Artin reciprocity

Let $L/K$ be a Galois extension of global fields and $C\_L$ stand for the idèle class group

of $L$. One of the statements of the Artin reciprocity law is that there is a canonical isomorphism called the global symbol mapNeukirch (1999) p.391Jürgen Neukirch, Algebraische Zahlentheorie, Springer, 1992, p. 408. In fact, a more precise version of the reciprocity law keeps track of the ramification.

: $\backslash theta:\; C\_K/\{N\_\{L/K\}(C\_L)\}\; \backslash to\; \backslash operatorname\{Gal\}(L/K)^\{\backslash text\{ab\}\},$

where $\backslash text\{ab\}$ denotes the abelianization of a group, and $\backslash operatorname\{Gal\}(L/K)$ is the Galois group of $L$ over $K$. The map $\backslash theta$ is defined by assembling the maps called the local Artin symbol, the local reciprocity map or the norm residue symbolSerre (1967) p.140Serre (1979) p.197

: $\backslash theta\_v:\; K\_v^\{\backslash times\}/N\_\{L\_v/K\_v\}(L\_v^\{\backslash times\})\; \backslash to\; G^\{\backslash text\{ab\}\},$

for different places $v$ of $K$. More precisely, $\backslash theta$ is given by the local maps $\backslash theta\_v$ on the $v$-component of an idèle class. The maps $\backslash theta\_v$ are isomorphisms. This is the content of the local reciprocity law, a main theorem of local class field theory.

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

: $(\backslash operatorname\{Gal\}(K^\{\backslash text\{sep\}\}/K),\backslash varinjlim\; C\_L)$

constitutes a class formation in the sense of Artin and Tate.Serre (1979) p.164 Then one proves that

: $\backslash hat\{H\}^\{0\}(\backslash operatorname\{Gal\}(L/K),\; C\_L)\; \backslash simeq\backslash hat\{H\}^\{-2\}(\backslash operatorname\{Gal\}(L/K),\; \backslash Z),$

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

{{See also|Quadratic reciprocity|Eisenstein reciprocity}}

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 Hasse 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 the 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 global class field theory. It can be used to prove that Artin L-functions are meromorphic, and also to prove the Chebotarev density theorem.Jürgen Neukirch, Algebraische Zahlentheorie, Springer, 1992, Chapter VII

Two years after the publication of his general reciprocity law in 1927, Artin rediscovered the transfer homomorphism of I. Schur and used the reciprocity law to translate the principalization problem for ideal classes of algebraic number fields into the group theoretic task of determining the kernels of transfers of finite non-abelian groups.{{citation|title=Idealklassen in oberkörpern und allgemeines reziprozitätsgesetz|first=Emil|last=Artin|authorlink=Emil Artin|journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg|date=December 1929|volume=7|issue=1|pages=46–51|doi=10.1007/BF02941159}}.

(See [https://math.stackexchange.com/questions/4131855/frobenius-elements#:~:text=A%20Frobenius%20element%20for%20P,some%20%CF%84%E2%88%88KP math.stackexchange.com] for an explanation of some of the terms used here)

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

If $\backslash mathfrak\{p\}$ is a prime of K then the decomposition groups of primes $\backslash mathfrak\{P\}$ above $\backslash mathfrak\{p\}$ are equal in Gal(L/K) since the latter group is abelian. If $\backslash mathfrak\{p\}$ is unramified in L, then the decomposition group $D\_\backslash mathfrak\{p\}$ is canonically isomorphic to the Galois group of the extension of residue fields $\backslash mathcal\{O\}\_\{L,\backslash mathfrak\{P\}\}/\backslash mathfrak\{P\}$ over $\backslash mathcal\{O\}\_\{K,\backslash mathfrak\{p\}\}/\backslash mathfrak\{p\}$. There is therefore a canonically defined Frobenius element in Gal(L/K) denoted by $\backslash mathrm\{Frob\}\_\backslash mathfrak\{p\}$ or $\backslash left(\backslash frac\{L/K\}\{\backslash mathfrak\{p\}\}\backslash right)$. 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, $I\_K^\backslash Delta$, by linearity:

:$\backslash begin\{cases\}$

\left(\frac{L/K}{\cdot}\right):I_K^\Delta \longrightarrow \operatorname{Gal}(L/K)\\

\prod_{i=1}^m\mathfrak{p}_i^{n_i} \longmapsto \prod_{i=1}^m\left(\frac{L/K}{\mathfrak{p}_i}\right)^{n_i}

\end{cases}

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

:$I\_K^\backslash mathbf\{c\}/i(K\_\{\backslash mathbf\{c\},1\})\backslash mathrm\{N\}\_\{L/K\}(I\_L^\backslash mathbf\{c\})\backslash overset\{\backslash sim\}\{\backslash longrightarrow\}\backslash mathrm\{Gal\}(L/K)$

where K_c,1 is the ray modulo c, N_L/K is the norm map associated to L/K and $I\_L^\backslash mathbf\{c\}$ 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 $\backslash mathfrak\{f\}(L/K).$

If $d\backslash neq1$ is a squarefree integer, $K=\backslash Q,$ and $L=\backslash Q(\backslash sqrt\{d\})$, then $\backslash operatorname\{Gal\}(L/\backslash Q)$ can be identified with {±1}. The discriminant Δ of L over $\backslash Q$ 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

:$p\backslash mapsto\backslash left(\backslash frac\{\backslash Delta\}\{p\}\backslash right)$

where $\backslash left(\backslash frac\{\backslash Delta\}\{p\}\backslash right)$ is the Kronecker symbol.{{harvnb|Lemmermeyer|2000|loc=§3.2}} More specifically, the conductor of $L/\backslash Q$ is the principal ideal (Δ) or (Δ)∞ according to whether Δ is positive or negative,{{harvnb|Milne|2008|loc=example 3.11}} and the Artin map on a prime-to-Δ ideal (n) is given by the Kronecker symbol $\backslash left(\backslash frac\{\backslash Delta\}\{n\}\backslash right).$ This shows that a prime p is split or inert in L according to whether $\backslash left(\backslash frac\{\backslash Delta\}\{p\}\backslash right)$ is 1 or −1.

Let m > 1 be either an odd integer or a multiple of 4, let $\backslash zeta\_m$ be a primitive mth root of unity, and let $L\; =\; \backslash Q(\backslash zeta\_m)$ be the mth cyclotomic field. $\backslash operatorname\{Gal\}(L/\backslash Q)$ can be identified with $(\backslash Z/m\backslash Z)^\{\backslash times\}$ by sending σ to a_σ given by the rule

:$\backslash sigma(\backslash zeta\_m)=\backslash zeta\_m^\{a\_\backslash sigma\}.$

The conductor of $L/\backslash Q$ is (m)∞,{{harvnb|Milne|2008|loc=example 3.10}} and the Artin map on a prime-to-m ideal (n) is simply n (mod m) in $(\backslash Z/m\backslash Z)^\{\backslash times\}.${{harvnb|Milne|2008|loc=example 3.2}}

Let p and $\backslash ell$ be distinct odd primes. For convenience, let $\backslash ell^*\; =\; (-1)^\{\backslash frac\{\backslash ell-1\}\{2\}\}\backslash ell$ (which is always 1 (mod 4)). Then, quadratic reciprocity states that

:$\backslash left(\backslash frac\{\backslash ell^*\}\{p\}\backslash right)=\backslash left(\backslash frac\{p\}\{\backslash ell\}\backslash right).$

The relation between the quadratic and Artin reciprocity laws is given by studying the quadratic field $F=\backslash Q(\backslash sqrt\{\backslash ell^*\})$ and the cyclotomic field $L=\backslash Q(\backslash zeta\_\backslash ell)$ as follows. First, F is a subfield of L, so if H = Gal(L/F) and $G=\; \backslash operatorname\{Gal\}(L/\backslash Q),$ then $\backslash operatorname\{Gal\}(F/\backslash Q)\; =\; G/H.$ Since the latter has order 2, the subgroup H must be the group of squares in $(\backslash Z/\backslash ell\backslash Z)^\{\backslash times\}.$ A basic property of the Artin symbol says that for every prime-to-ℓ ideal (n)

:$\backslash left(\backslash frac\{F/\backslash Q\}\{(n)\}\backslash right)=\backslash left(\backslash frac\{L/\backslash Q\}\{(n)\}\backslash right)\backslash pmod\; H.$

When n = p, this shows that $\backslash left(\backslash frac\{\backslash ell^*\}\{p\}\backslash right)=1$ if and only if, p modulo ℓ is in H, i.e. if and only if, p is a square modulo ℓ.

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.James Milne, [http://www.jmilne.org/math/CourseNotes/cft.html Class Field Theory]

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.{{citation | last = Gelbart | first = Stephen S. | authorlink = Stephen Gelbart | location = Princeton, N.J. | mr = 0379375 | publisher = Princeton University Press | series = Annals of Mathematics Studies | title = Automorphic forms on adèle groups | volume = 83| year = 1975}}.

Let $E/K$ be an abelian Galois extension with Galois group G. Then for any character $\backslash sigma:\; G\; \backslash to\; \backslash Complex^\{\backslash times\}$ (i.e. one-dimensional complex representation of the group G), there exists a Hecke character $\backslash chi$ of K such that

:$L\_\{E/K\}^\{\backslash mathrm\{Artin\}\}(\backslash sigma,\; s)\; =\; L\_\{K\}^\{\backslash mathrm\{Hecke\}\}(\backslash chi,\; s)$

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.

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

• List of eponymous laws

• Emil Artin (1924) "Über eine neue Art von L-Reihen", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 3: 89–108; Collected Papers, Addison Wesley (1965), 105–124
• Emil Artin (1927) "Beweis des allgemeinen Reziprozitätsgesetzes", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 5: 353–363; Collected Papers, 131–141
• Emil Artin (1930) "Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetzes", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 7: 46–51; Collected Papers, 159–164
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Category:Class field theory