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

Antonio Lei, Luca Mastella, Luochen Zhao

TL;DR

The paper advances the Iwasawa-theoretic study of Bloch–Kato Selmer groups for higher-weight modular forms over the anticyclotomic ${Z}_p$-tower of an imaginary quadratic field. By combining the bottom-up Matar–Nekovář framework with the BDP (plus/minus) approach and employing universal-norm/Perrin–Riou techniques, it proves the vanishing of the BDP Selmer groups, determines the growth of standard Selmer groups ${\rm Sel}(K_n,A)$, and establishes that ${\rm Sel}(K_\infty,A)^{\vee}$ is a free $\Lambda$-module of rank $1$ (when $f$ is ordinary at $p$) or rank $2$ (non-ordinary). Under the generalized Heegner hypotheses and local primitivity of the generalized Heegner class $z_{f,K}$, this yields the vanishing of Bloch–Kato–Shafarevich–Tate groups in the tower and extends prior elliptic-curve results to higher weight modular forms uniformly across ordinary/non-ordinary cases. The work highlights a robust structure for Selmer groups in anticyclotomic Iwasawa theory and connects the arithmetic of modular forms to generalized Heegner cycles and $p$-adic $L$-functions through a refined control-theoretic and norm-based framework.

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--Nekovář on elliptic curves. Furthermore, our proof applies uniformly to the ordinary and non-ordinary settings.

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

TL;DR

The paper advances the Iwasawa-theoretic study of Bloch–Kato Selmer groups for higher-weight modular forms over the anticyclotomic -tower of an imaginary quadratic field. By combining the bottom-up Matar–Nekovář framework with the BDP (plus/minus) approach and employing universal-norm/Perrin–Riou techniques, it proves the vanishing of the BDP Selmer groups, determines the growth of standard Selmer groups , and establishes that is a free -module of rank (when is ordinary at ) or rank (non-ordinary). Under the generalized Heegner hypotheses and local primitivity of the generalized Heegner class , this yields the vanishing of Bloch–Kato–Shafarevich–Tate groups in the tower and extends prior elliptic-curve results to higher weight modular forms uniformly across ordinary/non-ordinary cases. The work highlights a robust structure for Selmer groups in anticyclotomic Iwasawa theory and connects the arithmetic of modular forms to generalized Heegner cycles and -adic -functions through a refined control-theoretic and norm-based framework.

Abstract

Let be an odd prime number and let be an imaginary quadratic field in which is split. Let be a modular form with good reduction at . We study the variation of the Bloch--Kato Selmer groups and the Bloch--Kato--Shafarevich--Tate groups of over the anticyclotomic -extension of . In particular, we show that under the generalized Heegner hypothesis, if the -localization of the generalized Heegner cycle attached to is primitive and certain local conditions hold, then the Pontryagin dual of the Selmer group of over is free over the Iwasawa algebra. Consequently, the Bloch--Kato--Shafarevich--Tate groups of 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.
Paper Structure (22 sections, 28 theorems, 103 equations)

This paper contains 22 sections, 28 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Setting and notation
  3. Background
  4. Main result
  5. Strategy of proof
  6. Obiter dictum
  7. Acknowledgement
  8. Selmer groups
  9. Review of generalized Heegner cycles
  10. Control theorem
  11. Vanishing of $\mathrm{Sel}^\mathrm{BDP}$ and its consequences
  12. Results on universal norms
  13. Growth of Selmer groups: the case of $a_p=0$
  14. Summary of results
  15. Organization
  16. ...and 7 more sections

Key Result

Theorem A

Let $f$ be a normalized cuspidal eigen-newform of even weight $k$ and level $\Gamma_0(N)$. Let $K$ be an imaginary quadratic field with coprime-to-$Np$ discriminant $d_K\ne -3,-4$ and satisfying ass:genHeeg. Assume moreover $N\ge 5$ if $N^-=1$, and $N^+> 3$ and $k\ge4$ if $N^-\ne 1$. Let $p$ be a pr Then we have:

Theorems & Definitions (66)

  • Theorem A
  • Remark 1.1
  • Remark 1.2
  • Theorem 3.1
  • Remark 3.2
  • proof : Proof of Theorem \ref{['th:mastella-th-0.2']}
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Definition 3.5
  • ...and 56 more