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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2104.09200 |
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| _version_ | 1866910344378580992 |
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| author | Colmez, Pierre Wang, Shanwen |
| author_facet | Colmez, Pierre Wang, Shanwen |
| contents | We show that the modular symbol $(0,\infty)$, considered as an element of the dual of Emerton's completed cohomology, interpolates Kato's Euler system at classical points, and we deduce from this a factorisation of Beilinson-Kato's system as a product of two symbols $(0,\infty)$ (an algebraic analog of Rankin's method). The proof uses the $p$-adic local Langlands correspondence for ${\mathbf GL}_2({\mathbf Q}_p)$ and Emerton's factorization of the completed cohomology of the tower of modular curves for which we provide a new proof resting upon the construction of a Kirillov model for the completed cohomology, and which we refine by imposing conditions at classical points; the existence of such a refinement is a manifestation of an analyticity property for $p$-adic periods of modular forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_09200 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Une factorisation de la cohomologie complétée et du système de Beilinson-Kato Colmez, Pierre Wang, Shanwen Number Theory We show that the modular symbol $(0,\infty)$, considered as an element of the dual of Emerton's completed cohomology, interpolates Kato's Euler system at classical points, and we deduce from this a factorisation of Beilinson-Kato's system as a product of two symbols $(0,\infty)$ (an algebraic analog of Rankin's method). The proof uses the $p$-adic local Langlands correspondence for ${\mathbf GL}_2({\mathbf Q}_p)$ and Emerton's factorization of the completed cohomology of the tower of modular curves for which we provide a new proof resting upon the construction of a Kirillov model for the completed cohomology, and which we refine by imposing conditions at classical points; the existence of such a refinement is a manifestation of an analyticity property for $p$-adic periods of modular forms. |
| title | Une factorisation de la cohomologie complétée et du système de Beilinson-Kato |
| topic | Number Theory |
| url | https://arxiv.org/abs/2104.09200 |