| Project/Area Number |
13640046
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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 | Waseda University |
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Principal Investigator |
KOMATSU Keiichi Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (80092550)
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| Co-Investigator(Kenkyū-buntansha) |
HASHIMOTO Kiichiro Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (90143370)
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| Project Period (FY) |
2001 – 2003
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| Project Status |
Completed (Fiscal Year 2003)
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| Budget Amount *help |
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
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| Keywords | abelian extension / Birch-Swinnerton-Dyer conjecture / elliptic curve / unit / Jacobian variety / Siegel modular function / ray class field / Siegel modular function / L-関数 / 単数群 / 類数 / Kroneckerの極限公式 |
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| Research Abstract |
In 1976, Coates and Wiles gave large improvement to Birch-Swinnerton-Dyer conjecture for some elliptic curves with complex multiplication by using elliptic units in abelian extensions of imaginary quadratic fields. Main purpose of car investigation is to consider Birch-Swinnerton-Dyer conjecture of the Jacobian variety of some genus-2-curves with complex multiplications.
In our investigation, we obtained the following :
We put ζ=e^(2πi)/(13) and α=5+5^3+5^9. Then the field k=Q(α) is the CM-field
corresponding to the Jacobian variety J(C) of the curve C :
y^2=x^5-156x^4+10816x^3-421824x^2+8998912x-8042776.
We construct unit groups in abelian extensions of k by special values of Siegel modular functions at a CM-point corresponding to J(C). moreover we write the values of Hecke L-functions associated to the above abelian fields using units given by Siegel modular functions.
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