Genus one curves defined by separated variable polynomials and a polynomial Pell equation (Q2759130)

The authors consider the following problem. Determine all pairs \((G, H)\) of polynomials in one variable over a field \(K\) of characteristic zero such that the degrees \(m\) of \(G\) and \(n\) of \(H\) are coprime and the curve given by NEWLINE\[NEWLINEC: G(X) = H (Y)NEWLINE\]NEWLINE has genus one. They characterize all the solutions to the above problem. These are, apart from a finite number of cases, two infinite families whose elements correspond to the isomorphism classes of elliptic curves together with a torsion point. Finally the authors exploit this to compute explicit presentations of all the solutions over the rationals.