Weil pairing
In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) 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 (1940) 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.is known to be a Cartesian product of two cyclic groups of order n. The Weil pairing produces an n-th root of unity by means of Kummer theory, for any two pointsTherefore if we define we shall have an n-th root of unity (as translating n times must give 1) other than 1.With this definition it can be shown that w is alternating and bilinear,[1] giving rise to a non-degenerate pairing on the n-torsion.If A is equipped with a polarisation then composition gives a (possibly degenerate) 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 (see autoduality of Jacobians).