# Unlikely intersections with isogeny orbits in a product of elliptic schemes

## TL;DR

This paper characterizes subvarieties in a product of elliptic schemes that have potentially dense intersections with images of finite-rank subgroups under isogenies, using a combination of advanced Diophantine and o-minimal techniques.

## Contribution

It provides a new characterization of subvarieties with dense intersections in a non-isotrivial elliptic scheme setting, extending previous unlikely intersection results.

## Key findings

- Characterization of subvarieties with dense isogeny orbit intersections.
- Application of a generalized Vojta-Rémond inequality.
- Integration of Pila-Zannier strategy in the proof.

## Abstract

Fix an elliptic curve $E_0$ without CM and a non-isotrivial elliptic scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of a fixed finite-rank subgroup (of arbitrary rank) of $E_0^g$, also defined over the algebraic numbers, under all isogenies between $E_0^g$ and some fiber of the $g$-th fibered power $\mathcal{A}$ of the elliptic scheme, where $g$ is a fixed natural number. As a special case of a slightly more general result, we characterize the subvarieties (of arbitrary dimension) inside $\mathcal{A}$ that have potentially Zariski dense intersection with this set. In the proof, we combine a generalized Vojta-R\'emond inequality with the Pila-Zannier strategy.