Journal of the Institute of Mathematics of Jussieu

Table of Contents - 2010 - Volume 9, Issue 02  

Research Article

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

Hiroaki Nakamuraa1, Hiroshi Tsunogaia2 and Seidai Yasudaa3

a1 Department of Mathematics, Faculty of Science, Okayama University, Okayama 700-8530, Japan (h-naka@math.okayama-u.ac.jp)

a2 Department of Mathematics, Sophia University, Tokyo 102-8554, Japan (tsuno-h@cc.sophia.ac.jp)

a3 Research Institute of Mathematical Science, Kyoto University, Kyoto 606-8502, Japan (yasuda@kurims.kyoto-u.ac.jp)

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 $\smash{\hat{B}_4}$. 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.

(Received August 24 2008)

(Accepted September 29 2008)

Keywords

  • Grothendieck–Teichmüller group;
  • Galois actions on profinite braid groups;
  • adelic beta function

AMS 2010 Mathematics subject classification

  • Primary 14H30;
  • Secondary 20E18;
  • 12E12