Finiteness and cofiniteness of fine Selmer groups over function fields

2024/08/13 by Sohan Ghosh, Ghosh, Sohan, Jishnu Ray +3
Mathematics · #11R23 (Primary) 11G10 #11R58 #14G17 #14K15 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2408.06938

openalex publication_date 2024/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the dual fine Selmer group of an abelian variety over the unramified ℤ_p-extension of a function field is finitely generated over ℤ_p. This is a function field version of a conjecture of Coates--Sujatha. We further prove that the fine Selmer group is finite (respectively zero) if the separable p-primary torsion of the abelian variety is finite (respectively zero). These results are then generalized to certain ramified p-adic Lie extensions.

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