Iwasawa Main Conjecture for ordinary semistable elliptic curves over global function fields
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912963100672000 |
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| author | Tan, Ki-Seng Trihan, Fabien Tsoi, Kwok-Wing |
| author_facet | Tan, Ki-Seng Trihan, Fabien Tsoi, Kwok-Wing |
| contents | Let $A$ be an ordinary elliptic curve over a global function field $K$ of characteristic $p$, assumed semistable at every place, and let $L/K$ be a $\mathbb{Z}_p^d$-extension ramified only at finitely many places where $A$ has ordinary reduction. Building on the framework of [Tan26] (arXiv:2603.10576), we prove the Iwasawa Main Conjecture for $A$ over $L$, subject to a technical $μ$-invariant hypothesis that is already detected after specialization to the unramified $\mathbb{Z}_p$-extension. The principal new input is a `$χ$-formula' that compares appropriate $χ$-isotypic characteristic ideals of Selmer modules with the corresponding specializations of the $p$-adic $L$-function. Finally, to show that our $μ$-hypothesis is non-vacuous, we prove, for $p>3$, that the hypothesis holds on a Zariski open dense locus in the moduli of semistable elliptic curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11615 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Iwasawa Main Conjecture for ordinary semistable elliptic curves over global function fields Tan, Ki-Seng Trihan, Fabien Tsoi, Kwok-Wing Number Theory Algebraic Geometry 11R23, 11G10, 11S40, 14J27 Let $A$ be an ordinary elliptic curve over a global function field $K$ of characteristic $p$, assumed semistable at every place, and let $L/K$ be a $\mathbb{Z}_p^d$-extension ramified only at finitely many places where $A$ has ordinary reduction. Building on the framework of [Tan26] (arXiv:2603.10576), we prove the Iwasawa Main Conjecture for $A$ over $L$, subject to a technical $μ$-invariant hypothesis that is already detected after specialization to the unramified $\mathbb{Z}_p$-extension. The principal new input is a `$χ$-formula' that compares appropriate $χ$-isotypic characteristic ideals of Selmer modules with the corresponding specializations of the $p$-adic $L$-function. Finally, to show that our $μ$-hypothesis is non-vacuous, we prove, for $p>3$, that the hypothesis holds on a Zariski open dense locus in the moduli of semistable elliptic curves. |
| title | Iwasawa Main Conjecture for ordinary semistable elliptic curves over global function fields |
| topic | Number Theory Algebraic Geometry 11R23, 11G10, 11S40, 14J27 |
| url | https://arxiv.org/abs/2603.11615 |