Lang-Trotter phenomena and unlikely intersections

TL;DR

This paper connects the Lang-Trotter conjecture for elliptic curves to new cases of the Zilber-Pink conjecture in algebraic geometry, using Yves André's G-functions method, without boundary assumptions.

Contribution

It establishes a novel implication from the Lang-Trotter conjecture to Zilber-Pink for certain curves, expanding applicability to potentially compact cases.

Findings

Links Lang-Trotter conjecture to Zilber-Pink conjecture in new cases

Does not rely on boundary intersection assumptions

Applies to potentially compact curves in al_3

Abstract

We show that the Lang-Trotter conjecture for pairs of elliptic curves implies new cases of the Zilber-Pink conjecture for curves in $\mathcal{A}_3$. Unlike previous results for curves in $\mathcal{A}_g$, our result does not rely on any assumption on intersections with the boundary, and in particular applies to potentially compact curves. The argument is based on the $G$-functions method of Yves Andr\'e.