Galois groups and fundamental groups

如果X是k领域上的几何连接代数变异，而X-y D4-y D4 =X [交叉产品] k-y D4，k-y D4-y D4成为k的代数闭合，然后是绝对Galois集团G-D2 k-D2=Gal（k ― D4-y ― D4/k） of k acts outerly on the algebraic fundamental group π ― D21 y ― D2（Xiy D4-Xiy D4） of Xiy D4《自然杀手》中的D4：首席调查员Ihara继续对K=Q中这种行为的算术方面进行研究（计算逻辑数字）和X= P-D11-{0，1，∞} （预测线减去三个点）。在this case中，π D21イエD2 （XイイD4-イエD4） is a free profinite group （FイイD4^イエD4）イイD22イエD2 of rank2， on which GイD2QイD2 acts faithfully。G-D2 Q-D2， regarded as a subgroup of the automorphism group Aut（F-D4 ^-D4）G-D22-D2， is known to be contained in a subgroup GT of Aut（F-D4 ^-D4）G-D22-D2 （the Grothendieck-Teichmuller group）It is not known whether G2Q D2≠GT， but there are some properties known to be satisfied by elements of G2Q D2 but unkno ... More wn and doubt\whether they are satisfied by "any" element of GT.Ihara\eted his study of the GT-action on the max\meta-abelian quotient of （F-D4^-D4）-D22-D2 [1]。它与Adelic Beta函数及其理论有关-解构，继续G.的先前作品。安德森和伊哈拉自己。This contains a certain arithmetic necessary condition for an element的过程中，GT to belong to G-D2 Q-D2 in terms of the （hyper adelic） gamma function-D2 and certain quasi1-cocycles。在[2]中，更多基本算术条件（类1-循环术语）和几何条件（对X-D4-D4循环覆盖X-D4循环覆盖的循环的反思也可以作为X-D4-D4的开放子空间嵌入）被讨论，这些条件之间的逻辑依赖性也是明确的。Ihara Studed the G-D2Q-D2 action_（FイイD4^イエD4）イイD22イエD2 （p ; an arbitrary fixed prime）， in connection with （i） abelian extensions over the cyclotomic filed Q（イD2pイエD2），（ii） the stable derivation algebra。Among them， （i） concerns with the kernel of G1; asking which abelian extension of Q（D2G2 ∞） is contained in the field corresponding to this kernel， while （ii）is a sort of the“graded Lie algebra version over Z”of GT， and is a basic object in studying the image of G1（G1）。[3]的主要创新是（i） （ii）之间的连接的分类，以及一些数字结果显示出与（i） （ii）的异常现象相关的现象（i） （ii）时p是不正常的prime。Less（低）

If X is a geometrically connected algebraic variety over a field k, and XィイD4-ィエD4 =X 【cross product】 kィイD4-ィエD4, kィイD4-ィエD4 being an algebraic closure of k, then the absolute Galois group GィイD2kィエD2=Gal(kィイD4-ィエD4/k) of k acts outerly on the algebraic fundamental group πィイD21ィエD2(XィイD4-ィエD4) of XィイD4-ィエD4 in a natural manner.The head investigator Ihara continued his study on the arithmetic aspects of this action in the most basic case where k=Q (the rational number filed) and X=PィイD11ィエD1-{0,1,∞} (the projective line minus three points). In this case, πィイD21ィエD2 (XィイD4-ィエD4) is a free profinite group (FィイD4^ィエD4)ィイD22ィエD2 of rank 2, on which GィイD2QィエD2 acts faithfully. GィイD2QィエD2, regarded as a subgroup of the automorphism group Aut(FィイD4^ィエD4)ィイD22ィエD2, is known to be contained in a subgroup GT of Aut(FィイD4^ィエD4)ィイD22ィエD2 (the Grothendieck-Teichmuller group). It is not known whether GィイD2QィエD2≠GT, but there are some properties known to be satisfied by elements of GィイD2QィエD2 but unkno … More wn and doubtful whether they are satisfied by "any" element of GT.Ihara completed his study of the GT-action on the maximal meta-abelian quotient of (FィイD4^ィエD4)ィイD22ィエD2 [1]. It concerns with the theory of adelic beta functions and their г-decompositions, continuing previous works of G. Anderson and Ihara himself. This contains a certain arithmetic necessary condition for an element σ ∈ GT to belong to GィイD2QィエD2 in terms of the (hyper adelic) gamma function гィイD2σィエD2 and certain quasi 1-cocycles. In [2], more basic arithmeticconditions (in terms of quasi 1-cocycles) and geometric conditions (a reflection of the circumstances that cyclic covers of XィイD4-ィエD4 can also be embedded as open subspaces of XィイD4-ィエD4) are discussed, and logical dependencies among these conditions are clarified.In[3], Ihara studied the GィイD2QィエD2 action ψィイD1(pィエD1) on the maximal prop-p quotient FィイD3(pィエD3),ィイD22ィエD2 of (FィイD4^ィエD4)ィイD22ィエD2 (p ; an arbitrary fixed prime), in connection with (i) abelian extensions over the cyclotomic filed Q(μィイD2pィエD2∞),(ii) the stable derivation algebra. Among them, (i) concerns with the kernel of ψィイD1pィエD1; asking which abelian extension of Q(μィイD2pィエD2∞) is contained in the field corresponding to this kernel, while (ii)is a sort of the "graded Lie algebra version over Z" of GT, and is a basic object in studying the image of ψィイD1(pィエD1). The main innovation of [3] is the clarification of the connection between (i) (ii), and some numerical result which shows quite an exciting phenomenon related to (i) (ii) when p is an irregular prime. Less