Integral points and the rank of elliptic curves over imaginary quadratic fields (Q946992)

Let \(f(x,y)=ax^3+bx^2y+cxy^2+dy^3 \in \mathbb Z[x,y]\) be a homogeneous polynomial of degree 3 with non-zero discriminant and let \(C_m\) be the projective curve \[ C_m: f(x,y)=mz^3. \] By a theorem of \textit{J. H. Silverman} [Invent. Math. 66, 339--358 (1982; Zbl 0494.14008)] there exist constants \(k\) and \(m_0\) with \(k\) absolute and \(m_0\) depending on \(f\), such that for all cube-free integers \(m\) satisfying \(|m|<m_0\), \[ N_f(m)<k^{rank(C_m(\mathbb Q))+1}, \] where \(N_f(m)\) is the number of solutions \((x,y)\in \mathbb Z^2\) of the equation \(f(x,y)=m\). The main result of this paper proves that an analogue of Silverman's theorem is true over any imaginary quadratic field.