🔢Arithmetic Geometry Unit 10 – Class field theory

Key Concepts and Definitions

• Class field theory studies abelian extensions of local and global fields, providing a correspondence between such extensions and certain groups associated with the base field
• Local fields are complete fields with respect to a discrete valuation and have a finite residue field (examples include $\mathbb{Q}_p$ and $\mathbb{F}_p((t))$)
• Global fields are either algebraic number fields (finite extensions of $\mathbb{Q}$) or function fields in one variable over a finite field
• Abelian extensions are Galois extensions with an abelian Galois group
• The Artin map is a homomorphism from the idele class group of a global field to the Galois group of its maximal abelian extension
  • It generalizes the Frobenius element in Galois theory
• The idele group of a global field is the restricted direct product of the multiplicative groups of its completions with respect to all inequivalent absolute values
• The idele class group is the quotient of the idele group by the diagonal embedding of the global units

Historical Context and Development

• Class field theory originated in the early 20th century, with contributions from mathematicians such as Hilbert, Takagi, Artin, and Chevalley
• Hilbert's Theorem 90, which states that $H^1(G, L^\times) = 0$ for a cyclic extension $L/K$ and $G = \mathrm{Gal}(L/K)$, played a crucial role in the development of the theory
• Takagi's work in the 1920s established the main results of class field theory for algebraic number fields
• Artin's reciprocity law, formulated in the 1920s, provided a generalization of the quadratic reciprocity law and laid the foundation for the Artin map
• Chevalley's idele-theoretic approach in the 1940s provided a unified treatment of class field theory for both local and global fields
• Subsequent developments include the cohomological formulation of class field theory and its generalization to higher-dimensional schemes (e.g., Grothendieck's étale cohomology)

Fundamental Theorems

• The Existence Theorem states that for every open subgroup $H$ of the idele class group of a global field $K$, there exists a unique abelian extension $L/K$ such that the Artin map induces an isomorphism between $C_K/H$ and $\mathrm{Gal}(L/K)$
• The Isomorphism Theorem asserts that the Artin map provides a canonical isomorphism between the idele class group of a global field and the Galois group of its maximal abelian extension
• The Reciprocity Law, also known as Artin's Reciprocity Law, states that the Artin map sends an idele to the Frobenius element associated with its prime components
  • This generalizes the quadratic reciprocity law and other reciprocity laws in number theory
• The Kronecker-Weber Theorem, a special case of class field theory, states that every abelian extension of $\mathbb{Q}$ is contained in a cyclotomic field
• The Local Existence Theorem and Local Isomorphism Theorem provide analogues of the global theorems for local fields

Applications in Arithmetic Geometry

• Class field theory plays a central role in the study of abelian varieties over global fields, particularly in the context of the Tate-Shafarevich group and the Birch and Swinnerton-Dyer conjecture
• The theory is used to construct and classify abelian extensions of number fields and function fields, which is important for understanding the arithmetic properties of algebraic varieties
• Class field theory provides a framework for studying the étale fundamental group of a scheme, which captures information about the scheme's arithmetic and geometric properties
• The theory is used to prove the Weil conjectures for abelian varieties over finite fields, which relate the zeta function of the variety to its arithmetic properties
• Class field theory is also applied in the study of modular forms and elliptic curves, particularly in the context of the Langlands program and the Taniyama-Shimura conjecture (now a theorem)

Computational Techniques

• Computing class groups and unit groups of number fields is a fundamental problem in computational number theory, and class field theory provides algorithms for constructing abelian extensions with prescribed ramification
• The Artin map can be computed explicitly using the arithmetic of ideles and their prime components
• Algorithms for computing ray class groups and Hilbert class fields rely on the machinery of class field theory
• Computational class field theory is used in the construction of cryptographic protocols, such as the Buchmann-Williams key exchange and the ABR (Algebraic Buchmann-Williams Roquette) cryptosystem
• Efficient algorithms for working with ideles and adeles, such as the Pohlig-Hellman algorithm for discrete logarithms, are essential for computational applications of class field theory

Connections to Other Areas of Mathematics

• Class field theory has deep connections to algebraic number theory, as it provides a powerful tool for studying abelian extensions of number fields and their arithmetic properties
• The theory is closely related to the study of L-functions and zeta functions, which encode arithmetic information about number fields, function fields, and algebraic varieties
• Class field theory plays a crucial role in the Langlands program, which seeks to unify various areas of mathematics, including number theory, representation theory, and harmonic analysis
• The local-global principle in class field theory, embodied by the Hasse principle and the Grunwald-Wang theorem, has analogues in other areas of mathematics, such as quadratic forms and Galois cohomology
• The cohomological formulation of class field theory, using Galois cohomology and étale cohomology, provides a bridge between class field theory and algebraic geometry

Advanced Topics and Current Research

• Higher class field theory, also known as Milnor K-theory, extends the ideas of class field theory to higher-dimensional schemes and provides a framework for studying non-abelian extensions
• The Langlands program, which includes the Langlands reciprocity conjecture and the Langlands functoriality conjecture, aims to generalize class field theory to non-abelian extensions and connect it with representation theory and automorphic forms
• The Hilbert's 12th problem, which seeks to generalize the Kronecker-Weber theorem to non-abelian extensions, remains an active area of research in class field theory
• The equivariant Tamagawa number conjecture, formulated by Bloch and Kato, relates special values of L-functions to arithmetic invariants of motives and provides a unifying framework for various conjectures in number theory
• The study of higher-dimensional class field theory, in the context of arithmetic schemes and higher-dimensional local fields, is an active area of research with connections to K-theory and motivic cohomology

Problem-Solving Strategies

• When working with class field theory problems, it is essential to have a solid understanding of the basic definitions and theorems, such as the Artin map, the reciprocity law, and the existence and isomorphism theorems
• Identifying the type of field (local or global) and the relevant groups (idele group, idele class group, Galois group) is crucial for applying the appropriate tools and techniques
• Exploiting the relationship between the Artin map and the Frobenius elements can often simplify computations and provide insights into the structure of abelian extensions
• Using the local-global principle, such as the Hasse principle and the Grunwald-Wang theorem, can help reduce global problems to local ones, which are often more tractable
• Drawing analogies between class field theory and other areas of mathematics, such as Galois theory and algebraic geometry, can provide useful intuition and suggest new approaches to problem-solving
• When faced with a complex problem, breaking it down into smaller subproblems and applying the relevant theorems and techniques to each subproblem can make the overall problem more manageable
• Consulting examples and worked-out problems in textbooks and research papers can help solidify understanding and provide templates for solving similar problems