| Co-Investigator(Kenkyū-buntansha) |
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)
MIKI Hiroo Kyoto Institute of Technology, Faculty of Engineering and Design, professor, 工芸学部, 教授 (90107368)
CHIAKI Tsukamoto 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 professor, 工芸学部, 助教授 (40191077)
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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 l be a prime number. The absolute Galois group of k acts naturally on the algebraic fundamental group π^<alg>_1 of X【cross product】k^^- (or pro-l fundamental group (the maximal pro-l quotient of π^<alg>_1)) 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>. Let π_1(g, n) and Γ^n_g denote the fundamental group and the ma
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pping class group of a Riemann surface of genus g (【greater than or equal】 0) with n (【greater than or equal】 0) punctures respectively. Then the algebraic fundamental group of M_<g, n> 【cross product】 Q^^- and that of the general fiber are isomorphic to Γ^^<^>^n_g and π^^<^>_1 (g, n) respectively (^ : profinite completion). The natural action ρ_<g, n> of Γ^^<^>^n_g on π^^<^>_1 (g, n) is nothing but the (geometric part of) the monodromy representation. The group Γ^^<^>^n_g acts also on the pro-l fundamental group π^<(l)>_1(g, n), which is the pro-l completion of π_1(g, n), and we obtain a Galois representation ρ^<(l)>_<g, n>. In this research, we have investigated the kernels of the representations ρ_<g, n> and ρ^<(l)>_<g, n>. So far, the kernel of ρ^<(l)>_<g, n> has been known only in the case of g=0. In the case that l=2, by applying the method to prove the faithfulness of ρ_<1, 1>, the kernel of ρ^<(2)>_<1, 1> has been determined.
On the other hand, whether the center of any open subgroup of Γ^^<^>^n_g is trivial or not is an open problem. (This is related to whether M_<g, n > is "anabelian" or not). We have shown that, if the representation ρ_<g, n> is faithful, then the center of any open subgroup of Γ^^<^>^<n+1>_g is trivial. Less
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