Families of noncongruent numbers

Details Families of noncongruent numbers Families of noncongruent numbers

2002-02-13

arxiv.org/abs/math/0202311v1

Acta Arith. 110, No. 1, 15-36 (2003)

Mathematics

Number Theory

Scientific paper

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.

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