Properties of mapping class groups rolated to Gdois representations

设X是特征为0的域k上的一条非奇异代数曲线，它是亏格g([大于等于]0)的一条完备曲线去掉n([大于等于]0)k-有理点(2-2g-n&lt；0)而得到的，L是素数。K的绝对伽罗瓦群自然作用于X[叉积]k^-的代数基本群π^&lt；alg&gt；_1(或原L基本群(π^&lt；alg&gt；_1的极大原L商))，从而得到伽罗瓦表示.让我们考虑亏格的n点完全曲线的模空间M_&lt；g，n&gt；/q(q：有理数)和M_&lt；g，n&gt；上的泛曲线族.然后，M&lt；g，n&gt；的代数基本群自然地作用于一般纤维的代数基本群，因此我们有一个单行表示。(T.Oda已经提供了一个基金会。)这是在曲线X是泛曲线的情况下的伽罗瓦表示，k是M&lt；g，n&gt；的函数域。设π_1(g，n)和Γ^n_g表示基本群和ma…亏格g([大于或等于]0)具有n([大于或等于]0)个穿孔的Riemann曲面的多个p类群。则M_&lt；g，n&gt；[叉积]q^-的代数基本群和一般纤维的代数基本群分别同构于Γ^&lt；^&gt；^n_g和π^^&lt；^&gt；_1(g，n)(^：超定完备化)。ρ^&lt；g，n&gt；的Γ^&lt；g，n&gt；在π^&lt；^&gt；_1(g，n)上的自然作用只不过是单边表示的(几何部分)。群Γ^^&lt；^&gt；^n_g也作用于亲L基本群π^&lt；(L)&gt；_1(g，n)，它是π_1(g，n)的亲L完成，我们得到了ρ^&lt；(L)&gt；_&lt；g，n&gt；在本研究中，我们研究了ρ_&lt；g，n&gt；和ρ^&lt；(L)&gt；_&lt；g，n&gt；表示的核。到目前为止，ρ^&lt；(L)&gt；_&lt；g，n&gt；的核仅在g=0的情况下已知。在L=2的情况下，利用证明ρ^&lt；1，1&gt；的忠实性的方法，确定了ρ^&lt；(2)&gt；1，1&gt；的核.另一方面，Γ^&lt；^&gt；^n_g的任一开子群的中心是否平凡是一个公开的问题.(这与M_&lt；g，n&gt；是否是“Anabelian”有关)。我们证明了，如果表示ρ_&lt；g，n&gt；是忠实的，则Γ^&lt；^&gt；^&gt；n+1&gt；_g的任一开子群的中心是平凡的。较少

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 … More 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