On the analytic rank of the twin prime elliptic curve $y^2=x(x-2)(x-p)$
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| Format: | Preprint |
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2025
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| _version_ | 1866912540980674560 |
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| author | Joshi, Kirti |
| author_facet | Joshi, Kirti |
| contents | Let $p\geq 7$ and suppose $(p,p-2)$ are twin prime numbers, in [Hatley, 2009], the elliptic curve $E_p:y^2=x(x-2)(x-p)$ was considered in the context of a conjecture by Jason Beers about the Mordell-Weil ranks of $E_p/\mathbb{Q}$. I show that for $p\equiv 3,5\bmod 8$, the analytic rank of $E_p$ is at least one (Theorem 1.1.2) in line with Beers' predictions. This is done by finding a formula (Theorem 4.1.1) for the global root number of $E_p$ for all twin prime pairs. I also show that Beers' conjecture, that for $p\equiv 1\bmod 8$ the rank of $E_p$ is two, is false as stated because $E_{73}$ has rank zero. In the light of Theorem 4.1.1, Beers' conjecture needs to be modified: if $p\equiv 1\bmod 8$ then the rank of $E_p$ is zero or two (Conjecture 5.3.1). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12340 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the analytic rank of the twin prime elliptic curve $y^2=x(x-2)(x-p)$ Joshi, Kirti Number Theory Algebraic Geometry Let $p\geq 7$ and suppose $(p,p-2)$ are twin prime numbers, in [Hatley, 2009], the elliptic curve $E_p:y^2=x(x-2)(x-p)$ was considered in the context of a conjecture by Jason Beers about the Mordell-Weil ranks of $E_p/\mathbb{Q}$. I show that for $p\equiv 3,5\bmod 8$, the analytic rank of $E_p$ is at least one (Theorem 1.1.2) in line with Beers' predictions. This is done by finding a formula (Theorem 4.1.1) for the global root number of $E_p$ for all twin prime pairs. I also show that Beers' conjecture, that for $p\equiv 1\bmod 8$ the rank of $E_p$ is two, is false as stated because $E_{73}$ has rank zero. In the light of Theorem 4.1.1, Beers' conjecture needs to be modified: if $p\equiv 1\bmod 8$ then the rank of $E_p$ is zero or two (Conjecture 5.3.1). |
| title | On the analytic rank of the twin prime elliptic curve $y^2=x(x-2)(x-p)$ |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2508.12340 |