Tsunogai Hiroshi

In this article, we consider an analogue of Noether's problem for the fields of cross-ratios, and discuss on a rationality problem which connects this with Noether's problem. We show that the affirmative answer of the analogue implies the affirmative answer for Noether's Problem for any permutation group with odd degree. We also obtain some negative results for various permutation groups with even degree.

Suppose that a finite group G is realized in the Cremona group Cr-m(k), the group of k-automorphisms of the rational function field K of m variables over a constant field k. The most general version of Noether's problem is then to ask, whether the subfield K-G consisting of G-invariant elements is again rational or not. This paper treats Noether's problem for various subgroups G of G6, the symmetric group of degree 6, acting on the function field Q(s, t, z) over k = Q of the moduli space M-0,(6) of P-1 with ordered six marked points. We shall show that this version of Noether's problem has an affirmative answer for all but two conjugacy classes of transitive subgroups G of G6, by exhibiting explicitly a system of generators of the fixed field Q(s, t, z)G. In the exceptional cases G = 21(6), 21(5), the problem remains open.

We study behaviours of the 'equianharmonic' parameter of the Grothendieck-Teichmuller 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 (B) over cap (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.

In [LS], Lochak and Schneps introduced the "harmonic parameter g" of the Grothendieck-Teichmidller group (GT) over cap. We closely study the behavior of g on the absolute Galois group G(Q) using a family of lemniscate elliptic curves. We obtain a relationship of the adelic beta function and the harmonic parameter specialized in the matrix group SL2((Z) over cap).