Building Abelian Functions with Generalised Baker-Hirota Operators
We present a new systematic method to construct Abelian functions on Jacobian varieties of plane, algebraic curves. The main tool used is a symmetric generalisation of the bilinear operator defined in the work of Baker and Hirota. We give explicit formulae for the multiple applications of the operat...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2012 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2012
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/148444 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Building Abelian Functions with Generalised Baker-Hirota Operators / M. England, Ch. Athorne // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 36 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862589217253621760 |
|---|---|
| author | England, M. Athorne, C. |
| author_facet | England, M. Athorne, C. |
| citation_txt | Building Abelian Functions with Generalised Baker-Hirota Operators / M. England, Ch. Athorne // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 36 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present a new systematic method to construct Abelian functions on Jacobian varieties of plane, algebraic curves. The main tool used is a symmetric generalisation of the bilinear operator defined in the work of Baker and Hirota. We give explicit formulae for the multiple applications of the operators, use them to define infinite sequences of Abelian functions of a prescribed pole structure and deduce the key properties of these functions. We apply the theory on the two canonical curves of genus three, presenting new explicit examples of vector space bases of Abelian functions. These reveal previously unseen similarities between the theories of functions associated to curves of the same genus.
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| first_indexed | 2025-11-27T02:27:54Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-148444 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-27T02:27:54Z |
| publishDate | 2012 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | England, M. Athorne, C. 2019-02-18T12:25:23Z 2019-02-18T12:25:23Z 2012 Building Abelian Functions with Generalised Baker-Hirota Operators / M. England, Ch. Athorne // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 36 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14H40; 14H50; 14H70 DOI: http://dx.doi.org/10.3842/SIGMA.2012.037 https://nasplib.isofts.kiev.ua/handle/123456789/148444 We present a new systematic method to construct Abelian functions on Jacobian varieties of plane, algebraic curves. The main tool used is a symmetric generalisation of the bilinear operator defined in the work of Baker and Hirota. We give explicit formulae for the multiple applications of the operators, use them to define infinite sequences of Abelian functions of a prescribed pole structure and deduce the key properties of these functions. We apply the theory on the two canonical curves of genus three, presenting new explicit examples of vector space bases of Abelian functions. These reveal previously unseen similarities between the theories of functions associated to curves of the same genus. We acknowledge Mr. Lachlan Walker who contributed some preliminary work to proving that
 Q-functions are Abelian. In particular he proved equation (5.2), and derived a formula for the
 8-index Q-functions in terms of ℘-functions which helped motivate equation (5.7). We would
 also like to thank the two anonymous referees for their useful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Building Abelian Functions with Generalised Baker-Hirota Operators Article published earlier |
| spellingShingle | Building Abelian Functions with Generalised Baker-Hirota Operators England, M. Athorne, C. |
| title | Building Abelian Functions with Generalised Baker-Hirota Operators |
| title_full | Building Abelian Functions with Generalised Baker-Hirota Operators |
| title_fullStr | Building Abelian Functions with Generalised Baker-Hirota Operators |
| title_full_unstemmed | Building Abelian Functions with Generalised Baker-Hirota Operators |
| title_short | Building Abelian Functions with Generalised Baker-Hirota Operators |
| title_sort | building abelian functions with generalised baker-hirota operators |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148444 |
| work_keys_str_mv | AT englandm buildingabelianfunctionswithgeneralisedbakerhirotaoperators AT athornec buildingabelianfunctionswithgeneralisedbakerhirotaoperators |