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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 $2D \equiv P-\infty \pmod{\mathrm{Pic}(C)}$.
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참고문헌 (4)
참고문헌 신청
D. G. Cantor / 1987 / Computing in the Jacobian of a hyperelliptic curve / Math. Comp 48(177):95~101
D. G. Cantor / 1994 / On the analogue of the division polynomials for hyperelliptic curves / J. Reine Angew. Math 447:91~145
R. Feng / 2014 / A mean value formula for elliptic curves / Journal of Numbers 2014:5
R. Hartshorne / 1977 / Algebraic Geometry / Springer-Verlag
D. G. Cantor / 1994 / On the analogue of the division polynomials for hyperelliptic curves / J. Reine Angew. Math 447:91~145
R. Feng / 2014 / A mean value formula for elliptic curves / Journal of Numbers 2014:5
R. Hartshorne / 1977 / Algebraic Geometry / Springer-Verlag
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