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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2210.02013 |
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| _version_ | 1866910642522292224 |
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| author | Emerton, Matthew Pollack, Robert Weston, Tom |
| author_facet | Emerton, Matthew Pollack, Robert Weston, Tom |
| contents | We prove that the Mazur-Tate elements of an eigenform $f$ sit inside the Fitting ideals of the corresponding dual Selmer groups along the cyclotomic $\mathbb Z_p$-extension (up to scaling by a single constant). Our method begins with the construction of local cohomology classes built via the $p$-adic local Langlands correspondence. From these classes, we build algebraic analogues of the Mazur-Tate elements which we directly verify sit in the appropriate Fitting ideals. Using Kato's Euler system and explicit reciprocity laws, we prove that these algebraic elements divide the corresponding Mazur-Tate elements, implying our theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_02013 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Explicit reciprocity laws and Iwasawa theory for modular forms Emerton, Matthew Pollack, Robert Weston, Tom Number Theory We prove that the Mazur-Tate elements of an eigenform $f$ sit inside the Fitting ideals of the corresponding dual Selmer groups along the cyclotomic $\mathbb Z_p$-extension (up to scaling by a single constant). Our method begins with the construction of local cohomology classes built via the $p$-adic local Langlands correspondence. From these classes, we build algebraic analogues of the Mazur-Tate elements which we directly verify sit in the appropriate Fitting ideals. Using Kato's Euler system and explicit reciprocity laws, we prove that these algebraic elements divide the corresponding Mazur-Tate elements, implying our theorem. |
| title | Explicit reciprocity laws and Iwasawa theory for modular forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2210.02013 |