Vanishing of the p-part of the Shafarevich-Tate group of a modular form and its consequences for Anticyclotomic Iwasawa Theory

2023/07/24 by Luca Mastella, Mastella, Luca
Mathematics · #11F11 (Secondary) #11R23 (Primary) #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2307.13134

openalex publication_date 2023/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 (\mathfrakp-part of the) Shafarevich-Tate groups \widetildesha_\mathfrakp^∞(f/K) and \widetildesha_\mathfrakp^∞(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 ℤ_p-extension K_∞ and we show that if the basic generalized Heegner cycle z_{f, K} is non-torsion and not divisible by p, then \widetildesha_\mathfrakp^∞(f/K) = \widetildesha_\mathfrakp^∞(f/K_∞) = 0.

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