Arithmetic Teichmuller Theory

By Grothendieck’s anabelian conjectures, Galois representa- tions landing in outer automorphism group of the algebraic fundamen- tal group which are associated to hyperbolic smooth curves defined over number-fields encode all the arithmetic information of these curves. The Goal of this paper is to develop an arithmetic Teichmuller theory, by which we mean, introducing arithmetic objects summarizing the arith- metic information coming from all curves of the same topological type defined over number-fields. We also introduce Hecke-Teichmuller Lie al- gebra which plays the role of Hecke algebra in the anabelian framework.