Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866912043059118080 |
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| author | Caro, Jerson |
| author_facet | Caro, Jerson |
| contents | Watkins' conjecture asserts that the rank of an elliptic curve is upper bounded by the $2$-adic valuation of its modular degree. We show that this conjecture is satisfied when $E$ is any quadratic twist of an elliptic curve with rational $2$-torsion and prime power conductor. Furthermore, we give a lower bound of the congruence number for elliptic curves of the form $y^2=x^3-dx$, with $d$ a biquadratefree integer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_10008 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor Caro, Jerson Number Theory Watkins' conjecture asserts that the rank of an elliptic curve is upper bounded by the $2$-adic valuation of its modular degree. We show that this conjecture is satisfied when $E$ is any quadratic twist of an elliptic curve with rational $2$-torsion and prime power conductor. Furthermore, we give a lower bound of the congruence number for elliptic curves of the form $y^2=x^3-dx$, with $d$ a biquadratefree integer. |
| title | Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor |
| topic | Number Theory |
| url | https://arxiv.org/abs/2206.10008 |