The Weil pairing and the Hilbert symbol
Let \(C\) be a curve over a field \(k\), and suppose \(D\) and \(E\) are degree-zero divisors on \(C\) that represent \(m\)-torsion points on the Jacobian \(J\) of \(C\), so that \(mD= \text{div } f\) and \(mE=\text{div } g\) for some functions \(f\) and \(g\) on \(C\). The Weil pairing \(e_m\) on the \(m\)-torsion of \(J\), applied to the torsion points \([D]\) and \([E]\), can be calculated by the well-known formula \[ e_m ([D ],[ E])= \prod_P (-1 )^{m(\text{ord}_P D)(\text{ord}_P E)} {{g^{\text{ord}_P D}} \over {f^{\text{ord}_P E}}} (P), \] where \(P\) ranges over the geometric points of \(C\). We provide a new proof of this formula by reducing to the special case where \(k\) is finite. Our proof uses Kummer theory and class field theory to relate the Weil pairing to the Hilbert symbol, and in doing so explains the visual similarity between the formula above and Schmidt's explicit formula for the Hilbert symbol.
scientific article; zbMATH DE number 931965