AVERAGE VALUES ON THE JACOBIAN VARIETY OF A HYPERELLIPTIC CURVE

Abstract

We give explicitly an average value formula under the multiplication-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 infinity (mod Pic(C)) for P = (x(P),y(P)) is an element of C with y(P) not equal 0. Moreover, if g = 2, we give a more explicit formula for D such that 2D P - infinity (mod Pic(C)).