2017/02/28 by Damian Rössler
Mathematics · #Abelian group #Abelian variety #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Arithmetic of abelian varieties #Combinatorics #Conjecture #Divisibility rule #Elementary abelian group #Field (mathematics) #Finite field #Function field #Group (periodic table) #Mathematics #Meromorphic and Entire Functions #Perfect field #Physics #Pure mathematics #Rank of an abelian group #Torsion (gastropod) #Torsion subgroup #Variety (cybernetics) #math.AG #msc:14K05
paper · pdf · doi:10.2140/ant.2020.14.1123
published as Alg. Number Th. 14 (2020) 1123-1173 · This third version contains a complete proof of the conjecture of Esnault and Langer Verschiebung divisibility (not only over the algebraic closure of a finite field, as in the previous version)
arxiv created 2019/09/01 · openalex publication_date 2020/07/13 · arxiv updated 2020/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let [math] be an abelian variety over the function field [math] of a curve over a finite field. We describe several mild geometric conditions ensuring that the group [math] is finitely generated and that the [math] -primary torsion subgroup of [math] is finite. This gives partial answers to questions of Scanlon, Ghioca and Moosa, and Poonen and Voloch. We also describe a simple theory (used to prove our results) relating the Harder–Narasimhan filtration of vector bundles to the structure of finite flat group schemes of height one over projective curves over perfect fields. Finally, we use our results to give a complete proof of a conjecture of Esnault and Langer on Verschiebung divisibility of points in abelian varieties over function fields.