Multiplicative dependence in the denominators of points of elliptic curves

TL;DR

This work shows that multiplicative dependence among denominators or $x$-coordinates of points on a family of elliptic curves is rare. It develops a framework combining canonical-height analysis, congruence bounds, $S$-unit counting, and graph-cover arguments to bound the frequency of maximal-rank multiplicative dependence for tuples $(n_1P_1+Q_1, ldots,n_sP_s+Q_s)$. The main results provide nontrivial upper bounds $D^*_{oldsymbol P,oldsymbol Q}(M,N)$ and $X^*_{oldsymbol P,oldsymbol Q}(M,N)$ of order $N^{6s/7}$ (uniform in $M$), with height-based analogues $\widehat{D}^*(H),\widehat{X}^*(H) \ll H^{(r_1+\cdots+r_s)/2 - s/14}$. Special cases (e.g., points at infinity or small total rank) yield sharper bounds, and the methods extend to products with potential connections to Zsigmondy-type phenomena for elliptic divisibility sequences. The paper also raises open questions on extensions to broader fields and to more general functional settings.

Abstract

Let $E_1, \ldots, 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 $E_1(\mathbb{Q})\times \ldots \times E_s(\mathbb{Q})$ 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 $Q_i \in E_i(\mathbb{Q})$, $i =1, \ldots, s$, 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 $n_1, \ldots, n_s$ in an arbitrary interval of length $N$, and the second in which we count points $(P_1,\ldots,P_s) \in E_1(\mathbb{Q}) \times \ldots \times E_s(\mathbb{Q})$ of bounded canonical height.

Theorems & Definitions (26)

• Theorem 2.1
• Corollary 2.2
• Remark 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.4