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 ''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 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, 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 (over t ...

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 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, ''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 ...

to abelian varieties (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