Journal Article

Likely intersections

Eterović, Sebastian, Scanlon, Thomas

2025

Overview

We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient spaces definable in some fixed o-minimal expansion of the ordered field of real numbers. For an instance of our general result, consider the case of subvarieties of Shimura varieties. Let S be a Shimura variety. Let $\\pi :D \\to \\Gamma \\backslash D = S$ realize S as a quotient of D, a homogeneous space for the action of a real algebraic group G, by the action of $\\Gamma < G$ , an arithmetic subgroup. Let $S' \\subseteq S$ be a special subvariety of S realized as $\\pi (D')$ for $D' \\subseteq D$ a homogeneous space for an algebraic subgroup of G. Let $X \\subseteq S$ be an irreducible subvariety of S not contained in any proper weakly special subvariety of S. Assume that the intersection of X with $\\pi (gD')$ is persistently likely as g ranges through G with $\\pi (gD')$ a special subvariety of S, meaning that whenever $\\zeta :S_1 \\to S$ and $\\xi :S_1 \\to S_2$ are maps of Shimura varieties (regular maps of varieties induced by maps of the corresponding Shimura data) with $\\zeta $ finite, $\\dim \\xi \\zeta ^{-1} X + \\dim \\xi \\zeta ^{-1} \\pi (gD') \\geq \\dim \\xi S_1$ . Then $X \\cap \\bigcup _{g \\in G, \\pi (g D') \\text { is special }} \\pi (g D')$ is dense in X for the Euclidean topology.

• Publisher: Cambridge University Press
• Subject: 03C64 / 11G18 / 14D07 / 14G35 / Algebra / Quotients / Real numbers / Subgroups / Topology