The Birch--Swinnerton-Dyer exact formula for quadratic twists of elliptic curves
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866908738584051712 |
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| author | Zhai, Shuai |
| author_facet | Zhai, Shuai |
| contents | In the present paper, we obtain a general lower bound for the $2$-adic valuation of the algebraic part of the central value of the complex $L$-series for the quadratic twists of any elliptic curve over $\mathbb{Q}$, showing that when the $2$-part of the product of Tamagawa factors grows, the $2$-part of the algebraic central $L$-value grows as well, in accordance with the Birch--Swinnerton-Dyer exact formula. This generalises a result of Coates--Kim--Liang--Zhao to all elliptic curves defined over $\mathbb{Q}$. We also prove the existence of an explicit infinite family of quadratic twists with analytic rank $0$ for a large family of elliptic curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_11798 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Birch--Swinnerton-Dyer exact formula for quadratic twists of elliptic curves Zhai, Shuai Number Theory 11F67, 11G05, 11G40 In the present paper, we obtain a general lower bound for the $2$-adic valuation of the algebraic part of the central value of the complex $L$-series for the quadratic twists of any elliptic curve over $\mathbb{Q}$, showing that when the $2$-part of the product of Tamagawa factors grows, the $2$-part of the algebraic central $L$-value grows as well, in accordance with the Birch--Swinnerton-Dyer exact formula. This generalises a result of Coates--Kim--Liang--Zhao to all elliptic curves defined over $\mathbb{Q}$. We also prove the existence of an explicit infinite family of quadratic twists with analytic rank $0$ for a large family of elliptic curves. |
| title | The Birch--Swinnerton-Dyer exact formula for quadratic twists of elliptic curves |
| topic | Number Theory 11F67, 11G05, 11G40 |
| url | https://arxiv.org/abs/2102.11798 |