Average values on the Jacobian variety of a hyperelliptic curve

摘要

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≡2P?2∞(modPic(C)) for P=(xP,yP)∈C with yP≠0. Moreover, if g=2, we give a more explicit formula for D such that 2D≡P?∞(modPic(C)).