Properties of mapping class groups related to Galois representations

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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