A detailed algebraic-geometric background is presented for the tropical approach to enumeration of singular curves on toric surfaces, which consists of reducing the enumeration of algebraic curves to that of non-Archimedean amoebas, the images of algebraic curves by a real-valued non-Archimedean valuation. This idea was proposed by Kontsevich and recently realized by Mikhalkin, who enumerated t...

By Grothedieck'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 th...

‎The concept of self-similarity on subsets of algebraic varieties‎ ‎is defined by considering algebraic endomorphisms of the variety‎ ‎as `similarity' maps‎. ‎Self-similar fractals are subsets of algebraic varieties‎ ‎which can be written as a finite and disjoint union of‎ ‎`similar' copies‎. ‎Fractals provide a framework in which‎, ‎one can‎ ‎unite some results and conjectures in Diophantine g...

1.1. Sheaves. Let X be a topological space. A sheaf F on X associates to each open set U an abelian group F (U), called the sections of F over U , along with a restriction map rV,U : F (V ) → F (U) for any open sets U ⊂ V (for a section σ ∈ F (V ), we often write σ|U to denote rV,U (σ)), satisfying the following conditions: (1) For any open sets U ⊂ V ⊂W , rV,U ◦ rW,V = rW,U ; (2) For a collect...