On derivatives of Kato's Euler system for elliptic curves
aa r X i v : . [ m a t h . N T ] A p r ON DERIVATIVES OF KATO’S EULER SYSTEMFOR ELLIPTIC CURVES
DAVID BURNS, MASATO KURIHARA AND TAKAMICHI SANO
Abstract.
In this paper we study a new conjecture concerning Kato’s Euler system ofzeta elements for elliptic curves E over Q . This conjecture, which we refer to as the‘Generalized Perrin-Riou Conjecture’, predicts a precise congruence relation between a‘Darmon-type derivative’ of the zeta element of E over an arbitrary real abelian field andthe critical value of an appropriate higher derivative of the L -function of E over Q . Weprove that the conjecture specializes in the relevant case of analytic rank one to recoverPerrin-Riou’s conjecture on the logarithm of Kato’s zeta element. Under mild hypotheseswe also prove that the ‘order of vanishing’ part of the conjecture is valid in arbitraryrank. An Iwasawa-theoretic analysis of our approach leads to the formulation and proofof a natural higher rank generalization of Rubin’s formula concerning derivatives of p -adic L -functions. In addition, we establish a concrete and apparently new connection betweenthe p -part of the classical Birch and Swinnerton-Dyer Formula and the Iwasawa MainConjecture in arbitrary rank and for arbitrary reduction at p . In a forthcoming paperwe will show that the Generalized Perrin-Riou Conjecture implies (in arbitrary rank) theconjecture of Mazur and Tate concerning congruences for modular elements and, by usingthis approach, we are able to give a proof, under certain mild and natural hypotheses, thatthe Mazur-Tate Conjecture is valid in analytic rank one. Contents
1. Introduction 21.1. Background 21.2. Conjectures and results at finite level 31.3. Iwasawa-theoretic considerations 61.4. General notation 92. Formulation of the Generalized Perrin-Riou Conjecture 102.1. Kato’s Euler system 102.2. Birch and Swinnerton-Dyer elements 122.3. Bockstein regulator maps 152.4. The Generalized Perrin-Riou Conjecture 162.5. An algebraic analogue 183. Fitting ideals and order of vanishing 193.1. A ‘main conjecture’ at finite level 193.2. The proof of Theorem 1.3 194. Derivatives of Kato’s Euler system 214.1. Darmon derivatives 214.2. Iwasawa-Darmon derivatives 24 p -adic height pairings and the Bockstein regulator 285.1. Review of p -adic height pairings 285.2. A comparison result 315.3. Schneider’s height pairing 366. The Generalized Rubin Formula and consequences 376.1. Review of the p -adic L -function 376.2. The Generalized Rubin Formula 396.3. Review of the Coleman map 416.4. The proof of Theorem 6.2 427. The Iwasawa Main Conjecture and descent theory 507.1. Review of the Iwasawa Main Conjecture 507.2. Consequences of the Iwasawa Main Conjecture 517.3. The descent argument 527.4. The proof of Theorem 7.8 55References 581. Introduction
Background.
A central problem in modern number theory is to understand the arith-metic meaning of the values of zeta and L -functions.The Birch and Swinnerton-Dyer Conjecture and main conjecture in Iwasawa theory areimportant instances of this problem, being respectively related to the Hasse-Weil L -functionof an elliptic curve and to the p -adic L -function of an appropriate motive.For an elliptic curve E defined over Q , significant progress on the problem was madeby Kato in [23] who used Beilinson elements in the K -theory of modular curves to definecanonical ‘zeta elements’ in ´etale (Galois) cohomology groups that could be explicitly relatedto the values of Hasse-Weil L -functions.To be a little more precise we fix an odd prime p , a finite abelian extension F of Q , a finiteset of places S of Q that contains the archimedean place, p , all primes that ramify in F andall primes at which E has bad reduction. We write O F,S for the subring of F comprisingelements that are integral at all non-archimedean places whose residue characteristic doesnot belong to S and T p ( E ) for the p -adic Tate module of E .Then the zeta element z F constructed by Kato belongs to the ´etale cohomology group H ( O F,S , T p ( E )) and is explicitly related via the dual exponential map to the value at oneof the Hasse-Weil L -function of E (for more precise statements see § F varies over finite subfields of the cyclotomic Z p -extension of Q , these elements z F form a projective system that can be used to recover the p -adic L -function of E .In addition, as F varies more generally, the elements z F form an Euler system and socan be used to bound the p -adic Selmer group of E .In this way zeta elements have led to partial results on both the main conjecture andBirch and Swinnerton-Dyer Conjecture for E . For this reason, such elements have subsequently been much studied in the literature andhave led to numerous important results.Our main purpose in these articles is to investigate a conjectural property of Kato’selements that it seems has not been observed previously and to demonstrate that thisproperty, whenever valid, has significant applications.The conjecture itself predicts a precise link between a ‘Darmon-type’ derivative of z F forany given F and the value at the critical point of an appropriate higher derivative of the L -function of E over Q .This conjectural link constitutes a simultaneous refinement of well-known conjectures ofPerrin-Riou [34] and of Mazur and Tate [29] and will be described in more detail in thenext section.Although we shall not pursue it here, it seems reasonable to expect that the generalapproach we develop can also be applied to elliptic curves with complex multiplication,with the role of Kato’s zeta elements being replaced by elliptic units twisted by a Heckecharacter.We also expect that it should be possible to extend our approach to the setting of abelianvarieties and to modular forms and their families, and we hope to return to these questionsin a subsequent article.1.2. Conjectures and results at finite level.
We shall now give an overview of thecentral conjecture that we formulate and the evidence for it that we have so far obtained.1.2.1. At the outset we fix a finite real abelian extension F of Q and set G := Gal( F/ Q ).Then, following a general idea introduced by Darmon in [16], the key object of our studywill be the element N F/ Q ( z F ) := X σ ∈ G σ ( z F ) ⊗ σ − of H ( O F,S , T p ( E )) ⊗ Z p Z p [ G ].We write r for the rank of E ( Q ) and assume that r >
0, that E ( Q ) has no element oforder p and that the p -part of the Tate-Shafarevich group of E/ Q is finite.Then, under these hypotheses, in Definition 2.4 we shall use the leading term at s = 1 of L ( E, s ) to (unconditionally) define a canonical ‘Birch and Swinnerton-Dyer element’ η BSD in the dimension one vector space over C p that is spanned by V r Z p H ( Z S , T p ( E )).With I denoting the augmentation ideal of Z p [ G ], we shall also define (in § F : ^ r Z p H ( Z S , T p ( E )) −→ H ( Z S , T p ( E )) ⊗ Z p I r − /I r . Finally we note the Z p -module H ( O F,S , T p ( E )) is free and so H ( O F,S , T p ( E )) ⊗ Z p I r − identifies with a submodule of H ( O F,S , T p ( E )) ⊗ Z p Z p [ G ].Then, in terms of this notation, the central conjecture of this article can be stated asfollows. Conjecture 1.1 (The Generalized Perrin-Riou Conjecture) . (i) (‘Order of vanishing’) N F/ Q ( z F ) belongs to H ( O F,S , T p ( E )) ⊗ Z p I r − . (ii) (‘Integrality’) If r > , then η BSD belongs to V r Z p H ( Z S , T p ( E )) . DAVID BURNS, MASATO KURIHARA AND TAKAMICHI SANO (iii) (‘Leading term formula’) The image of N F/ Q ( z F ) in H ( O F,S , T p ( E )) ⊗ Z p I r − /I r is equal to Boc F ( η BSD ) . Remark 1.2.
A precise statement of Conjecture 1.1 will be given as Conjecture 2.12. Forthe moment, we note a key advantage of its formulation is that it uses a construction ofregulators that works in the same way for all reduction types. A further crucial advantageis that, in the case r = 1, the conjecture takes a particularly simple form and can be provedunder various natural hypotheses.In the rest of this section we outline the evidence that we have obtained for the aboveconjecture and also explain why it constitutes a simultaneous refinement and generalizationof conjectures of Perrin-Riou and of Mazur and Tate.1.2.2. We observe first that the containment predicted by Conjecture 1.1(i) can be studiedby using the equivariant theory of Euler systems that was recently described by Sakamotoand the first and third authors in [10].In particular, by using this approach we are able to prove that Conjecture 1.1(i) is validunder certain mild hypotheses.For example, the following concrete result will follow directly from stronger results thatwe prove in §
3. This result is a natural analogue for zeta elements of the main result ofDarmon [16, Th. 2.4] concerning Heegner points.
Theorem 1.3.
The containment of Conjecture 1.1(i) is valid if all of the following condi-tions are satisfied. (a) p > ; (b) the p -primary parts of X ( E/F ) and X ( E/ Q ) are finite; (c) the image of the representation G Q → Aut( T p ( E )) ≃ GL ( Z p ) contains SL ( Z p ) ; (d) for every prime number ℓ in S the group E ( Q ℓ ) contains no element of order p . Concerning Conjecture 1.1(ii), we can show in all cases that the predicted containmentis valid whenever the p -part of the Birch and Swinnerton-Dyer Formula for E over Q , or‘BSD p ( E )’ as we shall abbreviate it in the sequel, is valid. (In fact, a stronger version ofthis result will be proved in Proposition 2.6).Finally, to discuss the prediction of Conjecture 1.1(iii) we shall initially specialize to thecase that the analytic rank ord s =1 L ( E, s ) of E is equal to one.In this case, well-known results of Gross and Zagier and of Kolyvagin (amongst others)imply that r = 1 and so parts (i) and (ii) of Conjecture 1.1 are valid trivially.It is also straightforward to check in this case that the equality in Conjecture 1.1(iii) isvalid for any, and therefore every, choice of field F if and only if one has z Q = η BSD .By analysing the latter equality, we shall thereby obtain the explicit interpretation ofthis case of Conjecture 1.1 that is given in the next result. (A proof of this result will beexplained in Remark 2.13(ii)).In the sequel we write L S ( E, s ) for the S -truncated Hasse-Weil L -function of E . Theorem 1.4. If E has analytic rank one, then Conjecture 1.1 is valid for any, and there-fore every, field F if and only if one has z Q ∈ H f ( Q , T p ( E )) and log ω ( z Q ) = L ′ S ( E, + · h x, x i ∞ log ω ( x ) . Here log ω : H f ( Q , T p ( E )) → Q p is the formal logarithm associated to the (fixed) N´erondifferential ω , L ′ S ( E, denotes the value at s = 1 of the first derivative of L S ( E, s ) , Ω + isthe real N´eron period, x is a generator of E ( Q ) modulo torsion and h− , −i ∞ is the N´eron-Tate height pairing. The displayed equality in Theorem 1.4 is equivalent to the central conjecture formulatedby Perrin-Riou in [34, § E has good supersingular reduction at p and square-free conductor and in [43,Th. A] Venerucci has proved (a weak version of) the conjecture in the split multiplicativecase. In addition, analogous results in the good ordinary case are obtained by B¨uy¨ukboduk,Pollack, and Sasaki in [14] (Bertolini and Darmon have also announced a proof in [4]).To discuss Conjecture 1.1 in the case of arbitrary rank, we assume that BSD p ( E ), andhence also the containment of Conjecture 1.1(ii), is valid.In this case we will show in the second part of this article [9] that, under mild addi-tional hypotheses, the equality of Conjecture 1.1(iii) implies a refined, and in certain keyrespects better-behaved, version of the celebrated conjecture formulated by Mazur and Tatein [29] concerning congruence relations between modular symbols and the discriminants ofalgebraic height pairings that are defined in terms of the geometrical theory of bi-extensions.In particular, since there are by now many curves E over Q of analytic rank one andprimes p for which BSD p ( E ) is known to be valid (by recent work of Jetchev, Skinner andWan [21] and Castella [15]), we can thereby deduce the validity of the Mazur-Tate Con-jecture in this case from Theorems 1.3 and 1.4 and the results on Perrin-Riou’s conjecturethat are recalled above.In this way we shall obtain the first verifications of the Mazur-Tate Conjecture for anycurves E for which L ( E,
1) vanishes.This deduction gives a clear indication of the interest of, and new insight that can beobtained from, the general approach that underlies the formulation of Conjecture 1.1.In this regard, we also observe that one of the key motivations behind the developmentof this approach was an attempt to understand if there was a natural analogue for ellipticcurves of the conjecture formulated in [6, Conj. 5.4] in the setting of the multiplicativegroup.We finally recall that the latter conjecture was itself formulated as a natural strengtheningof the ‘refined class number formula for G m ’ that was previously conjectured by the thirdauthor [38], and (independently) by Mazur and Rubin [28]. DAVID BURNS, MASATO KURIHARA AND TAKAMICHI SANO
In the next section we shall focus on the conjecture in the important special case that F is contained in the cyclotomic Z p -extension of Q .1.3. Iwasawa-theoretic considerations.
We shall also show that the simultaneous studyof Conjecture 1.1 for the family of intermediate fields F of the cyclotomic Z p -extension Q ∞ of Q gives a more rigid framework that sheds light on a range of important problems.1.3.1. To explain this, for each natural number n we write Q n for the unique subfield of Q ∞ of degree p n over Q .We know the validity of Conjecture 1.1(i) with F = Q n (see Proposition 4.5), and wewrite κ n for the image of N Q n / Q ( z Q n ) under the natural projection H ( O Q n ,S , T p ( E )) ⊗ Z p I r − n → H ( O Q n ,S , T p ( E )) ⊗ Z p I r − n /I rn , where I n denotes the augmentation ideal of Z p [Gal( Q n / Q )].Then we can show the element κ n belongs to the subgroup H ( Z S , T p ( E )) ⊗ Z p I r − n /I rn of H ( O Q n ,S , T p ( E )) ⊗ Z p I r − n /I rn and, moreover, that as n varies the elements κ n are com-patible with the natural projection maps H ( Z S , T p ( E )) ⊗ Z p I r − n /I rn → H ( Z S , T p ( E )) ⊗ Z p I r − n − /I rn − . Hence, writing I for the augmentation ideal of Z p [[Gal( Q ∞ / Q )]], one obtains an elementof H ( Z S , T p ( E )) ⊗ Z p I r − /I r by setting κ ∞ := lim ←− n κ n ∈ lim ←− n H ( Z S , T p ( E )) ⊗ Z p I r − n /I rn ≃ H ( Z S , T p ( E )) ⊗ Z p I r − /I r (cf. Definition 4.6).In addition, the family of maps (Boc Q n ) n induces a canonical homomorphism C p · ^ r Z p H ( Z S , T p ( E )) → C p · H ( Z S , T p ( E )) ⊗ Z p I r − /I r and the fact that the Z p -module I r − /I r is torsion-free implies that the natural map H ( Z S , T p ( E )) ⊗ Z p I r − /I r → C p · H ( Z S , T p ( E )) ⊗ Z p I r − /I r is injective. In particular, this allows one to formulate Conjecture 1.1(iii) for the family ofelements N Q n / Q ( z Q n ) without having to assume the validity of Conjecture 1.1(ii).We shall show (in Proposition 4.15) that this version of Conjecture 1.1(iii) is equivalentto the following prediction.In the sequel we write L ( r ) S ( E,
1) for the coefficient of ( s − r in the Taylor expansion at s = 1 of L S ( E, s ). Conjecture 1.5 (Conjecture 4.9) . If r is also equal to the analytic rank ord s =1 L ( E, s ) of E , then one has κ ∞ = L ( r ) S ( E, + · R ∞ · R Boc ω , where Ω + is the real N´eron period, R ∞ is the N´eron-Tate regulator and R Boc ω is the ‘Bock-stein regulator’ in H ( Z S , T p ( E )) ⊗ Z p I r − /I r that is introduced in Definition 4.11. Remark 1.6. If r is equal to ord s =1 L ( E, s ), then the r -th derivative of L S ( E, s ) is holo-morphic at s = 1 and its (non-zero) value at s = 1 is equal to r ! · L ( r ) S ( E, Remark 1.7.
We will show that the Bockstein regulator that occurs in Conjecture 1.5 hasthe following properties.(i) If r = 1, then R Boc ω = log ω ( x ) · x for any element x of E ( Q ) that generates E ( Q ) modulo torsion (cf. Remark 4.13).(ii) Suppose that E does not have additive reduction at p and write h− , −i p for the classical p -adic height pairing. Then for any element x of E ( Q ) one has h x, R Boc ω i p = log ω ( x ) · R p , where R p denotes the p -adic regulator (cf. Theorems 5.6 and 5.11).If r = 1, then κ ∞ simply coincides with z Q and so Remark 1.7(i) implies that Conjecture1.5 is valid if and only if one has z Q = L ′ S ( E, + · R ∞ log ω ( x ) · x for any element x of E ( Q ) that generates E ( Q ) modulo torsion. This equality is equivalentto Perrin-Riou’s conjecture.In addition, whilst Remark 1.7(ii) implies that the Bockstein regulator R Boc ω is a variantof the classical p -adic regulator, a key role will be played in our approach by the fact that R Boc ω can be defined even in the case that E has additive reduction at p (in which case aconstruction of the p -adic regulator is still unknown).1.3.2. To interpret Conjecture 1.5 in terms of p -adic L -functions, we must first prove a‘Generalized Rubin Formula’ for the element κ ∞ .To discuss this result, and some of its consequences, we assume until further notice that E does not have additive reduction at p .If E has good reduction at p , then we write α for an allowable root of the Hecke polynomial X − a p X + p . We set β := p/α .If E has non-split multiplicative reduction at p , then we set α := − β := − p .We also write L ( r ) S,p for the ‘ r -th derivative’ of the S -truncated p -adic L -function L S,p of E (for a precise definition of this term see § Theorem 1.8 (The Generalized Rubin Formula, Theorem 6.2) . (i) If E has good or non-split multiplicative reduction at p , then for every element x of E ( Q ) one has h x, κ ∞ i p = (cid:18) − α (cid:19) − (cid:18) − β (cid:19) log ω ( x ) · L ( r ) S,p . (ii) If E has split multiplicative reduction at p , then for every element x of E ( Q ) onehas h x, κ ∞ i p · L = (cid:18) − p (cid:19) log ω ( x ) · L ( r +1) S,p , where L denotes the ‘ L -invariant’ of E (see Remark 6.4). DAVID BURNS, MASATO KURIHARA AND TAKAMICHI SANO
Remark 1.9. If r = 1, then one has κ ∞ = z Q and Theorem 1.8(i) recovers the formulathat is proved by Rubin in [36, Th. 1(ii)] in the case that E has good ordinary reductionat p .We shall then show that this result has the following consequences. Corollary 1.10 (Corollary 6.7) . The Generalized Perrin-Riou Conjecture of Conjecture1.5 implies the following ‘ p -adic Beilinson Formula’: one has (cid:18) − α (cid:19) − (cid:18) − β (cid:19) L ( r ) S,p = L ( r ) S ( E, + · R ∞ R p if E has good or non-split multiplicative reduction at p , and L ( r +1) S,p = L · L ( r ) S \{ p } ( E, + · R ∞ R p if E has split multiplicative reduction at p . In the next result we refer to the Iwasawa Main Conjecture for E and Q ∞ / Q that isformulated in Conjecture 7.1. Corollary 1.11 (Corollary 7.4) . Assume X ( E/ Q ) is finite. Then the Iwasawa Main Con-jecture for E and Q ∞ / Q implies the validity up to multiplication by elements of Z × p of the p -adic Birch and Swinnerton-Dyer Formula for E . Remark 1.12. If E has good reduction at p and its p -adic height pairing is non-degenerate,then the result of Corollary 1.11 was first proved by Perrin-Riou in [34, Prop. 3.4.6].1.3.3. Our general approach also allows a systematic analysis of descent arguments inIwasawa theory without having to make any restrictive hypotheses on the reduction typeof E at p .In particular, in this way we are able to prove the following result even in the case that E has additive reduction at p . Theorem 1.13 (Theorem 7.6) . Assume all of the following hypotheses: • X ( E/ Q ) is finite; • the analytic rank of E is equal to the rank r of E ( Q ) ; • the Iwasawa Main Conjecture of Conjecture 7.1 is valid; • the Generalized Perrin-Riou Conjecture of Conjecture 1.5 is valid; • the Bockstein regulator R Boc ω does not vanish.Then there exists an element u of Z × p such that L ( r ) ( E, + · R ∞ = u · X ( E/ Q ) · Tam( E ) E ( Q ) , where Tam( E ) denotes the product of the Tamagawa factors of E/ Q .In particular, the conjecture BSD p ( E ) is valid. As far as we are aware, this is the first result in which a concrete connection betweenthe p -part of the (classical) Birch and Swinnerton-Dyer Formula and the Iwasawa MainConjecture has been established in the context of either arbitrary analytic rank or arbitraryreduction at p .In this regard we also note that Theorem 1.13 is a natural analogue of the main result ofthe present authors in [7] in which a strategy for proving the equivariant Tamagawa NumberConjecture for G m is established (see Remark 7.7).1.4. General notation.
For the reader’s convenience we collect together some of the gen-eral notation that will be used throughout this article.At the outset we fix an odd prime number p . The symbol ℓ will also usually denote aprime number.For a field K , the absolute Galois group of K is denoted by G K .We fix an algebraic closure Q of Q . We also fix an algebraic closure Q p of Q p and fix anembedding Q ֒ → Q p .For a positive integer m , we denote by µ m ⊂ Q the group of m -th roots of unity.For an abelian group X , we use the following notations: • X tors : the subgroup of torsion elements; • X tf := X/X tors : the torsion-free quotient; • rank( X ) := rank Z ( X tf ); • X [ p ]: the subgroup of elements annihilated by p ; • X [ p ∞ ]: the subgroup of elements annihilated by a power of p .If X is endowed with an action of complex conjugation, we denote by X + the subgroupof X fixed by the action.If X is an R -module (with R a commutative ring), we set X ∗ := Hom R ( X, R ) . Note that this notation has ambiguity, since X may be regarded as an R ′ -module withanother ring R ′ and X ∗ can mean Hom R ′ ( X, R ′ ). However, this ambiguity would not makeany danger of confusion since the meaning is usually clear from the context.For an element x ∈ X , we denote by h x i R the submodule generated by x over R . Weabbreviate it to h x i when R is clear from the context.Suppose that X is a free R -module with basis { x , . . . , x r } . We denote by x ∗ i : X → R the dual of x i , i.e., the map defined by x j ( i = j, i = j. For a perfect complex C of R -modules, we denote by det R ( C ) the determinant moduleof C . This module is understood to be a graded invertible R -module (with the gradesuppressed from the symbol).For a number field F and a finite set S of places of Q , we denote by O F,S the ring of S F -integers of F , where S F denotes the set of places of F lying above a place in S . In particular, O Q ,S is denoted simply by Z S . We denote by R Γ( O F,S , − ) the etale cohomologycomplex R Γ ´et (Spec( O F,S ) , − ).As usual, the notation H if ( F, − ) indicates the Bloch-Kato Selmer group and H if ( F v , − )the Bloch-Kato local condition for a place v of F .For an elliptic curve E defined over Q , we denote by L ( E, s ) the Hasse-Weil L -functionof E . For a finite set S of places of Q , we denote by L S ( E, s ) the S -truncated L -functionof E . We denote by L ∗ S ( E,
1) the leading term at s = 1.The Tate-Shafarevich group of E over a number field F is denoted by X ( E/F ). Theproduct of Tamagawa factors of E/ Q is denoted by Tam( E ).We use some other standard notations concerning elliptic curves and modular curves,such as Γ( E, Ω E/ Q ), H ( E ( C ) , Q ), E ( Q p ), Y ( N ), X ( N ), etc.2. Formulation of the Generalized Perrin-Riou Conjecture
We fix a prime number p and assume throughout the article that p is odd.2.1. Kato’s Euler system.
Let E be an elliptic curve over Q of conductor N .Fix a modular parametrization φ : X ( N ) → E and write f = P ∞ n =1 a n q n for thenormalized newform of weight 2 and level N corresponding to E .Let T p ( E ) be the p -adic Tate module of E and set V := Q p ⊗ Z p T p ( E ). Let T be a G Q -stable sublattice of V that is given by the image of the following map: H ( Y ( N ) × Q Q , Z p (1)) ֒ → H ( Y ( N ) × Q Q , Q p (1))(2.1.1) ։ H ( X ( N ) × Q Q , Q p (1)) φ ∗ −→ H ( E × Q Q , Q p (1))= V ∗ (1) ≃ V, where the second arrow is the Manin-Drinfeld splitting, the third is induced by φ and thelast is induced by the Weil pairing.Note that T identifies with the maximal quotient of H ( Y ( N ) × Q Q , Z p (1)) on whichHecke operators T ( n ) act via a n and may be different from T p ( E ). If E [ p ] is an irreducible G Q -representation, we may assume T = T p ( E ).We fix the following data: • an embedding Q ֒ → C ; • a finite set S of places of Q such that {∞} ∪ { ℓ | pN } ⊂ S ; • integers c, d > cd is coprime to 6 and all primes in S , and that c ≡ d ≡ N ); • an element ξ ∈ SL ( Z ).For this data and any positive integer m that is coprime to cd , Kato constructed in [23,(8.1.3)] a ‘zeta element’ c,d z m ( ξ, S m ) := c,d z ( p ) m ( f, , , ξ, S m \ {∞} )in H ( O Q ( µ m ) ,S m , T ) , where S m denotes the set S ∪ { ℓ | m } . It is also known that the collection ( c,d z m ( ξ, S m )) m forms an Euler system (see [23, Ex.13.3]).For a finite abelian extension F of Q that is unramified outside S , we set c,d z F = c,d z F ( ξ, S ) := Cor Q ( µ m ) /F ( c,d z m ( ξ, S )) , where m = m F denotes the conductor of F .For later purposes we make a specific choice of ξ as follows. Just as in (2.1.1), the fixedmodular parametrization φ : X ( N ) → E induces a map H ( X ( N )( C ) , { cusps } , Z ) ≃ H ( Y ( N )( C ) , Z (1))(2.1.2) → H ( E ( C ) , Q (1)) ≃ H ( E ( C ) , Q ) , where the first and last isomorphisms are obtained by the Poincar´e duality.We write H for the image of this map (so H is a lattice of H ( E ( C ) , Q )) and let δ ( ξ ) ∈ H denote the image under the map (2.1.2) of the modular symbol { ξ (0) , ξ ( ∞ ) } ∈ H ( X ( N )( C ) , { cusps } , Z ) . Let g denote the complex conjugation and set e + := (1 + g ) / ξ so that the following condition is satisfied:(2.1.3) the element e + δ ( ξ ) of H ( E ( C ) , Q ) + is a Z ( p ) -basis of ( Z ( p ) ⊗ Z H ) + . The existence of such ξ ∈ SL ( Z ) is justified as follows. By a well-known theorem ofManin, we know that H ( X ( N )( C ) , { cusps } , Z ) is generated by the set {{ α (0) , α ( ∞ ) } | α ∈ SL ( Z ) } . This implies that the Z ( p ) -module ( Z ( p ) ⊗ Z H ) + is generated by the set { e + δ ( α ) | α ∈ SL ( Z ) } . Since ( Z ( p ) ⊗ Z H ) + ≃ Z ( p ) and Z ( p ) is local, Nakayama’s lemmaimplies the existence of ξ ∈ SL ( Z ) such that e + δ ( ξ ) generates ( Z ( p ) ⊗ Z H ) + .Throughout this article, we also fix a minimal Weierstrass model of E over Z and let ω ∈ Γ( E, Ω E/ Q )be the corresponding N´eron differential.We define the real period for ( ω, ξ ) by settingΩ ξ := Z e + δ ( ξ ) ω. (2.1.4)(In general, this integral need only agree with the usual real N´eron period Ω + up to multi-plication by an element of Q × . However, if E [ p ] is irreducible, then Ω ξ and Ω + will agreeup to multiplication by an element of Z × ( p ) .)Then Kato’s reciprocity law [23, Th. 6.6 and 9.7] gives the formulaexp ∗ ω ( c,d z Q ) = cd ( c − d − L S ( E, ξ in Q , (2.1.5)where exp ∗ ω : H ( Z S , T ) → H ( Q p , T ) → Q p is the dual exponential map associated to ω . Remark 2.1.
As in [23, Th. 12.5], one may normalize Kato’s zeta element in order toconstruct an element z of H ( Z S , V ) with the property that exp ∗ ω ( z ) = L { p } ( E, / Ω + , where the L -fucntion is truncated just at p rather than at all places in S . However, onedoes not in general know that this element z lies in H ( Z S , T ). This delicate integralityissue is the reason that we prefer to use c,d z Q = c,d z Q ( ξ, S ) rather than the normalizedelement. In addition, if H ( Z S , T ) is Z p -free, then one expects that the element z Q := 1 cd ( c − d − · c,d z Q of H ( Z S , V ) actually belongs to H ( Z S , T ) but, as far as we are aware, this has not beenproved in full generality.2.2. Birch and Swinnerton-Dyer elements.
In this subsection, we introduce a naturalnotion of ‘Birch and Swinnerton-Dyer element’.Such elements constitute an analogue for elliptic curves of the ‘Rubin-Stark elements’that are associated to the multiplicative group.In the sequel we shall denote the ‘algebraic rank’ rank( E ( Q )) of E over Q by r alg or often,for simplicity, by r .Throughout this section we shall then assume the following. Hypothesis 2.2. (i) H ( Z S , T ) is Z p -free;(ii) r alg > X ( E/ Q )[ p ∞ ] is finite. Remark 2.3. If E [ p ] is irreducible, then T = T p ( E ) and E ( Q )[ p ] = 0 so Hypothesis 2.2(i)is automatically satisfied.Following [10, Lem. 6.1], we note that these assumptions imply the existence of a canon-ical isomorphism H ( Z S , V ) ≃ Q p ⊗ Z E ( Q )(2.2.1)and also, since the image of the localization map H ( Z S , V ) → H ( Q p , V ) lies in H f ( Q p , V ) = Q p ⊗ Z p E ( Q p ), of a canonical short exact sequence0 → Q p ⊗ Z p E ( Q p ) ∗ → Q p ⊗ Z E ( Q ) ∗ → H ( Z S , V ) → . (2.2.2) We fix an embedding R ֒ → C p and consider the following canonical ‘period-regulator’isomorphism of C p -modules λ : C p ⊗ Z p ^ r Z p H ( Z S , T ) ≃ C p ⊗ Z ^ r Z E ( Q ) ≃ C p ⊗ Z ^ r Z E ( Q ) ∗ ≃ C p ⊗ Q p (cid:18) E ( Q p ) ∗ ⊗ Z p ^ r − Q p H ( Z S , V ) (cid:19) ≃ C p ⊗ Q p (cid:18) Γ( E, Ω E/ Q ) ⊗ Q ^ r − Q p H ( Z S , V ) (cid:19) ≃ C p ⊗ Q p (cid:18) H ( E ( C ) , Q ) + , ∗ ⊗ Q ^ r − Q p H ( Z S , V ) (cid:19) . Here the first isomorphism is induced by (2.2.1), the second by the N´eron-Tate height pairing h− , −i ∞ : E ( Q ) × E ( Q ) → R , the third by (2.2.2), the fourth by the dual exponential mapexp ∗ : E ( Q p ) ∗ → Q p ⊗ Q Γ( E, Ω E/ Q ) , the last by the period mapΓ( E, Ω E/ Q ) → H ( E ( C ) , R ) + , ∗ ; ω ( γ Z γ ω ) . Definition 2.4.
Fix an element x of the space V r − Q p H ( Z S , V ). Then the Birch andSwinnerton-Dyer element η BSD x = η BSD x ( ξ, S ) of the data ξ , S and x is the element of C p ⊗ Z p V r Z p H ( Z S , T ) obtained by setting η BSD x := λ − (cid:0) L ∗ S ( E, · ( e + δ ( ξ ) ∗ ⊗ x ) (cid:1) . The ‘( c, d )-modified Birch and Swinnerton-Dyer element’ for the given data is the element c,d η BSD x := cd ( c − d − · η BSD x . Remark 2.5.
Each choice of an ordered basis of E ( Q ) tf gives rise to a natural choice ofelement x as above (see § r = 1 and x = 1, the above definitionsimplifies to an equality η BSD x = L ∗ S ( E, ξ · R ∞ · log ω ( x ) · x in C p ⊗ Z E ( Q ) ≃ C p ⊗ Z p H ( Z S , T ), where R ∞ is the N´eron-Tate regulator, log ω : E ( Q ) → E ( Q p ) → Q p is the formal group logarithm associated to ω and x is any element of E ( Q )that generates E ( Q ) tf .The p -part of the Birch-Swinnerton-Dyer Formula for E asserts that there should be anequality of Z p -submodules of C p of the form L ∗ ( E, · Z p = ( X ( E/ Q )[ p ∞ ] · Tam( E ) · E ( Q ) − · Ω + · R ∞ ) · Z p , where Tam( E ) denotes the product of the Tamagawa factors of E/ Q . In the sequel we shallabbreviate this equality of lattices to ‘BSD p ( E )’.The next result explains the connection between this conjectural equality and the inte-grality properties of Birch and Swinnerton-Dyer elements. Proposition 2.6.
Set r := r alg and fix a Z p -basis x of the lattice V r − Z p H ( Z S , T ) tf . Then BSD p ( E ) is valid if and only if there is an equality of Z p -lattices Z p · η BSD x = H ( Z S , T ) tors · ^ r Z p H ( Z S , T ) . (2.2.3) In particular, the validity of
BSD p ( E ) implies that η BSD x belongs to V r Z p H ( Z S , T ) .Proof. It is well-known that the validity of BSD p ( E ) is equivalent to the equality of latticesthat underlies the statement of the Tamagawa Number Conjecture (or ‘TNC’ for short) forthe pair ( h ( E )(1) , Z p ) (this has been shown, for example, by Kings in [24]). It is thereforesufficient to show that the equality (2.2.3) is equivalent to the TNC and to do this we mustrecall the formulation of the latter conjecture.The statement of the TNC involves a canonical isomorphism of C p -modules(2.2.4) ϑ : C p ⊗ Z p det − Z p ( R Γ c ( Z S , T ∗ (1))) ∼ −→ C p that arises as follows. Firstly, global duality induces a canonical isomorphismdet − Z p ( R Γ c ( Z S , T ∗ (1))) ≃ det − Z p ( R Γ( Z S , T )) ⊗ Z p T ∗ (1) + (cf. [11, Prop. 2.22]) and hence also a canonical isomorphism C p ⊗ Z p det − Z p ( R Γ c ( Z S , T ∗ (1)))(2.2.5) ≃ C p ⊗ Q p (cid:18)^ r Q p H ( Z S , V ) ⊗ Q p ^ r − Q p H ( Z S , V ) ∗ ⊗ Q p V ∗ (1) + (cid:19) . The isomorphism ϑ in (2.2.4) is then obtained by combining the latter isomorphism withthe canonical ‘comparison’ isomorphism V ∗ (1) + ≃ Q p ⊗ Q H ( E ( C ) , Q (1)) + ≃ Q p ⊗ Q H ( E ( C ) , Q ) + and the period-regulator isomorphism λ : C p ⊗ Q p ^ r Q p H ( Z S , V ) ≃ C p ⊗ Q p (cid:18) H ( E ( C ) , Q ) + , ∗ ⊗ Q ^ r − Q p H ( Z S , V ) (cid:19) constructed earlier.If z is the unique element of C p ⊗ Z p det − Z p ( R Γ c ( Z S , T ∗ (1))) that satisfies ϑ ( z ) = L ∗ S ( E, , then the TNC predicts that Z p · z = det − Z p ( R Γ c ( Z S , T ∗ (1))) . Given this, the claimed result is a consequence of the fact that the isomorphism (2.2.5)sends the element z to η BSD x ⊗ x ∗ ⊗ e + δ ( ξ ) ∈ C p ⊗ Q p (cid:18)^ r Q p H ( Z S , V ) ⊗ Q p ^ r − Q p H ( Z S , V ) ∗ ⊗ Q p V ∗ (1) + (cid:19) , and the lattice det − Z p ( R Γ c ( Z S , T ∗ (1))) to H ( Z S , T ) tors · ^ r Z p H ( Z S , T ) ⊗ Z p ^ r − Z p H ( Z S , T ) ∗ tf ⊗ Z p T ∗ (1) + . (cid:3) Bockstein regulator maps.
In this subsection, we shall introduce a canonical con-struction of Bockstein regulator maps (see (2.3.3) below).We first set some notations. Let F/ Q be a finite abelian extension unramified outside S and G its Galois group. Since all results and conjectures we study are of p -adic nature, wemay assume that [ F : Q ] is a p -power. In particular, since p is odd, F is a totally real field.The augmentation ideal I F := ker( Z p [ G ] ։ Z p )and the augmentation quotients Q aF := I aF /I a +1 F for a non-negative integer a will play important roles. We remark that Q F is understoodto be Z p [ G ] /I F = Z p .For simplicity, in this subsection we shall abbreviate the ideal I F to I .At the outset we note that the tautological short exact sequence0 → I/I → Z p [ G ] /I → Z p → Z p -modules of the form R Γ( O F,S , T ) ⊗ L Z p [ G ] I/I → R Γ( O F,S , T ) ⊗ L Z p [ G ] Z p [ G ] /I → R Γ( O F,S , T ) ⊗ L Z p [ G ] Z p . Next we recall (from, for example, [18, Prop. 1.6.5]) that R Γ( O F,S , T ) is acyclic outsidedegrees one and two and that there exists a canonical isomorphism in the derived categoryof Z p -modules R Γ( O F,S , T ) ⊗ L Z p [ G ] Z p ≃ R Γ( Z S , T ) . (2.3.1)Taking account of these facts, the above triangle gives rise to a morphism of complexesof Z p -modules δ F : R Γ( Z S , T ) → (cid:0) R Γ( Z S , T ) ⊗ L Z p I/I (cid:1) [1]and hence to a composite homomorphism of Z p -modules β F : H ( Z S , T ) ( − × H ( δ F ) −−−−−−−−→ H ( R Γ( Z S , T ) ⊗ L Z p I/I )(2.3.2) = H ( Z S , T ) ⊗ Z p I/I ։ H ( Z S , T ) tf ⊗ Z p I/I , in which the equality is valid since R Γ( Z S , T ) is acyclic in degrees greater than two and thelast map is induced by the natural map from H ( Z S , T ) to H ( Z S , T ) tf .We writeBoc F : ^ r Z p H ( Z S , T ) → H ( Z S , T ) ⊗ Z p ^ r − Z p H ( Z S , T ) tf ⊗ Z p Q r − F for the homomorphism of Z p -modules with the property thatBoc F (cid:0) y ∧ · · · ∧ y r (cid:1) = r X i =1 ( − i +1 y i ⊗ (cid:0) β F ( y ) ∧ · · · ∧ β F ( y i − ) ∧ β F ( y i +1 ) ∧ · · · ∧ β F ( y r ) (cid:1) for all elements y i of H ( Z S , T ).Then, each choice of basis element x of the (free, rank one) Z p -module V r − Z p H ( Z S , T ) tf ,gives rise to a composite ‘Bockstein regulator’ homomorphismBoc F, x : ^ r Z p H ( Z S , T ) Boc F −−−→ H ( Z S , T ) ⊗ Z p ^ r − Z p H ( Z S , T ) tf ⊗ Z p Q r − F (2.3.3) id ⊗ φ x ⊗ id −−−−−−→ H ( Z S , T ) ⊗ Z p Q r − F , where φ x is the isomorphism V r − Z p H ( Z S , T ) tf ≃ Z p induced by the choice of x . Remark 2.7. If r = 1 and x = 1 is the canonical basis of V r − Z p H ( Z S , T ) tf = Z p , thenBoc F, x = Boc F is simply equal to the identity map on H ( Z S , T ).2.4. The Generalized Perrin-Riou Conjecture.
In the sequel we shall write r an for theanalytic rank ord s =1 L ( E, s ) of E .2.4.1. In [34], Perrin-Riou investigates relations between Kato’s Euler system and the p -adic Birch-Swinnerton-Dyer Conjecture. In particular, she formulates the following con-jecture. Conjecture 2.8 (Perrin-Riou [34], see also [13]) . (i) The element c,d z Q is non-zero if and only if r an is at most one. (ii) If r an = r alg = 1 , then in C p ⊗ Z p H ( Z S , T ) ≃ C p ⊗ Z E ( Q ) one has c,d z Q = cd ( c − d − L ′ S ( E, ξ · R ∞ log ω ( x ) · x, (2.4.1) where x is any element of E ( Q ) that generates E ( Q ) tf . Remark 2.9.
This conjecture is a slight modification of, but equivalent to, Perrin-Riou’soriginal formulation of the conjecture. By Kato’s reciprocity law (2.1.5), the element c,d z Q is explicitly related to L ( E,
1) and, in particular, does not vanish if r an = 0. Perrin-Riou’sconjecture predicts that c,d z Q does not vanish even if r an = 1 and, moreover, that it shouldbe explicitly related to the first derivative L ′ ( E,
1) via the formula (2.4.1).By Remark 2.5, we immediately obtain the following interpretation of Perrin-Riou’s con-jecture in terms of the BSD element.
Proposition 2.10. If r an = r alg = 1 and x = 1 , then Conjecture 2.8(ii) is valid if and onlyif one has c,d z Q = c,d η BSD x . Remark 2.11.
An interpretation of Perrin-Riou’s conjecture in the same style as Proposi-tion 2.10 was previously given by Sakamoto and the first and the third authors in [10, § I F and Q aF introduced in § r := r alg and write ι F : H ( Z S , T ) ⊗ Z p Q r − F → H ( O F,S , T ) ⊗ Z p Q r − F (2.4.2) → H ( O F,S , T ) ⊗ Z p Z p [ G ] /I rF for the composite homomorphism that is induced by the restriction map H ( Z S , T ) → H ( O F,S , T ) and the natural inclusion Q r − F ֒ → Z p [ G ] /I rF . (This map ι F is actually injective- see the discussion in § c,d z F to be the element of H ( O F,S , T ) ⊗ Z p Z p [ G ] obtained by setting N F/ Q ( c,d z F ) := X σ ∈ G σ ( c,d z F ) ⊗ σ − . We can now give a precise formulation of Conjecture 1.1. This prediction involves theBirch-Swinnerton-Dyer element c,d η BSD x and Bockstein regulator map Boc F, x that were re-spectively defined in § § Conjecture 2.12 (The Generalized Perrin-Riou Conjecture) . Set r := r alg . Then for each Z p -basis element x of V r − Z p H ( Z S , T ) tf the following claims are valid. (i) The element c,d η BSD x belongs to V r Z p H ( Z S , T ) . (ii) The image in H ( O F,S , T ) ⊗ Z p Z p [ G ] /I rF of the Darmon norm N F/ Q ( c,d z F ) of c,d z F is equal to ι F (cid:0) Boc F, x ( c,d η BSD x ) (cid:1) . Remark 2.13. (i) Proposition 2.6 shows that Conjecture 2.12(i) is implied by the validity of BSD p ( E ).(ii) Assume r alg = 1 and that x = 1 in V r − Z p H ( Z S , T ) tf = Z p . Then in this case one has N F/ Q ( c,d z F ) = N F/ Q ( c,d z F ) in H ( O F,S , T ) ⊗ Z p Z p [ G ] /I F ≃ H ( O F,S , T ) , where N F/ Q := P σ ∈ G σ . In particular, since Cor F/ Q ( c,d z F ) = c,d z Q and Boc F, x is theidentity map on H ( Z S , T ) (by Remark 2.7), Conjecture 2.12 is equivalent in this case toan equality c,d z Q = c,d η BSD x . From Proposition 2.10 it therefore follows that if r an = r alg = 1then Conjecture 2.12 is equivalent to Perrin-Riou’s conjecture (as stated in Conjecture2.8(ii)). This observation proves Theorem 1.4 and also motivates us to refer to Conjecture2.12 as the ‘Generalized Perrin-Riou Conjecture’. Remark 2.14.
The formulation of Conjecture 2.12 can also be regarded as a natural ana-logue for elliptic curves of the conjectural ‘refined class number formula for G m ’ concerningRubin-Stark elements that was originally formulated independently by Mazur and Rubin[28, Conj. 5.2] and by the third author [38, Conj. 3] and then subsequently refined by thepresent authors in [6, Conj. 5.4]. Remark 2.15.
It is straightforward to show that the element Boc F, x ( c,d η BSD x ), and hencealso the validity of Conjecture 2.12(ii), is independent of the choice of basis element x . Remark 2.16. In § c,d z F .2.5. An algebraic analogue.
We now formulate an analogue of Conjecture 2.12 that ismore algebraic, and elementary, in nature.To do this we recall that if X ( E/ Q ) is finite, then the Birch and Swinnerton-Dyer Formulafor E predicts that(2.5.1) L ∗ S ( E,
1) = Y ℓ ∈ S \{∞} L ℓ X ( E/ Q ) · Tam( E ) · Ω + · R ∞ E ( Q ) , where Ω + is the usual real N´eron period of E , L ℓ is the standard Euler factor at ℓ ofthe Hasse-Weil L -function (so that ( Q ℓ ∈ S \{∞} L ℓ ) L ∗ ( E,
1) = L ∗ S ( E, E ) is theproduct of Tamagawa factors. Definition 2.17.
Set r := r alg . Then for each element x of V r − Q p H ( Z S , V ) the algebraicBirch and Swinnerton-Dyer element η alg x = η alg x ( ξ, S ) of the data ξ , S and x is the elementof C p ⊗ Z p V r Z p H ( Z S , T ) obtained by setting η alg x := λ − Y ℓ ∈ S \{∞} L ℓ X ( E/ Q ) · Tam( E ) · Ω + · R ∞ E ( Q ) · ( e + δ ( ξ ) ∗ ⊗ x ) . The ‘( c, d )-modified algebraic Birch and Swinnerton-Dyer element’ of the given data is thendefined by setting c,d η alg x := cd ( c − d − · η alg x . Remark 2.18.
It is clear that, if x is non-zero, then the Birch and Swinnerton-DyerFormula (2.5.1) is valid for E if and only if the elements η alg x and c,d η alg x are respectivelyequal to the Birch and Swinnerton-Dyer elements η BSD x and c,d η BSD x from Definition 2.4.An easy exercise shows that if x is a Z p -basis element of V r − Z p H ( Z S , T ) tf , then there isan equality of lattices Z p · η alg x = H ( Z S , T ) tors · ^ r Z p H ( Z S , T )(2.5.2)and hence η alg x belongs to V r Z p H ( Z S , T ).Upon combining this fact with Remark 2.18, one is led to formulate the following algebraicanalogue of Conjecture 2.12. Conjecture 2.19 (The Refined Mazur-Tate Conjecture) . Set r := r alg . Then for each Z p -basis element x of V r − Z p H ( Z S , T ) tf the image in H ( O F,S , T ) ⊗ Z p Z p [ G ] /I rF of N F/ Q ( c,d z F ) is equal to ι F (cid:0) Boc F, x ( c,d η alg x ) (cid:1) . Remark 2.20.
We refer to this algebraic analogue of Conjecture 2.12 as a refined Mazur-Tate Conjecture since in the complementary article [9] we are able to prove that, undercertain mild and natural hypotheses, the equality predicted by Conjecture 2.19 is strictlyfiner than the celebrated congruences for modular elements that are conjectured by Mazur and Tate in [29]. This fact is in turn a key ingredient in the approach used in [9] to obtainthe first verifications of the conjecture of Mazur and Tate for elliptic curves of strictlypositive rank. 3. Fitting ideals and order of vanishing
In this section we shall discuss a further arithmetic property of Kato’s zeta elements and,in particular, use it to prove Theorem 1.3.Throughout we fix F , G and I F as in § H ( Z S , T ) is Z p -free. However, unless explicitly stated, in this subsection we do not need to assumeeither that r alg > X ( E/ Q )[ p ∞ ] is finite.3.1. A ‘main conjecture’ at finite level.
We write m for the conductor of F and set t c,d := cd ( c − σ c )( d − σ d ) ∈ Z p [ G ] , where σ a is the element of G obtained by restriction of the automorphism of Q ( µ m ) thatsends ζ m to ζ am .We then propose the following conjecture involving the initial Fitting ideal of the Z p [ G ]-module H ( O F,S , T ). Conjecture 3.1. (cid:8) Φ( c,d z F ) | Φ ∈ Hom Z p [ G ] ( H ( O F,S , T ) , Z p [ G ]) (cid:9) = t c,d · Fitt Z p [ G ] ( H ( O F,S , T )) . Remark 3.2.
Conjecture 3.1 is analogous to the ‘weak main conjecture’ for modular ele-ments that is formulated by Mazur and Tate [29, Conj. 3]. (In fact, since our conjecturepredicts an equality rather than simply an inclusion, it corresponds to a strengthening of[29, Conj. 3]). It is also an analogue of the conjectures [6, Conj. 7.3] and [8, Conj. 3.6(ii)]that were formulated by the present authors in the setting of the multiplicative group.The prediction in Conjecture 3.1 can be studied by using the equivariant theory of Eulersystems developed by Sakamoto and the first and third authors in [10]. In this way, thefollowing evidence for Conjecture 3.1 is obtained in [10, Th. 6.11].
Proposition 3.3.
Assume that the following conditions are all satisfied. (a) p > ; (b) X ( E/F )[ p ∞ ] and X ( E/ Q )[ p ∞ ] are finite; (c) the image of the representation G Q → Aut( T p ( E )) ≃ GL ( Z p ) contains SL ( Z p ) ; (d) E ( Q ℓ )[ p ] vanishes for all primes ℓ in S .Then for any homomorphism Φ : H ( O F,S , T ) → Z p [ G ] of Z p [ G ] -modules one has Φ( c,d z F ) ∈ Fitt Z p [ G ] ( H ( O F,S , T )) . The proof of Theorem 1.3.
In the rest of this section, we assume the conditions(a), (b), (c) and (d) in Theorem 1.3 (which are the same as those in Proposition 3.3). Inparticular, E [ p ] is irreducible by (c), and we may assume T = T p ( E ). Proposition 3.4.
Assume that E ( F )[ p ] vanishes and that X ( E/ Q )[ p ∞ ] is finite. Set a :=max { , r alg − } and define an ideal of Z p [ G ] by setting I F,S,a := H ( Z S , T ) tors · I aF + I a +1 F ⊂ I aF . Then N F/ Q ( z F ) belongs to H ( O F,S , T ) ⊗ Z p I F,S,a whenever one has Φ( c,d z F ) ∈ Fitt Z p [ G ] ( H ( O F,S , T )) for all Φ ∈ Hom Z p [ G ] (cid:0) H ( O F,S , T ) , Z p [ G ] (cid:1) .Proof. At the outset we fix a surjective homomorphism of Z p [ G ]-modules of the form P ′ F ։ H ( O F,S , T ) ∗ in which P ′ F is both finitely generated and free.Then the linear dual P F := ( P ′ F ) ∗ is a finitely generated free Z p [ G ]-module and the abovesurjection induces an injective homomorphism of Z p [ G ]-modules(3.2.1) H ( O F,S , T ) ∗∗ ֒ → P F , the cokernel of which is Z p -free. In addition, since E ( F )[ p ] is assumed to vanish the Z p -module H ( O F,S , T ) is free and so identifies with H ( O F,S , T ) ∗∗ .We may therefore use (3.2.1) to identify H ( O F,S , T ) as a submodule of the free module P F in such a way that the quotient P F /H ( O F,S , T ) is torsion-free.Having done this, the argument of [6, Prop. 4.17] shows that the validity of the displayedinclusion for all Φ in Hom Z p [ G ] (cid:0) H ( O F,S , T ) , Z p [ G ] (cid:1) is equivalent to asserting that one has N F/ Q ( c,d z F ) ∈ P F ⊗ Z p J F,S with J F,S := Fitt Z p [ G ] ( H ( O F,S , T )) and, moreover, that the projection of N F/ Q ( c,d z F ) to P F ⊗ Z p ( J F,S /I F · J F,S ) belongs to the submodule P GF ⊗ Z p ( J F,S /I F · J F,S ).To complete the proof it is therefore enough to show that(3.2.2) J F,S ⊂ I F,S,a . To do this we note that H i ( O F,S , T ) vanishes for all i > H ( O F,S , T ) ։ H ( Z S , T ) is surjective.In addition, since X ( E/ Q )[ p ∞ ] is assumed to be finite, the Z p -rank of H ( Z S , T ) is equalto a and so the Z p -module H ( Z S , T ) is isomorphic to H ( Z S , T ) tors ⊕ Z ap .The corestriction map therefore induces a surjective homomorphism of Z p [ G ]-modules H ( O F,S , T ) ։ H ( Z S , T ) tors ⊕ Z ap and hence an inclusion of Fitting ideals J F,S = Fitt Z p [ G ] ( H ( O F,S , T )) ⊂ Fitt Z p [ G ] ( H ( Z S , T ) tors ⊕ Z ap ) = Fitt Z p [ G ] ( H ( Z S , T ) tors ) · I aF . To deduce (3.2.2) from this it is thus enough to note the image of Fitt Z p [ G ] ( H ( Z S , T ) tors )under the natural map Z p [ G ] → Z p [ G ] /I F ≃ Z p is equal toFitt Z p (( H ( Z S , T ) tors ) G ) = Fitt Z p ( H ( Z S , T ) tors ) = H ( Z S , T ) tors · Z p . (cid:3) Remark 3.5.
The containment discussed in Proposition 3.4 would imply thatΦ( c,d z F ) ∈ I aF for all Φ ∈ Hom Z p [ G ] ( H ( O F,S , T ) , Z p [ G ]) . (3.2.3)This prediction constitutes an analogue for Kato’s Euler system c,d z F of the ‘weak vanishing’conjecture for modular elements that is formulated by Mazur and Tate in [29, Conj. 1].3.2.2. If the algebraic rank r := r alg of E over Q is strictly positive, then the integer a inProposition 3.4 is equal to r − I aF = I r − F .One therefore obtains a proof of Theorem 1.3 directly upon combining the results ofPropositions 3.3 and 3.4.4. Derivatives of Kato’s Euler system
In this section, we shall define a canonical ‘Darmon derivative’ c,d κ F of Kato’s zetaelement c,d z F and use it to reinterpret the conjectures formulated above.In particular, in this way we are able to formulate more explicit versions of the Conjectures2.12 and 2.19 for subfields F of the cyclotomic Z p -extension of Q . Throughout this section, we assume that H ( Z S , T ) is Z p -free and X ( E/ Q )[ p ∞ ] is finite. Darmon derivatives.
We use the notations in § ι F in (2.4.2) is injective.This follows easily from the facts that H ( Z S , T ) is Z p -free and that H ( Z S , T ) identifieswith the submodule H ( O F,S , T ) G of G -invariant elements in H ( O F,S , T ) (since H ( Z S , T )vanishes).Conjecture 2.12 is therefore equivalent to asserting the existence of a unique element c,d κ F ∈ H ( Z S , T ) ⊗ Z p Q r − F with the property that both ι F ( c,d κ F ) = N F/ Q ( c,d z F ) and c,d κ F = Boc F, x ( c,d η BSD x ) . (In particular, if r = 1, then c,d κ F is simply equal to c,d z Q .)We regard this element c,d κ F as a ‘Darmon-type derivative’ of the zeta element c,d z F andfirst consider conditions under which it can be unconditionally defined.4.1.1. To do this we fix a finitely generated free Z p [ G ]-module P F and an injective homo-morphism of Z p [ G ]-modules j F : H ( O F,S , T ) → P F as in the proof of Proposition 3.4.We use j F to regard H ( Z S , T ) = H ( O F,S , T ) G as a submodule of P Q := P GF . Then,just as in (2.4.2), there are natural injective homomorphisms ι F : P Q ⊗ Z p Q aF ֒ → P F ⊗ Z p Q aF ֒ → P F ⊗ Z p Z p [ G ] /I a +1 F (where, we recall, Q aF denotes I aF /I a +1 F ). Definition 4.1.
Set a := max { , r alg − } and assume that the containment (3.2.3) is validfor all Φ in Hom Z p [ G ] ( H ( O F,S , T ) , Z p [ G ]). Then the argument of Proposition 3.4 impliesthe existence of a unique element c,d κ F of P Q ⊗ Z p Q aF with the property that ι F ( c,d κ F ) = N F/ Q ( c,d z F )in P Q ⊗ Z p Q aF . We shall refer to c,d κ F as the Darmon derivative of c,d z F with respect to theembedding j F . Remark 4.2.
If one restricts the embeddings j F by requiring that the associated module P F has minimal possible rank, then the derivatives c,d κ F can be checked to be independent,in a natural sense, of the choice of j F .4.1.2. Conjecture 2.12 predicts that the element c,d κ F belongs to the image of the (injec-tive) homomorphism(4.1.1) H ( Z S , T ) ⊗ Z p Q aF → P Q ⊗ Z p Q aF induced by j F . At this stage, however, we can only verify this prediction in certain specialcases.In the next section we shall verify that it is valid if F is contained in the cyclotomic Z p -extension Q ∞ of Q . In the following result we record some evidence in the general case.Before stating the result we note that the hypothesis in its first paragraph is valid when-ever the data E, F, S and p satisfy the conditions (a), (b), (c) and (d) of Proposition 3.3and that, in general, its validity would follow from that of Conjecture 3.1.In particular, claim (ii) of this result is a natural analogue for zeta elements of one of themain results of Darmon in [16, Th. 2.5] concerning Heegner points. Theorem 4.3.
Set z := c,d z F and κ := c,d κ F . If one has Φ( z ) ∈ Fitt Z p [ G ] ( H ( O F,S , T )) forevery Φ in Hom Z p [ G ] (cid:0) H ( O F,S , T ) , Z p [ G ] (cid:1) , then the following claims are valid. (i) If p N is the minimum of the exponents of the groups H ( Z S , T ) tors · Q aF and H ( Z S , T ) tors , then p N · κ belongs to the image of the map (4.1.1). (ii) The image of κ under the natural map P Q ⊗ Z p Q aF → P Q ⊗ Z p Q aF ⊗ Z Z / ( p ) belongs to the image of the map H ( Z S , T ) ⊗ Z p Q aF ⊗ Z Z / ( p ) → P Q ⊗ Z p Q aF ⊗ Z Z / ( p ) induced by (4.1.1).Proof. The proof of claim (i) requires a refinement of the construction used to prove Propo-sition 3.4. This relies on the fact that the complex C F := R Hom Z p ( R Γ c ( O F,S , T ∗ (1)) , Z p [ − Z p [ G ]-modules that is acyclic outside degrees zero and one, and thatthere exists a canonical isomorphism H ( C F ) ≃ H ( O F,S , T )(4.1.2) and an exact sequence0 → H ( O F,S , T ) → H ( C F ) → Z p [ G ] ⊗ Z p T ∗ (1) + , ∗ → . (4.1.3)(See [11, Prop. 2.22].)In particular, by [11, Prop. A.11(i)], one finds that C F is represented by a complex ofthe form P F → P F , where P F is a finitely generated free Z p [ G ]-module and the first termis placed in degree zero.In this way we obtain an exact sequence0 → H ( O F,S , T ) → P F f F −→ P F → H ( C F ) → , Then (2.3.1) implies that C Q is represented by the complex P Q f Q −→ P Q obtained by taking G -invariants of the complex P F f F −→ P F and hence that there is an exact sequence(4.1.4) 0 → H ( Z S , T ) → P Q f Q −→ P Q → H ( C Q ) → . Set ˜ Q aF := H ( Z S , T ) tors · Q aF ⊂ Z p [ G ] /I a +1 F . Then the above sequences combine togive a commutative diagram / / H ( O F,S , T ) ⊗ Z p Z p [ G ] /I a +1 F / / P F ⊗ Z p Z p [ G ] /I a +1 F ˜ f F / / P F ⊗ Z p Z p [ G ] /I a +1 F / / H ( Z S , T ) ⊗ Z p ˜ Q aF / / ˜ ι F O O P Q ⊗ Z p ˜ Q aF ˜ f Q / / ˜ ι F O O P Q ⊗ Z p ˜ Q aF ˜ ι F O O in which the maps ˜ ι F are obtained by restricting ι F .Then the argument of Proposition 3.4 implies that(4.1.5) κ ∈ P Q ⊗ Z p ˜ Q aF ⊂ P Q ⊗ Z p Q aF , and so the commutativity of this diagram implies that˜ ι F ( ˜ f Q ( κ )) = ˜ f F (˜ ι F ( κ )) = ˜ f F ( N F/ Q ( z )) = 0and hence, since ˜ ι F is injective, that ˜ f Q ( κ ) = 0.Now, the exact sequence (4.1.4) induces exact sequences0 → H ( Z S , T ) ⊗ Z p ˜ Q aF → P Q ⊗ Z p ˜ Q aF µ −→ im( f Q ) ⊗ Z p ˜ Q aF → → Tor Z p (cid:0) H ( Z S , T ) tors , ˜ Q aF (cid:1) µ −→ im( f Q ) ⊗ Z p ˜ Q aF µ −→ P Q ⊗ Z p ˜ Q aF . with the property that µ ◦ µ is equal to ˜ f Q . (The first sequence here is exact since the Z p -module im( f Q ) is free and the second is exact as consequence of the fact that (4.1.3)identifies H ( Z S , T ) tors with H ( C Q ) tors .)These sequences combine with the equality ˜ f Q ( κ ) = 0 to imply µ ( κ ) belongs to theimage of µ in the lower sequence above.Thus, since the definition of p N ensures it annihilates the group Tor Z p (cid:0) H ( Z S , T ) tors , ˜ Q aF (cid:1) ,it follows that µ ( p N · κ ) vanishes, and hence that p N · κ belongs to H ( Z S , T ) ⊗ Z p ˜ Q aF ⊂ H ( Z S , T ) ⊗ Z p Q aF . This proves claim (i). Turning to claim (ii), we note first that if H ( Z S , T ) tors is trivial, then claim (i) implies κ belongs to the image of the map (4.1.1) and so claim (ii) follows immediately.On the other hand, if H ( Z S , T ) tors is non-trivial, then ˜ Q aF is contained in p · Q aF andso (4.1.5) implies that the projection of κ to P Q ⊗ Z p Q aF ⊗ Z / ( p ) vanishes. In this case,therefore, the result of claim (ii) is also clear. (cid:3) Remark 4.4. If G has exponent p and r alg >
0, then a > Q aF is annihilatedby p . In any such case, therefore, Theorem 4.3(ii) implies (under the stated hypotheses)that κ belongs to the image of the map (4.1.1). In general, the argument of Theorem 4.3shows that the group H ( Z S , T ) tors constitutes the obstruction to attempts to deduce thiscontainment from Euler system arguments (via the result of Proposition 3.3). To describethis obstruction more explicitly we assume that E [ p ] is an irreducible G Q -representation. Inthis case, one can assume T = T p ( E ) and then global duality gives rise to an exact sequence E ( Q ) ⊗ Z Z p → M ℓ ∈ S \{∞} lim ←− n E ( Q ℓ ) /p n → (cid:0) H ( Z S , T ) tors (cid:1) ∨ → X ( E/ Q )[ p ∞ ] → , in which the first arrow denotes the natural diagonal map.4.2. Iwasawa-Darmon derivatives.
To consider the above constructions in an Iwasawa-theoretic setting we shall use the following notations for non-negative integers n and i : • Q n : the n -th layer of the cyclotomic Z p -extension Q ∞ / Q (i.e., the subfield of Q ∞ such that [ Q n : Q ] = p n ), • G n := Gal( Q n / Q ), • I n := ker( Z p [ G n ] ։ Z p ) , • Q an := I an /I a +1 n with a := max { , r alg − } as above, • H in := H i ( O Q n ,S , T ), • c,d z n := c,d z Q n , • Γ := Gal( Q ∞ / Q ) , • Λ := Z p [[Γ]] , • I := ker( Z p [[Γ]] ։ Z p ) , • Q a = I a /I a +1 , • H i := lim ←− n H in .4.2.1. We first verify the prediction (3.2.3) in this setting. Proposition 4.5.
For any non-negative integer n , the element c,d z n belongs to I an · H n . In particular, the weak vanishing order prediction of (3.2.3) holds for the field F = Q n for every n .Proof. We use Kato’s result on the Iwasawa Main Conjecture [23, § Q ⊗ Z H is a free Q ⊗ Z Λ-module of rank one. This together with the injectivityof H /I H → H ( Z S , T ) and the assumption that H ( Z S , T ) is Z p -free implies that H is afree Λ-module of rank one. Since H ։ H ( Z S , T ) tf ≃ Z ap is surjective, the characteristic ideal of H is in I a . Therefore, the characteristic ideal of H / h c,d z ∞ i is also in it by [23, Th. 12.5(3)], where c,d z ∞ := ( c,d z n ) n (note that ( c,d z n ) n is in the inverse limit lim ←− n H n = H ). This shows that c,d z ∞ ∈ I a · H , which implies the conlusion of Proposition 4.5. (cid:3) By using Proposition 4.5, we can now explicitly construct the Darmon derivative of c,d z n .To do this we fix a topological generator γ of Γ and denote the image of γ in G n by thesame symbol. In view of Proposition 4.5 one has c,d z n = ( γ − a w n for some choice of element w n of H n .We then compute N Q n / Q ( c,d z n ) = X σ ∈ G n σ ( c,d z n ) ⊗ σ − = X σ ∈ G n σ ( γ − a w n ⊗ σ − = X σ ∈ G n σw n ⊗ σ − ( γ − a ∈ H n ⊗ Z p I an . Thus, in H n ⊗ Z p Q an , we have N Q n / Q ( c,d z n ) = X σ ∈ G n σw n ⊗ ( γ − a . Hence, the derivative in Definition 4.1 is explicitly given by c,d κ n := Cor Q n / Q ( w n ) ⊗ ( γ − a ∈ H ⊗ Z p Q an . (4.2.1)One easily sees that this element is well-defined, i.e., independent of the choice of w n .Furthermore, the collection ( c,d κ n ) n is an inverse system, so we can give the followingdefinition. Definition 4.6.
We define the
Iwasawa-Darmon derivative of Kato’s Euler system by c,d κ ∞ := ( c,d κ n ) n ∈ lim ←− n H ⊗ Z p Q an = H ( Z S , T ) ⊗ Z p Q a . We also define the normalized version κ ∞ := 1 cd ( c − d − · c,d κ ∞ ∈ H ( Z S , V ) ⊗ Z p Q a . Remark 4.7.
The Iwasawa-Darmon derivative can be regarded as a natural analogue ofthe ‘cyclotomic p -units’ that are defined by Solomon in [41] in the setting of the classicalcyclotomic unit Euler system. In a more general setting, it is an analogue of the derivative κ of the (conjectural) Rubin-Stark Euler system that occurs in [7, Conj. 4.2]. Remark 4.8. If r alg is at most one, then a = 0, Q a = Z p and in H ( Z S , V ) one has κ ∞ = z Q := 1 cd ( c − d − · c,d z Q so that Definition 4.6 gives nothing new in this case. The Generalized Perrin-Riou Conjecture at infinite level.
In this section weassume Hypothesis 2.2 in order to state an Iwasawa-theoretic version of Conjecture 2.12.We set r := r alg .4.3.1. To do this we fix a Z p -basis x of V r − Z p H ( Z S , T ) tf and writeBoc n, x = Boc Q n , x : ^ r Z p H ( Z S , T ) → H ( Z S , T ) ⊗ Z p I r − n /I rn for the Bockstein regulator map (2.3.3) for the field Q n , as defined in § n varies these maps combine to induce a homomorphismlim ←− n Boc n, x : ^ r Z p H ( Z S , T ) → H ( Z S , T ) ⊗ Z p lim ←− n I r − n /I rn = H ( Z S , T ) ⊗ Z p Q r − and hence also, by scalar extension, a homomorphismBoc ∞ , x : C p ⊗ Z p ^ r Z p H ( Z S , T ) → C p ⊗ Z p H ( Z S , T ) ⊗ Z p Q r − . (4.3.1)We recall from Definition 2.4 the Birch and Swinnerton-Dyer element η BSD x that is con-structed (unconditionally) in the space C p ⊗ Z p V r Z p H ( Z S , T ). Conjecture 4.9.
One has κ ∞ = Boc ∞ , x ( η BSD x ) in C p ⊗ Z p H ( Z S , T ) ⊗ Z p Q r − . Remark 4.10.
In contrast to the more general situation considered in Conjecture 2.12 wedo not here need to assume η BSD x belongs to V r Z p H ( Z S , T ). This is because the group Q r − is Z p -torsion-free and so one does not lose anything by defining the Bockstein homo-morphism Boc ∞ , x on C p -modules. In particular, if r = 1, then the discussion of Remark2.13 shows that Conjecture 4.9 is equivalent to Perrin-Riou’s original conjecture. Finally,we observe that Conjecture 4.9 is a natural analogue for elliptic curves of the conjectureformulated for the multiplicative group in [7, Conj. 4.2].4.3.2. We shall now give an explicit interpretation of Conjecture 4.9 in terms of the leadingterm L ∗ S ( E,
1) (see Proposition 4.15 below).Take a basis { x , . . . , x r } of E ( Q ) tf . We define an element x ∈ V r − Q p H ( Z S , V ) as theelement corresponding to1 ⊗ x ⊗ ( x ∗ ∧ · · · ∧ x ∗ r ) ∈ Q p ⊗ Z p (cid:16) E ( Q p ) ⊗ Z ^ r Z E ( Q ) ∗ (cid:17) under the isomorphism ^ r − Q p H ( Z S , V ) ≃ Q p ⊗ Z p (cid:16) E ( Q p ) ⊗ Z ^ r Z E ( Q ) ∗ (cid:17) induced by (2.2.2). We note that, by linearity, the definition of the Bockstein regulator map(4.3.1) is extended for any element in V r − Q p H ( Z S , V ), which is not necessarily a Z p -basisof V r − Z p H ( Z S , T ) tf . Thus Boc ∞ , x is defined for above x .Let ω be the fixed N´eron differential and log ω : E ( Q ) → E ( Q p ) → Q p the formallogarithm associated to ω . We give the following definition. Definition 4.11.
We define the
Bockstein regulator associated to ω by setting R Boc ω := log ω ( x ) · Boc ∞ , x ( x ∧ · · · ∧ x r ) ∈ ( Q p ⊗ Z E ( Q )) ⊗ Z p Q r − . (Here we identify H ( Z S , V ) = Q p ⊗ Z E ( Q ) by (2.2.1).) One can check that this does notdepend on the choice of the basis { x , . . . , x r } of E ( Q ) tf . Remark 4.12.
The Bockstein regulator defined above is closely related to classical p -adicregulators: for details, see Theorems 5.6 and 5.11 below. Remark 4.13.
When r = 1, then Boc ∞ , x is the identity map and one has R Boc ω = log ω ( x ) · x ∈ Q p ⊗ Z E ( Q )for any generator x of E ( Q ) tf . Remark 4.14.
Let Ω ξ be as in (2.1.4) and R ∞ the N´eron-Tate regulator. Then one cancheck that Boc ∞ , x ( η BSD x ) = L ∗ S ( E, ξ · R ∞ · R Boc ω . In fact, by the definition of the Birch and Swinnerton-Dyer element, one checks that η BSD x = L ∗ S ( E, ξ · R ∞ · log ω ( x ) · x ∧ · · · ∧ x r . (4.3.2)By Remark 4.14, we obtain the following interpretation of Conjecture 4.9. Proposition 4.15.
Conjecture 4.9 is valid if and only if one has κ ∞ = L ∗ S ( E, ξ · R ∞ · R Boc ω in C p ⊗ Z p H ( Z S , T ) ⊗ Z p Q r − ≃ ( C p ⊗ Z E ( Q )) ⊗ Z p Q r − . L ℓ denotes the Euler factor at a prime ℓ so that one has Y ℓ ∈ S \{∞} L ℓ · L ∗ ( E,
1) = L ∗ S ( E, . We also write v ξ for the non-zero rational number that is defined by the equalityΩ + = v ξ · Ω ξ (4.3.3)where Ω + is the real N´eron period that occurs in (2.5.1). Conjecture 4.16. If X ( E/ Q ) is finite, then in ( Q p ⊗ Z E ( Q )) ⊗ Z p Q r − one has κ ∞ = v ξ Y ℓ ∈ S \{∞} L ℓ X ( E/ Q ) · Tam( E ) E ( Q ) · R Boc ω . Remark 4.17.
One checks easily that Conjecture 4.16 is equivalent to an equality κ ∞ = Boc ∞ , x ( η alg x ) , where x is any non-zero element of V r − Q p H ( Z S , V ) and η alg x is the algebraic Birch andSwinnerton-Dyer element that is defined (unconditionally) in Definition 2.17. Remark 4.18.
In Corollary 6.6 below we will show that Conjecture 4.16 is a refinementof the p -adic Birch-Swinnerton-Dyer Formula (from [30, Chap. II, § p -adic L -function (which we will refer to as a ‘ p -adic Beilinson Formula’).A key advantage of the formulations of Conjectures 4.9 and 4.16 is that they do notinvolve the p -adic L -function and so are not in principle dependent on the precise reductiontype of E at p . In particular, the conjectures make sense (and are canonical) even when E has additive reduction at p .5. p -adic height pairings and the Bockstein regulator In this section, as an important preliminary to the proofs of Theorem 1.8 and Corollaries1.10 and 1.11, we shall make an explicit comparison of the Bockstein regulator R Boc ω definedin Definition 4.11 with the various notions of classical p -adic regulator (see Theorems 5.6and 5.11 below).In the following, we say ‘ p is − ’ if E has − reduction at p . For example, ‘ p is goodordinary’ means that E has good ordinary reduction at p . In this section, we assume that E does not have additive reduction at p . We shall use the same notations as in § § Review of p -adic height pairings. In this section, we give a review of the construc-tion of p -adic height pairing using Selmer complexes.5.1.1. The ordinary case.
Suppose first that p is ordinary, i.e., good ordinary or multiplica-tive. In this case we follow Nekov´aˇr’s construction of a p -adic height pairing in [32, § § ϕ, Γ)-modules.)We recall the definition of Nekov´aˇr’s Selmer complex.To do this we note that, since p is ordinary, we have a canonical filtration F + V ⊂ V of G Q p -modules (due to Greenberg, see [19]).We set F + T := T ∩ F + V . For any non-negative integer n , we also denote the unique p -adic place of Q n by p .Then, following the exact triangle given in (the third row of) [32, (6.1.3.2)], we definethe Selmer complex of T by setting f R Γ f ( Q n , T ) := Cone R Γ( O Q n ,S , T ) → R Γ( Q n, p , T /F + T ) ⊕ M v ∈ S Q n \{ p } R Γ /f ( Q n,v , T ) [ − . We set e H if ( Q n , T ) := H i ( f R Γ f ( Q n , T )) and e H if ( Q n , V ) := Q p ⊗ Z p e H if ( Q n , T ) . We have a natural isomorphism f R Γ f ( Q n , T ) ⊗ L Z p [ G n ] Z p ≃ f R Γ f ( Q , T )(see [32, Prop. 8.10.1] or [18, Prop. 1.6.5(3)]), and so we can define ( − e H f ( Q , T ) → e H f ( Q , T ) ⊗ Z p I n /I n associated to the complex f R Γ f ( Q n , T ) (in the same way as (2.3.2)). Taking lim ←− n and Q p ⊗ Z p − , we obtain a map e β : e H f ( Q , V ) → e H f ( Q , V ) ⊗ Z p I/I . (5.1.1)Combining this map with the global duality map e H f ( Q , V ) → e H f ( Q , V ) ∗ (see [32, § h− , −i p : e H f ( Q , V ) × e H f ( Q , V ) → Q p ⊗ Z p I/I . Noting that there is a natural embedding Q p ⊗ Z E ( Q ) ֒ → e H f ( Q , V ) (see Remark 5.1 below),we obtain the p -adic height pairing h− , −i p : E ( Q ) × E ( Q ) → Q p ⊗ Z p I/I . Remark 5.1. If p is good ordinary or non-split multiplicative, then e H f ( Q , V ) coincideswith the usual Selmer group H f ( Q , V ) (see [32, § p is split multiplicative, then wehave a canonical decomposition e H f ( Q , V ) ≃ H f ( Q , V ) ⊕ Q p (see [32, § Q p ⊗ Z E ( Q ) ֒ → e H f ( Q , V ). Remark 5.2.
For comparisons of the above p -adic height pairing with the classical ones,see [32, §§ The supersingular case.
Suppose that p is good supersingular. In this case we followthe construction of the p -adic height pairing due to Benois [2]. His construction uses Selmercomplexes associated to ( ϕ, Γ)-modules, which was studied by Pottharst [35]. See also thereview in [3].We fix one of the roots α ∈ Q p of the polynomial X − a p X + p . We set L := Q p ( α ) . We also set V L := L ⊗ Q p V and D L := D crys ( V L ) = D dR ( V L ) ≃ L ⊗ Q H ( E/ Q ) , which is endowed with an action of the Frobenius operator ϕ and also a natural decreasingfiltration { D iL } i ∈ Z such that D L ≃ L ⊗ Q Γ( E, Ω E/ Q ). We set t V,L := D L /D L ≃ L ⊗ Q Lie( E ) . Let N α be the subspace of D L on which ϕ acts via αp − . Explicitly, N α is the subspacegenerated by ϕ ( ω ) − α − ω ∈ D L . Then the natural projection D L ։ D L /D L = t V,L inducesan isomorphism N α ∼ −→ t V,L . (5.1.2)A subspace of D L with this property is called a ‘splitting submodule’ in [2, § p -adic height pairing h− , −i p = h− , −i p,α : E ( Q ) × E ( Q ) → L ⊗ Z p I/I . Since there is a natural embedding Q p ⊗ Z E ( Q ) ֒ → H f ( Q , V ), it is sufficient to construct apairing h− , −i p : H f ( Q , V ) × H f ( Q , V ) → L ⊗ Z p I/I . We recall some basic facts from the theory of ( ϕ, Γ)-modules. Let D † rig ( V L ) denote the( ϕ, Γ Q p )-module associated V L (where Γ Q p := Gal( Q p ( µ p ∞ ) / Q p )). (See [2, Th. 2.1.3].) By[2, Th. 2.2.3], there is a submodule D α ⊂ D † rig ( V L ) corresponding to N α ⊂ D L . For ageneral ( ϕ, Γ Q p )-module D , one can define a complex (the ‘Fontaine-Herr complex’) R Γ( Q p , D ) , which is denoted by C • ϕ,γ Q p ( D ) in [2, § D = D † rig ( V L ), this is naturally quasi-isomorphic to R Γ( Q p , V L ) (see [2, Prop. 2.5.2]). So there is a natural morphism in thederived category of L -vector spaces R Γ( Z S , V L ) → R Γ( Q p , V L ) ≃ R Γ( Q p , D † rig ( V L )) → R Γ( Q p , D † rig ( V L ) / D α ) . We define the Selmer complex by f R Γ f ( Q , V L ) := Cone R Γ( Z S , V L ) → R Γ( Q p , D † rig ( V L ) / D α ) ⊕ M ℓ ∈ S \{ p } R Γ /f ( Q ℓ , V L ) [ − . (We adopt [3, (2.6)] as the definition.) We set e H if ( Q , V L ) := H i ( f R Γ f ( Q , V L )). It is knownthat H f ( Q , V L ) ≃ e H f ( Q , V L ) . (See [2, Th. III].)We next study the Iwasawa theoretic version. We set H := ( f ( X ) = ∞ X n =0 c n X n ∈ L [[ X ]] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( X ) converges on the open unit disk ) . Then, for a general ( ϕ, Γ Q p )-module D , one can define an Iwasawa cohomology complex of H -modules R Γ Iw ( Q p , D ) . (See [2, § γ ∈ Γ. Then Γ acts on H by identifying X = γ −
1. We set V L := V L ⊗ L H , where G Q acts on H via G Q ։ Γ γ γ − −−−−→ Γ . When D = D † rig ( V L ), we have a natural quasi-isomorphism R Γ Iw ( Q p , D ) ≃ R Γ( Q p , V L ) (see[2, Th. 2.8.2]). Thus there is a natural morphism in the derived category of H -modules R Γ( Z S , V L ) → R Γ( Q p , V L ) ≃ R Γ Iw ( Q p , D † rig ( V L )) → R Γ Iw ( Q p , D † rig ( V L ) / D α ) . We define the Iwasawa Selmer complex by f R Γ f, Iw ( Q , V L ) := Cone R Γ( Z S , V L ) → R Γ Iw ( Q p , D † rig ( V L ) / D α ) ⊕ M ℓ ∈ S \{ p } R Γ /f ( Q ℓ , V L ) [ − . We know the following ‘control theorem’ f R Γ f, Iw ( Q , V L ) ⊗ L H L ≃ f R Γ f ( Q , V L ) . (5.1.3)(See [35, Th. 1.12].)We now give the definition of the p -adic height pairing. Let I := ( X ) be the augmentationideal of H . Note that I / I is identified with L ⊗ Z p I/I . From the exact sequence0 → I / I → H / I → L → , we obtain the exact triangle f R Γ f, Iw ( Q , V L ) ⊗ L H I / I → f R Γ f, Iw ( Q , V L ) ⊗ L H H / I → f R Γ f, Iw ( Q , V L ) ⊗ L H L. By the control theorem (5.1.3), we have f R Γ f ( Q , V L ) ⊗ L L I / I → f R Γ f, Iw ( Q , V L ) ⊗ L H H / I → f R Γ f ( Q , V L ) . The ( − e H f ( Q , V L ) → H ( f R Γ f ( Q , V L ) ⊗ L L I / I ) = e H f ( Q , V L ) ⊗ L I / I . Composing this map with the global duality map e H f ( Q , V L ) → e H f ( Q , V L ) ∗ (see [2, Th. 3.1.5]), we obtain e H f ( Q , V L ) → e H f ( Q , V L ) ∗ ⊗ L I / I . This gives the desired p -adic height pairing. Remark 5.3.
The above construction makes sense even when p is good ordinary. In thiscase, α is canonically chosen so that ord p ( α ) <
1, and we can take N α to be D crys ( F + V ).One sees that the p -adic height pairing with this choice coincides with that in § Remark 5.4.
Comparisons of this p -adic height pairing with the classical ones are studiedin detail by Benois [2]. In particular, this p -adic height pairing coincides with the oneconstructed by Nekov´aˇr in [31], which is used by Kobayashi in [26].5.2. A comparison result.
We shall define the p -adic regulator and compare it with theBockstein regulator R Boc ω . In this subsection, we assume Hypothesis 2.2. L be the splitting field of the polynomial X − a p X + p over Q p . Note that L = Q p unless p is supersingular.Let h− , −i p : E ( Q ) × E ( Q ) → L ⊗ Z p I/I be the p -adic height pairing defined above. (When p is supersingular, this depends on thechoice of a root α of X − a p X + p .) Definition 5.5.
The p -adic regulator R p = R p,α ∈ L ⊗ Z p Q r is defined to be the discriminant of the p -adic height pairing, i.e., R p := det( h x i , x j i p ) ≤ i,j ≤ r with { x , . . . , x r } a basis of E ( Q ) tf .The p -adic height pairing induces a map E ( Q ) × ( Q p ⊗ Z E ( Q )) ⊗ Z p Q r − → L ⊗ Z p Q r (5.2.1) ( x, ( a ⊗ y ) ⊗ b ) a · b · h x, y i p , which we denote also by h− , −i p .The following gives a relation between R p and R Boc ω . Theorem 5.6.
For any x ∈ E ( Q ) we have h x, R Boc ω i p = log ω ( x ) · R p . § Lemma 5.7.
The p -adic height pairing is symmetric, i.e., h x, y i p = h y, x i p for any x, y ∈ E ( Q ) .Proof. See [32, Cor. 11.2.2] and [2, Th. I] in the ordinary and supersingular cases respec-tively. (cid:3)
Lemma 5.8.
The following diagram is commutative. E ( Q ) / / lim ←− n β n ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ ( L ⊗ Z E ( Q )) ∗ ⊗ Z p I/I . . ) (cid:15) (cid:15) (cid:15) (cid:15) L ⊗ Q p H ( Z S , V ) ⊗ Z p I/I , where the horizontal arrow is the map induced by the p -adic height pairing x ( y
7→ h x, y i p ) . (For the definition of β n := β Q n , see (2.3.2).) Proof.
We first suppose that p is ordinary. We have the commutative diagram f R Γ f ( Q , T ) ⊗ L Z p I n /I n / / (cid:15) (cid:15) f R Γ f ( Q n , T ) ⊗ L Z p [ G n ] Z p [ G n ] /I n / / (cid:15) (cid:15) f R Γ f ( Q , T ) (cid:15) (cid:15) R Γ( Z S , T ) ⊗ L Z p I n /I n / / R Γ( O Q n ,S , T ) ⊗ L Z p [ G n ] Z p [ G n ] /I n / / R Γ( Z S , T ) , whose rows are exact triangles. The map β n is defined by the connecting homomorphismof the bottom triangle. On the other hand, the p -adic height pairing is defined by theconnecting homomorphism of the top triangle. Thus the claim follows from the functorialityof the connecting homomorphism, i.e., the commutativity of the diagram e H f ( Q , T ) / / (cid:15) (cid:15) e H f ( Q , T ) ⊗ Z p I n /I n (cid:15) (cid:15) H ( Z S , T ) / / H ( Z S , T ) ⊗ Z p I n /I n , where the horizontal arrows are connecting homomorphisms.Next, suppose that p is good supersingular. With the notations in § f R Γ f ( Q , V L ) ⊗ L L I / I / / (cid:15) (cid:15) f R Γ f, Iw ( Q , V L ) ⊗ L H H / I / / (cid:15) (cid:15) f R Γ f ( Q , V L ) (cid:15) (cid:15) R Γ( Z S , V L ) ⊗ L L I / I / / R Γ( Z S , V L ) ⊗ L H H / I / / R Γ( Z S , V L ) . Since the map lim ←− n β n coincides with the map defined by the connecting homomorphism ofthe bottom triangle (by Shapiro’s lemma), the claim follows by the same argument as inthe ordinary case. (cid:3) Lemma 5.9.
Let M and N be L -vector spaces of dimension r and r − respectively. Supposethat an exact sequence → N ι −→ M ℓ −→ L → and L -linear maps f : M → M ∗ and g : M → N ∗ are given. Assume the following. (a) The diagram M f / / g ! ! ❉❉❉❉❉❉❉❉ M ∗ ι ∗ (cid:15) (cid:15) (cid:15) (cid:15) N ∗ is commutative. (b) The map f satisfies f ( x )( y ) = f ( y )( x ) for any x, y ∈ M . Then for any x ∈ M the following diagram is commutative. V rL M V r f / / V r − g (cid:15) (cid:15) V rL M ∗ ℓ ( x ) × ' ' ◆◆◆◆◆◆◆◆◆◆◆ V rL M ∗ M ⊗ L V r − L N ∗ δ ≃ / / M ⊗ L V rL M ∗ . f ( x ) ⊗ id ♣♣♣♣♣♣♣♣♣♣♣ (5.2.3) Here δ is the natural isomorphism induced by (5.2.2), and the left vertical arrow is definedby (cid:16)^ r − g (cid:17) ( x ∧ · · · ∧ x r ) = r X i =1 ( − i +1 x i ⊗ g ( x ) ∧ · · · ∧ g ( x i − ) ∧ g ( x i +1 ) ∧ · · · ∧ g ( x r ) . Proof.
Let { x , . . . , x r } be a basis of M and fix x ∈ M . It is sufficient to prove f ( x ) ◦ δ ◦ (cid:16)^ r − g (cid:17) ( x ∧ · · · ∧ x r ) = ℓ ( x ) · f ( x ) ∧ · · · ∧ f ( x r ) . We shall describe the left hand side explicitly. Using assumption (a), we have δ ◦ (cid:16)^ r − g (cid:17) ( x ∧ · · · ∧ x r )(5.2.4) = r X i =1 x i ⊗ f ( x ) ∧ · · · ∧ f ( x i − ) ∧ ℓ ∧ f ( x i +1 ) ∧ · · · ∧ f ( x r ) . Thus we have f ( x ) ◦ δ ◦ (cid:16)^ r − g (cid:17) ( x ∧· · ·∧ x r ) = r X i =1 f ( x )( x i ) · f ( x ) ∧· · ·∧ f ( x i − ) ∧ ℓ ∧ f ( x i +1 ) ∧· · ·∧ f ( x r ) . Suppose first that f is bijective. Then { f ( x ) , . . . , f ( x r ) } is a basis of M ∗ and we canwrite ℓ = r X i =1 a i f ( x i ) in M ∗ with some a , . . . , a r ∈ L . By assumption (b), we have f ( x )( x i ) = f ( x i )( x ) and so wecompute r X i =1 f ( x )( x i ) · f ( x ) ∧ · · · ∧ f ( x i − ) ∧ ℓ ∧ f ( x i +1 ) ∧ · · · ∧ f ( x r )= r X i =1 a i f ( x i )( x ) · f ( x ) ∧ · · · ∧ f ( x r )= ℓ ( x ) · f ( x ) ∧ · · · ∧ f ( x r ) . This proves the lemma in this case. Suppose next that f is not bijective. Then { f ( x ) , . . . , f ( x r ) } is linearly dependent sowe may assume f ( x ) = r X i =2 a i f ( x i )with some a , . . . , a r ∈ L .We then compute r X i =1 f ( x )( x i ) · f ( x ) ∧ · · · ∧ f ( x i − ) ∧ ℓ ∧ f ( x i +1 ) ∧ · · · ∧ f ( x r )= r X i =1 f ( x i )( x ) · f ( x ) ∧ · · · ∧ f ( x i − ) ∧ ℓ ∧ f ( x i +1 ) ∧ · · · ∧ f ( x r )= r X i =2 a i f ( x i )( x ) ! · ℓ ∧ f ( x ) ∧ · · · ∧ f ( x r )+ r X i =2 f ( x i )( x ) · r X j =2 a j f ( x j ) ∧ f ( x ) ∧ · · · ∧ f ( x i − ) ∧ ℓ ∧ f ( x i +1 ) ∧ · · · ∧ f ( x r )= r X i =2 a i f ( x i )( x ) · ℓ ∧ f ( x ) ∧ · · · ∧ f ( x r )+ r X i =2 a i f ( x i )( x ) · f ( x i ) ∧ f ( x ) ∧ · · · ∧ f ( x i − ) ∧ ℓ ∧ f ( x i +1 ) ∧ · · · ∧ f ( x r )= 0 . Since V r f is also zero in this case, this proves the desired commutativity. (cid:3) M := L ⊗ Z E ( Q ), N := L ⊗ Q p H ( Z S , V ) ∗ andthe exact sequence 0 → L ⊗ Q p H ( Z S , V ) ∗ → L ⊗ Z E ( Q ) log ω −−→ L → , which is obtained from (2.2.2) (so we let ℓ in (5.2.2) be log ω ). We fix a Z p -basis of I/I and identify it with Z p . By letting f : M → M ∗ ; x ( y
7→ h x, y i p )and g := lim ←− n β n : M → N ∗ , we see that assumptions (a) and (b) in Lemma 5.9 are satisfied by Lemmas 5.8 and 5.7respectively.Let { x , . . . , x r } be a basis of E ( Q ) tf ⊂ M . By the definition of R p , we have (cid:16)^ r f (cid:17) ( x ∧ · · · ∧ x r ) = R p · x ∗ ∧ · · · ∧ x ∗ r ∈ ^ rL M ∗ . On the other hand, we have δ ◦ (cid:16)^ r − g (cid:17) ( x ∧ · · · ∧ x r ) = R Boc ω ⊗ ( x ∗ ∧ · · · ∧ x ∗ r ) ∈ M ⊗ L ^ rL M ∗ , (5.2.5)where δ is as in (5.2.3). This again follows from the definition of R Boc ω . Hence, for any x ∈ E ( Q ), the commutativity of (5.2.3) implies f ( x )( R Boc ω ) = ℓ ( x ) · R p , i.e., h x, R Boc ω i p = log ω ( x ) · R p . This completes the proof of Theorem 5.6.5.3.
Schneider’s height pairing.
We now consider the case that p is split multiplicative.In this case, the classical p -adic height pairing constructed by Schneider [39] is differentfrom that of Nekov´aˇr constructed above. Explicitly, Schneider’s p -adic height pairing h− , −i Sch p : E ( Q ) × E ( Q ) → Q p ⊗ Z p I/I is related to Nekov´aˇr’s height pairing h− , −i p by ℓ p ( h x, y i Sch p ) = ℓ p ( h x, y i p ) − log ω ( x ) log ω ( y )log p ( q E ) in Q p , (5.3.1)where ℓ p denotes the isomorphism ℓ p : Q p ⊗ Z p I/I γ − γ −−−−−→ Q p ⊗ Z p Γ χ cyc −−→ Q p ⊗ Z p (1 + p Z p ) log p −−→ Q p , (5.3.2)with χ cyc the cyclotomic character, and q E ∈ Q p is the p -adic Tate period of E . (See [32, Th.11.4.6], where Schneider’s height is denoted by h norm π .) Note that, by the so-called ‘SaintEtienne Theorem’ of Barr´e-Sirieix, Diaz, Gramain and Philibert [1], one has log p ( q E ) = 0and so the above formula makes sense. Since the relation (5.3.1) characterizes h− , −i Sch p ,we adopt it as the definition of Schneider’s p -adic height pairing. Definition 5.10.
We define Schneider’s p -adic regulator R Sch p ∈ Q p ⊗ Z p Q r by the discriminant of Schneider’s p -adic height pairing, i.e., R Sch p := det( h x i , x j i Sch p ) ≤ i,j ≤ r with { x , . . . , x r } a basis of E ( Q ) tf .We identify Q p ⊗ Z p I/I = Q p via the isomorphism ℓ p . By using the relation (5.3.1), onechecks that R Sch p = R p − p ( q E ) r X i =1 log ω ( x i ) det h x , x i p h x , x i p · · · log ω ( x ) · · · h x , x r i p h x , x i p · · · · · · log ω ( x ) · · · h x , x r i p ... ... ... h x r , x i p · · · · · · log ω ( x r ) · · · h x r , x r i p , where the vector (log ω ( x j )) j is put on the i -th column in the matrix on the right hand side.In fact, this follows from the elementary formuladet( a ij + cb i b j ) = det( a ij ) + c r X i =1 b i det a a · · · b · · · a r a · · · · · · b · · · a r ... ... ... a r · · · · · · b r · · · a rr (with the vector ( b j ) j put on the i -th column). Furthermore, by (5.2.4) and (5.2.5), we have R Boc ω = r X i =1 x i ⊗ det h x , x i p h x , x i p · · · log ω ( x ) · · · h x , x r i p h x , x i p · · · · · · log ω ( x ) · · · h x , x r i p ... ... ... h x r , x i p · · · · · · log ω ( x r ) · · · h x r , x r i p , and hence we have R Sch p = R p − log ω ( R Boc ω )log p ( q E ) . From this and Theorem 5.6, we obtain the following.
Theorem 5.11.
For any x ∈ E ( Q ) we have h x, R Boc ω i Sch p = log ω ( x ) · R Sch p . The Generalized Rubin Formula and consequences
In this section we relate Conjectures 4.9 and 4.16 to the p -adic analogue of the Birchand Swinnerton-Dyer conjecture formulated by Mazur, Tate and Teitelbaum in [30] (seeCorollaries 6.6 and 6.7).In particular, we continue to assume in this section that E does not have additive reduc-tion at p .6.1. Review of the p -adic L -function. In this subsection, we review the p -adic L -function of Mazur-Tate-Teitelbaum [30]. See also the review in [23, § p is good, let α ∈ Q p be a root of X − a p X + p such that ord p ( α ) < β (:= p/α ) the other root. Note that, when p is good ordinary, α is uniquelydetermined by this property.When p is split (resp. non-split) multiplicative, we set α := 1 (resp. −
1) and β := p (resp. − p ).We set L := Q p ( α ) . Note that L = Q p unless p is supersingular.Recall that Q ∞ / Q denotes the cyclotomic Z p -extension and Γ := Gal( Q ∞ / Q ). Let b Γdenote the set of Q -valued characters of Γ of finite order.Recall also that an embedding Q ֒ → C is fixed. For a positive integer m , let ζ m ∈ Q bethe element corresponding to e π √− /m ∈ C . We also fix an isomorphism C ≃ C p . From this, we obtain an embedding Q ֒ → Q p . Thus each character in b Γ is regarded both Q p and C -valued.As in §
2, we fix a N´eron differential ω ∈ Γ( E, Ω E/ Q ). Let ξ be the element of SL ( Z )used in the construction of Kato’s Euler system (and normalized as in (2.1.3). Let Ω ξ bethe real period associated to ( ω, ξ ) (see (2.1.4)).We fix a topological generator γ of Γ. Then we have a natural identification O L [[Γ]] = O L [[ γ − . Let | − | p : C p → R ≥ denote the p -adic absolute value normalized by | p | p = p − . For apositive integer h , we define H h := ( ∞ X n =0 c n ( γ − n ∈ L [[ γ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim n →∞ | c n | p n h = 0 ) and H ∞ := [ h H h . For any continuous character χ : Γ → Q × p and f = P n c n ( γ − n ∈ H ∞ , we can define anatural evaluation χ ( f ) := X n c n ( χ ( γ ) − n ∈ Q p . It is known that there is a unique element (the ‘ p -adic L -function’ of E ) L S,p = L S,p,α,ω,ξ ∈ H that has the following property: for any character χ ∈ b Γ one has χ ( L S,p ) = (cid:18) − α (cid:19) (cid:18) − β (cid:19) − L S ( E, ξ if χ = 1 ,τ ( χ ) α n L S ( E, χ − , ξ if χ has conductor p n > . Here in the latter case τ ( χ ) denotes the Gauss sum τ ( χ ) := X σ ∈ Gal( Q ( µ pn ) / Q ) χ ( σ ) ζ σp n , and L S ( E, χ − , s ) denotes the S -truncated Hasse-Weil L -function of E twisted by χ − . Forthe construction of L S,p from Kato’s Euler system, see Theorem 6.10 below.Let I := ( γ −
1) be the augmentation ideal of H ∞ . For a non-negative integer a , we set Q a := I a / I a +1 . Note that we have a natural identification Q a = L ⊗ Z p Q a . We know the following ‘order of vanishing’ (which is actually a consequence of Proposition4.5). Proposition 6.1 ([23, Th. 18.4]) . Set r := rank Z ( E ( Q )) . Then we have L S,p ∈ ( I r if p is good or non-split multiplicative, I r +1 if p is split multiplicative. The Generalized Rubin Formula.
Let L ( r ) S,p (resp. L ( r +1) S,p ) denote the image of L S,p ∈ I r (resp. I r +1 ) in Q r (resp. Q r +1 ) when p is good or non-split multiplicative (resp.split multiplicative).Recall some notations. Let h− , −i p = h− , −i p,α : E ( Q ) × ( Q p ⊗ Z E ( Q )) ⊗ Z p Q r − → L ⊗ Z p Q r = Q r be the map induced by the p -adic height pairing (see (5.2.1)). Let log ω : E ( Q p ) → Q p bethe formal logarithm associated to the fixed N´eron differential ω . Let κ ∞ ∈ H ( Z S , V ) ⊗ Z p Q r − ≃ ( Q p ⊗ Z E ( Q )) ⊗ Z p Q r − . be the Iwasawa-Darmon derivative in Definition 4.6.The following is a generalization of ‘Rubin’s formula’ for the higher rank case. Theorem 6.2 (The Generalized Rubin Formula) . Under Hypothesis 2.2, we have the fol-lowing. (i) If p is good or non-split multiplicative, then for any x ∈ E ( Q ) we have h x, κ ∞ i p = (cid:18) − α (cid:19) − (cid:18) − β (cid:19) log ω ( x ) · L ( r ) S,p in Q r . (ii) If p is split multiplicative, then for any x ∈ E ( Q ) we have h x, κ ∞ i Sch p · p ( q E ) (1 − rec p ( q E )) = (cid:18) − p (cid:19) log ω ( x ) · L ( r +1) S,p in Q r +1 . Here q E ∈ Q × p denotes the p -adic Tate period of E and rec p : Q × p → Γ the localreciprocity map. The proof of this theorem will be given in § Remark 6.3.
When r = 1, we have κ ∞ = z Q (see Remark 4.8), so Theorem 6.2(i) asserts h x, z Q i p = (cid:18) − α (cid:19) − (cid:18) − β (cid:19) log ω ( x ) · L (1) S,p in I / I . When p is good ordinary, this formula is proved by Rubin [36, Th. 1(ii)], which we call‘Rubin’s formula’ (following Nekov´aˇr [32, (11.3.14)]). (Note that ‘ L ′ z ,ω ( )’ in [36, Th. 1(ii)]corresponds to our (cid:0) − α (cid:1) L (1) S,p .) Thus Theorem 6.2(i) is regarded as a ‘higher rank’ gen-eralization of Rubin’s formula.
Remark 6.4.
The element 1ord p ( q E ) (1 − rec p ( q E )) ∈ Q p ⊗ Z p I/I
20 DAVID BURNS, MASATO KURIHARA AND TAKAMICHI SANO appearing in Theorem 6.2(ii) is essentially the ‘ L -invariant’. In fact, one checks that theimage of this element under the isomorphism Q p ⊗ Z p I/I γ − γ −−−−−→ Q p ⊗ Z p Γ χ cyc −−→ Q p ⊗ Z p (1 + p Z p ) log p −−→ Q p (see (5.3.2)) is the usual L -invariant log p ( q E )ord p ( q E ) . Remark 6.5.
When r = 1, Theorem 6.2(ii) is obtained by Venerucci [42, Th. 12.31] andB¨uy¨ukboduk [12, Th. 3.22].A proof of Theorem 6.2 will be given in § v ξ ∈ Q × is defined by Ω + = v ξ · Ω ξ (see (4.3.3)). Corollary 6.6.
Conjecture 4.16 implies the p -adic Birch-Swinnerton-Dyer Formula in [30,Chap. II, § , i.e., (cid:18) − α (cid:19) − (cid:18) − β (cid:19) · L ( r ) S,p = v ξ Y ℓ ∈ S \{∞} L ℓ X ( E/ Q ) · Tam( E ) E ( Q ) R p if p is good or non-split multiplicative, and L ( r +1) S,p = 1ord p ( q E ) (1 − rec p ( q E )) · v ξ Y ℓ ∈ S \{∞ ,p } L ℓ X ( E/ Q ) · Tam( E ) E ( Q ) R Sch p if p is split multiplicative.If R p = 0 (resp. R Sch p = 0 ), then the converse also holds when p is good or non-splitmultiplicative (resp. split multiplicative).Proof. We only treat the case when p is good or non-split multiplicative. The case when p is split multiplicative is treated similarly, by using Theorem 5.11.Conjecture 4.16 asserts κ ∞ = v ξ Y ℓ ∈ S \{∞} L ℓ X ( E/ Q )Tam( E ) E ( Q ) · R Boc ω in ( Q p ⊗ Z E ( Q )) ⊗ Z p Q r − . Take x ∈ E ( Q ) such that log ω ( x ) = 0. Taking h x, −i p to both sides, we obtain (cid:18) − α (cid:19) − (cid:18) − β (cid:19) log ω ( x ) · L ( r ) S,p = v ξ Y ℓ ∈ S \{∞} L ℓ X ( E/ Q )Tam( E ) E ( Q ) log ω ( x ) R p by Theorems 6.2 and 5.6. Since log ω ( x ) = 0, we can cancell it from both sides and obtainthe desired formula.If R p = 0, then the map y ( x
7→ h x, y i p ) is injective, and so the converse holds. (cid:3) Similarly, we also obtain the following. Corollary 6.7.
Conjecture 4.9 implies the p -adic Beilinson Formula, i.e., (cid:18) − α (cid:19) − (cid:18) − β (cid:19) · L ( r ) S,p = L ∗ S ( E, ξ · R ∞ R p (6.2.1) if p is good or non-split multiplicative, and L ( r +1) S,p = 1ord p ( q E ) (1 − rec p ( q E )) · L ∗ S \{ p } ( E, ξ · R ∞ R Sch p (6.2.2) if p is split multiplicative.If R p = 0 (resp. R Sch p = 0 ), then the converse also holds when p is good or non-splitmultiplicative (resp. split multiplicative).Proof. This follows by the same argument as the proof of Corollary 6.6, using Proposition4.15. (cid:3)
Remark 6.8.
When p is good and r an = r = 1, the formula (6.2.1) was proved by Perrin-Riou [33, Cor. 1.8] in the ordinary case, and by Kobayashi [26, Cor. 1.3] in the supersingularcase. (It is essentially the ‘ p -adic Gross-Zagier Formula’.) When p is split multiplicativeand r an = r = 0, the formula (6.2.2) was first proved by Greenberg and Stevens [20] andthen by Kobayashi [25].6.3. Review of the Coleman map.
As a preliminary of the proof of Theorem 6.2, wereview the construction of the Coleman map. We follow the explicit construction due toRubin [37, Appendix]. See also [27, § D := D crys ( V ) . Let ϕ denote the Frobenius operator acting on D . For a finite extension K/ Q p , we set D K := K ⊗ Q p D. Let [ − , − ] K : ( K ⊗ Q p D dR ( V )) × D K → K denote the natural pairing.We use the following fact. Lemma 6.9 ([23, Th. 16.6(1)]) . Set L := Q p ( α ) . There exists a unique ν = ν α,ω ∈ D L such that ϕ ( ν ) = αp − ν = β − ν and [ ω, ν ] L = 1 . Let Q n,p denote the completion of Q n at the unique prime lying above p . We set L n := L · Q n,p . Let ν ∈ D L be as in Lemma 6.9 and set δ n := 1 p n +1 Tr L ( µ pn +1 ) /L n n X i =0 ζ p n +1 − i ϕ i − n − ( ν ) + (1 − ϕ ) − ( ν ) ! (6.3.1) = 1 α n +1 Tr L ( µ pn +1 ) /L n n X i =0 ζ p n +1 − i − β i + ββ − ! ν ∈ D L n . This element satisfies Tr L n +1 /L n ( δ n +1 ) = δ n and for any character χ of G n X σ ∈ G n σ ( δ n ) χ ( σ ) = (cid:18) − α (cid:19) (cid:18) − β (cid:19) − ν if χ = 1, τ ( χ ) α m ν if χ has conductor p m > D L ( µ pn +1 ) (see [37, Lem. A.1] or [27, Lem. 3.1]).As in § H in := H i ( O Q n ,S , T ) and H i := lim ←− n H in . We define a map Col n : H n → L [ G n ]by Col n ( z ) := X σ ∈ G n Tr L n /L ([exp ∗ n ( z ) , σδ n ] L n ) σ, where exp ∗ n = exp ∗ Q n,p ,V : H n → H ( Q n,p , T ) → Q n,p ⊗ Q p D dR ( V )denotes the Bloch-Kato dual exponential map. This map induces a map on the inverse limitCol := lim ←− n Col n : H → H ∞ . This is the definition of the Coleman map.We set t c,d := cd ( c − σ c )( d − σ d ) ∈ Z p [[Γ]] . (6.3.3)Here σ a ∈ Γ is the restriction of the automorphism of Q ( µ p ∞ ) characterized by ζ σ a p n = ζ ap n for every n .The following result is well-known. Theorem 6.10 (Kato [23, Th. 16.6(2)]) . We have
Col(( c,d z n ) n ) = t c,d · L S,p . The proof of Theorem 6.2.
In this subsection, we prove Theorem 6.2.6.4.1. We first establish several important preliminary results.We initially suppose that p is good or non-split multiplicative, and give a proof of Theorem6.2(i).We shall use the derivative introduced by Nekov´aˇr in [32, § § F − V := ( V /F + V if p is ordinary , D † rig ( V L ) / D α if p is supersingular . For y ∈ H , we define ‘Rubin’s derivative’ D ( y ) ∈ H ( Q p , F − V ) ⊗ Z p I/I as follows. (Compare the definition given by Nekov´aˇr in [32, § Dx Iw ’ is used.)Suppose first that p is ordinary. We have a commutative diagram with exact rows andcolumns f R Γ f ( Q , V ) ⊗ L Z p I/I / / (cid:15) (cid:15) R Γ( Z S , V ) ⊗ L Z p I/I / / (cid:15) (cid:15) R Γ( Q p , F − V ) ⊗ L Z p I/I i (cid:15) (cid:15) f R Γ f, Iw ( Q , V ) ⊗ L Λ Λ /I / / (cid:15) (cid:15) R Γ Iw ( Z S , V ) ⊗ L Λ Λ /I p / / (cid:15) (cid:15) R Γ Iw ( Q p , F − V ) ⊗ L Λ Λ /I (cid:15) (cid:15) f R Γ f ( Q , V ) / / R Γ( Z S , V ) / / R Γ( Q p , F − V ) . (6.4.1)Here f R Γ f ( Q , V ) := f R Γ f ( Q , T ) ⊗ L Z p Q p and f R Γ f, Iw ( Q , V ) := lim ←− n f R Γ f ( Q n , T ) ! ⊗ L Z p Q p . R Γ Iw ( Z S , V ) and R Γ Iw ( Q p , F − V ) are defined in a similar way.We regard y ∈ H as an element of H ( R Γ Iw ( Z S , V ) ⊗ L Λ Λ /I ). Since y lies in e H f ( Q , V )and H ( Q p , F − V ) = 0, a diagram chasing shows that there exists a unique element D ( y ) ∈ H ( Q p , F − V ) ⊗ Z p I/I such thatloc p ( y ) = i ( D ( y )) in H ( R Γ Iw ( Q p , F − V ) ⊗ L Λ Λ /I ) . This gives the definition of Rubin’s derivative in this case.When p is supersingular, Rubin’s derivative is defined in the same way, by consideringthe commutative diagram with exact rows and columns f R Γ f ( Q , V L ) ⊗ L Z p I/I / / (cid:15) (cid:15) R Γ( Z S , V L ) ⊗ L Z p I/I / / (cid:15) (cid:15) R Γ( Q p , F − V ) ⊗ L Z p I/I (cid:15) (cid:15) f R Γ f, Iw ( Q , V L ) ⊗ L H H / I / / (cid:15) (cid:15) R Γ( Z S , V L ) ⊗ L H H / I / / (cid:15) (cid:15) R Γ Iw ( Q p , F − V ) ⊗ L H H / I (cid:15) (cid:15) f R Γ f ( Q , V L ) / / R Γ( Z S , V L ) / / R Γ( Q p , F − V ) . Let ( − , − ) p : H f ( Q p , V ) × H ( Q p , F − V ) → H ( Q p , L (1)) ≃ L be the pairing defined by the cup product. This pairing induces( − , − ) p : E ( Q ) × ( H ( Q p , F − V ) ⊗ Z p I/I ) → L ⊗ Z p I/I = I / I . (6.4.2) The following is an abstract version of Rubin’s formula.
Theorem 6.11 (Rubin, Nekov´aˇr) . Suppose that p is not split multiplicative. For any x ∈ E ( Q ) and y = ( y n ) n ∈ lim ←− n H n = H , we have h x, y i p = ( x, D ( y )) p in I / I . Proof.
This is proved in [32, Prop. 11.3.15]. We give a proof for the reader’s convenience.We treat only the ordinary case, since the supersingular case is treated in a similar way.Recall that the map e β : e H f ( Q , V ) → e H f ( Q , V ) ⊗ Z p I/I in (5.1.1) is defined to be( − δ : H ( Q p , F − V ) ⊗ Z p I/I → e H f ( Q , V ) be the connecting homomorphism of the upper hor-izontal triangle of (6.4.1). Then, by the compatibility of connecting homomorphisms (see[32, Lem. 1.2.19]), we have e β ( y ) = δ ( D ( y )) . We identify e H f ( Q , V ) = e H f ( Q , V ) ∗ = Q p ⊗ Z E ( Q ) ∗ by global duality. Then for any x ∈ E ( Q ) we have e β ( y )( x ) = h x, y i p by the definition of the p -adic height pairing. On the other hand, by the compatibilitybetween local and global duality, we have δ ( D ( y ))( x ) = ( x, D ( y )) p . Thus we have h x, y i p = ( x, D ( y )) p . (cid:3) We shall now apply Theorem 6.11 in our setting.
Lemma 6.12.
Let c,d κ ∞ ∈ H ⊗ Z p Q r − be the Iwasawa-Darmon derivative in Definition4.6. Then there exists a unique w = ( w n ) n ∈ lim ←− n H n = H such that c,d z n = ( γ − r − w n for every n and c,d κ ∞ = w ⊗ ( γ − r − . Proof.
By Proposition 4.5, one can take w n ∈ H n such that c,d z n = ( γ − r − w n . Thiselement is well-defined modulo H , so we see that the collection ( w n ) n is an inverse systemin lim ←− n H n /p n . However, since lim ←− n H n /p n is isomorphic to lim ←− n H n = H , we can take each w n ∈ H n so that ( w n ) n ∈ H . The description of c,d κ ∞ follows from (4.2.1). (cid:3) By Lemma 6.12, we can define the ‘Rubin’s derivative of the Iwasawa-Darmon derivative’ D ( c,d κ ∞ ) := D ( w ) · ( γ − r − ∈ H ( Q p , F − V ) ⊗ Z p Q r . Applying Theorem 6.11 to this element, we obtain the following. Corollary 6.13.
For any x ∈ E ( Q ) , we have h x, c,d κ ∞ i p = ( x, D ( c,d κ ∞ )) p in Q r , where ( − , − ) p : E ( Q ) × ( H ( Q p , F − V ) ⊗ Z p Q r ) → Q r is the map induced by (6.4.2). Lemma 6.14.
Let y ∈ H . Then we have Col( y ) ∈ I and Col( y ) = (exp ( δ ) , D ( y )) p in I / I , where exp = exp Q p ,V : D L → L ⊗ Q p H f ( Q p , V ) denotes the Bloch-Kato exponential map.Proof. We shall show the first claim. By the construction of the Coleman map, it is sufficientto show that X σ ∈ G n Tr L n /L ([exp ∗ n ( y n ) , σδ n ] L n ) = 0for every n . The left hand side is equal to [exp ∗ ( y ) , δ ] L . Since y lies in H f ( Q , V ), weknow that exp ∗ ( y ) = 0 and so we have proved the first claim.Next, we shall show the second claim. Note that, by construction, we haveCol n ( y n ) = X σ ∈ G n (exp n ( δ n ) , σy n ) L n σ − , where exp n : D L n → H f ( L n , V ) denotes the Bloch-Kato exponential map and( − , − ) L n : H f ( L n , V ) × H ( Q n,p , F − V ) → L denotes the cup product pairing. Noting this, one verifiesCol( y ) = (exp ( δ ) , D ( y )) p in I / I by the definition of D ( y ). (cid:3) Proof of Theorem 6.2(i).
Let w ∈ H be the element in Lemma 6.12. We compute t c,d · L S,p = Col(( c,d z n ) n ) (by Theorem 6.10)= Col( w ) · ( γ − r − (by Lemma 6.12) ∈ I r (by Lemma 6.14) . Hence, in the quotient Q r = I r / I r +1 , we compute t c,d · L ( r ) S,p = (exp ( δ ) , D ( w )) p · ( γ − r − (by Lemma 6.14)= (exp ( δ ) , D ( c,d κ ∞ )) p . By (6.3.2), note that δ = (cid:18) − α (cid:19) (cid:18) − β (cid:19) − ν. Since [ ω, ν ] L = 1 by Lemma 6.9, we have (cid:18) − α (cid:19) − (cid:18) − β (cid:19) log ω ( x ) exp ( δ ) = x in H f ( Q p , V )for any x ∈ E ( Q ). Thus we have (cid:18) − α (cid:19) − (cid:18) − β (cid:19) log ω ( x ) t c,d · L ( r ) S,p = ( x, D ( c,d κ ∞ )) p = h x, c,d κ ∞ i p (by Corollary 6.13) . Upon multiplying both sides by t − c,d we obtain the desired formula.This completes the proof of Theorem 6.2(i).6.4.3. We now suppose that p is split multiplicative and prepare for the proof of Theorem6.2(ii).Note first that, by Tate’s uniformization, we have an exact sequence of G Q p -modules0 → Z p (1) → T → Z p → . (6.4.3)This means that F + V ≃ Q p (1) and F − V := V /F + V ≃ Q p .Since H ( Q p , F − V ) does not vanish in this case, Rubin’s derivative D ( y ) is not deter-mined uniquely, so we impose more condition to define it. Let ρ p : H ( Q p , F − V ) → H ( Q p , F − V ) ⊗ Z p I/I be the connecting homomorphism obtained from the right vertical exact triangle in (6.4.1).We know that im( ρ p ) = h log p χ cyc i ⊗ Z p I/I , where we regard log p χ cyc : G Q p → Q p as an element of H ( Q p , F − V ) = H ( Q p , Q p ) =Hom cont ( G Q p , Q p ). (See the proof of [42, Lem. 15.1] for example.) Let π p : H ( Q p , V ) ⊗ Z p I/I → H ( Q p , F − V ) ⊗ Z p I/I be the map induced by V ։ F − V . Then one sees that im( ρ p ) ∩ im( π p ) = 0 (since log p ( q E ) =0), by which one can take a unique element D ( y ) ∈ im( π p )such that loc p ( y ) = i ( D ( y )) in H ( R Γ Iw ( Q p , F − V ) ⊗ L Λ Λ /I ). Compare Venerucci’s con-struction [42, Lem. 15.1] (where I/I is identified with Z p ).An analogue of Theorem 6.11 is as follows. Theorem 6.15.
Suppose that p is split multiplicative. For any x ∈ E ( Q ) and y = ( y n ) n ∈ lim ←− n H n = H , we have h x, y i Sch p = ( x, D ( y )) p in Q p ⊗ Z p I/I . Proof.
We identify Q p ⊗ Z p I/I = Q p via the isomorphism ℓ p in (5.3.2). By Venerucci’scomputation [42, Prop. 15.2], we havelog ω ( x ) · D ( y )(Fr p ) = − log p ( q E )ord p ( q E ) h x, y i Sch p . (See also [42, (127)].) Here D ( y )(Fr p ) means the evaluation of D ( y ) ∈ H ( Q p , Q p ) =Hom cont ( G Q p , Q p ) at the arithmetic Frobenius Fr p (this corresponds to Der p ( x ) in [42, § x corresponds to our y ). Since D ( y )(Fr p ) = − log p ( q E )ord p ( q E ) exp ∗ ω ( D ( y )) (see [25, (6)] or(6.4.4) below) and log p ( q E ) = 0, we havelog ω ( x ) exp ∗ ω ( D ( y )) = h x, y i Sch p . Since the left hand side is equal to ( x, D ( y )) p , we obtain the desired formula. (cid:3) The following is an analogue of Corollary 6.13
Corollary 6.16.
For any x ∈ E ( Q ) , we have h x, c,d κ ∞ i Sch p = ( x, D ( c,d κ ∞ )) p in Q p ⊗ Z p Q r . Since E over Q p is a Tate curve, we have an isomorphism E ( Q p ) ≃ Q × p / h q E i . We denoteby λ p the composite map λ p : Q × p → ( Q × p / h q E i ) ⊗ Q p ≃ E ( Q p ) ⊗ Q p → H ( Q p , V )where the final map is the Kummer map. This map λ p also coincides with the composite Q × p → H ( Q p , Q p (1)) = H ( Q p , F + V ) → H ( Q p , V ) where the first map is the Kummermap. Therefore, for any a ∈ Q × p and z ∈ H ( Q p , V ) we have( λ p ( a ) , z ) p = ( a, π p ( z )) G m where π p : H ( Q p , V ) → H ( Q p , Q p ) is the natural map induced by V ։ F − V = Q p ,and ( − , − ) G m is the pairing induced by the cup product H ( Q p , Q p (1)) × H ( Q p , Q p ) → H ( Q p , Q p (1)) ≃ Q p .The following result explains how the L -invariant occurs in our generalized version ofRubin’s formula. Lemma 6.17.
For any z ∈ H ( Q p , V ) we have ( λ p ( p ) , z ) p · ( γ −
1) = ( p, π p ( z )) G m · ( γ −
1) = − log p χ cyc ( γ )ord p ( q E ) exp ∗ ω ( z )(1 − rec p ( q E )) in Q p ⊗ Z p I/I .Proof. We write log q E : ( Q × p / h q E i ) ⊗ Q p → Q p for the logarithm that vanishes on h q E i andnote that this coincides with the formal logarithm via the isomorphism E ( Q p ) ≃ Q × p / h q E i .We also write exp q E for the inverse of log q E .Then, by using the equality of functionslog q E = log p − log p ( q E )ord p ( q E ) · ord p (cf. the proof of [43, Cor. 3.7]), one computes that λ p ( p ) = λ p (exp q E (log q E ( p )))= λ p (cid:18) exp q E (cid:18) − log p ( q E )ord p ( q E ) (cid:19)(cid:19) = − exp Q p ,V (cid:18) log p ( q E )ord p ( q E ) ν (cid:19) in E ( Q p ) ⊗ Q p . Thus we have( λ p ( p ) , z ) p = − log p ( q E )ord p ( q E ) exp ∗ ω ( z ) . (6.4.4)(See also [25, (6)].) The claim follows by noting1 − rec p ( q E ) = log p ( q E )log p χ cyc ( γ ) · ( γ − . (cid:3) Let U n be the group of principal local units in Q n,p . Let ( d n ) n ∈ lim ←− n U n be the systemconstructed by Kobayashi in [25, § δ n ) n defined in (6.3.1)by δ n = log p ( d n ) · ν in Q n,p ⊗ Q p D crys ( V ) . Since d = 1, Hilbert’s theorem 90 implies that there exists x n ∈ Q × n,p such that d n = γx n x n . We regard x n ∈ H ( Q n,p , Z p (1)) via the Kummer map. The element Cor Q n,p / Q p ( x n ) iswell-defined in H ( Q p , Z /p n (1)), i.e., independent of the choice of x n . We define d ′ := (Cor Q n,p / Q p ( x n )) n ∈ lim ←− n H ( Q p , Z /p n (1)) ≃ H ( Q p , Z p (1)) . For each field Q n,p with n ≥ − , − ) G m : H ( Q n,p , Z p (1)) × H ( Q n,p , Z p ) → H ( Q n,p , Z p (1)) ≃ Z p (6.4.5)for the pairing defined by the cup product. Let π p : H n = H ( O Q n ,S , T ) → H ( Q n,p , T ) → H ( Q n,p , Z p )be the map induced by the surjection T ։ Z p in (6.4.3).We define Col ′ n : H n → Z /p n [ G n ]by Col ′ n ( z ) := X σ ∈ G n ( σx n , π p ( z )) G m σ and set Col ′ := lim ←− n Col ′ n : H → lim ←− n Z /p n [ G n ] ≃ Λ . Lemma 6.18. (i)
The Coleman map
Col : H → Λ coincides with ( γ − − · Col ′ . (ii) Let y ∈ H . Then we have Col ′ ( y ) ∈ I and Col ′ ( y ) = ( d ′ , D ( y )) G m in Q p ⊗ Z p I/I , where ( − , − ) G m : H ( Q p , Q p (1)) × ( H ( Q p , Q p ) ⊗ Z p I/I ) → Q p ⊗ Z p I/I is induced by (6.4.5).Proof. Claim (i) follows directly from construction. (See also Kobayashi’s computation ofCol n ( z ) in [25, p. 573].)Claim (ii) is proved in the same way as Lemma 6.14 and so, for brevity, we omit theproof. (cid:3) Proof of Theorem 6.2(ii).
Let w ∈ H be the element in Lemma 6.12. We compute t c,d · L S,p = Col(( c,d z n ) n ) (by Theorem 6.10)= Col( w ) · ( γ − r − (by Lemma 6.12)= Col ′ ( w ) · ( γ − − γ − r − (by Lemma 6.18(i)) ∈ I r +1 (by Lemma 6.18(ii)).Thus, in I r +1 /I r +2 = Q r +1 , we further compute t c,d · L ( r +1) S,p = − Col ′ ( w ) · ( γ − r = − ( d ′ , D ( w )) G m · ( γ − r (by Lemma 6.18(ii))Since ( d ′ , D ( w )) G m = (cid:18) − p (cid:19) − (log p χ cyc ( γ )) − ( p, D ( w )) G m (see Kobayashi [25, p. 574, line 2], note that ‘N x n ’ in [25] is congruent to d ′ modulo p n ),Lemma 6.17 implies that − ( d ′ , D ( w )) G m · ( γ − r = (cid:18) − p (cid:19) − exp ∗ ω ( D ( w )) · p ( q E ) (1 − rec p ( q E )) · ( γ − r − . Note that, for any x ∈ E ( Q ) and y ∈ H ( Q p , V ), we havelog ω ( x ) exp ∗ ω ( y ) = ( x, y ) p . Hence we have (cid:18) − p (cid:19) log ω ( x ) t c,d · L ( r +1) S,p = ( x, D ( w )) p · p ( q E ) (1 − rec p ( q E )) · ( γ − r − = ( x, D ( c,d κ ∞ )) p · p ( q E ) (1 − rec p ( q E ))= h x, c,d κ ∞ i Sch p · p ( q E ) (1 − rec p ( q E )) (by Corollary 6.16) . Upon multiplying both sides by t − c,d we obtain the desired formula.This thereby completes the proof of Theorem 6.2.7. The Iwasawa Main Conjecture and descent theory
The aim of this section is to directly relate Conjectures 4.9 and 4.16 with a natural mainconjecture of Iwasawa theory. The main results in this section are Theorems 7.3 and 7.6.As before, we always assume that p is odd and that H ( Z S , T ) is Z p -free.7.1. Review of the Iwasawa Main Conjecture.
We use the notations in § C n := R Hom Z p ( R Γ c ( O Q n ,S , T ∗ (1)) , Z p [ − C ∞ := lim ←− n C n . Then we have a canonical isomorphism H ( C ∞ ) ≃ H and an exact sequence0 → H → H ( C ∞ ) f −→ Λ ⊗ Z p T ∗ (1) + , ∗ → . (7.1.1)(See (4.1.2) and (4.1.3).) Let Q (Λ) denote the quotient field of Λ. Kato proved that Q (Λ) ⊗ Λ H i ( ≃ Q (Λ) if i = 1 , = 0 if i = 2 . (See [23, Th. 12.4].) Hence, we have a canonical isomorphism Q (Λ) ⊗ Λ det Λ ( C ∞ ) ≃ Q (Λ) ⊗ Λ ( H ⊗ Z p T ∗ (1) + ) . (7.1.2)We set c,d z ∞ := ( c,d z n ) n ∈ lim ←− n H n = H and z ∞ := t − c,d · c,d z ∞ ∈ Q (Λ) ⊗ Λ H , where t c,d ∈ Λ is as in (6.3.3). We then define z ∞ ∈ Q (Λ) ⊗ Λ det Λ ( C ∞ ) to be the element corresponding to z ∞ ⊗ e + δ ( ξ ) ∈ Q (Λ) ⊗ Λ ( H ⊗ Z p T ∗ (1) + )under the isomorphism (7.1.2), where δ ( ξ ) ∈ Z p ⊗ Z H ≃ T ∗ (1) is defined in § Conjecture 7.1 (Iwasawa Main Conjecture) . We have h z ∞ i Λ = det Λ ( C ∞ ) . Remark 7.2.
Since Λ is a regular local ring, we see by [22, Chap. I, Prop. 2.1.5] thatConjecture 7.1 is equivalent to the equalitychar Λ ( H / h z ∞ i Λ ) = char Λ ( H ) . Thus Conjecture 7.1 is equivalent to [23, Conj. 12.10] (by letting f in loc. cit. be thenormalized newform corresponding to E ).7.2. Consequences of the Iwasawa Main Conjecture.
We now state main results ofthis section.
Theorem 7.3.
Assume Hypothesis 2.2. Then Conjecture 7.1 (Iwasawa Main Conjecture)implies Conjecture 4.16 up to Z × p , i.e., there exists u ∈ Z × p such that κ ∞ = u · v ξ Y ℓ ∈ S \{∞} L ℓ X ( E/ Q ) · Tam( E ) E ( Q ) · R Boc ω in ( Q p ⊗ Z E ( Q )) ⊗ Z p Q r − . Combining this theorem with Corollary 6.6, we immediately obtain the following.
Corollary 7.4.
Assume Hypothesis 2.2. Then Conjecture 7.1 (Iwasawa Main Conjecture)implies the p -adic Birch-Swinnerton-Dyer Formula up to Z × p , i.e., there exists u ∈ Z × p suchthat (cid:18) − α (cid:19) − (cid:18) − β (cid:19) · L ( r ) S,p = u · v ξ Y ℓ ∈ S \{∞} L ℓ X ( E/ Q ) · Tam( E ) E ( Q ) R p if p is good or non-split multiplicative, and L ( r +1) S,p = u · p ( q E ) (1 − rec p ( q E )) · v ξ Y ℓ ∈ S \{∞ ,p } L ℓ X ( E/ Q ) · Tam( E ) E ( Q ) R Sch p if p is split multiplicative. Remark 7.5.
Corollary 7.4 improves upon results of Schneider [40, Th. 5] and Perrin-Riou[34, Prop. 3.4.6] in which it is shown that the Iwasawa Main Conjecture and non-degeneracyof the p -adic height pairing together imply the p -adic Birch-Swinnerton-Dyer Formula upto Z × p under the restrictive hypothesis that the reduction of E at p is good ordinary andgood respectively. Theorem 7.6.
Assume Hypothesis 2.2. Assume also that • Conjecture 7.1 (Iwasawa Main Conjecture) is valid, • Conjecture 4.9 (Generalized Perrin-Riou Conjecture at infinite level) is valid, and • the Bockstein regulator R Boc ω in Definition 4.11 does not vanish.Then the p -part of the Birch-Swinnerton-Dyer Formula is valid so that there is an equality L ∗ ( E, · Z p = (cid:0) X ( E/ Q )[ p ∞ ] · Tam( E ) · E ( Q ) − · Ω + · R ∞ (cid:1) · Z p of Z p -sublattices of C p . Remark 7.7.
Theorem 7.6 explains the precise link between the natural main conjectureof Iwasawa theory and the classical Birch-Swinnerton-Dyer Formula, even in the case ofadditive reduction. We note also that this result is, in effect, an analogue of the main result[7, Th. 5.2] of the current authors, where, roughly speaking, the following result is provedin the setting of the multiplicative group: if one assumes the validity of • the Iwasawa Main Conjecture for G m (cf. [7, Conj. 3.1]), • the Iwasawa-Mazur-Rubin-Sano Conjecture for G m (cf. [7, Conj. 4.2]), and • the injectivity of a certain Bockstein homomorphism (which is implied by the con-dition ‘(F)’ in [7, Th. 5.2]: see [7, Prop. 5.16]),then the equivariant Tamagawa Number Conjecture for G m is also valid.7.3. The descent argument.
In the following, we assume both Hypothesis 2.2 and thevalidity of Conjecture 7.1.7.3.1.
A key commutative diagram.
We shall first give quick proofs of Theorems 7.3 and 7.6by using the following key result.
Theorem 7.8.
Let x be a Z p -basis of V r − Z p H ( Z S , T ) tf . Then there is a commutativediagram det Λ ( C ∞ ) Π ∞ / / N ∞ (cid:15) (cid:15) (cid:15) (cid:15) I r − · H N ∞ ' ' ◆◆◆◆◆◆◆◆◆◆◆◆ H ⊗ Z p Q r − det Z p ( C ) Π x / / V r Z p H
10 Boc ∞ , x ♣♣♣♣♣♣♣♣♣♣♣ (7.3.1) with the following properties: (a) Π ∞ ( z ∞ ) = z ∞ ; (b) N ∞ ( z ∞ ) = κ ∞ ; (c) h η Kato x i Z p = H ( Z S , T ) tors · V r Z p H , where η Kato x := Π x (N ∞ ( z ∞ )) ; (d) h Boc ∞ , x ( η Kato x ) i Z p = Z p · v ξ (cid:16)Q ℓ ∈ S \{∞} L ℓ (cid:17) X ( E/ Q )[ p ∞ ]Tam( E ) E ( Q ) − · R Boc ω . Admitting this, we give proofs of Theorems 7.3 and 7.6.
Proof of Theorem 7.3.
It is sufficient to show that h κ ∞ i Z p = Z p · v ξ Y ℓ ∈ S \{∞} L ℓ X ( E/ Q )[ p ∞ ]Tam( E ) E ( Q ) − · R Boc ω . By the commutativity of (7.3.1) and properties (a) and (b), we have κ ∞ = Boc ∞ , x ( η Kato x ) . (7.3.2)Hence the claim follows from the property (d). (cid:3) Proof of Theorem 7.6.
We assume Conjecture 4.9 and R Boc ω = 0, in addition to Hypothesis2.2 and Conjecture 7.1. Recall that Conjecture 4.9 asserts the equality κ ∞ = Boc ∞ , x ( η BSD x ) . Combining this with (7.3.2), we haveBoc ∞ , x ( η BSD x ) = Boc ∞ , x ( η Kato x ) . Since the non-vanishing of R Boc ω is equivalent to the injectivity of Boc ∞ , x by construction,we have η BSD x = η Kato x . By the property (c) in Theorem 7.8, we have Z p · η BSD x = H ( Z S , T ) tors · ^ r Z p H . By Proposition 2.6, this is equivalent to the p -part of the Birch-Swinnerton-Dyer Formula,so we have completed the proof. (cid:3) The rest of this section is devoted to the proof of Theorem 7.8.7.3.2.
Definitions of maps.
First, we give definitions of the maps Π ∞ , N ∞ , N ∞ and Π x inthe diagram (7.3.1). • The map Π ∞ : det Λ ( C ∞ ) → I r − · H is induced by Q (Λ) ⊗ Λ det Λ ( C ∞ ) ( . . ) ≃ Q (Λ) ⊗ Λ ( H ⊗ Z p T ∗ (1) + ) ≃ Q (Λ) ⊗ Λ H , where the second isomorphism is induced by T ∗ (1) + ≃ Z p ; e + δ ( ξ ) . By Remark 7.2, the image of det Λ ( C ∞ ) under this isomorphism is char Λ ( H ) · H .Since char Λ ( H ) ⊂ I r − , we see that the image of det Λ ( C ∞ ) is contained in I r − · H and thus Π ∞ is defined. By this construction, it is obvious that Π ∞ ( z ∞ ) = z ∞ , i.e.,the property (a) of Theorem 7.8 holds. • The construction of the map N ∞ : I r − · H → H ⊗ Z p Q r − is done in the same way as the construction of c,d κ ∞ from ( c,d z n ) n in § N ∞ is defined to be the limit of theDarmon norm N Q n / Q .) It is obvious that N ∞ ( z ∞ ) = κ ∞ , i.e., the property (b) inTheorem 7.8 holds. • The surjection N ∞ : det Λ ( C ∞ ) ։ det Z p ( C )is defined to be the augmentation mapdet Λ ( C ∞ ) ։ det Λ ( C ∞ ) ⊗ Λ Z p ≃ det Z p ( C ) , where the last isomorphism follows from the fact C ∞ ⊗ L Λ Z p ≃ C . • The map Π x : det Z p ( C ) → ^ r Z p H is induced by Q p ⊗ Z p det Z p ( C ) ≃ Q p ⊗ Z p (cid:16) det Z p ( H ( C )) ⊗ Z p det − Z p ( H ( C )) (cid:17) ≃ Q p ⊗ Z p (cid:18)^ r Z p H ⊗ Z p ^ r − Z p H ( Z S , T ) ∗ tf ⊗ Z p T ∗ (1) + (cid:19) ≃ Q p ⊗ Z p ^ r Z p H , where the second isomorphism follows from (4.1.2) and (4.1.3), and the last isomor-phism is induced by ^ r − Z p H ( Z S , T ) ∗ tf ⊗ Z p T ∗ (1) + ≃ Z p ; x ∗ ⊗ e + δ ( ξ ) . Since the image of det Z p ( C ) under this isomorphism is H ( Z S , T ) tors · V r Z p H ,the map Π x is defined. This also shows that the property (c) in Theorem 7.8 holds.7.3.3. The property (d).
We have already seen that the properties (a), (b) and (c) in The-orem 7.8 are satisfied.We shall now verify property (d), i.e., that there is an equality of Z p -lattices Z p · (cid:0) Boc ∞ , x ( η Kato x ) (cid:1) = Z p · c E · R Boc ω , where c E := v ξ · Y ℓ ∈ S \{∞} L ℓ · X ( E/ Q )[ p ∞ ] · Tam( E ) · E ( Q ) − . One checks that the element Boc ∞ , x ( η Kato x ) is independent of the choice of x . So we take x to be as in § { x , . . . , x r } of E ( Q ) tf . Note that this element x belongs to V r − Q p H ( Z S , V ) and may not be a Z p -basis of V r − Z p H ( Z S , T ) tf . However, bothBoc ∞ , x and η Kato x are defined for this x by linearity.By the definition of R Boc ω (see Definition 4.11), it is sufficient to show that h η Kato x i Z p = Z p · c E · log ω ( x ) · x ∧ · · · ∧ x r . (7.3.3)By the property (c) and (2.5.2), we have h η Kato x i Z p = h η alg x i Z p . (Here η alg x is defined in Definition 2.17, where the finiteness of X ( E/ Q ) is assumed. But wemay define η alg x , replacing X ( E/ Q ) by X ( E/ Q )[ p ∞ ] since we only consider the Z p -modules here. Then we need only the finiteness of X ( E/ Q )[ p ∞ ].) On the other hand, by (4.3.2), wehave h η alg x i Z p = Z p · c E · log ω ( x ) · x ∧ · · · ∧ x r . From this, we obtain the desired equality (7.3.3). Hence we have proved that the property(d) holds.7.4.
The proof of Theorem 7.8.
In this subsection, we prove the commutativity of thediagram (7.3.1) and thus complete the proof of Theorem 7.8. Our argument is similar to[6, Lem. 5.22], [7, Lem. 5.17] and [11, Th. 4.21].Fix a non-negative integer n . It is sufficient to show the commutativity of the following‘ n -th layer version’ of (7.3.1):det Z p [ G n ] ( C n ) Π n / / N n (cid:15) (cid:15) (cid:15) (cid:15) I r − n · H n N n ' ' ◆◆◆◆◆◆◆◆◆◆◆◆ H ⊗ Z p Q r − n det Z p ( C ) Π x / / V r Z p H . Boc n, x ♣♣♣♣♣♣♣♣♣♣♣ (7.4.1)We shall describe maps Π ∞ , Π n , Π x and Boc n, x explicitly by choosing a useful represen-tative of the complex C ∞ . Then the commutativity of the diagram is checked by an explicitcomputation.7.4.1. Choice of the representative.
We make a similar argument to [6, § C ∞ is represented by P ψ −→ P , where P is a free Λ-module of rank, say, d . We have an exact sequence0 → H → P ψ −→ P π −→ H ( C ∞ ) → . (7.4.2)Also, setting P n := P ⊗ Λ Z p [ G n ], we have an exact sequence0 → H n → P n ψ n −−→ P n π n −→ H ( C n ) → . (7.4.3)Let { b , . . . , b d } be a basis of P . We denote the image of b i in P n by b i,n . We set x i := π ( b i ) ∈ H ( C ∞ ) and x i,n := π n ( b i,n ) ∈ H ( C n ) . By the argument of [11, Prop. A.11(i)], one may assume(i) f ( x ) = 1 ⊗ e + δ ( ξ ) ∗ , where f : H ( C ∞ ) → Λ ⊗ Z p T ∗ (1) + , ∗ is as in (7.1.1);(ii) h x , . . . , x d i Λ = H ⊂ H ( C ∞ );(iii) { x , , . . . , x r, } is a Z p -basis of H ( Z S , T ) tf ⊂ H ( C ). We set ψ i := b ∗ i ◦ ψ : P → Λand ψ i,n := b ∗ i,n ◦ ψ n : P n → Z p [ G n ] . Note that the property (iii) implies thatim ψ i,n ⊂ I n for every 1 < i ≤ r. (7.4.4)7.4.2. Explicit descriptions of Π ∞ , Π n and Π x . With the above representative of C ∞ , wehave an identification det Λ ( C ∞ ) = ^ d Λ P ⊗ Λ ^ d Λ P ∗ . We define a map Π ∞ : ^ d Λ P ⊗ Λ ^ d Λ P ∗ → P by a ⊗ ( b ∗ ∧ · · · ∧ b ∗ d ) ( − d − (cid:16)^
We shall describe the Bockstein regulator map Boc n, x ex-plicitly.For an integer i with 1 < i ≤ r , we define a map β i,n : P → I n /I n by β i,n ( a ) := ψ i,n ( e a ) (mod I n ) , where for a ∈ P we take an element e a ∈ P n such that P σ ∈ G n σ ( e a ) = a (we regard P ⊂ P n by identifying P with P G n n ). Note that ψ i,n ( e a ) ∈ I n by (7.4.4) and its image in I n /I n isindependent of the choice of e a . Hence the map β i,n is well-defined.Let β Q n : H → H ( Z S , T ) tf ⊗ Z p I n /I n be the Bockstein map defined in (2.3.2). Onechecks by the definition of the connecting homomorphism that − β i,n = x ∗ i, ◦ β Q n on H . From this, we see that the mapBoc n, x := ( − r − ^
We prove the commutativity of (7.4.1). We may assume x = x , ∧ · · · ∧ x r, .In view of the explicit descriptions (7.4.6), (7.4.7) and (7.4.8), it is sufficient to provethat(7.4.9) ( − d − N n ◦ (cid:16)^
The third author would like to thank Kazim B¨uy¨ukboduk for helpfuldiscussions, especially about Rubin’s formula. The authors would like to thank TakenoriKataoka for discussions with him and for his comments on the first draft of this paper, whichwere very helpful. The authors also would like to thank Christian Wuthrich for carefullyreading the manuscript and giving them helpful comments.
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