Everett W. Howe: The Weil pairing and the Hilbert symbol, Math. Ann. 305 (1996) 387–392, MR 97b:11084, Zbl 0854.11031.
An official version is available online.
The Weil pairing on the m-torsion of the Jacobian of a curve can be defined in two ways, one using the canonical polarization of a Jacobian, and the other using an explicit formula. The explicit formula bears a striking resemblance to H. L. Schmidt's formula for the tame Hilbert symbol. I show how Schmidt's formula can be used to prove that the two definitions of the Weil pairing are equivalent.