Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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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.

Average values on the Jacobian variety of a hyperelliptic curve

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

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.

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