Quadratic Twists as Random Variables
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910299485896704 |
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| author | Paterson, Ross |
| author_facet | Paterson, Ross |
| contents | Let $D\neq 1$ be a fixed squarefree integer. For elliptic curves $E/\mathbb{Q}$, writing $E_D$ for the quadratic twist by $D$, we consider the question of how often $E(\mathbb{Q})$ and $E_D(\mathbb{Q})$ generate $E(\mathbb{Q}(\sqrt{D}))$. We bound the proportion of $E/\mathbb{Q}$, ordered by height, for which this is not the case, showing that it is very small for typical $D$.
The central theorem is concerned with intersections of 2-Selmer groups of quadratic twists. We establish their average size in terms of a product of local densities. We additionally propose a heuristic model for these intersections, which explains our result and similar results in the literature. This heuristic predicts further results in other families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08836 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quadratic Twists as Random Variables Paterson, Ross Number Theory 11G05 (Primary) 11G07, 11R45 (Secondary) Let $D\neq 1$ be a fixed squarefree integer. For elliptic curves $E/\mathbb{Q}$, writing $E_D$ for the quadratic twist by $D$, we consider the question of how often $E(\mathbb{Q})$ and $E_D(\mathbb{Q})$ generate $E(\mathbb{Q}(\sqrt{D}))$. We bound the proportion of $E/\mathbb{Q}$, ordered by height, for which this is not the case, showing that it is very small for typical $D$. The central theorem is concerned with intersections of 2-Selmer groups of quadratic twists. We establish their average size in terms of a product of local densities. We additionally propose a heuristic model for these intersections, which explains our result and similar results in the literature. This heuristic predicts further results in other families. |
| title | Quadratic Twists as Random Variables |
| topic | Number Theory 11G05 (Primary) 11G07, 11R45 (Secondary) |
| url | https://arxiv.org/abs/2401.08836 |