[Paper Review] One-dimensional abstract local class field theory

Ivan D. Chipchakov|arXiv (Cornell University)|Jun 24, 2005

Algebraic Geometry and Number Theory36 references5 citations

TL;DR

This paper establishes a one-dimensional abstract local class field theory for fields E satisfying specific Brauer group conditions: nontrivial p-component of Br(E) when E is properly included in its maximal p-extension, and Br(L/E) equaling the maximal p-exponent subgroup of Br(E) for every cyclic p-extension L/E. It proves that finite abelian extensions of E are uniquely determined by their norm groups, with analogues of the fundamental correspondence, local reciprocity law, and Hasse symbol, generalizing classical local class field theory in an abstract setting.

ABSTRACT

Abstract. Let E be a field satisfying the following conditions: (i) the p-component of the Brauer group Br(E) is nontrivial whenever p is a prime number for which E is properly included in its maximal p-extension; (ii) the relative Brauer group Br(L/E) equals the maximal subgroup of Br(E) of exponent p, for every cyclic extension L/E of degree p. The paper proves that finite abelian extensions of E are uniquely determined by their norm groups and related essentially as in the classical local class field theory. This includes analogues to the fundamental correspondence, the local reciprocity law and the local Hasse symbol.

Motivation & Objective

• To develop an abstract framework for local class field theory in one-dimensional settings using Brauer group conditions.
• To generalize classical local class field theory by replacing the local field assumption with cohomological and group-theoretic axioms.
• To prove that finite abelian extensions of E are uniquely determined by their norm groups under the given Brauer group constraints.
• To establish analogues of the fundamental correspondence, local reciprocity law, and Hasse symbol in this abstract setting.
• To characterize the structure of relative Brauer groups in cyclic p-extensions as maximal p-torsion subgroups of the absolute Brauer group.

Proposed method

• Assumes that for every prime p, the p-component of the Brauer group Br(E) is nontrivial whenever E is properly contained in its maximal p-extension.
• Imposes the condition that for every cyclic extension L/E of degree p, the relative Brauer group Br(L/E) is the maximal subgroup of Br(E) of exponent p.
• Uses these axioms to derive properties of norm groups of finite abelian extensions of E.
• Applies cohomological techniques to relate the structure of Br(E) to Galois cohomology and class field theory duality.
• Establishes a bijective correspondence between finite abelian extensions of E and open subgroups of finite index in the absolute Galois group of E via norm groups.
• Constructs the local reciprocity map and the Hasse symbol as abstract analogues of classical invariants in local class field theory.

Experimental results

Research questions

• RQ1 Under what Brauer group conditions is the norm group of a finite abelian extension of E sufficient to determine the extension uniquely?
• RQ2 How can the classical local reciprocity law be reconstructed in an abstract setting without assuming E is a local field?
• RQ3 To what extent do the relative Brauer groups Br(L/E) for cyclic p-extensions L/E reflect the structure of Br(E)?
• RQ4 Can the fundamental correspondence and Hasse symbol be generalized to fields satisfying only cohomological axioms on the Brauer group?
• RQ5 What abstract conditions on Br(E) ensure that the norm groups of abelian extensions fully classify the extensions?

Key findings

• Finite abelian extensions of E are uniquely determined by their norm groups under the given Brauer group axioms.
• The local reciprocity map exists and is an isomorphism from the profinite completion of the multiplicative group of E to the abelianization of the absolute Galois group of E.
• The Hasse symbol is well-defined and takes values in the p-torsion subgroup of Br(E), generalizing the classical local symbol.
• The relative Brauer group Br(L/E) is isomorphic to the maximal subgroup of Br(E) of exponent p for every cyclic extension L/E of degree p.
• The fundamental correspondence between finite abelian extensions and open subgroups of finite index in Gal(E^s/E) holds via norm groups.
• The theory recovers the classical local class field theory when E is a local field, showing the axioms are consistent with the classical case.