Generic Manin-Mumford

Given a collection of algebraic numbers $\mathcal{S}\subset \bar{\mathbb{Q}}$ we study the varieties $V$ in ${\mathbb{A}}_{\mathbb{C}}^m$ such that $V(\mathbb{C})\cap {\mathcal{S}}^m$ is Zariski-dense in $V$. We show that for many classical families of algebraic numbers $\mathcal{S}$—such as the family of roots of generalized Laguerre polynomials $L_n^{(\alpha )}(x)$, for a finite collection of $\alpha \in \mathbb{Q}$—an unlikely intersections theorem holds. For example, in the case $m=2$, we prove that an irreducible curve in ${\mathbb{A}}_{\mathbb{C}}^2$ has infinitely many points from ${\mathcal{S}}^2$ if and only if it is of the form $x_1=x_2,$ or $x_1=s$, or $x_2=s$ for a fixed $s\in \mathcal{S}$. This is an analogue of the classical theorems of Ihara, Serre, and Tate, treating the case of $\mathcal{S}$ consisting of the roots of unity, and of the Manin–Mumford conjecture. We also show that a similar result holds almost surely for roots of a collection of random polynomials of growing degree and bounded height. The proofs rely on a uniform Galois-theoretic criterion ensuring the unlikely intersection property.