2005 Fiscal Year Final Research Report Summary
Galois groups and fundamental groups in anabelian geometry
| Project/Area Number |
14340017
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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 | OKAYAMA UNIVERSITY |
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Principal Investigator |
NAKAMURA Hiroaki OKAYAMA UNIVERSITY, Graduate School of Natural Science, Department of Mathematics, Professor, 大学院・自然科学研究科, 教授 (60217883)
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| Co-Investigator(Kenkyū-buntansha) |
YAMADA Hiro-fumi OKAYAMA UNIVERSITY, Department of Mathematics, Professor, 大学院・自然科学研究科, 教授 (40192794)
YOSHINO Yuji OKAYAMA UNIVERSITY, Department of Mathematics, Professor, 大学院・自然科学研究科, 教授 (00135302)
TANAKA Katsumi OKAYAMA UNIVERSITY, Admission Center, Associate Professor, アドミッションセンター, 助教授 (60207082)
KATSUDA Atsushi OKAYAMA UNIVERSITY, Department of Mathematics, Associate Professor, 大学院・自然科学研究科, 助教授 (60183779)
HIROKAWA Masao OKAYAMA UNIVERSITY, Department of Mathematics, Professor, 大学院・自然科学研究科, 教授 (70282788)
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| Project Period (FY) |
2002 – 2005
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| Keywords | anabelian geometry / outer Galois representation / Teichmueller space / mapping class group / braid group / absolute Galois group / Grothendieck Conjecture / covers of Riemann surfaces |
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| Research Abstract |
We studied a measure function that describes the meta-abelian quotient of the monodromy representation associated with the universal family of elliptic curves and its relation with generalized Dedekind sums.
In particular, we showed a congruence formula that describes moment integrals of the measure function along variation of weights. Equations in the Grothendieck-Teichmueller group satisfied by the Galois image were investigated.
Using genus zero non-Galois covers, we found a new type equation. Utilizing the Magnus-Gassner type representation, another new type equation was found to hold in the topological matrix ring in two variables.
In a collaboration with H.Tsunogai, using a characterization of the lemniscate elliptic curve as a Grothendieck dessin, we studied the behavior of Galois parameters of the Grothendieck-Teichmueller group, and described the decomposition of the standard parameter into a product of mutually transposed harmonic parameters in terms of adelic beta functions. In a collaboration with P.Lochak and L.Schneps, we replaced a toplogical path from the standard tangential basepoint to the five cyclic point by a composition of algebraic paths that are transformed by the Galois group with Grothendieck-Teichmueller parameters. Then, we succeeded in interpreting the five cyclic decomposition of the standard parameter in the fundamental group of the moduli spaces of the 5-pointed projective lines. Comparing the method of Ihara-Matsumoto with a paper by Gerritzen-Herrlich-Put about stable compactification of moduli spaces of n-pointed projective lines, we obtained a natural interpretation of tangential base points on those moduli spaces. Through discussions with Wojtkowiak at Nice University, a new direction of investigation and perspectives about 1-adic itereated integrals was obtained.
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