Non-vanishing theorems for central $L$-values of some elliptic curves with complex multiplication

The paper uses Iwasawa theory at the prime $p=2$ to prove non-vanishing theorems for the value at $s=1$ of the complex $L$-series of the family of elliptic curves, first studied systematically by B. Gross, with complex multiplication by the full ring of integers of the field $K = \mathbb{Q} (\sqrt{-q})$, where $q$ is any prime $\equiv 7 \mod 8$. Some of these non-vanishing theorems were proven earlier by D. Rohrlich using complex analytic methods, but others are completely new. It is essential for the proofs to study the Iwasawa theory of the higher dimensional abelian variety with complex multiplication which is obtained by taking the restriction of scalars to $K$ of the Gross elliptic curve.


Introduction
Let K = Q( √ −q) be an imaginary quadratic field, where q is any prime number with q ≡ 7 mod 8. Fix an embedding of K into C. Let O K be the ring of integers of K, and write h for the class number of K. Note that, since q ≡ 7 mod 8, the prime 2 splits in K, say 2O K = pp * , a fact which will underly all of our subsequent arguments with Iwasawa theory. We fix one of these primes p, and we assume from now on that we have chosen the sign of √ −q so that ord p ((1 − √ −q)/2) > 0. Let H denote the Hilbert class field of K. Gross ([18], Theorem 12.2.1) has proven that there exists a unique elliptic curve A defined over Q(j(O K )), with complex multiplication by O K , minimal discriminant (−q 3 ), and which is a Q-curve in the sense that it is isogenous over H to all of its conjugates. An explicit equation for A over H is given by where m 3 = j(O K ), and r 2 = ((12) 3 − j(O K ))/q with r > 0 (see [19]). Let L(A/H, s) be the complex L-series of A/H, and write L(A/H, η, s) for the twist of this L-series by any finite order abelian character η of H. For 1 ≤ n ≤ ∞, let A p n be the Galois module of p n -division points on A, and define F ∞ = H(A p ∞ ). Let G denote the Galois group of F ∞ over H. The action of G on A p ∞ defines an isomorphism ρ p : G ≃ O × p = Z × 2 , where O p denotes the ring of integers of the completion of O K at p. Theorem 1.1. Assume that q is any prime such that q ≡ 7 mod 16. Then, for all characters ν of finite order of G = Gal(F ∞ /H), we have L(A/H, ν, 1) = 0.
We remark that, in the special case when ν is the trivial character, D. Rohrlich [22] has proven that L(A/H, 1) = 0 for all primes q ≡ 7 mod 8, by a completely different method using complex analytic arguments. However, it does not seem that his method can easily be extended to proving the nonvanishing of the L(A/H, ν, 1) for all characters ν of finite order of G when q ≡ 7 mod 16, and we do not know if this stronger assertion is even true for the primes q ≡ 15 mod 16 (but see [23], where it is shown, in particular, that L(A/H, ν, 1) = 0 for all but a finite of characters ν of finite order of G for all primes q ≡ 7 mod 8).
h-dimensional abelian abelian variety B/K, which is the restriction of scalars from H to K of the elliptic curve A/H, rather than for the elliptic curve A/H itself. As far as we know, this has not been exploited before in the literature, and we suspect that this idea can be used in other situations. We establish all of our analytic results for the Iwasawa theory by employing the Euler system of elliptic units as defined in the Appendix of [7], showing that this Euler system works beautifully even for the prime p = 2.
The present paper grew out of [4], which discussed only the case q = 7. We would like to thank the organizers of the conference "Iwasawa 2017" held at Tokyo University in July 2017 for providing us with an excellent opportunity to initially discuss the ideas developed here. The first author would also like to thank Tsinghua University and the Morningside Center of the Chinese Academy of Sciences for generous hospitality while much of the subsequent detailed work was being developed. Finally, we thank Zhibin Liang for making the numerical computations disussed in §8 of this paper.

Preliminaries
It is essential for our subsequent arguments that we work with the the abelian variety which is obtained from A by restriction of scalars from H to K (see [18], §15). We define Then T is a CM field of degree h over K. Let ψ A/H : A × H → K × , φ : A × K → T × be the Serre-Tate characters (see [24], Theorem 10) attached to A/H and B/K, respectively, where A × H (resp. A × K ) denotes the idele group of H (resp. of K). Then we have where N H/K is the norm map from the idele group of H to the idele group of K. The conductor of φ is equal to q = √ −qO K . For a more detailed discussion of the following facts, see [3], §1, and [18], §13.
The endomorphism ring B is generated over O K by the values φ(c) for c running over all integral ideals of K prime to q, and has discriminant h h O K as an O K -module. Thus B is an order in the field T , which is not necessarily maximal. However, it is always maximal when localized away from h. Also, the field T contains only 2 roots of unity, and we have H ∩ T = K. We fix an embedding of T in C (2.2) i : T → C.
which extends our given embedding of K in C. When there is no danger of confusion, we will omit the embedding i from the notation.
Lemma 2.1. The prime 2 is unramified in the field T , and there is a prime P of T above p with residue field of order 2.
Proof. Since the class number h of K is odd, it follows that from the above remarks that 2 will be unramified in the field T . Furthermore, the finitely generated abelian group A(H) = B(K) is a module for the algebra B, and the torsion subgroup of this abelian group is O K /2O K (see [18], §13). This action of B stabilizes the torsion, and thus gives an O K -algebra surjection The kernel of this homomorphism is the product of two conjugate primes P, P * in T , and P has the desired property.
We shall use the following notation throughout the rest of this paper. Gross [19] has proved the existence of a global minimal Weierstrass equation for A over H, and we fix one such minimal equation (2.3) y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 , whose coefficients are algebraic integers in H. For each integral ideal c of K prime to q, we write A c for the equation obtained by applying the Artin symbol of c to (2.3). Throughout the rest of this paper, w will denote any fixed prime of H above p. We write H w for the completion of H at w and O H,w for its ring of integers. If c is any integral ideal of K prime to q, we obtain a global minimal Weierstrass equation for the conjugate curve A c /H by applying the Artin symbol of c to the coefficients of (2.3). We write (x c , y c ) for a generic point on A c . Let A c w be the formal group of A c at w. It is a formal group defined over O H,w with local parameter t c,w = −x c /y c . Lemma 2.2. For each integral ideal c of K prime to q, A c w is a relative Lubin-Tate formal group in the sense of [14] for the unramified extension H w /K p .
Proof. The canonical isogeny η A c (p) : A c → A cp induces a map of formal groups defined over O H,w (2.4) η A c ,p,w : A c w → A cp w which can be realized by a formal power series η A c ,p,w (t c,w ) lying in O H,w [[t c,w ]]. Moreover, since φ is the Serre-Tate character of B, it is not difficult to see that (2.5) η A c ,p,w (t c,w ) ≡ t 2 c,w mod w, η A c ,p,w (t c,w ) ≡ m c t c,w mod t 2 c,w O H,w [[t c,w ]], where N Hw /Kp (m c ) has p-order equal to [H w : K p ]. These are precisely the conditions required to define a Lubin-Tate formal group relative to H w /K p . Corollary 2.3. The prime w is totally ramified in the extension H w (A p ∞ )/H w , and the Galois group of this extension is isomorphic to O × p .
For α ∈ B, let B α to be the kernel of the endomorphism α on B(K), and, for each integer n ≥ 1, define B P n = ∩ α∈P n B α .
Theorem 2.4. The abelian variety B has good reduction everywhere over the field F = K(B P 2 ), and the elliptic curve A has good reduction everywhere over the field F = H(A p 2 ).
Proof. We give the proof for B, and the proof for A is entirely similar. The conductor of φ is prime to 2, and thus the abelian variety B has good reduction at the primes of K lying above 2. Let φ F be the Serre-Tate homomorphism attached to B over the field F , so that φ F = φ • N F/K , where N F/K denotes the norm map from the idele group of F to the idele group of K. Let v be any place of F which does not lie above 2, and let U v be the group of units of the completion F v of F at v. As is shown in [24], B will have good reduction where G ab K denotes the Galois group of the maximal abelian extension of K, be Artin's global reciprocity map. Note that ξ K (J v ′ ) fixes F. We write τ P : G ab K → O × p for the character giving the action of G ab K on B P ∞ . Now B has potential good reduction everywhere, since the same assertion is true for the elliptic curve A and its conjugate curves over H. Hence, since v ′ does not lie above 2, by the criterion of Neron-Ogg-Shafarevich, we must have that τ P (ξ K (x)) is a root of unity for each x in the local units at v ′ . Moreover, by another basic property of the Serre-Tate homomorphism (see [24], Theorem 11), we have τ P (ξ K (x)) = φ(x) for every x in the local units at v ′ . Now assume that x lies in J v ′ , so that τ P (ξ K (x)) must belong to the subgroup 1 + P 2 of O × p . But this subgroup contains no non-trivial roots of unity, whence we must have that τ P (ξ K (x)) = 1, and so φ(x) = 1. Hence B has good reduction everywhere over F , as claimed.
In the present paper, we shall mainly be studying the Iwasawa theory of B over the tower of fields and B P ∞ = ∪ n≥1 B P n . Lurking in the background, we will also consider the Iwasawa theory of A over the tower of fields Moreover, as in the Introduction, let K ∞ be the unique Z 2 -extension of K which is unramified outside p, and write K n for the unique intermediate field with [K n : K] = 2 n . Put and let be the characters giving the action of these two Galois groups on B P ∞ and A p ∞ , respectively.
Lemma 2.5. We have strict inclusions K ∞ ⊂ F ∞ ⊂ F ∞ , and F ∞ = F K ∞ , F ∞ = HF ∞ . Moreover, the two characters (2.8) are both isomorphisms, the prime p of K is totally ramified in F ∞ , and all primes of H above p are totally ramified in F ∞ .
Proof. We first remark that the classical theory of complex multiplication shows that K ∞ ⊂ F ∞ . Let w be any prime of H above p. Then, by Lemma 2.2, w is totally ramified in F ∞ , and the character ρ p is an isomorphism. Thus we must have F ∞ = FK ∞ . Also, we must have F ∞ ⊂ F ∞ since B is isomorphic over H to the product of the h curves conjugate to A under the action of Gal(H/K). It then follows that p must be totally ramified in F ∞ . Furthermore, since B has good reduction everywhere over F , we have F = K, and thus ρ P must be an isomorphism. This completes the proof.
If b is any ideal of K prime to pq, we shall write τ b for the Artin symbol of b in Gal(F ∞ /K). Note that, since φ is the Serre-Tate character of B, the Artin symbol τ b will fix the field F n if and only if For each n ≥ 0, we fix a set C n of integral ideals of K, prime to pq such that Thus the elements of C n satisfy (2.9), and also give a complete set of representatives of the ideal class group of K since the restriction map from Gal(F n /F n ) to Gal(H/K) is an isomorphism. For each integral ideal b of K, which is prime to q, let σ b denote the Artin symbol of b in Gal(H/K). The restriction map defines an isomorphism from Gal(F ∞ /F ∞ ) to Gal(H/K), and we define δ to be the unique element of Gal(F ∞ /F ∞ ) whose restriction to H is the Artin symbol σ p of p. Finally, we fix a set {V n : n ≥ 0} of primitive p n+2 -division points on A, which are compatible in the following sense. If b is any integral ideal of K prime to q, it is well known [16] that the endomorphism φ(b) of B gives rise to a canonical isogeny η A (b) : A → A b with kernel precisely the group A b of b-division points on A; here A b denotes the conjugate curve obtained by applying σ b to the coefficients of any equation for A/H. Then the compatibility relation which we require between the V n is that 3. Iwasawa theory for the abelian variety B over the field The aim of this section is to use some very elementary arguments from Iwasawa theory to study descent theory on B over F ∞ = K(B P ∞ ). We define M (F ∞ ) to be the maximal abelian 2-extension of F ∞ = K(B P ∞ ), which is unramified outside the primes lying above p, and put We recall that we have chosen the sign of √ −q so that ord p ( √ −q − 1)/2) > 0.
We now closely follow the arguments of elementary Iwasawa theory given in [4], §2 to prove Theorem 3.1. Of course, M (F ∞ ) is Galois over K by maximality, and thus G = Gal(F ∞ /K) has the usual natural continuous action of Iwasawa theory on X(F ∞ ). We also remark that, since ρ P is an isomorphism, the Galois group G is of the form G = ∆ × Γ, where ∆ is cyclic of order 2 and Γ is isomorphic to Z 2 , and all of our arguments will be based on Nakayama's lemma for either of the natural ∆-actions or Γ-actions. Let R = Z 2 [∆] be the group ring of ∆ over Z 2 . If V is any R-module, we write as usual V ∆ for the largest quotient of V on which ∆ acts trivially. Similarly, if V is a compact Γ-module which is a Z 2 -module, (V ) Γ will be the largest quotient of V on which Γ acts trivially.
Lemma 3.2. The field K ∞ has no non-trivial abelian 2-extension, which is unramified outside the unique prime of K ∞ above p.
Proof. Denote by M (K ∞ ) the maximal abelian 2-extension over K ∞ which is unramified outside the unique prime above p, and let X(K ∞ ) be the Galois group Gal(M (K ∞ )/K ∞ ). Since M (K ∞ ) is Galois over K by maximality, the Galois group Γ = Gal(K ∞ /K) acts on it continuously by lifting inner automorphisms. We claim that (X(K ∞ )) Γ = 0, which will suffice to prove what we want by Nakayama's lemma. Denote by J the maximal abelian extension of K in M (K ∞ ), so that Gal(J/K ∞ ) = (X(K ∞ )) Γ . Now, since the class number of K is odd, class field theory shows immediately that K ∞ itself is the maximal abelian 2-extension of K which is unramified outside p. Hence J = K ∞ , and the proof is complete.
Lemma 3.3. Let r q denote the number of primes of K ∞ lying above the prime q = √ −qO K of K. Then r q = 2 k−2 , where k = ord p ( √ −q − 1).
Proof. It follows from the definition of k that √ −q ∈ 1 + p k but √ −q / ∈ 1 + p k+1 , where k ≥ 2. Noting that K n is the 2-part of the ray class field of K modulo p n+2 for all n ≥ 0, we then conclude easily from class field theory that q = √ −qO K splits completely in the extension K k−2 , and that each prime of K k−2 above q is inert in K ∞ . Thus there are precisely r q primes of K ∞ above q , and the proof is complete.
Lemma 3.4. We have (X(F ∞ )) ∆ is an F 2 -vector space of dimension at most r q − 1, where F 2 denotes the field with 2 elements.
Proof. By the definition of the ∆ action, we have Gal(I/F ∞ ) = (X(F ∞ )) ∆ , where I is the maximal abelian extension of K ∞ contained in M (F ∞ ). But the only primes of K ∞ which ramify in I are the unique prime above p, and the r q primes above q. Moreover, the ramification index of each of these primes above q in I is precisely 2, because this is the ramification index of q in F . Let D denote the subgroup of Gal(I/K ∞ ) generated by the inertial subgroups of these primes above q. Thus D is a vector space of dimension at most r q over F 2 . Now the fixed field of D is an abelian 2-extension of K ∞ unramified outside p. Thus, by Lemma 3.2, this fixed field must be equal to K ∞ . Hence D = Gal(I/K ∞ ), and the assertion of the lemma follows because [F ∞ : K ∞ ] = 2 Corollary 3.5. X(F ∞ ) is a finitely generated R-module, which is generated by at most 2 k−2 −1 elements over R. In particular, X(F ∞ )) is a finitely generated Z 2 -module.
Proof. As X(F ∞ ) is a compact R-module, the corollary follows immediately from Lemma 3.4 and the Nakayama lemma.
Proof. We have the exact sequence of finitely generated Z 2 -modules where the middle map is given by multiplication 1 − ǫ, where ǫ denotes the non-trivial element of ∆. Since (X(F ∞ )) ∆ is finite by Lemma 3.4, it follows that X(F ∞ ) ∆ is also finite. But X(F ∞ ) ∆ is also a Γ-module, whence by the theorem of Greenberg [17] asserting that X(F ∞ ) has no nonzero finite Γsubmodule, we conclude that (X(F ∞ )) ∆ = 0, and also that the torsion subgroup of X(F ∞ ) must be zero. This completes the proof.
We omit the proof (see [4], Lemma 2.8) of the following simple algebraic lemma, whose proof was pointed out to one of us by Romyar Sharifi. Lemma 3.7. Let Y be a free Z 2 -module of finite rank, which is also a ∆-module, and assume that (Y ) ∆ = (Z/2Z) r (r ≥ 0). Then Y is a free Z 2 -module of rank r.
Combining Corollary 3.5, and Lemmas 3.6 and 3.7, the proof of Theorem 3.1 is complete.
Remarkably, the following result is valid for all primes q with q ≡ 7 mod 8.
Theorem 3.8. For all primes q with q ≡ 7 mod 8, the field F n = K(B P n+2 ) has odd class number for all n ≥ 0, and F ∞ has no unramified abelian 2-extension.
Proof. We first show that F has odd class number. Let L(F ) be the 2-Hilbert class field of F , and put Y (F ) = Gal(L(F )/F ). By maximality, L(F ) is Galois over K, and so ∆ acts on Y (F ) in the usual fashion. Thus Y (F ) ∆ = Gal(J/F ), where J is the maximal abelian extension of K contained in L(F ). Now the only primes of K which are ramified in J are p and q. Let Φ be the inertial subgroup of q in Gal(J/K). Then the fixed field J Φ of Φ must be an abelian 2-extension of K which is unramified outside of p. But, since K has odd class number, the only abelian 2-extensions of K unramified outside of p are the fields K n (n ≥ 0), and so we must have J Φ = K m for some m. But then it follows that K m F ⊂ J, and so the extension K m F/F is unramified. But the unique prime above p is totally ramified in the extension K m F/F . Thus we must have K m ⊂ F . However, K m = F because q is ramified in F , and so we conclude that m = 0. Hence we have shown that K is the fixed field of Φ, and so Gal(J/K) = Φ. But q has ramification index 2 in the extension J/K, so that Φ has order 2, whence also Gal(J/K) has order 2. It follows that necessarily J = F , and so Y (F ) ∆ = 0. But then by Nakayama's lemma, we must have Y (F ) = 0, proving that F has odd class number. Now let L(F ∞ ) be the maximal unramified abelian 2-extension of F ∞ , and put Y (F ∞ ) = Gal(L(F ∞ )/F ∞ ). Recall that Γ = Gal(F ∞ /F ). Now the Z 2 -extension F ∞ /F is totally ramified at the unique prime of F above p, and no other prime of F is ramified in it. A classical argument in Iwasawa theory then proves that Y (F ∞ ) Γ ≃ Gal(L(F )/F ), where again L(F ) denotes the 2-Hilbert class field of F . But we have just shown that L(F ) = F . Hence Y (F ∞ ) Γ = 0, and so by the topological Nakayama's lemma, we must have Y (F ∞ ) = 0. But, if we write L(F n ) for the 2-Hilbert class field of F n for any n ≥ 0, the same classical argument in Iwasawa theory shows that Y (F ∞ ) Γn ≃ Gal(L(F n )/F n ), where Γ n denotes the unique closed subgroup of Γ of index 2 n . Hence L(F n ) = F n , and so F n has odd class number for all n ≥ 0. This completes the proof.
The above arguments make essential use of the fact that we are working with Galois groups. However, for the arithmetic applications, it is important that we translate all into assertions about Selmer groups, as is done in [4] in the special case q = 7. We make use of the standard notation for the Galois cohomology of Galois modules and abelian varieties. Recall that B is the ring of K-endomorphisms of the abelian variety B. We fix any non-zero element π of B such that the ideal factorization of π in the ring of integers of T is P r for some integer r ≥ 1. Now let L be any algebraic extension of K. As usual, we define, for each integer n ≥ 1, the Selmer group Sel π n (B/L) by the exact sequence where v runs over all finite places of L, and L v is the compositum of the completions at v of all finite extensions of K contained in L. Passing to the inductive limit over all n ≥ 1, and noting that B π ∞ = B P ∞ , we then define the Selmer group Sel P ∞ (B/L) to be the inductive limit of the Selmer groups Sel π n (B/L), so that we have here, for any B-module V, we write V (P) for the submodule of elements which are annihilated by some power of π. In an entirely similar manner, the modified Selmer group Sel ′ P ∞ (B/L) is defined by where now the product is taken over all primes v of L which do not lie above the prime p of K.
Theorem 3.9. We have Proof. Since B has good reduction everywhere over F , the G F -module B P ∞ is unramified outside the set of primes of F lying above p. Combining this with the fact that B P ∞ is fixed by Gal(F /F ∞ ), an entirely similar argument to that given in the proof of Theorem 12 of [6] shows that . Hence the assertion of the theorem will follow once we have shown that, for the unique place v of F ∞ above p, we have Since F ∞ /K is totally ramified at the unique prime above p, it follows that the residue field of v restricted to F n is always equal to k v = Z/2Z. Let B ′ denote the dual abelian variety of B over K, so that B ′ also has good reduction everywhere over F . We write B ′ v for the reduction of B ′ modulo v. Fix at first the integer n ≥ 1. Then Tate local duality at v shows that, for all integers m ≥ 1, where π * denotes the complex conjugate of π for the CM field T . Now v lies above p, and so π * m is an automorphism of the formal group of B ′ at v, whence , and so H 1 (F ∞,v , B)(P) will be dual to the projective limit of the B ′ v (k v )(P * ) with respect to the norm maps up the tower F ∞ /F . But the Galois group of F ∞ /F acts trivially on the finite group B ′ v (k v )(P * ), and so the projective limit 7 of these groups with respect to the norm map is clearly zero. Thus completes the proof of (3.1) and the theorem.
As before, let ǫ be the non-trivial element of ∆, so that ǫ acts on and this latter group is zero by Lemma 3.6, whence the assertion of the proposition follows.
Proposition 3.11. For all n ≥ 0, the restriction map yields an isomorphism Proof. By the definition of the Selmer group, the restriction maps gives rise to the following commutative diagram with exact rows: Now the middle vertical map is an isomorphism. Indeed H 2 (Γ n , B P ∞ ) = 0 because Γ n has 2-cohomological dimension equal to 1, and a well known argument (see the proof of Lemma 2.11 in [4]) shows that also H 1 (Γ n , B P ∞ ) = 0. Moreover, the right vertical map is injective because B has good reduction at all primes v, and the extension F ∞,w /F n,v is unramified when w does not lie above the prime p of K. The assertion of the lemma now follows.
Theorem 3.12. For all n ≥ 0, we have Rank Z (B(F n )) = Rank Z (B(K n )), and the Z 2 -corank of X(B/F n )(P) is equal to the Z 2 -corank of X(B/K n )(P), for all n ≥ 0.
Proof. Note first that it follows immediately from Propositions 3.10 and 3.11 that, for all n ≥ 0, we have . Now since ∆ is of order 2, one sees easily that the kernel and cokernel of the restriction map from Sel ′ P ∞ (B/K n ) to Sel ′ P ∞ (B/F n ) ∆ are annihilated by 2, and so it follows from (3.2) that, for all n ≥ 0, we have . Define the modified Shafarevich-Tate group via the exactness of the sequence where v runs over all the finite places of K n distinct from p. Note that we then have the exact sequence We have an entirely similar exact sequence for the field F n . Denote by g Kn , t Kn the Z 2 -corank of B(K n ) ⊗ B (T P /B P ), and of X ′ (B/K n )(P), respectively. Define g Fn , t Fn in an entirely analogous fashion for the field F n . It follows immediately from (3.3) that we have Note further that t Kn ≤ t Fn and g Kn ≤ g Fn , because the restriction maps have finite kernels. Therefore, we conclude from (3.5) that g Kn = g Fn , t Kn = t Fn . The first assertion of the theorem now follows easily. For the second assertion, note that we have the exact sequence where w denotes the unique prime of K n above p. But an entirely similar argument with Tate local duality to that used above in the proof of Theorem 3.9 shows that the group on the extreme right of this last exact sequence is finite. Hence t Kn is equal to the Z 2 -corank of X(B/K n )(P). Similarly, we find that t Fn is the Z 2 -corank of X(B/F n )(P), and the proof of the theorem is now complete.

Ellitpic units for the field
The aim of the present section is to define, for every n ≥ 0 a suitable group of elliptic units for the field F n = K(B P n+2 ), which we will denote by C(F n ), and to prove the existence of a suitable interpolating power series for them. We use a variant of the method first pioneered in [11].
We recall that we have fixed a global minimal generalized Weierstrass equation (2.3) for A/H. Now j(A) = j(O K ) and H = K(j(O K )), so that our fixed embedding of K in C induces an embedding of H into C. The Neron differential dx/(2y + a 1 x + a 3 ) then has a complex period lattice of the form where Ω ∞ (A) is uniquely determined up to sign. Let I denote the ring of integers of the completion of the maximal unramified extension of K p . We fix an embedding of H into the fraction field of I , which will determine a place of H above p which we will denote by w throughout this section.
We first determine the conductors of some of the abelian extensions of K which arise in the rest of the section. Proof. The classical theory of complex multiplication shows that the ray class field of K modulo f is contained in the field H(A f ). Conversely, suppose that α is an element of K with ord v (α − 1) ≥ ord v (f) for all places v of K dividing f, and write (α) K and (α) H for the respective principal ideals of K and H generated by α. Note that the abelian extension H(A f )/K is unramified outside the set of primes of K dividing f, because all primes of bad reduction for A/H must divide the ideal of H generated by f. Write m K for the Artin symbol of (α) K in Gal(H(A f )/K). We must show that m K fixes A f . To do this, let us consider the Artin symbol m H of (α) H for the abelian extension H(A f )/H. By the definition of the Grossencharacter, the Artin symbol m H acts on A f by multiplication by the endomorphism But by the functorality of the Artin symbol, we know that m K = m h H , and so it follows that m K also fixes A f , as required.
Recall that F n = H(A p n+2 ) = HF n . Lemma 4.2. For all n ≥ 0, the conductors of the abelian extensions F n /K and F n /K are both equal to f n = qp n+2 .
Proof. By the previous lemma, we know that H(A fn ) is equal to the ray class field of K modulo f n . But and so the conductors of F n /K and F n /K must both divide f n . But K n /K has conductor equal to p n+2 , since otherwise it would be contained in the field HK n−1 , and this is impossible because p has ramification index 2 n in K n . Since K n ⊂ F n , it follows that the conductors of F n /K and F n /K must both be divisible by p n+2 . Moreover, B has good reduction everywhere over the field F n , and thus its Grossencharacter over this field, which is φ • N Fn/K , must have trivial conductor. Hence the conductor q of φ must divide the conductor of F n /K. Similarly, A has good reduction everywhere over F n , whence again q must divided the conductor of F n /K. As p and q are relatively prime, this completes the proof.
While our aim is to define a group of elliptic units for each of the fields F n = K(B P n+2 ), we need first to discuss the appropriate group of elliptic units for the fields F n = H(A p n+2 ). Let us introduce the index set The congruence α ≡ 1 mod p 2 imposed on the elements of J is not strictly necessary, but we will use it to avoid some technical complications in later arguments. For each α ∈ J , we defined the rational function r α,A (P ) on A/H by where V runs over any set of representatives of the set of non-zero elements of the Galois module A α modulo ±1. Here P = (x, y) is a generic point on (2.3). As is shown by a very elementary argument in the Appendix of [7], there exists a unique non-zero c α (A) ∈ H such that the normalized function for all non-zero β in O K with (β, α) = 1; here the symbol ⊕ denotes the group law on the elliptic curve A.
Recall that we have a natural embedding of H in C, and that L = Ω ∞ (A)O K is the period lattice of the Neron differential on our generalized Weierstrass equation (2.3) for A/H. We then have the Weierstrass isomorphism W(z, L) : C/L → A(C) given by Following an idea already introduced in [10], we then define which is thus a rational function on A with coefficients in H. Plainly R α,A (P ) depends only on the orbit of Q under the action of Gal(H(A q )/H), but note that, in view of Lemma 4.1, this Galois group does not act transitively on the set of all primitive q-division points of A. It is this which guarantees the all important fact that R α,A (P ) = R α,A (⊖P ), where ⊖ denotes the subtraction on the elliptic curve. It is also important to define intrinsic analogues of the rational function R α,A (P ) for every conjugate curve of A/H under the action of the Galois group G = Gal(H/K). In what follows, a, b, · · · will denote integral ideals of K, which will always be assumed to satisfy (a, q) = 1. Given such an ideal a, for simplicity we write A a for the elliptic curve over H which is obtained by applying the Artin symbol of a in G to the coefficients of the equation (2.3) for A. Now the endomorphism φ(a) of the abelian variety B defines also a canonical H-isogeny with kernel precisely A a . Then, as is explained in §4 of [16], the pull back by η A (a) of the Neron differential ω a on A a must be of the form ξ(a)ω, where ω denotes the Neron differential on A and ξ(a) is a uniquely determined non-zero element of H. Further, always with our fixed embedding of H into C, the complex period lattice L a of ω a is then equal to ξ(a) for the Weierstass isomorphism, and let Q(a) be the q-division point on A a defined by We then define the rational function on A a /H by Thanks to Lemma 4.1, the fields H(A a q ) are all equal to H(A q ), and then (see §4 of [16]) the Artin symbol of a in Gal(H(A q )/K) maps Q to Q(a). It follows easily that, whenever a and b are integral ideals of K, which are prime to q and lie in the same ideal class, then necessarily R α,A a (P ) = R α,A b (P ).
We first define the group of elliptic units for the field F n = H(A p n+2 ) for any integer n ≥ 0. Recall that V n denotes a primitive p n+2 -division point on the curve A. We then define C(F n ) to be the subgroup of F × n which is generated by all conjugates of R α,A (V n ) under the action of the Gal(F n /K) and all α ∈ J . We shall show below that the elements of C(F n ) are indeed global units. We recall that, if c is any integral ideal of K prime to pq, τ c denotes the Artin symbol of c in Gal(F ∞ /K). Since τ c (A) = A c and τ c (V n ) = η A (c)(V n ), the action of Gal(F n /K) on R α,A (V n ) is given by the following lemma.
Lemma 4.3. For all n ≥ 0, and all integral ideals c of K prime to pq, we have We next discuss the behaviour of C(F n ) under the norm map N Fn/Fn−1 from F n to F n−1 . Note that the conjugate curve A p is perfectly well defined, but of course the Artin symbol of p cannot be defined in Gal(F ∞ /K). However, we recall that Gal(F ∞ /F ∞ ) is isomorphic to Gal(H/K) under restriction, and that we use δ to denote the unique element of Gal(F ∞ /F ∞ ) whose restriction is the Artin symbol of p in Gal(H/K). Now we have the H-isogeny whose kernel is precisely A p . Thus η A (p)(V n ) must be a primitive p n+1 -division point on the curve A p . On the other hand, it is shown in Theorem 4 of the Appendix of [7] that the rational functions R α,A (P ) behave nicely with respect to isogenies of degree prime to α. Since (p, α) = 1 because (α, 6) = 1, the following lemma then follows easily.
Lemma 4.4. We have Now for n ≥ 1, we have [F n : F n−1 ] = 2 and the conjugates of V n under the action of Gal(F n /F n−1 ) are precisely the V n ⊕ V with V ∈ A p . Hence we immediately obtain On the other hand, we have already fixed in section 2 (see (2.11)) the convention that we have chosen the primitive p n+1 -division point V n−1 so that δ(V n−1 ) = η A (p)(V n ). Hence the above corollary can be rewritten as giving N Fn/Fn−1 (R α,A (V n )) = δ(R α,A (V n−1 )) (n ≥ 1). Hence we immediately obtain the theorem.  Indeed, since every prime of F n which does not lie above p is unramified in F ∞ and its decomposition group is of finite index, it follows easily from the universal norm property of Theorem 4.6 (see Lemma 5 of [7]) that the only primes which can possibly divide R α,A (V n ) must divide p. On the other hand, the following classical lemma shows that R α,A (V n ) is a unit at each prime w of H above p since the power series in the lemma, in the special case when b = O K , will certainly converge at t w (V n ) to a unit at w.
Proof. This is a well known classical argument (see the proof of Lemma 23 of [10] or Lemma 8 of [7]) and we omit the details. Note that we also use the fact that c A b (λ) is a unit in O H,w because it is shown in the Appendix to [7] is the discriminant of our global minimal equation for A b , and we have (p, ∆(A b )α) = 1.
We now turn to the fields F n = K(B P n+2 ), where, of course, the only natural thing to do is to define the group of elliptic units C(F n ) of F n by (4.13) C(F n ) = N Fn/Fn (C(F n )), where N Fn/Fn denotes the norm map from F n to F n . Thus, writing (4.14) u α,n = N Fn/Fn (R α,A (V n )), these u α,n are global units in F n , and, since the restriction of δ to F n lies in Gal(F n /F n ), we conclude immediately from Theorem 4.6 that (4.15) N Fn/Fn−1 (u α,n ) = u α,n−1 (n ≥ 1).
We write (4.16) u α,∞ = (u α,n ) for this norm compatible system of elliptic units. However, unlike the situation for the field F ∞ , it seems that these units u α,n cannot be obtained for all n ≥ 0 by evaluating a single rational function on A at the point V n . All we can do in this direction is the following. Recall that C n denotes a set of integral ideals of K prime to pq whose Artin symbols in Gal(F n /K) give precisely Gal(F n /F n ), and define the rational function D α,n (P ) on A/H by Lemma 4.9. For all integers n and k with n ≥ 0 and 0 ≤ k ≤ n, we have D α,n (V k ) = u α,k .
Proof. This clear from Lemma 4.3 and the fact that Gal(F n /F n ) is isomorphic under restriction to Gal(F k /F k ) for 0 ≤ k ≤ n.  Proof. For all m 2 ≥ m 1 , it follows from Lemma 4.9 that the power series vanishes at the points t w (V k ) (k = 0, · · · , m 1 ) and all conjugates of these points over H w . Thus ] by the monic polynomial P m1 (t w ) whose roots are given by all conjugates of the t w (V k )(k = 0, · · · , m 1 ) over H w . Since it is easily seen that P m1 (t w ) tends to zero in the m-adic topology as m 1 → ∞, it follows by completeness that the limit power series d α,∞ (t w ) = lim m→∞ d α,m (t w ) exists, and satisfies d α,∞ (t w (V n )) = u α,n for all n ≥ 0. This completes the proof.

Canonical measures attached to elliptic units.
The aim of this section is to show how to relate norm compatible systems of elliptic units in the tower F ∞ /F to the the complex L-values L(φ k , k) for all integers k ≥ 1, again broadly following the method introduced in [11]. We recall that δ always denotes the unique element of Gal(F ∞ /F ∞ ) whose restriction to Gal(H/K) is the Artin symbol of p. Thus, for any place w of H above p, we can view δ as a generator of Gal(H w /K p ), where H w denotes the completion of H at w. If J(P ) is any rational function on A/H, we shall write J δ (P p ) for the rational function on A p /H obtained by applying δ to the coefficients of J(P ).
We first establish the following analogue of (4.12).
Lemma 5.1. For all n ≥ 0, we have Proof. For each c ∈ C n , we have the equality of isogenies whence it follows easily that Since p is prime to α, the good behaviour of the R-functions with respect to isogeny (see Theorem 4 of [7]) shows that The assertion of the lemma then follows on noting that the isogeny η A (c) maps A p isomorphically to A c p because of our assumption that (c, p) = 1 for c ∈ C n . This completes the proof.
As in the previous section, we write d α,n (t w ) for the formal power series expansion of the rational function D α,n (P ) on A/H in terms of the parameter t w of the formal group of A at w, say d α,n (t w ) = k≥0 e α,n (k)t k w . It is then clear that the rational function D δ α,n (P p ) on A p has the expansion in terms of the parameter t p,w of the formal group of A p at w given by d δ α,n (t p,w ) = k≥0 δ(e α,n (k))t k p,w . Moreover, since d α,∞ (t w ) = lim n→∞ d α,n (t w ), we see that lim n→∞ d δ α,n (t p,w ) also exists, and we denote this limit by d δ α,∞ (t p,w ). All of these formal power series are, of course, units. Then, for 0 ≤ n ≤ ∞, we can define the power series where, as before, η A,p,w : A w → A p w is the map between formal groups induced by the isogeny η A (p). Proof. Since φ is the Serre-Tate character of B/K, the reduction of the endomorphism φ(p) of B must be the Frobenius endomorphism of the reduction of B modulo p, from which it follows easily that δ(e α,n (k))( η A,p,w (t w )) k ≡ ∞ n=0 e α,n (k) 2 t 2k w mod m H,w .
On the other hand, d α,n (t w ) 2 has t w -expansion  Proof. The first assertion follows immediately from Lemma 5.2 because H w is an unramified extension of Q 2 . Moreover, for 0 ≤ n ≤ ∞, Lemma 5.1 shows that But then we conclude that this identity must also hold for n = ∞ by passage to the limit as n → ∞. The second assertion of the lemma now follows on taking logarithms, and noting that the points in A p do indeed lie on the formal group A w .
We now fix an embedding of H into the maximal unramified extension of K p which induces the prime w of H. Let I denote the ring of integers of the completion of the maximal unramified extension of K p . Since the formal group A w has height 1, a classical theorem asserts that it is isomorphic over I to the formal multiplicative group G m , and we fix such an isomorphism Writing W for the parameter of the formal multiplicative group, the isomorphism j w can be viewed as being given by a formal power series t w = j w (W ) in W with coefficients in I of the form j w (W ) = Ω w (A)W + · · · , where Ω w (A) is a unit in I . For 0 ≤ n ≤ ∞, we can then define the power series In particular, we conclude from (5.9) that, for all n with 0 ≤ n ≤ ∞ there exists a unique measure µ α,n,w in Λ I (O × p ) such that M(i(µ α,n,w )) = J α,n,w (W ). Recalling that we can canonically identify the Galois group G with O × p via the character ρ P , we shall in what follows always view the µ α,n,w as I -valued measures on G. Moreover, we have (5.12) µ α,∞,w = lim n→∞ µ α,n,w .
In the following, when there is no danger of confusion about the place w of H lying above p, we shall simply write µ α,∞ for µ α,∞,w .
We recall that we have fixed an embedding of the field T into C which extends our embedding of K into C, so that we can then consider the complex Hecke L-functions L(φ k , s) for all integers k ≥ 1. Now the Hecke characterφ k has conductor q or O K , according as k is odd or even. For all k ≥ 1, we shall write L q (φ k , s) for the Euler product with the Euler factor at q removed (so that this L-series is imprimitive when k is even). Finally, we fix an embedding of the compositum T H into the fraction field of I which induces the prime w of H and the prime P of T . This is possible because H ∩ T = K.
Theorem 5.4. For all integers k ≥ 1, the values Ω ∞ (A) −k L q (φ k , k) belong to T H, and we have The crucial step in proving this theorem is the following classical result. If b is any integral ideal of K prime to q, we write γ b for the class of b. We then define the partial Hecke L-function L q (φ k , γ b , s) by where the sum is taken over all integral ideals c of K, which are prime to q, and which lie in the class γ b .
Proposition 5.5. Let b be any integral ideal of K prime to q. Then, for all α ∈ J , we have 14 Proof. The proof (see [10]) rests upon a miraculous product formula from the 19th century theory of elliptic functions. We recall rapidly this product formula, and its link with our rational functions R α,A b , without giving a fully detailed, but essentially straightforward, proof of the Proposition. Let L be any lattice in C, say L = uZ + vZ, with v/u having positive imaginary part. We define where the limit is taken over real values of s > 0. For each integer k ≥ 1, we have the Kronecker-Eisenstein series where we assume that z / ∈ L. This series converges in the half plane R(s) > 1 + k/2, but it has a holomorphic continuation to the whole s-plane. We then define . The all important classical infinite product which we use is and we then define θ(z, L) = exp(−s 2 (L)z 2 /2)σ(z, L). Taking logarithmic derivatives, we easily obtain from the infinite product that, for all z 0 ∈ C with z 0 / ∈ L, we have We now take these functions for the lattice L b , and we have the following equality (see [16], Theorem 1.9): Now take E to be any set of integral of K, prime to q, whose Artin symbols in Gal(H(A q )/K) give precisely Gal(H(A q )/H). Recalling that H(A q ) = H(A b q ), and taking the q-division point on A b given by Q(b) = W(ξ(b)Ω ∞ (A)/ √ −q, L e ), we then have (see [16], Proposition 5.5) On combining equations (5.16), (5.17), (5.18), recalling the definition (4.10), and noting that the formula (5.15) follows easily. This completes the proof.
Corollary 5.6. For all integral ideals b of K prime to q and all integers k ≥ 1, belongs to H and is integral at w.
) is a rational function on A/H, the first assertion is clear from (5.15). Moreover, we have already seen (see Lemma 4.8) that this rational function has an expansion, in terms of the parameter t w of the formal group A w , which is a unit in O H,w [[t w ]]. Now we can interpret the differential operator d/dz in terms of the formal group A w as follows. The exponential map of A w is given by expanding t w = ν w (z), where ν w (z) is given by expanding the right hand side of (5.20) as a power series in z. The logarithm map of A w is given by the inverse series under composition, say We shall simply write λ ′ w (t w ) for the formal derivative of the power series λ w (t w ) with respect to t w . By one of the basic properties of such a logarithm map, λ ′ w (t w ) is in fact a unit power series in ], we conclude that, for all integers k ≥ 1, the t w -expansion of (d/dz) k log R α,A b (η A (b)(P )) will lie in O H,w [[t w ]]. In particular, its constant term will lie in O H,w , proving the second assertion of the corollary.
Proof. The exponential map of the formal group G m is given by W = e z − 1. Thus, by the uniqueness of the exponential map for A w , it follows that we must have Now it is very well known (see, for example, [8]) that, for all integers k ≥ 1, we have . In view of (5.23), we conclude from (5.8) that Hence the conclusion of the proposition follows immediately from (5.25) and applying Proposition 5.5 with b = c and b = cp. This completes the proof.
We can now prove Theorem 5.4. In view of (5.12), we have, for all integers k ≥ 1, We recall that we have fixed an embedding of the compositum HT into the fraction field of I which induces the prime w on H and the prime P on T . Recall also that for c ∈ C n , we have φ(c) ≡ 1 mod P n+2 . Moreover, for each n ≥ 0, C n gives a complete set of representatives of the ideal class group of K. It is therefore clear that Theorem 5.4 will follow from on passing to the limit as n → ∞ from (5.22). This completes the proof.
6. Iwasawa theory of the local tower K p (B P ∞ )/K p We need to establish the local theory at the prime p for the tower F ∞ /K in order to prove the main conjecture. Of course, the local extension K p (B P ∞ )/K p is not given by the points of finite order on a Lubin-Tate group. However, we take our fixed prime w of H above p, and we show that, since the class number of K is odd, we can fairly easily derive what is needed from the classical theory for the Lubin-Tate extension H w (A p ∞ )/H w .
Put r = [H w : K p ], so that r is the order of both δ, and the ideal class of p, and, of course, r is odd because it divides the class number of K. Define and put Of course, H w /K p is unramified, p is totally ramified in F ∞,p , and w is totally ramified in H w,∞ . Let F n,p = K p (B P n+2 ) and H n,w = H w (A p n+2 ), and write U (F n,p ) and U (H n,w ) for their respective groups of local units. Define where the projective limits are taken with respect to the local norm maps. We recall that we have fixed a set {V n } of primitive p n+2 -division points on A, which are compatible in the sense that (2.11) holds for all n ≥ 0. We write, as before, t w for the parameter of the formal group A w of A at w, and O H,w for the ring of integers of H w . The following is de Shalit's extension of Coleman's theorem [15].
] such that u n = δ −n (C u∞ (t w (V n ))) for all n ≥ 0.
We shall also need the following corollary of this theorem.
To deduce the corollary from the theorem, we note that, for all m ≥ n, Gal(F m,w /F n,w ) is isomorphic to Gal(F m,p /F n,p ) under restriction. Hence u ∞ clearly can be viewed as an element of U (H ∞,w ), and so has its Coleman power series C u∞ (t w ). This Coleman power series then has the interpolation property stated in the corollary because δ n (u n ) = u n for all n ≥ 0. The uniqueness in both results is obvious from the Weierstrass Preparation Theorem. Let u ∞ = (u n ) be any element of U (H ∞,w ), with Coleman power series C u∞ (t w ). Define , and satisfies the identity where [+] denotes the group law of the formal group of A w .
Proof. An entirely similar argument to that given in the proof of Lemma 5.2 shows that C u∞ (t w ) belongs to 1 + m H,w [[t w ]], and so the first assertion of the lemma follows. Moreover, it is shown in [15] (see Proposition 2.1 on p. 12 and Corollary 2.3, (ii) on p. 14) that whence the second assertion of the lemma also follows easily on taking logarithms.
Recall (5.7) that we have fixed an I -isomorphism j w : G m ≃ A w , and so, writing W for the parameter of the formal multiplicative group, we can then define the power series (6.6) B u∞ (W ) = A u∞ (j w (W )).
In view of (6.5), it follows, on applying Mahler's theorem to the power series B u∞ (W ), that there exists an I -valued measure µ(u ∞ ) on O × p such that M(i(µ(u ∞ ))) = B u∞ (W ). However, we now view µ(u ∞ ) as a measure on the Galois group G w via the canonical isomorphism ρ p : G w → O × p . Recall that G w,∞ = G w × ∆ w . Following [15], we then define the measureμ(u ∞ ) in Λ I (G w,∞ ) by (6.7)μ(u ∞ ) = σ∈∆w σµ(σ −1 (u ∞ )); here, and in what follows, for any profinite group H, Λ I (H) will denote the ring of I -valued measures on H (which is, of course, equal to the Iwasawa algebra of H with coefficients in the ring I ). This enables us to define the I -linear G w,∞ -homomorphism De Shalit (see Theorem 3.6, Chap. 1, of [15]) then goes on to prove that j Hw,∞ is injective, and has a cokernel of the form W w ⊗ Op I , where W w is a finite G w,∞ -module. On the other hand, if we take any u ∞ ∈ U (F ∞,p ), and define the power series B u∞ (W ) attached to u ∞ by exactly the same procedure as above, we again obtain by Mahler's theorem a measure µ(u ∞ ) on O × p . However, in this case, we view µ(u ∞ ) as a measure on the Galois group G via the canonical isomorphism ρ P : G → O × p . This in turn enables us to define the canonical I -linear G-homomorphism (6.9) Theorem 6.4. The map j F∞ is an injective Λ I (G)-homomorphism, and its cokernel is of the form W ⊗ Op I , where W is a finite G-module.
Proof. We first prove that j F∞ is injective. If µ(u ∞ ) = 0, then we must C u∞ (t w ) = 1, and so (6.10) C u∞ (t w ) 2 = C δ u∞ ( η A,p,w (t w )). Putting t w = t w (V n ) in this equation, and recalling the compatibility relation (2.11), we conclude that u 2 n = u n−1 for all n ≥ 1. Putting t w = t w (V 0 ) in the equation, we also conclude that u 2 0 = C u∞ (0) δ . Finally, putting t w = 0 in (6.8), we obtain C u∞ (0) 2 = C u∞ (0) δ , and so C u∞ (0) 2 r −1 = 1, whence finally C u∞ (0) = 1, because F p has residue class field F 2 . But the group µ 2 ∞ of all 2-power roots of unity, cannot belong to the completion of F ∞ at the unique prime above p. Indeed, if it did, then the field H w (A p ∞ ) would have to also contain A p * ∞ by the Weil pairing, and this is impossible because this latter group of points would map injectively under reduction modulo w, which cannot be the case because the field H w (A p ∞ ) has a finite residue field. Thus we must have u n = 1 for all n ≥ 0, and the proof of injectivity is complete.
We next show how the remaining assertion of the theorem can be derived from Theorem 3.6, Chap. 1, of [15]. The local norm map N H∞,w/F∞,p extends by I -linearity to a map θ ∞ : U (H ∞,w ) ⊗I → U (F ∞,p ) ⊗ Op I . We note that θ ∞ is surjective, because if u ∞ = (u n ) is any element of U (F ∞ ), then, noting that raising to the r-th power is an automorphism of U (F ∞,p ) because r is odd, we can simply define the element u ∞ = (u 1/r n ) of U (H ∞,w ), whence clearly N H∞,w/F∞,p (u ∞ ) = u ∞ . Noting that every element ζ of Λ I (G w,∞ ) can be written uniquely in the form ζ = σ∈∆w σ −1 A(σ) with A(σ) in Λ I (G w ), we can define the map λ ∞ : Λ I (G w,∞ ) → Λ I (G) by where A(σ) is the image of A(σ) under the isomorphism from Λ I (G w ) to Λ I (G) given by the restriction map. We then have the following commutative diagram with exact rows The commutativity of the left hand square is easily verified from the explicit description we have given of the maps θ ∞ and λ ∞ . Since the middle vertical map is clearly surjective because r is odd, it follows that the right hand vertical map is also surjective. Thus the above diagram shows that Theorem 6.4 does indeed follow from Theorem 3.6, Chap. 1, of [15].

Proof of the main conjecture
We establish in this section the analogue for B/F ∞ of Iwasawa's [20] celebrated theorem on cyclotomic fields, which led to the discovery of the main conjectures in general. Recall that G = Γ × ∆, where For each n ≥ 0, letC(F n ) denote the closure of C(F n ) in U (F n,p ) in the p-adic topology, and definē C(F ∞ ) = lim ← −nC (F n ), where the projective limit is taken with respect to the local norm maps. Recalling that U (F ∞,p ) = lim ← −n U (F n,p ), we then define the G-module We also define where N n denotes the local norm map from F n,p to F p . Thanks to (4.15), we see that we haveC(F ) ⊂ U ′ (F p ).
We first establish several preliminary results needed for the proof of this thoerem. Write N F/K for the global norm map from F to K, or the local norm map from F p to K p . Proof. We first note that an element z of U (K p ) is a norm from F n,p for all n ≥ 0 if and only if z = 1. Indeed, since p is totally ramified in F n and Gal(F n,p /K p ) = (O p /p n+2 ) × , it follows easily from local class field theory that N Fn/Kp (U (F n,p )) = 1 + p n+2 O p , whence the previous assertion is clear. Now assume that u is any element of U (F p ). If u lies in U ′ (F p ), then z = N F/K (u) is an element of U (K p ) which is a norm from U (F n,p ) for all n ≥ 0, and so z must be 1 by our first remark. Conversely, suppose u is any element of U (F p ) with N F/K (u) = 1. Now the restriction map from Gal(F n,p /F p ) to Gal(K n,p /K p ) is an isomorphism for all n ≥ 0. Moreover, the local Artin symbol of u for the extension F n,p /F p , say σ restricts to the local Artin symbol of N F/K (u) for the extension K n,p /K p , which is equal to the identity. Hence σ = 1, and so by local class field theory, u must be a norm from U (F n,p ), and the proof is complete.
Proof. This is essentially an exercise in classical Iwasawa theory, whose proof we briefly sketch. Let Y (F ∞,p ) be the Galois group over F ∞,p of the maximal abelian 2-extension of F ∞,p , and let Y (F p ) be the Galois group over F ∞,p of the maximal abelian 2-extension of F p . As usual, Γ = Gal(F ∞,p /F p ) acts on Y (F ∞,p ), and we have On the other hand, Lubin-Tate theory shows that where Z 2 denotes the Galois group of the maximal unramified 2-extension of F ∞,p . Since clearly U (F ∞,p ) Γ = 0, we conclude that Y (F ∞,p ) Γ is equal to the Galois group of the maximal unramified 2-extension of F ∞,p . Viewing (7.5) as a short exact sequence for the Γ-module Y (F ∞,p ) with U (F ∞,p ) as submodule and Z 2 as quotient, and taking Γ-cohomology of this exact sequence, we obtain from the snake lemma the exact sequence On the other hand, we have the short exact sequence of Z 2 -modules There is now an obvious commutative diagram with exact rows mapping the sequence (7.6) to the sequence (7.7) in which the left hand vertical map is obviously surjective, and the middle map is an isomorphism by (7.4). Hence the left vertical map must be an isomoprhism, and the proof is complete.
We can now prove Theorem 7.1. We have the obvious commutative diagram with exact rows Now the left vertical map is surjective by (4.15), and the middle vertical map is an isomorphism by Lemma 7.3, whence the right vertical map must be an isomorphism. This completes the proof of Theorem 7.1.
As before, let X(F ∞ ) = Gal(M (F ∞ )/F ∞ ), where M (F ∞ ) is the maximal abelian 2-extension of F ∞ , which is unramified outside p. For each n ≥ 0, let E(F n ) be the group of all global units of F n , and letĒ(F n ) be their closure in U (F n,p ) in the p-adic topology. DefineĒ(F ∞ ) = lim ← −nĒ (F n ), where the projective limit is taken with respect to the norm maps. We have already shown in Theorem 3.8 that F ∞ has no unramified abelian 2-extension. Hence global class field theory provides the following explicit description of X(F ∞ ), Finally, defineĒ ′ (F ) = U ′ (F p ) ∩Ē(F ).
Proof. Let M (F ) be the maximal abelian 2-extension of F , which is unramified outside p. Then where the first equality is elementary Iwasawa theory, and the second equality is by global class field theory. Obviously the group on the right is finite because E ′ (F ) has rank 1 as an abelian group. Hence, since X(F ∞ ) is a free finitely generated Z 2 -module by Theorem 3.1, we conclude that we must have , so that we have the exact sequence of G-modules Proof. Since (X(F ∞ )) Γ = 0, we have the exact sequence The corollary then follows immediately from Theorems 7.1 and 7.4.
Recall that ∆ = {1, j} is the Galois group of F ∞ over K ∞ . In the group ring Z 2 [∆], we define the two elements (7.11) ǫ The proof is based on the analytic class number formula and Kronecker's limit formula. By Dirichlet's theorem, the unit group of F has has rank 1, and, as above, its torsion subgroup is just µ 2 = {±1}. Write η for a generator of the unit group of F modulo torsion. We fix an embedding of F into C. If z is any element of F , |z| C will denote the square of the ordinary complex absolute value of z. Let h F denote the class number of F , and, as before, h will denote the class number of K. Let ω denote the non-trivial character of ∆, and write L(ω, s) for its complex L-series. Then Dirichlet's formula for the residue of the complex zeta functions immediately gives where D F denotes the discriminant of F . Now, by Lemma 4.1, the conductor f of F/K is equal to qp 2 . and so the conductor discriminant formula tells us that D F = q 2 N f, where N f denotes the absolute norm of f. Moreover, the functional equation for L(ω, s) gives that where W (ω) = ±1 is the sign in the functional equation for L(ω, s) Combining (7.14) and (7.15), we obtain finally We recall that, for each α ∈ J , the elliptic unit u α,0 is defined by u α,0 = N F0/F (R α,A (V 0 )). Moreover, by Lemma 7.3, we have N F/K u α,0 = 1, so that j(u α,0 ) = u −1 α,0 . Write γ α for the Artin symbol of the ideal αO K in G. Proof. The proof, which we only sketch, rests crucially on Kronecker's second limit formula. For each integral ideal g of K, we write K(g) for the ray class group of K modulo g. By Lemma 4.1, we know that K(g) = H(A g ) whenever the ideal g is divisible by the conductor q of φ. Now the conductor f of ω is equal to qp 2 , and so K(f) is the compositum of its two subfields F 0 = H(A p 2 ) and H(A q ). Moreover, F 0 ∩ H(A q ) = H because the primes of H above p are totally ramified in F 0 and unramified in H(A q ). Thus it is then clear from (4.7) that R α, where Q is the primitive q-division point on A defined by (4.6). Hence we obtain On the other hand, writing z Q = Ω ∞ (A)/ √ −q and V 0 = W(z 0 , L), we have the fundamental identity (see (5.17)) R α,A (V ⊕ Q) 12 = (c α (A)θ(z 0 + z Q , L) N α /θ(z 0 + z Q , α −1 L)) 12 . In view of this equality, the classical Kronecker's formula asserts that, for our character ω of Gal(K f /K), we have Recalling (7.17), we see that the right hand side of this last formula is simply equal to log |u α,0 | C − log |τ (u α,0 )| C , where we have written τ for the non-trivial element of Gal(F/K). But Lemma 7.3 shows that u α,0 .τ (u α,0 ) = 1 (because u α,0 is a norm from every finite layer of the Z 2 -extension F ∞ /F ). Hence log |u α,0 | C − log |τ (u α,0 )| C = 2 log |u α,0 | C , and so the assertion of the lemma follows from (7.18). This completes the proof.
For the next lemma, we recall that we have chosen the sign of √ −q so that ord p (( √ −q − 1)/2) > 0.
Proof. The field K(B 4 ) is an abelian extension of K, whose Galois group is a product of two cyclic groups of order 2, corresponding to the two subfields F = K(B P 2 ), and F * = K(B P * 2 ), where P * denotes the unramified degree 1 prime of T above p * . Moreover, by the Weil pairing, K(B 4 ) contains the group µ 4 of 4-th roots of unity. Thus we must have K(B 4 ) = K(i, √ b), where b is some non-zero element of K, and three quadratic extensions of K contained in K(B 4 ) are then K(i), K( √ b), and K( √ −b). Now the prime q = √ −qO K of K does not ramify in K(i), and so, making a choice of the sign of b, we can assume that F = K( √ b) and F * = K( √ −b). Now, by Theorem 2.4, q ramifies in both F and F * , but p * does not ramify in F , and p does not ramify in F * . It follows that we must have bO K = qb 2 for some fractional ideal b of K. Since the ideal q is principal, it follows that b 2 is principal. But K has odd class number, and so the ideal b itself must be principal, whence, modifying b by a square in K × , we must have b = ±β, where, as above β = √ −q. To see which choice of sign to take, we note that from which it follows easily that p * is unramified in the extension K( √ −β). Hence we must have F = K( √ −β), and the proof of the lemma is complete.
Corollary 7.12. The curve A (−β) has good reduction outside the set of primes of H dividing p.
Proof. Thanks to the equation (1.2), we know that all bad primes of A (−β) must divide either 2 or 3. However, combining the previous lemma, and Theorem 2.4, we see that every prime of H where A (−β) has bad reduction must ramify in the field F = F H. But the only primes of K which ramify in F are q and p, and so the only primes of H which ramify in F must lie above p or q. But the primes of H above q are not bad primes because of (1.2), completing the proof.
We can now complete the proof of Proposition 7.9. It suffices to show that there exists α ∈ J such that the index of the subgroup of E ′ (F ) generated by u α,0 is odd. Take α = a 2 + b 2 √ −q, where a is an even positive rational integer which is prime to 3, and b is an odd rational integer which is divisible by 3. Plainly, we then have (α, 6f) = 1, and α ≡ 1 mod p 2 because √ −q ≡ 1 mod p 2 . Also, it is clear that α is the norm from F to K of the the element a + b √ −β, so that the restriction of the Artin symbol of the ideal αO K to Gal(F/K) is trivial. Noting that N α ≡ q mod 4 ≡ 3 mod 4, we conclude that, for this choice of α, we have N α − ω(γ α ) ≡ 2 mod 4.
But now, combining this congruence with the formulae of Lemmas (7.16) and (7.10), and recalling that both h F and h are odd, and W (ω) = ±1, we conclude that the index of the subgroup of C(F ) generated by u α,0 in E ′ (F ) modulo torsion is indeed odd, and the proof of Proposition 7.9 is complete.
We can now prove the "main conjecture" for ǫ − X(F ∞ ). We recall that Γ = Gal(F ∞ /F ), and that Λ I (Γ) denotes the Iwasawa algebra of Γ with coefficients in I . Write ρ P,Γ for the restriction of ρ P to Γ. We recall that, in the results which follow concerning the link between "main conjectures" and complex L-values, we have fixed until further notice the embedding i : T → C given by (2.2).
Theorem 7.13. For all primes q ≡ 7 mod 8, we have the exact sequence of Λ I (Γ)-modules where M is a finite Γ-module, and µ A is the unique element of Λ I (Γ) such that, for all odd positive integers k = 1, 3, 5, · · · , we have We note that, since Λ I (G) is the Iwasawa algebra of Γ with coefficients in the group ring I [∆], and ǫ − I [∆] = ǫ − I , we have If m is any element of Λ I (G), we write m(−) for the measure in Λ I (Γ) such that ǫ − m = ǫ − m(−) under the equality (7.21). Now for a character of a p-adic Lie group the integral against a measure of the character coincides with evaluation of the character at the measure, it follows that, for every continuous homomorphism ξ from G to the multiplicative group of Q 2 such that ξ(j) = −1, we must have where ξ Γ denotes the restriction of ξ to Γ. For each α ∈ J , we recall that γ α is the Artin symbol of αO K in G. We note that in fact γ α always belongs to Γ because of our hypothesis that α ≡ 1 mod p 2 for α in I . Recall that µ α,∞ is the measure in Λ I (G) satisfying (5.13).
We can now prove Theorem 7.13. Since the map j F∞ in (6.8) is a Λ I (G)-homomorphism, then recalling (7.21), we see that Theorem 6.4 shows immediately j F∞ gives rise to an exact sequence of Λ I (Γ)-modules where W is some finite Γ-module. Recall thatC(F ∞ ) is, by definition, generated as a Λ(G)-module by the the norm compatible system of elliptic units u α,∞ defined by (4.16) for α running over J . It then follows from the results of §5 that j F∞ ((ǫ −C (F ∞ )) ⊗ Op I ) will be generated as a Λ I (Γ)-module by the µ α,∞ (−) for α ∈ J , which by Lemma 7.14 is equal to the ideal of Λ I (Γ) generated by the v α µ A for α ∈ J , where, as before, v α = N α − γ α . However, we claim that the v α for α ∈ J generate the maximal ideal (2, T ) of Λ(Γ) = Z 2 [[T ]]. Indeed, the ideal they generate is just the annihilator of the group of all 2-power roots of unity in F ∞ , and this group just consists of {±1} since the prime p * of K is not ramified in F ∞ . Thus we have finally (7.28) j F∞ ((ǫ −C (F ∞ )) ⊗ Op I ) = (2I , T )µ A .
Proof. Since q ≡ 7 mod 16, we have ǫ − X(F ∞ ) = 0 by Theorem 3.1, and so it follows from the exact sequence (7.19) that µ A must be a unit in the Iwasawa algebra Λ I (Γ). Hence´Γ(ρ P χ) Γ dµ A will be a unit in I , and so, noting that L(ρχ, 1) is the complex conjugate of L(ρχ, 1), the result follows from the theorem in the Appendix and the fact that (ρ P χ)(j) = −1.
Theorem 7.16. For all primes q with q ≡ 7 mod 8, the measure ν A is a unit in Λ I (Γ).
Proof. We simply apply (7.30) with ξ = ρ P χ, and m = µ α,∞ , noting that (ρ P χ)(j) = +1. Since ν A is a unit in Λ I (Γ),´Γ(ρ P χ) Γ dν A will be a unit in I , and we then finally apply the theorem in the Appendix to give the value of´G ξdµ α,∞ .
Finally, we note that the proofs we have given in §5 and above are valid for every choice of the embedding i : T → C made at the begining of section §2. In fact, there are precisely h such embeddings lying above our fixed embedding of K in C, and we now denote these distinct embeddings by i (r) (r = 1, . . . , h). Let ρ (r) denote the complex Grossencharacter of B/K relative to the embedding i (r) , and let ψ A/H denote the complex Grossencharacter of the elliptic curve A/H. Finally, for the proof of Theorem 1.2, we note that, writing L(A/J, s) for the complex L-series of a finite extension J of H, Theorem 1.1 tells us that, when q ≡ 7 mod 16, we have L(A/J, 1) = 0 whenever J ⊂ F ∞ . But then a standard argument, whose details we omit, shows that the finiteness of both A(J) and X(A/J)(p) follow from this non-vanishing result and the main conjecture for A/F ∞ , which is proven in [12].

Related results and some numerical data
Combining Theorems (3.1) and (3.9), we have shown in §3 that, for all primes q ≡ 7 mod 16, we have Sel P ∞ (B/F ∞ ) = 0, where we recall that F ∞ = K(B P ∞ ). However, the following theorem shows that nothing like this is valid for the p ∞ -Selmer group of the elliptic curve A over the field If L is any algebraic extension of H, we write Sel p ∞ (A/L) for the classical p ∞ -Selmer group of A/L. We also write M (F ∞ ) for the maximal abelian 2-extension of F ∞ , which is unramified outside the primes of F ∞ lying above p, and put X(F ∞ ) = Gal(M (F ∞ )/F ∞ ). Since A has good reduction everywhere over F ∞ by Theorem (2.4), we have (8.2) Sel p ∞ (A/F ∞ ) = Hom(X(F ∞ ), A p ∞ ).
Theorem 8.1. Assume q ≡ 7 mod 16. Then, provided there is more than one prime of H lying above p, or equivalently provided the ideal class of p does not have order exactly equal to h, we have Sel p ∞ (A/F ∞ ) = X(A/F ∞ )(p) = (K p /O p ) mq , for some integer m q > 0.