Bull. Korean Math. Soc. 2019; 56(2): 333-349
Online first article March 19, 2019 Printed March 1, 2019
https://doi.org/10.4134/BKMS.b180167
Copyright © The Korean Mathematical Society.
Jiman Chung, Bo-Hae Im
Chung-Ang University; KAIST
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
Supported by: Bo-Hae Im was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT & Future Planning(NRF-2017R1A2B4002619). Jiman Chung was supported by the Chung-Ang University Graduate Research Scholarship.
2009; 46(4): 789-802
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