Galois representations in arithmetic geometry
نویسنده
چکیده
Takeshi SAITO When he formulated an analogue of the Riemann hypothesis for congruence zeta functions of varieties over finite fields, Weil predicted that a reasonable cohomology theory should lead us to a proof of the Weil conjecture. The dream was realized when Grothendieck defined etale cohomology. Since then, -adic etale cohomology has been a fundamental object in arithmetic geometry. It enables us to investigate the arithmetic of algebraic varieties and also to construct representations of the absolute Galois group of a field of arithmetic interest. With etale cohomology, geometric problems can be studied using linear algebra as in the diagram
منابع مشابه
Deformation of Outer Representations of Galois Group
To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than t...
متن کاملDeformation of Outer Representations of Galois Group II
This paper is devoted to deformation theory of "anabelian" representations of the absolute Galois group landing in outer automorphism group of the algebraic fundamental group of a hyperbolic smooth curve defined over a number-field. In the first part of this paper, we obtained several universal deformations for Lie-algebra versions of the above representation using the Schlessinger criteria for...
متن کاملArithmetic Teichmuller Theory
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...
متن کاملGalois deformations and arithmetic geometry of Shimura varieties
Shimura varieties are arithmetic quotients of locally symmetric spaces which are canonically defined over number fields. In this article, we discuss recent developments on the reciprocity law realized on cohomology groups of Shimura varieties which relate Galois representations and automorphic representations. Focus is put on the control of -adic families of Galois representations by -adic fami...
متن کاملPolarization Measurement aboard the Satellite and Solution of the Emission Mechanism of the Gamma-Ray Bursts
Tetsushi Ito (Kyoto University, Graduate School of Science, Assistant Professor) 【Outline of survey】 Shimura varieties are algebraic varieties (geometric objects defined by equations), which are generalizations of modular curves. Previously, several mathematical objects in arithmetic geometry, Galois representations, automorphic representations were studied from individual perspectives. However...
متن کامل