Rank stability of elliptic curves in certain non-abelian extensions
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909490156142592 |
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| author | Pathak, Siddhi Ray, Anwesh |
| author_facet | Pathak, Siddhi Ray, Anwesh |
| contents | Let $E_{/\mathbb{Q}}$ be an elliptic curve with rank $E(\mathbb{Q})=0$. Fix an odd prime $p$, a positive integer $n$ and a finite abelian extension $K/\mathbb{Q}$ with rank $E(K) = 0$. In this paper, we show that there exist infinitely many extensions $L/K$ such that $L/\mathbb{Q}$ is Galois with $\operatorname{Gal}(L/\mathbb{Q}) \simeq \operatorname{Gal}(K/\mathbb{Q}) \ltimes \mathbb{Z}/p^n\mathbb{Z}$, and rank $E(L)=0$. This is an extension of earlier results on rank stability of elliptic curves in cyclic extensions of prime power order to a non-abelian setting. We also obtain an asymptotic lower bound for the number of such extensions, ordered by their absolute discriminant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13582 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rank stability of elliptic curves in certain non-abelian extensions Pathak, Siddhi Ray, Anwesh Number Theory 11R45, 11G05 Let $E_{/\mathbb{Q}}$ be an elliptic curve with rank $E(\mathbb{Q})=0$. Fix an odd prime $p$, a positive integer $n$ and a finite abelian extension $K/\mathbb{Q}$ with rank $E(K) = 0$. In this paper, we show that there exist infinitely many extensions $L/K$ such that $L/\mathbb{Q}$ is Galois with $\operatorname{Gal}(L/\mathbb{Q}) \simeq \operatorname{Gal}(K/\mathbb{Q}) \ltimes \mathbb{Z}/p^n\mathbb{Z}$, and rank $E(L)=0$. This is an extension of earlier results on rank stability of elliptic curves in cyclic extensions of prime power order to a non-abelian setting. We also obtain an asymptotic lower bound for the number of such extensions, ordered by their absolute discriminant. |
| title | Rank stability of elliptic curves in certain non-abelian extensions |
| topic | Number Theory 11R45, 11G05 |
| url | https://arxiv.org/abs/2401.13582 |