Weil Pairing - Formulation

Formulation

Choose an elliptic curve E defined over a field K, and an integer n > 0 (we require n to be prime to char(K) if char(K) > 0) 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 unity

by 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 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 (as translating n times must give 1) other than 1. With this definition it can be shown that w is antisymmetric 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 (the direct limit of n-torsion points) because the pairings for different n are not the same. However they do fit together to give a pairing T_ℓ(E) × T_ℓ(E) → T_ℓ(μ) on the Tate module T_ℓ(E) of the elliptic curve E (the inverse limit of the ℓn-torsion points) to the Tate module T_ℓ(μ) of the multiplicative group (the inverse limit of ℓn roots of unity).