Motivic aspect of moduli space of algebraic curves

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1. Teichmueller groupoids are fundamental groupoids of the moduli space of pointed Riemann surfaces, and are studied in topology and mathematical physics. Using the arithmetic Schottky-Mumford uniformization theory on algebraic curves given by the head investigator, we constructed Teichmueller groupoids in the category of arithmetic geometry.2. Using the result in 1, we verified Grothendieck's conjecture on motives attached to Teichmueller groupoids for Galois representations and monodromy representations given by conformal field theory, i.e. showed that these objects are generated by basic ones.3. We described explicitly the Chow-forms of elliptic normal curves of degree 4, and showed that certain projective algebraic varieties admit Chow-forms having a special property.4. We studied unit groups, class numbers and integer rings for some abelian number fields.5. We discussed the infinite level asymptotics of the perturbative Chern-Simons integral without renormalization, by introducing a Gaussian kernel increasing along the level tends to infinity.6. We determined the minimum distances of evaluation codes of the Hermite type, and constructed the bases of a part of trace-norm codes.7. Using complex of curves, we obtained Gervais' symmetric presentation for the mapping class group of a surface. Furthermore, we determined the virtual cohomological dimension and the Euler number of the mapping class group of a 3-dimensional handlebody.