Finiteness properties of torsion fields of abelian varieties

2024/01/11 by Wojciech Gajda, Gajda, Wojciech, Sebastian Petersen +1 · 1 citation
Mathematics · #11G10 #14K15 #15K05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2401.05805

openalex publication_date 2024/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A be an abelian variety defined over a field K. We study finite generation properties of the profinite group Gal(Ω/K) and of certain closed normal subgroups thereof, where Ω is the torsion field of A over K. In fact, we establish more general finite generation properties for monodromy groups attached to smooth projective varieties via étale cohomology. We apply this in order to give an independent proof and generalizations of a recent result of Checcoli and Dill about small exponent subfields of Ω/K in the number field case. We also give an application of our finite generation results in the realm of permanence principles for varieties with the weak Hilbert property.

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• Hilbert properties of varieties

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