Reciprocity law

Last updated September 10, 2023

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

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.

Related Research Articles

In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of an odd prime number p: its value at a (nonzero) quadratic residue mod p is 1 and at a non-quadratic residue (non-residue) is −1. Its value at zero is 0.

<span class="mw-page-title-main">Quadratic reciprocity</span> Gives conditions for the solvability of quadratic equations modulo prime numbers

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is:

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that:

In mathematics, the Weil conjectures were highly influential proposals by André Weil (1949). They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.

In number theory, the Kronecker symbol, written as $or, is a generalization of the Jacobi symbol to all integers . It was introduced by Leopold Kronecker.$

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζ_K(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the field of rational numbers Q). It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its values encode arithmetic data of K. The extended Riemann hypothesis states that if ζ_K(s) = 0 and 0 < Re(s) < 1, then Re(s) = 1/2.

In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number $p$ , then this root can be lifted to a unique root modulo any higher power of $p$ . More generally, if a polynomial factors modulo $p$ into two coprime polynomials, this factorization can be lifted to a factorization modulo any higher power of $p$ .

Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity.

The Artin reciprocity law, which was established by Emil Artin in a series of papers, is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.

In mathematics, a Kloosterman sum is a particular kind of exponential sum. They are named for the Dutch mathematician Hendrik Kloosterman, who introduced them in 1926 when he adapted the Hardy–Littlewood circle method to tackle a problem involving positive definite diagonal quadratic forms in four as opposed to five or more variables, which he had dealt with in his dissertation in 1924.

In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred proofs of the law of quadratic reciprocity have been published.

In mathematics, the Hilbert symbol or norm-residue symbol is a function from K^× × K^× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers. It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by David Hilbert in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields.

Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x³ ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x³ ≡ p is solvable if and only if x³ ≡ q is solvable.

Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x⁴ ≡ p is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x⁴ ≡ p to that of x⁴ ≡ q.

In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension. The definition of the conductor is related to the Artin map.

In algebraic number theory the n-th power residue symbol is a generalization of the (quadratic) Legendre symbol to n-th powers. These symbols are used in the statement and proof of cubic, quartic, Eisenstein, and related higher reciprocity laws.

In algebraic number theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers. It is one of the earliest and simplest of the higher reciprocity laws, and is a consequence of several later and stronger reciprocity laws such as the Artin reciprocity law. It was introduced by Eisenstein (1850), though Jacobi had previously announced a similar result for the special cases of 5th, 8th and 12th powers in 1839.

In mathematics, an explicit reciprocity law is a formula for the Hilbert symbol of a local field. The name "explicit reciprocity law" refers to the fact that the Hilbert symbols of local fields appear in Hilbert's reciprocity law for the power residue symbol. The definitions of the Hilbert symbol are usually rather roundabout and can be hard to use directly in explicit examples, and the explicit reciprocity laws give more explicit expressions for the Hilbert symbol that are sometimes easier to use.

In the mathematical field of algebraic number theory, the concept of principalization refers to a situation when, given an extension of algebraic number fields, some ideal of the ring of integers of the smaller field isn't principal but its extension to the ring of integers of the larger field is. Its study has origins in the work of Ernst Kummer on ideal numbers from the 1840s, who in particular proved that for every algebraic number field there exists an extension number field such that all ideals of the ring of integers of the base field become principal when extended to the larger field. In 1897 David Hilbert conjectured that the maximal abelian unramified extension of the base field, which was later called the Hilbert class field of the given base field, is such an extension. This conjecture, now known as principal ideal theorem, was proved by Philipp Furtwängler in 1930 after it had been translated from number theory to group theory by Emil Artin in 1929, who made use of his general reciprocity law to establish the reformulation. Since this long desired proof was achieved by means of Artin transfers of non-abelian groups with derived length two, several investigators tried to exploit the theory of such groups further to obtain additional information on the principalization in intermediate fields between the base field and its Hilbert class field. The first contributions in this direction are due to Arnold Scholz and Olga Taussky in 1934, who coined the synonym capitulation for principalization. Another independent access to the principalization problem via Galois cohomology of unit groups is also due to Hilbert and goes back to the chapter on cyclic extensions of number fields of prime degree in his number report, which culminates in the famous Theorem 94.

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

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[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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