Eigenloci of 5 Point Configurations on the Riemann Sphere and the Grothendieck-Teichm・ler Group

Let GQ be the absolute Galois group of the rational number field Q. In this paper we closely study the action of GQ on an element of the Teichmuller modular group which can be viewed simply as the order 5 rotation of the Riemann sphere marked at the 5th-roots of unity. Especially, we explicitly compute the conjugating factor of the action in terms of the Galois representation in π1(P−{0, 1,∞}, −→ 01). The overall meaning of this computation can be explained most naturally from the point of view, or the framework, of Grothendieck-Teichmuller theory. In this introduction we will content ourselves with recalling the least necessary background, relying on references for technical detail. We postpone until §6 of the present paper a short discussion of the why and what for. Let M0,n (resp. M0,[n]) be the fine moduli space of sphere with n labeled (resp. unlabeled) marked points, viewed as a Q-scheme (resp. stack). Let Γ0 (resp. Γ [n] 0 ) be the topological (resp. orbifold) fundamental group of M0,n (resp. M0,[n]) as a complex manifold (resp. orbifold), regarding Q as embedded in C. Finally, let Γ0 and Γ [n] 0 be the profinite completions of these groups, which one can regard as the geometric fundamental groups of M0,n and M0,[n] respectively. There is a canonical outer action of GQ on Γ [n] 0 , which preserves the pure subgroup Γ0 .