Editing Weil pairing (section)

==Generalisation to abelian varieties==
For [[abelian varieties]] over an algebraically closed field ''K'', the Weil pairing is a nondegenerate pairing

:<math>A[n] \times A^\vee[n] \longrightarrow \mu_n</math>

for all ''n'' prime to the characteristic of '' K''.<ref>[[James Milne (mathematician)|James Milne]], ''Abelian Varieties'', available at www.jmilne.org/math/</ref> Here <math>A^\vee</math> denotes the [[dual abelian variety]] of ''A''. This is the so-called ''Weil pairing'' for higher dimensions. If ''A'' is equipped with a [[Abelian variety#Polarisation and dual abelian variety|polarisation]]

:<math>\lambda: A \longrightarrow A^\vee</math>,
then composition gives a (possibly degenerate) pairing

:<math>A[n] \times A[n] \longrightarrow \mu_n.</math>

If ''C'' is a projective, nonsingular curve of genus ≥&nbsp;0 over ''k'', and ''J'' its [[Jacobian variety|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

:<math> J[n]\times J[n] \longrightarrow \mu_n</math>

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''.