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

TL;DR

This paper investigates the structure of Bloch--Kato Selmer groups for modular forms over anticyclotomic $ extbf{Z}_p$-extensions of imaginary quadratic fields, establishing conditions under which these groups are free and their associated Tate groups vanish.

Contribution

It generalizes previous results on elliptic curves to modular forms, providing a uniform proof applicable to both ordinary and non-ordinary cases.

Findings

Pontryagin dual of the Selmer group is free over the Iwasawa algebra.

Bloch--Kato--Shafarevich--Tate groups of the modular form vanish.

Results hold under generalized Heegner hypothesis and certain local conditions.

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.