On elliptic curves \(y^{2} = x^{3}-n^{2}x\) with rank zero
A positive square-free integer \(n\) is called a congruent number (CN) if \(n\) is the area of some rational right triangular. It is a classical problem to determine when \(n\) is CN or not CN. By Tunnell's work, such problem is equivalent to determine the rank of the rational points of the elliptic curvs \(E_n: y^2=x^3-n^2x\) is zero or not. In this paper, all elliptic curves \(E_n\) of such type with trivial 2-Selmer group and 2-dual Selmer group \(\{ \pm 1, \pm n\}\) are determined in terms of graph language. The rank of rational points of such elliptic curves is always zero. As an application, the non CN with arbitarily many prime divisors can be given, which also extends the first author's previous result. The proof of the paper uses the 2-descent method and reduces the problem to study the Legendre symbol between prime divisors of \(n\). The graph language shows its advantage for this complicated combinatoric question.
scientific article; zbMATH DE number 2136916