| Project/Area Number |
13640020
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Kyoto Institute of Technology |
|---|
Principal Investigator |
ASADA Mamoru Kyoto Institute of Technology, Faculty of Engineering and Design, associate professor, 工芸学部, 助教授 (30192462)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
YAGASAKI Tatsuhiko Kyoto Institute of Technology, Faculty of Engineering and Design, associate professor, 工芸学部, 助教授 (40191077)
MAITANI Fumio Kyoto Institute of Technology, Faculty of Engineering and Design, professor, 工芸学部, 教授 (10029340)
MIKI Hiroo Kyoto Institute of Technology, Faculty of Engineering and Design, professor, 工芸学部, 教授 (90107368)
TSUKAMOTO Chiaki Kyoto Institute of Technology, Faculty of Textile Science, associate professor, 繊維学部, 助教授 (80155340)
NAKAOKA Akira Kyoto Institute of Technology, Faculty of Engineering and Design, professor, 工芸学部, 教授 (90027920)
岩塚 明 京都工芸繊維大学, 繊維学部, 教授 (40184890)
|
|---|
| Project Period (FY) |
2001 – 2002
|
| Project Status |
Completed (Fiscal Year 2002)
|
| Budget Amount *help |
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2002: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2001: ¥1,700,000 (Direct Cost: ¥1,700,000)
|
|---|
| Keywords | Galois representation / fundamental group / mapping class group |
|---|
| Research Abstract |
Let us consider the moduli space M_<g, n>/Q (Q : the rationals) of n-pointed complete curves of genus g and the universal family of curves over M_<g, n>. The algebraic fundamental group π^<alg>_1 (M_<g, n>) of M_<g, n> acts naturally on the pro-l fundamental group (l : prime) of the general fiber so that we have a monodromy representation. Let Γ^n_g denote the mapping class group of a n-pointed Riemann surface of genus g. The algebraic fundamental group of M_<g, n> 【cross product】 Q^^- is isomorphic to Γ^^^^n_g (^ : profinite completion). In this research, for the purpose of investigating the monodromy representation in the case that (g, n) = (1, 1), we have tried to determine the weighted completion of π^<alg>_1 (M_<1, 1>). We have applied the general theory of the weighted completion (Hain-Matsumoto) to a former result of Ihara, the structure theorem of the projective limit of l-adic Tate modules of Jacobian varieties of modular curves. This leads us to the determination of weighted
… More
completion of the subgroup π^<alg>_1 (M<1, 1>【cross product】 Q^^-) of π^<alg>_1 (M<1, 1>).
Let X be a non-singular algebraic curve over a field k which is obtained from a complete curve of genus g by removing n k-rational points. In the case 2 - 2g- n < 0, the algebraic fundamental group π^<alg>_1 (【cross product】 k^^-) of 【cross product】 k^^- has the following property ; every subgroup with finite index is centerfree. Whether the group Γ^^^^n_g also has this property or not is an open problem. In order for it to have this poperty, it is necessary that its dense subgroup Γ^n_g also has the same property, and this is known. We have given, under the assumption that n 【greater than or similar】 1, an alternative proof of this fact.
On the other hand, let k be a finite algebraic number field and k_∞ denote the field obtained by adjoining all roots of unity to k. Let M be the maximum unramified Galois extension of k_∞. The Galois group Gal (M/k_∞) is regarded as an analogue, in algebraic number fields, to the group π^<alg>_1 (X 【cross product】 k^^-). In this research, we have shown that Gal (M/k_∞) and Gal (M/k) both have the above property. Less
|
