Isogeny relations in products of families of elliptic curves
Luca Ferrigno
TL;DR
The paper addresses Isogeny relations in products of Legendre-family elliptic curves by applying a Pila–Zannier–style strategy in an o-minimal setting. It constructs a uniformization to translate Diophantine constraints into definable-analytic data and proves a sharp counting bound $\#\mathcal{Z}(T)\ll_{\varepsilon} T^{\varepsilon}$ for the set of potential relations, leveraging Habegger–Pila results along with height and CM theory. Under the asymmetry and non-generic-isogeny hypotheses, it shows that there are only finitely many parameter values on an irreducible curve $\mathcal{C}$ for which a suitable isogeny $\phi$ yields a nontrivial End$(E_{\lambda(\mathbf{c})})$-linear relation among the $m+n$ points, providing evidence toward the Zilber–Pink conjecture in this setting. The work integrates modular-curve bounds, CM height estimates, and o-minimal definability to derive effective finiteness results, contributing to the broader understanding of unlikely intersections in families of elliptic curves.
Abstract
Let $E_λ$ be the Legendre family of elliptic curves with equation $Y^2=X(X-1)(X-λ)$. Given a curve $\mathcal{C}$, satisfying a condition on the degrees of some of its coordinates and parametrizing $m$ points $P_1, \ldots, P_m \in E_λ$ and $n$ points $Q_1, \ldots, Q_n \in E_μ$ and assuming that those points are generically linearly independent over the generic endomorphism ring, we prove that there are at most finitely many points $\mathbf{c}_0$ on $\mathcal{C}$, such that there exists an isogeny $φ: E_{μ(\mathbf{c}_0)} \rightarrow E_{λ(\mathbf{c}_0)}$ and the $m+n$ points $P_1(\mathbf{c}_0), \ldots, P_m(\mathbf{c}_0), φ(Q_1(\mathbf{c}_0)), \ldots, φ(Q_n(\mathbf{c}_0)) \in E_{λ(\mathbf{c}_0)}$ are linearly dependent over $\mathrm{End}(E_{λ(\mathbf{c}_0)})$.