) in the sense of [25].

Proposition 2.3.1. Let 𝔮 = 𝔭 or 𝔭^{∗}. There is an Euler system c = c_{𝔞} for (T_{𝔭}, 𝔣p, K_{𝔞}) and an injection (the Coleman map)

Col_{𝔮} : H^{1}(K(𝔣p^{∞})_{𝔮}, T_{𝔭})_{R_{0}} ⟶ Λ(K(𝔣p^{∞}))_{R_{0}}

with the following property: if we set

z = lim_{←} c(K(𝔣p^{k})) ∈ H^{1}(K(𝔣p^{∞}), T_{𝔭}),

and let loc_{𝔮}(z) be its image in H^{1}(K(𝔣p^{∞})_{𝔮}, T_{𝔭}), then Col_{𝔮} sends

loc_{𝔮}(z) ⊗ 1 ↦ μ_{𝔮}(K(𝔣p^{∞}), ψ_{𝔭^{∗}}, 𝔞).

The image of Col_{𝔮} is Tw_{ψ_{𝔭^{∗}}} (J), the ideal generated by all elements of the form σ_{𝔟} − ψ_{𝔭}(σ_{𝔟}) with 𝔟 prime to 𝔣p.

Proof. This follows from the “twisting” theorems of Chapter 6 of [25]. The G_{K}- module T_{𝔭} is isomorphic to the twist of Z_{p}(1) by the character ω^{−1}ψ_{𝔭} = ψ_{𝔭^{∗}}^{−1}, where ω is the cyclotomic character. A choice of such an isomorphism determines an isomorphism of Z_{p}-modules

H^{1}(K(𝔣p^{∞}), Z_{p}(1)) ⟶^{ϕ} H^{1}(K(𝔣p^{∞}), T_{𝔭})

satisfying ϕ ∘ Tw_{ψ_{𝔭^{∗}}}^{−1} (λ) = λ ∘ ϕ for any λ ∈ Λ(K(𝔣p^{∞})). Similarly, if 𝔮 = 𝔭 or 𝔭^{∗}, there is an isomorphism of Z_{p}-modules

U_{𝔮}(K(𝔣p^{∞})) ≅ H^{1}(K(𝔣p^{∞})_{𝔮}, Z_{p}(1)) ⟶^{ϕ_{𝔮}} H^{1}(K(𝔣p^{∞})_{𝔮}, T_{𝔭})

which is compatible with ϕ and the localization map loc_{𝔮}.

The Euler system c_{𝔞} is the twist of ϑ_{𝔞} by ψ_{𝔭^{∗}}, and by construction

z = ϕ(lim_{←} ϑ_{𝔞}(𝔣p^{k})) = ϕ(lim_{←} Θ(1; 𝔣p^{k}, 𝔞)).

See Section 6.3 of [25], especially Theorem 6.3.5. In particular, loc_{𝔮}(z) = ϕ_{𝔮}(β_{𝔞}), and if we define Col_{𝔮} to be the composition

H^{1}(K(𝔣p^{∞})_{𝔮}, T_{𝔭})_{R_{0}} ⟶^{ϕ_{𝔮}^{−1}} U_{𝔮}(K(𝔣p^{∞}))_{R_{0}} ≅ J ⟶^{Tw_{ψ_{𝔭^{∗}}}} Tw_{ψ_{𝔭^{∗}}} (J)

then Col_{𝔮} is an isomorphism of Λ(K(𝔣p^{∞}))_{R_{0}}-modules with the desired properties.

□

Definition 2.3.2. We say that the prime p is anomalous if p divides

(1 − ψ(𝔭))(1 − ψ(𝔭^{∗})),

or equivalently if there is any p-torsion defined over Z/pZ on the reduction of E at p.

Lemma 2.3.3. Let F ⊂ K_{∞} contain K and let 𝔮 = 𝔭 or 𝔭^{∗}. The natural corestriction map

(2.6)  H^{1}(K_{∞,𝔮}, T_{𝔭}) ⊗_{Λ(K_{∞})} Λ(F) ⟶ H^{1}(F_{𝔮}, T_{𝔭})

is an isomorphism if either 𝔮 = 𝔭^{∗} or if 𝔮 = 𝔭 and p is not anomalous. If F = C_{m}D_{n} for some 0 ≤ m, n ≤ ∞, then the map is injective with finite cokernel.

Proof. Let v be any place of K_{∞} above 𝔭^{∗}, and denote also by v the place of F below it. By Lemma 1.1.2, E(K_{∞,v})[𝔭^{∗}] = 0, and the inflation-restriction sequence shows that

H^{1}(F_{v}, W_{𝔭^{∗}}) ⟶ H^{1}(K_{∞,v}, W_{𝔭^{∗}})^{Gal(K_{∞,v}/F_{v})}

is an isomorphism. This implies that the restriction map

H^{1}(F_{𝔭^{∗}}, W_{𝔭^{∗}}) ⟶ H^{1}(K_{∞,𝔭^{∗}}, W_{𝔭^{∗}})^{Gal(K_{∞}/F)}

is an isomorphism. By local duality, the map

H^{1}(K_{∞,𝔭^{∗}}, T_{𝔭}) ⊗_{Λ(K_{∞})} Λ(F) ⟶ H^{1}(F_{𝔭^{∗}}, T_{𝔭})

is an isomorphism. If p is not anomalous then E(L)[𝔭] = 0 for any p-power extension L/K_{𝔭}, and the same argument as above shows that

H^{1}(K_{∞,𝔭}, T_{𝔭}) ⊗_{Λ(K_{∞})} Λ(F) ⟶ H^{1}(F_{𝔭}, T_{𝔭})

is an isomorphism.

Now suppose F = C_{m}D_{n}, let v be a place of K_{∞} above 𝔭, and suppose also that p is anomalous; this implies that all 𝔭^{∗}-power torsion of E is defined over K_{∞,v}, and in fact is defined over the unique unramified Z_{p}-extension of K_{v}. Set L = C_{∞}D_{n}, and define

B = H^{0}(L_{v}, W_{𝔭^{∗}}) = H^{0}(K_{v}^{unr} ∩ L_{v}, W_{𝔭^{∗}}).

Note that B is finite, since K_{v}^{unr} ∩ L_{v} is a finite extension of Q_{p}. We have the inflation-restriction sequence

H^{1}(L_{v}/F_{v}, B) ⟶ H^{1}(F_{v}, W_{𝔭^{∗}}) ⟶^{res} H^{1}(L_{v}, W_{𝔭^{∗}})^{Gal(L_{v}/F_{v})} ⟶ H^{2}(L_{v}/F_{v}, B).

Since Gal(L_{v}/F_{v}) ≅ Z_{p} is of cohomological dimension one, the final term of this sequence is trivial, and so the map res is surjective. The first term of the sequence is isomorphic to B/(γ − 1)B for any generator γ of Gal(L_{v}/F_{v}), and this is finite since B is.

Now consider the restriction map

H^{1}(L_{v}, W_{𝔭^{∗}}) ⟶ H^{1}(K_{∞,v}, W_{𝔭^{∗}})^{Gal(K_{∞,v}/L_{v})}.

Since K_{∞,v}/L_{v} is a Z_{p}-extension, the restriction is surjective, exactly as above. Similarly, the kernel is isomorphic to W_{𝔭^{∗}}/(σ − 1)W_{𝔭^{∗}} for any generator σ of Gal(K_{∞,v}/L_{v}). Such a σ acts on W_{𝔭^{∗}} through some scalar ≠ 1, and since W_{𝔭^{∗}} is divisible the kernel of restriction is trivial. As above, local duality gives the stated results. □

Proposition 2.3.4. Let F ⊂ K_{∞} be a (possibly infinite) extension of K, and let 𝔮 = 𝔭 or 𝔭^{∗}. Suppose that one of the following holds

(1) 𝔮 = 𝔭^{∗},

(2) 𝔮 = 𝔭 and p is not anomalous,

(3) 𝔮 = 𝔭 and F = D_{m}C_{n} for 0 ≤ m, n ≤ ∞. There is an injection of Λ_{R}(F)-modules

H^{1}(F_{𝔮}, T_{𝔭})_{R} ⟶ Λ(F)_{R}

taking loc_{𝔮}(c(F)) ⊗ 1 to μ_{𝔮}(F; ψ_{𝔭^{∗}}, 𝔞). The image of this map is the ideal generated by λ(F; ψ_{𝔭^{∗}}, 𝔞) as 𝔞 varies, and if F = D_{∞} the map is an isomorphism. If p does not divide [K(𝔣) : K] and if either (1) or (2) holds, the result is true with R replaced by R_{0}.

Proof. Let w be a place of K(𝔣p^{∞}) above p, set H = Gal(K(𝔣p^{∞})_{w}/K_{∞,w}) and consider the inflation-restriction sequence

H^{1}(H, W_{𝔭^{∗}}) ⟶ H^{1}(K_{∞,w}, W_{𝔭^{∗}}) ⟶ H^{1}(K(𝔣p^{∞})_{w}, W_{𝔭^{∗}}) ⟶ H^{2}(H, W_{𝔭^{∗}}).

The first and last terms are finite. If p does not divide [K(𝔣) : K] then p does not divide |H|, and so the kernel and cokernel of restriction are trivial. Applying local duality and considering the semi-local cohomology, we see that the map

H^{1}(K(𝔣p^{∞})_{𝔮}, T_{𝔭}) ⊗_{Λ(K(𝔣p^{∞}))} Λ(K_{∞}) ⟶ H^{1}(K_{∞,𝔮}, T_{𝔭})

has finite kernel and cokernel, and is an isomorphism if [K(𝔣) : K] is prime to p. By Lemma 2.3.3 the natural map

H^{1}(K(𝔣p^{∞})_{𝔮}, T_{𝔭}) ⊗_{Λ(K(𝔣p^{∞}))} Λ(F) ⟶ H^{1}(F_{𝔮}, T_{𝔭})

has finite kernel and cokernel, and so becomes an isomorphism upon applying first ˆ⊗R_{0} and then ⊗R. Tensoring the Coleman map of Proposition 2.3.1 with Λ(F) yields the desired map. If F = D_{∞} then F is disjoint from the extension of K cut out by ψ_{𝔭}, and it follows that the image of the map above is the ideal of Λ(F)_{R} generated by all elements of the form γ − α, where γ runs over Gal(D_{∞}/K) and α runs over Z_{p}^{×}. This ideal is all of Λ(F)_{R}. □

2.4. Main conjectures. Throughout this subsection we fix a topological generator γ ∈ Gal(K_{∞}/D_{∞}), and we let I = (γ − 1)Λ(K_{∞}).

Denote by W the sign in the functional equation of L(ψ, s). Let 𝔞 be an integral ideal of O_{K} prime to p𝔣, and recall that K_{𝔞} is the union of all ray class fields of K of conductor prime to 𝔞. Let c_{𝔞} be the Euler system for (T_{𝔭}, 𝔣p, K_{𝔞}) of Proposition 2.3.1, and for any F ⊂ K_{∞}, let

c_{𝔞}(F) = lim_{←} c_{𝔞}(F′) ∈ H^{1}(F, T_{𝔭}).

be the limit as F′ ranges over subfields of F finite over K. Let C_{𝔞}(F) be the Λ(F)-submodule of S_{rel}(F, T_{𝔭}) generated by c_{𝔞}(F), and let C(F) be the submodule generated by C_{𝔞}(F) as 𝔞 varies over all ideals prime to p𝔣. Define

Z(F) = S_{rel}(F, T_{𝔭})/C(F),

and define Z_{𝔞}(F) similarly, replacing C by C_{𝔞}.

Remark 2.4.1. It is clear from the definitions that Sel_{rel}(F, T_{𝔭}) = H^{1}(F, T_{𝔭}) for every extension F/K. In particular c_{𝔞}(F) ∈ S_{rel}(F, T_{𝔭}).

Lemma 2.4.2. For every F ⊂ K_{𝔞} finite over K, the class c_{𝔞}(F) is unramified at every prime of F not dividing p.

Proof. This follows from Corollary B.3.5 of [25], and the fact that the class c_{𝔞}(F) is a universal norm in the cyclotomic direction. □

Proposition 2.4.3. The submodule loc_{𝔭^{∗}}(C(D_{∞})) ⊂ H^{1}(D_{∞,𝔭^{∗}}, T_{𝔭}) is nontrivial if and only if W = 1. The submodule loc_{𝔭}(C(D_{∞})) ⊂ H^{1}(D_{∞,𝔭}, T_{𝔭}) is nontrivial if and only if W = −1. In particular, C(D_{∞}) ≠ 0 regardless of the value of W.

Proof. Suppose W = 1. By Corollary 2.1.5,

μ_{𝔭}(D_{∞}, ψ_{𝔭^{∗}}) = 0 μ_{𝔭^{∗}}(D_{∞}, ψ_{𝔭^{∗}}) ≠ 0.

We may choose 𝔞 so that λ(D_{∞}; ψ_{𝔭^{∗}}, 𝔞) ≠ 0, and then Proposition 2.3.4 implies that loc_{𝔭^{∗}}(C_{𝔞}(D_{∞})) ≠ 0. Therefore loc_{𝔭^{∗}}(C(D_{∞})) ≠ 0. On the other hand, for every choice of 𝔞, μ_{𝔭}(D_{∞}; ψ_{𝔭^{∗}}, 𝔞) = 0, and so Proposition 2.3.4 implies that loc_{𝔭}(C_{𝔞}(D_{∞})) is trivial in H^{1}(D_{∞,𝔭}, T_{𝔭})_{R} By Proposition 1.1.6, H^{1}(D_{∞,𝔭}, T_{𝔭}) is torsion-free, and so loc_{𝔭}(C_{𝔞}(D_{∞})) is trivial in H^{1}(D_{∞,𝔭}, T_{𝔭}). The case W = −1 is entirely similar. □

Now armed with a nontrivial Euler system, we may apply the general theory introduced by Kolyvagin and developed by Kato, Perrin-Riou, and Rubin.

Proposition 2.4.4. The Λ(D_{∞})-module X_{str}^{∗}(D_{∞}) is torsion and S_{rel}(D_{∞}, T_{𝔭}) is torsion free of rank one. Furthermore, we have the divisibility of characteristic ideals

char(X_{str}^{∗}(D_{∞})) divides char(Z(D_{∞})).

If W = 1 then the Λ(D_{∞})-module X^{∗}(D_{∞}) is torsion. If W = −1 then X^{∗}(D_{∞}) has rank one and S(D_{∞}, T_{𝔭}) = S_{rel}(D_{∞}, T_{𝔭}).

Proof. By Proposition 2.4.3, we may choose some 𝔞 so that C_{𝔞}(D_{∞}) ≠ 0. The first claim follows from Theorem 2.3.2 of [25] (together with Lemma 2.4.2 and the remarks of Section 9.2 of [25]), and the second then follows from Lemmas 1.1.9 and 1.2.1 and Theorem 1.2.2. Applying Theorem 2.3.3 of [25] as 𝔞 varies over all integral ideals prime to p𝔣, one obtains the divisibility of characteristic ideals.

Suppose W = 1. The image of C(D_{∞}) under

loc_{𝔭^{∗}} : S_{rel}(D_{∞}, T_{𝔭}) ⟶ H^{1}(D_{∞,𝔭^{∗}}, T_{𝔭})

is nontrivial by Proposition 2.4.3, and since both modules are torsion-free of rank one this map must be injective with torsion cokernel. By Proposition 1.1.7 we obtain the exact sequence

0 ⟶ S_{rel}(D_{∞}, T_{𝔭}) ⟶ H^{1}(D_{∞,𝔭^{∗}}, T_{𝔭}) ⟶ X^{∗}(D_{∞}) ⟶ X_{str}^{∗}(D_{∞}) ⟶ 0

which shows that X^{∗}(D_{∞}) is torsion.

Suppose W = −1. By Proposition 2.4.3 and the exact sequence (1.3), C(D_{∞}) ⊂ S(D_{∞}, T_{𝔭}). Since C(D_{∞}) ≠ 0 and S(D_{∞}, T_{𝔭}) ⊂ S_{rel}(D_{∞}, T_{𝔭}), it follows that S(D_{∞}, T_{𝔭}) is torsion free of rank one. From Lemma 1.2.1 and Proposition 1.2.2, we see that X^{∗}(D_{∞}) has rank one. Furthermore, the image of loc_{𝔭^{∗}} in the exact sequence (1.3) (with F = D_{∞}) must be a torsion module, and hence must be trivial. Therefore S(D_{∞}, T_{𝔭}) = S_{rel}(D_{∞}, T_{𝔭}). □

Let K = K(E[p^{∞}]) and define abelian extensions of K as follows:

• M_{rel} is the maximal abelian pro-p-extension of K unramified outside of p,

• M is the maximal abelian pro-p-extension of K unramified outside of 𝔭^{∗},

• M_{str} is the maximal abelian pro-p-extension of K unramified everywhere. Let E be the inverse limit of the groups O_{F}^{×}⊗Z_{p} over subfields F ⊂ K containing K. For any integral ideal 𝔞 of K prime to p𝔣, let U_{𝔞} ⊂ E be the submodule generated by the (untwisted) elliptic unit Euler system ϑ_{𝔞} of §2.3. Let U be the submodule generated by all such U_{𝔞}. By Kummer theory we may view U ⊂ E ⊂ H^{1}(K, Z_{p}(1)).

The following result is essentially due to Coates (see [3] Theorem 12).

Lemma 2.4.5. There is a group isomorphism

α : H^{1}(K, Q_{p}/Z_{p}) ≅ H^{1}(K, W_{𝔭^{∗}})

satisfying α ∘ Tw_{ψ^{∗}}(λ) = λ ∘ α for every λ ∈ Λ(K), where Tw_{ψ_{𝔭^{∗}}} : Λ(K) ⟶ Λ(K) is the ring automorphism of §2.1. This map restricts to an isomorphism (of groups, not Λ(K)-modules)

Hom(Gal(M/K), Q_{p}/Z_{p}) ≅ Sel(K, W_{𝔭^{∗}})

and similarly for the relaxed and restricted Selmer groups, replacing M by M_{rel} and M_{str}, respectively. Similarly, there is a group isomorphism

β : H^{1}(K, Z_{p}(1)) ≅ H^{1}(K, T_{𝔭})

satisfying β ∘ Tw_{ψ_{𝔭^{∗}}}^{−1} (λ) = λ ∘ β for every λ ∈ Λ(K). This isomorphism identifies E with S_{rel}(K, T_{𝔭}) and U with C(K).

Proof. The existence of

α : H^{1}(K, Q_{p}/Z_{p}) ≅ H^{1}(K, W_{𝔭^{∗}})

follows from the twisting theorems of [25] §6.2, once one fixes an isomorphism W_{𝔭^{∗}} ≅ (Q_{p}/Z_{p})(ψ_{p^{∗}}). From the definitions, together with Proposition 1.1.7, we have the following characterizations of our Selmer groups in H^{1}(K, W_{𝔭^{∗}}):

• Sel_{rel}(K, W_{𝔭^{∗}}) consists of the classes locally trivial away from p,

• Sel(K, W_{𝔭^{∗}}) consists of the classes locally trivial away from 𝔭^{∗},

• Sel_{str}(K, W_{𝔭^{∗}}) consists of the classes everywhere locally trivial. The isomorphism α identifies each of these Selmer groups with the subgroup of classes in H^{1}(K, Q_{p}/Z_{p}) satisfying the same local conditions, and so it suffices to check that for every place v of K, the condition “locally trivial at v” agrees with the condition “unramified at v”. For v not dividing p this is [25] Lemma B.3.3, and the case v|p is identical: fix a place v of K and note that regardless of the rational prime below v, K_{v} always contains the unique unramified Z_{p}-extension of K_{v}. In particular, if K^{unr} is the maximal unramified extension of K, then Gal(K^{unr}/K) has trivial pro-p-part, and so H^{1}(K^{unr}/K, Q_{p}/Z_{p}) = 0.

For the compact cohomology group H^{1}(K, T_{𝔭}) the existence of β is proved in the same fashion, using the fact that ψ_{𝔭}ψ_{𝔭^{∗}} is the cyclotomic character. That β identifies the relaxed Selmer group with the unit group is a consequence of the discussion above, since local duality shows that the unramified conditions agree with the relaxed conditions everywhere locally. The identification of U_{𝔞} with C_{𝔞}(K) is immediate from the construction of the twisted Euler system c_{𝔞} from the elliptic units ϑ_{𝔞} in Proposition 2.3.1. □

Decompose

Gal(K/K) ≅ Δ × Gal(K_{∞}/K)

where Δ ≅ Gal(K(E[p])/K), and let ψ_{𝔮} = χ_{𝔮}η_{𝔮} be the associated decomposition of ψ_{𝔮}, for 𝔮 = 𝔭 of 𝔭^{∗}.

Corollary 2.4.6. We have the equality of characteristic ideals in Λ(K_{∞})

Tw_{η_{𝔭}^{∗}}^{−1}(char(Gal(M/K)^{χ_{𝔭^{∗}}})) = char(X^{∗}(K_{∞}))

and similarly for the relaxed and restricted Selmer groups. Also,

Tw_{η_{𝔭}^{∗}}^{−1}(char(E/U)^{χ_{𝔭^{∗}}} ) = char(Z(K_{∞})).

Proof. This follows easily by taking Δ-invariants of the Λ(K/K)-modules of the Lemma above (cf. e.g. [25] Lemma 6.1.2). One must remember our convention, Remark 1.1.8, about the Λ(K)-action on X^{∗}(K). □

Theorem 2.4.7. (Rubin) The Λ(K_{∞})-module X^{∗}(K_{∞}) is torsion, S_{rel}(K_{∞}, T_{𝔭}) has rank one, and

char(X^{∗}(K_{∞})) = char(H^{1}(K_{∞,𝔭^{∗}}, T_{𝔭})/loc_{𝔭^{∗}}C(K_{∞}))
char(X_{str}^{∗}(K_{∞})) = char(Z(K_{∞})).

Proof. In view of Lemma 2.4.5 and its corollary, this is a twisted form of the main results of [21]. □

Remark 2.4.8. The fact that X^{∗}(K_{∞}) is torsion is originally due to Coates [3].

The following proposition follows from a deep result of Greenberg. Strictly speaking, it is not needed to prove the main result of this section, Theorem 2.4.17 below, but it is helpful for understanding the case W = 1. See Remark 2.4.18.

Proposition 2.4.9. The Λ(K_{∞})-modules X_{rel}^{∗}(K_{∞}) and X^{∗}(K_{∞}) have no nonzero pseudo-null submodules.

Proof. It is a theorem of Greenberg [5] that Gal(M_{rel}/K) has no non-zero pseudonull submodules, and so Lemma 2.4.5 implies that X_{rel}^{∗}(K_{∞}) also has none. By [17] §II.2 Théorème 23, X^{∗}(K_{∞}) also has no nontrivial pseudo-null submodules. □

Lemma 2.4.10. For 𝔮 = 𝔭 or 𝔭^{∗}, the kernel of the restriction map

H^{1}(D_{∞,𝔮}, W_{𝔭^{∗}}) ⟶ H^{1}(K_{∞,𝔮}, W_{𝔭^{∗}})

is finite.

Proof. The kernel of the restriction map is isomorphic to

H^{0}(K_{∞,𝔮}, W_{𝔭^{∗}})/IH^{0}(K_{∞,𝔮}, W_{𝔭^{∗}})

by the inflation-restriction sequence and [25] Lemma B.2.8. The finiteness follows from Lemma 2.3.3 and local duality. □

Lemma 2.4.11. The semi-local restriction map

⨁_{w|𝔣} H^{1}(D_{∞,w}, W_{𝔭^{∗}}) ⟶ ⨁_{w|𝔣} H^{1}(K_{∞,w}, W_{𝔭^{∗}}).

is injective.

Proof. This is [17] II.7, Lemme 13 (or [4], proof of Lemma IV.3.5), together with the isomorphism

H^{1}(L, W_{𝔭^{∗}}) ≅ H^{1}(L, E)[𝔭^{∗∞}]

for any algebraic extension L/K_{w} (it suffices to prove this isomorphism for finite extensions, where it is a consequence of the Kummer sequence and the fact that E(L) has a finite index pro-ℓ subgroup, where ℓ ≠ p is the residue characteristic of w). □

We will need the following slight generalization of the control theorems of Mazur and Perrin-Riou.

Proposition 2.4.12. The dual to the restriction map

(2.7)  Sel_{rel}(D_{∞}, W_{𝔭^{∗}}) ⟶ Sel_{rel}(K_{∞}, W_{𝔭^{∗}})[I]

is an isomorphism of Λ(D_{∞})-modules

X_{rel}^{∗}(K_{∞})/IX_{rel}^{∗}(K_{∞}) ⟶ X_{rel}^{∗}(D_{∞}).

The analogous maps for X^{∗} and X_{str}^{∗} are surjective with finite cokernel.

Proof. Let S be the set of places of K consisting of the archimedean place and the primes dividing p𝔣, and denote by K_{S}/K the maximal extension of K unramified outside S. By Lemma 1.1.2 H^{0}(K_{S}/K_{∞}, W_{𝔭^{∗}}) = 0, and so the inflation-restriction sequence shows that the restriction map

H^{1}(K_{S}/D_{∞}, W_{𝔭^{∗}}) ⟶ H^{1}(K_{S}/K_{∞}, W_{𝔭^{∗}})[I]

is an isomorphism. As in the proof of Lemma 1.1.5, for w a prime of D_{∞} not lying above a prime of S the local condition H_{f}^{1}(D_{∞,w}, W_{𝔭^{∗}}) is equal to the unramified condition, and similarly for K_{∞}. From the definition of the relaxed Selmer group, we have the commutative diagram with exact rows

\xymatrix{0 \ar[r] & Sel_{rel}(D_{∞}, W_{𝔭^{∗}}) \ar[r] \ar[d] & H^{1}(K_{S}/D_{∞}, W_{𝔭^{∗}}) \ar[r] \ar[d] & ⨁ H^{1}(D_{∞,w}, W_{𝔭^{∗}}) \ar[d] \\ 0 \ar[r] & Sel_{rel}(K_{∞}, W_{𝔭^{∗}})[I] \ar[r] & H^{1}(K_{S}/K_{∞}, W_{𝔭^{∗}})[I] \ar[r] & ⨁ H^{1}(K_{∞,w}, W_{𝔭^{∗}})}

where the direct sums are over places w|𝔣. In particular, since the middle vertical arrow is an isomorphism, the restriction map (2.7) is injective, and to bound the cokernel of this map it suffices to bound the kernel of the right vertical arrow in the diagram above. This kernel is trivial by Lemma 2.4.11. This completes the proof for the relaxed Selmer groups.

In order to prove the result for the true Selmer groups, we replace the top and bottom rows of the commutative diagram above with the exact sequence (1.6) applied with F = D_{∞} and F = K_{∞}, respectively. Again by the snake lemma, it then suffices to bound the kernel of restriction

H^{1}(D_{∞,𝔭}, W_{𝔭^{∗}}) ⟶ H^{1}(K_{∞,𝔭}, W_{𝔭^{∗}}),

and this is the content of Lemma 2.4.10. Similarly, one deduces the result for the strict Selmer group from the result for the true Selmer group by using the exact sequence (1.4), together with another application of Lemma 2.4.10. □

Definition 2.4.13. We define the descent defect 𝔇 ⊂ Λ(D_{∞}) by

𝔇 = char_{Λ(D_{∞})}(X_{str}^{∗}(K_{∞})[I]).

Corollary 2.4.14. The descent defect 𝔇 is nonzero, and we have the equality of ideals in Λ(D_{∞})

char_{Λ(D_{∞})}(X_{str}^{∗}(D_{∞})) = char_{Λ(K_{∞})}(X_{str}^{∗}(K_{∞})) · 𝔇

If W = 1 then

char_{Λ(D_{∞})}(X^{∗}(D_{∞})) = char_{Λ(K_{∞})}(X^{∗}(K_{∞})).

Proof. By Proposition 2.4.4 X_{str}^{∗}(D_{∞}) is a torsion Λ(D_{∞})-module, and so by Proposition 2.4.12, the same is true of X_{str}^{∗}(K_{∞})/IX_{str}^{∗}(K_{∞}). The claim now follows from [21] Lemma 6.2 (i).

When W = 1, X^{∗}(D_{∞}) is a torsion module and the proof is identical, except that now [21] Lemma 6.2 (i) and Proposition 2.4.9 above show that X^{∗}(K_{∞})[I] = 0. □

Recall from Propositions 2.4.3 and 2.4.4 that Z(D_{∞}) is a torsion Λ(D_{∞})-module. The following proposition gives the other half of the descent from K_{∞} to D_{∞}.

Proposition 2.4.15. The natural maps of Λ(D_{∞})-modules

S_{rel}(K_{∞}, T_{𝔭})/IS_{rel}(K_{∞}, T_{𝔭}) ⟶ S_{rel}(D_{∞}, T_{𝔭})
Z(K_{∞})/IZ(K_{∞}) ⟶ Z(D_{∞})

are injective, and their cokernels have characteristic ideal 𝔇. The same holds with Z replaced by Z_{𝔞} for any ideal 𝔞 prime to p𝔣.

Proof. Let L ⊂ K_{∞} be finite over K. As always, let S the set of places of K consisting of the archimedean place and the primes dividing p𝔣. Let K_{S}/K be the maximal extension of K unramified outside S. From the Poitou-Tate nine-term exact sequence we extract the exact sequence

0 ⟶ X_{str}^{∗}(L) ⟶ H^{2}(K_{S}/L, T_{𝔭}) ⟶ ⨁_{v∈S} H^{0}(L_{v}, W_{𝔭^{∗}})^{∨},

(for example, by taking B_{v} = 0 in [19] Proposition 4.1). Passing to the limit as L varies and taking I-torsion gives

0 ⟶ X_{str}^{∗}(K_{∞})[I] ⟶ H^{2}(K_{S}/K, T_{𝔭} ⊗ Λ(K_{∞}))[I] ⟶ ⨁_{v|p𝔣} H^{0}(K_{∞,v}, W_{𝔭^{∗}})^{∨}[I],

where we have used Shapiro’s lemma to identify

H^{2}(K_{S}/K, T_{𝔭} ⊗ Λ(K_{∞})) ≅ lim_{←} H^{2}(K_{S}/L, T_{𝔭}).

The final term in the exact sequence is the Pontryagin dual of

⨁_{v|p𝔣} H^{0}(K_{∞,v}, W_{𝔭^{∗}})/(γ − 1) ⨁_{v|p𝔣} H^{0}(K_{∞,v}, W_{𝔭^{∗}})

≅ ⨁_{v|p𝔣} H^{1}(K_{∞,v}/D_{∞,v}, H^{0}(K_{∞,v}, W_{𝔭^{∗}})),

which is the kernel of restriction

⨁_{v|p𝔣} H^{1}(D_{∞,v}, W_{𝔭^{∗}}) ⟶ ⨁_{v|p𝔣} H^{1}(K_{∞,v}, W_{𝔭^{∗}}).

Lemmas 2.4.10 and 2.4.11 (for v|p and v|𝔣, respectively) show that this kernel is finite. By Corollary 2.4.14 we conclude that H^{2}(K_{S}/K, T_{𝔭} ⊗Λ(K_{∞}))[I] is a torsion Λ(D_{∞})-module with characteristic ideal equal to 𝔇.

From the Gal(K_{S}/K)-cohomology of

0 ⟶ T_{𝔭} ⊗ Λ(K_{∞}) ⟶^{γ−1} T_{𝔭} ⊗ Λ(K_{∞}) ⟶ T_{𝔭} ⊗ Λ(D_{∞}) ⟶ 0

we deduce that the map

H^{1}(K_{S}/K, T_{𝔭} ⊗ Λ(K_{∞})) ⊗_{Λ(K_{∞})} Λ(D_{∞}) ⟶ H^{1}(K_{S}/K, T_{𝔭} ⊗ Λ(D_{∞}))

is injective with torsion cokernel of characteristic ideal 𝔇. Again using Shapiro’s lemma, together with Lemma 1.1.5, we see that the map

S_{rel}(K_{∞}, T_{𝔭})/IS_{rel}(K_{∞}, T_{𝔭}) ⟶ S_{rel}(D_{∞}, T_{𝔭})

is injective with cokernel of characteristic ideal 𝔇. The map

C(K_{∞})/IC(K_{∞}) ⟶ C(D_{∞})

is visibly surjective, since this merely asserts that the twisted elliptic units are universal norms in the cyclotomic direction. The snake lemma now proves the claim. □

Proposition 2.4.16. We have the equality of characteristic ideals

char(X_{str}^{∗}(D_{∞})) = char(Z(D_{∞})).

Proof. Let 𝔞 be an ideal of K prime to p𝔣. Using the fact that I is principal, the snake lemma gives the exactness of

S_{rel}(K_{∞}, T_{𝔭})[I] ⟶ Z_{𝔞}(K_{∞})[I] ⟶ C_{𝔞}(K_{∞})/IC_{𝔞}(K_{∞}).

The leftmost term is trivial by Lemma 1.1.9, and the term on the right is isomorphic to Λ(D_{∞}), since C_{𝔞}(K_{∞}) is free of rank one over Λ(K_{∞}). Therefore Z_{𝔞}(K_{∞})[I] is a torsion-free Λ(D_{∞})-module. On the other hand, the quotient Z_{𝔞}(K_{∞})/IZ_{𝔞}(K_{∞}) is a torsion Λ(D_{∞})-module (by Propositions 2.4.4 and 2.4.15), and so [21] Lemma 6.2 (i) tells us that Z(K_{∞})[I] is a torsion Λ(D_{∞})-module. We conclude that Z_{𝔞}(K_{∞})[I] = 0. Now by [21] Lemma 6.2 (ii),

char_{Λ(K_{∞})}(Z_{𝔞}(K_{∞})) · Λ(D_{∞}) = char_{Λ(D_{∞})}(Z_{𝔞}(K_{∞})/IZ_{𝔞}(K_{∞})).

Applying Proposition 2.4.15 gives

char_{Λ(K_{∞})}(Z_{𝔞}(K_{∞})) · 𝔇 = char_{Λ(D_{∞})}(Z_{𝔞}(D_{∞})).

Now let 𝔞 vary and apply Theorem 2.4.7 and Proposition 2.4.14 to get

char_{Λ(D_{∞})}(X_{str}^{∗}(D_{∞})) = char_{Λ(K_{∞})}(X_{str}^{∗}(K_{∞})) · 𝔇
= char_{Λ(K_{∞})}(Z(K_{∞})) · 𝔇

proving the claim. □

Theorem 2.4.17.

(1) If W = 1, then

(a) S(D_{∞}, T_{𝔭}) = 0,
(b) X^{∗}(D_{∞}) is a torsion Λ(D_{∞})-module,

(c) the ideal of Λ_{R}(D_{∞}) generated by char(X^{∗}(D_{∞})) is equal to the ideal generated by the p-adic L-function μ_{𝔭^{∗}}(D_{∞}, ψ_{𝔭^{∗}}). If p does not divide [K(𝔣) : K] the same holds with R replaced by R_{0}.

(2) If W = −1, then

(a) S(D_{∞}, T_{𝔭}) is a torsion-free Λ(D_{∞})-module of rank one,
(b) X^{∗}(D_{∞}) has rank one,
(c) char(X_{str}^{∗}(D_{∞})) = char(S(D_{∞}, T_{𝔭})/C(D_{∞})).

Proof. The first two claims of (1) and (2) all follow from Lemma 1.1.9, Theorem 1.2.2, and Proposition 2.4.4. When W = 1 the determination of the characteristic ideal follows from Propositions 2.3.4 and 2.4.16, using the exact sequence

0 ⟶ Z(D_{∞}) ⟶ H^{1}(D_{∞,𝔭^{∗}}, T_{𝔭})/loc_{𝔭^{∗}}C(D_{∞}) ⟶ X^{∗}(D_{∞}) ⟶ X_{str}^{∗}(D_{∞}) ⟶ 0.

When W = −1 the claim follows from the final statement of Proposition 2.4.4 and from Proposition 2.4.16. □

Remark 2.4.18. The case W = 1 can be deduced more directly from the first equality of Theorem 2.4.7 and the second part of Corollary 2.4.14 (which requires Proposition 2.4.9), by using Lemma 2.3.3, the local analogue of Proposition 2.4.15. This avoids the application of the Euler system machinery directly over D_{∞} (Proposition 2.4.4) needed to prove Proposition 2.4.15, or, more precisely, to prove the nontriviality of 𝔇. When W = −1, the ideals appearing in the first equality of Theorem 2.4.7 have trivial image in Λ(D_{∞}), so one seems to have no recourse but to prove some form of Proposition 2.4.15.

3. The p-adic height pairing

Throughout this section we assume W = −1 and we set

Δ = Gal(D_{∞}/K) Γ = Gal(C_{∞}/K).

We will frequently identify Γ ≅ Gal(K_{∞}/D_{∞}). For any nonnegative integer n, let Δ_{n} = Δ/Δ^{p^{n}} and similarly for Γ. Let I be the kernel of the natural projection Λ(C_{∞}) ⟶ Λ(K) and set J = I/I^{2}. Many authors use some choice of “logarithm” λ : Γ ⟶ Z_{p} to define the p-adic height pairing. Following the fashion of the day, we instead take

λ : Γ ⟶ J

to be the isomorphism γ ↦ γ − 1, and so obtain a J -valued height pairing.

3.1. The linear term. Choose a generator γ ∈ Γ and fix some integral ideal 𝔞 ⊂ O_{K} prime to 𝔣p. For every K ⊂ L ⊂ K_{∞} we set

c_{𝔞}(L) = lim_{←} c_{𝔞}(L′),

where c_{𝔞} is the Euler system of Proposition 2.3.1 and the limit is taken over all subfields L′ ⊂ L finite over K. We may identify

Λ(K_{∞})_{R_{0}} ≅ Λ(D_{∞})_{R_{0}}[[Γ]]

and expand μ_{𝔭^{∗}}(K_{∞}, ψ_{𝔭^{∗}}, 𝔞) as a power series in γ − 1,

μ_{𝔭^{∗}}(K_{∞}, ψ_{𝔭^{∗}}, 𝔞) = L_{𝔞,0} + L_{𝔞,1}(γ − 1) + L_{𝔞,2}(γ − 1)^{2} + · · · .

Similarly we may expand

μ_{𝔭^{∗}}(K_{∞}, ψ_{𝔭^{∗}}) = L_{0} + L_{1}(γ − 1) + L_{2}(γ − 1)^{2} + · · · .

By Corollary 2.1.5 and the assumption that W = −1, we have

L_{0} = μ_{𝔭^{∗}}(D_{∞}, ψ_{𝔭^{∗}}) = 0.

It follows that also L_{𝔞,0} = 0. By Proposition 2.3.4 the image of c_{𝔞}(D_{∞}) in H^{1}(D_{∞,𝔭^{∗}}, T_{𝔭}) is trivial, and so by Proposition 1.1.7, c_{𝔞}(D_{∞}) ∈ S(D_{∞}, T_{𝔭}).

Lemma 3.1.1. Set F_{n} = D_{n}C_{∞}. For every n there is a unique element

β_{n} ∈ H^{1}(F_{n,𝔭^{∗}}, T_{𝔭})_{R}

such that

(γ − 1)β_{n} = loc_{𝔭^{∗}}(c_{𝔞}(F_{n})).

Let α_{n} be the image of β_{n} in H^{1}(D_{n,𝔭^{∗}}, T_{𝔭})_{R}. The elements α_{n} are norm-compatible and they define an element α_{∞} ∈ H^{1}(D_{∞,𝔭^{∗}}, T_{𝔭})_{R}. The Coleman map of Proposition 2.3.4 identifies

H^{1}(D_{∞,𝔭^{∗}}, T_{𝔭})_{R} ≅ Λ(D_{∞})_{R}

and takes α_{∞} to L_{𝔞,1}.

Proof. This is immediate from Proposition 2.3.4, the fact that L_{𝔞,0} = 0, and the definition of L_{𝔞,1}. □

Tate local duality defines a pairing

⟨ , ⟩_{n} : H^{1}(D_{n,𝔭^{∗}}, T_{𝔭}) × H^{1}(D_{n,𝔭^{∗}}, T_{𝔭^{∗}}) ⟶ Z_{p}

whose kernel on either side is the Z_{p}-torsion submodule, and the induced pairing on the quotients by the torsion submodules is perfect.

The height pairing of the following theorem has been studied by many authors, including Mazur-Tate, Nekovář, Perrin-Riou, and Schneider. The fourth property of the pairing, the height formula, is due to Rubin, and plays a crucial role in what follows.

Theorem 3.1.2. For every nonnegative integer n there is a canonical (up to sign) p-adic height pairing

h_{n} : Sel(D_{n}, T_{𝔭}) × Sel(D_{n}, T_{𝔭^{∗}}) ⟶ Q_{p} ⊗ J

satisfying the following properties

(1) there is a positive integer k, independent of n, such that h_{n} takes values in p^{−k}Z_{p} ⊗ J

(2) if a ∈ Sel(D_{n}, T_{𝔭}), b ∈ Sel(D_{n}, T_{𝔭^{∗}}), and σ ∈ Δ_{n}, then

h_{n}(a^{σ}, b^{σ}) = h_{n}(a, b)

(3) if a_{n} ∈ Sel(D_{n}, T_{𝔭}), b_{n+1} ∈ Sel(D_{n+1}, T_{𝔭^{∗}}), and res and cor are the restriction and corestriction maps relative to D_{n+1}/D_{n}, then

h_{n+1}(res(a_{n}), b_{n+1}) = h_{n}(a_{n}, cor(b_{n+1}))

(4) (height formula) for every b ∈ Sel(D_{n}, T_{𝔭^{∗}}), we have (up to sign)

h_{n}(c_{𝔞}(D_{n}), b) = ⟨α_{n}, loc_{𝔭^{∗}}(b)⟩_{n} ⊗ (γ − 1).

Proof. This will be proved in the next section. □

Define the Λ(D_{∞})-adic Tate pairing

⟨ , ⟩_{∞} : H^{1}(D_{∞,𝔭^{∗}}, T_{𝔭})_{R} ⊗_{Λ(D_{∞})_{R}} H^{1}(D_{∞,𝔭^{∗}}, T_{𝔭^{∗}})_{R}^{ι} ≅ Λ(D_{∞})_{R}

by

⟨a_{∞}, b_{∞}⟩_{∞} = lim_{←} ∑_{σ∈Δ_{n}}⟨a_{n}^{σ}, b_{n}⟩_{n} · σ^{−1}

and define the Λ(D_{∞})-adic height pairing

h_{∞} : S(D_{∞}, T_{𝔭})_{R} ⊗_{Λ(D_{∞})_{R}} S(D_{∞}, T_{𝔭^{∗}})_{R}^{ι} ⟶ Λ(D_{∞})_{R} ⊗_{Z_{p}} J

similarly. The element α_{∞} ∈ H^{1}(D_{∞,𝔭^{∗}}, T_{𝔭}) satisfies

(3.1)  h_{∞}(c_{𝔞}(D_{∞}), b_{∞}) = ⟨α_{∞}, loc_{𝔭^{∗}}(b_{∞})⟩_{∞} ⊗ (γ − 1)

for every b_{∞} ∈ S(D_{∞}, T_{𝔭^{∗}}).

Definition 3.1.3. Define the anticyclotomic regulator, R, to be the characteristic ideal of the cokernel of h_{∞}.

Define R(C_{𝔞}) to be the characteristic ideal of the cokernel of

h_{∞}|_{C_{𝔞}} : C_{𝔞} ⊗_{Λ(D_{∞})_{R}} S(D_{∞}, T_{𝔭^{∗}})^{ι} ⟶ Λ(D_{∞})_{R} ⊗_{Z_{p}} J ,

and let η be the ideal

η = char(H^{1}(D_{∞,𝔭^{∗}}, T_{𝔭^{∗}})/loc_{𝔭^{∗}}(S(D_{∞}, T_{𝔭^{∗}}))).

From Proposition 2.4.4 and the results of Section 1.2 we have that S_{str}(D_{∞}, T_{𝔭^{∗}}) is trivial. The exactness of (1.5) then shows that η ≠ 0.

Proposition 3.1.4. There is an equality of ideals in Λ(D_{∞})_{R}

R · char(S(D_{∞}, T_{𝔭})/C_{𝔞}) = R(C_{𝔞})
= (L_{𝔞,1}) · η^{ι}.

Proof. The first equality is clear. The height formula (3.1) implies that the image of h_{∞}|_{C_{𝔞}} is equal to

⟨α_{∞}, loc_{𝔭^{∗}}(S(D_{∞}, T_{𝔭^{∗}})_{R})^{ι}⟩_{∞} ⊗ J ⊂ Λ(D_{∞})_{R} ⊗_{Z_{p}} J

The second equality now follows from Lemma 3.1.1 and the fact that the Λ(D_{∞})- adic Tate pairing is an isomorphism. □

Theorem 3.1.5. Let X denote the ideal of Λ(D_{∞})_{R} generated by the characteristic ideal of the Λ(D_{∞})-torsion submodule of X^{∗}(D_{∞}). Then we have the equality of ideals

R · X = (L_{1}).

Proof. If we replace 𝔭 by 𝔭^{∗} and take F = D_{∞} in the second pair of exact sequences of Proposition 1.1.7, we obtain the exact sequence

0 ⟶ H^{1}(D_{∞,𝔭^{∗}}, T_{𝔭^{∗}})/loc_{𝔭^{∗}}(S(D_{∞}, T_{𝔭^{∗}})) ⟶ X_{rel}(D_{∞}) ⟶ X(D_{∞}) ⟶ 0.

Taking Λ(D_{∞})-torsion and applying Lemma 1.2.1 and Theorem 1.2.2 we obtain

char(X_{str}^{∗}(D_{∞})) = η^{ι} · X .

Letting 𝔞 vary in Proposition 3.1.4, the claim follows from Theorem 2.4.17 (2), part (c). □

3.2. The height formula. In this section we sketch Perrin-Riou’s construction of the p-adic height pairing

h_{n} : Sel(D_{n}, T_{𝔭}) × Sel(D_{n}, T_{𝔭^{∗}}) ⟶ Q_{p} ⊗ J

of Theorem 3.1.2, as well Rubin’s proof of the height formula. Our exposition closely follows that of [23], to which we refer the reader for details.

For every 0 ≤ k ≤ ∞, let L_{k} = D_{n}C_{k}.

Lemma 3.2.1. Fix a place v of D_{n} and some extension of it to L_{∞}. The submodule of H_{f}^{1}(D_{n,v}, T_{𝔭}) defined by

H_{f}^{1}(D_{n,v}, T_{𝔭})^{univ} = ⋂_{k} cor H_{f}^{1}(L_{k,v}, T_{𝔭})

has finite index, and the index is bounded as v and n vary.

Proof. First assume that v divides 𝔭^{∗}, so that H_{f}^{1}(D_{n,v}, T_{𝔭}) is the torsion submodule of H^{1}(D_{n,v}, T_{𝔭}), which is in turn isomorphic to H^{0}(D_{n,v}, W_{𝔭}). It clearly suffices to show that this is bounded as n varies. But K_{v}(W_{𝔭}) is unramfied, and D_{∞,v} is a ramified Z_{p}-extension of K_{v}, so this is clear.

From now on suppose that v does not divide 𝔭^{∗}, so that

H_{f}^{1}(L_{k,v}, T_{𝔭}) = H^{1}(L_{k,v}, T_{𝔭}).

By local duality, it suffices to bound the kernel of restriction

H^{1}(D_{n,v}, W_{𝔭^{∗}}) ⟶ H^{1}(L_{∞,v}, W_{𝔭^{∗}}),

which is H^{1}(L_{∞,v}/D_{n,v}, M), where M = E(L_{∞,v})[𝔭^{∗∞}].

If v does not divide 𝔭 then L_{∞,v} is the unique unramified Z_{p}-extension of K_{v} (in particular it does not vary with n). If E does not have any 𝔭^{∗}-torsion defined over K_{v}, then K_{v}(E[𝔭^{∗}])/K_{v} is a nontrivial extension of degree dividing p − 1, so E has no 𝔭^{∗}-torsion defined over any p-extension of K_{v}, and so M = 0. Assume E[𝔭^{∗}] is defined over K_{v}, and that E has good reduction at v. Then K_{v}(E[𝔭^{∗∞}]) = L_{∞,v} and so M = W_{𝔭^{∗}} is p-divisible. If γ is a generator of Gal(L_{∞,v}/D_{n,v}) then

H^{1}(L_{∞,v}/D_{n,v}, M) ≅ M/(γ − 1)M.

Since γ acts as a scalar ≠ 1 on M, this group is trivial. If E has bad reduction at v then M is finite by the criterion of Néron-Ogg-Shafarevich, and the order of M does not vary with n (since L_{∞,v} does not vary).

Now assume that v divides 𝔭. The extension of K_{v} generated by E[𝔭^{∗∞}] is unramified, and since K_{v}^{unr} ∩ L_{∞,v} is a finite extension of K_{v}, M is finite. If γ generates Gal(L_{∞,v}/D_{n,v}) then using the exactness of

0 ⟶ M^{γ=1} ⟶ M ⟶^{γ−1} M ⟶ M/(γ − 1)M ⟶ 0

we see that the order of H^{1}(L_{∞,v}/D_{n,v}, M) is equal to the order of

M^{γ=1} = E(D_{n,v})[𝔭^{∗∞}].

Since D_{∞,v} is a ramified Z_{p}-extension of K_{v}, this order is bounded as n varies. □

Fix a ∈ Sel(D_{n}, T_{𝔭}) and b ∈ Sel(D_{n}, T_{𝔭^{∗}}). In view of the above lemma, it suffices to define the height pairing of a and b under the assumption that both are everywhere locally contained in H_{f}^{1}(D_{n,v}, T_{𝔭})^{univ}. Viewing b as an element of the larger group H^{1}(D_{n}, T_{𝔭^{∗}}), b defines an extension of Galois modules

0 ⟶ T_{𝔭^{∗}} ⟶ M_{b}^{∗} ⟶ Z_{p} ⟶ 0,

and taking Z_{p}(1)-duals we obtain an exact sequence

(3.2)  0 ⟶ Z_{p}(1) ⟶ M_{b} ⟶ T_{𝔭} ⟶ 0.

If L/D_{n} is any finite extension, we may consider the global and local Galois cohomology

\xymatrix{H^{1}(L, Z_{p}(1)) \ar[r] \ar[d] & H^{1}(L, M_{b}) \ar[r]^{π_{L}} \ar[d] & H^{1}(L, T_{𝔭}) \ar[r]^{δ_{L}} \ar[d] & H^{2}(L, Z_{p}(1)) \ar[d] \\ H^{1}(L_{w}, Z_{p}(1)) \ar[r] & H^{1}(L_{w}, M_{b}) \ar[r]^{π_{Lw}} & H^{1}(L_{w}, T_{𝔭}) \ar[r]^{δ_{Lw}} & H^{2}(L_{w}, Z_{p}(1)).}

Lemma 3.2.2. Let L be a finite Galois extension of D_{n} and suppose a′ ∈ H^{1}(L, T_{𝔭}) satisfies cor(a′) = a. Then a′ is in the image of π_{L}. For every place w of L, H_{f}^{1}(L_{w}, T_{𝔭}) is contained in the image of π_{L_{w}}

Proof. Let res be the restriction map from D_{n} to L. The connecting homomorphism δ_{L} is given (up to sign) by ∪res(b). If w is any place of L and v is the place of D_{n} below it,

loc_{w}(δ(a′)) = loc_{w}(a′ ∪ res(b)) = loc_{v}(a ∪ b) = 0,

since a and b are everywhere locally orthogonal under the Tate pairing. Thus δ_{L}(a′) is everywhere locally trivial, and by a fundamental fact of class field theory it is globally trivial. The proof of the second claim is similar. □

Class field theory gives a homomorphism

ρ : A_{D_{n}}^{×} ⟶ Gal(D_{n}C_{∞}/D_{n}) ≅ Γ ⟶^{λ} J

and we factor ρ = ∑_{v} ρ_{v}, the sum over all places of D_{n}. By local Kummer theory we may view ρ_{v} as a map

ρ_{v} : H^{1}(D_{n,v}, Z_{p}(1)) ⟶ J ,

which can also be described as follows: the homomorphism

Gal(D_{n}C_{∞}/D_{n}) ≅ Γ ⟶^{λ} J

defines a class λ ∈ H^{1}(D_{n}, J) (we always regard J as having trivial Galois action), and cup product with loc_{v}(λ) defines a map

H^{1}(D_{n,v}, Z_{p}(1)) ⟶ H^{2}(D_{n,v}, J (1)) ≅ J

which agrees (up to sign) with ρ_{v}.

Taking L = D_{n} and a′ = a in Lemma 3.2.2, we may choose some y^{glob} ∈ H^{1}(D_{n}, M_{b}) with π_{D_{n}}(y^{glob}) = a. Fix a place v of D_{n} and an extension of v to L_{∞}, and for every k choose y_{k,v} ∈ H_{f}^{1}(L_{k,v}, T_{𝔭}) which corestricts to loc_{v}(a). By Lemma 3.2.2 we may choose some y′_{k,v} ∈ H^{1}(L_{k,v}, M_{b}) such that π_{L_{k,v}}(y′_{k,v}) = y_{k,v}. Let cor(y′_{k,v}) be the image of y′_{k,v} in H^{1}(D_{n,v}, M_{b}). Then loc_{v}(y^{glob}) − cor(y′_{k,v}) comes from some w_{k,v} ∈ H^{1}(D_{n,v}, Z_{p}(1)), and we define

h_{n}(a, b) = lim_{k→∞} ∑_{v} ρ_{v}(w_{k,v})

This limit exists and is independent of all choices made.

We now sketch the proof of the height formula. Suppose that a = c_{𝔞}(D_{n}), and set a_{k} = c_{𝔞}(L_{k}),

a_{∞} = lim_{←} a_{k} ∈ lim_{←} Sel_{rel}(L_{k}, T_{𝔭}).

By Lemma 3.2.2 there is a sequence z_{k} ∈ H^{1}(L_{k}, M_{b}) with π_{L_{k}}(z_{k}) = a_{k}. Working semi-locally above 𝔮 = 𝔭 or 𝔭^{∗}, we have defined in the preceeding paragraph a sequence y′_{k,𝔮} ∈ H^{1}(L_{k,𝔮}, M_{b}) which lifts y_{k,𝔮}. The image of

t_{k,𝔮} = loc_{𝔮}(z_{k}) − y′_{k,𝔮} ∈ H^{1}(L_{k,𝔮}, M_{b})

in H^{1}(D_{n,𝔮}, M_{b}) comes from some s_{k,𝔮} ∈ H^{1}(D_{n,𝔮}, Z_{p}(1)).

Proposition 3.2.3. With notation as above

h_{n}(a, b) = lim_{k→∞}[ρ_{𝔭}(s_{k,𝔭}) + ρ_{𝔭^{∗}}(s_{k,𝔭^{∗}})].

Proof. This is Proposition 5.3 of [23]. □

Define H_{k,𝔮} by the exactness of

0 ⟶ H_{k,𝔮} ⟶ H^{1}(L_{k,𝔮}, T_{𝔭}) ⟶^{cor} H^{1}(D_{n,𝔮}, T_{𝔭}).

In [23] Section 4, one finds the definition of a derivative operator

Der_{k,𝔮} : H_{k,𝔮} ⟶ H^{1}(D_{n,𝔮}, T_{𝔭}/p^{k}T_{𝔭}) ⊗ J .

From the definition of t_{k,𝔮}, it is immediate that π_{L_{k,𝔮}}(t_{k,𝔮}) ∈ H_{k,𝔮}. We set

t′_{k,𝔮} = Der_{k,𝔮}(π_{L_{k,𝔮}}(t_{k,𝔮})) ∈ H^{1}(D_{n,𝔮}, T_{𝔭}/p^{k}T_{𝔭}) ⊗ J .

Proposition 4.3 of [23] then reads

λ ∪ s_{k,𝔮} = δ_{D_{n,𝔮}}(t′_{k,𝔮}) ∈ H^{2}(D_{n,𝔮}, (Z/p^{k}Z)(1)) ⊗ J

and so up to sign

(3.3)  ρ_{𝔮}(s_{k,𝔮}) ≡ inv_{𝔮}(δ_{D_{n,𝔮}}(t′_{k,𝔮})) (mod p^{k})
= inv_{𝔮}(t′_{k,𝔮} ∪ loc_{𝔮}(b))

where inv_{𝔮} is the semi-local invariant

H^{2}(D_{n,𝔮}, (Z/p^{k}Z)(1)) ⊗ J ⟶ J /p^{k}J .

From the definition of Sel(D_{n}, T_{𝔭^{∗}}), we see that loc_{𝔭}(b) is a torsion element. If p^{ℓ}loc_{𝔭}(b) = 0 then (3.3) implies that ρ_{𝔭}(s_{k,𝔭}) is divisible by p^{k−ℓ}. Letting k → ∞, we have ρ_{𝔭}(s_{k,𝔭}) → 0, leaving

(3.4)  h_{n}(a, b) = lim_{k→∞} ρ_{𝔭^{∗}}(s_{k,𝔭^{∗}}).

Lemma 3.2.4. Suppose d_{k} ∈ H_{k,𝔭^{∗}} is such that d_{k} ∪ z = loc_{𝔭^{∗}}(a_{k}) ∪ z for every z ∈ H_{f}^{1}(L_{k,𝔭^{∗}}, T_{𝔭^{∗}}). Then for any sequence x_{k} ∈ H_{f}^{1}(L_{k,𝔭^{∗}}, T_{𝔭^{∗}}) such that the image of x_{k} in H_{f}^{1}(D_{n,𝔭^{∗}}, T_{𝔭^{∗}}) is constant,

lim_{k→∞} inv_{𝔭^{∗}}(Der_{k,𝔭^{∗}}(d_{k}) ∪ x_{0}) = lim_{k→∞} ∑_{σ∈Γ_{k}}⟨loc_{𝔭^{∗}}(a_{k}), σ · x_{k}⟩_{L_{k},𝔭^{∗}} ⊗ λ(σ)

where λ is viewed as a character Γ_{k} ⟶ J /p^{k}J.

Proof. This is Lemma 5.1 of [23]. □

Recall that Lemma 3.1.1 provides, for some choice of generator γ ∈ Γ, a β ∈ H^{1}(L_{∞,𝔭^{∗}}, T_{𝔭}) such that

(γ − 1)β = loc_{𝔭^{∗}}(a_{∞}).

Let α be the image of β in H^{1}(D_{n,𝔭^{∗}}, T_{𝔭}). Write

β = lim_{←} β_{k} ∈ lim_{←} H^{1}(L_{k,𝔭^{∗}}, T_{𝔭})

so that β_{0} = α. Fix some sequence x_{k} ∈ H_{f}^{1}(L_{k,𝔭^{∗}}, T_{𝔭^{∗}}) lifting loc_{𝔭^{∗}}(b). Applying Lemma 3.2.4 with d_{k} = π_{L_{k,𝔭^{∗}}} (t_{k,𝔭^{∗}}) and comparing with (3.3) and (3.4) gives

h_{n}(a, b) = lim_{k→∞} ρ_{𝔭^{∗}}(s_{k,𝔭^{∗}})
= lim_{k→∞} ∑_{σ∈Γ_{k}}⟨loc_{𝔭^{∗}}(a_{k}), x_{k}^{σ}⟩_{L_{k},𝔭^{∗}} ⊗ λ(σ)
= lim_{k→∞} ∑_{σ∈Γ_{k}}⟨(γ − 1)β_{k}, σ · x_{k}⟩_{L_{k},𝔭^{∗}} ⊗ λ(σ)

= lim_{k→∞} ∑_{i=1}^{p^{k}−1}⟨(γ − 1)β_{k}, iγ^{i} · x_{k}⟩_{L_{k},𝔭^{∗}} ⊗ (γ − 1)
= lim_{k→∞}⟨β_{k}, Norm_{Γ_{k}}x_{k}⟩_{L_{k},𝔭^{∗}} ⊗ (γ − 1)
= ⟨α, loc_{𝔭^{∗}}(b)⟩_{D_{n,𝔭^{∗}}} ⊗ (γ − 1).

This completes the proof of the height formula.

Appendix A. Proof of Theorem B by Karl Rubin

In this appendix we prove Theorem B of the introduction. Essentially what we need to prove is that the anticyclotomic regulator R of Definition 3.1.3 is nonzero. The key tool is Theorem A.1 of Bertrand below, which says that on a CM elliptic curve the 𝔭-adic height of a point of infinite order is nonzero. This is much weaker than saying that the p-adic height is nondegenerate, but it suffices for our purposes.

We assume throughout this appendix that the sign in the functional equation of L(E/Q, s) is −1.

We need to consider a slightly more general version of p-adic heights than appears in the main text. If F is a finite extension of K in K_{∞}, then there is a p-adic height pairing

h_{F} : Sel(F, T_{𝔭}) ⊗ Sel(F, T_{𝔭^{∗}}) → Gal(K_{∞}/F) ⊗ Q_{p}.

We are interested in three specializations of this pairing. Namely,

h_{F,cycl} : Sel(F, T_{𝔭}) ⊗ Sel(F, T_{𝔭^{∗}}) → Gal(F C_{∞}/F) ⊗ Q_{p},
h_{F,anti} : Sel(F, T_{𝔭}) ⊗ Sel(F, T_{𝔭^{∗}}) → Gal(F D_{∞}/F) ⊗ Q_{p},
h_{F,𝔭} : Sel(F, T_{𝔭}) ⊗ Sel(F, T_{𝔭^{∗}}) → Gal(F L_{∞}/F) ⊗ Q_{p}

where L_{∞} is the unique Z_{p}-extension of K which is unramified outside of 𝔭, are defined by restricting the image of h_{F} to the appropriate group. If P ∈ E(F), we will write h_{F} (P) = h_{F} (a, b) where a is the image of P in Sel(F, T_{𝔭}) and b is the image of P in Sel(F, T_{𝔭^{∗}}).

The p-adic height pairing h_{n} of Theorem 3.1.2 is the composition of h_{D_{n},cycl} with the isomorphism Gal(D_{n}C_{∞}/D_{n}) ≅ Gal(C_{∞}/K) ≅ J.

Theorem A.1 (Bertrand [2]). Suppose F is a number field and P ∈ E(F) is a point of infinite order. Then the 𝔭-adic height h_{F,𝔭}(P) is nonzero.

Lemma A.2. Suppose a ∈ Sel(F, T_{𝔭}) and b ∈ Sel(F, T_{𝔭^{∗}}). If two of h_{F,cycl}(a, b), h_{F,anti}(a, b), h_{F,𝔭}(a, b) are zero, then so is the third.

Proof. Since Gal(K_{∞}/F) ≅ Z_{p}^{2}, the projections Gal(K_{∞}/F) → Gal(F C_{∞}/F), Gal(K_{∞}/F) → Gal(F D_{∞}/F), and Gal(K_{∞}/F) → Gal(F L_{∞}/F) are linearly dependent. It follows that each of the three heights is a linear combination of the other two. □

Definition A.3. For every n define a submodule of Sel(D_{n}, T_{𝔭}), the universal norms, by

Sel(D_{n}, T_{𝔭})^{univ} = ⋂_{m>n} cor_{D_{m}/D_{n}}Sel(D_{m}, T_{𝔭}).

Define Sel(D_{n}, T_{𝔭^{∗}})^{univ} similarly.

Lemma A.4. For every n we have

h_{D_{n},anti}(Sel(D_{n}, T_{𝔭})^{univ} ⊗ Sel(D_{n}, T_{𝔭^{∗}})^{univ}) = 0.

Proof. This is a basic property of the p-adic height (see for example Proposition 4.5.2 of [16]). □

Proposition A.5. The natural maps

X(D_{∞}) ⊗ Λ(D_{n}) → X(D_{n})

have kernel and cokernel which are finite and bounded independently of n.

Proof. This is the standard “Control Theorem”, see for example [11]. □

Recall that Δ_{n} = Gal(D_{n}/K).

Lemma A.6. For every n, Sel(D_{n}, T_{𝔭})^{univ} ⊗Q_{p} and Sel(D_{n}, T_{𝔭^{∗}})^{univ} ⊗Q_{p} are free of rank one over Q_{p}[Δ_{n}].

Proof. This is proved in exactly the same way as Theorem 4.2 of [13] (which proves a stronger statement about universal norms in Mordell-Weil groups, assuming that all relevant Tate-Shafarevich groups are finite), using Theorem 2.4.17. For completeness we give a proof here.

The exact sequence 0 → E[𝔭^{k}] → W_{𝔭} → W_{𝔭} → 0 and the fact that E(D_{n}) ∩ E[𝔭] = 0 show that Sel(D_{n}, E[𝔭^{k}]) = Sel(D_{n}, W_{𝔭})[𝔭^{k}] for every k. This and the Control Theorem (Proposition A.5) give us maps

Sel(D_{n}, T_{𝔭}) = \underleftarrow{lim}_{k} Sel(D_{n}, E[𝔭^{k}]) = \underleftarrow{lim}_{k} Hom(Sel(D_{n}, E[𝔭^{k}])^{∧}, Z/p^{k}Z)

= \underleftarrow{lim}_{k} Hom(X(D_{n}), Z/p^{k}Z) = Hom(X(D_{n}), Z_{p})

= Hom_{Λ(D_{n})}(X(D_{n}), Λ(D_{n})) → Hom_{Λ(D_{∞})}(X(D_{∞}), Λ(D_{n}))

with finite kernel and cokernel bounded independently of n. This gives the bottom row, and passing to the inverse limit over n gives the top row, of the commutative diagram with horizontal isomorphisms

S(D_{∞}, T_{𝔭}) ⊗ Q_{p} ⟶^{∼} Hom_{Λ(D_{∞})}(X(D_{∞}), Λ(D_{∞})) ⊗ Q_{p}
↓  ↓
Sel(D_{n}, T_{𝔭}) ⊗ Q_{p} ⟶^{∼} Hom_{Λ(D_{∞})}(X(D_{∞}), Λ(D_{n})) ⊗ Q_{p}

By Theorem 2.4.17, the upper modules are free of rank one over Λ(D_{∞}) ⊗ Q_{p}. The kernel of the right-hand vertical map is I_{n}Hom_{Λ(D_{∞})}(X(D_{∞}), Λ(D_{∞})) ⊗ Q_{p} where I_{n} is the kernel of the map Λ(D_{∞}) → Λ(D_{n}), and so the kernel of the left-hand vertical map is I_{n}S(D_{∞}, T_{𝔭}) ⊗ Q_{p}. Hence the image of the left-hand vertical map is free of rank one over Q_{p}[Δ_{n}]. But that image is precisely Sel(D_{n}, T_{𝔭})^{univ} ⊗ Q_{p}.

The proof for Sel(D_{n}, T_{𝔭^{∗}})^{univ} ⊗ Q_{p} is the same. □

Proposition A.7. If the anticyclotomic regulator R is zero, then the p-adic height pairing h_{n} is identically zero on Sel(D_{n}, T_{𝔭})^{univ} ⊗ Sel(D_{n}, T_{𝔭^{∗}})^{univ}.

Proof. Suppose n ≥ 0, a = (a_{n}) ∈ S(D_{∞}, T_{𝔭}) and b = (b_{n}) ∈ S(D_{∞}, T_{𝔭}^{∗}). Recall that Δ_{n} = Gal(D_{n}/K). Using property (3) of Theorem 3.1.2 and the definition of h_{∞}, we see that projecting h_{∞}(a, b) to Z_{p}[Δ_{n}] ⊗ J gives ∑_{σ∈Δ_{n}} h_{n}(a_{n}^{σ}, b_{n})σ^{−1}.

Now suppose v_{n} ∈ Sel(D_{n}, T_{𝔭})^{univ} and v_{n}^{∗} ∈ Sel(D_{n}, T_{𝔭^{∗}})^{univ}. Since v_{n} and v_{n}^{∗} are universal norms we can choose a = (a_{n}) ∈ S(D_{∞}, T_{𝔭}) and b = (b_{n}) ∈ S(D_{∞}, T_{𝔭}^{∗}) with a_{n} = v_{n} and b_{n} = v_{n}^{∗}. If R = 0, then h_{∞}(a, b) = 0, and projecting to Z_{p}[Δ_{n}] ⊗ J shows that h_{n}(v_{n}, v_{n}^{∗}) = 0. □

Proposition A.8. If n is sufficiently large then there are points P ∈ E(D_{n}) of infinite order such that the image of P in Sel(D_{n}, T_{𝔭}) lies in Sel(D_{n}, T_{𝔭})^{univ} and the image of P in Sel(D_{n}, T_{𝔭^{∗}}) lies in Sel(D_{n}, T_{𝔭^{∗}})^{univ}. I.e., there are D_{n}-rational points of infinite order which are universal norms in the Selmer group.

Proof. Let ψ denote the Hecke character of K attached to E, so that L(E/Q, s) = L(ψ, s). Fix an integer n.

Choose a character χ of Δ_{n} of order p^{n} (any two such characters are conjugate under G_{Q}, so the choice will not matter). The Hecke L-function L(ψχ, s) is the L-function of a modular form f_{χ} on Γ_{0}(Np^{2n}), where N is the conductor of E. Let A_{n} denote the simple factor over Q of the Jacobian J_{0}(Np^{2n}) corresponding to f_{χ}. Comparing L-functions we see that there is an isogeny of abelian varieties over K

A_{n} × Res_{D_{n−1}/K}E ∼ Res_{D_{n}/K}E

where Res stands for the restriction of scalars. Passing to Mordell-Weil groups we get

(A_{n}(K) ⊗ Q) × (E(D_{n−1}) ⊗ Q) ≅ E(D_{n}) ⊗ Q,

and therefore, if γ is a topological generator of Δ_{∞},

(A.1)  A_{n}(K) ⊗ Q ≅ (1 − γ^{p^{n−1}})E(D_{n}) ⊗ Q.

It follows from our assumption about the sign in the functional equation of L(E/Q, s) that L(ψρ, 1) = 0 for every character ρ of finite order of Δ_{∞}. By a theorem of Rohrlich [20], there are only finitely many characters ρ of Δ_{∞} such that the derivative L′(ψρ, 1) = 0. Suppose now that n is large enough so that

(a) L′(ψχ, 1) ≠ 0,

(b) char(X(D_{∞})_{tor}) is relatively prime to (γ^{p^{n}} − 1)/(γ^{p^{n−1}} − 1).

Since L′(ψχ, 1) ≠ 0, the theorem of Gross and Zagier [7] shows that

(A.2)  rank_{Z}A_{n}(Q) ≥ dim A_{n} = p^{n} − p^{n−1}.

On the other hand, using (A.1) and the Control Theorem (Proposition A.5) we get

(A.3) rank_{Z}A_{n}(Q) ≤ rank_{Z_{p}}(1 − γ^{p^{n−1}})X(D_{n})

= rank_{Z_{p}}(1 − γ^{p^{n−1}})X(D_{∞})/(1 − γ^{p^{n}})X(D_{∞})

Since X(D_{∞}) has Λ(D_{∞})-rank one (Theorem 2.4.17(2)), we conclude from condition (b) on n that

rank_{Z_{p}}(1 − γ^{p^{n−1}})X(D_{∞})/(1 − γ^{p^{n}})X(D_{∞}) = p^{n} − p^{n−1}.

It follows that we must have equality in (A.2) and (A.3), and

dim_{Q_{p}}(1 − γ^{p^{n−1}})Sel(D_{n}, T_{𝔭}) ⊗ Q_{p} = rank_{Z_{p}}(1 − γ^{p^{n−1}})X(D_{n}) = p^{n} − p^{n−1}.

Since (1 − γ^{p^{n−1}})Q_{p}[Δ_{n}] is a simple Q_{p}[Δ_{n}]-module it follows that

(1 − γ^{p^{n−1}})Sel(D_{n}, T_{𝔭}) ⊗ Q_{p} ≅ (1 − γ^{p^{n−1}})Q_{p}[Δ_{n}]

By Lemma A.6 we now see that

(1 − γ^{p^{n−1}})Sel(D_{n}, T_{𝔭})^{univ} ⊗ Q_{p} = (1 − γ^{p^{n−1}})Sel(D_{n}, T_{𝔭}) ⊗ Q_{p}.

In particular if P is any point of infinite order in (1 − γ^{p^{n−1}})E(D_{n}) (and we know that such points exist by (A.1)) then some multiple of the image of P in Sel(D_{n}, T_{𝔭}) lies in Sel(D_{n}, T_{𝔭})^{univ}. In exactly the same way some multiple of the image of P in Sel(D_{n}, T_{𝔭^{∗}}) lies in Sel(D_{n}, T_{𝔭^{∗}})^{univ}, and the proposition is proved. □

Proof of Theorem B. Using Proposition A.8, find an n and a point of infinite order P ∈ E(D_{n}) whose images in Sel(D_{n}, T_{𝔭}) and Sel(D_{n}, T_{𝔭^{∗}}) are universal norms.

By Bertrand’s Theorem A.1, we have h_{D_{n},𝔭}(P) ≠ 0. By Lemma A.4 we have h_{D_{n},anti}(P) = 0, and therefore by Lemma A.2 we have that h_{D_{n},cycl}(P) (and hence h_{n}(P)) is nonzero.

It now follows from Proposition A.7 that the anticyclotomic regulator R is nonzero, and so by Theorem 3.1.5 the leading term L_{1} is nonzero. This is Theorem B. □

Remark A.9. In the notation of Proposition A.8, the abelian variety A_{n} is isogenous to its twist by the quadratic character of K/Q, and so there are isomorphisms

A_{n}(K) ⊗ Q ≅ (Res_{K/Q}A_{n})(Q) ⊗ Q ≅ (A_{n} × A_{n})(Q) ⊗ Q.

Thus

p^{n} − p^{n−1} = \frac{1}{2} rank_{Z}A_{n}(K)
= rank_{O_{K}} E(D_{n}) − rank_{O_{K}} E(D_{n−1})

for n ≫ 0. This implies that there is a constant c such that

rank_{O_{K}} E(D_{n}) = p^{n} + c

for n ≫ 0. The same asymptotic formula holds for the corank of the 𝔭-primary Selmer group (by Theorem 2.4.17 and Proposition A.5), and so the O_{𝔭}-corank of of � (E/D_{n})_{𝔭^{∞}} is bounded as n varies.

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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, Boston College, Chestnut Hill, MA 02467

   E-mail address: howardbe@bc.edu

Department of Mathematics, Stanford University, Stanford, CA 94305 Current address: Department of Mathematics, UC Irvine, Irvine, CA 92697

E-mail address: krubin@math.uci.edu