Arithmetic Teichmuller Theory

TL;DR

This paper develops an arithmetic Teichmuller theory to encode arithmetic information of hyperbolic curves over number fields, introducing new algebraic structures like the Hecke-Teichmuller Lie algebra within an anabelian framework.

Contribution

It introduces the concept of arithmetic Teichmuller objects and the Hecke-Teichmuller Lie algebra, advancing the understanding of Galois representations and anabelian geometry.

Findings

Defined arithmetic objects summarizing all curves of a given topological type.

Introduced the Hecke-Teichmuller Lie algebra as an analog of the Hecke algebra.

Connected Galois representations with arithmetic Teichmuller theory.

Abstract

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.