A Universal Genus-Two Curve from Siegel Modular Forms
Let p be any point in the moduli space of genus-two curves M2 and K its field of moduli. We provide a universal equation of a genus-two curve Cα,β defined over K(α,β), corresponding to p, where α and β satisfy a quadratic α²+bβ²=c such that b and c are given in terms of ratios of Siegel modular form...
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| Datum: | 2017 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2017
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| Schriftenreihe: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/149268 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A Universal Genus-Two Curve from Siegel Modular Forms / A. Malmendier, T. Shaska // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 19 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Zusammenfassung: | Let p be any point in the moduli space of genus-two curves M2 and K its field of moduli. We provide a universal equation of a genus-two curve Cα,β defined over K(α,β), corresponding to p, where α and β satisfy a quadratic α²+bβ²=c such that b and c are given in terms of ratios of Siegel modular forms. The curve Cα,β is defined over the field of moduli K if and only if the quadratic has a K-rational point (α,β). We discover some interesting symmetries of the Weierstrass equation of Cα,β. This extends previous work of Mestre and others. |
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