## 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
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.