arxiv:2103.11535v1 [math.nt] 22 mar 2021

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021 ON DERIVATIVES OF KATO’S EULER SYSTEM

AND THE MAZUR-TATE CONJECTURE

DAVID BURNS, MASATO KURIHARA AND TAKAMICHI SANO

Abstract. We provide a new interpretation of the Mazur-Tate Conjecture and then useit to obtain the first (unconditional) theoretical evidence in support of the conjecture forelliptic curves of strictly positive rank.

Contents

1. Introduction 11.1. Discussion of the main results 11.2. General notation 42. Review of the Mazur-Tate Conjecture 52.1. The Birch and Swinnerton-Dyer Conjecture 52.2. Modular elements and the Mazur-Tate Conjecture 63. Review of the Generalized Perrin-Riou Conjecture 103.1. The Bockstein regulator 103.2. Kato’s zeta elements 113.3. The Generalized Perrin-Riou Conjecture 124. The main result 134.1. Statement of the main result and the deduction of Theorem 1.1 134.2. Determinantal zeta elements 145. The proof of Theorem 4.6 and the deduction of Corollary 1.2 175.1. Construction of the modular element 175.2. Bloch-Kato Selmer complexes 195.3. Bockstein maps for Selmer complexes 215.4. The algebraic Birch and Swinnerton-Dyer element 225.5. Completion of the proof of Theorem 4.6 245.6. The deduction of Corollary 1.2 24References 25

1. Introduction

1.1. Discussion of the main results. Let E be an elliptic curve over Q and write r forthe rank of its Mordell-Weil group E(Q).

In [19] Mazur and Tate formulated a ‘refined conjecture of Birch and Swinnerton-Dyertype’ for E relative to a finite real abelian extension F of Q. Set G := Gal(F/Q) and

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http://arxiv.org/abs/2103.11535v1

2 DAVID BURNS, MASATO KURIHARA AND TAKAMICHI SANO

write I(G) for the augmentation ideal of the integral group ring Z[G]. Then, under theassumption that E has good reduction at ramifying primes in F/Q, the conjecture of Mazurand Tate predicts, roughly speaking, that an element of Q[G] constructed from modularsymbols associated to E belongs to the r-th power of I(G) and, further, that its image inthe quotient group I(G)r/I(G)r+1 is equal to the product of the discriminant of a canonicalG-valued pairing on E(Q) defined using the geometrical theory of biextensions by a suitableratio of the orders of finite groups that occur in the Birch and Swinnerton-Dyer Conjecturefor E over Q. For convenience, we refer to the prediction that the modular element belongsto I(G)r and to the predicted formula for its image in I(G)r/I(G)r+1 as respectively the‘order of vanishing’ and ‘leading-term’ components of the Mazur-Tate Conjecture.

If r = 0, then the full Mazur-Tate Conjecture (which is stated precisely as Conjecture 2.3below) is easily seen to be equivalent to the original conjecture of Birch and Swinnerton-Dyer (see Remark 2.5) and if r > 0, then the order of vanishing component of the conjecturehas proved amenable to analysis via Euler system methods by using Kato’s zeta elements(see the recent article of Ota [20]). However, if r > 0, then the leading-term component ofthe conjecture has remained stubbornly mysterious and, even now, there is no conceptualunderstanding of it or theoretical evidence for it and the only supporting numerical evidenceis for the case r = 1 (for details of which see, for example, the recent article of Portillo-Bobadilla [23]).

Notwithstanding these difficulties, Mazur and Tate’s celebrated conjecture has been veryinfluential and led to the study of a range of similar conjectures, both in the setting ofelliptic curves (for example, by Darmon and by Bertolini and Darmon) and in the settingof the multiplicative group (for example, by Gross, by Tate, by Darmon and by Mazur andRubin).

In the sequel we assume r > 0. In this case, we formulated in an earlier article anexplicit refinement of Perrin-Riou’s conjecture (from [22]) concerning the logarithm of Kato’szeta element at a fixed prime p. We recall that the ‘Generalized Perrin-Riou Conjecture’formulated in [7] predicts a precise congruence relation between a natural (r − 1)-st order‘Darmon derivative’ of the zeta element at p of E over an arbitrary real abelian field andthe value at s = 1 of the r-th derivative of the Hasse-Weil L-function L(E, s) of E over Q.In particular, the special case of the conjecture relating to subfields of the cyclotomic Zp-extension of Q (which is stated explicitly as Conjecture 3.1 below) is known to be equivalentin the case r = 1 to Perrin-Riou’s original conjecture (see [7, Rem. 4.10]) and in the caseof general r to follow, up to multiplication by an element of Z×

p , from the validities of therelevant cases of the Birch and Swinnerton-Dyer conjecture and generalized Iwasawa mainconjecture (see [7, Th. 7.3]), and hence to follow essentially from the relevant case of theequivariant Tamagawa number conjecture.

Building on this earlier approach, in the current article we now develop techniques thatallow us to prove results of the following sort.

To state the result we write Tam(E) for the product of the Tamagawa factors of E ateach prime of bad reduction (see §2.1). In addition, for each prime p we regard the groupE[p] of Q-rational points of E of order dividing p as a module over Gal(Q/Q) in the natural

3

way, and we write Ens(Fp) for the group of non-singular Fp-rational points of the reductionof E modulo p.

Theorem 1.1. Let E be an elliptic curve over Q and F a finite real abelian extension ofQ that satisfies all of the following conditions.

(a) The conductor m of F is square-free and coprime to the conductor N of E.(b) [F : Q] is coprime to 6 ·m ·N · Tam(E) and to #Ens(Fp) for every prime divisor p

of m ·N · [F : Q].(c) For every prime divisor p of [F : Q] the action of Gal(Q/Q) on E[p] is surjective.

Then the Mazur-Tate Conjecture is valid for E relative to F/Q whenever both of the fol-lowing conditions are satisfied:

(d) E validates the Birch and Swinnerton-Dyer Conjecture over Q;(e) If r > 0, then for every prime divisor p of [F : Q], the Generalized Perrin-Riou

Conjecture is valid for E relative to the cyclotomic Zp-extension of Q (see Conjecture3.1 for the Generalized Perrin-Riou Conjecture).

This result shows that, under certain mild technical hypotheses, the ‘refined’ nature ofthe Mazur-Tate Conjecture in comparison with the Birch and Swinnerton-Dyer Conjectureis accounted for by our Generalized Perrin-Riou conjecture in [7]. Since the latter conjecturecan itself be deduced from the validity of certain standard conjectures (as recalled above),Theorem 1.1 therefore gives a new conceptual interpretation of the Mazur-Tate Conjectureand, even better, can be combined with existing results on the conjectures of Birch andSwinnerton-Dyer and Perrin-Riou to obtain concrete theoretical evidence in support of theMazur-Tate Conjecture for elliptic curves with strictly positive rank.

In this way, for example, we are able to give unconditional evidence in support of theMazur-Tate Conjecture of the sort described in the next result. Before stating this resultwe recall that if r = 1, then the group I(G)r/I(G)r+1 is naturally isomorphic to G, and wesay that an equality x = y of elements of G is ‘valid up to an automorphism of G’ if thereexists an automorphism α of G such that x = α(y).

Corollary 1.2. Assume that E and F/Q satisfy the conditions (a), (b) and (c) in Theorem1.1. Assume also that L(E, s) vanishes to order one at s = 1 (which implies r = 1 by Gross,Zagier and Kolyvagin), that the conductor of E is square-free and that E has supersingularreduction at each prime divisor of [F : Q].

Then the order of vanishing component of the Mazur-Tate Conjecture for E and F/Q isvalid and the leading-term component of the Mazur-Tate Conjecture for E and F/Q is validup to an automorphism of Gal(F/Q).

Further, if the Mazur-Tate Conjecture is not valid for E and F/Q, then the Birch andSwinnerton-Dyer Conjecture is not valid for E over Q.

This result gives the first theoretical (and unconditional) evidence in support of theleading term component of the Mazur-Tate Conjecture for any elliptic curve of positiverank. Its proof will be given in §5.6 and combines calculations that are used in the proofof Theorem 1.1 with recent results of Buyukboduk [12] on Perrin-Riou’s conjecture and ofJetchev, Skinner, and Wan [15] on the Birch and Swinnerton-Dyer Conjecture. In fact, forprime divisors p of [F : Q] at which E has ordinary reduction, our approach can also be


similarly combined with the recently announced results of Bertolini and Darmon [1], and ofBuyukboduk, Pollack and Sasaki [13], concerning Perrin-Riou’s conjecture to obtain furthertheoretical evidence in support of the Mazur-Tate Conjecture.

It is interesting to note that, whilst the Mazur-Tate Conjecture predicts an equality inthe group I(G)r/I(G)r+1, the corresponding case of the Generalized Perrin-Riou Conjecturepredicts an equality in E(Q) ⊗ I(G)r−1/I(G)r and so is in natural sense ‘finer’. Anotherinteresting feature of Theorem 1.1 is that, for a given curve E, the Generalized Perrin-RiouConjecture over the cyclotomic Zp-extension Q∞ of Q can be used to prove the p-primarycomponent of the Mazur-Tate Conjectures relative to fields F that are disjoint from Q∞.

To prove Theorem 1.1 we shall in fact first formulate for each prime p an ‘algebraic’analogue of the Generalized Perrin-Riou Conjecture for E relative to the cyclotomic Zp-extension of Q (see Conjecture 3.4). This conjecture is we feel of some independent interestand, in particular, can be seen to be equivalent to the relevant case of the Generalized Perrin-Riou Conjecture precisely when E validates the Birch and Swinnerton-Dyer Conjecture overQ. Theorem 1.1 thereby follows directly from Theorem 4.1 which asserts that, under thestated technical hypotheses, Conjecture 3.4 implies the p-primary component of the Mazur-Tate Conjecture for E relative to F/Q.

Our main task is therefore to prove Theorem 4.1 and the argument we use for thishas several key steps (that are split across §4 and §5). Firstly, we shall apply the theory ofequivariant Kolyvagin and Stark systems (as developed by Sakamoto and the first and thirdauthors in [8], and already used in this setting by Kataoka in [16]) to Kato’s zeta elementsat p in order to study certain canonical p-adic ‘determinantal zeta elements’ (see Definition4.3). Then we shall show Conjecture 3.4 implies that the p-adic determinantal zeta elementassociated toQ is equal to a p-adic ‘algebraic Birch and Swinnerton-Dyer element’ that arisesin the approach of [7] (see Proposition 4.4). Finally, we shall show, by explicit computation,that the latter equality implies the validity of the p-primary component of the Mazur-TateConjecture (see Theorem 4.6). We remark that this last computation relies on the preciserelation between modular elements and Kato’s zeta elements (as previously discussed by thesecond author in [18], by Otsuki in [21] and by Ota in [20]), on an explicit description of therelevant Bloch-Kato Selmer complexes and on a Galois-cohomological interpretation of thebiextension-pairing of Mazur and Tate in terms of Bockstein homomorphisms associated toBloch-Kato Selmer complexes that is proved by Macias-Castllo and the first author in [3](and relies on earlier cohomological calculations of Tan and of Bertolini and Darmon).

Finally, we note that, whilst the hypothesis that m is square-free (in Theorem 1.1) isequivalent to the condition that F/Q is tamely ramified, we believe that our general ap-proach should also work in the setting of arbitrary real abelian fields F .

1.2. General notation. For the reader’s convenience we collect together some notationsand conventions that will be used in the sequel (and are, in general, consistent with thoseused in [7]).

We write Q for the algebraic closure of Q in C and regard any algebraic extension of Qas a subfield of Q. We set GQ := Gal(Q/Q). For each natural number n we also set

ζn := e2πi/n ∈ Q.

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For an abelian group M , we writeMtors for its torsion subgroup and Mtf for the quotientof M by Mtors (which we regard as a subgroup of Q⊗ZM in the natural way). For a primenumber p, we denote byM [p] and M [p∞] the subgroups of M comprising elements that arerespectively annihilated by p and by some power of p. We also write Z(p) for the localizationof Z at p and Zp and Qp for the ring of p-adic integers and field of p-adic rationals (so thatZ(p) = Q ∩ Zp ⊂ Qp).

If M is a module over a commutative ring R, we set

M∗ := HomR(M,R).

If M is a free R-module with basis x1, . . . , xr, then the dual basis of M∗ is denoted byx∗1, . . . , x

∗r.

For a Zp-module M , the Pontryagin dual is defined by

M∨ := HomZp(M,Qp/Zp).

For a finite group Γ, we set Γ := Hom(Γ,C×) = Hom(Γ,Q×). For χ ∈ Γ, we define a

primitive idempotent of Q[Γ] by setting

eχ :=1

#Γ

∑

σ∈Γ

χ(σ)σ−1.

If E has good reduction at a rational prime ℓ, then we write E(Fℓ) in place of Ens(Fℓ).We also use several standard notations for elliptic curves such as E0, E1 and, for any finiteset S of prime numbers, LS(E, s), LS(E,χ, s), etc.

For a number field F and a prime number p, we set

Fp := F ⊗Q Qp ≃⊕

p|p

Fp,

where in the direct sum p runs over all p-adic places of F .We use standard notations for Galois (etale) cohomology complexes such as RΓ(OF,S,−),

RΓf (F,−), etc. The notation E1(Fp) indicates⊕

p|pE1(Fp) and we use similar notation to

denote ‘semi-local’ Bloch-Kato complexes such as RΓf (Fp,−), RΓ/f (Fp,−), etc.

2. Review of the Mazur-Tate Conjecture

In this section, we quickly review the statement of the Birch and Swinnerton-Dyer Con-jecture for elliptic curves over Q and then describe a reformulation of the Mazur-TateConjecture that is convenient for our approach.

2.1. The Birch and Swinnerton-Dyer Conjecture. Let E be an elliptic curve over Qfor which the Tate-Shafarevich group X(E/Q) of E over Q is finite.

We write N for the conductor of E, consider the product of Tamagawa factors

Tam(E) :=∏

ℓ|N

#(E(Qℓ)/E0(Qℓ))


and define the ‘algebraic Birch and Swinnerton-Dyer constant’ for E over Q to be therational number

Lalg = L(E)alg :=#X(E/Q) · Tam(E)

(#(E(Q)tors))2.

We next fix a minimal Weierstrass model of E over Z and write ω for the correspondingNeron differential. We also fix a generator γ of H1(E(C),Z)+, write c∞ for the number ofconnected components of E(R) and define the real period of E by setting

Ω+ = Ω+ω := c∞

∣∣∣∣∫

γω

∣∣∣∣ .

(We note that this period is twice the period Ω+E defined in [19].) We finally write RegNT

for the classical Neron-Tate regulator of E.We then recall that the Birch and Swinnerton-Dyer Conjecture for E over Q predicts

that if one defines the ‘analytic Birch and Swinnerton-Dyer constant’ to be the real number

Lan = L(E)an :=L(r)(E, 1)

r! · Ω+ ·RegNT,

where we set r := rank(E(Q)) and L(r)(E, s) denotes the r-th derivative of L(E, s), thenthere should be an equality

(1) Lan = Lalg.

Remark 2.1. The value at s = 1 of the Euler factor of E at a rational prime ℓ is equalto #Ens(Fℓ)/ℓ. Thus, if for any finite set of rational primes Σ one defines a real numberLanΣ = LΣ(E)an just as with Lan except that L(E, s) is replaced by the Σ-truncated Hasse-Weil L-function LΣ(E, s), then the conjectural equality (1) is valid if and only if LanΣ isequal to the Σ-truncated algebraic Birch and Swinnerton-Dyer constant that is defined bysetting

(2) LalgΣ := Lalg ·∏

ℓ∈Σ

#Ens(Fℓ)

ℓ=

#X(E/Q) · Tam(E)

(#(E(Q)tors))2·∏

ℓ∈Σ

#Ens(Fℓ)

ℓ.

2.2. Modular elements and the Mazur-Tate Conjecture. We fix a finite real abelianextension F of Q of conductor m, so that m is the smallest natural number for which F iscontained in the maximal real subfield Q(ζm)

+ of Q(ζm). We consider the Galois groups

Gm := Gal(Q(ζm)/Q), G+m := Gal(Q(ζm)

+/Q) and G := Gal(F/Q).

We fix a prime p and consider the following assumptions on the data E,F and p.

Hypothesis 2.2.

(i) F/Q is tamely ramified, i.e., m is square-free.(ii) E has good reduction at every prime divisor of m, i.e., (m,N) = 1.(iii) p does not divide 6mN ·#(E(Q)tors) · Tam(E) ·

∏ℓ|pmN #Ens(Fℓ).

(iv) The action of GQ on E[p] is surjective.

We define a finite set of primes

S := ℓ | F/Q is ramified at ℓ or E has bad reduction at ℓ = ℓ | mN

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and note that Hypothesis 2.2(iii) implies that S does not contain p.We write

θF,S = θ(E)F,S ∈ Q[G]

for the ‘S-truncated modular element’ that is characterized by the interpolation property

χ(θF,S) = τm(χ)LS(E,χ

−1, 1)

Ω+for every χ ∈ G,(3)

where τm(χ) is the Gauss sum

τm(χ) :=∑

σ∈Gm

χ(σ)ζσm.

Here we regard χ as a character of Gm via the natural surjection Gm ։ G. Note thatthis S-truncated modular element is slightly different from the original element defined in[19], but the comparison is not difficult. One can construct θF,S directly from modularsymbols, namely from the integrals of the corrsponding modular form, but we will latergive a construction from Kato’s Euler system (see Proposition 5.1 below). Since, underHypothesis 2.2, p does not divide #(E(Q)tors) one knows that

θF,S ∈ Z(p)[G]

and, in the sequel, we use this containment to regard θF,S as an element of Zp[G].We write ǫG for the Zp-linear ‘augmentation’ map Zp[G] → Zp that sends each element

of G to 1 and write

IF := ker(ǫG)

for the associated augmentation ideal of Zp[G]. We write I for IF when there is no confusion.We recall, in particular, that there exists a canonical isomorphism of abelian groups

G⊗ Zp ≃ I/I2; σ 7→ σ − 1.(4)

Let

〈−,−〉m : E(Q)×ES(Q)→ G+m

be the Mazur-Tate pairing constructed in [19, Chap. II], where

ES(Q) := ker

E(Q)→

⊕

ℓ|m

E(Fℓ)⊕⊕

ℓ|N

E(Qℓ)/E0(Qℓ)

.

Composing this pairing with the natural surjection

G+m ։ G⊗ Zp

(4)≃ I/I2,

we obtain the pairing

〈−,−〉F : E(Q)× ES(Q)→ I/I2.

Hypothesis 2.2(iii) ensures that Zp ⊗Z ES(Q) = Zp ⊗Z E(Q), so we obtain

〈−,−〉F : (Zp ⊗Z E(Q))× (Zp ⊗Z E(Q))→ I/I2.(5)

In §5.3 below we will describe an alternative construction of this pairing in terms of anatural Bockstein homomorphism on Galois cohomology.


We set r := rank(E(Q)) and define the Mazur-Tate regulator

RF = R(E)F ∈ Ir/Ir+1

to be the discriminant of the pairing 〈−,−〉F , i.e.,

(6) RF := det(〈xi, xj〉F )1≤i,j≤r

with x1, . . . , xr a basis of E(Q)tf .Finally, let ν(m) denote the number of prime divisors of m.We can now recall (the p-component of) the Mazur-Tate Conjecture. We note that this

statement uses the algebraic Birch and Swinnerton-Dyer constant LalgS defined in (2).

Conjecture 2.3 (Mazur-Tate Conjecture). We have

θF,S ∈ IrF

andθF,S = (−1)ν(m)LalgS · RF in IrF/I

r+1F .

Remark 2.4. Hypothesis 2.2(iii) implies that p does not divide m · N · #(E(Q)tors) and

hence that the rational number LalgS belongs to Z(p). This shows that the right hand sideof the ‘leading term formula’ in Conjecture 2.3 is well-defined.

Remark 2.5. When r = 0, Conjecture 2.3 is equivalent to the classical leading term formulapredicted by Birch and Swinnerton-Dyer. To see this we note that the interpolation formula(3) combines with the elementary equality τm(1) = (−1)ν(m) to imply the image of θF,S in

Zp[G]/I ≃ Zp is equal to (−1)ν(m)LS(E, 1)/Ω+ and hence that Conjecture 2.3 is equivalent

in this case to an equality LS(E, 1)/Ω+ = LalgS . This equality coincides precisely with the

relevant case of the Birch-Swinnerton-Dyer formula, as recalled in Remark 2.1.

Remark 2.6. It suffices to verify Conjecture 2.3 after replacing F by its maximal subex-tension F (p) of p-power degree over Q. This is true since it is enough (following Remark2.5) to verify Conjecture 2.3 in the case r > 0 and, in this case, the natural projection map

Zp[G] → Zp[Gal(F (p)/Q)] sends θF,S to θF (p),S and also induces an isomorphism of finite

groups IrF/Ir+1F ≃ Ir

F (p)/Ir+1F (p) .

Remark 2.7. Under the given hypotheses on p, Conjecture 2.3 is equivalent to the ‘p-primary component’ of the original Mazur-Tate conjecture [19, Conj. 4]. To check this, itis enough to show Conjecture 2.3 is equivalent to Darmon’s reformulation [14, Conj. 2.4]of the p-component of the Mazur-Tate Conjecture and, following Remark 2.6, we may, andwill, assume that #G is a power of p. To proceed we write

θF,m ∈ Q[G]

for c(ϕ)−1 times the projection of the element θMTm of Q[G+

m], where θMTm is defined in [14,

§2.3.1] by using a modular parameterization ϕ : X0(N)→ E with Manin constant c(ϕ). Werecall (from [14, Prop. 2.3]) that the modular element θF,m is characterized by the explicitinterpolation formula

χ(θF,m) = τm(χ)m

fχ

Lm(E,χ−1, 1)

Ω+for every χ ∈ G,(7)

9

where fχ denotes the conductor of χ and Lm(E,χ, s) the m-truncated L-function.Then, for this modular element, [14, Conj. 2.4] predicts the following equality

θF,m = (−1)ν(m)#X(E/Q) · JS ·RS in Ir/Ir+1,(8)

where JS is the order of the finite group

coker

E(Q)→

⊕

ℓ|m

E(Fℓ)⊕⊕

ℓ|N

E(Qℓ)/E0(Qℓ)

and RS is the discriminant in Ir/Ir+1 of the pairing

〈−,−〉F : E(Q)× ES(Q)→ I/I2

(in the sense of [19, (2.5)]). In particular, these definitions imply directly that

JS · RS =

∏

ℓ|m

#E(Fℓ)

Tam(E)

(#(E(Q)tors))2·RF .(9)

On the other hand, Hypothesis 2.2(ii) and (iii) combine to imply that the productEulF/Q,N of the G-equivariant Euler factors at each prime divisor ℓ of N belongs to Z(p)[G].

It is also clear that the augmentation map ǫG sends EulF/Q,N to∏ℓ|N

(#Ens(Fℓ)/ℓ

)and

hence, since the latter element is a unit of Zp (as a consequence of Hypothesis 2.2(iii)) and#G is a power of p, that EulF/Q,N is a unit in Z(p)[G].

In addition, the interpolation formulas (3) and (7) together imply an equality

EulF/Q,N · θF,m =

∑

χ∈G

m

fχeχ

θF,S,(10)

We also note that ∑

χ∈G

m

fχeχ belongs to Z(p)[G]

×

(by [2, Prop. 3.1]) and that ǫG sends this element to m.Now suppose that (8) holds. Then in Ir/Ir+1 one computes that

θF,S(10)= m−1

∏

ℓ|N

#Ens(Fℓ)

ℓ

· θF,m

(8)= (−1)ν(m)m−1

∏

ℓ|N

#Ens(Fℓ)

ℓ

·#X(E/Q) · JS · RS

(9)= (−1)ν(m)LalgS ·RF .

This shows that the formula predicted in Conjecture 2.3 is implied by (8). In addition,since EulF/Q,N is a unit in Z(p)[G]

×, the same argument also shows that the equality (8) isimplied by that in Conjecture 2.3.


Remark 2.8. Assume (following Remark 2.5) that r > 0. Then, in this case, the Mazur-Tate Conjecture is valid if and only if its p-primary component is valid for every primep that does not divide #(E(Q)tors) (see the discussion in §4.1). Thus, taking account ofRemark 2.7, the study of Conjecture 2.3 will in principal allow one to verify the Mazur-TateConjecture modulo its components at the finitely many primes p that do not satisfy theconditions in Hypothesis 2.2(iii) and (iv). This means, in particular, that our methodsalways neglect the 2-primary and 3-primary components of the Mazur-Tate Conjecture(whenever they arise). A further restriction on our approach that appears difficult toremove is that, following Hypothesis 2.2(ii), we can only consider the situation in which therespective conductors of the curve E and field F are coprime.

3. Review of the Generalized Perrin-Riou Conjecture

In this section we review the construction of ‘Bockstein regulator’ elements from [7] andthen formulate a natural ‘algebraic’ analogue of (a special case of) the Generalized Perrin-Riou Conjecture studied in loc. cit.

3.1. The Bockstein regulator. We keep the notations in the previous subsection. Wedenote the p-adic Tate module of E by T and set V := Qp ⊗Zp T . We set

Σ := S ∪ p = ℓ | pmN.

We assume r := rank(E(Q)) > 0 in the rest of this section. In this case, the naturallocalization map E(Q)⊗ZQp → E1(Qp)⊗Zp Qp is surjective so that H1(ZΣ, V ) = H1

f (Q, V )and hence the Kummer map induces an isomorphism

(11) E(Q)⊗Z Zp ≃ H1(ZΣ, T ).

(See [7, (2.2.1)].)Write D(Zp) for the derived category of Zp-modules and let F/Q be a finite abelian

extension with Galois group G. (For the moment, we do not need to assume Hypothesis2.2.) Then the tautological exact sequence

0→ IF /I2F → Zp[G]/I

2F → Zp → 0

combines with the canonical projection isomorphism RΓ(OF,Σ, T )⊗L

Zp[G] Zp ≃ RΓ(ZΣ, T ) in

D(Zp) to give a canonical exact triangle in D(Zp)

RΓ(ZΣ, T )⊗L

ZpIF/I

2F → RΓ(OF,Σ, T )⊗

L

Zp[G] Zp[G]/I2F → RΓ(ZΣ, T )→ .

We consider the composite homomorphism

(12) β = βF : E(Q)⊗Z Zp(11)−−−→ H1(ZΣ, T )→ H2(ZΣ, T )⊗Zp IF /I

2F

where the last arrow denotes (−1)-times the connecting homomorphism that is induced bythe above exact triangle (cf. [7, §2.3]) and then define a homomorphism

BocF :(∧r

ZE(Q)

)⊗Z Zp → E(Q)⊗Z

∧r−1

Zp

H2(ZΣ, T )⊗Zp Ir−1F /IrF

11

by

x1 ∧ · · · ∧ xr 7→r∑

i=1

(−1)i+1xi ⊗ β(x1) ∧ · · · ∧ β(xi−1) ∧ β(xi+1) ∧ · · · ∧ β(xr).

Since p is fixed we write Q∞ for the cyclotomic Zp-extension of Q and Qn for eachnatural number n for the unique subfield of Q∞ of degree pn over Q. We note that theaugmentation ideal I∞ of Zp[[Gal(Q∞/Q)]] identifies with the inverse limit over the idealsIQn (with respect to the natural projection maps) and so by applying the above constructionwith F taken to be each field Qn and then passing to the inverse limit over n we obtain acanonical homomorphism of Zp-modules

BocQ∞:(∧r

ZE(Q)

)⊗Z Zp → E(Q)⊗Z

∧r−1

Zp

H2(ZΣ, T )⊗Zp Ir−1∞ /Ir∞.

Next we note that the natural exact sequence

0→ H2(ZΣ, V )∗ → H1(ZΣ, V )→ E1(Qp)⊗Zp Qp → 0

(cf. [7, (2.2.2)]) gives rise to a canonical composite isomorphism of Qp-spaces

εΣ :(∧r

ZE(Q)

)⊗Z Qp ≃

(∧r

Qp

H1(ZΣ, V ))⊗Qp

(E1(Qp)⊗Zp Qp

)∗(13)

≃∧r−1

Qp

H2(ZΣ, V )∗

≃(∧r−1

Zp

H2(ZΣ, T ))∗⊗Zp Qp

in which the first isomorphism is induced by (11) and the isomorphism E1(Qp)⊗Zp Qp ≃ Qp

induced by the logarithm logω associated to the fixed Neron differential ω, and the lastisomorphism is the obvious identification.

Any choice of basis element x of the Z-module∧r

ZE(Q)tf therefore gives rise to anisomorphism of Qp-spaces

εΣ(x) :(∧r−1

Zp

H2(ZΣ, T ))⊗Zp Qp ≃ Qp

and hence to a ‘Bockstein regulator’ element

RBoc :=((1⊗ εΣ(x)⊗ 1) (BocQ∞

⊗Zp Qp))(x⊗ 1) ∈ E(Q)⊗Z I

r−1∞ /Ir∞ ⊗Zp Qp

that is easily seen to be independent of the choice of x.

3.2. Kato’s zeta elements. In the following, we fix a finite abelian p-extension F of Qthat satisfies Hypothesis 2.2 (cf. Remark 2.6).

We recall that if Kato’s zeta element (Euler system)

zF = zF,Σ ∈ H1(OF,Σ, V )

is normalized as in [9, Def. 6.8], then for every χ in G one has

∑

σ∈G

χ(σ) exp∗ω(σzF ) =LΣ(E,χ, 1)

Ω+,(14)


where exp∗ω : H1(OF,Σ, V ) → H1/f (Fp, V ) → Fp denotes the dual exponential map that is

associated to the fixed Neron differential ω (see [17, Th. 6.6 and 9.7]).Now Hypothesis 2.2(iii) implies that H1(OF,Σ, T ) is torsion-free and hence identifies with

a sublattice of H1(OF,Σ, V ). Further, since E has good reduction at p (by Hypothesis2.2(iii)) and the GQ-representation E[p] is irreducible (by Hypothesis 2.2(iv)) one knowsthat zF belongs to the lattice H1(OF,Σ, T ) (cf. [9, Rem. 6.9 and §6.4]).

3.3. The Generalized Perrin-Riou Conjecture. In our earlier article [7, §4.2] we con-structed a ‘wild Kolyvagin derivative’ element

κ ∈ H1(ZΣ, T )⊗Zp Ir−1∞ /Ir∞ ≃ E(Q)⊗Z I

r−1∞ /Ir∞,

with the property that for each natural number n one has

(15) ιQn(n(κ)) =∑

σ∈Γn

σ(zQn)⊗ σ−1

Here Γn denotes Gal(Qn/Q),

n : H1(ZΣ, T )⊗Zp Ir−1∞ /Ir∞ → H1(ZΣ, T )⊗Zp I

r−1Qn

/IrQn

is the natural projection map,

ιQn : H1(ZΣ, T )⊗Zp Ir−1Qn

/IrQn→ H1(OQn,Σ, T )⊗Zp Zp[Γn]/I

rQn

is the homomorphism induced by the restriction map H1(ZΣ, T ) → H1(OQn,Σ, T ) and

inclusion Ir−1Qn

/IrQn→ Zp[Γn]/I

rQn

and σ denotes the image of σ in Zp[Γn]/IrQn

.

The central conjecture of [7] predicts an explicit formula for κ. To state this conjecturewe regard the Σ-truncated analytic Birch and Swinnerton-Dyer constant LanΣ = LΣ(E)an

defined in Remark 2.1 as an element of Cp by means of a fixed embedding R → Cp.

Conjecture 3.1 (Generalized Perrin-Riou Conjecture for E and Q∞/Q). One has

κ = LanΣ · RBoc and RBoc 6= 0

in E(Q)⊗Z

(Ir−1∞ /Ir∞ ⊗Zp Cp

).

Remark 3.2. If r = 1, then BocQ∞is the identity automorphism of E(Q) ⊗Z Zp and one

can check that RBoc = logω(x) · x for any choice of a generator x of E(Q)tf . This factimplies directly that, if r = 1, then RBoc 6= 0 and can also be used to show that, in thiscase, Conjecture 3.1 coincides with the conjecture formulated by Perrin-Riou in [22] (fordetails of this deduction see [7, Prop. 4.15, Rem. 4.13 and Rem. 4.10]).

Remark 3.3. The Generalized Perrin-Riou Conjecture [7, Conj. 2.12] does not predictnon-vanishing of the Bockstein regulator for E relative to every extension F/Q. However,the Bockstein regulator for E relative to Q∞/Q can be explicitly described in terms ofclassical p-adic regulators that have been conjectured not to vanish (see [7, Th. 5.6 and5.11]) and this accounts for the prediction RBoc 6= 0 in Conjecture 3.1.

To state an ‘algebraic’ analogue of Conjecture 3.1 we use the Σ-truncated algebraic Birch

and Swinnerton-Dyer constant LalgΣ defined in (2).

13

Conjecture 3.4 (Algebraic Generalized Perrin-Riou Conjecture for E and Q∞/Q). Onehas

κ = LalgΣ ·RBoc and RBoc 6= 0

in E(Q)⊗Z

(Ir−1∞ /Ir∞ ⊗Zp Qp

).

The next result follows directly from the explicit interpretation of the Birch and Swinnerton-Dyer Conjecture described in Remark 2.1.

Lemma 3.5. If E validates the Birch and Swinnerton-Dyer Conjecture over Q, then Con-jecture 3.1 is equivalent to Conjecture 3.4.

4. The main result

4.1. Statement of the main result and the deduction of Theorem 1.1. The mainresult that we shall prove in the remainder of this article is the following.

Theorem 4.1. Assume that the data E,F and p satisfies Hypothesis 2.2. Assume also thatr := rank(E(Q)) > 0 and that the Algebraic Generalized Perrin-Riou Conjecture (Conjec-ture 3.4) is valid. Then the Mazur-Tate Conjecture (Conjecture 2.3) is valid.

In the next section we shall describe the basic strategy that will be used to prove thisresult. However, before doing so, we first explain how it implies Theorem 1.1.

At the outset we note that, since Theorem 1.1 assumes E validates the Birch andSwinnerton-Dyer Conjecture over Q, Remark 2.5 allows us to assume r > 0 and Lemma 3.5implies that for every prime p the Generalized Perrin-Riou Conjecture for E relative to thecyclotomic Zp-extension of Q is equivalent to Conjecture 3.4. We further note that, underthese hypotheses, LS(E, 1) vanishes and so the interpolation property (3) implies that themodular element θF,S belongs to I(G)⊗Z Q.

To proceed we write Z′ for the subring of Q generated by the inverse of

D := 6 ·#(E(Q)tors).

Then it is known that θF,S belongs to Z′[G] and hence to I(G) ⊗Z Z′ and the centralconjecture formulated by Mazur and Tate in [19] further predicts that θF,S belongs toI(G)r ⊗Z Z′ and that its image in the quotient module

(I(G)r/I(G)r+1

)⊗Z Z′ is equal to

a precise multiple of the Mazur-Tate regulator.Now θF,S belongs to the lattice I(G)r ⊗Z Z′ if and only if it belongs to I(G)r ⊗Z Zℓ for

every prime ℓ that does not divide D. Further, since r > 0, the module I(G)r/I(G)r+1 isisomorphic to a quotient of the finite group Symr(G) and so decomposes as a direct sum

I(G)r

I(G)r+1≃⊕

ℓ|#G

(I(G) ⊗Z Zℓ)r

(I(G) ⊗Z Zℓ)r+1

where ℓ runs over all prime divisors of #G. In particular, if ℓ does not divide #G, thenI(G)r ⊗Z Zℓ = I(G)⊗Z Zℓ and so the containment θF,S ∈ I(G)

r ⊗Z Zℓ follows directly fromthe observations made above.

These facts combine to imply that it is enough to verify the Mazur-Tate Conjecture afterreplacing Z′ by Zp for each prime divisor p of #G that does not divide D.


In addition, for any such prime p, the hypotheses (a), (b) and (c) in Theorem 1.1 togetherimply that the data E,F and p verify all of the conditions in Hypothesis 2.2. Thus, if theGeneralized Perrin-Riou Conjecture for E relative to the cyclotomic Zp-extension of Q isvalid, then the validity of the Mazur-Tate Conjecture after replacing Z′ by Zp follows directlyfrom Theorem 4.1 (and Lemma 3.5).

This completes the proof of Theorem 1.1.

4.2. Determinantal zeta elements. In this section, we introduce (in Definition 4.3) anatural notion of ‘determinantal zeta element’ and discuss the key role that such elementswill play in the proof of Theorem 4.1. In particular, in this way we shall establish animportant reduction step in the proof of the latter result.

We remark at the outset that determinantal zeta elements have implicitly played a keyrole in several of our earlier articles. For example, the ‘determinantal zeta element for Gm’is the central object of study in [4] (where it is simply referred to as the ‘zeta elementfor Gm’) and is also used to formulate and study a natural main conjecture of higher rankIwasawa theory in [5]. In addition, in the respective articles [6] and [7] we studied analogouselements in the determinant of the Galois cohomology of Tate twists Zp(j) (for arbitraryintegers j) and of the p-adic Tate modules of elliptic curves and, more recently, Kataoka[16] has made a systematic study of determinantal zeta elements in the setting of generalGalois representations.

To be more precise, we henceforth write F∞ for the cyclotomic Zp-extension of F . Foreach subfield K of F∞ we also write ΛK for the algebra Zp[[Gal(K/Q)]] and regard thetensor product

TK := T ⊗Zp ΛK

as a module over Zp[[GQ]]⊗Zp ΛK in the natural way.Then, as a first step in the proof of Theorem 4.1, we shall use the equivariant theory of

Kolyvagin and Stark systems in the setting of Kato’s zeta elements in order to construct foreach such field K a ‘determinantal zeta element’ zK in the linear dual of the ΛK-determinantof the Galois cohomology of TK over ZΣ (for details see Definition 4.3).

Next we prove (in Proposition 4.4) that the validity of Conjecture 3.4 implies zQ coin-cides with an ‘algebraic Birch and Swinnerton-Dyer element’ ηalg that is defined using theapproach of [7].

Using this last observation, the proof of Theorem 4.1 is reduced to showing that thevalidity of Conjecture 2.3 is implied by the equality zQ = ηalg (see Theorem 4.6) and thisdeduction will then be proved in §5.

To proceed we write Ω(K) for the set of subfields of K that are of finite degree overQ, regarded as partially ordered by inclusion, and we use the compatibility under norm ofKato’s zeta elements to define an element

zK = (zM,Σ)M∈Ω(K) ∈ H1(ZΣ, TK) = lim

←−M∈Ω(K)

H1(OM,Σ, T ).

In the sequel we also write detΛK(C) for the ΛK-equivariant determinant (in the sense

of Knudsen and Mumford) of any perfect complex of ΛK-modules C and we set

det−1ΛK

(C) := HomΛK(detΛK

(C),ΛK).

15

Then it is well-known that the definition of Σ ensures the Galois cohomology complexRΓ(ZΣ, TK) is a perfect complex of ΛK-modules. In Proposition 5.2 below we will define acanonical homomorphism of ΛK-modules

πK : det−1ΛK

(RΓ(ZΣ, TK))→ H1(ZΣ, TK),

and by taking the limit of these homomorphisms over all K in Ω(F∞) we thereby obtain acanonical homomorphism of ΛF∞

-modules

πF∞: det−1

ΛF∞

(RΓ(ZΣ, TF∞))→ H1(ZΣ, TF∞

).

The following observation regarding the link between the element zF∞and map πF∞

relies on the equivariant theory of Euler, Kolyvagin and Stark systems and will play a keyrole in our approach.

Lemma 4.2 (Kataoka). If Hypothesis 2.2 is satisfied, then zF∞belongs to the image of

πF∞.

Proof. This has recently been proved in [16] and, for the reader’s convenience, we sketchthe argument.

In [10, Th. 3.12(ii)], the first and the third authors showed that, for each pair of naturalnumbers n and m, there exists a natural isomorphism of Z/pm[Gal(Fn/Q)]-modules

πn,m : det−1Z/pm[Gal(Fn/Q)](RΓ(ZΣ, TFn/p

m))∼−→ SS1(TFn/p

m),

where SS1(−) denotes the module of Stark systems of rank one (see also Theorem 5.6 in[16]).

In addition, in [8, Th. 5.2], Sakamoto and the first and the third authors showed thatthere is a natural isomorphism of Z/pm[Gal(Fn/Q)]-modules

Reg : SS1(TFn/pm)

∼−→ KS1(TFn/p

m),

where KS1(−) denotes the module of Kolyvagin systems of rank one.Now, it is well-known that one obtains a Kolyvagin system from an Euler system, and

that Kato’s Euler system yields a Kolyvagin system xn,m for which one has

(xn,m)1 = zFn ∈ H1(OFn,Σ, T/p

m) = H1(ZΣ, TFn/pm).

The required claim is therefore true since the homomorphism πF∞coincides, by its very

construction, with the inverse limit (over n and m) of the composite maps

det−1Z/pm[Gal(Fn/Q)](RΓ(ZΣ, TFn/p

m))πn,m−−−→ SS1(TFn/p

m)

Reg−−→ KS1(TFn/p

m)κ 7→κ1−−−→ H1(ZΣ, TFn/p

m).

It is clear that the ΛF∞-module det−1

ΛF∞

(RΓ(ZΣ, TF∞)) is free of rank one. Thus, since

the annihilator of zF∞in ΛF∞

vanishes (as a consequence of the argument in [16, §6.1]),the result of Lemma 4.2 implies that πF∞

is injective, and hence that there exists a uniqueelement zF∞

of det−1ΛF∞

(RΓ(ZΣ, TF∞)) such that

πF∞(zF∞

) = zF∞.


This observation motivates us to make the following definition.

Definition 4.3. For any subfield K of F∞, the ‘(p-adic) determinantal zeta element of K’is the image zK of zF∞

under the canonical projection map

det−1ΛF∞

(RΓ(ZΣ, TF∞)) ։ det−1

ΛK(RΓ(ZΣ, TK)).

Then the main observation we wish to make in this section is that Conjecture 3.4 impliesan explicit formula for the determinantal zeta element of Q.

To state this result we use the canonical ‘passage to cohomology’ isomorphism

det−1Zp

(RΓ(ZΣ, T )) ≃ #H2(ZΣ, T )tors ·∧r

Zp

H1(ZΣ, T )⊗Zp

∧r−1

Zp

H2(ZΣ, T )∗tf .(16)

We also recall (from [7, (2.5.2)]) that, if we fix a basis element x of∧r

ZE(Q)tf and usethe isomorphisms (11) and (13), then the ‘algebraic Birch and Swinnerton-Dyer element’

(17) ηalg := LalgΣ · (x⊗ εΣ(x))

belongs to the codomain of the isomorphism (16) and is independent of the choice of x.In the sequel we can (and will) therefore use (16) to regard ηalg as an element of the

lattice det−1Zp

(RΓ(ZΣ, T )).

Proposition 4.4. If Conjecture 3.4 is valid, then zQ = ηalg.

Proof. The key point in this argument is that for any real abelian extension F/Q, withG = Gal(F/Q), the map πF lies in a commutative diagram of the form

det−1Zp[G](RΓ(OF,Σ, T ))

πF //

νF

H1(OF,Σ, T )NF/Q // H1(OF,Σ, T )⊗Zp Zp[G]/I

rF

det−1Zp

(RΓ(ZΣ, T ))(16)

// ∧r

ZpH1(ZΣ, T )⊗Zp

∧r−1Zp

H2(ZΣ, T )∗tf Boc′F

// H1(ZΣ, T )⊗Zp Ir−1F /IrF .

ιF

OO

Here νF is the natural projection map, NF/Q is the ‘Darmon norm’ that sends each ele-

ment x to the class of∑

σ∈G σ(x) ⊗ σ−1, ιF is the map induced by the restriction map

H1(ZΣ, T ) → H1(OF,Σ, T ) and inclusion Ir−1/Ir → Zp[G]/Ir and Boc′F denotes the com-

posite homomorphism∧r

Zp

H1(ZΣ, T )⊗Zp

∧r−1

Zp

H2(ZΣ, T )∗tf

BocF⊗1−−−−−→

(H1(ZΣ, T )⊗Zp

∧r−1

Zp

H2(ZΣ, T )⊗Zp Ir−1F /IrF

)⊗Zp

∧r−1

Zp

H2(ZΣ, T )∗tf

→H1(ZΣ, T )⊗Zp Ir−1F /IrF

where the last map is induced by the natural isomorphism(∧r−1

Zp

H2(ZΣ, T )tf)⊗Zp

(∧r−1

Zp

H2(ZΣ, T )∗tf

)≃ Zp.

In addition, the commutativity of the above diagram follows from the same explicit com-putation that verifies commutativity of the diagram (7.4.1) in [7].

17

We also note that the map ιF is injective. Indeed, this follows easily from the facts thatH1(ZΣ, T ) is Zp-free and that H1(ZΣ, T ) identifies with the submodule H1(OF,Σ, T )

G ofG-invariant elements in H1(OF,Σ, T ) (since H

0(ZΣ, T ) vanishes).

In addition, a comparison of the definitions of ηalg and RBoc combines with a directcalculation to show that the homomorphism

Boc′∞ :∧r

Zp

H1(ZΣ, T )⊗Zp

∧r−1

Zp

H2(ZΣ, T )∗tf → H1(ZΣ, T )⊗Zp I

r−1∞ /Ir∞

given by the limit of the maps Boc′Qnfor n ≥ 1 sends ηalg to LalgΣ · R

Boc.In particular, if Conjecture 3.4 is valid, then the commutativity of the above diagram

(for each field F = Qn) combines with the property (15) of the element κ to imply that

Boc′∞(zQ) = LalgΣ · R

Boc = Boc′∞(ηalg).

Thus, since Boc′∞ is injective (as Conjecture 3.4 predicts RBoc 6= 0), one has zQ = ηalg, asrequired.

Remark 4.5. The equality zQ = ηalg that occurs in Proposition 4.4 implies the ‘refinedMazur-Tate conjecture’ formulated in [7, Conj. 2.19] for any abelian p-extension F/Q ofthe form considered in loc. cit. This is because if one applies the commutative diagramin the proof of Proposition 4.4 to F and the element zF , then one obtains an equalityNF/Q(zF ) = ιF (Boc

′F (η

alg)), which can be shown to agree precisely with the formulation of[7, Conj. 2.19].

In view of Proposition 4.4, it is clear that Theorem 4.1 is a direct consequence of thefollowing result (that will be proved in the next section).

Theorem 4.6. If zQ = ηalg, then Conjecture 2.3 is valid.

Remark 4.7. If r = 1, then one can use the isomorphism (16) to regard ηalg as an elementof H1(ZΣ, T ) and hence consider the possibility that

zQ?= ηalg,(18)

where zQ = zQ,Σ is Kato’s zeta element in H1(ZΣ, T ). This displayed equality is a natural‘algebraic’ variant of Perrin-Riou’s conjecture (as discussed in [7, Conj. 2.8 and Prop.2.10]) and is easily seen to be equivalent to the equality zQ = ηalg assumed in Theorem 4.6.Hence, if r = 1, then Theorem 4.6 implies that the p-primary component of the Mazur-TateConjecture is directly implied by the algebraic Perrin-Riou Conjecture given by (18).

5. The proof of Theorem 4.6 and the deduction of Corollary 1.2

5.1. Construction of the modular element. We first review the construction of themodular element θF,S from Kato’s zeta element zF (see [18] by the second author, [21] byOtsuki and especially [20] by Ota).

As is shown in [20, Lem. 5.5], we have an isomorphism

E1(Fp)∼−→ OF ⊗Z Zp(19)


defined by

c 7→

(1−

appFrp +

1

pFr2p

)logω(c),

where ap := p + 1 − #E(Fp), Frp ∈ G is the arithmetic Frobenius at p, and logω is theformal logarithm E1(Fp) → pOF ⊗Z Zp associated to ω. (The assumptions in loc. cit. aresatisfied, since Hypothesis 2.2 ensures that p is unramified in F and p ∤ 6N#E(Fp).) Wedefine

cF ∈ E1(Fp)

to be the element corresponding to

TrQ(ζm)/F (ζm) ∈ OF

under the isomorphism (19). We regard cF as an element of H1f (Fp, T ) by means of the

Kummer map E1(Fp)→ H1f (Fp, T ).

We write

(−,−)F : H1f (Fp, T )×H

1/f (Fp, T )→ H2(Fp,Zp(1)) ≃

⊕

p|p

Zp(ap)p 7→

∑pap

−−−−−−−−→ Zp

for the pairing induced by the cup product (after identifying T with T ∗(1) by means of theWeil pairing). By composing it with the localization map H1(OF,Σ, T ) → H1

/f (Fp, T ), we

obtain a pairing

(−,−)F : H1f (Fp, T )×H

1(OF,Σ, T )→ Zp,

which we denote by the same symbol.

Proposition 5.1. In Zp[G] one has

θF,S =∑

σ∈G

(cF , σzF )Fσ−1.

Proof. By (3), it is sufficient to show

∑

σ∈G

(cF , σzF )Fχ−1(σ) = τm(χ)

LS(E,χ−1, 1)

Ω+

for every χ ∈ G. By the computation in [20, (5.10)], the left hand side is equal to(∑

σ∈G

logω(σcF )χ(σ)

)×

(∑

σ∈G

exp∗ω(σzF )χ−1(σ)

).

Since we have

∑

σ∈G

logω(σcF )χ(σ) =

(1−

appχ−1(Frp) +

1

pχ−2(Frp)

)−1

τm(χ)

by [20, Prop. 5.8], the desired equality follows from Kato’s formula (14).

19

5.2. Bloch-Kato Selmer complexes. In this section we shall use the local point cF ∈E1(Fp) that was constructed in §5.1 to construct an isomorphism of Zp[G]-modules of theform

ϕF : det−1Zp[G](RΓ(OF,Σ, T ))

∼−→ det−1

Zp[G](RΓf (F, T )).

To do this we note first that if ℓ ∈ Σ and ℓ 6= p (i.e., ℓ | mN), then Hypothesis 2.2(iii)implies E(Fℓ)[p] vanishes and hence that the complex RΓ/f (Fℓ, T ) is acyclic. In this case,therefore, there exists a canonical isomorphism of Zp[G]-modules.

det−1Zp[G](RΓ/f (Fℓ, T )) ≃ det−1

Zp[G](0) = Zp[G].(20)

Next we note that Hypothesis 2.2(iii) implies the group

H2(Fp, T ) ≃ E(Fp)[p∞]∨

vanishes, and hence that there is a natural identification

RΓ/f (Fp, T ) = H1/f (Fp, T )[−1] = E1(Fp)

∗[−1].(21)

In addition, since F/Q is tamely ramified, the Zp[G]-module E1(Fp) ≃ OF ⊗Z Zp isisomorphic to Zp[G] (by the Hilbert-Speiser theorem) and is generated by the element cF .The identification (21) therefore induces an isomorphism of Zp[G]-modules

det−1Zp[G](RΓ/f (Fp, T )) = det−1

Zp[G](E1(Fp)∗[−1]) = E1(Fp)

∗ ≃ Zp[G],(22)

in which the last map sends c∗F to 1.Now we note that there is a canonical exact triangle of Zp[G]-modules

RΓf (F, T )→ RΓ(OF,Σ, T )→⊕

ℓ∈Σ

RΓ/f (Fℓ, T ),(23)

and that the above argument shows that (each direct summand of) the last term in thistriangle is a perfect complex of Zp[G]-modules. Since RΓ(OF,Σ, T ) is also a perfect complexof Zp[G]-modules, this exact triangle therefore implies that RΓf (F, T ) is a perfect complexof Zp[G]-modules and also induces a canonical isomorphism of Zp[G]-modules

(24) det−1Zp[G](RΓ(OF,Σ, T )) ≃ det−1

Zp[G](RΓf (F, T )) ⊗Zp[G]

⊗

ℓ∈Σ

det−1Zp[G](RΓ/f (Fℓ, T )).

Then, upon combining this isomorphism with the maps (20) (for each ℓ ∈ Σ \ p) and(22) we obtain the desired isomorphism ϕF : det−1

Zp[G](RΓ(OF,Σ, T ))→ det−1Zp[G](RΓf (F, T )).

The key property of this map ϕF that we shall use in the sequel is established by thefollowing result.

Proposition 5.2. There exist canonical maps πF and πF,f that lie in a commutative dia-gram of Zp[G]-modules


πF //

ϕF

H1(OF,Σ, T )

∑σ∈G(cF ,σ(·))F σ

−1

det−1

Zp[G](RΓf (F, T )) πF,f

// Zp[G].


Proof. One knows that the complex RΓ(OF,Σ, T ) is represented by

PFψ−→ QF ,

where PF and QF are free Zp[G]-modules of rank d and d− 1 respectively for some d > r,and PF is placed in degree one (see the proof of [7, Th. 4.3]). Since RΓ/f (Fℓ, T ) is acyclicif ℓ ∈ S = Σ \ p, we see by the triangle (23) that

RΓf (F, T ) ≃ Cone

(RΓ(OF,Σ, T )

locp−−→ RΓ/f (Fp, T )

)[−1].

By the definition of mapping cones and (21), we see that

RΓf (F, T ) =

[PF

ψf−→ E1(Fp)

∗ ⊕QF

],

where PF is placed in degree one and ψf is given by

ψf (a) := (locp(a),−ψ(a)).

Fix a basis c2, . . . , cd of QF . For each i with 2 ≤ i ≤ d, we set

ψi := c∗i ψ : PF → Zp[G].

Similarly, for each i with 1 ≤ i ≤ d, we set

ψf,i := c∗i ψf : PF → Zp[G],

where c1 := c∗F ∈ E1(Fp)∗.

We define

πF : det−1Zp[G](RΓ(OF,Σ, T )) =

∧d

Zp[G]PF ⊗Zp[G]

∧d−1

Zp[G]Q∗F → PF

by

πF (a⊗ c∗2 ∧ · · · ∧ c

∗d) :=

(∧2≤i≤d

ψi

)(a).

One checks that this is well-defined and the image is contained in H1(OF,Σ, T ).Similarly, we define

πF,f : det−1Zp[G](RΓf (F, T )) =

∧d

Zp[G]PF ⊗Zp[G]

∧d

Zp[G](E1(Fp)

∗ ⊕QF )∗ → Zp[G]

by

πF,f (a⊗ c∗1 ∧ · · · ∧ c

∗d) :=

(∧1≤i≤d

ψf,i

)(a).

We now prove the commutativity of the diagram in the claim. One checks by definitionthat ϕF is explicitly given by

ϕF (a⊗ c∗2 ∧ · · · ∧ c

∗d) = a⊗ c∗1 ∧ · · · ∧ c

∗d.

Noting that ψf,i = −ψi for 2 ≤ i ≤ d, we have(∧

1≤i≤dψf,i

)(a) = (−1)d−1ψf,1

(∧2≤i≤d

ψf,i

)(a) = ψf,1

(∧2≤i≤d

ψi

)(a).

Hence we haveπF,f ϕF = ψf,1 πF .

21

So it is sufficient to prove

ψf,1(a) =∑

σ∈G

(cF , σ(a))Fσ−1

for any a ∈ H1(OF,Σ, T ). By the definition of ψf,1, we have

ψf,1(a) = c∗1(locp(a)).

Here locp(a) is an element of H1/f (Fp, T ), which we identify with E1(Fp)

∗ via the pairing

(−,−)F , and c∗1 is by definition the map E1(Fp)

∗ → Zp[G] sending c∗F to 1. It is now easy

to check

c∗1(locp(a)) =∑

σ∈G

(cF , σ(a))Fσ−1,

which completes the proof.

One can define maps πFn and πF∞similarly for the n-th layer Fn and the cyclotomic

Zp-extension F∞. In particular, since the following diagram (in which both vertical arrowsare the natural projection maps)

det−1ΛF∞

(RΓ(ZΣ, TF∞))

πF∞ //

H1(ZΣ, TF∞)

det−1

Zp[G](RΓ(ZΣ, TF )) = det−1Zp[G](RΓ(OF,Σ, T )) πF

// H1(OF,Σ, T ),

is commutative, one has

πF (zF ) = zF .

We can therefore deduce the following consequence of Propositions 5.1 and 5.2.

Corollary 5.3. In Zp[G] one has

θF,S = πF,f (ϕF (zF )).

5.3. Bockstein maps for Selmer complexes. As we have seen before, Hypothesis 2.2ensures that the Bloch-Kato Selmer complex RΓf (F, T ) is a perfect complex of Zp[G]-modules which is acyclic outside degrees one and two and satisfies the base change property

RΓf (F, T )⊗L

Zp[G] Zp ≃ RΓf (Q, T ).

Using this, we obtain an exact triangle

RΓf (Q, T )⊗L

ZpI/I2 → RΓf (F, T )⊗

L

Zp[G] Zp[G]/I2 → RΓf (Q, T ),

and hence an induced connecting homomorphism in its long exact cohomology sequence

H1f (Q, T )→ H2(RΓf (Q, T )⊗Zp I/I

2) = H2f (Q, T )⊗Zp I/I

2,

where the displayed equality follows directly from the fact that RΓf (Q, T ) is acyclic indegrees greater than two.

It is convenient to use (−1)-times the above ‘Bockstein map’, which we denote by

βf : H1f (Q, T )→ H2

f (Q, T )⊗Zp I/I2.


Upon combining this map with the canonical isomorphisms

H1f (Q, T ) ≃ Zp ⊗Z E(Q) and H2

f (Q, T ) ≃ Selp(E/Q)∨,

where Selp(E/Q) denotes the classical p-Selmer group, we obtain a composite map (whichwe denote by the same symbol)

βf : Zp ⊗Z E(Q)→ Selp(E/Q)∨ ⊗Zp I/I2։ (Zp ⊗Z E(Q))∗ ⊗Zp I/I

2,

where the second map is the natural surjection. We recall that the associated pairing

〈−,−〉F : (Zp ⊗Z E(Q))× (Zp ⊗Z E(Q))→ I/I2; (x, y) 7→ βf (y)(x)

is known to coincide with the Mazur-Tate pairing (5) (this is proved by Macias Castillo andthe first author in [3, Th. 10.3]).

Let

RegF : Zp ⊗Z

(∧r

ZE(Q)⊗Z

∧r

ZE(Q)

)→ Ir/Ir+1

be the map defined by

RegF (x1 ∧ · · · ∧ xr ⊗ y1 ∧ · · · ∧ yr) := det(〈xi, yj〉F )1≤i,j≤r.

Proposition 5.4. Let πF,f : det−1Zp[G](RΓf (F, T ))→ Zp[G] be the map constructed in Propo-

sition 5.2. Then the following claims are valid.

(i) The image of πF,f is contained in Ir.(ii) The following diagram is commutative:

det−1Zp[G](RΓf (F, T ))

νf

πF,f // Ir

)) ))

Ir/Ir+1

det−1Zp

(RΓf (Q, T )) πQ,f

// Zp ⊗Z (∧r

ZE(Q)⊗Z∧r

ZE(Q)) ,

RegF

55

where both νf and the unlabelled arrow denote the natural projection maps and πQ,fis induced by the canonical isomorphism

det−1Zp

(RΓf (Q, T )) ≃ detZp(H1f (Q, T ))⊗Zp det

−1Zp

(H2f (Q, T )).

Proof. This follows directly from the argument of [11, Th. 3.3.7] after fixing the data(R,C, J, a, a′) in loc. cit. to be (Zp,RΓf (F, T ), G, 0, r).

5.4. The algebraic Birch and Swinnerton-Dyer element. Let

ϕQ : det−1Zp

(RΓ(ZΣ, T ))∼−→ det−1

Zp(RΓf (Q, T ))

23

be the isomorphism induced by ϕF , which fits into the commutative diagram


ϕF //

ν

det−1Zp[G](RΓf (F, T ))

νf

det−1Zp

(RΓ(ZΣ, T )) ϕQ

// det−1Zp

(RΓf (Q, T )).

(25)

In the following result we use the element ηalg of det−1Zp

(RΓ(ZΣ, T )) defined in (17), the

S-truncated algebraic Birch and Swinnerton-Dyer constant LalgS ∈ Q× defined in (2) andthe Mazur-Tate regulator RF ∈ I

r/Ir+1 defined in (6).

Lemma 5.5. In Ir/Ir+1 one has

(RegF πQ,f ϕQ)(ηalg) = (−1)ν(m)LalgS ·RF .

Proof. The inverse of the isomorphism εΣ from (13) induces a composite isomorphism ofQp-spaces

ψQ : det−1Zp

(RΓ(ZΣ, T ))⊗Zp Qp ≃(Qp ⊗Z

∧r

ZE(Q)

)⊗Qp

∧r−1

Qp

H2(ZΣ, V )∗

1⊗ε−1Σ−−−−→Qp ⊗Z

(∧r

ZE(Q)⊗Z

∧r

ZE(Q)

)

in which the first isomorphism is induced by (16). This isomorphism is such that

(26) ψQ(ηalg) = LalgΣ · (x⊗ x)

for any choice of basis element x of∧r

ZE(Q)tf (where the displayed equality follows directly

from the explicit definition of ηalg).The composite of ϕQ and the map πQ,f that occurs in the lower row of the diagram in

Proposition 5.4(ii) also induces a similar isomorphism of Qp-spaces

πQ,f ϕQ : det−1Zp

(RΓ(ZΣ, T )) ⊗Zp Qp ≃ Qp ⊗Z

(∧r

ZE(Q)⊗Z

∧r

ZE(Q)

).

In addition, if we write NF/Q : E1(Fp) → E1(Qp) for the natural norm map, then therelation

#E(Fp)

p· logω(NF/Q(cF )) = TrQ(ζm)/Q(ζm) = (−1)ν(m)

can be used to show that

ψQ = (−1)ν(m)#E(Fp)

p· (πQ,f ϕQ).

From the equality (26), one can therefore deduce that

(πQ,f ϕQ)(ηalg) = (−1)ν(m)

(#E(Fp)

p

)−1

LalgΣ · (x⊗ x) = (−1)ν(m)LalgS · (x⊗ x),

where the last equality is valid because both Σ = S ∪ p and p /∈ S and hence(#E(Fp)

p

)−1

LalgΣ = Lalg(#E(Fp)

p

)−1(∏

ℓ∈Σ

#Ens(Fℓ)

ℓ

)= Lalg

(∏

ℓ∈S

#Ens(Fℓ)

ℓ

)= LalgS .


The above equality in turn implies that

(RegF πQ,f ϕQ)(ηalg) = (−1)ν(m)LalgS · RegF (x⊗ x)

and this implies the claimed result since the definition of RF implies directly that it is equalto RegF (x⊗ x).

5.5. Completion of the proof of Theorem 4.6. Upon combining the equality θF,S =πF,f(ϕF (zF )) from Corollary 5.3 with the commutativity of the diagram in Proposition5.4(ii), one derives an equality

θF,S = (RegF πQ,f νf ϕF )(zF ) in Ir/Ir+1.(27)

In addition, if one assumes zQ = ηalg, then the commutative diagram (25) implies that

(νf ϕF )(zF ) = ϕQ(zQ) = ϕQ(ηalg)

and hence, by Lemma 5.5, that

(28) (RegF πQ,f νf ϕF )(zF ) = (−1)ν(m)LalgS ·RF .

Since the equalities (27) and (28) combine to imply the equality θF,S = (−1)ν(m)LalgS ·RFthat is predicted by Conjecture 2.3, this argument therefore completes the proof of Theorem4.6 (and hence also, by Proposition 4.4, of Theorem 4.1), as required.

5.6. The deduction of Corollary 1.2. We finally explain the deduction of Corollary 1.2from the proof of Theorem 4.1. To do this we assume the hypotheses of Corollary 1.2 (butno longer require that #G is a prime power).

Then, under these hypotheses, the main result of Jetchev, Skinner and Wan in [15] impliesthat the quotient

u := LanS /LalgS

is a (non-zero) rational number that is coprime to every prime divisor of #G.From Remark 2.1 we also know that u = 1 if and only if E validates the Birch and

Swinnerton-Dyer Conjecture over Q.Hence, from the discussion of §4.1, the result of Corollary 1.2 will follow if we can show

that for each prime divisor p of #G the displayed equality in Conjecture 2.3 is valid afterone multiplies the right hand side by u (which acts invertibly on I/I2 ≃ G).

To do this we fix such a prime p. Then, since, by assumption, E has square-free conductorand supersingular reduction at p, the validity of the Generalized Perrin-Riou Conjecturefor E and Q∞/Q (Conjecture 3.1) follows directly from Remark 3.2 and the result [12, Th.2.4(iv)] of Buyukboduk. From the commutative diagram in the proof of Proposition 4.4 ittherefore follows that zQ = u · ηalg.

The latter equality then combines with the argument of §5.5 to imply that the equality(28) is unconditionally valid provided that one multiplies its right hand side by u and, bycomparing this fact to the equality (27), we deduce that the displayed equality in Conjecture2.3 is also unconditionally valid after one multiplies its right hand side by u, as required.

This completes the proof of Corollary 1.2.

25

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[2] W. Bley, The equivariant Tamagawa number conjecture and modular symbols, Math. Ann. 356no.1 (2013) 179-190.

[3] D. Burns, D. Macias Castillo, On refined conjectures of Birch and Swinnerton-Dyer type forHasse-Weil-Artin L-series, preprint, arXiv:1909.03959.

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Soc. Mat. Mex. 25 (2019) 529-541.

King’s College London, Department of Mathematics, London WC2R 2LS, U.K.

Email address: [email protected]

http://arxiv.org/abs/1909.03959






Keio University, Department of Mathematics, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-

8522, Japan


Osaka City University, Department of Mathematics, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka,

558-8585, Japan


arxiv:2103.11535v1 [math.nt] 22 mar 2021

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