Lang-Trotter phenomena and unlikely intersections

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 André.

Revised and submitted version