JournalscmhVol. 90, No. 1pp. 23–32

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

  • Damian Rössler

    Université Paul Sabatier, Toulouse, France
On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic cover
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Abstract

Let KK be the function field of a smooth and proper curve SS over an algebraically closed field kk of characteristic p>0p>0. Let AA be an ordinary abelian variety over KK. Suppose that the Néron model A\mathcal A of AA over SS has some closed fibre As\mathcal A_s, which is an abelian variety of pp-rank 00.

We show that in this situation the group A(Kperf)A(K^{perf}) is finitely generated (thus generalizing a special case of the Lang-Néron theorem). Here Kperf=KpK^{perf}=K^{p^{-\infty}} is the maximal purely inseparable extension of KK. 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. 23–32

DOI 10.4171/CMH/344