Zeta elements for elliptic curves and applications
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910600947302400 |
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| author | Burungale, Ashay Skinner, Christopher Tian, Ye Wan, Xin |
| author_facet | Burungale, Ashay Skinner, Christopher Tian, Ye Wan, Xin |
| contents | Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with conductor $N$ and $p\nmid 2N$ a prime. Let $L$ be an imaginary quadratic field with $p$ split. We prove the existence of $p$-adic zeta element for $E$ over $L$, encoding two different $p$-adic $L$-functions associated to $E$ over $L$ via explicit reciprocity laws at the primes above $p$. We formulate a main conjecture for $E$ over $L$ in terms of the zeta element, mediating different main conjectures in which the $p$-adic $L$-functions appear, and prove some results toward them.
The zeta element has various applications to the arithmetic of elliptic curves. This includes a proof of main conjecture for semistable elliptic curves $E$ over $\mathbb{Q}$ at supersingular primes $p$, as conjectured by Kobayashi in 2002. It leads to the $p$-part of the conjectural Birch and Swinnerton-Dyer (BSD) formula for such curves of analytic rank zero or one, and enables us to present the first infinite families of non-CM elliptic curves for which the BSD conjecture is true. We provide further evidence towards the BSD conjecture: new cases of $p$-converse to the Gross--Zagier and Kolyvagin theorem, and $p$-part of the BSD formula for ordinary primes $p$. Along the way, we give a proof of a conjecture of Perrin-Riou connecting Beilinson--Kato elements with rational points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01350 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zeta elements for elliptic curves and applications Burungale, Ashay Skinner, Christopher Tian, Ye Wan, Xin Number Theory Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with conductor $N$ and $p\nmid 2N$ a prime. Let $L$ be an imaginary quadratic field with $p$ split. We prove the existence of $p$-adic zeta element for $E$ over $L$, encoding two different $p$-adic $L$-functions associated to $E$ over $L$ via explicit reciprocity laws at the primes above $p$. We formulate a main conjecture for $E$ over $L$ in terms of the zeta element, mediating different main conjectures in which the $p$-adic $L$-functions appear, and prove some results toward them. The zeta element has various applications to the arithmetic of elliptic curves. This includes a proof of main conjecture for semistable elliptic curves $E$ over $\mathbb{Q}$ at supersingular primes $p$, as conjectured by Kobayashi in 2002. It leads to the $p$-part of the conjectural Birch and Swinnerton-Dyer (BSD) formula for such curves of analytic rank zero or one, and enables us to present the first infinite families of non-CM elliptic curves for which the BSD conjecture is true. We provide further evidence towards the BSD conjecture: new cases of $p$-converse to the Gross--Zagier and Kolyvagin theorem, and $p$-part of the BSD formula for ordinary primes $p$. Along the way, we give a proof of a conjecture of Perrin-Riou connecting Beilinson--Kato elements with rational points. |
| title | Zeta elements for elliptic curves and applications |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.01350 |