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

TL;DR

The paper advances anticyclotomic Iwasawa theory for modular forms by defining and analyzing the $\mathfrak{p}$-part of Shafarevich--Tate groups via generalized Heegner cycles and $p$-adic Abel–Jacobi maps. Under the hypothesis that the basic generalized Heegner cycle $z_{f,K}$ is non-torsion and not divisible by $p$, it proves that $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K)=0$ and that $\mathop{\mathrm{H}}^1_f(K,A)$ is generated by $z_{f,K}$, while $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K_\infty)=0$ and $\mathop{\mathrm{H}}^1_f(K_\infty,A)$ is cofree of corank $1$ over the Iwasawa algebra $\Lambda$. The argument combines an Euler system of generalized Heegner cycles with Iwasawa control theorems to deduce a rank-one, non-torsion structure for the $\Lambda$-adic Selmer group $\mathcal{X}_\infty$, with broad consequences for anticyclotomic Iwasawa theory of modular forms.

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 $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K) = \widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K_\infty) = 0$.

Theorems & Definitions (103)

• Theorem 1: Theorem \ref{['th:main']}
• Theorem 2: Theorem \ref{['th:besser']}
• Remark 3
• Definition 1.1
• Remark 1.2
• Example 1.3
• Definition 1.4
• Example 1.5
• Proposition 1.6
• proof