Galois groups and fundamental groups in anabelian geometry

研究了与泛椭圆曲线族有关的单列表示的亚阿贝尔商的度量函数及其与广义Dedekin和的关系，得到了描述度量函数随权变化的矩积分的同余公式.研究了Galois映象所满足的Grothendieck-Teichmueller群中的方程，利用亏格为零的非Galois覆盖，得到了一类新的方程.利用Magnus-Gassner型表示，在拓扑矩阵环上发现了另一类新的二元方程.与H.Tsunogai合作，利用交错椭圆曲线的一个特征作为Grothendieck dessin，研究了Grothendieck-Teichmueller群的Galois参数的性态，并用adelic beta函数描述了标准参数分解为调和参数互换的乘积.在与P.Lochak和L.Schneps的合作中，我们用具有Grothendieck-Teichmueller参数的Galois群变换的代数路径的组合来代替从标准切向基点到五个循环点的拓扑路。然后，我们成功地解释了五点射影线模空间的基本群中标准参数的五次循环分解。将Ihara-MatSumoto的方法与Gerritzen-Herrlich-Put关于n点射影线模空间的稳定紧化的方法进行了比较，得到了模空间上切向基点的一个自然解释。通过与尼斯大学的Wojtkowiak的讨论，得到了关于1维迭代积分的一个新的研究方向和观点。

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.