Isogeny relations in products of families of elliptic curves

Abstract

Let $E_{\lambda }$ be the Legendre family of elliptic curves with equation $Y^2=X(X-1)(X-\lambda )$. Given a curve $\mathcal{C}$, satisfying a condition on the degrees of some of its coordinates and parametrizing $m$ points $P_1,\dots ,P_m\in E_{\lambda }$ and $n$ points $Q_1,\dots ,Q_n\in E_{\mu }$ 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 $\varphi :E_{\mu ({\mathbf{c}}_0)}\to E_{\lambda ({\mathbf{c}}_0)}$ and the $m+n$ points $P_1({\mathbf{c}}_0),\dots ,P_m({\mathbf{c}}_0),\varphi (Q_1({\mathbf{c}}_0)),\dots ,\varphi (Q_n({\mathbf{c}}_0))\in E_{\lambda ({\mathbf{c}}_0)}$ are linearly dependent over $\mathrm{End}(E_{\lambda ({\mathbf{c}}_0)})$.

20 pages. v3: Final version. To appear in Forum Mathematicum. This version improves the exposition and the results of Section 5. It also corrects several inaccuracies, in particular in Section 4. Comments are welcome!

References in corpus (4)

• Height bounds and the Siegel property
• CM relations in fibered powers of elliptic families
• Unlikely Intersections in families of abelian varieties and the polynomial Pell equation
• Unlikely intersections between isogeny orbits and curves