In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity. 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 generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between and. Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert norm residue symbols, taking values in roots of unity, is equal to 1. Artin reformulated the reciprocity laws as a statement that the Artin symbol from ideals 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.
Quadratic reciprocity
In terms of the Legendre symbol, the law of quadratic reciprocity for positive odd primes states
Cubic reciprocity
The law of cubic reciprocity for Eisenstein integers states that if α and β are primary then
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 Gaussian primes then
Octic reciprocity
Eisenstein reciprocity
Suppose that ζ is an th root of unity for some odd prime. The power character is the power of ζ such that for any prime ideal of Z. It is extended to other ideals by multiplicativity. The Eisenstein reciprocity law states that for a any rational integer coprime to and α any element ofZ that is coprime to a and and congruent to a rational integer modulo 2.
Kummer reciprocity
Suppose that ζ is an lth root of unity for some odd regular primel. Since l is regular, we can extend the symbol to ideals in a unique way such that The Kummer reciprocity law states that for p and q any distinct prime ideals of Z other than.
Hilbert reciprocity
In terms of the Hilbert symbol, Hilbert's reciprocity law for an algebraic number field states that 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 But p is equal to the Legendre symbol, ∞ is 1 if one of p and q is positive and –1 otherwise, and 2 is /4. So for p and q positive odd primes Hilbert's law is the law of quadratic reciprocity.
In the language of ideles, the Artin reciprocity law for a finite extensionL/K states that the Artin map from the idele class groupCK to the abelianization Galab of the Galois group vanishes on NL/K, and induces an isomorphism 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 is a Kummer extension of K, the fact that the Artin map vanishes on NL/K 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 onto the Galois group.
Explicit reciprocity laws
In order to get a classical style reciprocity law from the Hilbert reciprocity law Πp=1, one needs to know the values of 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
Rational reciprocity laws
A rational reciprocity law is one stated in terms of rational integers without the use of roots of unity.
The Langlands program includes several conjectures for general reductive algebraic groups, which for the special of the group GL1 imply the Artin reciprocity law.
