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| Local reciprocity law | |
|---|---|
| Name | Local reciprocity law |
| Field | Number theory |
| Related | Local class field theory, Global class field theory, Galois theory |
Local reciprocity law is a cornerstone result in number theory and algebraic number theory that describes the relationship between multiplicative groups of local fields and abelian extensions of those fields. It provides an explicit isomorphism between the profinite completion of the multiplicative group of a non-archimedean local field and the Galois group of its maximal abelian extension, connecting objects studied by Carl Friedrich Gauss, Ernst Eduard Kummer, Richard Dedekind, and Helmut Hasse. The law is a local analogue of results in global class field theory such as the Artin reciprocity law and underpins advances by John Tate, Emil Artin, Richard Brauer, and Claude Chevalley.
The statement identifies a canonical continuous homomorphism, the local reciprocity map or local Artin map, from the multiplicative group K^× of a non-archimedean local field K (for example Q_p, finite extensions of Q_p, or F_q((T))) to the Galois group Gal(K^ab/K) of the maximal abelian extension K^ab. For finite abelian extensions L/K, the map induces an isomorphism K^×/N_{L/K}(L^×) ≅ Gal(L/K), echoing constructions from Artin in global class field theory and formalized in the framework used by Hasse and Zassenhaus. The law is compatible with the norm residue symbol, local symbols used by Hilbert in the Hilbert reciprocity law, and with cohomological interpretations due to Tate and Nakayama.
Origins trace to reciprocity phenomena observed by Gauss in his quadratic reciprocity law and further to work of Kummer on cyclotomic fields and Leopoldt on p-adic properties. Emil Artin formulated the global reciprocity law that suggested a local counterpart, which was developed by Hasse and contemporaries to handle completions such as Q_p and extensions studied by Eisenstein and Hilbert. The modern cohomological perspective emerged from work by John Tate, Claude Chevalley, and Serre who connected local reciprocity with Galois cohomology and duality theorems, while explicit descriptions for local symbols were refined by Iwasawa and Lubin alongside Tate's thesis.
Local reciprocity law is formulated within local class field theory, which classifies abelian extensions of a local field K via the topological group K^×. The central objects include the inertia subgroup and decomposition subgroup in Gal(L/K) for finite L/K, the reciprocity map, and the local Artin map constructed via Frobenius elements when K has finite residue field as in extensions of F_q((T)) or Q_p. Cohomological tools invoked include Tate cohomology groups H^i(G,M) for Galois groups G and modules M encountered in analyses by Tate and Serre, and duality theorems like local Tate duality which parallel Pontryagin duality used in Pontryagin-style classifications by Pontryagin.
Proofs of the local reciprocity law follow several routes: classical explicit constructions using cyclic extensions and norm groups as in approaches by Hasse and Artin; cohomological proofs leveraging group cohomology, cup products, and Tate duality as developed by Tate and Serre; and analytic or rigid-analytic methods using formal groups and Lubin–Tate theory introduced by Lubin and Tate. Lubin–Tate theory furnishes explicit local reciprocity via formal Lubin–Tate formal group laws, while cohomological proofs connect cup-product pairings with the local invariant map appearing in the Brauer group studies by Wedderburn and Brauer.
Concrete cases include K = Q_p and cyclotomic extensions studied by Kummer and Kronecker, where the reciprocity map sends p-adic units and uniformizers to Frobenius automorphisms in Gal(K(μ_{p^n})/K). For finite fields of constants like F_q((T)), explicit computations reduce to powers of the geometric Frobenius element studied by Weil in his work on zeta functions, while Lubin–Tate formal groups provide explicit generators for totally ramified abelian extensions. Computations in quadratic and cyclic extensions rely on classical symbols from Hilbert and explicit norm index calculations used by Hasse and Chevalley.
The law underlies the proofs of global reciprocity theorems including Artin reciprocity and the Tchebotarev density theorem applications to splitting of primes in abelian extensions examined by Chebotarev. It is used in local-global principles, local duality statements like Tate local duality, and the description of local components of automorphic representations in the Langlands program as articulated by Langlands and applied by Deligne and Jacquet. Local reciprocity is essential in explicit class field theory computations for fields in Iwasawa theory studied by Iwasawa and in the study of local factors of global L-functions in works of Grothendieck and Serre.
Generalizations include non-abelian reciprocity conjectures central to the Langlands correspondence for local fields, higher local class field theory for higher-dimensional local fields studied by Kato and Parshin, and analogues in arithmetic geometry such as the study of the étale fundamental group in Grothendieck's anabelian geometry. Related structures encompass the Brauer group of local fields, explicit Lubin–Tate module structures, and reciprocity maps appearing in Milnor K-theory and the norm residue isomorphism conjecture proved by Voevodsky and Rost.