Hilbert's irreducibility theorem via random walks

2022/02/10 by Lior Bary‐Soroker, Bary-Soroker, Lior, Daniele Garzoni +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Mathematical Dynamics and Fractals #Advanced Topology and Set Theory

paper · pdf · doi:10.48550/arxiv.2202.05010

Abstract

Let G be a connected linear algebraic group over a number field K, let Γ be a finitely generated Zariski dense subgroup of G(K) and let Z⊆ G(K) be a thin set, in the sense of Serre. We prove that, if G/R_u(G) is semisimple and Z satisfies certain necessary conditions, then a long random walk on a Cayley graph of Γ hits elements of Z with negligible probability. We deduce corollaries to Galois covers, characteristic polynomials, and fixed points in group actions. We also prove analogous results in the case where K is a global function field.

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• Hilbert properties of varieties

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