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Action of the Grothendieck-Teichmüller group on torsion elements of full Teichmüller modular groups in genus zero

Benjamin Collas (2012)

Journal de Théorie des Nombres de Bordeaux

In this paper we establish the action of the Grothendieck-Teichmüller group G T ^ on the prime order torsion elements of the profinite fundamental group π 1 g e o m ( 0 , [ n ] ) . As an intermediate result, we prove that the conjugacy classes of prime order torsion of π ^ 1 ( 0 , [ n ] ) are exactly the discrete prime order ones of the π 1 ( 0 , [ n ] ) .

Characteres and Galois invariants of regular dessins.

Manfred Streit, Jürgen Wolfart (2000)

Revista Matemática Complutense

We describe a new invariant for the action of the absolute Galois groups on the set of Grothendieck dessins. It uses the fact that the automorphism groups of regular dessins are isomorphic to automorphism groups of the corresponding Riemman surfaces and define linear represenatations of the space of holomorphic differentials. The characters of these representations give more precise information about the action of the Galois group than all previously known invariants, as it is shown by a series...

Characters of the Grothendieck-Teichmüller group through rigidity of the Burau representation

Ivan Marin (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

We present examples of characters of absolute Galois groups of number fields that can be recovered through their action by automorphisms on the profinite completion of the braid groups, using a “rigidity” approach. The way we use to recover them is through classical representations of the braid groups, and in particular through the Burau representation. This enables one to extend these characters to Grothendieck-Teichmüller groups.

Comparaison de deux notions de rationalité d'un dessin d'enfant

Layla Pharamond dit d'Costa (2001)

Journal de théorie des nombres de Bordeaux

Soit f un revêtement ramifié de 𝐏 1 défini sur 𝐐 ¯ . Lorsqu’on s’intéresse aux propriétés de rationalité de f sur les les corps de nombres, on peut soit exiger que la base soit 𝐏 1 , soit l’autoriser à être une courbe de genre 0 . Nous comparons ces deux points de vue pour les revêtements non ramifiés en dehors de 0 , 1 ,

Multiple zeta values and periods of moduli spaces 𝔐 ¯ 0 , n

Francis C. S. Brown (2009)

Annales scientifiques de l'École Normale Supérieure

We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces 𝔐 0 , n of Riemann spheres with n marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on 𝔐 0 , n and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes’ formula iteratively to reduce each period integral to multiple zeta values. We also give a geometric interpretation of the double shuffle...

On systems of linear inequalities

Masami Fujimori (2003)

Bulletin de la Société Mathématique de France

We show in detail that the category of general Roth systems or the category of semi-stable systems of linear inequalities of slope zero is a neutral Tannakian category. On the way, we present a new proof of the semi-stability of the tensor product of semi-stable systems. The proof is based on a numerical criterion for a system of linear inequalities to be semi-stable.

Prime to p fundamental groups and tame Galois actions

Mark Kisin (2000)

Annales de l'institut Fourier

We show that for a local, discretely valued field F , with residue characteristic p , and a variety 𝒰 over F , the map ρ : Gal ( F sep / F ) Out ( π 1 , geom ( p ' ) ( 𝒰 ) ) to the outer automorphisms of the prime to p geometric étale fundamental group of 𝒰 maps the wild inertia onto a finite image. We show that under favourable conditions ρ depends only on the reduction of 𝒰 modulo a power of the maximal ideal of F . The proofs make use of the theory of logarithmic schemes.

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