1998 Fiscal Year Final Research Report Summary
Properties of mapping class groups related to Galois representations
| Project/Area Number |
09640033
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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 | Kyoto Institute of Technology |
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Principal Investigator |
ASADA Mamoru Kyoto Institute of Technology, Faculty of Engineering and Design, associate professor, 工芸学部, 助教授 (30192462)
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| Co-Investigator(Kenkyū-buntansha) |
TSUKAMOTO Chiaki Kyoto Institute of Technology, Faculty of Textile Science, associate professor, 繊維学部, 助教授 (80155340)
YAGASAKI Tatsuhiko Kyoto Institute of Technology, Faculty of Engineering and Design, associate prof, 工芸学部, 助教授 (40191077)
NAKAOKA Akira Kyoto Institute of Technology, Faculty of Engineering and Design, professor, 工芸学部, 教授 (90027920)
MAITANI Fumio Kyoto Institute of Technology, Faculty of Engineering and Design, professor, 工芸学部, 教授 (10029340)
HIROO Miki Kyoto Institute of Technology, Faculty of Engineering and Design, professor, 工芸学部, 教授 (90107368)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Galois representation / fundamental group / mapping class group |
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| Research Abstract |
Let X be a non-singular algebraic curve over a field k of characteristic 0 which is obtained from a complete curve of genus g(<greater than or equal> 0) by removing n(<greater than or equal> 0) k-rational points (2- 2g-n.<0), and 1 be a prime number. The absolute Galois group of k acts naturally on the algebraic fundamental group pi_1^<alg> of X <cross product> k (or pro-1 fundamental group (the maximal pro-l quotient of pi_1^<alg>)) so that we obtain a Galois representation.
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>. Then the algebraic fundamental group of M_<g, n> acts naturally on that of the general fiber so that we have a monodromy representation. (A foundation has been given by T.Oda.) This is the Galois representation in the case that the curve X is the universal curve, k being the function field of M_<g, n>. In this research, we have investigated the faithfulness of this monodromy representation and have shown that, in the case that g = 0, 1, it is faithful for all n.
In investigating Galois representations, one of basic properties of the group pi_1^<alg> (*pi_1^<alg>(g, n) ) is that it is center free (M.P.Anderson). In this research we have shown, by a comparatively elementary way, a property which is stronger than this.
The braid group of a finitely generated free nilpotent pro-I group can be regarded as approximating the genus 0 pro-l mapping class groups. In this research we have shown that this group has a structure an algebraic group which is independent of l.
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Research Products
(8 results)
