Unlikely and just likely intersections for high dimensional families of elliptic curves

TL;DR

This paper investigates the density of points in high-dimensional modular varieties that are related through Hecke translations, providing new results on unlikely intersections and their necessity in certain cases.

Contribution

It offers an affirmative answer to a question about the density of Hecke-translated points in varieties, with specific results for divisors and curves, and explores the necessity of assumptions in unlikely intersection scenarios.

Findings

Points in certain varieties are dense under Hecke translations.

High-dimensional spaces over finite fields can have infinitely many intersections of curves via Hecke translations.

Assumptions on dimensions are necessary for the density results.

Abstract

Given two varieties V,W in the n-fold product of modular curves, we answer affirmatively a question (formulated by Shou-Wu Zhang's AIM group) on whether the set of points in V that are Hecke translations of some point on W is dense in V. We need to make some (necessary) assumptions on the dimensions of V,W but for instance, when V is a divisor and W is a curve, no further assumptions are needed. We also examine the necessity of our assumptions in the case of unlikely intersections and show that, contrary to exceptions, two curves in a high dimensional space over a finite field can intersect infinitely often up to Hecke translations.