2001 Fiscal Year Final Research Report Summary
The study of modular equations and elliptic curves.
| Project/Area Number |
12640007
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Akita University |
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Principal Investigator |
ITO Hideji Akita University, Faculty of Education and Human Studies, Assistant Professor, 教育文化学部, 助教授 (70091783)
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| Co-Investigator(Kenkyū-buntansha) |
FUKUHARA Kenzo Akita University, Faculty of Education and Human Studies, Assistant Professor, 教育文化学部, 助教授 (00006561)
TATEOKA Atsushi Akita University, Faculty of Education and Human Studies, Professor, 教育文化学部, 教授 (40006565)
UDA Toshio Akita University, Faculty of Education and Human Studies, Professor, 教育文化学部, 教授 (20006589)
TORISU Ichiro Akita University, Faculty of Education and Human Studies, Assistant, 工学資源学部, 助手 (50323134)
MIKAMI Kentaro Akita University, Faculty of Education and Human Studies, Professor, 工学資源学部, 教授 (70006592)
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| Project Period (FY) |
2000 – 2001
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| Keywords | elliptic modular function / modular equation / modular polyromial / elliptic curve / complex multipluction / monster simple group / one-parametrization |
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| Research Abstract |
1. We obtain the explicit forms of the modular polynomials Φ_p(X, Y) of j(Z) of prime order p as far as p = 199.
2. The modular function corresponding to a conjugacy class of the monster simple group hs modular eguations. We compute explicit forms of several of them and find that behaviours of their coefficients when reducing modulo p^2 are much more complicated than previously imagined.
3. We discover that the solutions of the one-parametrization of the modular ezuation Φ_p(X, Y^p) = 0 have some peculiar properties (the shape of their distribution is very striking and the unique real root seems to converge to the ??? -5.54590087【triple bond】 when the order becomes big.)
4. We discover some rules of the vanishing of coefficients of the modular polynomials of j(Z).
5. We clarify the decompusition and the number of real roots of Φ_p(X, j) = 0 where j is a j-invariant of CM curve / Q.
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