Iwasawa theory and ranks of elliptic curves in quadratic twist families
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909425187422208 |
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| author | Hatley, Jeffrey Ray, Anwesh |
| author_facet | Hatley, Jeffrey Ray, Anwesh |
| contents | We study the distribution of ranks of elliptic curves in quadratic twist families using Iwasawa-theoretic methods, contributing to the understanding of Goldfeld's conjecture. Given an elliptic curve $ E/\mathbb{Q} $ with good ordinary reduction at $ 2 $ and $ λ_2(E/\mathbb{Q}) = 0 $, we use Matsuno's Kida-type formula to construct quadratic twists $ E^{(d)} $ such that $ λ_2(E^{(d)}/\mathbb{Q}) $ remains unchanged or increases by $ 2 $. When the root number of $E^{(d)}$ is $-1$ and the Tate-Shafarevich group $Sha(E^{(d)}/\mathbb{Q})[2^\infty] $ is finite, this yields quadratic twists with Mordell--Weil rank $ 1 $. These results support the conjectural expectation that, on average, half of the quadratic twists in a family have rank $ 0 $ and half have rank $ 1 $. In the cases we consider we obtain asymptotic lower bounds for the number of twists by squarefree numbers $d\leq X$ which match with the conjectured value up to an explicit power of $\log X$. They complement recent groundbreaking results of Smith on Goldfeld's conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07308 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Iwasawa theory and ranks of elliptic curves in quadratic twist families Hatley, Jeffrey Ray, Anwesh Number Theory 11R23, 11G05, 11R45 We study the distribution of ranks of elliptic curves in quadratic twist families using Iwasawa-theoretic methods, contributing to the understanding of Goldfeld's conjecture. Given an elliptic curve $ E/\mathbb{Q} $ with good ordinary reduction at $ 2 $ and $ λ_2(E/\mathbb{Q}) = 0 $, we use Matsuno's Kida-type formula to construct quadratic twists $ E^{(d)} $ such that $ λ_2(E^{(d)}/\mathbb{Q}) $ remains unchanged or increases by $ 2 $. When the root number of $E^{(d)}$ is $-1$ and the Tate-Shafarevich group $Sha(E^{(d)}/\mathbb{Q})[2^\infty] $ is finite, this yields quadratic twists with Mordell--Weil rank $ 1 $. These results support the conjectural expectation that, on average, half of the quadratic twists in a family have rank $ 0 $ and half have rank $ 1 $. In the cases we consider we obtain asymptotic lower bounds for the number of twists by squarefree numbers $d\leq X$ which match with the conjectured value up to an explicit power of $\log X$. They complement recent groundbreaking results of Smith on Goldfeld's conjecture. |
| title | Iwasawa theory and ranks of elliptic curves in quadratic twist families |
| topic | Number Theory 11R23, 11G05, 11R45 |
| url | https://arxiv.org/abs/2412.07308 |