Construction of a Homomorphism Concerning Euler Systems for an Elliptic Curve

Otsuki, Rei

doi:10.3836/tjm/1249648421

Cited by 5 publications

(12 citation statements)

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“…Proof. This is a straightforward generalization of (one case of) Proposition 2.5 of [Ots09]. We define H ℓ (X) = G ℓ (X)−F ℓ (X) X , so we have…”

Section: 2mentioning

confidence: 96%

“…Otsuki's functionals. We now explain a construction due to Otsuki [Ots09], who has shown how to construct canonical linear functionals on cohomology groups by composing the dual exponential map with an appropriate "weighted trace". These maps do not satisfy the compatibility properties of a Perrin-Riou functional, and thus give rise to systems of elements of group rings satisfying a modified compatibility property; we shall show that this modification is consistent with the results we have shown for our generalized Beilinson-Flach classes.…”

Section: 2mentioning

confidence: 99%

“…We regard D mp ∞ as a Γ m -module, via the usual action of G m on Q(µ m ) and of Γ on Λ E (Γ). Following [Kur02] and [Ots09], we make the following definition: Here we extend , cris to be Γ-linear in the second variable and Γ-antilinear in the first. One checks that t m (σx, τ z) = [σ −1 τ ] · t m (x, z) for all σ, τ ∈ Γ m (not just in Γ).…”

Section: 2mentioning

confidence: 99%

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Euler systems for Rankin-Selberg convolutions of modular forms

2014

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We construct an Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use these elements to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin-Selberg L-function is non-vanishing at s = 1. Dedicated to Kazuya KatoContents 1. Outline 1 2. Generalized Beilinson-Flach elements 2 3. Norm relations for generalized Beilinson-Flach elements 14 4. Relation to complex L-values 37 5. Relation to p-adic L-values 43 6. Families of cohomology classes 49 7. Bounding strict Selmer groups 62 8. Conjectures on higher-rank Euler systems 71 Appendix A. Ancillary results 74 References 77 1. OutlineIn [BDR12], Bertolini, Darmon and Rotger have studied certain canonical global cohomology classes (the "Beilinson-Flach elements", obtained from the constructions of [Beȋ84] and [Fla92]) in the cohomology of the tensor products of the p-adic Galois representations of pairs of weight 2 modular forms, and related their image under the Bloch-Kato logarithm maps to the values of p-adic Rankin-Selberg L-functions. These Beilinson-Flach elements are constructed as the image of elements of the higher Chow group of a product of modular curves.In this paper, we construct a form of Euler system -a compatible system of cohomology classes over cyclotomic fields -of which the Beilinson-Flach elements are the bottom layer. We first define elements of higher Chow groups of the product of two (affine) modular curves over a cyclotomic field,for integers m ≥ 1, N ≥ 5, and j ∈ Z/mZ (cf. Definition 2.7.3). These are obtained by considering the images of various maps from higher level modular curves to the surface Y 1 (N ) 2 , together with modular units (Siegel units) on these curves. For m = 1 our elements reduce to those considered in [BDR12], and as in op.cit., we show that after tensoring with Q we can construct preimages of our elements in the higher Chow group of the self-product of the projective modular curve X 1 (N ); however, in this paper (as in [Kat04]) we shall take the affine versions as the principal objects of study. We next turn to the relation between our elements and L-values. Theorem 4.3.7 shows, following an argument due to Beilinson, that the images of the elements c Ξ m,N,j under the Beilinson regulator map into complex de Rham cohomology are related to the derivatives at s = 1 of Rankin-Selberg L-functions of weight 2 modular forms. Theorem 5.6.4 is a p-adic analogue of this result, generalizing a theorem of Bertolini-Darmon-Rotger [BDR12]; it gives a formula for the image of our element for m = 1 under the p-adic syntomic regulator, for a prime p ∤ N , in terms of Hida's p-adic Rankin-Selberg L-functions.Next we consider the images of our elements inétale cohomology. Applying Huber's "continuousétale realization" functor and the Hochschild-Serre exact sequence, and projecting into the isotypical component corresponding to a pair of eige...

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“…Proof. This is a straightforward generalization of (one case of) Proposition 2.5 of [Ots09]. We define H ℓ (X) = G ℓ (X)−F ℓ (X) X , so we have…”

Section: 2mentioning

confidence: 96%

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

Euler systems for Rankin-Selberg convolutions of modular forms

2014

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“…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).…”

mentioning

confidence: 73%

“…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).…”

Section: Determinantal Zeta Elementsmentioning

confidence: 99%

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

Burns¹,

Kurihara²,

Sano³

2021

Preprint

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We provide a new interpretation of the Mazur-Tate Conjecture and then use it to obtain the first (unconditional) theoretical evidence in support of the conjecture for elliptic curves of strictly positive rank. Contents 1. Introduction 1.1. Discussion of the main results 1.2. General notation 2. Review of the Mazur-Tate Conjecture 2.1. The Birch and Swinnerton-Dyer Conjecture 2.2. Modular elements and the Mazur-Tate Conjecture 3. Review of the Generalized Perrin-Riou Conjecture 3.1. The Bockstein regulator 3.2. Kato's zeta elements 3.3. The Generalized Perrin-Riou Conjecture 4. The main result 4.1. Statement of the main result and the deduction of Theorem 1.1 4.2. Determinantal zeta elements 5. The proof of Theorem 4.6 and the deduction of Corollary 1.2 5.1. Construction of the modular element 5.2. Bloch-Kato Selmer complexes 5.3. Bockstein maps for Selmer complexes 5.4. The algebraic Birch and Swinnerton-Dyer element 5.5. Completion of the proof of Theorem 4.6 5.6. The deduction of Corollary 1.2 References

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

Ota¹

2018

American Journal of Mathematics

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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...

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Construction of a Homomorphism Concerning Euler Systems for an Elliptic Curve

Cited by 5 publications

References 9 publications

Euler systems for Rankin-Selberg convolutions of modular forms

Euler systems for Rankin-Selberg convolutions of modular forms

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

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

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