= Weil pairing for an abelian variety
{c}
{title2=$e_N:A[N]\times\widehat A[N]\to\mu_N$}
For an <abelian variety> and an integer $N$ invertible in the base <field>, a torsion <line bundle> $M\in\widehat A[N]$ has trivial pullback by $[N]$. Choose a trivialization. The descent action of the kernel $A[N]$ on that trivial <line bundle> acts through a character $\chi_M:A[N]\to\mu_N$. Define $e_N(P,M)=\chi_M(P)$. This is independent of rescaling the trivialization and bilinear under translation and <tensor product>. A trivial character descends the trivial bundle, so $M$ is trivial. Since both groups have order $N^{2\dim A}$, the character map is an isomorphism and the <Weil pairing for an abelian variety> is perfect. The <theta autoduality of a Jacobian> turns it into a pairing on two copies of the <Jacobian variety>'s torsion group.
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