Galois groups and fundamental groups in anabelian geometry

点击翻译按钮获取中文摘要

We studied a measure function that describes the meta-abelian quotient of the monodromy representation associated with the universal family of elliptic curves and its relation with generalized Dedekind sums.In particular, we showed a congruence formula that describes moment integrals of the measure function along variation of weights. Equations in the Grothendieck-Teichmueller group satisfied by the Galois image were investigated.Using genus zero non-Galois covers, we found a new type equation. Utilizing the Magnus-Gassner type representation, another new type equation was found to hold in the topological matrix ring in two variables.In a collaboration with H.Tsunogai, using a characterization of the lemniscate elliptic curve as a Grothendieck dessin, we studied the behavior of Galois parameters of the Grothendieck-Teichmueller group, and described the decomposition of the standard parameter into a product of mutually transposed harmonic parameters in terms of adelic beta functions. In a collaboration with P.Lochak and L.Schneps, we replaced a toplogical path from the standard tangential basepoint to the five cyclic point by a composition of algebraic paths that are transformed by the Galois group with Grothendieck-Teichmueller parameters. Then, we succeeded in interpreting the five cyclic decomposition of the standard parameter in the fundamental group of the moduli spaces of the 5-pointed projective lines. Comparing the method of Ihara-Matsumoto with a paper by Gerritzen-Herrlich-Put about stable compactification of moduli spaces of n-pointed projective lines, we obtained a natural interpretation of tangential base points on those moduli spaces. Through discussions with Wojtkowiak at Nice University, a new direction of investigation and perspectives about 1-adic itereated integrals was obtained.