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).

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)

f^ag^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^ag^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 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)
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.

Topics in algebraic curves

Rational curves

Elliptic curves

Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form

Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm

Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

Plane curves

Riemann surfaces

Constructions

Structure of curves

Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law

Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve

Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem

Singularities	Acnode Crunode Cusp Delta invariant Tacnode

Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves

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