mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Weil reciprocity law is a result of

André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...

holding in the function field ''K''(''C'') of an

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...

''C'' over an

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...

''K''. Given functions ''f'' and ''g'' in ''K''(''C''), i.e. rational functions on ''C'', then :''f''((''g'')) = ''g''((''f'')) where the notation has this meaning: (''h'') is the

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

of the function ''h'', or in other words the

formal sum In mathematics, a formal sum, formal series, or formal linear combination may be: *In group theory, an element of a free abelian group, a sum of finitely many elements from a given basis set multiplied by integer coefficients. *In linear algebra, an ...

of its zeroes and poles counted with

multiplicity Multiplicity may refer to: In science and the humanities * Multiplicity (mathematics), the number of times an element is repeated in a multiset * Multiplicity (philosophy), a philosophical concept * Multiplicity (psychology), having or using multi ...

; and a function applied to a formal sum means the product (with multiplicities, poles counting as a negative multiplicity) of the values of the function at the points of the divisor. With this definition there must be the side-condition, that the divisors of ''f'' and ''g'' have disjoint support (which can be removed). In the case of the

projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...

, this can be proved by manipulations with the

resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (ove ...

of polynomials. To remove the condition of disjoint support, for each point ''P'' on ''C'' a ''local symbol'' :(''f'', ''g'')_''P'' is defined, in such a way that the statement given is equivalent to saying that the product over all ''P'' of the local symbols is 1. When ''f'' and ''g'' both take the values 0 or ∞ at ''P'', the definition is essentially in limiting or

removable singularity In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourh ...

terms, by considering (up to sign) :''f''^''a''''g''^''b'' with ''a'' and ''b'' such that the function has neither a zero nor a pole at ''P''. This is achieved by taking ''a'' to be the multiplicity of ''g'' at ''P'', and −''b'' the multiplicity of ''f'' at ''P''. The definition is then ::(''f'', ''g'')_''P'' = (−1)^''ab'' ''f''^''a''''g''^''b''. See for example

Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...

, ''Groupes algébriques et corps de classes'', pp. 44–46, for this as a special case of a theory on mapping algebraic curves into commutative groups. There is a generalisation of

Serge Lang Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...

abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular function ...

(Lang, ''Abelian Varieties'').

References

*André Weil, ''Oeuvres Scientifiques I'', p. 291 (in ''Lettre à Artin'', a 1942 letter to Artin, explaining the 1940 ''Comptes Rendus'' note ''Sur les fonctions algébriques à corps de constantes finis'') * for a proof in the

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

case * * {{Algebraic curves navbox Algebraic curves Theorems in algebraic geometry