### Families of noncongruent numbers, by Franz Lemmermeyer

Let E_{k} denote the elliptic curve
defined by y^{2} = x(x^{2} - k^{2}).
We consider the curves with k = pl, p = l = 1 mod 8
primes, and show that the density of rank-0 curves among
them is at least 1/2 by explicitly constructing nontrivial
elements in the 2-part of the Tate-Shafarevich group of
E_{k}.

Franz Lemmermeyer <franzl@csusm.edu>