Congruences for critical values of higher derivatives of twisted Hasse-Weil L-functions
aa r X i v : . [ m a t h . N T ] M a y Congruences for critical values of higherderivatives of twisted Hasse-Weil L -functions Werner Bley and Daniel Macias CastilloNovember 6, 2018
Abstract
Let A be an abelian variety over a number field k and F a finite cyclicextension of k of p -power degree for an odd prime p . Under certain technicalhypotheses, we obtain a reinterpretation of the equivariant Tamagawa numberconjecture (‘eTNC’) for A , F/k and p as an explicit family of p -adic congru-ences involving values of derivatives of the Hasse-Weil L -functions of twists of A , normalised by completely explicit twisted regulators. This reinterpretationmakes the eTNC amenable to numerical verification and furthermore leads toexplicit predictions which refine well-known conjectures of Mazur and Tate. Let A be an abelian variety of dimension d defined over a number field k . Wewrite A t for the dual abelian variety. Let F/k be a finite Galois extension withgroup G := Gal( F/k ) . We let A F denote the base change of A through F/k andconsider the motive M F := h ( A F )(1) as a motive over k with a natural action ofthe semi-simple Q -algebra Q [ G ] .We will study the equivariant Tamagawa number conjecture as formulated byBurns and Flach in [9] for the pair ( M F , Z [ G ]) . This conjecture asserts the validityof an equality in the relative algebraic K -group K ( Z [ G ] , R [ G ]) . If p is a prime, werefer to the image of this equality in K ( Z p [ G ] , C p [ G ]) as the ‘eTNC p for ( M F , Z [ G ]) ’(here C p denotes the completion of an algebraic closure of Q p ). If p does not dividethe order of G then the ring Z p [ G ] is regular and one can use the techniques describedin [8, §1.7] to give an explicit interpretation of this projection. In this manuscriptwe will focus on primes p dividing the order of G , for which such an interpretationis in general very difficult to obtain.In [12], a close analysis of the finite support cohomology of Bloch and Katofor the base change of the p -adic Tate module of A t is carried out under certaintechnical hypotheses on A and F . A consequence of this analysis is an explicitreinterpretation of the eTNC p in terms of a natural ‘equivariant regulator’ (see [12,1h. 5.1]). The main results of the present manuscript are based on the computationof this equivariant regulator in the special case where F/k is cyclic of degree p n for an odd prime p . Under certain additional hypotheses on the structure of Tate-Shafarevich groups of A over the intermediate fields of F/k we obtain a completelyexplicit interpretation of the eTNC p (see Theorem 2.9). Whilst this is of independenttheoretical interest, it also makes the eTNC p amenable to numerical verifications.One of the main motivations behind our study of the equivariant Tamagawanumber conjecture for the pair ( M F , Z [ G ]) is the hope that this conjecture mayprovide a coherent overview of and a systematic approach to the study of propertiesof leading terms and values at s = 1 of Hasse-Weil L -functions. In order to describeour current steps in this direction, we first recall the general philosophy of ‘refinedconjectures of the Birch and Swinnerton-Dyer type’ that originates in the work ofMazur and Tate in [20]. These conjectures concern, for elliptic curves A defined over Q and certain abelian groups G , the properties of ‘modular elements’ θ A,G belonginga priori to the rational group ring Q [ G ] and constructed from the modular symbolsassociated to A , therefore interpolating the values at s = 1 of the twisted Hasse-Weil L -functions associated to A and G . More precisely, the aim is to predict the precisepower r (possibly infinite) of the augmentation ideal I of the integral group ring Z [ G ] with the property that θ A,G belongs to I r but not to I r +1 , and furthermoreto describe the image of θ A,G in the quotient I r /I r +1 (whenever such an integer r exists). In the process of studying the modular element θ A,G , Mazur and Tate alsopredict that it should belong to the Fitting ideal over Z [ G ] of their ‘integral Selmergroup’ S ( A/F ) (and refer to such a statement as a ‘weak main conjecture’) andask for a ‘strong main conjecture’ predicting a generator of the Fitting ideal of anexplicitly described natural modification of S ( A/F ) (see [20, Remark after Conj.3]).However, it is well-known that in many cases of interest the modular element θ A,G vanishes, thus rendering any such properties trivial, and it would therefore bedesirable to carry out an analogous study for elements interpolating leading termsrather than values at s = 1 of the relevant Hasse-Weil L -functions, normalised byappropriate regulators. Although the aim to study such elements already underliesthe results of [12], one of the main advantages of confining ourselves to the specialcase in which the given extension of number fields F/k is cyclic of prime-powerdegree is that we are led to defining completely explicit ‘twisted regulators’ fromour computation of the aforementioned equivariant regulator of [12]. Furthermore,we arrive at very explicit statements without having to restrict ourselves to situationsin which the relevant Mordell-Weil groups are projective when considered as Galoismodules. In particular, we derive predictions of the following nature for such anelement L that interpolates leading terms at s = 1 of twisted Hasse-Weil L -functionsnormalised by our twisted regulators from the assumed validity of the eTNC p for ( M F , Z [ G ]) : • a formula for the precise power h ∈ Z ≥ of the augmentation ideal I G,p of theintegral group ring Z p [ G ] with the property that L belongs to I hG,p but not to2 h +1 G,p (expressed in terms of the ranks of the Mordell-Weil groups of A over theintermediate fields of F/k ), and a formula for the image of L in the quotient I hG,p /I h +1 G,p (see Corollary 2.11); • the statement that the element L of Z p [ G ] (resp. a straightforward modifica-tion of L ) annihilates the p -primary Tate-Shafarevich group of A t (resp. A )over F as a Galois module (see Theorem 2.12 and Corollary 2.14); • and the explicit description of a natural quotient of (the Pontryagin dual of)the p -primary Selmer group of A over F whose Fitting ideal is generated by L (see Theorem 2.12).The structure of the paper is as follows. In Section 2 we present our main resultsand in Section 4 we supply the proofs. In order to prepare for the proofs we recallin Section 3 the relevant material from [12]. In the final Section 5 we present somenumerical computations.We would like to thank David Burns and Christian Wuthrich for some help-ful discussions concerning this project, and the referee for making several usefulsuggestions. We mostly adapt the notations from [12].For a finite group Γ we write D ( Z p [Γ]) for the derived category of complexes ofleft Z p [Γ] -modules. We also write D p ( Z p [Γ]) for the full triangulated subcategory of D ( Z p [Γ]) comprising complexes that are perfect (that is, isomorphic in D ( Z p [Γ]) toa bounded complex of finitely generated projective Z p [Γ] -modules).We also write ˆΓ for the set of irreducible E -valued characters of Γ , where E denotes either C or C p (we will throughout our arguments have fixed an isomorphismof fields j : C → C p and use it to implicitly identify both sets, with the intendedmeaning of ˆΓ always clear from the context). We let Γ denote the trivial characterof Γ and write ˇ ψ for the contragrediant character of each ψ ∈ ˆΓ . We write e ψ = ψ (1) | Γ | X γ ∈ Γ ψ ( γ ) γ − for the idempotent associated with ψ ∈ ˆΓ and also set Tr Γ := P γ ∈ Γ γ .For any abelian group M we let M tor denote its torsion subgroup and M tf thetorsion-free quotient M/M tor . We also set M p := Z p ⊗ Z M and, if M is finitelygenerated, we set rk( M ) := dim Q ( Q ⊗ Z M ) .For any Z p [Γ] -module M we write M ∨ for the Pontryagin dual Hom Z p ( M, Q p / Z p ) and M ∗ for the linear dual Hom Z p ( M, Z p ) , each endowed with the natural contra-gredient action of Γ . Explicitly, for a homomorphism f and elements m ∈ M and γ ∈ Γ , one has ( γf )( m ) = f ( γ − m ) . 3or any Galois extension of fields L/K we abbreviate
Gal(
L/K ) to G L/K . Wefix an algebraic closure K c of K and abbreviate G K c /K to G K . For each non-archimedian place v of a number field we write κ v for the residue field.Throughout this paper, we will consider the following situation. We have fixedan odd prime p and a Galois extension F/k of number fields with group G = G F/k .Except in Section 3, the extension
F/k will always be cyclic of degree p n . We giveourselves an abelian variety A of dimension d defined over k . For each intermediatefield L of F/k we write S Lp , S Lr and S Lb for the set of non-archimedean places of L which are p -adic, which ramify in F/L and at which
A/L has bad reductionrespectively. Similarly, we write S L ∞ , S L R and S L C for the sets of archimedean, realand complex places of L respectively. If L = k we simply write S p , S r , S b , S ∞ , S R and S C .Finally, we write A ( L ) for the Mordell-Weil group and X p ( A L ) for the p -primaryTate-Shafarevich group of A over L . Recall that A is an abelian variety of dimension d defined over the number field k .Furthermore, F/k is cyclic of degree p n where p is an odd prime.We assume throughout this section that A/k and
F/k are such that(a) p ∤ | A ( k ) tor | · | A t ( k ) tor | ,(b) p ∤ Q v ∈ S b c v ( A, k ) , where c v ( A, k ) denotes the Tamagawa number of A at v ,(c) A has good reduction at all p -adic places of k ,(d) p is unramified in F/ Q ,(e) No place of bad reduction for A is ramified in F/k , i.e. S b ∩ S r = ∅ ,(f) p ∤ Q v ∈ S r | A ( κ v ) | ,(g) The Tate-Shafarevich group X ( A F ) is finite,(h) X p ( A F H ) = 0 for all non-trivial subgroups H of G . Remarks 2.1.
Our assumptions (a) - (g) recover the hypotheses (a) - (h) of [12].For a fixed abelian variety
A/k , the hypotheses (a), (b) and (c) clearly exclude onlyfinitely many choices of odd prime p , while the additional hypotheses (d), (e) and(f ) constitute a mild restriction on the choice of cyclic field extension F of k of odd,prime-power degree. In order to further illustrate this point, we let S denote anyfinite set of places of k at which A has good reduction. We then define a set Σ( S ) of rational primes as the union of the set of all prime divisors ℓ of | A ( k ) tor | · | A t ( k ) tor | · Y v ∈ S b c v ( A, k ) · Y v ∈ S | A ( κ v ) | nd the set of all primes ℓ with the property that A has bad reduction at an ℓ -adicplace of k . The set Σ( S ) is then clearly finite and, for any odd prime p Σ( S ) andany cyclic field extension F of k of p -power degree which is unramified outside S and with the property that p is unramified in F/ Q , all of the hypotheses (a)-(f ) aresatisfied.The hypothesis (g) is famously conjectured to be true in all cases, and it isstraightforward to produce specific examples for which all of the other hypotheses,including the additional hypothesis (h), are satisfied (see also Section 5 and Remark2.13 in this regard). We emphasize that in (h) we allow X p ( A F ) to be non-trivial. An understanding of the G -module structure of the relevant Mordell-Weil groupsis key to our approach. We hence begin by applying a result of Yakovlev [22] in orderto obtain such explicit descriptions. This approach is inspired by work of Burns,who first obtained a similar result in [7, Prop. 7.2.6(i)]. For a non-negative integer m and a Z p [ G ] -module M we write M
There exist isomorphisms of Z p [ G ] -modules of the form A ( F ) p ∼ = M J ≤ G Z p [ G/J ]
Corollary 2.3.
For any subgroup H of G we have rk( A ( F H )) = rk( A t ( F H )) == X J>H | G/J | m J + | G/H | X J ≤ H m J ≤ | G/H | rk( A ( k )) . Proposition 2.2 combines with Roiter’s Lemma (see [13, (31.6)]) to imply theexistence of points P ( J,j ) ∈ A ( F ) and P t ( J,j ) ∈ A t ( F ) for J ≤ G and j ∈ [ m J ] withthe property that A ( F ) p = L J ≤ G L j ∈ [ m J ] Z p [ G/J ] P ( J,j ) , Z p [ G/J ] P ( J,j ) ∼ = Z p [ G/J ] ,A t ( F ) p = L J ≤ G L j ∈ [ m J ] Z p [ G/J ] P t ( J,j ) , Z p [ G/J ] P t ( J,j ) ∼ = Z p [ G/J ] . (1)Furthermore, our choice of points as in (1) guarantees that one also has Q ⊗ Z A ( F ) = L J ≤ G L j ∈ [ m J ] Q [ G/J ] P ( J,j ) , Q [ G/J ] P ( J,j ) ∼ = Q [ G/J ] , Q ⊗ Z A t ( F ) = L J ≤ G L j ∈ [ m J ] Q [ G/J ] P t ( J,j ) , Q [ G/J ] P t ( J,j ) ∼ = Q [ G/J ] . (2)5e now fix sets P = { P ( J,j ) ∈ A ( F ) : J ≤ G, j ∈ [ m J ] } , P t = { P t ( J,j ) ∈ A t ( F ) : J ≤ G, j ∈ [ m J ] } , such that (2) holds. For ≤ t ≤ n we write H t for the (unique) subgroup of G of order p n − t and set P ( t,j ) := P ( H t ,j ) , P t ( t,j ) := P t ( H t ,j ) . We also put m t := m H t and e H t := | H t | Tr H t = | H t | P g ∈ H t g . We write h , i F for the Néron-Tate height pairing A ( F ) × A t ( F ) → R defined relative to the field F and define a matrix with entriesin C [ G ] by setting R ( P , P t ) := | H u | X τ ∈ G/H u h τ · P ( u,k ) , P t ( t,j ) i F ( τ · e H u ) ( u,k ) , ( t,j ) , where ( u, k ) is the row index with ≤ u ≤ n , k ∈ [ m u ] , and ( t, j ) is the columnindex with ≤ t ≤ n , j ∈ [ m t ] (we always order sets of the form { ( t, j ) : 0 ≤ t ≤ n, j ∈ [ m t ] } lexicographically). We note that, since each point P ( u,k ) belongs to A ( F H u ) , the action of G/H u on P ( u,k ) is well-defined.For any matrix A = (cid:0) a ( u,k ) , ( t,j ) (cid:1) ( u,k ) , ( t,j ) indexed as above we define A t := (cid:0) a ( u,k ) , ( t,j ) (cid:1) ( u,k ) , ( t,j ) ,u,t ≥ t , with the convention A t = 1 whenever no entries a ( u,k ) , ( t,j ) with u, t ≥ t exist. If A is a matrix with coefficients a ij in C [ G ] or C p [ G ] , then for any ψ ∈ b G we write ψ ( A ) for the matrix with coefficients ψ ( a ij ) . We also set R t ( P , P t ) = R ( P , P t ) t . Definition 2.4.
For each character ψ ∈ ˆ G we define t ψ ∈ { , . . . , n } by the equality ker( ψ ) = H t ψ and call λ ψ ( P , P t ) := det (cid:0) ψ (cid:0) R t ψ ( P , P t ) (cid:1)(cid:1) . the ’lower ψ -minor’ of R ( P , P t ) . Remark 2.5.
It is easy to see that the element P ψ ∈ b G λ ψ ( P , P t ) e ψ ∈ C [ G ] de-pends upon the choice of points P and P t satisfying (2) only modulo Q [ G ] × . Sim-ilarly, for any given isomorphism of fields j : C → C p , it is clear that the element P ψ ∈ b G j ( λ ψ ( P , P t )) e ψ ∈ C p [ G ] depends upon the choice of points P and P t satisfying(1) only modulo Z p [ G ] × . For any order Λ in Q [ G ] that contains Z [ G ] we let C ( A, Λ) denote the integralitypart of the equivariant Tamagawa number conjecture (‘eTNC’ for brevity) for thepair ( h ( A F )(1) , Λ) as formulated by Burns and Flach in [9, Conj. 4(iv)]. Similarly,we let C ( A, Q [ G ]) denote the rationality part as formulated in [9, Conj. 4(iii) orConj. 5]. We recall that, under the assumed validity of hypothesis ( g ) , C ( A, Λ) takesthe form of an equality in the relative K -group K (Λ , R [ G ]) . For each embedding6 : R −→ C p we denote by C p,j ( A, Λ) the image of this conjectural equality under theinduced map K (Λ , R [ G ]) −→ K (Λ p , C p [ G ]) . We then say that C p ( A, Λ) is valid ifC p,j ( A, Λ) is valid for every isomorphism j : C → C p .The eTNC is an equality between analytic and algebraic invariants associatedwith A/k and
F/k . In the following we describe and define the analytic part. Wefirst recall the definition of periods and Galois Gauss sums of [12, Sec. 4.4]. We fixNéron models A t for A t over O k and A tv for A tk v over O k v for each v in S p and thenfix a k -basis { ω b } b ∈ [ d ] of the space of invariant differentials H ( A t , Ω A t ) which gives O k v -bases of H (cid:0) A tv , Ω A tv (cid:1) for each such v and is also such that each ω b extends toan element of H (cid:0) A t , Ω A t (cid:1) .For each v in S C we fix a Z -basis { γ v,a } a ∈ [2 d ] of H (cid:0) σ v ( A t )( C ) , Z (cid:1) . For each v in S R we let c denote complex conjugation and fix a Z -basis { γ + v,a } a ∈ [ d ] of H (cid:0) σ v ( A t )( C ) , Z (cid:1) c =1 .For each v in S R , resp. S C , we then define periods by setting Ω v ( A/k ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:18)Z γ + v,a ω b (cid:19) a,b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , resp. Ω v ( A/k ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:18)Z γ v,a ω b , c (cid:16)Z γ v,a ω b (cid:17)(cid:19) a,b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where in the first matrix ( a, b ) runs over [ d ] × [ d ] and in the second matrix ( a, b ) runs over [2 d ] × [ d ] .In our special case all characters are one-dimensional and, moreover, | G | is odd.Therefore the definitions of [12] simplify and we set Ω( A/k ) := Y v ∈ S ∞ Ω v ( A/k ) ,w ∞ ( k ) := i | S C | . For each place v in S r we write ¯ I v ⊆ G for the inertia group of v and Fr v for thenatural Frobenius in G/ ¯ I v . We define the ‘non-ramified characteristic’ u v by u v ( ψ ) := ( − ψ (Fr − v ) , ψ | ¯ I v = 1 , , ψ | ¯ I v = 1 . and u ( ψ ) := Y v ∈ S r u v ( ψ ) . For each character ψ ∈ b G we then define the modified Galois-Gauss sum by setting τ ∗ ( Q , ind Q k ( ψ )) := u ( ψ ) τ ( Q , ind Q k ( ψ )) ∈ ( Q c ) × , where each individual Galois-Gauss sum τ ( Q , · ) is as defined by Martinet in [19].For each ψ ∈ b G we set L ∗ ψ = L ∗ A,F/k,ψ := L ∗ S r ( A, ˇ ψ, τ ∗ ( Q , ind Q k ( ψ )) d Ω( A/k ) w ∞ ( k ) d ∈ C × , Σ of places of k we write L ∗ Σ ( A, ψ, for the leadingterm in the Taylor expansion at s = 1 of the Σ -truncated ψ -twisted Hasse-Weil- L -function of A . Without any further mention we will always assume that the functions L Σ ( A, ψ, s ) have analytic continuation to s = 1 (as conjectured in [9, Conj. 4 (i)]) andrecall that they are then expected to have a zero of order r ψ := dim C ( e ψ ( C ⊗ Z A ( F ))) (this is the rank conjecture [9, Conj. 4 (ii)]).We finally define L ∗ = L ∗ A,F/k := X ψ ∈ b G L ∗ A,F/k,ψ e ψ ∈ C [ G ] × and note that the element L ∗ defined in [12, Th. 5.1] specialises precisely to ourdefinition. Theorem 2.6. C ( A, Q [ G ]) is valid if and only if L ∗ ψ λ ψ ( P , P t ) − ∈ Q ( ψ ) for all ψ ∈ b G and furthermore, for any γ ∈ Gal( Q ( ψ ) / Q ) , L ∗ ψ γ λ ψ γ ( P , P t ) − = γ (cid:0) L ∗ ψ λ ψ ( P , P t ) − (cid:1) , for any, or equivalently every, choice of points P and P t such that (2) holds. Remarks 2.7. (i) From the definitions of u ( ψ ) , w ∞ ( k ) and the definition of localEuler factors it is immediately clear that in the statement of Theorem 2.6 we canreplace L ∗ ψ by ˜ L ∗ ψ := L ∗ ( A, ˇ ψ, τ ( Q , ind Q k ( ψ )) d Ω( A/k ) . (ii) The explicit conditions on elements of the form L ∗ ψ λ ψ ( P , P t ) − given in The-orem 2.6 generalise and refine the predictions given by Fearnley and Kisilevsky in[16, 17]. For details see [2, Ex. 5.2]. In particular, we note that the numericalcomputations performed by Fearnley and Kisilevsky can be interpreted via Theorem2.6 as supporting evidence for conjecture C ( A, Q [ G ]) . We fix a generator σ of G and define Σ to be the diagonal matrix indexed bypairs ( t, j ) , ( s, i ) with σ p t − at the diagonal entry associated to ( t, j ) and zeroselsewhere. For any matrix A = (cid:0) a ( u,k ) , ( t,j ) (cid:1) ( u,k ) , ( t,j ) indexed by tuples ( u, k ) and ( t, j ) as above we define A t := (cid:0) a ( u,k ) , ( t,j ) (cid:1) ( u,k ) , ( t,j ) ,u,t ≤ t , once again with the convention A t = 1 whenever no entries a ( u,k ) , ( t,j ) with u, t ≤ t exist. We recall that for each character ψ ∈ b G we defined t ψ such that ker( ψ ) = H t ψ .We define the the ’upper ψ -minor’ of Σ by δ ψ := det (cid:0) ψ (cid:0) Σ t ψ − (cid:1)(cid:1) .
8t is easy to see that for another choice of generator of G , say τ , one has X ψ ∈ b G δ ψ ( σ ) δ ψ ( τ ) e ψ ∈ Z p [ G ] × . Under our current hypotheses on the data ( A, F/k, p ) and the additional hy-pothesis that X p ( A F ) = 0 , and for any intermediate field L of F/k , we shall saythat BSD p ( L ) holds if, for any choice of Z -bases { Q i } and { R j } of A ( L ) and A t ( L ) respectively and of isomorphism j : C → C p , one has that j L ∗ ( A/L, · ( p | d L | ) d det( h Q i , R j i L ) · Q v ∈ S L ∞ Ω v ( A/L ) ! ∈ Z × p . Here d L denotes the discriminant of the field L and each period Ω v ( A/L ) is asdefined above but relative to the field L rather than k . It will become apparentin the proof of Theorem 2.8 below that the validity of BSD p ( L ) is equivalent tothe validity of the p -part of the eTNC for the pair ( h ( A L )(1) , Z ) . We recall thathypotheses (a), (b) and (h) justify the fact that no orders of torsion subgroups ofMordell-Weil groups, Tamagawa numbers or orders of Tate-Shafarevich groups occurin this formulation, and furthermore note that, by explicitly computing integrals,the periods Ω v ( A/L ) can be related to those obtained by integrating measures asoccurring in the classical formulation of the Birch and Swinnerton-Dyer conjecture– see, for example, Gross [18, p. 224].For the remainder of this section, we assume that C ( A, Q [ G ]) is valid. It is theneasy to see that, for any order Λ in Q [ G ] that contains Z [ G ] , the validity of C p,j ( A, Λ) is independent of the choice of isomorphism j : C → C p , and so we fix such a j forthe remainder of this section. In fact, all relevant elements of C [ G ] appearing inthe statements of our results will actually belong to Q [ G ] (as a consequence of aneasy application of Theorem 2.6) and so we will consider them simultaneously aselements of Q p [ G ] ⊂ C p [ G ] in the natural way without any explicit mention of j .Let M denote the maximal Z -order in Q [ G ] . For any ψ ∈ b G , let O ψ be thevaluation ring of Q p ( ψ ) . Let p ψ be the (unique) prime ideal of O ψ above p . Wewrite v p ψ for the normalised valuation defined by p ψ . Theorem 2.8.
Let P and P t be any choice of points such that (1) holds. We assumethat X p ( A F ) = 0 . Then the following are equivalent.(i) C p ( A, M ) is valid.(ii) BSD p ( L ) is valid for all intermediate fields L of F/k .(iii) For each ψ ∈ b G one has v p ψ (cid:18) L ∗ ψ λ ψ ( P , P t ) (cid:19) = b ψ where b ψ := t ψ − X s =0 p s m s . iv) X ψ ∈ b G L ∗ ψ λ ψ ( P , P t ) δ ψ e ψ ∈ M × p . To describe the full range of implications of the validity of C p ( A, Z [ G ]) requiresyet more work and some further notations.For each finite extension L/k and natural number n we write Sel ( p n ) ( A L ) for theSelmer group associated to the isogeny [ p n ] . We define the p -primary Selmer groupby Sel p ( A L ) := lim −→ Sel ( p n ) ( A L ) . We recall that one then obtains a canonical short exact sequence −→ Q p / Z p ⊗ Z A ( F ) −→ Sel p ( A F ) −→ X p ( A F ) −→ of Z p [ G ] -modules, from which upon taking Pontryagin duals one derives a canonicalshort exact sequence −→ X p ( A F ) ∨ −→ Sel p ( A F ) ∨ −→ A ( F ) ∗ p −→ . (3)We will throughout use this canonical short exact sequence to fix identifications of (Sel p ( A F ) ∨ ) tor with X p ( A F ) ∨ and of (Sel p ( A F ) ∨ ) tf with A ( F ) ∗ p .In [12] a suitable integral model R Γ f ( k, T p,F ( A )) of the finite support cohomologyof Bloch and Kato for the base change through F/k of the p -adic Tate module of A t is defined and then used in order to define an ‘equivariant regulator’ which isessential to the explicit reformulation of C p ( A, Z [ G ]) (see [12, Th. 5.1]). We willrecall this reformulation in Section 3.By [12, Lem. 4.1], R Γ f ( k, T p,F ( A )) is under our current hypotheses a perfectcomplex of Z p [ G ] -modules which is acyclic outside degrees 1 and 2 and whose coho-mology groups in degrees 1 and 2 canonically identify with A t ( F ) p and Sel p ( A F ) ∨ respectively. We recall that given any complex E with just two nonzero cohomologymodules H m ( E ) and H n ( E ) , n > m , the complex τ ≤ n τ ≥ m E represents an element inthe Yoneda ext-group Ext n − m +1 Z p [ G ] ( H n ( E ) , H m ( E )) . Here τ is the truncation of com-plexes preserving cohomology in the indicated degrees. In this way, R Γ f ( k, T p,F ( A )) uniquely determines a class δ A,K,p in Ext Z p [ G ] (Sel p ( A F ) ∨ , A t ( F ) p ) . The element δ A,K,p is furthermore perfect, meaning that it can be represented as a Yoneda 2-extensionby a four term exact sequence in which each of the two middle modules is perfectwhen considered as an object of D ( Z p [ G ]) . We will use Proposition 2.2 to fix anexplicit -syzygy of the form → M ι → F → F → A ( F ) ∗ p → , (4)in which we set M := M ( t,j ) Z p [ G/H t ] F and F are finitely generated free Z p [ G ] -modules and then use the exactsequence (4) to compute Ext Z p [ G ] ( A ( F ) ∗ p , A t ( F ) p ) via the canonical isomorphism Ext Z p [ G ] ( A ( F ) ∗ p , A t ( F ) p ) ≃ Hom Z p [ G ] ( M, A t ( F ) p ) /ι ∗ (cid:0) Hom Z p [ G ] ( F , A t ( F ) p ) (cid:1) . If we now assume that X p ( A F ) vanishes, we may identify Sel p ( A F ) ∨ and A ( F ) ∗ p ,so that δ A,F,p uniquely determines an element of the above quotient. We willprove (see Lemmas 4.2 and 4.3 below) that we may choose a representative Φ ∈ Hom Z p [ G ] ( M, A t ( F ) p ) of δ A,F,p with the following properties:(P1) Φ is bijective,(P2) for every j ∈ [ m n ] , Φ restricts to send an element x ( n,j ) of the ( n, j ) -th directsummand Z p [ G ] to x ( n,j ) P t ( n,j ) .For a fixed choice of points P and P t such that (1) holds and of Φ ∈ Hom Z p [ G ] ( M, A t ( F ) p ) as above, we fix a canonical Z p [ G/H t ] -basis element e ( t,j ) of each direct summand Z p [ G/H t ] of M and fix any elements Φ ( t,j ) , ( s,i ) of Z p [ G ] with the property that Φ( e ( s,i ) ) = X ( t,j ) Φ ( t,j ) , ( s,i ) P t ( t,j ) . (5)We thus obtain an invertible matrix (cid:0) Φ ( t,j ) , ( s,i ) (cid:1) ( t,j ) , ( s,i ) with entries in Z p [ G ] , whichby abuse of notation we shall also denote by Φ . The matrix Φ is of the form (cid:0) Φ ( t,j ) , ( s,i ) (cid:1) t,s Let P and P t be any choice of points such that (1) holds. Assumethat X p ( A F ) = 0 . Let Φ ∈ Hom Z p [ G ] ( M, A t ( F ) p ) be any representative of δ A,F,p suchthat (P1) and (P2) hold. Then C p ( A, Z [ G ]) is valid if and only if X ψ ∈ b G L ∗ ψ λ ψ ( P , P t ) · ε ψ (Φ) · δ ψ e ψ ∈ Z p [ G ] × . (7)11 emark 2.10. Theorem 2.9 can be reformulated in terms of explicit congruences. Via Theorem 2.9, we now obtain completely explicit predictions concerning con-gruences in the augmentation filtration of the integral group ring Z p [ G ] for leadingterms at s = 1 of the relevant Hasse-Weil- L -functions of A normalised by our twistedregulators. We recall that such predictions constitute a refinement and generalisa-tion of the congruences for modular symbols that are conjectured by Mazur andTate in [20].In order to state such conjectural congruences, we require the following notation:if the inequality rk( A ( F J )) ≤ | G/J | rk( A ( k )) of Corollary 2.3 is strict for somesubgroup J of G , we may and will denote by H = H t the smallest non-trivialsubgroup of G with the property that m H = 0 . Hence t is the maximal index withthe properties m t = 0 and t < n . We then define L := P ψ ∈ b G L ∗ ψ det( ψ ( R ( P , P t ))) e ψ , if rk( A ( F J )) = | G/J | rk( A ( k )) for every J, P ψ | H =1 L ∗ ψ λ ψ ( P , P t ) e ψ , otherwise . We also let I G,p denote the kernel of the augmentation map Z p [ G ] −→ Z p . Corollary 2.11. Let P and P t be any choice of points such that (1) holds. Assumethat X p ( A F ) = 0 . Let Φ ∈ Hom Z p [ G ] ( M, A t ( F ) p ) be any representative of δ A,F,p suchthat (P1) and (P2) hold. If C p ( A, Z [ G ]) is valid, then(i) L belongs to the ideal I hG,p of Z p [ G ] , where h := P t L ∈ I hG,p of Corollary 2.11(i) fromthe assumed validity of conjecture C p ( A, Z [ G ]) in situations in which X p ( A F ) is non-trivial. In this greater level of generality, it furthermore leads to explicit statementsconcerning annihilation of Tate-Shafarevich groups and (generalised) ‘strong mainconjectures’ of the kind that Mazur and Tate ask for in [20, Remark after Conj. 3].Namely, we obtain the following result: Theorem 2.12. Let P and P t be any choice of points such that (1) holds. If C p ( A, Z [ G ]) is valid, then(i) L belongs to the ideal I hG,p of Z p [ G ] , where h := P t It will become clear in the course of the proof that, provided that thereexist sets of points P and P t such that (1) holds from which one may construct theelement L , Theorem 2.12 remains valid even if hypothesis (h) fails to hold. Thisfact is relevant because, as we will see in Section 5, it allows us to obtain numericalsupporting evidence for C p ( A, Z [ G ]) (via verifying the explicit assertions of Theorem2.12) in a wider range of situations. Let Z p [ G ] −→ Z p [ G ] denote the involution induced by g g − . Recallingthat the Cassels-Tate pairing induces a canonical isomorphism between X p ( A F ) ∨ and X p ( A tF ) , we immediately obtain the following corollary: Corollary 2.14. Under the assumptions of Theorem 2.12 one has that the element L of I hG,p annihilates the Z p [ G ] -module X p ( A F ) . p ( A, Z [ G ]) Let R be either Z or Z p and, for the moment, let G be any finite group. We write K for the quotient field of R and let E be a field extension of K . Let Λ be an R -orderin K [ G ] . We recall that there is a canonical exact sequence of algebraic K -groups K (Λ) −→ K ( E [ G ]) ∂ , E −→ K (Λ , E [ G ]) −→ K (Λ) −→ K ( E [ G ]) (8)where K (Λ , E [ G ]) is the relative algebraic K -group, as defined by Swan in [21,p. 215], associated to the ring inclusion Λ ⊆ E [ G ] .For any ring Σ we write ζ (Σ) for its center. We let nr E [ G ] : K ( E [ G ]) −→ ζ ( E [ G ]) × denote the (injective) homomorphism induced by the reduced norm map. If Λ is a Z -order in Q [ G ] we write δ G : ζ ( R [ G ]) × −→ K (Λ , R [ G ]) ,δ G,p : ζ ( C p [ G ]) × −→ K (Λ p , C p [ G ]) for the extended boundary homomorphisms as defined in [9, Sec. 4.2]. Recall that δ G ◦ nr R [ G ] = ∂ , R , δ G,p ◦ nr C p [ G ] = ∂ p , C p . By the general construction described in [9, Prop. 2.5] (and [5, Lem. 5.1]) each pair ( C • , λ ) consisting of a complex C • ∈ D p (Λ p ) and an isomorphism of C p [ G ] -modules λ : C p ⊗ Z p M i ∈ Z H i ( C • ) ! −→ C p ⊗ Z p M i ∈ Z H i +1 ( C • ) ! χ G,p ( C • , λ ) ∈ K (Λ p , C p [ G ]) . For anexplicit example of the computation of χ G,p ( C • , λ ) in a special case, which is alsorelevant for the computations in this paper, we refer the reader to [3, Sec. 3].It is well known that ∂ p , C p is onto and that nr C p [ G ] is an isomorphism. Wetherefore deduce from (8) that K (Λ p , C p [ G ]) ≃ ζ ( C p [ G ]) × / nr C p [ G ] ( K (Λ p )) . (9)Since Λ p is semilocal, we can replace K (Λ p ) by Λ × p in (9). Moreover, it followsfrom (9) that for an element ξ ∈ ζ ( C p [ G ]) × one has that δ G,p ( ξ ) = 0 if and only if ξ ∈ nr C p [ G ] (cid:0) Λ × p (cid:1) . Finally, if G is abelian, we have that K (Λ p , C p [ G ]) ≃ C p [ G ] × / Λ × p , and hence δ G,p ( ξ ) = 0 if and only if ξ ∈ Λ × p .In this context we also recall [2, Lem. 2.5]. We naturally interpret K (Λ , Q [ G ]) and K (Λ p , Q p [ G ]) as subgroups of K (Λ , R [ G ]) and K (Λ p , C p [ G ]) respectively, andrecall that if ξ ∈ ζ ( R [ G ]) × , then δ G ( ξ ) ∈ K (Λ , Q [ G ]) ⇐⇒ ξ ∈ ζ ( Q [ G ]) × while if ξ ∈ ζ ( C p [ G ]) × , then δ G,p ( ξ ) ∈ K (Λ p , Q p [ G ]) ⇐⇒ ξ ∈ ζ ( Q p [ G ]) × . We finally recall that, for any isomorphism j : C ∼ = C p , there is an inducedcomposite homomorphism of abelian groups j G, ∗ : K (cid:0) Λ , R [ G ] (cid:1) → K (cid:0) Λ , C [ G ] (cid:1) ∼ = K (cid:0) Λ , C p [ G ] (cid:1) → K (cid:0) Λ p , C p [ G ] (cid:1) (where the first and third arrows are induced by the inclusions R [ G ] ⊂ C [ G ] and Λ ⊂ Λ p respectively). We also write j ∗ : ζ ( C [ G ]) × → ζ ( C p [ G ]) × for the obvious mapinduced by j , and note that it is straightforward to check that one has j G, ∗ ◦ δ G = δ G,p ◦ j ∗ . Conjecture C ( A, Z [ G ]) is formulated as the vanishing of the ‘equivariant Tamagawanumber’ T Ω (cid:0) h ( A F )(1) , Z [ G ] (cid:1) of K ( Z [ G ] , R [ G ]) that is defined in [9, Conj. 4]and constructed (unconditionally under the assumed validity of hypothesis (g)) viathe formalism of virtual objects from the various canonical comparison morphismsbetween the relevant realisations and cohomology spaces associated to the motive h ( A F )(1) (for more details see [9]).Motivated by work of Bloch and Kato, and in order to isolate the main arithmeticdifficulties involved in making T Ω (cid:0) h ( A F )(1) , Z [ G ] (cid:1) explicit, the approach of [12]14elies upon the definition of a suitable (global) finite support cohomology complex C f, • A,F := R Γ f ( k, T p,F ( A )) (see [12, Sec. 4.2]). Under the hypotheses of [12] thecomplex C f, • A,F is perfect and acyclic outside degrees one and two. Moreover, thereare canonical identifications of H ( C f, • A,F ) and H ( C f, • A,F ) with A t ( F ) p and Sel p ( A F ) ∨ respectively (see [12, Lemma 4.1]). Hence, for a given isomorphism j : C → C p , the C -linear extension of the Néron-Tate height pairing of A defined relative to the field F induces a canonical trivialisation λ NT ,jA,F : C p ⊗ Z p H (cid:0) C f, • A,F (cid:1) ∼ = C p ⊗ Z p A t ( F ) p ∼ = C p ⊗ C ,j ( C ⊗ Z A t ( F )) ∼ = C p ⊗ C ,j Hom C (cid:0) C ⊗ Z A ( F ) , C (cid:1) ∼ = C p ⊗ Z p Hom Z p (cid:0) A ( F ) p , Z p (cid:1) ∼ = C p ⊗ Z p H (cid:0) C f, • A,F (cid:1) . It is finally proved in [12, Th. 5.1] that j G, ∗ (cid:0) T Ω (cid:0) h ( A /F )(1) , Z [ G ] (cid:1)(cid:1) = δ G,p (cid:0) j ∗ ( L ∗ A,F/k ) (cid:1) + χ G,p (cid:16) C f, • A,F , ( λ NT ,jA,F ) − (cid:17) , (10)or equivalently that conjecture C p,j ( A, Z [ G ]) is valid if and only if δ G,p (cid:0) j ∗ ( L ∗ A,F/k ) (cid:1) = − χ G,p (cid:16) C f, • A,F , ( λ NT ,jA,F ) − (cid:17) . (11)In order to prove our results stated in Section 2 we must therefore computethe refined Euler characteristic χ G,p (cid:16) C f, • A,F , ( λ NT ,jA,F ) − (cid:17) in terms of the heights of thechosen sets of points P and P t . In this subsection we will prove Proposition 2.2. The existence of global points P ( t,j ) and P t ( t,j ) such that (1) holds is then an immediate consequence of Roiter’s lemma(see [13, (31.6)]).To ease notation we set H := H ( C f, • A,F ) = A t ( F ) p and H := H ( C f, • A,F ) =Sel p ( A F ) ∨ . We recall that, for any intermediate field L of F/k , we may and will usethe relevant canonical short exact sequence of the form (3) to identify (Sel p ( A L ) ∨ ) tor with X p ( A L ) ∨ and (Sel p ( A L ) ∨ ) tf with A ( L ) ∗ p .Under the assumed validity of hypotheses (a)-(e), the result of [11, Prop. 3.1]directly combines with hypothesis (h) to imply that, for every non-trivial subgroup J of G , the Tate cohomology group b H − ( J, H ) vanishes and the module ( H ) J istorsionfree. By the definition of Tate cohomology, we have that the finite group b H − ( J, H ) identifies with a submodule of ( H ) J and therefore vanishes too.Furthermore, since the complex C f, • A,F is perfect and acyclic outside degrees 1 and2, for each subgroup J of G the group b H ( J, H ) is isomorphic to b H − ( J, H ) . In15ddition, since G is cyclic, the Tate cohomology of each J is periodic of order 2 andso b H − ( J, H ) also vanishes.We next note that, since G is a p -group, hypothesis (a) implies that A t ( F ) p = H is torsion-free.We now apply the main result [22, Th. 2.4] of Yakovlev to see that both A t ( F ) p = H and A ( F ) ∗ p = H are Z p [ G ] -permutation modules, that is, that thereexist isomorphisms of the form A t ( F ) p ≃ M J ≤ G Z p [ G/J ] Ext Z p [ G ] ( A ( F ) ∗ p , A t ( F ) p ) ≃ Hom Z p [ G ] ( M, A t ( F ) p ) /ι ∗ (Hom Z p [ G ] ( F , A t ( F ) p )) under which an element φ of Hom Z p [ G ] ( M, A t ( F ) p ) corresponds to the element ǫ ( φ ) of Ext Z p [ G ] ( A ( F ) ∗ p , A t ( F ) p ) which has the bottom row of the commutative diagramwith exact rows −−−→ M ι −−−→ X Θ −−−→ X π −−−→ A ( F ) ∗ p −−−→ φ y y (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) −−−→ A t ( F ) p −−−→ X ( φ ) −−−→ F π −−−→ A ( F ) ∗ p −−−→ , (15)as a representative. In this diagram X ( φ ) is defined as the push-out of ι and φ . We now proceed to prove that, when considering perfect elements ǫ ( φ ) of Ext Z p [ G ] ( A ( F ) ∗ p , A t ( F ) p ) , one may without loss of generality restrict attention toa special class of elements φ of Hom Z p [ G ] ( M, A t ( F ) p ) .17 emma 4.2. For all subgroups J of G one has(i) Ext Z p [ G ] ( Z p [ G ] , Z p [ G/J ]) = 0 ,(ii) Ext Z p [ G ] ( Z p [ G/J ] , Z p [ G ]) = 0 Proof. Claim (i) is clear. Concerning claim (ii), we first note that since Z p [ G/J ] is Z p -torsion-free, there is an isomorphism of the form Ext Z p [ G ] ( Z p [ G/J ] , Z p [ G ]) ∼ = H ( G, Hom Z p ( Z p [ G/J ] , Z p [ G ])) . But the G -module Hom Z p ( Z p [ G/J ] , Z p [ G ]) is cohomologically-trivial and therefore Ext Z p [ G ] ( Z p [ G/J ] , Z p [ G ]) vanishes, as required.Lemma 4.2 now implies that we can without loss of generality restrict attentionto those elements φ of Hom Z p [ G ] ( M, A t ( F ) p ) which satisfy (P2) and, in addition, bythe argument of [3, Lemma 4.3], which are furthermore injective. Lemma 4.3. Suppose that φ ∈ Hom Z p [ G ] ( M, A t ( F ) p ) has all of the properties de-scribed in the previous paragraph. Then the element ǫ ( φ ) of Ext Z p [ G ] ( A ( F ) ∗ p , A t ( F ) p ) is perfect if and only if φ is an isomorphism.Proof. The fact that φ restricts to send an element x ( n,j ) of the ( n, j ) -th directsummand Z p [ G ] to x ( n,j ) P t ( n,j ) immediately implies that cok( φ ) = cok( φ ′ ) where φ ′ : M ( t,j ) ,t Proposition 4.4. There exist splittings s and s as above with the property that det C p [ G ] ( h λ NT ,jA,F ◦ Φ C p , Θ , s , s i ) = P ψ ∈ b G j ( λ ψ ( P , P t )) ǫ ψ (Φ) δ ψ e ψ .Proof. Let { w ( s,i ) : s = 0 , . . . , n, i ∈ [ m s ] } be the standard basis of X . For each pair ( s, i ) we write W s = W ( s,i ) for the kernel of the canonical map C p [ G ] −→ C p [ G/H s ] , 19o that W s = ( σ p s − C p [ G ] = (1 − e H s ) C p [ G ] . We then have a commutativediagram / / C p [ G/H s ] ι s / / C p [ G ] σ ps − / / " " " " ❊❊❊❊❊❊❊❊ C p [ G ] π s,i / / C p [ G/H s ] P ∗ ( s,i ) / / W s -(cid:13) < < ②②②②②②②② (16)with furthermore L ( s,i ) W ( s,i ) equal to im(Θ) C p = ker( π ) C p . We now fix the requiredsplittings s and s by summing over all pairs ( s, i ) the splittings of the short exactsequences in (16) given by C p [ G ] −→ C p [ G/H s ] ⊕ W s , (cid:18) | H s | , σ p s − (cid:19) (17)and C p [ G ] −→ C p [ G/H s ] P ∗ ( s,i ) ⊕ W s , (cid:0) P ∗ ( s,i ) , − e H s (cid:1) (18)respectively. Note that for the inverse map in (18) we have ( P ∗ ( s,i ) , e H s and (0 , σ p s − σ p s − .After these preparations we proceed to compute the matrix Λ NT (Φ) which rep-resents h λ NT ,jA,F ◦ Φ C p , Θ , s , s i with respect to the fixed C p [ G ] -basis { w ( s,i ) } of X C p .From (17) and (5) it follows easily that the composite of s and (Φ C p , id) maps w ( s,i ) to | H s | X ( t,j ) Φ ( t,j ) , ( s,i ) P t ( t,j ) , (cid:0) . . . , σ p s − , . . . (cid:1) in A t ( F ) C p ⊕ im(Θ) C p = (cid:16)L ( t,j ) C p [ G/H t ] P t ( t,j ) (cid:17) ⊕ (cid:16)L ( t,j ) W t (cid:17) with the only non-zero component in ⊕ ( t,j ) W t at the ( s, i ) -spot. By Lemma 4.5 below this is furthermapped by ( λ NT ,jA,F , id) to | H s | X ( t,j ) Φ ( t,j ) , ( s,i ) X ( u,k ) X τ ∈ G/H u j ( h τ P ( u,k ) , P t ( t,j ) i F ) τ e H u P ∗ ( u,k ) , (cid:0) . . . , σ p s − , . . . (cid:1) . Rearranging the summation and applying the map s − as described in (18) we obtain X ( u,k ) | H s | X ( t,j ) Φ ( t,j ) , ( s,i ) X τ ∈ G/H u j ( h τ P ( u,k ) , P t ( t,j ) i F ) τ e H u w ( u,k ) + ( σ p s − w ( s,i ) . 20e now fix a character ψ ∈ b G . We have that ψ | H s | X ( t,j ) Φ ( t,j ) , ( s,i ) X τ ∈ G/H u j ( h τ P ( u,k ) , P t ( t,j ) i F ) τ e H u = | H s | P ( t,j ) Φ ( t,j ) , ( s,i ) P τ ∈ G/H u j ( h τ P ( u,k ) , P t ( t,j ) ) i F ψ ( τ ) , u ≥ t ψ , , u < t ψ , while ψ ( σ p s − is equal to 0 if and only if s ≥ t ψ .We immediately obtain that det (cid:0) ψ (cid:0) Λ NT (Φ) (cid:1)(cid:1) = j ( λ ψ ( P , P t )) · ε ψ (Φ) · δ ψ , as required.We finally provide the relevant Lemma used in the course of the above proof. Lemma 4.5. λ NT ,jA,F (cid:0) P t ( t,j ) (cid:1) = X ( u,k ) X τ ∈ G/H u j ( h τ P ( u,k ) , P t ( t,j ) i F ) τ e H u P ∗ ( u,k ) . Proof. We recall that λ NT ,jA,F is induced by h , i F : A ( F ) × A t ( F ) −→ C . For P t ∈ A t ( F ) we explicitly have λ NT ,jA,F ( P t ) = j ( h , P t i F ) . Let f ∈ A ( F ) ∗ p denote the mapdefined by the right hand side of the equation in Lemma (4.5). From (14) weimmediately see that e H u P ∗ ( u,k ) = P ∗ ( u,k ) . For each pair ( v, l ) and γ ∈ G/H v we hencehave that f ( γP ( v,l ) ) = X ( u,k ) X τ ∈ G/H u j ( h τ P ( u,k ) , P t ( t,j ) i F ) (cid:0) τ P ∗ ( u,k ) (cid:1) (cid:0) γP ( v,l ) (cid:1) = j ( h γP ( v,l ) , P t ( t,j ) i F )= (cid:16) λ NT ,jA,F ( P t ( t,j ) ) (cid:17) ( γP ( v,l ) ) . We set M A,F = h ( A F )(1) and recall that the approach used in [9] in order toformulate the conjecture C ( A, Q [ G ]) relies upon the theory of categories of virtualobjects. Although focused on the study of p -parts of the relevant equivariant Tama-gawa numbers for prime numbers p , the exact same techniques involved in the proofof [12, Prop. 4.2] allow one to translate this more technical language into the oneof refined Euler characteristics employed throughout this article. For this reason,21e will avoid any explicit mention of categories of virtual objects throughout thisproof. Furthermore, since we only need to consider the relevant realisations and co-homology spaces associated to M A,F as modules over the semisimple algebras Q [ G ] or R [ G ] , we may directly reformulate C ( A, Q [ G ]) in terms of the determinants ofcertain endomorphisms of free R [ G ] -modules, which is what we do in the sequel.The leading term L ∗ ( M A,F , at s = 0 of the Q [ G ] -equivariant motivic L -functionof M A,F is given by P χ ∈ Ir( G ) e χ L ∗ ( A, ˇ χ, . We let once again λ NTA,F : R ⊗ Z A t ( F ) → R ⊗ Z A ( F ) ∗ denote the canonical isomorphism induced by the Néron-Tate heightpairing and α A,F : R ⊗ Q M v ∈ S F ∞ H ( F v , H v ( M A,F )) → R ⊗ Q H dR ( M A,F ) /F denote the canonical period isomorphism described by Deligne in [14] (see also [9,Sec. 3] for a general description of the modules involved and [12, Sec. 4.3] for moredetails in the relevant special case).We now note that the Q [ G ] -modules X := Q ⊗ Z A t ( F ) and Y := Q ⊗ Z A ( F ) ∗ ,resp. Z := L v ∈ S F ∞ H ( F v , H v ( M A,F )) and W := H dR ( M A,F ) /F , are isomorphic (asa consequence, for instance, of [1, p. 110]) and hence, since Q [ G ] is semisimple,there exist Q [ G ] -modules M and N with the property that both X ⊕ M ∼ = Y ⊕ M and Z ⊕ N ∼ = W ⊕ N are free Q [ G ] -modules. In the sequel we (choose bases andso) fix identifications of X ⊕ M , Y ⊕ M , Z ⊕ N and W ⊕ N with direct sums ofcopies of Q [ G ] and hence regard λ NTA,F ⊕ id R ⊗ Q M and α A,F ⊕ id R ⊗ Q N as elements of K ( R [ G ]) .The validity of conjecture C ( A, Q [ G ]) is then equivalent to the containment L ∗ ( M A,F , / (det R [ G ] ( α A,F ⊕ id R ⊗ Q N )det R [ G ] ( λ NTA,F ⊕ id R ⊗ Q M )) ∈ Q [ G ] × . But it is clear from the proof of Lemma 4.5 that, independently of our choice offixed identifications, det R [ G ] ( λ NTA,F ⊕ id R ⊗ Q M ) / X χ ∈ Ir( G ) e χ λ χ ( P , P t ) ∈ Q [ G ] × , (19)and it is straightforward to deduce from the proof of [12, Lemma 4.5] that, indepen-dently of our choice of fixed identifications, det R [ G ] ( α A,F ⊕ id R ⊗ Q N ) / X χ ∈ Ir( G ) e χ w ∞ ( k ) d · Ω( A/k ) τ ∗ ( Q , ind Q k ( χ )) d ∈ Q [ G ] × . (20)The equalities (19) and (20), combined with the fact that the Euler factors involvedin the truncation of each of the leading terms L ∗ S r ( A, ˇ ψ, live by definition in Q [ G ] × ,therefore imply that the validity of C ( A, Q [ G ]) is equivalent to the containment X ψ ∈ b G L ∗ ψ λ ψ ( P , P t ) e ψ ∈ Q [ G ] × . 22y [2, Lem. 2.9] this containment is equivalent to the explicit condition describedin Theorem 2.6, as required. We assume now that C ( A, Q [ G ]) is valid and proceed to prove the explicit inter-pretation of C p ( A, M ) claimed in Theorem 2.8. We begin by noting that, for anyfixed isomorphism of fields j : C → C p , the respective maps j G, ∗ restrict to give thevertical arrows in a natural commutative diagram with exact rows of the form K ( Z [ G ] , Q [ G ]) tor −−−→ K ( Z [ G ] , Q [ G ]) µ −−−→ K ( M , Q [ G ]) j G, ∗ y j G, ∗ y j G, ∗ y K ( Z p [ G ] , Q p [ G ]) tor −−−→ K ( Z p [ G ] , Q p [ G ]) µ p −−−→ K ( M p , Q p [ G ]) . (21)We note that the exactness of the rows follows from [9, Lemma 11]. We now proceedto prove several useful results. Lemma 4.6. C p ( A, M ) holds if and only if j G, ∗ (cid:0) T Ω (cid:0) h ( A F )(1) , Z [ G ] (cid:1)(cid:1) belongs to K ( Z p [ G ] , Q p [ G ]) tor .Proof. The equality T Ω (cid:0) h ( A F )(1) , M (cid:1) = µ (cid:0) T Ω (cid:0) h ( A F )(1) , Z [ G ] (cid:1)(cid:1) proved in [9,Th. 4.1] combines with the commutativity of the right-hand square of diagram (21)to imply that j G, ∗ (cid:0) T Ω (cid:0) h ( A F )(1) , M (cid:1)(cid:1) = µ p (cid:0) j G, ∗ (cid:0) T Ω (cid:0) h ( A F )(1) , Z [ G ] (cid:1)(cid:1)(cid:1) . The exactness of the bottom row of diagram (21) thus completes the proof. Lemma 4.7. C p ( A, M ) holds if and only if P ψ ∈ b G L ∗ ψ j ( λ ψ ( P , P t )) ǫ ψ (Φ) δ ψ e ψ ∈ M × p .Proof. Lemma 4.6 combines with equalities (10) and (12) to imply that C p ( A, M ) holds if and only if µ p δ G,p X ψ ∈ b G L ∗ ψ j ( λ ψ ( P , P t )) ǫ ψ (Φ) δ ψ e ψ = 0 . We next note that the respective maps δ G,p induce vertical (bijective) arrows ina commutative diagram of the form Q p [ G ] × / Z p [ G ] × −−−→ Q p [ G ] × / M × p y y K ( Z p [ G ] , Q p [ G ]) µ p −−−→ K ( M p , Q p [ G ]) . This completes the proof of the Lemma.23 emma 4.8. ε ψ (Φ) ∈ Z p [ ψ ] × .Proof. The map Φ ⊗ M p : M ⊗ Z p [ G ] M p −→ A t ( F ) p ⊗ Z p [ G ] M p is an isomorphism of M p -modules. Since M p contains the Q p [ G ] -rational idempotents, Φ ⊗ Z p [ ψ ] : M ⊗ Z p [ G ] Z p [ ψ ] −→ A t ( F ) p ⊗ Z p [ G ] Z p [ ψ ] is an isomorphism of Z p [ ψ ] -modules. It is easy to see that Φ ⊗ Z p [ ψ ] is representedby ψ (cid:0) Φ t ψ (cid:1) .We now proceed to give the proof of Theorem 2.8.The equivalence of (i) and (iv) follows directly upon combining Lemmas 4.7 and4.8.Furthermore it is straightforward to compute the valuation of each element δ ψ .One has v p ψ ( δ ψ ) = b ψ with b ψ defined as in Theorem 2.8, and hence (iii) and (iv)are clearly equivalent.In order to prove the equivalence of (i) and (ii), we will use (a special caseof) a general fact which we now describe. If H is any subgroup of G , we write ρ GH : K ( Z p [ G ] , Q p [ G ]) −→ K ( Z p [ H ] , Q p [ H ]) for the natural restriction map and q H : K ( Z p [ H ] , Q p [ H ]) −→ K ( Z p , Q p ) for the natural map induced by sending anelement [ P, φ, Q ] of K ( Z p [ H ] , Q p [ H ]) to the element [ P H , φ H , Q H ] of K ( Z p , Q p ) .By [6, Thm. 4.1] one then has that K ( Z p [ G ] , Q p [ G ]) tor = \ H ≤ G ker( q H ◦ ρ GH ) . (22)The functoriality properties of the element T Ω (cid:0) h ( A F )(1) , Z [ G ] (cid:1) with respect tothe maps ρ GH and q H proved in [9, Prop. 4.1] then imply that, for any subgroup H of G , ( q H ◦ ρ GH ) (cid:0) j G, ∗ (cid:0) T Ω (cid:0) h ( A F )(1) , Z [ G ] (cid:1)(cid:1)(cid:1) = j , ∗ (cid:0) T Ω (cid:0) h ( A F H )(1) , Z (cid:1)(cid:1) , and so Lemma 4.6 combines with (22) to imply that C p ( A, M ) holds if and only if, forevery intermediate field L of F/k , the element j G L/L , ∗ (cid:0) T Ω (cid:0) h ( A L )(1) , Z (cid:1)(cid:1) vanishes,that is, if and only if the p -part of the eTNC holds for the pair (cid:0) h ( A L )(1) , Z (cid:1) .Noting that it is easy to check that the set of data ( A/L, L/L, p ) satisfies all thehypotheses of Theorem 2.9 for any such field L (see for instance [11, Lem. 3.4]for a proof of a more general assertion), all that is left to do in order to prove theequivalence of (i) and (ii) is to apply Theorem 2.9. Indeed, any choice of Z -bases { Q i } and { R j } of A ( L ) and A t ( L ) respectively satisfy condition (1) for the set of data ( A/L, L/L, p ) , while an explicit computation proves that τ ∗ ( Q , ind Q L ( GL/L )) w ∞ ( L ) = p | d L | .24 .5 The proof of Corollary 2.11 For brevity we set λ ψ := λ ψ ( P , P t ) , ǫ ψ := ǫ ψ (Φ) , u := X ψ ∈ b G L ∗ ψ λ ψ ǫ ψ δ ψ e ψ . By Theorem 2.9 the validity of C p ( A, Z [ G ]) is equivalent to the containment u ∈ Z p [ G ] × , which we assume holds throughout the proof. We also let ε : Z p [ G ] −→ Z p denote the augmentation map.We begin by noting that claim (ii) is just the ψ = G special case of Lemma 4.8,and proceed now to deduce claim (iii) from it. One clearly has that L ∗ G / ( λ G ǫ G δ G ) = ε ( u ) ∈ Z × p with δ G equal by definition to 1, while a straightforward computationshows that τ ∗ ( Q , ind Q k ( G )) w ∞ ( k ) = ( − | S r | p | d k | . Claim (ii) therefore indeed implies that v = L ∗ G /λ G = ε ( u ) · ǫ G = ε ( u ) · ǫ (23)belongs to Z × p , as required.In order to prove the remaining claims, we first note that, if rk( A ( F J )) = | G/J | rk( A ( k )) for every subgroup J of G , then h = 0 by Proposition 2.2 while Φ can be chosen to be the identity matrix by property (P2) and each element δ ψ issimply equal to 1 by convention. In any such case, claim (i) therefore reduces to thetrivial statement L = u ∈ Z p [ G ] while claim (iv) simply reads u ≡ v (mod I G,p ) andfollows directly from (23). We therefore may and will henceforth assume that theinequality rk( A ( F J )) ≤ | G/J | rk( A ( k )) of Corollary 2.3 is strict for some subgroup J of G . We recall that H = H t denotes the smallest non-trivial subgroup of G withthe property that m H = 0 .In order to prove claim (i), we note first that for each ψ ∈ b G we have ψ | H = 1 ⇐⇒ ker( ψ ) ⊆ H t and ker( ψ ) = H t ⇐⇒ t ψ > t . From the definitions of ǫ ψ and δ ψ we immediately deduce that, for each ψ ∈ b G suchthat ψ | H = 1 , ǫ ψ = 1 , δ ψ = δ := t Y j =0 (cid:16) σ p j − (cid:17) m j . Since δe ψ = 0 for each ψ such that ψ | H = 1 we deduce that L = δu ∈ δ Z p [ G ] ⊆ I hG,p ,as required.Finally, claim (iv) follows from (23) because u is clearly congruent to ε ( u ) = v/ǫ modulo I G,p and therefore L = δu is congruent to δ vǫ modulo I h +1 G,p , as required. We begin by defining a (free) Z p [ G ] -submodule P := M j ∈ [ m n ] Z p [ G ] P ∗ ( n,j ) A ( F ) ∗ p and then fix, as we may, an injective lift κ : P −→ Sel p ( A F ) ∨ of theinclusion P ⊆ A ( F ) ∗ p through the canonical projection of (3). We also fix, as wemay, a representative of the perfect complex C f, • A,F of the form C → C in whichboth C and C are finitely generated, cohomologically-trivial Z p [ G ] -modules. Wethen obtain a commutative diagram with exact rows and columns of the form y y y y −→ L j Z p [ G ] P t ( n,j ) L j Z p [ G ] P t ( n,j ) 0 −→ im( κ ) im( κ ) −→ y y y y −→ A t ( F ) p −→ C −→ C −→ Sel p ( A F ) ∨ −→ y y y y −→ N −→ D −→ D −→ cok( κ ) −→ y y y y (24)in which we have set N := M t 0) = − χ e N Z p [ G ] ,e N C p [ G ] ( e N C f, • A,F , e N ( λ NT ,jA,F ) − )+ χ e N Z p [ G ] ,e N C p [ G ] ( e N B • , λ ′ ) (26)where λ ′ denotes the canonical isomorphism e N ( C p ⊗ Z p im( κ )) = e N ( C p ⊗ Z p Sel p ( A F ) ∨ ) e N ( λ NT ,jA,F ) − −→ e N ( C p ⊗ Z p A t ( F ) p ) = e N ( C p ⊗ Z p M j Z p [ G ] P t ( n,j ) ) . If we now write ϕ : L j Z p [ G ] P t ( n,j ) → L j Z p [ G ] P ∗ ( n,j ) for the canonical isomorphismthat maps an element P t ( n,j ) to the element P ∗ ( n,j ) , then one finds that χ e N Z p [ G ] ,e N C p [ G ] ( e N B • , λ ′ ) = δ e N Z p [ G ] ,e N C p [ G ] (det e N C p [ G ] ( λ ′ ◦ e N ( C p ⊗ Z p ( κ ◦ ϕ ))))= − δ e N Z p [ G ] ,e N C p [ G ] ( X ψ ∈ Υ N j ( λ ψ ( P , P t )) e ψ ) , (27)where the last equality follows from Lemma 4.5.The assumed validity of C p ( A, Z [ G ]) therefore combines via (11) with equalities(26) and (27) to imply that, in the terminology of [10, §2.3.2], the element j ∗ ( X ψ ∈ Υ N L ∗ ψ λ ψ ( P , P t ) − e ψ ) = j ∗ ( L ) of e N C p [ G ] is a characteristic element for e N D • . The result [10, Lem. 2.6] thereforeimplies that there exists a characteristic element L ′ for D • in C p [ G ] with the propertythat e N L ′ = j ∗ ( L ) . Since D • is clearly an admissible complex of Z p [ G ] -modules (inthe terminology of [10, §2.1.1]), the results of [10, Cor. 3.3] therefore imply that theelement j ∗ ( L ) belongs to the ideal I ˜ hG,p of Z p [ G ] , with ˜ h := dim Q p ( Q p ⊗ Z p cok( κ ) G ) ,and furthermore generates Fitt Z p [ G ] (cok( κ )) . To proceed with the proof, we firstnote that ˜ h = dim Q p ( Q p ⊗ Z p cok( κ ) G ) = dim Q p ( Q p ⊗ Z p M t 27s injective and hence that one has that Fitt Z p [ G ] (cok( κ )) ⊆ Ann Z p [ G ] ( X p ( A F ) ∨ ) . Recalling finally that the Cassels-Tate pairing induces a canonical isomorphism be-tween X p ( A F ) ∨ and X p ( A tF ) completes the proof of claim (ii) and thus of Theorem2.12. In this section we gather some evidence, mostly numerical, in support of conjectureC p ( A, Z [ G ]) . Our aim is to verify statements that would not follow in an straight-forward manner from the validity of the Birch and Swinnerton-Dyer conjecture forall intermediate fields of F/k . Because of the equivalence of statements (i) and (ii)in Theorem 2.8 we therefore choose not to focus on presenting evidence for conjec-ture C p ( A, M ) (although we also used our MAGMA programs to produce numericalevidence for C p ( A, M ) by verifying statement (iii) of Theorem 2.8).Throughout this section A will always denote an elliptic curve. p ( A, Z [ G ]) For the verification of C p ( A, Z [ G ]) using Theorem 2.9 it is necessary to have explicitknowledge of a map Φ that represents the extension class δ A,F,p . Whenever A ( F ) p isnot projective as a Z p [ G ] -module we are currently not able to numerically compute Φ , so we only deal with examples in which A ( F ) p is projective. To the best of ourknowledge there are currently three instances of theoretical evidence (in situationsin which our fixed cyclic extension F/k is not trivial): • In [4], it is shown that for each elliptic curve A/ Q with L ( A/ Q , = 0 thereare infinitely many primes p and for each such prime p infinitely many (cyclic) p -extensions F/ Q such that C p ( A, Z [Gal( F/ Q )]) holds. All of these examplessatisfy our hypotheses and are such that A ( F ) p vanishes. • In [12, Th. 1.1], C p ( A, Z [Gal( F/ Q )]) is proved for certain elliptic curves A/ Q ,where F denotes the Hilbert p -classfield of an imaginary quadratic field k .This result combines with the functoriality properties of the eTNC to implythe validity of C p ( A, Z [Gal( F/k )]) . In these examples one has that A ( F ) p is afree Z p [Gal( F/k )] -module of rank one. • In [12, Cor. 6.2], certain S -extensions F/K are considered. Let k and L denote the quadratic and cubic subfield of F/K respectively. Under certainadditional assumptions it is then shown that the validity of the Birch andSwinnerton-Dyer conjecture for A over the fields k, K and L implies the valid-ity of C p ( A, Z [Gal( F/K )]) . Again by functoriality arguments, the validity of28 p ( A, Z [Gal( F/k )]) follows. We note that the assumptions are such that oneagain has that A ( F ) p is a free Z p [Gal( F/k )] -module of rank one.In the rest of this section we are concerned with numerical evidence. In [2,Sec. 6] there is a list of examples of elliptic curves A/ Q and dihedral extensions F/ Q of order p for which C p ( A, Z [Gal( F/ Q )]) is numerically verified. Here thequadratic subfield k is real and A ( F ) p vanishes. Again by functoriality argumentswe obtain examples where C p ( A, Z p [Gal( F/k )]) is numerically verified. There aretwo further analogous numerical verifications in dihedral examples in [12, Sec. 6.3],one of degree and one of degree , both of them with the property that A ( F ) p is a free Z p [Gal( F/k )] -module of rank one.In the following we fix an odd prime p and let q denote a prime such that q ≡ p ) . We let F denote the unique subfield of Q ( ζ q ) / Q of degree p andtake k to be Q . For p ∈ { , , } and q < we went through the list of semistableelliptic curves of rank one and conductor N < and checked numerically whether L ( A/ Q , χ, 1) = 0 and L ′ ( A/ Q , χ, = 0 for a non-trivial character χ of G , and inaddition, whether our hypotheses are satisfied. This resulted in a list of examples( for p = 3 , for p = 5 and for p = 7 ). In each of these examples we could finda point R such that A ( F ) p = Z p [ G ] R and numerically verify conjecture C p ( A, Z [ G ]) .We now describe in detail an example with [ F : Q ] = 7 . Let A be the ellipticcurve A : y + xy + y = x + x − x. This is the curve 79a1 in Cremona’s notation. It is known that A ( Q ) is free ofrank one generated by P = (0 , and that X ( A Q ) = 0 . Moreover it satisfies thehypotheses used throughout the paper.We take p = 7 and let F be the unique subfield of Q ( ζ ) of degree . Explicitly, F is the splitting field of f ( x ) = x + x − x − x + 28 x + 14 x − x + 1 and we let α denote a root of f . Using the MAGMA command Points it is easy tofind a point R of infinite order in A ( F ) \ A ( Q ) , R = (cid:18) 117 (31 α + 23 α − α − α + 814 α + 372 α − , 117 ( − α − α + 380 α + 771 α − α − α + 232) (cid:19) . By Proposition 2.2 we know that A ( F ) p is a permutation module, hence A ( F ) p ≃ Z p [ G ] . Furthermore, [11, Prop. 3.1] now implies that X p ( A F ) = 0 .We set Q := Tr F/ Q ( R ) = ( , − ) and easily verify that Q = − P . We checkednumerically that Z p [ G ] R = A ( F ) p . 29omputing numerical approximations to the leading terms using Dokchitser’sMAGMA implementation of [15] we obtain the following vector for (cid:0) L ∗ χ /λ χ ( P , P t ) (cid:1) χ ∈ ˆ G ( − . , − . . i, − . . i, − . − . i, − . − . i, − . − . i, − . . i ) This is very close to ( − / , ζ + ζ + ζ , − ζ − ζ − ζ − , − ζ − ζ − ζ − , − ζ − ζ − ζ − , ζ + ζ + ζ , ζ + ζ + ζ ) It is now easy to verify the rationality conjecture C ( A, Q [ G ]) by the criterion ofTheorem 2.6. Moreover, the valuations of − / and ζ + ζ + ζ at p χ are , sothat by Theorem 2.8 we deduce the validity of C p ( A, M ) . Finally, one easily checksthat − / ≡ ζ + ζ + ζ (mod (1 − ζ )) , so that the element in (7) is actually aunit in Z p [ G ] , thus (numerically) proving C p ( A, Z [ G ]) . p ( A, Z [ G ]) In this subsection we collect evidence for statements that we have shown to followfrom the validity of C p ( A, Z [ G ]) and focus on situations in which A ( F ) p is not Z p [ G ] -projective. In particular, we aim to verify claim (i) of Theorem 2.12. Since we canneither compute the module X p ( A F ) nor a map Φ as required, we are not able toverify any other claim of either Corollary 2.11 or Theorem 2.12.Again we want to focus on evidence which goes beyond implications of the Birchand Swinnerton-Dyer conjecture for A over all intermediate fields of F/k . We assumethe notation of Theorem 2.12, so in particular set h = P t for some sets of non-negative integers { r J } and { s J } . But the Néron-Tate heightpairing induces an isomorphism of C p [ G ] -modules between C p ⊗ Z p A t ( F ) p and C p ⊗ Z p A ( F ) ∗ p and so by rank considerations we find that r J = s J =: m J for every J .Finally, it is easy to see that the Z p -linear dual of a permutation module is againa permutation module of the same form. Therefore the canonical isomorphism A ( F ) ∗∗ p ≃ A ( F ) p shows that one also has that A ( F ) p ≃ M J ≤ G Z p [ G/J ]