On the structure of Selmer and Shafarevich–Tate groups of even weight modular forms

Masoero, Daniele

doi:10.1090/tran/7407

Cited by 3 publications

(7 citation statements)

References 33 publications

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“…Let r ± π (f /K) be the O π -corank of Sel(K, W f (1)) ± . Combining Theorem B.2 with Kolyvagin's theorem [Kol91,Mas19], the following result on the relation between Selmer coranks and the vanishing order of Kolyvagin systems easily follows.…”

Section: A Local Computation Let Loc Smentioning

confidence: 85%

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Refined applications of Kato's Euler systems for modular forms

Kim¹

2022

Preprint

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We discuss refined applications of Kato's Euler systems for modular forms of higher weight at good primes (with more emphasis on the non-ordinary ones) beyond the one-sided divisibility of the main conjecture and the finiteness of Selmer groups. These include a proof of the Mazur-Tate conjecture on Fitting ideals of Selmer groups over p-cyclotomic extensions and a new interpretation of the Iwasawa main conjecture with structural applications. Contents 1. Introduction 1 2. Galois and Iwasawa cohomologies 11 3. Mazur-Tate elements I: Betti construction 13 4. Construction of canonical Kato's Euler systems 17 5. Review of p-adic Hodge theory 22 6. Local Iwasawa theory and explicit reciprocity law à la Perrin-Riou 24 7. Mazur-Tate elements II: étale construction 29 8. Towards the "weak" main conjecture of Mazur-Tate 40 9. Towards the Iwasawa main conjecture for modular forms 45 10. Towards the µ = 0 conjecture of Coates-Sujatha 50 Appendix A. On the p-part of the BSD formula for elliptic curves of analytic rank one 53 Appendix B. Kato's Kolyvagin systems and Heegner point Kolyvagin systems 53 References 58

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Section: A Local Computation Let Loc Smentioning

confidence: 85%

“…Indeed, this is [Zha14, Theorem 1.2 and Theorem 11.2] where the statements are given only for elliptic curves over Q since so is the setting of [Kol91]. The generalization of the setting to (even higher weight) modular forms is now developed in [Mas19,Theorem 8.4].…”

Section: A Local Computation Let Loc Smentioning

confidence: 99%

Refined applications of Kato's Euler systems for modular forms

Kim¹

2022

Preprint

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“…The Galois representation T p linked to f can be defined as follows [21,23]: For the p-adic sheaf F = lim ← −n F n over Y (N ) with the sheaves…”

Section: Galois Representationsmentioning

confidence: 99%

“…As described by [21, sections 3.1-3.3] and [23, chapters 2-4], for any field F containing Q, there is a p-adic Abel-Jacobi map (extending the map Φ p,L in [21] by Z p -linearity)…”

Section: The P-adic Abel-jacobi Mapmentioning

confidence: 99%

A BSD formula for high-weight modular forms

Thackeray

2022

Journal of Number Theory

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“…Remark 3.8. Several papers [see for instance Nek92]; [Bes97]; [LV17]; [Mas19], use classical Heegner cycles instead of the generalized ones, they consider therefore a different Abel-Jacobi map,…”

Section: ]) the Cycle ∆ ϕ Is Homologically Trivialmentioning

confidence: 99%

Kolyvagin’s result on the vanishing of $\protect \Sha(E/K)[p^\infty ]$ and its consequences for anticyclotomic Iwasawa theory

Matar¹,

Nekovář²

2019

Journal De Théorie Des Nombres De Bordeaux

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In this article we prove a refinement of a theorem of Longo and Vigni in the anticyclotomic Iwasawa theory for modular forms. More precisely we give a definition for the (p-part of the) Shafarevich-Tate groups X p ∞ (f /K) and X p ∞ (f /K∞) of a modular form f of weight k > 2, over an imaginary quadratic field K satisfying the Heegner hypothesis and over its anticyclotomic Zp-extension K∞ and we show that if the basic generalized Heegner cycle z f,K is non-torsion and not divisible by p, thenLet f = n>0 a n q n be a cusp-newform of even weight k > 2 and level Γ 0 (N ), fix an odd prime p ∤ N and an embedding i p : Q ֒→ Qp . Denote by F the totally real field generated over Q by the Fourier coefficients a n of f and let O F be its ring of integers. The embedding F ֒→ Qp induced by i p defines a prime ideal p of O F above p: let K := F p be the completion of F at p and let O be its valuation ring. Deligne [see Del71] attached to f and p a p-adic representation W p ofFix moreover an imaginary quadratic field K of discriminant d K = −3, −4 coprime with N p and satisfying the so called Heegner hypothesis, i.e. such that the prime factors of N split in K, and let K ∞ be its anticyclotomic Z p -extension, that is the unique Z p -extension of K pro-dihedral over Q. Put Γ = Gal(K ∞ /K) ∼ = Z p and let Λ = O Γ ∼ = O T be the Iwasawa algebra.The main result of [LV19] is a structure theorem (as Λ-module) for the Pontryagin dual X ∞ of the Bloch-Kato Selmer group H 1 f (K ∞ , A) of A over K ∞ : they show, under some hypothesis on (f, K, p) as in particular the p-ordinariety of f and the big image property for V p := W * p , that X ∞ is pseusoisomorphic to Λ ⊕ M ⊕ M , for a torsion Λ-module M , moreover they formulate an anticyclotomic main conjecture in this setting and they prove one divisibility of it.In particular this shows that H 1 f (K ∞ , A) has corank 1 as Λ-module, the aim of this paper is to show that (under some thechnical assumptions) if the basic generalized Heegner cycle z f,K (see Sec. 3.4) is not divisible by p in H 1 (K, T ), then H 1 f (K ∞ , A) is in fact cofree of corank 1 over Λ. We give moreover a suitable definition of the (p-part of the) Shafarevich-Tate groups), where K[n] denote the ring class field of conductor n > 1 and X k−2 is the generalized Kuga-Sato variety of [BDP13]: the product of the Kuga-Sato variety Ẽk−2 Γ1(N ) of level Γ 1 (N ) and weight k − 2 and the (k − 2)-fold selfproduct of a fixed CM elliptic curve A defined over K[1].

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On the structure of Selmer and Shafarevich–Tate groups of even weight modular forms

Cited by 3 publications

References 33 publications

Refined applications of Kato's Euler systems for modular forms

Refined applications of Kato's Euler systems for modular forms

A BSD formula for high-weight modular forms

Kolyvagin’s result on the vanishing of $\protect \Sha(E/K)[p^\infty ]$ and its consequences for anticyclotomic Iwasawa theory

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