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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2308.10241 |
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| _version_ | 1866909515491835904 |
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| author | Kaya, Enis Maex, Michaël Waeterschoot, Art |
| author_facet | Kaya, Enis Maex, Michaël Waeterschoot, Art |
| contents | Given a smooth, proper curve $C$ over a discretely valued field $k$, we equip the $k$-vector space $H^{0}(C,ω_{C/k})$ with a canonical discrete valuation $v_{\mathrm{can}}$ which measures how canonical forms degenerate on regular integral models of $C$. More precisely, $v_{\mathrm{can}}$ maps a canonical form to the minimal value of its associated weight function, as introduced by Mustaţă--Nicaise. Our main result states that $v_{\mathrm{can}}$ computes Edixhoven's jumps of the Jacobian of $C$ when evaluated in an orthogonal basis. As a byproduct, we deduce a short proof for the rationality of the jumps of Jacobians. We also show how $v_{\mathrm{can}}$ and the jumps can be computed efficiently for the class of $Δ_v$-regular curves introduced by Dokchitser. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10241 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Jumps of Jacobians via orthogonal canonical forms Kaya, Enis Maex, Michaël Waeterschoot, Art Algebraic Geometry Number Theory 14H25 (Primary) 14D10, 14E22 (Secundary) Given a smooth, proper curve $C$ over a discretely valued field $k$, we equip the $k$-vector space $H^{0}(C,ω_{C/k})$ with a canonical discrete valuation $v_{\mathrm{can}}$ which measures how canonical forms degenerate on regular integral models of $C$. More precisely, $v_{\mathrm{can}}$ maps a canonical form to the minimal value of its associated weight function, as introduced by Mustaţă--Nicaise. Our main result states that $v_{\mathrm{can}}$ computes Edixhoven's jumps of the Jacobian of $C$ when evaluated in an orthogonal basis. As a byproduct, we deduce a short proof for the rationality of the jumps of Jacobians. We also show how $v_{\mathrm{can}}$ and the jumps can be computed efficiently for the class of $Δ_v$-regular curves introduced by Dokchitser. |
| title | Jumps of Jacobians via orthogonal canonical forms |
| topic | Algebraic Geometry Number Theory 14H25 (Primary) 14D10, 14E22 (Secundary) |
| url | https://arxiv.org/abs/2308.10241 |