Artin reciprocity law

The Artin reciprocity law, established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of the global class field theory.^[1] The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.

Significance

Artin's reciprocity law implies a description of the abelianization of the absolute Galois group of a global field K which is based on the local–global principle and the use of the Frobenius elements. Together with the Takagi existence theorem, it is used to describe the abelian extensions of K in terms of arithmetic of K and to understand the behavior of the nonarchimedean places in them. Therefore, the Artin reciprocity law can be interpreted as one of the main theorems of the global class field theory. It can be used to prove that Artin L-functions are meromorphic and for the proof of the Chebotarev density theorem.^[2]

Finite extensions of global fields

The definition of the Artin map for a finite (abelian) extension L/K of global fields (such as a finite abelian extension of Q) has a concrete description in terms of prime ideals and Frobenius elements.

If $\mathfrak{p}$ is a prime of K then the decomposition groups of primes $\mathfrak{P}$ above $\mathfrak{p}$ are equal in Gal(L/K) since the latter group is abelian. If $\mathfrak{p}$ is unramified in L, then the decomposition group $D_\mathfrak{p}$ is canonically isomorphic to the Galois group of the extension of residue fields $\mathcal{O}_{L,\mathfrak{P}}/\mathfrak{P}$ over $\mathcal{O}_{K,\mathfrak{p}}/\mathfrak{p}$. There is therefore a canonically defined Frobenius element in Gal(L/K) denoted by $\mathrm{Frob}_\mathfrak{p}$ or $\left(\frac{\mathfrak{p}}{L/K}\right)$. If Δ denotes the relative discriminant of L/K, the Artin symbol (or Artin map, or (global) reciprocity map) of L/K is defined on the group of prime-to-Δ fractional ideals, $I_K^\Delta$, by linearity:

$\begin{matrix} \left(\frac{\cdot}{L/K}\right):&I_K^\Delta&\longrightarrow&\mathrm{Gal}(L/K)\\ &\displaystyle{\prod_{i=1}^m\mathfrak{p}_i^{n_i}}&\mapsto&\displaystyle{\prod_{i=1}^m\left(\frac{\mathfrak{p}_i}{L/K}\right)^{n_i}.} \end{matrix}$

The Artin reciprocity law (or global reciprocity law) states that there is a modulus c of K such that the Artin map induces an isomorphism

$I_K^\mathbf{c}/i(K_{\mathbf{c},1})\mathrm{Nm}_{L/K}(I_L^\mathbf{c})\overset{\sim}{\longrightarrow}\mathrm{Gal}(L/K)$

where K_c,1 is the ray modulo c, Nm_L/K is the norm map associated to L/K and $I_L^\mathbf{c}$ is the fractional ideals of L prime to c. Such a modulus c is called a defining modulus for L/K. The smallest defining modulus is called the conductor of L/K and typically denoted $\scriptstyle\mathfrak{f}(L/K)$.

Example: quadratic fields

If d is a squarefree integer, K = Q, and $\scriptstyle L=\mathbf{Q}(\sqrt{d})$, then the Galois group Gal(L/Q) can be identified with {±1}. The discriminant of L over Q is d or 4d depending on whether d ≡ 1 (mod 4) or not. The Artin map is then defined on primes p that do not divide Δ by

$p\mapsto\left(\frac{d}{p}\right)$

where $\left(\frac{d}{p}\right)$ is the Legendre symbol.^[3]

Cohomological interpretation

Let L_v⁄K_v be a Galois extension of local fields with Galois group G. The local reciprocity law describes a canonical isomorphism

$\theta_v: K_v^{\times}/N_{L_v/K_v}(L_v^{\times}) \to G^{\text{ab}},$

called the local Artin symbol.

Let L⁄K be a Galois extension of global fields and C_L stand for the idèle class group of L. The maps θ_v for different places v of K can be assembled into a single global symbol map by multiplying the local components of an idèle class. One of the statements of the Artin reciprocity law is that this results in the canonical isomorphism^[4]

$\theta: C_K/{N_{L/K}(C_L)} \to \text{Gal}(L/K)^{\text{ab}}.$

A cohomological proof of the global reciprocity law can be achieved by first establishing that

$(\text{Gal}(K^{sep}/K),\varinjlim C_L)$

constitutes a class formation in the sense of Artin and Tate. Then one proves that

$\hat{H}^{0}( \text{Gal}(L/K), C_L) \simeq\hat{H}^{-2}( \text{Gal}(L/K), \mathbb{Z}),$

where $\hat{H}^{i}$ denote the Tate cohomology groups. Working out the cohomology groups establishes that θ is an isomorphism.

Alternative statement

An alternative version of the reciprocity law, leading to the Langlands program, connects Artin L-functions associated to abelian extensions of a number field with Hecke L-functions associated to characters of the idèle class group.^[5]

A Hecke character (or Größencharakter) of a number field K is defined to be a quasicharacter of the idèle class group of K. Robert Langlands interpreted Hecke characters as automorphic forms on the reductive algebraic group GL(1) over the ring of adeles of K.^[6]

Let E⁄K be an abelian Galois extension with Galois group G. Then for any character σ: G  → C^× (i.e. one-dimensional complex representation of the group G), there exists a Hecke character χ of K such that

$L_{E/K}^{\mathrm{Artin}}(\sigma, s) = L_{K}^{\mathrm{Hecke}}(\chi, s)$

where the left hand side is the Artin L-function associated to the extension with character σ and the right hand side is the Hecke L-function associated with χ, Section 7.D of.^[6]

The formulation of the Artin reciprocity law as an equality of L-functions allows formulation of a generalisation to n-dimensional representation, though a direct correspondence still lacking.

Notes

1. ^ Helmut Hasse, History of Class Field Theory, in Algebraic Number Theory, edited by Cassels and Frölich, Academic Press, 1967, pp. 266–279
2. ^ Jürgen Neukirch, Algebraische Zahlentheorie, Springer, 1992, Chapter VII
3. ^ Milne 2008, example 3.1
4. ^ Jürgen Neukirch, Algebraische Zahlentheorie, Springer, 1992, p. 408. In fact, a more precise version of the reciprocity law keeps track of the ramification.
5. ^ James Milne, Class Field Theory
6. ^ ^{a b} Stephen Gelbart, Automorphic Forms on Adele Groups, Annals of Mathematics Studies, No. 83, Princeton University Press, 1975. ISBN 0-691-08156-5

References

• Emil Artin, Über eine neue Art von L-Reihen, Abh. Math. Semin. Univ. Hamburg, 3 (1924), 89–108; Collected Papers, Addison Wesley, 1965, 105–124

• Emil Artin, Beweis des allgemeinen Reziprozitätsgesetzes, Abh. Math. Semin. Univ. Hamburg, 5 (1927), 353–363; Collected Papers, 131–141

• Emil Artin, Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetzes, Abh. Math. Semin. Univ. Hamburg, 7 (1930), 46–51; Collected Papers, 159–164

• Frei, Günther (2004), "On the history of the Artin reciprocity law in abelian extensions of algebraic number fields: how Artin was led to his reciprocity law", in Olav Arnfinn Laudal; Ragni Piene, The legacy of Niels Henrik Abel, Berlin: Springer-Verlag, pp. 267–294, ISBN 978-3-540-43826-7, MR2077576 

• Lang, Serge (1994), Algebraic number theory, Graduate Texts in Mathematics, 110 (2 ed.), New York: Springer-Verlag, ISBN 978-0-387-94225-4, MR1282723 
• Milne, James (2008), Class field theory (v4.0 ed.), http://jmilne.org/math/CourseNotes/cft.html, retrieved 2010-02-22