On Frobenius' Theta Formula
Mumford's well-known characterization of the hyperelliptic locus of the moduli space of ppavs in terms of vanishing and non-vanishing theta constants is based on Neumann's dynamical system. Poor's approach to the characterization uses the cross ratio. A key tool in both methods is Fro...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2020 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2020
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210693 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On Frobenius' Theta Formula. Alessio Fiorentino and Riccardo Salvati Manni. SIGMA 16 (2020), 057, 14 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862577027336372224 |
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| author | Fiorentino, Alessio Salvati Manni, Riccardo |
| author_facet | Fiorentino, Alessio Salvati Manni, Riccardo |
| citation_txt | On Frobenius' Theta Formula. Alessio Fiorentino and Riccardo Salvati Manni. SIGMA 16 (2020), 057, 14 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Mumford's well-known characterization of the hyperelliptic locus of the moduli space of ppavs in terms of vanishing and non-vanishing theta constants is based on Neumann's dynamical system. Poor's approach to the characterization uses the cross ratio. A key tool in both methods is Frobenius' theta formula, which follows from Riemann's theta formula. In a 2004 paper, Grushevsky gives a different characterization in terms of cubic equations in second-order theta functions. In this note, we first show the connection between the methods by proving that Grushevsky's cubic equations are strictly related to Frobenius' theta formula, and we then give a new proof of Mumford's characterization via Gunning's multisecant formula.
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| first_indexed | 2025-12-17T12:03:39Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-210693 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:03:39Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Fiorentino, Alessio Salvati Manni, Riccardo 2025-12-15T15:18:28Z 2020 On Frobenius' Theta Formula. Alessio Fiorentino and Riccardo Salvati Manni. SIGMA 16 (2020), 057, 14 pages 1815-0659 2020 Mathematics Subject Classification: 14H42; 14H45; 14K25; 14K12; 14H40 arXiv:2004.05099 https://nasplib.isofts.kiev.ua/handle/123456789/210693 https://doi.org/10.3842/SIGMA.2020.057 Mumford's well-known characterization of the hyperelliptic locus of the moduli space of ppavs in terms of vanishing and non-vanishing theta constants is based on Neumann's dynamical system. Poor's approach to the characterization uses the cross ratio. A key tool in both methods is Frobenius' theta formula, which follows from Riemann's theta formula. In a 2004 paper, Grushevsky gives a different characterization in terms of cubic equations in second-order theta functions. In this note, we first show the connection between the methods by proving that Grushevsky's cubic equations are strictly related to Frobenius' theta formula, and we then give a new proof of Mumford's characterization via Gunning's multisecant formula. The authors would like to thank Bert van Geemen for drawing their attention to the result in [5]. They are also grateful to Sam Grushevsky for many helpful discussions and explanations. The authors are greatly indebted to an anonymous referee for the careful reading and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Frobenius' Theta Formula Article published earlier |
| spellingShingle | On Frobenius' Theta Formula Fiorentino, Alessio Salvati Manni, Riccardo |
| title | On Frobenius' Theta Formula |
| title_full | On Frobenius' Theta Formula |
| title_fullStr | On Frobenius' Theta Formula |
| title_full_unstemmed | On Frobenius' Theta Formula |
| title_short | On Frobenius' Theta Formula |
| title_sort | on frobenius' theta formula |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210693 |
| work_keys_str_mv | AT fiorentinoalessio onfrobeniusthetaformula AT salvatimanniriccardo onfrobeniusthetaformula |