Weil reciprocity law
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).
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)
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.
- 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)
- Griffiths, Phillip; Harris, Joseph (1994). Principles of Algebraic Geometry. Wiley Classics Library. New York, NY: John Wiley & Sons Ltd. pp. 242–3. ISBN 0-471-05059-8. Zbl 0836.14001. for a proof in the Riemann surface case
- Arbarello, E.; De Concini, C.; Kac, V.G. (1989). "The infinite wedge representation and the reciprocity law for algebraic curves". In Ehrenpreis, Leon; Gunning, Robert C. Theta functions, Bowdoin 1987. (Proceedings of the 35th Summer Research Institute, Bowdoin Coll., Brunswick/ME July 6-24, 1987). Proceedings of Symposia in Pure Mathematics. 49.1. Providence, RI: American Mathematical Society. pp. 171–190. ISBN 0-8218-1483-4. Zbl 0699.22028.
- Serre, Jean-Pierre (1988). Algebraic groups and class fields. Graduate Texts in Mathematics. 117 (Translation of the French 2nd ed.). New York, etc.: Springer-Verlag. ISBN 3-540-96648-X. Zbl 0703.14001.