Average values on the Jacobian variety of a hyperelliptic curve
Bull. Korean Math. Soc. 2019 Vol. 56, No. 2, 333-349
Published online March 1, 2019
Jiman Chung, Bo-Hae Im
Chung-Ang University; KAIST
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 $2D \equiv P-\infty \pmod{\mathrm{Pic}(C)}$.
Keywords : Jacobian variety, hyperelliptic curve
MSC numbers : Primary 11G05
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