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- Weil_reciprocity_law 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, thenf((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)Pis 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)fagbwith 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).".
- Weil_reciprocity_law wikiPageID "2968782".
- Weil_reciprocity_law wikiPageRevisionID "556133930".
- Weil_reciprocity_law hasPhotoCollection Weil_reciprocity_law.
- Weil_reciprocity_law subject Category:Algebraic_curves.
- Weil_reciprocity_law type Abstraction100002137.
- Weil_reciprocity_law type AlgebraicCurves.
- Weil_reciprocity_law type Attribute100024264.
- Weil_reciprocity_law type Curve113867641.
- Weil_reciprocity_law type Line113863771.
- Weil_reciprocity_law type Shape100027807.
- Weil_reciprocity_law 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.".
- Weil_reciprocity_law label "Weil reciprocity law".
- Weil_reciprocity_law sameAs m.08gryk.
- Weil_reciprocity_law sameAs Q7980189.
- Weil_reciprocity_law sameAs Q7980189.
- Weil_reciprocity_law sameAs Weil_reciprocity_law.
- Weil_reciprocity_law wasDerivedFrom Weil_reciprocity_law?oldid=556133930.
- Weil_reciprocity_law isPrimaryTopicOf Weil_reciprocity_law.