ON RUBIN S VARIANT OF THE p-adic BIRCH AND SWINNERTON-DYER CONJECTURE II

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1 ON RUBIN S VARIANT OF THE -ADIC BIRCH AND SWINNERTON-DYER CONJECTURE II A. AGBOOLA Abstract. Let E/Q be an ellitic curve with comlex multilication by the ring of integers of an imaginary quadratic field K. In 1991, by studying a certain secial value of the Katz two-variable -adic L-function lying outside the range of -adic interolation, K. Rubin formulated a -adic variant of the Birch and Swinnerton-Dyer conjecture when E(K) is infinite, and he roved that his conjecture is true for E(K) of rank one. When E(K) is finite, however, the statement of Rubin s original conjecture no longer alies, and the relevant secial value of the aroriate -adic L-function is equal to zero. In this aer we extend our earlier work and give an unconditional roof of an analogue of Rubin s conjecture when E(K) is finite. 1. Introduction The goal of this article is to extend the results of [1] to give an unconditional roof of a certain variant of the -adic Birch and Swinnerton-Dyer conjecture for ellitic curves with comlex multilication. Let E/Q be an ellitic curve with comlex multilication by O K, the ring of integers of an imaginary quadratic field K; this imlies that K is necessarily of class number one. Let > 3 be a rime of good, ordinary reduction for E; then we may write O K =, with = πo K and = π O K. Set K := K(E π ), K := K(E π ), and K := K K. Let O denote the comletion of the ring of integers of K,. For any extension L/K we set Λ(L) := Λ(Gal(L/K)) := Z [[Gal(L/K)]], and Λ(L) O := O[[Gal(L/K)]]. Date: Final version. Aril 1, Mathematics Subject Classification. 11G05, 11R23, 11G16. Key words and hrases. Rubin, -adic L-function, Birch and Swinnerton-Dyer conjecture, ellitic curve. Partially suorted by NSA grant no. H I am very grateful to the anonymous referee for a number of very helful comments. University, Berlin. generous hositality. This aer was comleted while I was visiting the Humboldt I would like to thank the members of the Mathematics Deartment there for their 1

2 2 A. AGBOOLA Let ψ : Gal(K/K) Aut(E π ) O K, Z, ψ : Gal(K/K) Aut(E π ) O K, Z denote the natural Z -valued characters of Gal(K/K) arising via Galois action on E π E π resectively. We may identify ψ with the Grossecharacter associated to E (and ψ with the comlex conjugate ψ of this Grossencharacter), as described, for examle, in [12,. 325]. We write T and T for the -adic and -adic Tate modules of E resectively. We now recall that the Katz two-variable -adic L-function L Λ(K ) O satisfies a -adic interolation formula that may be described as follows (see [12, Theorem 7.1] for the version given here, and also [5, Theorem II.4.14]. Note also that, as the notation indeicates, L deends uon a choice of rime lying above cf. Remark 1.2 below). For all airs of integers j, k Z with 0 j < k, and for all characters χ : Gal(K(E )/K) K, we have and L (ψ k ψ j χ) = A L(ψ k ψ j χ 1, 0). (1.1) Here L(ψ k ψ j χ 1, s) denotes the comlex Hecke L-function, and A denotes an exlicit, non-zero factor whose recise descrition we shall not need. Z Write ψ : Gal(K/K) 1 + Z for the comosition of ψ with the natural rojection 1 + Z, and define ψ in a similar manner. Define L (s) := L (ψ ψ s 1 ), L (s) := L (ψ ψ s 1 ) for s Z. The character ψ lies within the range of interolation of L, and the behaviour of L at ψ is redicted by the -adic Birch and Swinnerton-Dyer conjecture for E (see [3, ages ], [10, Theorem V.8]). Conjecturally, ord s=1 L (s) is equal to the rank r of E(Q), and the exact value of lim s 1 L (s)/[s 1] r may be described in terms of various arithmetic invariants associated to E. The character ψ, however, lies outside the range of interolation of L, and the function L (s) has not been studied nearly as much as L (s). The first results concerning the behaviour of L (s) were obtained by Karl Rubin (see [12], [13]). When r 1, Rubin formulated a variant of the -adic Birch and Swinnerton-Dyer conjecture for L (s) which redicts that that ord s=1 L (s) is equal to r 1, and which gives a formula for lim s 1 [L (s)/(s 1) r 1 ]. Under suitable hyotheses, Rubin showed that his conjecture is equivalent to the usual -adic Birch and Swinnerton-Dyer conjecture, and he roved both conjectures when r = 1. In the case r = 1, he then used these results to give the first examles of a -adic construction of a global oint of infinite order in E(Q) directly from the secial value of a -adic L-function.

3 BIRCH AND SWINNERTON-DYER CONJECTURE II 3 On the other hand, when r = 0, matters become quite different, and the results of [12], [13] do not aly. It is not hard to show that the functional equation satisfied by L (see [5, 6]) imlies that ord s=1 L (s) 1. Rubin seculated that erhas ord s=1 L (s) = 1, and assuming that this was true, he also raised the question of determining the value of lim s 1 [L (s)/(s 1)] (see [13, Remark on.74]). A natural aroach to studying the behaviour of L at ψ is to try to find a suitable Selmer grou that governs the behaviour of L (s) at s = 1. This may be done by using the two-variable main conjecture to study certain Iwasawa modules associated naturally to L (s), as described in detail in [1, 1 3]. In [1], we defined a restricted Selmer grou ˇΣ (T ) H 1 (K, T ). This restricted Selmer grou is defined by reversing the Selmer conditions above and that are used to define the usual Selmer grou Sel(K, T ). The O K, -module ˇΣ (K, T ) is free of rank r 1, and if r 1, then in fact ˇΣ (K, T ) Sel(K, T ) (see [1, Lemma 3.6, Proosition 6.5, and Proosition 6.7]). We also defined a similar grou ˇΣ (K, T ) H 1 (K, T ), and we constructed a -adic height airing [, ] K, : ˇΣ (K, T ) ˇΣ (K, T ) O K,. If r 1, and if the -adic Birch and Swinnerton-Dyer conjecture is true, then [, ] K, non-degenerate. We conjectured that [, ] K, is also non-degenerate when r = 0. It was shown in [1] that if [, ] K, is finite, then is non-degenerate and the -rimary art of X(E/K) ord s=1 L (s) = rank OK, (ˇΣ (K, T )) = r 1. Under these assumtions, we also determined the value of lim s 1 L (s)/(s 1) r 1 u to multilication by a -adic unit (thereby recovering a weak form of [12, Corollary 11.3] in the case r 1). In articular, our results imlied that if r = 0 (in which case X(E/K) is known to be finite; see [11]) and [, ] K, guessed by Rubin in [13]. is non-degenerate, then ord s=1 L (s) = 1, as was Suose now that r = 0. In this aer, we strengthen the results of [1] by giving an unconditional roof of the fact that ord s=1 (L (s)) = 1, and we also determine the exact value of the first derivative of L (s) at s = 1. We do this via an aroach involving ellitic units and exlicit recirocity laws (cf. [13]), rather than the two-variable main conjecture and Galois cohomology, as in [1]. In order to state our main result, we must introduce some further notation. Suose that y ˇΣ (K, T ) and y ˇΣ (K, T ) are of infinite order, and let is ex : H 1 (K, T ) Q, ex : H1 (K, T ) Q

4 4 A. AGBOOLA denote the Bloch-Kato dual exonential mas. Via localisation, these induce mas (which we denote by the same symbols): ex : ˇΣ (K, T ) Q, ex : ˇΣ (K, T ) Q. Write f O K for the conductor of the Grossencharacter associated to E, and let N(f) denote the norm of this ideal. Set L (s) L (ψ ) := lim s 1 s 1. The following result may erhas be viewed as being an analogue of a similar excetional zero henomenon observed in the work of Mazur, Tate and Teitelbaum concerning -adic Birch and Swinnerton-Dyer conjectures for ellitic curves without comlex multilication (see [6, esecially Conjecture 2], [7, esecially age 38]). It relates the value of L at a oint within the range of -adic interolation to the derivative of L at a oint lying outside the range of -adic interolation. Theorem A. Suose that L (ψ) 0. (a) The -adic height airing [, ] K, is non-degenerate and ord s=1 L (s) = 1. (b) We have that ( ( 1) 1 1 ) ψ( ) where Ω and Ω N(f) E (see Section 4 below). ( 1 1 ( 1 ψ () ) Ω L (ψ ) = ψ () [y, y ] K, ) ( ) 1 ψ( ) L (ψ) Ω ex (y ) ex (y), are certain -adic eriods defined using the formal grou associated to It is interesting to comare Theorem A with [12, Theorem 10.1]. Both of these results are examles of the following more general henomenon concerning certain secial values of the Katz two-variable -adic L-function L. Let k 0 be an integer, and set φ k := ψ k+1 ψ k, φ k := ψ k ψ k+1. Then we see from (1.1) that φ k lies within the range of interolation of L ; for k 1, the behaviour of L at φ k is redicted by various conjectures due to Bloch, Beilinson, Kato and Perrin-Riou. On the other hand, the character φ k lies outside the range of interolation of L, and as far the resent author is aware, the behaviour of L at φ k for k 1 does not aear to have reviously been studied. Write φ k (resectively φ k ) for the comosition of φ k (resectively φ k ) with the natural rojection Z 1 + Z, and define L (φ k, s) := L (φ k φ k s 1 ), L (φ k, s) := L (φ k φ k s 1 )

5 for s Z. BIRCH AND SWINNERTON-DYER CONJECTURE II 5 Using the techniques of [12], [1] and the resent aer, it may be shown that the orders of vanishing of L (φ k, s) and L (φ k, s) at s = 1 are of oosite arity, and that the values of the first non-vanishing derivatives of L (φ k, s) and L (φ k, s) at s = 1 are related in a manner similar to [12, Theorem 10.1] (if L (φ k ) = 0) or to Theorem A above (if L (φ k ) 0). We shall discuss this more fully in a future article (see [2]). The strategy of the roof of Theorem A is similar to that emloyed in [12]; however, because we work with restricted Selmer grous rather than true Selmer grous, the details are rather different. airing [, ] K, These differences mainly arise from the fact that the -adic height on restricted Selmer grous (when r = 0) is somewhat more difficult to work with than is the -adic height airing on true Selmer grous (when r = 1). The basic ideas involved in the roof of Theorem A may be described as follows. Using ellitic units, we construct canonical elements s ˇΣ (K, T ), s ˇΣ (K, T ). It follows from the roof of Theorem 8.4(a) below that s is of infinite order only if L (ψ ) 0. By analysing certain Kummer and cu roduct airings, and using Wiles s exlicit recirocity for formal grous, we rove that ex (s ). = L (ψ), (1.2) where the symbol. = denotes equality u to multilication by a non-zero factor. Hence we see that if L (ψ) 0, then s at s = 1. is indeed of infinite order, and so L (s) has a first-order zero We then comute [s, s ] K, using Kummer theory, Hilbert symbols, and Wiles s exlicit recirocity law, and we see that [s, s ] K,. = L (ψ) L (ψ ) (1.3) The roof of Theorem A is then comleted by showing that if y ˇΣ (K, T ) and y ˇΣ (K, T ) are both of infinite order, then ex (y ) ex (y) [y, y ] K, = ex (s ) ex (s ), [s, s ] K, and so the desired result follows by alying (1.2) and (1.3). An outline of the contents of this aer is as follows. In Section 2 we recall some basic roerties of restricted Selmer grous. In Section 3 we describe a local decomosition of the -adic height airing [, ] K, in terms of local Artin symbols. We recall some basic facts concerning the formal grou Ê associated to E in Section 4, and we establish a number of

6 6 A. AGBOOLA conventions for use in subsequent calculations. We discuss roerties of various Kummer airings in Section 5, and we use these airings to comute the value of [, ] K, on certain cohomology classes in restricted Selmer grous that are constructed using global units. In Section 6, we use the results of Section 5 to comute certain secial values of the dual exonential ma in terms of logarithmic derivatives of certain Coleman ower series. In Section 7, we describe the construction of the canonical elements s and s via ellitic units. Finally, in Section 8, we aly our revious results to these canonical elements, and we rove Theorem A. We conclude this Introduction by remarking that the methods and results of this aer remain valid if we assume that E is defined over its field of comlex multilication K rather than over Q. Notation and conventions. Throughout this aer, K denotes an imaginary quadratic field of class number one. If L is any field, we write L ab for the maximal abelian extension of L, and L for an algebraic closure of L. For each integer n 1, we write K n := K(E π n), Kn := K(E π n), K n := K(E n) = K n Kn, and K := K(E π ), K := K(E π ), K := K(E ). We also ut N n := K n K,, and we write m n, for the maximal ideal of the comletion of the ring of integers of N n,. The symbol O denotes the comletion of the ring of integers of K,. For any extension L/K we set Λ(L) := Λ(Gal(L/K)) := Z [[Gal(L/K)]], and Λ(L) O := O[[Gal(L/K)]]. We write I := Ker(ψ : Λ(K ) Z ), I := Ker(ψ : Λ(K ) Z ), and let ϑ and ϑ be the generators of I and I fixed in [12, 6]; so ϑ = γψ (γ 1 ) 1, where γ is any toological generator of Gal(K /K) satisfying log (ψ (γ)) =, and ϑ is defined analogously. We set D := K /O K, and D := K /O K,. If M is any Z -module, we write M for the Pontryagin dual of M. For each integer n 1, we let e n : E π n E π n µ n

7 BIRCH AND SWINNERTON-DYER CONJECTURE II 7 denote the Weil airing, normalised as decribed in [8, 3.1.2] (the reader should note that this is not the same normalisation as that used in [12]). This airing satisfies the identities for ς n E π n, ς n E π n, and for ς n E π n, We set W := E π ς n+1 E π (n+1). e n (π ς n, ς n) = e n (ς n, πς n ) e n+1 (ς n, ς n+1) = e n (ς n, π ς n+1) and W := E π. We write T and T for the π-adic and π -adic Tate modules of E resectively. Let w = [w n ] and w = [w n] denote generators of T and T resectively. We use the following notation to denote various unit grous: U n, := units in K n, congruent to 1 modulo ; U n, := units in K n, congruent to 1 modulo ; U, := lim U n,, U, := lim U n, ; Un, := units in Kn, congruent to 1 modulo ; U n, := units in K n, congruent to 1 modulo ; U, := lim U n,, U, := lim U n,, where all inverse limits are taken with resect to the obvious norm mas. We also set E n := global units of K n, E n := global units of K n; E n := the closure of the rojection of E n into U n, ; E n := the closure of the rojection of E n into U n, ; E := lim E n, E := lim E n. Remark 1.1. Note that since the strong Leooldt conjecture holds for all abelian extensions of K (see [4]), we have that E n E n Z Z, E n E n Z Z, and so we may also view E as being a submodule of U, and E as being a submodule of U,. We shall do this without further comment several times in what follows. Remark 1.2. It is imortant for the reader to bear in mind that every theorem or construction in this aer that deends uon a choice of rime of K lying above also has a corresonding version in which the roles of and are interchanged. We shall sometimes make use of this fact without stating it exlicitly.

8 8 A. AGBOOLA 2. Restricted Selmer grous In this section we shall recall some basic roerties of restricted Selmer grous. We refer the reader to [1, Sections 3 and 4] for more comlete details. Suose that F/K is any finite extension. For any lace v of F, we define H 1 f (F v, W ) to be the image of E(F v ) D under the Kummer ma E(F v ) OK D H 1 (F v, W ), and we define H 1 f (F v, W ) in a similar manner. Note that H 1 f (F v, W ) = 0 if v. We also set H 1 f (F v, E π n) := Im[E(F v )/π n E(F v ) H 1 (F v, E π n)], H 1 f (F v, E π n) := Im[E(F v )/π n E(F v ) H 1 (F v, E π n)]. Suose that M {W, W, E π n, E π n} and that q {, }. If c H 1 (F, M), then we write loc v (c) for the image of c in H 1 (F v, M). We define the true Selmer grou Sel(F, M) by Sel(F, M) = { c H 1 (F, M) loc v (c) H 1 f (F v, M) for all v } ; the relaxed Selmer grou Sel rel (F, M) by Sel rel (F, M) = { c H 1 (F, M) loc v (c) H 1 f (F v, M) for all v not dividing } ; the q-restricted Selmer grou (or simly restricted Selmer grou for short when q is understood) Σ q (F, M) by Σ q (F, M) = {c Sel rel (F, M) loc v (c) = 0 for all v dividing q}. (The terminology restricted Selmer grou is meant to reflect a choice of a combination of relaxed and strict Selmer conditions at laces above.) We also define ˇΣ q (F, T ) := lim Σ q (F, E π n), ˇΣq (F, T ) := lim Σ q (F, E π n). n n If L/K is an infinite extension, we define Σ q (L, M) = lim Σ q (L, M), where the direct limits are taken with resect to restriction over all subfields L L finite over K. We record the following standard cohomological result that will be used later.

9 BIRCH AND SWINNERTON-DYER CONJECTURE II 9 Lemma 2.1. Let n 0 be an integer, and suose that L and M are fields with K L M N n. Then, for every integer m 1, the restriction mas H 1 (L, E π m) H 1 (M, E π m), H 1 (L, µ m) H 1 (M, µ m) are injective and they induce isomorhisms H 1 (L, E π m) H 1 (M, E π m) Gal(M/L), H 1 (L, µ m) H 1 (M, µ m) Gal(M/L). A similar result holds if L and M are relaced by L and M with K L M N n,. Proof. This is quite standard, and may be roved via the argument given in [10,.40], for examle. We now exlain how elements in restricted Selmer grous may be constructed by combining Kummer theory on E with Kummer theory on the multilicative grou (see Proosition 2.6 below). In order to do this, we require several rearatory lemmas. Our starting oint is the following result of Perrin-Riou. Lemma 2.2. There is an O K -linear isomorhism of Gal(K n/k)-modules H 1 (Kn, E π n) Hom(E π n, Kn /Kn n ); f f. (2.1) For each lace v of K n, there is also a corresonding local O K -linear isomorhism H 1 (K n,v, E π n) Hom(E π n, K n,v/k n n,v ). Proof. See [8, Lemme 3.8]. The isomorhism (2.1) is defined as follows. Let f H 1 (K n, E π n), and recall that e n : E π n E π n µ n denotes the Weil airing. We identify Kn /Kn n with H 1 (Kn, µ n) via Kummer theory. If ς E π n, then f(ς ) H 1 (Kn, µ n) is defined to be the element reresented by the cocycle σ e n (f(σ), ς ) for all σ Gal(K/K n). Corollary 2.3. There are isomorhisms r n : H 1 (K, E π n) Hom(E π n, K n /Kn n ) Gal(K n/k) r n : H 1 (K, E π n) Hom(E π n, K n /K n n ) Gal(Kn/K). Proof. This follows from Lemmas 2.1 and 2.2 (cf. also [12, Lemma 2.1] or [8, Lemme 3.8]).

10 10 A. AGBOOLA Lemma 2.4. For each lace v of K n with v, there is an O K -linear isomorhism E(K n,v)/π n E(K n,v) Hom(E π n, O K n,v /O n K n,v ). Proof. See [8, Lemme 3.11]. Corollary 2.5. Suose that h H 1 (K n, E π n). Then h Σ (K n, E π n) if and only if, for each ς E π n, the following local conditions are satisfied: (a) h(ς ) K n n,v for all v ; (b) n v K n ( h(ς )) for all v. (Note that we imose no local conditions at laces lying above.) Proof. This follows from Lemmas 2.2 and 2.4, using the definition of the isomorhism 2.1 (see also [8, Lemme 3.8]). Proosition 2.6. There are natural injections ρ : Hom(T, (U, Q)/E ) Gal(K /K) ˇΣ (K, T ), ρ : Hom(T, (U, Q)/E ) Gal(K /K) ˇΣ (K, T ) Proof. The roof of this result is essentially the same, mutatis mutandis, as that of [12, Proosition 2.4]. The ma ρ is defined as follows. For any f Hom(T, (U, Q)/E ) Gal(K /K) and any integer n 1, we define f n Hom(E π n, En/E n n ) Gal(K /K) to be the image of f under the following comosition of mas: Hom(T, (U, Q)/E ) Gal(K /K) Hom(T, (U n, Q)/E n) Gal(K /K) Hom(E π n, E n/e n n ) Gal(K /K), where the first arrow is the ma induced by the natural rojection U, U n,, and the second arrow is induced by raising to the n -th ower in U n,. We define ρ(f) := [( 1)(π ) n rn 1 (f n )] lim H 1 (K, E π n). (2.2) n It follows from [8, Lemma 3.16] that ρ(f) does indeed lie in lim n H 1 (K, E π n). It is not hard to check from the definition that ρ is injective. It follows from [1, Theorem 3.1, Proosition 3.2 and Corollary 3.3] that rn 1 (f n ) Σ (K, E π n) if and only if the restriction of rn 1 (f n ) to H 1 (K, E π n) is unramified outside. It may be shown via an argument very similar to that given in [12, Lemmas 2.1 and 2.3] that this in fact the case. We shall use Proosition 2.6 to roduce canonical elements in restricted Selmer grous by alying ρ and ρ to certain Galois-equivariant homomorhisms that are constructed using norm-coherent sequences of ellitic units (see Section 7 below).

11 BIRCH AND SWINNERTON-DYER CONJECTURE II The -adic height airing on restricted Selmer grous Our goal in this section is to describe a local decomosition of the -adic height airing [, ] K, : ˇΣ (K, T ) ˇΣ (K, T ) O K,. (3.1) in terms of Artin symbols; this is analogous to the local decomosition of the standard -adic height airing on true Selmer grous described in [8, Lemme 3.19]. This local decomosition will be used in Section 5 to determine certain values of [, ] K, exlicitly; these will in turn lay a key rôle in showing that the airing (3.1) is non-degenerate when r = 0 in Section 8. We begin by recalling the outlines of the main stes in the construction of [, ] K,. For more comlete details, we refer the reader to [1, 4]. Let Y(K ) denote the Galois grou over K of the maximal abelian ro- extension of K that is unramified away from and totally slit at all laces of K lying above. The first ste in the construction of the airing [, ] K, is the construction of an isomorhism Ψ K : ˇΣ (K, T ) Hom(T, Y(K )) Gal(K /K). (3.2) This isomorhism is constructed as follows. Write J n for the grou of finite ideles of K n, and let V n denote the subgrou of J n whose comonents are equal to 1 at all laces dividing and are units at all laces not dividing. Set C n := J n /(V n Kn ), Ω n := µ n(kn,v), v and write C n () for the -rimary subgrou of C n. Note that the order of Ω n remains bounded as n varies. We view Ω n as being a subgrou of C n () via the obvious embedding of Ω n into J n. Using Kummer theory (cf. Corollary 2.5 above), one constructs an exact sequence 0 Hom(E π n, Ω n ) Gal(K n/k) Hom(E π n, C n ) Gal(K n/k) ηn Σ (K, E π n) 0 (see [1, Proosition 4.6]). Let η n denote the ma obtained from η n via assage to the quotient by Ker(η n ). It may be shown that assing to inverse limits over the mas η n 1 isomorhism Ξ K : lim Σ (K, E π n) = ˇΣ (K, T ) Hom(T, lim C n ()) Gal(K /K), yields an where the inverse limit lim C n () is taken with resect to the norm mas Kn Kn 1. One then shows via class field theory (along with the fact that the weak -adic Leooldt conjecture holds for K) that there is an isomorhism Hom(T, lim C n ()) Gal(K /K) Hom(T, Y(K )) Gal(K /K), (3.3)

12 12 A. AGBOOLA and comosing this last isomorhism with with Ξ K yields the desired isomorhism Ψ K. Next, by suitably interreting restricted Selmer grous in terms of certain Galois grous (see [1, Theorem 3.1]), one shows that there is a natural homomorhism We thus obtain a ma β K : Hom(T, Y(K )) Gal(K /K) Hom OK, (ˇΣ (K, T ), O K, ). β K Ψ K : ˇΣ (K, T ) Hom OK, (ˇΣ (K, T ), O K, ), and this yields the -adic height airing airing on restricted Selmer grous. [, ] K, : ˇΣ (K, T ) ˇΣ (K, T ) O K, In order to describe the local decomosition of [, ] K,, we must introduce some further notation. Suose that y = [y n ] ˇΣ (K, T ), y = [y n] ˇΣ (K, T ). For each ositive integer n, we define q n to be the ma q n : ˇΣ (K, T ) Ψ K Hom(T, Y(K )) Gal(K /K) Hom(E π n, C n ) Gal(K /K), where the second arrow is the natural quotient ma afforded by the isomorhism (3.3). For each ς E π n, let n(ς ) denote the exact ower of π that kills ς. Let S n,ς (y n ) denote any reresentative of ηn 1 (y n )(ς ) in J n. For each finite lace v of K, define {y, y } (ς ) n,v to be the unique element of O K /π n(ς ) O K such that where [S n,ς (y n ) v, K ab v {y, y } (ς ) n,v Proosition 3.1. (cf. [8, Lemma 3.19]) (a) For any ς E π n, we have ς = y n([s n,ς (y n ) v, K ab v /K n,v ]), /K n,v ] Gal(Kv ab /K n,v ) is the obvious local Artin symbol. [y, y ] K, ς = y n([q n (y)(ς ), K ab /K n]), (3.4) where [q n (y)(ς ), K ab /K n] Gal(K ab /K n) is the obvious global Artin symbol. (b) We have [y, y ] K, {y, y } (ς ) n,v v (mod π n(ς ) O K, ), (3.5) where the sum is over all finite laces v of Kn.

13 BIRCH AND SWINNERTON-DYER CONJECTURE II 13 Proof. (a) This follows immediately from the following commutative diagram: ˇΣ (K, T ) Σ (K, E π n) Ψ K Hom(T, Y(K )) Gal(K /K) HomOK, (ˇΣ (K, T ), O K, ) η 1 n Hom(E π n, C n) Gal(K /K) Ker(η n ) β K Hom(Σ (K, E π n), O K /π n O K ) In this diagram, the second arrow on the bottom row is induced by the ma f (c {ς c([f(ς ), K ab /K n])}), f Hom(E π n, C n ) Gal(K /K) and is well-defined because [z, K ab /K n] = 0 for all z Ω n. Note also that here we have canonically identified O K /π n O K with Hom(E π n, E π n) via the ma β {ς β ς }. (b) This follows from the local decomosition of the global Artin symbol afforded via class field theory, viz. if α J n, then [α, K ab /K n] = v [α v, Kv ab /Kn,v], where the roduct is over all finite laces v of K n. 4. Formal Grous The urose of this section is to recall a number of facts concerning formal grous, and to establish certain conventions that we shall use in Section 5. We fix a minimal Weierstrass model of E over O K,, and we write Ê for its associated formal grou. Let Ĝm denote the formal grou over O K, associated to the multilicative grou G m. If x is a oint on G m or on E, then we write ˆx for the corresonding value of the arameter on Ĝm or on Ê. We denote the formal grou logarithm associated to Ê by λ be (Z) = Z + (higher order terms) O K, [[Z]], and we write log E, for the corresonding -adic logarithm associated to E. We denote the -adic logarithm associated to G m by log. Recall that O denotes the comletion of the ring of integers of K,. Since Ê is a height one Lubin-Tate formal grou, we may fix an isomorhism η : Ĝm Ê, η O[[Z]]. As exlained in [12, 6], this choice of isomorhism then yields: (a) A generator w = [w n] of T such that for every n 1 and ς E π n, we have η( en (π n ς, w n)) = ˆς. (4.1)

14 14 A. AGBOOLA (b) A -adic eriod Ω := η (0) O which is such that Ω σ = ψ (σ 1 )Ω for every σ Gal(K/K). We fix a generator w = [w n ] of T, and for each n 0, we set ζ n := e n (π n w n, wn); so from (4.1) above, we have that η(ˆζ n ) = ŵ n. We note that for each integer n 1, our choice of η induces an isomorhism χ n : µ n E π n; ζ n w n which is Gal(K/Kn)-equivariant; this in turn induces an isomorhism (which we denote by the same symbol) χ n : H 1 (N n,, µ n) H 1 (N n,, E π n). (4.2) Proosition 4.1. Recall that for each integer n 1, m n, denotes the maximal ideal in the comletion of the ring of integers of N n,. (a) With notation as above, the following diagram commutes: H 1 (N n,, µ n) Ĝ m (m n, ) Ĝ m (m n, ) n O N n, O n N n, χ n H 1 (N n,, E π n) η Ê(m n, ) nê(m n,). (Here the vertical arrows denote the natural mas afforded by Kummer theory on Ĝm and Ê.) (b) If ˆx Ĝm(m n, ), then log E, (η(ˆx)) Ω log (ˆx) (mod m n n,) on Ĝm(m n, )/Ĝm(m n, ) n. Proof. (a) This follows directly from the definitions of η and χ n. (b) See the roof of [12, Corollary 9.2].

15 BIRCH AND SWINNERTON-DYER CONJECTURE II Kummer airings and -adic heights In this section we shall comute the value of the -adic height airing [, ] K, on elements of restricted Selmer grous that are constructed via Proosition 2.6 (see Theorem 5.11 below). This is accomlished by using Proosition 3.1 to exress these values in terms of certain Kummer airings, and then alying Wiles s exlicit recirocity law. Definition 5.1. For each integer n 1, we define a airing by (, ),Eπ n : U n, H 1 (K, E π n) E π n (5.1) (u n, c n),eπ n = c n([u n, K ab /K n,]) =: α,eπ n(u n, c n) w n, with α,eπ n(u n, c n) O K / n O K Z / n Z. The airings (5.1) give a airing (, ),T : U, H 1 (K, T ) T ; (u, c ) α,t (u, c ) w (5.2) that is defined as follows. Suose that u = [u n ] U, and Then we set c = [c n] lim H 1 (K, E π n) H 1 (K, T ). (u, c ),T = [(u n, c n),eπ n] = [α,eπ n(u n, c n)] w =: α,t (u, c ) w. The airing (5.2) is related to the -adic height airing [, ] K, in the following way. Recall from Proosition 2.6 above that there is a natural injection ρ : Hom(T, (U, Q)/E ) Gal(K /K) ˇΣ (K, T ). Proosition 5.2. Suose that ξ Hom(T, U,/E ) Gal(K /K). Let ξ(w ) U, denote any lift of ξ(w ), and suose that y ˇΣ (K, T ). Then we have [ρ(ξ), y ] K, w = (ξ(w ), loc (y )),T, and so it follows that [ρ(ξ), y ] K, = [α,eπ n((ξ(w )) n, loc (yn))] = α,t (ξ(w ), loc (y )). Proof. This follows directly from Proosition 3.1 and Definition 5.1.

16 16 A. AGBOOLA Our goal in this section is to evaluate [ρ(ξ), y ] K, (see Theorem 5.11 below). The difficulty in doing this arises from the fact that that the airing (5.2) cannot be directly evaluated using exlicit recirocity laws. In order to overcome this obstacle, we shall first exress the Kummer airings (5.1) and (5.2) in terms of certain Hilbert symbols. This will then enable us to relate these airings to two other airings (namely (5.5) and (5.6) below) that can be evaluated using Wiles s exlicit recirocity law for formal grous. Definition 5.3. We define the Hilbert airing by (, ),µ n : N n, N n n, N n, N n n, µ n (5.3) (a, b),µ n = (b1/n ) σa, b 1/n where σ a denotes the local Artin symbol [a, K ab /N n, ] and b 1/n is any n -th root of b in K ab. We remark that it is a standard roerty of the Hilbert airing (see e.g. [17, Chater XIV, 2]) that (a, b),µ n Lemma 5.4. Suose that u U, and c H 1 (K, T ). Then = (b, a) 1,µ n. (5.4) (u n, r n(c n)(π n w n )),µ n = α,eπ n(u n, c n) ζ n. Proof. It follows from the definition of r n (see Corollary 2.3) that (u n, r n(c n)(π n w n )),µ n = e n (π n w n, c n([u n, K ab /N n, ])) = e n (π n w n, c n([u n, K ab /K n,])) (since u n K n,) = e n (π n w n, α,eπ n(u n, c n) w n) = α,eπ n(u n, c n) e n (w n, π n w n) = α,eπ n(u n, c n) ζ n, as claimed. Definition 5.5. For each integer n 1, we define a airing (, ),Eπ n : N n, H 1 (N n,, E π n) E π n (5.5) by (ν, y),eπ n = y([ν, K ab /N n, ]).

17 BIRCH AND SWINNERTON-DYER CONJECTURE II 17 It is not hard to check (using Lemma 2.1) that the airings (, ),Eπ n airing combine to yield a (, ),T : U, H 1 (K, T ) T ; (β, c) α,t (β, c) w (5.6) that is defined as follows. Suose that β = [β n ] U,, and that Then we set c = [c n ] lim H 1 (K, E π n) H 1 (K, T ). (β, c),t = [(β n, c n ),Eπ n] = [α,eπ n(β n, c n )] w =: α,t (β, c) w. The advantage of the airings (5.5) and (5.6) is that they can be evaluated using exlicit recirocity laws. The following result gives the relationshi between the airings (5.1), (5.3) and (5.5). Lemma 5.6. (a) Suose that c H 1 (K, T ), and that u U,. Then (u n, r n (c n )(π n w n)),µ n = α,eπ n(u n, c n ) ζ n. (b) Suose that u U, and c H 1 (K, T ). Then (u n, χ n (r n(c n(π n w n )))),Eπ n = (r n(c n)(π n w n ), χ n (u n )),Eπ n = α,eπ n(u n, c n ) w n. (Recall that the isomorhism χ n is defined in (4.2).) Proof. Part (a) may be roved in exactly the same way as Lemma 5.4, while Part (b) is an immediate consequence of the same lemma. In order to evaluate the airings in Lemma 5.6(b) above using exlicit recirocity laws (and thereby also comute the value of the [ρ(ξ), y ] K, require the following result (cf. Corollary 2.3). Lemma 5.7. There are injective homomorhisms in Proosition 5.2 above), we shall κ : H 1 (K, T ) lim H 1 (K n, Z (1)) (5.7) κ : H 1 (K, T ) lim H 1 (K n, Z (1)), (5.8) where the inverse limits are taken with resect to the obvious corestriction mas. A similar result holds if K is relaced by K or K. Proof. We shall just exlain the construction of κ, since that of κ is very similar. Recall that w = [w n ] denotes a generator of T. Suose that c = [c n] lim H 1 (K, E π n) H 1 (K, T ),

18 18 A. AGBOOLA and consider the comosition of mas H 1 (K n+1,, µ n+1) H 1 (K n,, µ n+1) H 1 (K n,, µ n), (5.9) where the first arrow is given by corestriction and the second arrow is induced by the natural ma µ n+1 µ n. Recall from Corollary 2.3 that there is an isomorhism r n : H 1 (K, E π n) Hom(E π n, K n /K n n ) Gal(Kn/K). (5.10) It is not hard to check that (5.9) mas {r n+1(c n+1)}(π (n+1) w n+1 ) to {r n(c n)}(π n w n ). We may therefore define c (w) := [{r n(c n)}(π n w n )] lim H 1 (K n,, µ n), where the second inverse limit is taken with resect to the mas (5.9). It is a standard fact (see e.g. [15, Aendix B, Section B.3]) that lim H1 (K n,, µ n) lim H 1 (K n,, Z (1)), where the right-hand inverse limit is taken with resect to the obvious corestriction mas. We may therefore view c (w) as being an element of lim H 1 (K n,, Z (1)), and we write κ (c ) = [κ (c ) n ] lim H 1 (K n,, Z (1)) for this element. We remark that it follows from the construction of κ that, for each integer n 1, we have: κ (c ) n {r n(c n)}(π n w n ) (mod K n n, ) (5.11) in H 1 (K n,, µ n) K n,/k n n,. This imlies that κ is injective. Suose now that u U, and that c H 1 (K, T ) with κ (c ) U, lim H 1 (K n,, Z (1)). Lemma 5.6(b) imlies that, for each integer n 1, α,eπ n(u n, c n) may be determined by calculating the value of (κ (c ) n, χ n (u n )),E n; this in turn may be done by using Wiles s exlicit recirocity law alied to the formal grou describe. Ê associated to E, as we shall now For any β = [β n ] U,, let g β,w (Z) O K, [[Z]] denote the Coleman ower series of β; so, for every integer n > 0, we have Set and write δg β,w (Z) := δ w (β) = δg β,w (0) := g β,w (ŵ n ) = β n. (5.12) g β,w (Z) g β,w (Z) λ b E (Z), (5.13) g β,w (0) g β,w (0) λ b E (0) = g β,w (0) g β,w (0), (5.14) (where for the last equality we have used the fact that λ be (Z) = Z + (higher order terms)).

19 BIRCH AND SWINNERTON-DYER CONJECTURE II 19 Remark 5.8. Suose that c H 1 (K, T ) with κ (c ) U, lim H 1 (K n,, Z (1)). In Proosition 6.2, we shall exress δ w (κ (c )) in terms of ex (c ), and in Proosition 8.2(a), we shall show that, for suitably chosen β, we have that δ w (β) is equal to a non-zero multile of L (ψ). This will enable us to deduce that the canonical class s ˇΣ (K, T ) is of infinite order when L (ψ) 0 (see Theorem 8.3). Proosition 5.9. Suose that u U, and that c H 1 (K, T ) with κ (c ) U, lim H1 (K n,, Z (1)). Then, with notation as above, we have ( ψ α,eπ n(u n, c () n) σ Gal(K n,/k ) ) 1 δ w (κ (c )) Ω where Ω O is the -adic eriod described in Section 4. ψ (σ 1 ) log (u σ n) (mod n ), (5.15) Hence we have (u, c ),T = α,t (u, c ) w, where ( ) ψ α,t (u, c () ) = 1 δ w (κ (c )) Ω lim ψ (σ 1 ) log n (u σ n). (5.16) σ Gal(K n,/k ) Proof. Alying Wiles s exlicit recirocity law (see [5, Chater I, Theorem 4.2]) to evaluate (κ (c ) n, χ n (u n )),Eπ n, we see (using Lemma 5.6) that the value of α,eπ n(u n, c n) is given by: ( α,eπ n(u n, c n) Tr K n, /K {π n Tr Kn,/K n, log E, ( χ )} n (u n ))) δg κ (c ),w(ŵ n ) (mod n ). (5.17) Since u n U n, and χ n is Gal(K/K n)-equivariant, it follows that log E, ( χ n (u n )) K n,. Also, we have that π n Tr Kn,/K n, (δg κ (c ),w(ŵ n )) K because ŵ n K n,. Hence (5.17) imlies that α,eπ n(u n, c n) { π n Tr Kn,/K n, (δg κ (c )(ŵ n )) } Tr K n, /K (log E, ( χ n (u n ))) (mod n ). (5.18) From Proosition 4.1(b), we see that log E, ( χ n (u n )) Ω log (u n ) (mod n ), and this imlies that Tr K n, /K (log E, ( χ n (u n )) Ω ψ (σ 1 log (u σ n)) (mod n+1 ). (5.19) σ Gal(K n,/k )

20 20 A. AGBOOLA We also observe that [5, Chater II, Proosition 4.5(iii)] imlies that ( ) π n Tr Kn,/K n, (δg κ (c ),w(ŵ n )) = 1 ψ () δ w (κ (c )), (5.20) which is indeendent of n. The congruence (5.15) now follows from (5.18), (5.19) and (5.20). The equality (5.16) is a direct consequence of (5.15). Suose now that and set ξ Hom(T, U,/E ) Gal(K /K), ξ Hom(T, U, /E ) Gal(K /K), c := ρ(ξ) ˇΣ (K, T ), c := ρ (ξ ) ˇΣ (K, T ). Then the definition of ρ imlies that c = [c n] = [( 1)π n rn 1 (ξn)] lim H 1 (K, E π n). n For any ς E π n, a routine comutation shows that and so setting ς = π n w n, we see that Hence it follows from (5.11) that r n(c n)(ς) = ξ n(( 1)π n ς), r n(c n)(π n w n ) = ξ n(w n ) 1. κ (c ) n ξ n(w n ) 1 (mod K n n, ) (5.21) in H 1 (K n,, µ n) K n,/k n n,. Recall from Section 1 that ϑ is a certain generator of the ideal I of Λ(K ). We now observe that as ξ is Gal(K /K)-equivariant, it follows that ξ (w) ϑ E /IE. The following result describes the relationshi between κ (c ) and ξ (w) ϑ. Proosition With the above notation, we have δ w (κ (c )) = δ w (ξ (w) ϑ ), where δ w (ξ (w) ϑ ) denotes δ w alied to any lift in E of ξ (w) ϑ E /IE. Proof. Suose that ξ (w) = α = [α n ], with α n U n, /E n, and set ξ (w) ϑ = β = [β n ], with β n E n /IE n. It follows from the definition of ξ n (see Proosition 2.6) that ξ n(w n ) = α n E n /E n n = E n /E n n. For each n 1, let ϑ n denote the rojection of ϑ to Z [Gal(K n /K)]. It follows from [12, Lemma 6.3] that ϑ n σ Gal(K n/k) ψ 1 (σ)σ ( 1) n (mod n IZ [Gal(K n /K)]), (5.22)

21 BIRCH AND SWINNERTON-DYER CONJECTURE II 21 where σ denotes any lift of σ Gal(K n /K) to Gal(K /K). Hence we have (using additive notation for Galois action): ϑξn(w n ) = n β n ( 1) 1 ϑ ψ 1 (σ)σ β n (mod n ϑe n ). σ Gal(K n/k) Since E n has no ϑ-torsion (see [5, Chater III, Proosition 1.3]), we can divide this relation by ϑ and aly (5.21) to obtain ( 1)ξn(w n ) ψ 1 (σ)σ β n (mod n E n ) σ Gal(K n/k) κ (c ) n (mod n E n ). It now follows from [5, Chater II, Lemma 4.8] that δ w (κ (c )) = δ w (β), and this imlies the desired result. Let ξ(w ) U, be any lift of ξ(w ). Theorem (a) For any y ˇΣ (K, T ), we have ( ) ψ [ρ(ξ), y () ] K, = ( 1) 1 δ w (κ (y )) Ω lim ψ (σ 1 ) log n ((ξ(w )) σ n). (b) We have σ Gal(K n,/k ) ( ψ [ρ(ξ), ρ (ξ () )] K, = ( 1) lim n σ Gal(K n,/k ) ) 1 δ w (ξ (w) ϑ ) Ω ψ (σ 1 ) log ((ξ(w )) σ n). Proof. This result follows directly from Proositions 5.2, 5.9 and 5.10.

22 22 A. AGBOOLA 6. The Dual Exonential Ma Suose that c H 1 (K, T ) with κ (c ) U, lim H 1 (K n,, Z (1)), and let ex : H 1 (K, T ) Q denote the Bloch-Kato dual exonential ma (see e.g. [16, Section 5] for an account of the dual exonential ma associated to an ellitic curve). In this section we shall evaluate ex (c ) in terms of the δ w (κ (c )) (cf. Remark 5.8 above and Proosition 6.2 below). We begin by describing the relationshi between the cu roduct airing : H 1 (K, T ) H 1 (K, T ) Z (6.1) and the Kummer airing (, ),T of the revious section. This result is robably well known, but we are not aware of a written reference, and so we include a roof here. Proosition 6.1. Suose that c = [c n ] H 1 (K, T ) and c = [c n] H 1 (K, T ). Then i.e. Proof. Let (κ (c ), c),t = (c c ) w, α,t (κ (c ), c) = [α,eπ n(κ (c ) n, c n )] = (c c ). : H 1 (K, E π n) H 1 (K, E π n) Z/ n Z denote the cu roduct airing at level n afforded by (6.1). To rove the desired result, it suffices to show that, for each integer n 1, we have (r n(c n)(π n w n ), c n ),Eπ n = (c n c n) w n. In order to do this, we first recall (see e.g. [17, Chater XIV]) that the Hilbert airing (, ),µ n may be identified with a cu roduct airing so that : H 1 (N n,, µ n) H 1 (N n,, µ n) Z/ n Z (a, b),µ n = (a b) ζ n. We also note that the Gal(K/N n )-equivariant isomorhisms E π n E π n µ n; w n e n (w n, π n w n), µ n; w n e n (π n w n, w n)

23 BIRCH AND SWINNERTON-DYER CONJECTURE II 23 induce isomorhisms of cohomology grous κ n : H 1 (N n,, E π n) H 1 (N n,, µ n); y n r n (y n )(π n wn); κ n : H 1 (N n,, E π n) H 1 (N n,, µ n); yn rn(y n)(π n w n ), and via functoriality of cu roduct airings, we have c n c n = κ n (c n ) κ n(c n) = (κ n(c n) κ n (c n )). Hence, from Lemma 5.6(b), we see that α,eπ n(rn(c n)(π n w n ), c n ) = α,µ n(rn(c n)(π n w n ), r n (c n )(π n wn)) = κ n(c n ) κ n (c n ) = (κ n (c n ) κ n(c n)) = (c n c n), as required. We can now state the main result of this section. Proosition 6.2. Suose that c H 1 (K, T ) with κ (c ) U, lim H 1 (K n,, Z (1)). Then ( ) ψ ex (c () ) = 1 δ w (κ (c )). Proof. It follows from the definition of the dual exonential ma (see [16, Section 5]) that x c = log E, (x) ex (c ) for all x H 1 (K, T ). (Here we have extended the logarithm ma log E, from E(K ) Z to H 1 (K, T ) via linearity.) From Wiles s exlicit recirocity law (see [5, Chater IV, 2.5] for the version we require), we have (κ (c ), x),t = The result now follows from Proosition 6.1. { ( ) } log E, (x) 1 ψ () δ w (κ (c )) w. 7. Ellitic units and canonical elements We shall now exlain how ellitic units may be used, following the methods described in [12], to construct canonical elements s ˇΣ (K, T ), s ˇΣ (K, T )

24 24 A. AGBOOLA when r = 0. These are the analogues in the resent situation of the elements x (1) and x (1) ˇ Sel(K, T ) constructed in [12, 4] when r = 1. ˇ Sel(K, T ) Let C E and C E denote the norm-coherent systems of ellitic units constructed in [12, 3], and write C and C for the closures of C in E and C in E resectively. Recall that f O K denotes the conductor of the Grossencharacter associated to E, and fix B E f / Gal(K/K) of exact order f as described in [12, 6]. For each generator w = [w n ] of T and w = [w n] of T, let θ B (w) = [θ B (w n )] C U, Q θ B (w ) = [θ B (w n)] C U, Q denote the norm-coherent sequences of ellitic units constructed in [12, 3]. In articular, we have that θ B (w) σ = θ B (w σ ), θ B (w ) σ = θ B (w σ ) (7.1) for all σ Gal(K/K), and for each n > 0, we have that C n = Z [Gal(K n /K)] θ B (w n ), C n = Z [Gal(K n/k)] θ B (w n) (7.2) (see [13, Proositions 3.1 and 3.2]). The following result is due to Rubin (see [13, Theorem 7.2]) and it rovides a crucial link between ellitic units and -adic L-functions. It is roved via a careful analysis of the construction of the Katz two-variable -adic L-function L using -adic measures arising from ellitic units as described in [5, Chater II, 4]. We refer the reader to [13, 7] for the details of the roof. Theorem 7.1. Suose that w is any generator of T. Then there are natural Galoisequivariant injections ι w : U, Λ(K ) O, ι : U, Λ(K ) O, (where ι w deends uon the choice of w, but ι does not), satisfying the following roerties. Suose that w is a generator of T fixed as in Section 4. Then (a) We have that (1 Fr )ι w (θ B (w)) = L K, ι (θ B ( N(f) 1 w )) = L K, (7.3) where Fr denotes the Frobenius of in Gal(K /K), and L K and L K denote the images of L under the natural quotient mas Λ(K ) O Λ(K ) O and Λ(K ) O Λ(K ) O resectively.

25 (b) For every β U,, we have BIRCH AND SWINNERTON-DYER CONJECTURE II 25 ι w (β) = ( ) 1 ψ () Ω δ w (β). (7.4) (c) For every u = [u n ] U,, we have ( ) ι (u)(ψ ) = 1 ψ () lim n τ Gal(K n/k) Now suose that r = 0. Then L (1) 0, and so we see from (7.3) that C E U, Q and C I(U, Q). ψ 1 (τ) log (u τ n). (7.5) In articular, we have that C IE U, Q. Similar remarks imly that also C I E U, Q. Alying Remark 1.1, we deduce that C I E U, Q. (7.6) Also, if L (1) 0, then this imlies that L (1) = 0, and so again from (7.3), it follows that we have C I (U, Q). (7.7) Recall from Section 1 that ϑ is a certain generator of the ideal I of Λ(K ). Proosition 7.2. There exists a unique homomorhism σ Hom(T, (U, Q)/E ) Gal(K /K) such that in E /I E. σ (w ) ϑ = θ B (w ) Proof. The roof of this result is very similar to that of [12, Theorem 4.2]. We first observe that it follows from (7.1) and (7.2) that Gal(K /K) acts on T and C /I C via the same character ψ. Since Gal(K /K) also acts transitively on T T, it follows that the ma w θ B (w ) induces a well-defined homomorhism σ (0) Hom(T, C /I C ) Gal(K /K). Next, we note that it follows from [5, Chater III, Proosition 1.3] that U, contains no ϑ -torsion elements, and so, if α C, then ϑ 1 α is a well-defined element of U, Q. It is not hard to show that the image of ϑ 1 α in (U, Q)/E deends only uon the image of α in C /I C. The ma σ is defined by σ := ϑ 1 σ (0).

26 26 A. AGBOOLA Definition 7.3. We set s := ρ(σ ) ˇΣ (K, T ), s := ρ (σ ) ˇΣ (K, T ), where of course the definition σ Hom(T, (U, Q)/E ) Gal(K /K) is the same, mutatis mutandis, as that of σ. Remark 7.4. We shall see from the roof of Theorem 8.4(a) below that s order only if L (ψ ) 0. is of infinite s 8. Proof of Theorem A In this section we shall establish a number of roerties of the cohomology classes s and constructed in Section 7, and thereby rove Theorem A of the Introduction. We begin by recalling some basic facts concerning derivatives of elements in Iwasawa algebras (see [13, Section 7] or [1, Section 2], for examle). Suose that F Λ(K ) O, and consider the function s F (ψ ψ s ), s Z ; this is a -adic analytic function on Z. For any integer m 0, define D (m) F (ψ ) = 1 d m! ds F (ψ ψ s ) ; s=0 then, by definition, we have It is easy to check that L (ψ ) = D (1) L (ψ ). D (m) ϑ m (ψ ) = m for every integer m 1. The following result is [13, Lemma 7.3]. It is roved via a routine comutation which we shall omit. Lemma 8.1. With notation as above, we have D (m) (ϑ m F )(ψ ) = m F (ψ ) for every integer m 1. We now turn to the recise relationshi between the canonical elements s and s, and certain secial values of the -adic L-function L.

27 BIRCH AND SWINNERTON-DYER CONJECTURE II 27 Proosition 8.2. Let σ (w ) U, be any lift of σ (w ). We have ( L (ψ) = 1 1 ) ( 1 1 ) Ω ψ( δ w (θ B (w)). (8.1) ) ψ() and L (ψ ) = N(f) 1 ( ) 1 ψ () lim n τ Gal(K n/k) Proof. The equality (8.1) follows directly from (7.3) and (7.4). ψ 1 (τ) log (σ (w ) τ n ). (8.2) To rove (8.2), we first observe that, since there is no ϑ -torsion in U, (see [5, Chater III, Proosition 1.3]), there exists a unique β = [β n ] U, Z Q such that Alying Theorem 7.1(a), we see that β ϑ = θ B (w ). ϑ N(f) 1 ι (β) = ι (θ B (N(f) 1 w ) = L K. It therefore follows from Lemma 8.1 and Theorem 7.1(c) that L (ψ ) = N(f) 1 ι (β)(ψ ) ( ) = N(f) 1 1 ψ () lim n τ Gal(K n/k) ψ 1 (τ) log (βn) τ. (8.3) Now suose that σ (w ) U, is any lift of σ (w ). It follows from Proosition 7.2 that we have σ (w ) ϑ = θ B (w ) = β ϑ (8.4) in E /I E. For each integer n 1, let ϑ n denote the rojection of ϑ to Z [Gal(cK n/k)]. It is shown in [13, Lemma 6.3] that ψ 1 (τ)τ ( 1) n ϑ n τ Gal(K n/k) (mod n IZ [Gal(K n/k)]). We therefore deduce from (8.4) that, for each n 1, we have (writing Galois action additively) ϑ ψ 1 (τ)τ β n ϑ ψ 1 (τ)τ σ (w ) τ Gal(Kn /K) τ Gal(Kn /K) (mod n ϑ E n).

28 28 A. AGBOOLA Since E n has no ϑ -torsion (see [13, Chater III, Proosition 1.3]), we can divide both sides of this last relation by ϑ, and then take -adic logarithms and let n to deduce that lim ψ 1 (τ) log n (βn) τ = lim ψ 1 (τ) log n (σ (w ) τ n ). (8.5) τ Gal(K n/k) τ Gal(K n/k) The desired equality (8.2) now follows from (8.3) and (8.5). Theorem 8.3. We have ( ψ ex (s ) () = ( ψ () = Hence, if L (ψ) 0, then s ) 1 δ w (θ B (w)) ) ( ) 1 ( 1 1 ) 1 L(ψ). ψ( ) ψ() Ω is of infinite order. Proof. This follows from Proositions 5.10, 6.2 and 8.2. We can now rove the first art of Theorem A. Theorem 8.4. Suose that L (ψ) 0. (a) We have L (ψ ) 0. (b) The height airing [, ] K, is non-degenerate, and we have ( [s, s ] K, = ( 1) 1 N(f) ψ( ) ) 1 ( 1 1 ψ() ) 1 L (ψ) L (ψ ). Proof. (a) Suose that L (ψ ) = 0, and set t = ord s=1 L (s), so t 2. Then it follows from (7.3) that C I t (U, Q). Hence we have that σ (w ) I (t 1) (U, Q) E, and this in turn imlies that s = 0. This contradicts Theorem 8.3 because by hyothesis L (ψ) 0. It therefore follows that L (ψ ) 0, as claimed. (b) The equality follows directly from Theorem 5.11(b), Proosition 8.2, and Theorem 8.3. As [s, s ] K, 0, we deduce that [, ] K, is non-degenerate because ˇΣ (K, T ) and ˇΣ (K, T ) are both free, rank one Z -modules (see [1, Lemma 3.6 and Proosition 6.7]). The second art of Theorem A is a direct consequence of the following result. Theorem 8.5. (cf. [12, Theorem 10.1])

29 BIRCH AND SWINNERTON-DYER CONJECTURE II 29 Suose that L (ψ) 0, and that y ˇΣ (K, T ) and y ˇΣ (K, T ) are both non-zero. Then ex (y ) ex (y) [y, y ] K, Proof. Since L (ψ) 0, it follows that ( ψ = ( 1) 1 () N(f) ) ( ) ψ( ) 1 1 L (ψ) (1 ψ 1 ()) (1 ψ 1 ( )) Ω Ω L (ψ ). Hom(ˇΣ (K, T ) Z ˇΣ (K, T ), Z ) is a free Z -module of rank one (see [1, Lemma 3.6 and Proosition 6.7]), and that both ex ( ) ex ( ) and [, ] K, are of infinite order, we have that are non-zero elements of this module. Hence, since s and s ex (y) ex (y ) [y, y ] K, = ex (s ) ex (s ). [s, s ] K, Alying Theorems 8.3 and 8.4, we see that ex (s ) ex ( (s ) ψ = ( 1) 1 () N(f) [s, s ] K, ) ( ) ψ( ) 1 1 L (ψ ) (1 ψ 1 ()) (1 ψ 1 ( )) Ω Ω L (ψ ). The desired result now follows from the fact that L (ψ) = L (ψ ) (cf. Remark 1.2). This comletes the roof of Theorem A.

30 30 A. AGBOOLA References [1] A. Agboola, On Rubin s variant of the -adic Birch and Swinnerton-Dyer conjecture, Com. Math. 143 (2007), [2] A. Agboola, On certain secial values of the Katz two-variable -adic L-function, in rearation. [3] D. Bernardi, C. Goldstein, N. Stehens, Notes -adiques sur les courbes ellitiques, Crelle 351 (1985), [4] A. Brumer, On the units of algebraic number fields, Mathematika 14 (1967), [5] E. de Shalit, Iwasawa theory of ellitic curves with comlex multilication, Academic Press (1987). [6] R. Greenberg, Trivial zeros of -adic L-functions, In: -adic monodromy and the Birch and Swinnerton- Dyer conjecture (Boston, MA, 1991), , Contem. Math., 165, Amer. Math. Soc., Providence, RI, [7] B. Mazur, J. Tate, J. Teitelbaum, On -adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), [8] B. Perrin-Riou, Déscent infinie et hauteurs -adiques sur les courbes ellitiques à multilication comlexe, Invent. Math. 70 (1983), [9] B. Perrin-Riou, Théorie d Iwasawa et hauteurs -adiques, Invent. Math. 109 (1992), [10] B. Perrin-Riou, Arithmétique des courbes ellitiques et théorie d Iwasawa, Memoire de la Société Mathématique de France, 17 (1984). [11] K. Rubin, Tate-Shafarevich grous and L-functions of ellitic curves with comlex multilication, Invent. Math. 89 (1987), [12] K. Rubin, -adic L-functions and rational oints on ellitic curves with comlex multilication, Invent. Math. 107 (1992), [13] K. Rubin, -adic variants of the Birch and Swinnerton-Dyer conjecture for ellitic curves with comlex multilication. In: -adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), 71 80, Contem. Math., 165, Amer. Math. Soc., Providence, RI, [14] K. Rubin, The main conjectures of Iwasawa theory for imaginary quadratic fields, Invent. Math. 109 (1992), [15] K. Rubin, Euler Systems, Princeton University Press (2000). [16] K. Rubin, Euler systems and modular ellitic curves. In: Galois reresentations in arithmetic geometry, A.J. Scholl, R. L. Taylor (Eds.), CUP [17] J.-P. Serre, Local Fields, Sringer Verlag (1979). Deartment of Mathematics, University of California, Santa Barbara, CA address: agboola@math.ucsb.edu