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

Asvin G

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arXiv · 2203.16420

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

Asvin G

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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.

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Asvin G. 2022-03-30. Unlikely and just likely intersections for high dimensional families of elliptic curves. https://arxiv.org/abs/2203.16420

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