Geometric Galois Theory, Nonlinear Number Fields and a Galois Group Interpretation of the Idele Class Group
نویسندگان
چکیده
This paper concerns the description of holomorphic extensions of algebraic number fields. After expanding the notion of adele class group to number fields of infinite degree over Q, a hyperbolized adele class group ŜK is assigned to every number field K/Q. The projectivization of the Hardy space PH•[K] of graded-holomorphic functions on ŜK possesses two operations ⊕ and ⊗ giving it the structure of a nonlinear field extension of K. We show that the Galois theory of these nonlinear number fields coincides with their discrete counterparts in that Gal(PH•[K]/K) = 1 and Gal(PH•[L]/PH•[K]) ∼= Gal(L/K) if L/K is Galois. If Kab denotes the maximal abelian extension of K and CK is the idele class group, it is shown that there are embeddings of CK into Gal⊕(PH•[K ]/K) and Gal⊗(PH•[K ]/K), the “Galois groups” of automorphisms preserving ⊕ resp. ⊗ only.
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 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 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 کاملA History of Selected Topics in Categorical Algebra I: From Galois Theory to Abstract Commutators and Internal Groupoids
This paper is a chronological survey, with no proofs, of a direction in categorical algebra, which is based on categorical Galois theory and involves generalized central extensions, commutators, and internal groupoids in Barr exact Mal’tsev and more general categories. Galois theory proposes a notion of central extension, and motivates the study of internal groupoids, which is then used as an a...
متن کاملGalois Groups of Schubert Problems of Lines Are at Least Alternating
We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableau...
متن کامل