On fine Mordell-Weil groups over $\mathbb{Z}_p$-extensions of an imaginary quadratic field
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910914476769280 |
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| author | Lim, Meng Fai |
| author_facet | Lim, Meng Fai |
| contents | Let $E$ be an elliptic curve over $\mathbb{Q}$. Greenberg has posed a question whether the structure of the fine Selmer group over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ can be described by cyclotomic polynomials in a certain precise manner. A recent work of Lei has made progress on this problem by proving that the fine Mordell-Weil group (in the sense of Wuthrich) does have this required property. The goal of this paper is study the analogous question of Greenberg over various $\mathbb{Z}_p$-extensions of an imaginary quadratic field $F$. In particular, when the elliptic curve has complex multiplication by the ring of integers of the imaginary quadratic field, we obtain analogous results of Lei over the cyclotomic $\mathbb{Z}_p$-extension and anti-cyclotomic $\mathbb{Z}_p$-extension of $F$. In the event that the elliptic curve has good ordinary reduction at the prime $p$, we further obtain a result over the $\mathbb{Z}_p$-extension of $F$ unramified outside precisely one of the prime of $F$ above $p$. Finally, we study the situation of an elliptic curve over the anticyclotomic $\mathbb{Z}_p$-extension under the generalized Heegner hypothesis. Along the way, we establish an analogous result for the BDP-Selmer group. This latter result is then applied to obtain a relation between the BDP $p$-adic $L$-function and the Mordell-Weil rank growth in the anticyclotomic $\mathbb{Z}_p$-extension which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_04096 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On fine Mordell-Weil groups over $\mathbb{Z}_p$-extensions of an imaginary quadratic field Lim, Meng Fai Number Theory Let $E$ be an elliptic curve over $\mathbb{Q}$. Greenberg has posed a question whether the structure of the fine Selmer group over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ can be described by cyclotomic polynomials in a certain precise manner. A recent work of Lei has made progress on this problem by proving that the fine Mordell-Weil group (in the sense of Wuthrich) does have this required property. The goal of this paper is study the analogous question of Greenberg over various $\mathbb{Z}_p$-extensions of an imaginary quadratic field $F$. In particular, when the elliptic curve has complex multiplication by the ring of integers of the imaginary quadratic field, we obtain analogous results of Lei over the cyclotomic $\mathbb{Z}_p$-extension and anti-cyclotomic $\mathbb{Z}_p$-extension of $F$. In the event that the elliptic curve has good ordinary reduction at the prime $p$, we further obtain a result over the $\mathbb{Z}_p$-extension of $F$ unramified outside precisely one of the prime of $F$ above $p$. Finally, we study the situation of an elliptic curve over the anticyclotomic $\mathbb{Z}_p$-extension under the generalized Heegner hypothesis. Along the way, we establish an analogous result for the BDP-Selmer group. This latter result is then applied to obtain a relation between the BDP $p$-adic $L$-function and the Mordell-Weil rank growth in the anticyclotomic $\mathbb{Z}_p$-extension which may be of independent interest. |
| title | On fine Mordell-Weil groups over $\mathbb{Z}_p$-extensions of an imaginary quadratic field |
| topic | Number Theory |
| url | https://arxiv.org/abs/2308.04096 |