Kolyvagin's conjecture for modular forms at non-ordinary primes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912272571432960 |
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| author | Da Ronche, Enrico |
| author_facet | Da Ronche, Enrico |
| contents | In this article we prove a version of Kolyvagin's conjecture for modular forms at non-ordinary primes. In particular, we generalize the work of Wang on a converse to a higher weight Gross-Zagier-Kolyvagin theorem in order to prove the conjecture under the hypothesis that some Selmer group has rank one. The main ingredients that we use in non-ordinary setting are the signed Selmer groups introduced by Lei, Loeffler and Zerbes. We will also use a result of Wan, i.e., the $p$-part of the Tamagawa number conjecture for non-ordinary modular forms with analytic rank zero. Starting from the rank one case we will show how to prove the full version of the conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09955 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kolyvagin's conjecture for modular forms at non-ordinary primes Da Ronche, Enrico Number Theory 11F11, 14C15 In this article we prove a version of Kolyvagin's conjecture for modular forms at non-ordinary primes. In particular, we generalize the work of Wang on a converse to a higher weight Gross-Zagier-Kolyvagin theorem in order to prove the conjecture under the hypothesis that some Selmer group has rank one. The main ingredients that we use in non-ordinary setting are the signed Selmer groups introduced by Lei, Loeffler and Zerbes. We will also use a result of Wan, i.e., the $p$-part of the Tamagawa number conjecture for non-ordinary modular forms with analytic rank zero. Starting from the rank one case we will show how to prove the full version of the conjecture. |
| title | Kolyvagin's conjecture for modular forms at non-ordinary primes |
| topic | Number Theory 11F11, 14C15 |
| url | https://arxiv.org/abs/2503.09955 |