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

TL;DR

This paper proves that under certain conditions, the p-part of the Shafarevich-Tate group for a modular form vanishes over both an imaginary quadratic field and its anticyclotomic extension, impacting Iwasawa theory.

Contribution

It refines previous results by defining the p-part of Shafarevich-Tate groups for modular forms and proves their vanishing under specific non-torsion conditions.

Findings

p-part of Shafarevich-Tate groups vanishes under non-torsion conditions

Establishes a link between Heegner cycles and group vanishing

Implications 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 ($\mathfrak{p}$-part of the) Shafarevich-Tate groups $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K)$ and $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K_\infty)$ of a modular form $f$ of weight $k >2$, over an imaginary quadratic field $K$ satisfying the Heegner hypothesis and over its anticyclotomic $\mathbb{Z}_p$-extension $K_\infty$ and we show that if the basic generalized Heegner cycle $z_{f, K}$ is non-torsion and not divisible by $p$, then $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K) = \widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K_\infty) = 0$.