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

Howe, Everett W.

Math. Ann. 305, No. 2, 387-392 (1996).

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.

Reviewer: E.W.Howe (Ann Arbor)

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

Keywords:

curves over finite fields; Jacobian; torsion points; Weil pairing; Hilbert symbol

References:

[1]	Husem?ller, D.: Elliptic curves (Grad. Texts Math., vol. 111) Berlin Heidelberg New York: Springer 1987
[2]	Lang, S.: Abelian varieties. New York: Interscience 1959 · Zbl 0099.16103
[3]	Milne, J. S.: Abelian varieties. In: Cornell, G., Silverman, J. H. (eds.): Arithmetic geometry (pp. 103-150) Berlin Heidelberg New York: Springer 1986
[4]	Milne, J. S.: Jacobian varieties. In: Cornell, G., Silverman, J. H. (eds.): Arithmetic geometry (pp. 167-212) Berlin Heidelberg New York: Springer 1986
[5]	Schmidt, H. L.: ?ber das Reziprozit?tsgesatz in relativ-zyklischen algebraischen Funktionk?rpern mit endlichem Konstantenk?rper. Math. Z.40, 94-109 (1936) · JFM 61.0121.03
[6]	Serre, J.-P.: Algebraic groups and class fields (Grad. Texts Math., vol. 117) Berlin Heidelberg New York: Springer 1988
[7]	Silverman, J. H.: The arithmetic of elliptic curves (Grad. Texts Math., vol. 106) Berlin Heidelberg New York: Springer 1986 · Zbl 0585.14026