Selmer groups and anticyclotomic ℤ_p-extensions II

2017/09/19 by Ahmed Matar, Matar, Ahmed
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1709.06455

openalex publication_date 2017/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E/ℚ be an elliptic curve, p a prime where E has ordinary reduction and K_∞/K the anticyclotomic ℤ_p-extension of a quadratic imaginary field K satisfying the Heegner hypothesis. We give sufficient conditions on E and p in order to ensure that Sel_p^∞(E/K_∞) is a cofree Λ-module of rank one. We also show that these conditions imply that rank(E(K_n))=pⁿ for all n ≥ 0 and that the p-primary subgroup of the Tate-Shafarevich group of E/K_n is trivial for all n ≥ 0.

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