Families of noncongruent numbers, by Franz Lemmermeyer

Families of noncongruent numbers, by Franz Lemmermeyer

Let E_k 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 E_k.

noncongr.dvi (120888 bytes) [2002 Feb 13]
noncongr.dvi.gz (42241 bytes)
noncongr.pdf (296114 bytes)
noncongr.ps.gz (108285 bytes)

Franz Lemmermeyer <franzl@csusm.edu>