Low rank specializations of elliptic surfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910555503067136 |
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| author | Melistas, Mentzelos |
| author_facet | Melistas, Mentzelos |
| contents | Let $E/\mathbb{Q}(T)$ be a non-isotrivial elliptic curve of rank $r$. A theorem due to Silverman implies that the rank $r_t$ of the specialization $E_t/\mathbb{Q}$ is at least $r$ for all but finitely many $t \in \mathbb{Q}$. Moreover, it is conjectured that $r_t \leq r+2$, except for a set of density $0$. In this article, when $E/\mathbb{Q}(T)$ has a torsion point of order $2$, under an assumption on the discriminant of a Weierstrass equation for $E/\mathbb{Q}(T)$, we produce an upper bound for $r_t$ that is valid for infinitely many $t$. We also present two examples of non-isotrivial elliptic curves $E/\mathbb{Q}(T)$ such that $r_t \leq r+1$ for infinitely many $t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02419 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Low rank specializations of elliptic surfaces Melistas, Mentzelos Number Theory Let $E/\mathbb{Q}(T)$ be a non-isotrivial elliptic curve of rank $r$. A theorem due to Silverman implies that the rank $r_t$ of the specialization $E_t/\mathbb{Q}$ is at least $r$ for all but finitely many $t \in \mathbb{Q}$. Moreover, it is conjectured that $r_t \leq r+2$, except for a set of density $0$. In this article, when $E/\mathbb{Q}(T)$ has a torsion point of order $2$, under an assumption on the discriminant of a Weierstrass equation for $E/\mathbb{Q}(T)$, we produce an upper bound for $r_t$ that is valid for infinitely many $t$. We also present two examples of non-isotrivial elliptic curves $E/\mathbb{Q}(T)$ such that $r_t \leq r+1$ for infinitely many $t$. |
| title | Low rank specializations of elliptic surfaces |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.02419 |