Galois groups and fundamental groups

伽罗瓦群和基本群

• 批准号: 09440011
• 负责人: IHARA Yasutaka
• 金额: $ 5.5万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440011/
• 关键词: Galois group; Fundamental group; The projective line minus 3 points; Grothendieck-Teichmuller group; Adelic beta function; Hyperadelic gamma function; The stable derivation algebra; Arithmetic of cyclotomic fields; 絶対ガロア群; 射影直線マイナス3点の; 円分体のアーベル拡

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

如果X是域k上的几何连通代数簇，且X D4-D4 =X [叉积] k D4-D4，k D4-D4是k的代数闭包，则绝对伽罗瓦群G ∈ D2 k ∈ D2=Gal k的（k D4-D4/k）外作用于代数基本群π D21 D2上（X条D4-X条D4），其中X条D4-以自然的方式计算D4。首席研究员Ihara继续他的研究，在最基本的情况下，k=Q，这个行动的算术方面（有理数域）和X=P D11 D1-{0，1，∞}（射影直线减三点）。在这种情况下，π D 21 D 2（X D 4-D 4）是秩为2的自由profinite群（F D 4 ^D 4）D 22 D 2，G D 2 Q D 2忠实地作用在其上。G D2 Q D2，被认为是自同构群Aut（F D4 ^D4）D22 D2的子群，已知包含在Aut（F D4^D4）D22 D2的子群GT（Grothendieck-Teichmuller群）中。不知道G是否满足D2 Q的条件，但G的元素满足一些性质，但不知道G是否满足D2 Q的条件。 ...更多信息 Ihara完成了GT作用在（F_n_D_4 ^_n_D_4）_n_D_22_n_D_2的最大亚阿贝尔商上的研究[1]。本文是G.安德森和伊原本人。这包含了元素σ ∈ GT属于G D2 Q D2的一个算术必要条件，即（超顶点）Gamma函数D2σ D2和某些拟1-上圈.文[2]给出了更基本的算术条件（用拟1-上圈表示）和几何条件（反映了X D4-D4的循环覆盖也可以嵌入为X D4-D4的开子空间的情况）进行了讨论，并阐明了这些条件之间的逻辑依赖关系。Ihara研究了GイD2 Q D2作用ψイD1（p D1）对最大prop-p商FイD3（p D3）、イD22 D2的影响（FイD4^D4）イD22 D2（p ;任意固定素数），与（i）分圆域Q上的阿贝尔扩张（μイD2 p D2∞），（ii）稳定导子代数。其中，（i）涉及到G_n_D_1（p_n_D_1）的核，即Q（μ_n_D_2 p_n_D_2 ∞）的哪一个阿贝尔扩张包含在与该核对应的域中;（ii）是GT的一种“Z上的分次李代数版本”，是研究G_n_D_1（p_n_D_1）象的基本对象。文[3]的主要创新之处在于阐明了（i）（ii）之间的联系，并给出了一些数值结果，这些结果表明当p为非正则素数时与（i）（ii）有关的一个令人兴奋的现象。少

项目成果

Nakamura, Hiroaki (with L.Sehneps): "On a subgkroup of the Grothendieck-Teichmuller group acting on the tower of profinite Teichmuller modular group"Preprint, to appear in Inv. Math. (2000)

Nakamura, Hiroaki (与 L.Sehneps)：“论 Grothendieck-Teichmuller 群的子群作用于 profinite Teichmuller 模群的塔”预印本，出现在 Inv 中。

Nakamura, Hiroaki: "Limits of Galois representations in fundamental groups along maximal degeneration of marked curves, I"American Journal of Mathematics. 121. 315-358 (1999)

Nakamura、Hiroaki：“沿标记曲线最大退化的基本群中伽罗瓦表示的极限”，《美国数学杂志》。

M.Matsumoto: "A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities"Math. Ann. 印刷中.

M.Matsumoto：“用 Artin 群和奇点的几何单性来表示类群”，出版中。

S.Mochizuki: "Foundations of p-adic Teichmuller Theory" International Press社より出版予定,

S. Mochizuki：“p-adic Teichmuller 理论的基础”将由国际出版社出版，

Y.Ihara: "On Shimura curves and their rational points" AMS conferencd "Finite fields and Applications". (1997)

Y.Ihara：“论志村曲线及其有理点”AMS 会议“有限域和应用”。