On the structure of the Bloch--Kato Selmer groups of modular forms over anticyclotomic $\mathbf{Z}_p$-towers
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| Format: | Preprint |
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2024
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| _version_ | 1866917306543636480 |
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| author | Lei, Antonio Mastella, Luca Zhao, Luochen |
| author_facet | Lei, Antonio Mastella, Luca Zhao, Luochen |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_11966 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the structure of the Bloch--Kato Selmer groups of modular forms over anticyclotomic $\mathbf{Z}_p$-towers Lei, Antonio Mastella, Luca Zhao, Luochen Number Theory 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. |
| title | On the structure of the Bloch--Kato Selmer groups of modular forms over anticyclotomic $\mathbf{Z}_p$-towers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.11966 |