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

Abstract

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].

Elliptic curves with exceptionally large analytic order of the Tate–Shafarevich groups

“…A second way is to use a result of Gross (see [15,Props. 2.1,2.3] or [19,Theorem 0.5]) and some Magma computations. Let y K ∈ E(K) be the basic Heegner point attached to E and a suitable imaginary quadratic field K. If y K / ∈ 3E(K), then we obtain Ш(E/K)[3 ∞ ] = 0 as desired.…”

Correction to the paper “Kolyvagin’s result on the vanishing of Ш(E/K)[p ∞ ] and its consequences for anticyclotomic Iwasawa theory”

“…Journal de Théorie des Nombres de Bordeaux 33 (2021), 627-628 Correction to the paper "Kolyvagin's result on the vanishing of X(E/K)[p ∞ ] and its consequences for anticyclotomic Iwasawa theory" par Ahmed MATAR et Jan NEKOVÁŘ Résumé. Nous corrigeons une inexactitude mineure dans l'article [2] Abstract. We correct a minor inaccuracy in the article [2].…”

“…Nous corrigeons une inexactitude mineure dans l'article [2] Abstract. We correct a minor inaccuracy in the article [2].…”

“…It has come to our attention that condition (C5) in Proposition 6.4 of [2] needs to be made more precise by requiring the restriction of ρ to G K to be absolutely irreducible rather than irreducible. The point is that the proof of [1,Prop.…”

“…9.3] relies crucially on the fact that End Fp[G K ] (E[p]) = F p . As a result, the word "irreducible" in Proposition 6.5 of [2] needs to be replaced by "absolutely irreducible".…”