Generic Manin-Mumford
Abstract
Given a collection of algebraic numbers 𝒮⊂ ℚ̅ we study the varieties V in 𝔸mC such that V(ℂ)∩𝒮m is Zariski-dense in V. We show that for many classical families of algebraic numbers 𝒮---such as the family of roots of generalized Laguerre polynomials Ln(α)(x), for a finite collection of α∈ ℚ---an unlikely intersections theorem holds. For example, in the case m=2, we prove that an irreducible curve in 𝔸2C has infinitely many points from 𝒮2 if and only if it is of the form x1=x2, or x1=s, or x2=s for a fixed s ∈ 𝒮. This is an analogue of the classical theorems of Ihara, Serre, and Tate, treating the case of 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.
Related
BibTeX
@misc{barysoroker2026generic,
title = {{Generic Manin-Mumford}},
author = {Lior Bary-Soroker and Borys Kadets},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09354},
eprint = {2609.09354},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09354},
}
RIS
TY - UNPB
TI - Generic Manin-Mumford
AU - Lior Bary-Soroker
AU - Borys Kadets
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09354
UR - https://arxiv.org/abs/2609.09354
AB - Given a collection of algebraic numbers 𝒮⊂ ℚ̅ we study the varieties V in 𝔸^m_C such that V(ℂ)∩𝒮^m is Zariski-dense in V. We show that for many classical families of algebraic numbers 𝒮---such as the family of roots of generalized Laguerre polynomials L_n^(α)(x), for a finite collection of α∈ ℚ---an unlikely intersections theorem holds. For example, in the case m=2, we prove that an irreducible curve in 𝔸^2_C has infinitely many points from 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 ∈ 𝒮. This is an analogue of the classical theorems of Ihara, Serre, and Tate, treating the case of 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.
ER -