2003 Fiscal Year Final Research Report Summary
Units groups generated by special values of Siegel modular functions
| Project/Area Number |
13640046
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Waseda University |
|---|
Principal Investigator |
KOMATSU Keiichi Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (80092550)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
HASHIMOTO Kiichiro Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (90143370)
|
|---|
| Project Period (FY) |
2001 – 2003
|
| Keywords | abelian extension / Birch-Swinnerton-Dyer conjecture / elliptic curve / unit / Jacobian variety / Siegel modular function |
|---|
| 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.
|
