Zilber-Pink in $Y(1{)}^n$: Beyond multiplicative degeneration

Abstract

We establish Large Galois orbits conjectures for points of unlikely intersections of curves in $Y(1{)}^n$, upon assumptions on the intersection of such curves with the boundary $X(1{)}^n\backslash Y(1{)}^n$, in the Zilber-Pink setting. As a corollary, building on work of Habegger-Pila and Daw-Orr, we obtain new cases of the Zilber-Pink conjecture for curves in $Y(1{)}^n$.

This content formerly appeared in arXiv:2310.04943v1, which has been split into two papers. Version accepted for publication in the Israel Journal of Mathematics. The author thanks Bijay Bhatta and Martin Orr for pointing out inaccuracies in the statements of Theorem 4.2, Corollary 4.4, and Theorem 4.5 in an earlier version