Families of noncongruent numbers
Franz Lemmermeyer
Abstract
Let Ek denote the elliptic curve defined by y2 = x(x2 - k2). 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 Ek.
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