Reciprocity law

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

$f(x)\equiv f_{p}(x)=(x-n_{p})(x-m_{p}){\text{ }}({\text{mod }}p)$

holds. For a general reciprocity law^[1]^{pg 3}, it is defined as the rule determining which primes $p$ the polynomial $f_{p}$ splits into linear factors, denoted ${\text{Spl}}\{f(x)\}$ .

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,^[2] because odd primes reciprocate or not in the sense of quadratic reciprocity stated below according to their residue classes ${\bmod {4}}$ . 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

In terms of the Legendre symbol, the law of quadratic reciprocity states

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

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

For positive odd primes  $p,q$  the solubility of  $n^{2}-p\equiv 0{\bmod {q}}$  for  $n$  determines the solubility of  $m^{2}-q\equiv 0{\bmod {p}}$  for  $m$  and vice versa by the comparatively simple criterion whether  $(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}$  is  $1$  or  $-1$ .

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  $p,q$  the splitting of the polynomial  $x^{2}-p$  in  ${\bmod {q}}$ -residues determines the splitting of the polynomial  $x^{2}-q$  in  ${\bmod {p}}$ -residues and vice versa through the quantity  $(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}\in \{\pm 1\}$ .

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

The law of cubic reciprocity for Eisenstein integers states that if α and β are primary (primes congruent to 2 mod 3) then

{\Bigg (}{\frac {\alpha }{\beta }}{\Bigg )}_{3}={\Bigg (}{\frac {\beta }{\alpha }}{\Bigg )}_{3}.

Quartic reciprocity

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

{\Bigg [}{\frac {\pi }{\theta }}{\Bigg ]}\left[{\frac {\theta }{\pi }}\right]^{-1}=(-1)^{{\frac {N\pi -1}{4}}{\frac {N\theta -1}{4}}}.

Octic reciprocity

Eisenstein reciprocity

Suppose that ζ is an $l$ th root of unity for some odd prime $l$ . The power character is the power of ζ such that

\left({\frac {\alpha }{\mathfrak {p}}}\right)_{l}\equiv \alpha ^{\frac {N({\mathfrak {p}})-1}{l}}{\pmod {\mathfrak {p}}}

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

\left({\frac {a}{\alpha }}\right)_{l}=\left({\frac {\alpha }{a}}\right)_{l}

for a any rational integer coprime to $l$ and α any element of Z[ζ] that is coprime to a and $l$ 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

\left\{{\frac {p}{q}}\right\}^{n}=\left\{{\frac {p^{n}}{q}}\right\}

where n is some integer prime to l such that pⁿ is principal.

The Kummer reciprocity law states that

\left\{{\frac {p}{q}}\right\}=\left\{{\frac {q}{p}}\right\}

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

Hilbert reciprocity

In terms of the Hilbert symbol, Hilbert's reciprocity law for an algebraic number field states that

\prod _{v}(a,b)_{v}=1

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 $(p,q)_{\infty }(p,q)_{2}(p,q)_{p}(p,q)_{q}=1$ 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

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

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

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 $K^{\times }/N_{L/K}(L^{\times })$ onto the Galois group $Gal(L/K)$ .

Explicit reciprocity laws

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

A power reciprocity law may be formulated as an analogue of the law of quadratic reciprocity in terms of the Hilbert symbols as^[3]

\left({\frac {\alpha }{\beta }}\right)_{n}\left({\frac {\beta }{\alpha }}\right)_{n}^{-1}=\prod _{{\mathfrak {p}}|n\infty }(\alpha ,\beta )_{\mathfrak {p}}\ .

Rational reciprocity laws

A rational reciprocity law is one stated in terms of rational integers without the use of roots of unity.

Scholz's reciprocity law

Shimura reciprocity

Weil reciprocity law

Langlands reciprocity

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

Yamamoto's reciprocity law is a reciprocity law related to class numbers of quadratic number fields.

References

^ Hiramatsu, Toyokazu; Saito, Seiken (2016-05-04). An Introduction to Non-Abelian Class Field Theory. Series on Number Theory and Its Applications. WORLD SCIENTIFIC. doi:10.1142/10096. ISBN 978-981-314-226-8.
^ Chandrasekharan, K. (1985). Elliptic Functions. Grundlehren der mathematischen Wissenschaften. Vol. 281. Berlin: Springer. p. 152f. doi:10.1007/978-3-642-52244-4. ISBN 3-540-15295-4.
^ Neukirch (1999) p.415

Frei, Günther (1994), "The reciprocity law from Euler to Eisenstein", in Chikara, Sasaki (ed.), The intersection of history and mathematics. Papers presented at the history of mathematics symposium, held in Tokyo, Japan, August 31 - September 1, 1990, Sci. Networks Hist. Stud., vol. 15, Basel: Birkhäuser, pp. 67–90, doi:10.1090/S0002-9904-1972-12997-5, ISBN 9780817650292, MR 0308080, Zbl 0818.01002
Hilbert, David (1897), "Die Theorie der algebraischen Zahlkörper", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 4: 175–546, ISSN 0012-0456
Hilbert, David (1998), The theory of algebraic number fields, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03545-0, ISBN 978-3-540-62779-1, MR 1646901
Lemmermeyer, Franz (2000), Reciprocity laws. From Euler to Eisenstein, Springer Monographs in Mathematics, Berlin: Springer-Verlag, doi:10.1007/978-3-662-12893-0, ISBN 3-540-66957-4, MR 1761696, Zbl 0949.11002
Lemmermeyer, Franz, Reciprocity laws. From Kummer to Hilbert
Neukirch, Jürgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, vol. 322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN 3-540-65399-6, Zbl 0956.11021
Stepanov, S. A. (2001) [1994], "Reciprocity laws", Encyclopedia of Mathematics, EMS Press
Wyman, B. F. (1972), "What is a reciprocity law?", Amer. Math. Monthly, 79 (6): 571–586, doi:10.2307/2317083, JSTOR 2317083, MR 0308084. Correction, ibid. 80 (1973), 281.

Survey articles

Reciprocity laws and Galois representations: recent breakthroughs

[1] Hiramatsu, Toyokazu; Saito, Seiken (2016-05-04). An Introduction to Non-Abelian Class Field Theory. Series on Number Theory and Its Applications. WORLD SCIENTIFIC. doi:10.1142/10096. ISBN 978-981-314-226-8.

[EllFct-2] Chandrasekharan, K. (1985). Elliptic Functions. Grundlehren der mathematischen Wissenschaften. Vol. 281. Berlin: Springer. p. 152f. doi:10.1007/978-3-642-52244-4. ISBN 3-540-15295-4.

[Neu415-3] Neukirch (1999) p.415

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