Complementary Modules of Weierstrass Canonical Forms
The Weierstrass curve is pointed (, ∞) with a numerical semigroup , which is a normalization of the curve given by the Weierstrass canonical form, ʳ + ₁()ʳ⁻¹ + ₂()ʳ⁻² +⋯+ ᵣ₋₁() + ᵣ() = 0 where each ⱼ is a polynomial in of degree ≤ / for certain coprime positive integers and , < , such that...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2022
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211806 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Complementary Modules of Weierstrass Canonical Forms. Jiryo Komeda, Shigeki Matsutani and Emma Previato. SIGMA 18 (2022), 098, 39 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862739223515234304 |
|---|---|
| author | Komeda, Jiryo Matsutani, Shigeki Previato, Emma |
| author_facet | Komeda, Jiryo Matsutani, Shigeki Previato, Emma |
| citation_txt | Complementary Modules of Weierstrass Canonical Forms. Jiryo Komeda, Shigeki Matsutani and Emma Previato. SIGMA 18 (2022), 098, 39 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The Weierstrass curve is pointed (, ∞) with a numerical semigroup , which is a normalization of the curve given by the Weierstrass canonical form, ʳ + ₁()ʳ⁻¹ + ₂()ʳ⁻² +⋯+ ᵣ₋₁() + ᵣ() = 0 where each ⱼ is a polynomial in of degree ≤ / for certain coprime positive integers and , < , such that the generators of the Weierstrass non-gap sequence at ∞ include and . The Weierstrass curve has the projection ϖᵣ: → ℙ, (, ) ↦ , as a covering space. Let := ⁰(, (∗∞)) and ℙ := ⁰(ℙ, ℙ(∗∞)) whose affine part is ℂ[]. In this paper, for every Weierstrass curve , we show the explicit expression of the complementary module ᶜ of the ℙ-module X as an extension of the expression of the plane Weierstrass curves by Kunz. The extension naturally leads to the explicit expressions of the holomorphic one form ∞, except ⁰(ℙ, ℙ(∗∞)) in terms of . Since for every compact Riemann surface, we find a Weierstrass curve that is bi-rational to the surface, we also comment that the explicit expression of ᶜ naturally leads to the algebraic construction of generalized Weierstrass' sigma functions for every compact Riemann surface and is also connected with the data on how the Riemann surface is embedded into the universal Grassmannian manifolds.
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| first_indexed | 2026-04-17T17:22:35Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211806 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T17:22:35Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Komeda, Jiryo Matsutani, Shigeki Previato, Emma 2026-01-12T10:12:56Z 2022 Complementary Modules of Weierstrass Canonical Forms. Jiryo Komeda, Shigeki Matsutani and Emma Previato. SIGMA 18 (2022), 098, 39 pages 1815-0659 2020 Mathematics Subject Classification: 14H55; 14H50; 16S36; 13H10 arXiv:2207.01905 https://nasplib.isofts.kiev.ua/handle/123456789/211806 https://doi.org/10.3842/SIGMA.2022.098 The Weierstrass curve is pointed (, ∞) with a numerical semigroup , which is a normalization of the curve given by the Weierstrass canonical form, ʳ + ₁()ʳ⁻¹ + ₂()ʳ⁻² +⋯+ ᵣ₋₁() + ᵣ() = 0 where each ⱼ is a polynomial in of degree ≤ / for certain coprime positive integers and , < , such that the generators of the Weierstrass non-gap sequence at ∞ include and . The Weierstrass curve has the projection ϖᵣ: → ℙ, (, ) ↦ , as a covering space. Let := ⁰(, (∗∞)) and ℙ := ⁰(ℙ, ℙ(∗∞)) whose affine part is ℂ[]. In this paper, for every Weierstrass curve , we show the explicit expression of the complementary module ᶜ of the ℙ-module X as an extension of the expression of the plane Weierstrass curves by Kunz. The extension naturally leads to the explicit expressions of the holomorphic one form ∞, except ⁰(ℙ, ℙ(∗∞)) in terms of . Since for every compact Riemann surface, we find a Weierstrass curve that is bi-rational to the surface, we also comment that the explicit expression of ᶜ naturally leads to the algebraic construction of generalized Weierstrass' sigma functions for every compact Riemann surface and is also connected with the data on how the Riemann surface is embedded into the universal Grassmannian manifolds. The second author is grateful to Professor Yohei Komori for valuable discussions in his seminar and to Professor Takao Kato for letting him know about the paper [6]. He has been supported by the Grant-in-Aid for Scientific Research (C) of the Japan Society for the Promotion of Science, no. 21K03289. At the end of May 2022, since this paper was almost completed, the authors decided that it would be submitted at the beginning of July and would go through a month of checks for typos and other errors. In the meantime, Emma Previato, the third author of this paper, passed away on June 29, 2022. The first two authors sincerely wish that the great mathematician and their friend and collaborator, Emma Previato, rest in peace. Thus, the revised version was written only by the first two authors. They thank the anonymous reviewers for their helpful and valuable comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Complementary Modules of Weierstrass Canonical Forms Article published earlier |
| spellingShingle | Complementary Modules of Weierstrass Canonical Forms Komeda, Jiryo Matsutani, Shigeki Previato, Emma |
| title | Complementary Modules of Weierstrass Canonical Forms |
| title_full | Complementary Modules of Weierstrass Canonical Forms |
| title_fullStr | Complementary Modules of Weierstrass Canonical Forms |
| title_full_unstemmed | Complementary Modules of Weierstrass Canonical Forms |
| title_short | Complementary Modules of Weierstrass Canonical Forms |
| title_sort | complementary modules of weierstrass canonical forms |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211806 |
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