Weil Pairing - Generalisation To Abelian Varieties

Generalisation To Abelian Varieties

For abelian varieties over an algebraically closed field K, the Weil pairing is a nondegenerate pairing

for 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 (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). 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.