🔢Algebraic Number Theory Unit 15 – Class Field Theory

Key Concepts and Definitions

• Class field theory studies abelian extensions of global and local fields, focusing on the relationship between the Galois group and the arithmetic of the base field
• Global fields include number fields (finite extensions of $\mathbb{Q}$) and function fields (finite extensions of $\mathbb{F}_q(t)$)
• Local fields are completions of global fields with respect to a prime or a place (e.g., $\mathbb{Q}_p$, $\mathbb{F}_q((t))$)
• Abelian extensions are Galois extensions with an abelian Galois group, meaning the Galois group is commutative under composition
• Ideal class group of a number field $K$, denoted by $Cl_K$, is the quotient group of fractional ideals modulo principal ideals
  • Measures the failure of unique factorization in the ring of integers of $K$
• Idele class group of a global field $K$, denoted by $C_K$, is the quotient group of ideles (invertible adeles) modulo the principal ideles
  • Generalizes the ideal class group and plays a central role in class field theory
• Hilbert class field of a number field $K$ is the maximal unramified abelian extension of $K$

Historical Context and Development

• Class field theory originated from attempts to generalize quadratic reciprocity and higher reciprocity laws in the late 19th and early 20th centuries
• David Hilbert's Zahlbericht (1897) laid the foundation for the study of algebraic number fields and their extensions
• Teiji Takagi (1920s) developed the main theorems of class field theory for number fields, establishing the correspondence between abelian extensions and ideal class groups
• Emil Artin (1927) formulated the Artin reciprocity law, providing a powerful tool for studying abelian extensions
• Helmut Hasse (1930s) extended class field theory to local fields and introduced the concept of the local reciprocity map
• Claude Chevalley (1940s) gave a cohomological interpretation of class field theory using ideles and adeles
• John Tate (1950s) reformulated class field theory in terms of Galois cohomology and introduced the Tate cohomology groups
• Class field theory has since been generalized to function fields and other settings, with ongoing research in areas such as higher class field theory and non-abelian extensions

Fundamental Theorems

• Existence Theorem: For every modulus $\mathfrak{m}$ of a number field $K$, there exists a unique abelian extension $K_\mathfrak{m}$ of $K$, called the ray class field, such that the Artin map induces an isomorphism between the ray class group modulo $\mathfrak{m}$ and the Galois group of $K_\mathfrak{m}/K$
• Isomorphism Theorem: The Artin map induces an isomorphism between the idele class group of $K$ and the Galois group of the maximal abelian extension of $K$
• Kronecker-Weber Theorem: Every abelian extension of $\mathbb{Q}$ is contained in a cyclotomic field $\mathbb{Q}(\zeta_n)$ for some $n$, where $\zeta_n$ is a primitive $n$-th root of unity
• Artin Reciprocity Law: Generalizes quadratic and higher reciprocity laws, establishing a correspondence between abelian extensions and characters of the idele class group
• Chebotarev Density Theorem: Describes the distribution of prime ideals with a given Frobenius element in a Galois extension, generalizing Dirichlet's theorem on primes in arithmetic progressions

Class Field Extensions

• Hilbert class field of a number field $K$ is the maximal unramified abelian extension of $K$, corresponding to the ideal class group $Cl_K$
• Ray class fields are abelian extensions of $K$ corresponding to ray class groups, which generalize the ideal class group by considering congruence conditions modulo a modulus $\mathfrak{m}$
• Conductor of an abelian extension $L/K$ is the smallest modulus $\mathfrak{f}$ such that $L$ is contained in the ray class field $K_\mathfrak{f}$
• Genus field of $K$ is the maximal unramified extension of $K$ that is abelian over $\mathbb{Q}$, corresponding to the genus class group (a subgroup of $Cl_K$)
• Narrow class field of $K$ is the maximal abelian extension of $K$ unramified at all finite primes, corresponding to the narrow class group (a generalization of $Cl_K$ considering real embeddings)
• Hilbert's Theorem 94: Characterizes the discriminant and conductor of an abelian extension in terms of ramification

Artin Reciprocity Law

• Artin reciprocity law establishes a correspondence between abelian extensions of a global field $K$ and continuous characters of the idele class group $C_K$
• For an abelian extension $L/K$, the Artin map $\psi_{L/K}: C_K \to \text{Gal}(L/K)$ is a surjective homomorphism with kernel $N_{L/K}(C_L)$, where $N_{L/K}$ is the norm map
• The Artin map is functorial: for abelian extensions $L/K$ and $M/L$, we have $\psi_{M/K} = \psi_{L/K} \circ \psi_{M/L}$
• Local reciprocity law: For a local field $K_v$, the local Artin map $\psi_{L_w/K_v}: K_v^\times \to \text{Gal}(L_w/K_v)$ is an isomorphism, where $L_w$ is the completion of $L$ at a prime $w$ above $v$
• Product formula: The product of local Artin maps over all places of $K$ is trivial on the idele group $\mathbb{I}_K$, inducing the global Artin map on $C_K$
• Compatibility with class field theory: The Artin map induces isomorphisms between ray class groups and Galois groups of ray class fields

Applications in Number Theory

• Solving Diophantine equations: Class field theory can be used to study the solvability of certain Diophantine equations, such as the generalized Fermat equation $x^p + y^q = z^r$
• Langlands program: Class field theory is a key ingredient in the Langlands correspondence, which relates Galois representations to automorphic forms and has far-reaching implications in number theory and representation theory
• Stark's conjectures: Relate the values of L-functions at $s=0$ to units in abelian extensions, generalizing the analytic class number formula
• Kronecker's Jugendtraum: Hilbert's 12th problem, seeking to generate abelian extensions of arbitrary number fields using special values of transcendental functions (analogous to the Kronecker-Weber theorem for $\mathbb{Q}$)
• Elliptic curves and abelian varieties: Class field theory plays a role in the study of the Mordell-Weil group and the Tate-Shafarevich group of elliptic curves and abelian varieties over number fields
• Iwasawa theory: Studies the behavior of class groups and related objects in infinite towers of number fields, using techniques from class field theory and p-adic analysis

Computational Techniques

• Computation of class groups and unit groups: Algorithms such as the Buchmann-Lenstra algorithm and the Hafner-McCurley algorithm can be used to compute the ideal class group and unit group of a number field
• Computation of ray class groups: Generalized algorithms can be used to compute ray class groups modulo a given modulus, which are essential for constructing ray class fields
• Computation of Hilbert class fields: Various methods, such as the Stark-Heegner method and the complex multiplication method, can be used to construct Hilbert class fields of imaginary quadratic fields
• Computation of Artin representations: Algorithms based on the Artin reciprocity law can be used to compute Artin representations and their properties, such as conductors and L-functions
• Numerical verification of conjectures: Computational techniques can be used to test and provide evidence for conjectures in class field theory, such as the Cohen-Lenstra heuristics on the distribution of class groups
• Explicit class field theory: Develops explicit methods for constructing class fields and studying their properties, using techniques from algebraic number theory and computer algebra

Advanced Topics and Open Problems

• Higher class field theory: Generalizes class field theory to non-abelian extensions, using the language of group cohomology and Galois cohomology
• Langlands program: Seeks to unify various areas of mathematics, including class field theory, representation theory, and automorphic forms, through a web of conjectures and correspondences
• Stark's conjectures: Generalize the analytic class number formula and relate the values of L-functions to units in abelian extensions, with some cases still unproven
• Non-abelian reciprocity laws: Attempts to generalize the Artin reciprocity law to non-abelian extensions, such as the Langlands reciprocity conjecture and the Shimura-Taniyama-Weil conjecture (now a theorem)
• Iwasawa theory: Studies the behavior of class groups and related objects in infinite towers of number fields, with open problems such as the main conjecture and the non-commutative main conjecture
• Anabelian geometry: Investigates the extent to which arithmetic and geometric properties of objects (such as number fields and algebraic varieties) are determined by their absolute Galois groups, using techniques from class field theory and Galois theory
• Explicit methods and algorithms: Developing efficient algorithms for computing class groups, ray class groups, and constructing class fields remains an active area of research, with implications for cryptography and other applications