Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions
The article is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus 2. It contains new results, as well as a derivation from them of well-known results on these issues. Our research was motivated by applications to the theory of eq...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2022 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2022
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211535 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions. Takanori Ayano and Victor M. Buchstaber. SIGMA 18 (2022), 010, 30 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862730459506540544 |
|---|---|
| author | Ayano, Takanori Buchstaber, Victor M. |
| author_facet | Ayano, Takanori Buchstaber, Victor M. |
| citation_txt | Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions. Takanori Ayano and Victor M. Buchstaber. SIGMA 18 (2022), 010, 30 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The article is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus 2. It contains new results, as well as a derivation from them of well-known results on these issues. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions of genus 2. We consider a hyperelliptic curve of genus 2 that admits a morphism of degree 2 to an elliptic curve. Then there exist two elliptic curves ᵢ, i = 1, 2, and morphisms of degree 2 from to ᵢ. We construct hyperelliptic functions associated with from the Weierstrass elliptic functions associated with ᵢ and describe them in terms of the fundamental hyperelliptic functions defined by the logarithmic derivatives of the two-dimensional sigma functions. We show that the restrictions of hyperelliptic functions associated with to the appropriate subspaces in ℂ² are elliptic functions and describe them in terms of the Weierstrass elliptic functions associated with ᵢ. Further, we express the hyperelliptic functions associated with on ℂ² in terms of the Weierstrass elliptic functions associated with ᵢ. We derive these results by explicitly describing the homomorphisms between the Jacobian varieties of the curves and ᵢ induced by the morphisms from to ᵢ.
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| first_indexed | 2026-04-17T15:03:17Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211535 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T15:03:17Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ayano, Takanori Buchstaber, Victor M. 2026-01-05T12:28:45Z 2022 Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions. Takanori Ayano and Victor M. Buchstaber. SIGMA 18 (2022), 010, 30 pages 1815-0659 2020 Mathematics Subject Classification: 14H40; 14H42; 14K25; 32A20; 33E05 arXiv:2106.06764 https://nasplib.isofts.kiev.ua/handle/123456789/211535 https://doi.org/10.3842/SIGMA.2022.010 The article is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus 2. It contains new results, as well as a derivation from them of well-known results on these issues. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions of genus 2. We consider a hyperelliptic curve of genus 2 that admits a morphism of degree 2 to an elliptic curve. Then there exist two elliptic curves ᵢ, i = 1, 2, and morphisms of degree 2 from to ᵢ. We construct hyperelliptic functions associated with from the Weierstrass elliptic functions associated with ᵢ and describe them in terms of the fundamental hyperelliptic functions defined by the logarithmic derivatives of the two-dimensional sigma functions. We show that the restrictions of hyperelliptic functions associated with to the appropriate subspaces in ℂ² are elliptic functions and describe them in terms of the Weierstrass elliptic functions associated with ᵢ. Further, we express the hyperelliptic functions associated with on ℂ² in terms of the Weierstrass elliptic functions associated with ᵢ. We derive these results by explicitly describing the homomorphisms between the Jacobian varieties of the curves and ᵢ induced by the morphisms from to ᵢ. The authors would like to thank the referees for reading our manuscript carefully and giving useful comments. The work of Takanori Ayano was supported by JSPS KAKENHI Grant Number JP21K03296 and was partly supported by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions Article published earlier |
| spellingShingle | Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions Ayano, Takanori Buchstaber, Victor M. |
| title | Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions |
| title_full | Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions |
| title_fullStr | Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions |
| title_full_unstemmed | Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions |
| title_short | Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions |
| title_sort | relationships between hyperelliptic functions of genus 2 and elliptic functions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211535 |
| work_keys_str_mv | AT ayanotakanori relationshipsbetweenhyperellipticfunctionsofgenus2andellipticfunctions AT buchstabervictorm relationshipsbetweenhyperellipticfunctionsofgenus2andellipticfunctions |