Reciprocity law

In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials <math>f(x)</math> with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial <math>f(x) = x^2 + ax + b</math> splits into linear terms when reduced mod <math>p</math>. That is, it determines for which prime numbers the relation

<math>f(x) \equiv f_p(x) = (x-n_p)(x-m_p) \text{ } (\text{mod } p)</math>

holds. For a general reciprocity law<ref>Template:Cite book</ref>^{pg 3}, it is defined as the rule determining which primes <math>p</math> the polynomial <math>f_p</math> splits into linear factors, denoted <math>\text{Spl}\{f(x) \}</math>.

There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert norm residue symbols (a,b/p), taking values in roots of unity, is equal to 1. Artin reformulated the reciprocity laws as a statement that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see.

The name reciprocity law was coined by Legendre in his 1785 publication Recherches d'analyse indéterminée,<ref name=EllFct>Template:Cite book</ref> because odd primes reciprocate or not in the sense of quadratic reciprocity stated below according to their residue classes <math>\bmod 4</math>. This reciprocating behavior does not generalize well, the equivalent splitting behavior does. The name reciprocity law is still used in the more general context of splittings.

Quadratic reciprocity

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In terms of the Legendre symbol, the law of quadratic reciprocity states

for positive odd primes <math>p,q</math> we have <math> \left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}.</math>

Using the definition of the Legendre symbol this is equivalent to a more elementary statement about equations.

For positive odd primes <math>p,q</math> the solubility of <math>n^2-p\equiv0\bmod q</math> for <math>n</math> determines the solubility of <math>m^2-q\equiv0\bmod p</math> for <math>m</math> and vice versa by the comparatively simple criterion whether <math>(-1)^{\frac{p-1}{2}\frac{q-1}{2}}</math> is <math>1</math> or <math>-1</math>.

By the factor theorem and the behavior of degrees in factorizations the solubility of such quadratic congruence equations is equivalent to the splitting of associated quadratic polynomials over a residue ring into linear factors. In this terminology the law of quadratic reciprocity is stated as follows.

For positive odd primes <math>p,q</math> the splitting of the polynomial <math>x^2-p</math> in <math>\bmod q</math>-residues determines the splitting of the polynomial <math>x^2-q</math> in <math>\bmod p</math>-residues and vice versa through the quantity <math>(-1)^{\frac{p-1}{2}\frac{q-1}{2}}\in\{\pm 1\}</math>.

This establishes the bridge from the name giving reciprocating behavior of primes introduced by Legendre to the splitting behavior of polynomials used in the generalizations.

Cubic reciprocity

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The law of cubic reciprocity for Eisenstein integers states that if α and β are primary (primes congruent to 2 mod 3) then

<math>\Bigg(\frac{\alpha}{\beta}\Bigg)_3 = \Bigg(\frac{\beta}{\alpha}\Bigg)_3. </math>

Quartic reciprocity

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In terms of the quartic residue symbol, the law of quartic reciprocity for Gaussian integers states that if π and θ are primary (congruent to 1 mod (1+i)³) Gaussian primes then

<math>\Bigg[\frac{\pi}{\theta}\Bigg]\left[\frac{\theta}{\pi}\right]^{-1}=

(-1)^{\frac{N\pi - 1}{4}\frac{N\theta-1}{4}}.</math>

Octic reciprocity

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Eisenstein reciprocity

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Suppose that ζ is an <math>l</math>th root of unity for some odd prime <math>l</math>. The power character is the power of ζ such that

<math>\left(\frac{\alpha}{\mathfrak{p}}\right)_l \equiv \alpha^{\frac{N(\mathfrak{p})-1}{l}} \pmod {\mathfrak{p}}</math>

for any prime ideal <math>\mathfrak{p}</math> of Z[ζ]. It is extended to other ideals by multiplicativity. The Eisenstein reciprocity law states that

<math> \left(\frac{a}{\alpha}\right)_l=\left(\frac{\alpha}{a}\right)_l </math>

for a any rational integer coprime to <math>l</math> and α any element of Z[ζ] that is coprime to a and <math>l</math> and congruent to a rational integer modulo (1–ζ)².

Kummer reciprocity

Suppose that ζ is an lth root of unity for some odd regular prime l. Since l is regular, we can extend the symbol {} to ideals in a unique way such that

<math> \left\{\frac{p}{q}\right\}^n=\left\{\frac{p^n}{q}\right\} </math> where n is some integer prime to l such that pⁿ is principal.

The Kummer reciprocity law states that

<math> \left\{\frac{p}{q}\right\}=\left\{\frac{q}{p}\right\} </math>

for p and q any distinct prime ideals of Z[ζ] other than (1–ζ).

Hilbert reciprocity

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In terms of the Hilbert symbol, Hilbert's reciprocity law for an algebraic number field states that

<math>\prod_v (a,b)_v = 1</math>

where the product is over all finite and infinite places. Over the rational numbers this is equivalent to the law of quadratic reciprocity. To see this take a and b to be distinct odd primes. Then Hilbert's law becomes <math>(p,q)_\infty(p,q)_2(p,q)_p(p,q)_q=1</math> But (p,q)_p is equal to the Legendre symbol, (p,q)_∞ is 1 if one of p and q is positive and –1 otherwise, and (p,q)₂ is (–1)^{(p–1)(q–1)/4}. So for p and q positive odd primes Hilbert's law is the law of quadratic reciprocity.

Artin reciprocity

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In the language of ideles, the Artin reciprocity law for a finite extension L/K states that the Artin map from the idele class group C_K to the abelianization Gal(L/K)^ab of the Galois group vanishes on N_L/K(C_L), and induces an isomorphism

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

Although it is not immediately obvious, the Artin reciprocity law easily implies all the previously discovered reciprocity laws, by applying it to suitable extensions L/K. For example, in the special case when K contains the nth roots of unity and L=K[a^1/n] is a Kummer extension of K, the fact that the Artin map vanishes on N_L/K(C_L) implies Hilbert's reciprocity law for the Hilbert symbol.

Local reciprocity

Hasse introduced a local analogue of the Artin reciprocity law, called the local reciprocity law. One form of it states that for a finite abelian extension of L/K of local fields, the Artin map is an isomorphism from <math> K^{\times}/N_{L/K}(L^{\times})</math> onto the Galois group <math> Gal(L/K) </math>.

Explicit reciprocity laws

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In order to get a classical style reciprocity law from the Hilbert reciprocity law Π(a,b)_p=1, one needs to know the values of (a,b)_p for p dividing n. Explicit formulas for this are sometimes called explicit reciprocity laws.

Power reciprocity laws

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A power reciprocity law may be formulated as an analogue of the law of quadratic reciprocity in terms of the Hilbert symbols as<ref name=Neu415>Neukirch (1999) p.415</ref>

<math>\left({\frac{\alpha}{\beta}}\right)_n \left({\frac{\beta}{\alpha}}\right)_n^{-1} = \prod_{\mathfrak{p} | n\infty} (\alpha,\beta)_{\mathfrak{p}} \ . </math>

Rational reciprocity laws

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A rational reciprocity law is one stated in terms of rational integers without the use of roots of unity.

Scholz's reciprocity law

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Shimura reciprocity

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Weil reciprocity law

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Langlands reciprocity

Template:Further The Langlands program includes several conjectures for general reductive algebraic groups, which for the special of the group GL₁ imply the Artin reciprocity law.

Yamamoto's reciprocity law

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Yamamoto's reciprocity law is a reciprocity law related to class numbers of quadratic number fields.

See also

• Hilbert's ninth problem
• Stanley's reciprocity theorem

References

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• Template:Citation. Correction, ibid. 80 (1973), 281.

Survey articles

• Reciprocity laws and Galois representations: recent breakthroughs