Iwasawa Main Conjecture for Supersingular Elliptic Curves and BSD conjecture
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2014
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| _version_ | 1866917770855186432 |
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| author | Wan, Xin |
| author_facet | Wan, Xin |
| contents | In this paper we prove the $\pm$-main conjecture of Iwasawa theory formulated by Kobayashi for elliptic curves with supersingular reduction at an odd prime $p$ such that $a_p=0$, using a key new observation that it can be reduced to another Iwasawa-Greenberg main conjecture, which is more accessible and proved here as a first step. Then we develop some generalized $\pm$ local theory and deduce the main conjecture. The argument uses in an essential way the recent study on explicit reciprocity law for Beilinson-Flach elements by Kings-Loeffler-Zerbes. We also prove as corollaries the $p$-part of the BSD formula at supersingular primes when the analytic rank is $0$ or $1$. The main result enables us to present in the Appendix a number of explicit infinite families of elliptic curves without complex multiplications for which we can now prove the full Birch-Swinnerton-Dyer conjecture. No such infinite families of curves without complex multiplication were known previously. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1411_6352 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Iwasawa Main Conjecture for Supersingular Elliptic Curves and BSD conjecture Wan, Xin Number Theory In this paper we prove the $\pm$-main conjecture of Iwasawa theory formulated by Kobayashi for elliptic curves with supersingular reduction at an odd prime $p$ such that $a_p=0$, using a key new observation that it can be reduced to another Iwasawa-Greenberg main conjecture, which is more accessible and proved here as a first step. Then we develop some generalized $\pm$ local theory and deduce the main conjecture. The argument uses in an essential way the recent study on explicit reciprocity law for Beilinson-Flach elements by Kings-Loeffler-Zerbes. We also prove as corollaries the $p$-part of the BSD formula at supersingular primes when the analytic rank is $0$ or $1$. The main result enables us to present in the Appendix a number of explicit infinite families of elliptic curves without complex multiplications for which we can now prove the full Birch-Swinnerton-Dyer conjecture. No such infinite families of curves without complex multiplication were known previously. |
| title | Iwasawa Main Conjecture for Supersingular Elliptic Curves and BSD conjecture |
| topic | Number Theory |
| url | https://arxiv.org/abs/1411.6352 |