Extending finite-subgroup schemes of semistable abelian varieties via log-abelian varieties

We show-for a semistable abelian varietyAK over a complete discrete valuation fieldK-that every finite-subgroup scheme ofAK extends to a log finite-flat group scheme over the valuation ring ofK endowedwith the canonical log structure.To achieve this, we first give a positive answer to a question of Nakayama, namely whether every weak log-abelian variety over an fs (fine and saturated) log scheme with its underlying scheme locally noetherian is a sheaf for the Kummer-flat topology.We also give several equivalent conditions defining isogenies of log-abelian varieties.