  By Grothendieck's anabelian conjectures, Galois representations landing in
outer automorphism group of the algebraic fundamental 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 arithmetic information coming from all
curves of the same topological type defined over number-fields. We also
introduce Hecke-Teichmuller Lie algebra which plays the role of Hecke algebra
in the anabelian framework.
