Class field theory, its three main generalisations, and applications

2021/08/31 by Ivan Fesenko · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Advanced Algebra and Geometry #Algebraic number theory #Class field theory #Class (philosophy) #Mathematics #Field (mathematics) #Galois theory #Algebraic number #Algebra over a field #Pure mathematics #Computer science #Artificial intelligence

paper · pdf · doi:10.4171/emss/45

openalex publication_date 2021/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

This work presents branches of class field theory. Special and general approaches to class field theory, and their roles, are discussed. Three main generalisations of class field theory: higher class field theory, Langlands correspondences and anabelian geometry, and their further developments are discussed. Several directions of unification of generalisations of class field theory are proposed. New fundamental open problems are included.

Citations

• Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki
• Integration on product spaces and GL_n of a valuation field over a local field
• On the automorphic sheaves for GSp₄
• Outline of the proof of the geometric Langlands conjecture for GL(2)

Cited by

• Class field theory [wikipedia]

Related

• Geometric Class Field Theory
• Notes on abelian class field theory
• Derived class field theory
• Computational class field theory
• Higher ideles and class field theory