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

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 $K_{\mathrm{\infty }}$ 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 $K_{\mathrm{\infty }}$ is free over the Iwasawa algebra. Consequently, the Bloch–Kato–Shafarevich–Tate groups of $f$ vanish. This generalizes earlier works of Matar and Matar–Nekovář on elliptic curves. Furthermore, our proof applies uniformly to the ordinary and non-ordinary settings.