The parity of the rank of the mordell-weil group

Abstract

L(s) = i(s)% l)/&(s). Here c(s) is the Riemann zeta-function, and the series L,(s) is convergent for Re(s) > $. It is conjectured that L,(s) may be continued analytically over the whole plane; certainly this is so when r has complex multiplication (in particular, when ab = 0), for then L,(s) is a Hecke L-function with Grossencharaktere (see [6]); it is also so for a much larger dynasty ofcurves founded by Eichler [7] and Shimura [IO]. As is well known, the points of IY form a group; by the Mordell-Weil theorem, the subgroup d consisting of the rational points of r is finitely generated. Write g for the number of independent generators of infinite order of &, and write y for the order of the zero of L,(s) at s = 1 (so y = 0 if&(l) # 0). This makes sense when&.(s) can be continued-from now on we restrict ourselves to such cases. It has been conjectured [l, 2,4] that