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Harmonic and equianharmonic equations in the Grothendieck–Teichmüller group. III

Part of: Curves General field theory Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  11 August 2009

Hiroaki Nakamura ,
Hiroshi Tsunogai  and
Seidai Yasuda
Hiroaki Nakamura
Affiliation:
Department of Mathematics, Faculty of Science, Okayama University, Okayama 700-8530, Japan (h-naka@math.okayama-u.ac.jp)
Hiroshi Tsunogai
Affiliation:
Department of Mathematics, Sophia University, Tokyo 102-8554, Japan (tsuno-h@cc.sophia.ac.jp)
Seidai Yasuda
Affiliation:
Research Institute of Mathematical Science, Kyoto University, Kyoto 606-8502, Japan (yasuda@kurims.kyoto-u.ac.jp)
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Abstract

We study behaviours of the ‘equianharmonic’ parameter of the Grothendieck–Teichmüller group introduced by Lochak and Schneps. Using geometric construction of a certain one-parameter family of quartics, we realize the Galois action on the fundamental group of a punctured Mordell elliptic curve in the standard Galois action on a specific subgroup of the braid group . A consequence is to represent a matrix specialization of the ‘equianharmonic’ parameter in terms of special values of the adelic beta function introduced and studied by Anderson and Ihara.

Keywords

Grothendieck–Teichmüller groupGalois actions on profinite braid groupsadelic beta function

MSC classification

Secondary: 14H30: Coverings, fundamental group 20E18: Limits, profinite groups 12E12: Equations

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 2 , April 2010 , pp. 431 - 448
Copyright
Copyright © Cambridge University Press 2010

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