Kato's Euler system and the Mazur-Tate refined conjecture of BSD type

Abstract

Mazur and Tate proposed a conjecture which compares the Mordell-Weil rank of an elliptic curve over Q with the order of vanishing of Mazur-Tate elements, which are analogues of Stickelberger elements. Under some relatively mild assumptions, we prove this conjecture. Our strategy of the proof is to study divisibility of certain derivatives of Kato's Euler system. CONTENTS 1. Introduction 1 2. Mazur-Tate elements 4 3. Darmon-Kolyvagin derivatives and Euler systems 6 4. Divisibility of Euler systems for elliptic curves 15 5. Local study of Mazur-Tate elements 28 6. Proof of the main result 36 7. Trivial zeros 39 References 40 1 2 + 1    and R = Z[p −1 ; p is a prime less than d].Then, for every square-free product S of good supersingular primes, θ S ∈ R[G S ] andIn this paper, we also give a partial evidence (Theorem 6.4) of the part of the Mazur-Tate refined conjecture which relates arithmetic invariants such as the Tate-Shafarevich group X to the leading coefficient ofThe following is a special case of Theorem 6.4. THEOREM 1.5. Let p be a prime not invertible in R and S a square-free product of good primes ℓ such that ℓ ≡ 1 mod p and1.3. The plan of proof. We briefly explain how to prove Theorem 1.2. By a group ring theoretic argument (Lemma 5.4), we are reduced to proving thatLet p be a such prime andOur strategy of the proof of (1.1) is to show that Darmon-Kolyvagin derivatives of Kato's Euler system are divisible by a power of p. In order to investigate the divisibility, we modify an argument of Darmon [6], who proposed a refined conjecture for Heegner points and proved an analogue of Conjecture 1.1 in many cases (see [16] for a generalization to Heegner cycles). Next, by modifying ideas of Kurihara [14], Kobayashi [13] and Otsuki [23], we relate Kato's Euler system with Mazur-Tate elements. Then, the derivatives of Kato's Euler system appear in the coefficients of the Taylor expansion of θ S . By the divisibility of the derivatives and a group-ring theoretic argument, we show that θ S belongs to a power of the augmentation ideal. However, our modification of Darmon's argument implies only thatp} S . 4 K. OTA One might expect that Darmon's argument implies that θ S ∈ Z p ⊗ R I min{r p ∞ ,p} S. The obstruction is the difference between the local condition at p of Heegner points and that of Kato's Euler system. The localization of Heegner points at p obviously comes from local rational points (i.e. it is crystalline at p), and then Heegner points are related to the usual Selmer group. However, the localization of Kato's Euler system does not necessarily come from a local rational point, and then we can relate Kato's Euler system only with the strict Selmer group H 1 f,p (Q, E[p ∞ ]), whose local condition at p is zero. Since the corank of H 1 f,p (Q, E[p ∞ ]) is not necessarily greater than r p ∞ − 1, we have only (1.2).Our idea for deducing (1.1) from (1.2) is to apply the p-parity conjecture, which is now a theorem (cf.[8], [12], [22]). It asserts that r p ∞ ≡ ord s=1 (L(E, s)) mod 2. On the other hand, the...

On derivatives of Kato's Euler system and the Mazur-Tate Conjecture

“…Finally, we shall show, by explicit computation, that the latter equality implies the validity of the p-primary component of the Mazur-Tate Conjecture (see Theorem 4.6). We remark that this last computation relies on the precise relation between modular elements and Kato's zeta elements (as previously discussed by the second author in [18], by Otsuki in [21] and by Ota in [20]), on an explicit description of the relevant Bloch-Kato Selmer complexes and on a Galois-cohomological interpretation of the biextension-pairing of Mazur and Tate in terms of Bockstein homomorphisms associated to Bloch-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).…”

“…Construction of the modular element. We first review the construction of the modular element θ F,S from Kato's zeta element z F (see [18] by the second author, [21] by Otsuki and especially [20] by Ota).…”

“…If r = 0, then the full Mazur-Tate Conjecture (which is stated precisely as Conjecture 2.3 below) 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 conjecture has 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 of the conjecture has remained stubbornly mysterious and, even now, there is no conceptual understanding of it or theoretical evidence for it and the only supporting numerical evidence is for the case r = 1 (for details of which see, for example, the recent article of Portillo-Bobadilla [23]).…”

Equivariant Iwasawa theory for elliptic curves

“…Hence Corollary 1.9 proves the weak vanishing conjecture in that case. On the other hand, Ota [32] proved the weak vanishing conjecture for the trivial character under certain hypotheses. The nature of his hypotheses is very close to ours, but he does not require the condition (a) or (e).…”

On the indivisibility of derived Kato’s Euler systems and the main conjecture for modular forms

“…See [Kur14a, (21) (page 190) and (65)] and [Kur14b, (2) and (31), (32) (page 346)] for detail. For the expansion of higher degree terms, see[Ota18].7.4. Proof of Theorem 1.1.…”