Vol. 14, No. 5, 2020

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On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic, II

Damian Rössler

Vol. 14 (2020), No. 5, 1123–1173

Let A be an abelian variety over the function field K of a curve over a finite field. We describe several mild geometric conditions ensuring that the group A(Kperf) is finitely generated and that the p-primary torsion subgroup of A(Ksep) 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.

abelian varieties, rational points, purely inseparable extensions, Frobenius, Verschiebung
Mathematical Subject Classification 2010
Primary: 11J95
Secondary: 11G10, 14G25
Received: 20 September 2018
Revised: 19 November 2019
Accepted: 17 December 2019
Published: 13 July 2020
Damian Rössler
Mathematical Institute
University of Oxford
United Kingdom