en عمومى General پژوهشي Research paper <p>By Grothedieck&#39;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 arithmetic information of these curves. The goal of this paper is to develope and 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.</p> Anabelian geometry, Grothendieck conjectures, Huperbolic curves, Outer automorphism, Galois representation. 157 171 http://ijmsi.ir/browse.php?a_code=A-10-2299-1&slc_lang=en&sid=1 A. Rastegar rastegar1352@gmail.com 10031947532846007814 10031947532846007814 Yes Department of Mathematics, Sharif University of Technology, Tehran, Iran.