About: Weil reciprocity law

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In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field K(C) of an algebraic curve C over an algebraically closed field K. Given functions f and g in K(C), i.e. rational functions on C, then f((g)) = g((f)) In the case of the projective line, this can be proved by manipulations with the resultant of polynomials. To remove the condition of disjoint support, for each point P on C a local symbol (f, g)P fagb (f, g)P = (−1)ab fagb. There is a generalisation of Serge Lang to abelian varieties (Lang, Abelian Varieties).

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rdfs:label	Weil reciprocity law (en)
rdfs:comment	In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field K(C) of an algebraic curve C over an algebraically closed field K. Given functions f and g in K(C), i.e. rational functions on C, then f((g)) = g((f)) In the case of the projective line, this can be proved by manipulations with the resultant of polynomials. To remove the condition of disjoint support, for each point P on C a local symbol (f, g)P fagb (f, g)P = (−1)ab fagb. There is a generalisation of Serge Lang to abelian varieties (Lang, Abelian Varieties). (en)
dct:subject	Theorems in algebraic geometry Algebraic curves
Wikipage page ID	2968782 (xsd:integer)
Wikipage revision ID	1086477717 (xsd:integer)
Link from a Wikipage to another Wikipage	Multiplicity (mathematics) Algebraically closed field Resultant Riemann surface Mathematics André Weil Function field of an algebraic variety Algebraic curve American Mathematical Society Abelian varieties Projective line Removable singularity Jean-Pierre Serre Theorems in algebraic geometry Algebraic curves Picard group Serge Lang Formal sum Springer-Verlag
sameAs	Weil reciprocity law Weil reciprocity law Weil reciprocity law Weil reciprocity law
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:Algebraic_curves_navbox
has abstract	In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field K(C) of an algebraic curve C over an algebraically closed field 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 of the function h, or in other words the formal sum of its zeroes and poles counted with multiplicity; 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 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) fagb 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 fagb. 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 to abelian varieties (Lang, Abelian Varieties). (en)
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foaf:isPrimaryTopicOf	wikipedia-en:Weil_reciprocity_law
is Link from a Wikipage to another Wikipage of	List of algebraic geometry topics Reciprocity André Weil List of things named after André Weil Weil's reciprocity law Weil reciprocity Weil reciprocity for algebraic curves
is Wikipage redirect of	Weil's reciprocity law Weil reciprocity Weil reciprocity for algebraic curves
is known for of	André Weil
is foaf:primaryTopic of	wikipedia-en:Weil_reciprocity_law

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