Weil pairing
In mathematics, the Weil pairing is a pairing on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity. More generally there is a similar Weil pairing between points of order n of an abelian variety and its dual. It was introduced by André Weil for Jacobians of curves, who gave an abstract algebraic definition; the corresponding results for elliptic functions were known, and can be expressed simply by use of the Weierstrass sigma function.
Formulation
Choose an elliptic curve E defined over a field K, and an integer n > 0 if char such that K contains a primitive nth [root of unity]. Then the n-torsion on is known to be a Cartesian product of two cyclic groups of order n. The Weil pairing produces an n-th root of unityby means of Kummer theory, for any two points, where and.
A down-to-earth construction of the Weil pairing is as follows. Choose a function F in the field of an algebraic variety|function field] of E over the algebraic closure of K with divisor
So F has a simple zero at each point P + kQ, and a simple pole at each point kQ if these points are all distinct. Then F is well-defined up to multiplication by a constant. If G is the translation of F by Q, then by construction G has the same divisor, so the function G/F is constant.
Therefore if we define
we shall have an n-th root of unity other than 1. With this definition it can be shown that w is alternating and bilinear, giving rise to a non-degenerate pairing on the n-torsion.
The Weil pairing does not extend to a pairing on all the torsion points because the pairings for different n are not the same. However
they do fit together to give a pairing Tℓ × Tℓ → Tℓ on the Tate module Tℓ of the elliptic curve E to the Tate module Tℓ of the multiplicative group.
Generalisation to abelian varieties
For abelian varieties over an algebraically closed field K, the Weil pairing is a nondegenerate pairingfor all n prime to the characteristic of K. Here denotes the dual abelian variety of A. This is the so-called Weil pairing for higher dimensions. If A is equipped with a polarisation
then composition gives a pairing
If C is a projective, nonsingular curve of genus ≥ 0 over k, and J its Jacobian, then the theta-divisor of J induces a principal polarisation of J, which in this particular case happens to be an isomorphism. Hence, composing the Weil pairing for J with the polarisation gives a nondegenerate pairing
for all n prime to the characteristic of k.
As in the case of elliptic curves, explicit formulae for this pairing can be given in terms of divisors of C.