The size of Selmer groups for the congruent number problem, II

(1) have positive rank. As in the first paper (5) of this series we shall be interested in the distribution of the rank r(D) of these curves. Clearly one may, and we shall, restrict attention to square-free numbers D. Our results would take essentially the same form if one were to abandon this restriction, but it would still be necesary for the proofs. To estimate r(D) one may use the method of descent. We are concerned in particular with the "full 2-descent". The number of 2-descents is the order of the Selmer group S(2). This is a power of 2, and