Galois groups and fundamental groups
伽罗瓦群和基本群
基本信息
- 批准号:09440011
- 负责人:
- 金额:$ 5.5万
- 依托单位:
- 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:
- 关键词:
项目摘要
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 D 4 ^D 4)D 22 D 2(p ;任意固定素数),与(i)分圆域Q(μ D 2 p D 2 ∞)上的阿贝尔扩张,(ii)稳定导子代数有关。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
期刊论文数量(30)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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。
- DOI:
- 发表时间:
- 期刊:
- 影响因子:0
- 作者:
- 通讯作者:
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》。
- DOI:
- 发表时间:
- 期刊:
- 影响因子:0
- 作者:
- 通讯作者:
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。
- DOI:
- 发表时间:
- 期刊:
- 影响因子:0
- 作者:
- 通讯作者:
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,
- DOI:
- 发表时间:
- 期刊:
- 影响因子:0
- 作者:
- 通讯作者:
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”。
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