On the structure of the Bloch--Kato Selmer groups of modular forms over anticyclotomic $\mathbf{Z}_p$-towers

Abstract

Let $p$ be an odd prime number and let $K$ be an imaginary quadratic field in which $p$ is split. Let $f$ be a modular form with good reduction at $p$. We study the variation of the Bloch--Kato Selmer groups and the Bloch--Kato--Shafarevich--Tate groups of $f$ over the anticyclotomic $\mathbf{Z}_p$-extension $K_\infty$ of $K$. In particular, we show that under the generalized Heegner hypothesis, if the $p$-localization of the generalized Heegner cycle attached to $f$ is primitive and certain local conditions hold, then the Pontryagin dual of the Selmer group of $f$ over $K_\infty$ is free over the Iwasawa algebra. Consequently, the Bloch--Kato--Shafarevich--Tate groups of $f$ vanish. This generalizes earlier works of Matar and Matar--Nekov\'a\v{r} on elliptic curves. Furthermore, our proof applies uniformly to the ordinary and non-ordinary settings.