Generic Manin-Mumford

Abstract: Given a collection of algebraic numbers $\mathcal{S}\subset \overline{\mathbb{Q}}$ we study the varieties $V$ in $\mathbb{A}<sup>m_\mathbb{C}$ such that $V(\mathbb{C})\cap\mathcal{S}<sup>m$ 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 $L_n<sup>{(α)}(x)$, for a finite collection of $α\in \mathbb{Q}$---an unlikely intersections theorem holds. For example, in the case $m=2$, we prove that an irreducible curve in $\mathbb{A}<sup>2_\mathbb{C}$ has infinitely many points from $\mathcal{S}<sup>2$ 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.