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
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).
To remove the condition of disjoint support, for each point P on C a local symbol
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)
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
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.