Let be the function field of a smooth and proper curve over an algebraically closed field of characteristic . Let be an ordinary abelian variety over . Suppose that the Néron model of over has some closed fibre , which is an abelian variety of -rank .
We show that in this situation the group is finitely generated (thus generalizing a special case of the Lang-Néron theorem). Here is the maximal purely inseparable extension of . This result implies in particular that the "full" Mordell-Lang conjecture is verified in the situation described above. The proof relies on the theory of semistability (of vector bundles) in positive characteristic and on the existence of the compactification of the universal abelian scheme constructed by Faltings-Chai.
Cite this article
Damian Rössler, On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic. Comment. Math. Helv. 90 (2015), no. 1, pp. 23–32DOI 10.4171/CMH/344