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The Weil pairing and the Hilbert symbol. (English) Zbl 0854.11031
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.

MSC:
11G20 Curves over finite and local fields
11R37 Class field theory
11R58 Arithmetic theory of algebraic function fields
14H25 Arithmetic ground fields for curves
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References:
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