Average values on the Jacobian variety of a hyperelliptic curve
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초록
We give explicitly an average value formula under the multi\-plication-by-$2$ map for the $x$-coordinates of the $2$-division points $D$ on the Jacobian variety $J(C)$ of a hyperelliptic curve $C$ with genus $g$ if $2D \equiv 2P-2\infty \pmod{\mathrm{Pic}(C)}$ for $P=(x_P, y_P) \in C$ with $y_P \ne 0$. Moreover, if $g=2$, we give a more explicit formula for $D$ such that $2D \equiv P-\infty \pmod{\mathrm{Pic}(C)}$.
Abstract
We give explicitly an average value formula under the multi\-plication-by-$2$ map for the $x$-coordinates of the $2$-division points $D$ on the Jacobian variety $J(C)$ of a hyperelliptic curve $C$ with genus $g$ if $2D \equiv 2P-2\infty \pmod{\mathrm{Pic}(C)}$ for $P=(x_P, y_P) \in C$ with $y_P \ne 0$. Moreover, if $g=2$, we give a more explicit formula for $D$ such that $2D \equiv P-\infty \pmod{\mathrm{Pic}(C)}$.
참고문헌 (4)
- D. G. Cantor, "Computing in the Jacobian of a hyperelliptic curve", Math. Comp, 48(177), pp.95~101 (1987)
- D. G. Cantor, "On the analogue of the division polynomials for hyperelliptic curves", J. Reine Angew. Math, 447, pp.91~145 (1994)
- R. Feng, "A mean value formula for elliptic curves", Journal of Numbers, 2014, pp.5~ (2014)[DOI]
- R. Hartshorne, "Algebraic Geometry" (1977), Springer-Verlag