Néron models over arbitrary valuation rings

Determine whether every abelian variety over the fraction field of an arbitrary valuation ring admits a Néron model over that valuation ring.

Background

Section 5 proves existence of Néron models for semi-abelian varieties under additional hypotheses, including Henselian valuation rings whose residue fields at non-generic points are algebraically closed. The paper also derives existence for abelian varieties when the fraction field is algebraically closed and for certain complete affinoid fields with algebraically closed residue field.

The authors explicitly describe these results as only a partial answer to the broader question of whether abelian varieties admit Néron models over arbitrary valuation rings. Thus, the unrestricted existence problem remains unresolved beyond the cases covered by the paper.