Elliptic curves with rank 0 over number fields
For any nonzero integer \(b\) define the elliptic curve \(E_b:y^2=x^3+bx\). The paper studies the group \(E_b(k)\) of \(k\)-rational points of \(E_b\) over number fields \(k\) with \([k:\mathbb{Q}]\not\equiv 0\pmod{4}\). By direct computation, the author provides all torsion subgroups of \(E_b(k)\) (\(k\) as above) showing that all points are of even order \(\leqslant 4\). Finally, using the well known relation between the curve \(E_b\) and the diophantine equation \(X^4+bY^4=Z^4\), one checks that nontrivial solutions of \(X^4+bY^4=Z^4\) cannot correspond to points of order 2 or 4; thus proving that \(E_b(k)\) has rank 0 if and only if \(X^4+bY^4=Z^4\) has no nontrivial solutions in \(k^3\).
scientific article; zbMATH DE number 6864143
scientific article