Geometry of Moduli Spaces and Galois Actions

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We investigated arithmetic geometry using a nonabelian invariant, the fundamental group. Head investigator proved the following : the derivation Lie algebra of Galois action on the fundamental group of projective line minus three points is generated by the Soule elements. This solves the generation conjecture by Deligne and Ihara. The result was published in Compositio Mathematicae. Using a similar construction, Head investigator proved the following result on the action of the arithmetic fundamental group of the moduli space on the fundamental group of curves. The image of the action of the absolute Galois group on the unipotent fundamental group of a curve C is maximal, if and only if the algebraic cycle C-C^-in the Jacobian of C has non-torsion image by l-adic Abel Jacobi map in the Galois cohomology. The result is to appear in J. Inst. Math. Jussieu. Tamagawa researched the reconstruction of a curve in the positive characteristic case from its geometric fundamental groups. He proved that it is possible if genus is zero, and is possible up to finite isomorphic classes in a general case. The result was published in J. Alg. Geom. Mochizuki is generalizing the anabelian geometry which reconstructs a scheme from its etale fundamental group, and is constructing a theory of categorical arithmetic geometry, which is expected to have a good contribution to ABC conjecture. Tsuzuki established the descent theory of the rigid cohomology and prove the finiteness of its dimension and degeneration of the weight spectral sequences. The result is published in Rend. Sem. Mat. Univ Padova. Kimura defined the notion of finite dimensionality of pure motives using symmetric and exterior products, and proved them in the case of curves. S. Morita made a good advance towards Faber's conjecture on the cohomology ring of the moduli space. The result was published in Topology.