2018 Fiscal Year Annual Research Report
Research on characterization of the fundamental function sigma in the theory of Abelian functions via heat equations and general addition formulae
| Project/Area Number |
16K05082
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| Research Institution | Meijo University |
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Principal Investigator |
大西 良博 名城大学, 理工学部, 教授 (60250643)
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | algebraic curves / Jacobian varieties / sigma functions / heat equations / Abelian functions |
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| Outline of Annual Research Achievements |
I researched heat equations satisfied by the sigma function attached to any plane telescopic curve of genus there or smaller, which is a natural generalization of Weierstrass' original work and might be a foundational research in the theory of Abelian functions. This research was done in collaboration with J.C.Eilbeck, J.Gibbons, and S.Yasuda. I and coworkers wrote up a paper which includes a detailed exposition of the pioneering work by Buchetaber-Leykin which is published as a short paper in 2008, and gave a proof of one-dimensionality of the solution space of the system of heat equations satisfied by the sigma function associated to the given curve. Our paper was submitted for publiation and is under refereed. Moreover, I and Eilbeck did another improvement on the theory, in which the we shown that BL-theory is applicable for any plane telescopic curves of genus three or smaller without transforming to Weierstrass form. I and Eilbeck submitted a paper which shows this result for genus one case and was eventually accepted. We two will write another paper on higher genus case as well.
Besides the works above, I gave some investigation on the Hurwitz integrality on the power series expansion of the sigma function for any plane telescopic curve, which includes an ultimate result. This work was published as a paper in Proceedings of Edinburgh Mathematical Society.
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