Tamagawa number conjecture for CM modular forms and Rankin--Selberg convolutions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916980812939264 |
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| author | Castella, Francesc |
| author_facet | Castella, Francesc |
| contents | Let $E/F$ be an elliptic curve defined over a number field $F$ with complex multiplication by the ring of integers of an imaginary quadratic field $K$ such that the torsion points of $E$ generate over $F$ an abelian extension of $K$. In this paper we prove the $p$-part of the Birch--Swinnerton-Dyer formula for $E/F$ in analytic rank $1$ for primes $p>3$ split in $K$. This was previously known for $F=\mathbb{Q}$ by work of Rubin as a consequence of his proof of Mazur's Main Conjecture for rational CM elliptic curves, but the problem for $[F:\mathbb{Q}]>1$ remained wide open.
The approach introduced in this paper also yields a proof of similar results for CM abelian varieties $A/K$ and for CM modular forms, as well as an analogue in this setting of Skinner's $p$-converse to the theorem of Gross--Zagier and Kolyvagin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_11891 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tamagawa number conjecture for CM modular forms and Rankin--Selberg convolutions Castella, Francesc Number Theory Let $E/F$ be an elliptic curve defined over a number field $F$ with complex multiplication by the ring of integers of an imaginary quadratic field $K$ such that the torsion points of $E$ generate over $F$ an abelian extension of $K$. In this paper we prove the $p$-part of the Birch--Swinnerton-Dyer formula for $E/F$ in analytic rank $1$ for primes $p>3$ split in $K$. This was previously known for $F=\mathbb{Q}$ by work of Rubin as a consequence of his proof of Mazur's Main Conjecture for rational CM elliptic curves, but the problem for $[F:\mathbb{Q}]>1$ remained wide open. The approach introduced in this paper also yields a proof of similar results for CM abelian varieties $A/K$ and for CM modular forms, as well as an analogue in this setting of Skinner's $p$-converse to the theorem of Gross--Zagier and Kolyvagin. |
| title | Tamagawa number conjecture for CM modular forms and Rankin--Selberg convolutions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.11891 |