Families of noncongruent numbers

Abstract

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.

References in corpus (1)

• A Note on Pépin's counter examples to the Hasse principle for curves of genus 1

Cited by in corpus (3)

• Higher Descent on Pell Conics III. The First 2-Descent
• Binary Quadratic Forms and Counterexamples to Hasse's Local-Global Principle
• A Supplement to Scholz's Reciprocity Law