The Weil pairing and the Hilbert symbol

Abstract

Let C be a geometrically irreducible curve over a field k, let k be an algebraic closure of k, and let m be any positive integer not divisible by the characteristic of k. The Jacobian variety J of C comes equipped with a principal p~ar ization A, which is in particular an isomorphism from J to its dual variety J . The polarization A gives us an isomorphism between the m-torsion J m of J and its Cartier dual, and this isomorphism turns the natural pairing Jm • Jm ~ l~m into the Weilpairing em:Jm • ~ I~,n. Suppose D and E are k-divisors on C whose mth powers are principal, say m D = d i v f and mE = div g, where f and y are k-functions on C. The following well-known theorem tells how the Weil pairing on the classes of D and E in Jm(k) can be calculated.