Multiplicative dependence in the denominators of points of elliptic curves

Abstract

Let $E_1,\dots ,E_s$ be $s$, not necessary distinct, elliptic curves over $\mathbb{Q}$. We give upper bounds on the frequency of $s$-tuples of points in $E_1(\mathbb{Q})\times \dots \times E_s(\mathbb{Q})$ whose denominators or $x$-coordinates are multiplicatively dependent. More precisely, we give such bounds in two scenarios: one in which we fix $s$ non-torsion $\mathbb{Q}$-rational points $P_i\in E_i(\mathbb{Q})$ and arbitrary $\mathbb{Q}$-rational points $Q_i\in E_i(\mathbb{Q})$, $i=1,\dots ,s$, and we count $s$-tuples \[ (n_1P_1+Q_1,\ldots, n_sP_s+Q_s) \in E_1(\mathbb{Q}) \times \ldots \times E_s(\mathbb{Q}) \] with $n_1,\dots ,n_s$ in an arbitrary interval of length $N$, and the second in which we count points $(P_1,\dots ,P_s)\in E_1(\mathbb{Q})\times \dots \times E_s(\mathbb{Q})$ of bounded canonical height.