The rank of a CM elliptic curve and a recurrence formula

Abstract

Let p be a prime number and Ep denote the elliptic curve y2=x3+px. It is known that for p which is congruent to 1,9 modulo 16, the rank of Ep over Q is equal to 0,2. Under the condition that the Birch and Swinnerton-Dyer conjecture is true, we give a necessary and sufficient condition that the rank is 2 in terms of the constant term of some polynomial that is defined by a recurrence formula.