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Weil pairing and tame symbols. (English) Zbl 1092.14052
Math. Notes 77, No. 5, 735-738 (2005); translation from Mat. Zametki 77, No. 5, 797-800 (2005).
Let \(J\) be the Jacobian variety of a curve \(C\) over an algebraically closed field. For every integer \(n>0\), the Weil pairing on the \(n\)-torsion of \(J\) is a perfect pairing \[ {\mathbf e}_n : J[n]\times J[n] \to\mathbf{G}_m \] that takes values in the multiplicative group. The author provides a short proof of a formula for the Weil pairing in terms of tame symbols. In particular, if \((f_x)_{x\in C}\) and \((g_x)_{x\in C}\) are idèles corresponding to \(n\)-torsion bundles \(L\) and \(M\), chosen so that \((f_x^n)\) and \((g_x^n)\) are principal, then the value of the Weil pairing \(e_n([L],[M])\) is \[ \prod_{x\in C} (f_x,g_x)_x^n, \] where \((f_x,g_x)_x\) denotes the tame symbol at \(x\).
This result has been known to experts for a long time, but the first published proof for curves of arbitrary genus appeared in a paper by the reviewer [Math. Ann. 305, No. 2, 387–392 (1996; Zbl 0854.11031)]. This earlier proof involves a reduction to the case of curves over finite fields, and uses class field theory for curves. The author’s proof avoids this arithmetic reduction, and involves comparing various fibers of two different pullbacks to \(J\times J\) of the Poincaré bundle.

14K05 Algebraic theory of abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
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