| Project/Area Number |
09440011
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
IHARA Yasutaka RIMS, KYOTO UNIVERSITY PROFESSOR, 数理解析研究所, 教授 (70011484)
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| Co-Investigator(Kenkyū-buntansha) |
TSUJI Takeshi RIMS, KYOTO UNIVERSITY INSTRUCTOR, 数理解析研究所, 助手 (40252530)
MOCHIZUKI Shinichi RIMS, KYOTO UNIVERSITY ASSOCOATE PROFESSOR, 数理解析研究所, 助教授 (10243106)
TAMAGAWA Akio RIMS, KYOTO UNIVERSITY ASSOCOATE PROFESSOR, 数理解析研究所, 助教授 (00243105)
MATSUMOTO Makoto GRADUATE SCHOOL OF SCIENCES, KYUSHU UNIV., ASSOCOATE PROFESSOR, 大学院・数理学研究科, 助教授 (70231602)
NAKAMURA Hiroaki DEPT. OF MATH., TOKYO METROPOLITAN UNIV., ASSOCIATE PROFESSOR, 理学部, 助教授 (60217883)
斎藤 盛彦 京都大学, 数理解析研究所, 助教授 (10186968)
斎藤 恭司 京都大学, 数理解析研究所, 教授 (20012445)
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| Project Period (FY) |
1997 – 1999
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| Project Status |
Completed (Fiscal Year 1999)
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| Budget Amount *help |
¥8,600,000 (Direct Cost: ¥8,600,000)
Fiscal Year 1999: ¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 1998: ¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 1997: ¥3,000,000 (Direct Cost: ¥3,000,000)
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| Keywords | 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大 / 整数論 / Soule元 / pro-p基本群 |
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| Research Abstract |
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
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