Reciprocity law
Encyclopedia
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 (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 law of quadratic reciprocity for positive odd primes states
s states that if π and θ are primary (congruent to 1 mod (1+i)3) Gaussian primes then
The power character is the power of ζ such that
for any prime ideal p of Z[ζ]. It is extended to other ideals in by multiplicativity.
The Eisenstein reciprocity law states that
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.
where n is some integer prime to l such that pn is principal.
The Kummer reciprocity law states that
for p and q any distinct prime ideals of Z[ζ] other than (1–ζ).
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,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.
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[a1/n] is a Kummer extension of K, the fact that the Artin map vanishes on NL/K(CL) implies Hilbert's reciprocity law for the Hilbert symbol.
from L*/N(K*) onto the Galois group Gal(L/K).
includes several conjectures for general reductive algebraic groups, which for the special of the group GL1 imply the Artin reciprocity law.
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
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
is an nth power residue modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
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.
Quadratic reciprocity
In terms of the Legendre symbolLegendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic non-residue is −1....
, 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 (primes congruent to 2 mod 3) thenQuartic reciprocity
In terms of the quartic residue symbol, the law of quartic reciprocity for Gaussian integerGaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...
s states that if π and θ are primary (congruent to 1 mod (1+i)3) Gaussian primes then
Eisenstein reciprocity
Suppose that ζ is an lth root of unity for some odd prime l.The power character is the power of ζ such that
for any prime ideal p of Z[ζ]. It is extended to other ideals in by multiplicativity.
The Eisenstein reciprocity law states that
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 where n is some integer prime to l such that pn is principal.The Kummer reciprocity law states that
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 thatwhere 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,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 if ideles, the Artin reciprocity law for a finite extension L/K state that the Artin map from the idele class group CK to the abelianization Gal(L/K)ab of the Galois group vanishes on NL/K(CL), 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[a1/n] is a Kummer extension of K, the fact that the Artin map vanishes on NL/K(CL) 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 isomorphismfrom L*/N(K*) 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.Langlands reciprocity
The Langlands programLanglands program
The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
includes several conjectures for general reductive algebraic groups, which for the special of the group GL1 imply the Artin reciprocity law.