Leila Schneps

The universal Ptolemy-Teichmuller groupoid

We define the universal Ptolemy-Teichmüller groupoid, a generalization of Penner’s universal Ptolemy groupoid, on which the Grothendieck-Teichmüller group – and thus also the absolute Galois group – acts naturally as automorphism group. The essential new ingredient added to the definition of the universal Ptolemy groupoid is the profinite local group of pure ribbon braids of each tesselation.

On a new version of the Grothendieck-Teichmuller group

Abstract In this Note we introduce a certain subgroup Г of the Grothendieck-Teichmuller group GT, obtained by adding two new relations to the definition of GT. We show that Г gives an automorphism group of the profinite completions of certain surface mapping class groups with geometric compatibility conditions, and that the absolute Galois group (ℚ/ℚ) is embedded into Г.

On reduction of p-groups

We define the notion of irreducibility of a pgroup and show how any pgroup G can be reduced to an irreducible group H. We show that G is realizable as the Galois group of a regular extension of Q(T) if H is. Finally, we give some sufficient conditions on the number of generators of a pgroup and the structure of its Frattini subgroup for it to be reducible to the trivial group.

On galois groups and their maximal 2-subgroups

LetG be a finite group of even order, having a central element of order 2 which we denote by −1. IfG is a 2-group, letG be a maximal subgroup ofG containing −1, otherwise letG be a 2-Sylow subgroup ofG. LetH=G/{±1} andH=G/{±1}. Suppose there exists a regular extensionL1 of ℚ(T) with Galois groupG. LetL be the subfield ofL1 fixed byH. We make the hypothesis thatL1 admits a quadratic extensionL2 which is Galois overL of Galois groupG. IfG is not a 2-group we show thatL1 then admits a quadratic extension which is Galois over ℚ(T) of Galois groupG and which can be given explicitly in terms ofL2. IfG is a 2-group, we show that there exists an element α ε ℚ(T) such thatL1 admits a quadratic extension which is Galois over ℚ(T) of Galois groupG if and only if the cyclic algebra (L/ℚ(T).a) splits. As an application of these results we explicitly construct several 2-groups as Galois groups of regular extensions of ℚ(T).

A complete parametrization of cyclic field extensions of 2-power degree

Letq be a power of 2 at least equal to 8 and ζ be a primitiveq-th root of unity, and letK be any field of characteristic zero. We define the group of special projective conormsSK as a quotient of the group of elements ofK(ζ) of norm 1:SK is obviously trival if the groul Gal (K(ζ)/K) is cyclic. We prove that for some fieldsK, the groupSK is finite, and it is even trivial for certain fields such as ℚ or ℚ(X1,...,Xm). We then prove that the groupSK completely paramatrizes the cycle extensions ofK of degreeq. We exhibit an explicit polynomial defined over ℚ(T0,...,Tq/2) which parametrizes all cyclic extensions ofK of degreeq associated to the trivial element ofSK. In particular, this polynomial parametrizes all cyclic extensions ofK of degreeq whenever the groupSK is trivial.