Vanishing of the p-part of the Shafarevich-Tate group of a modular form 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 ($\mathfrak{p}$-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 ${\widetilde{\mathrm{sha}}}_{{\mathfrak{p}}^{\infty }}(f/K)={\widetilde{\mathrm{sha}}}_{{\mathfrak{p}}^{\infty }}(f/K_{\infty })=0$.

34 pages; final version, accepted for publication in Kyoto Journal of Mathematics