A CRITERION FOR ELLIPTIC-CURVES WITH LOWEST 2-POWER IN L(1)

Let D = pi(1)...pi(n), where pi(1),...,pi(n), are distinct Gaussian pr imes = 1(mod 4) and n is any positive integer. In this paper, we prove that the value of the Hecke L-function attached to the elliptic curve E-D2: y(2) = x(3)-D(2)x at s = 1, divided by the period omega defined below, is always divisible by 2(n-1). Moreover, we give a simple comb inatorial criterion for this value to be exactly divisible by 2(n-1). Our results are in accord with the predictions of the conjecture of Bi rch and Swinnerton-Dyer, and, when D is rational, enable us to prove t he conjecture of Birch and Swinnerton-Dyer for E-D2 when the value at s = 1 is exactly divisible by 2(n-1).