We show—for a semistable abelian variety over a complete discrete valuation field —that every finite-subgroup scheme of extends to a log finite-flat group scheme over the valuation ring of endowed with 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.
"Extending finite-subgroup schemes of semistable abelian varieties via log-abelian varieties." Kyoto J. Math. 60 (3) 895 - 910, September 2020. https://doi.org/10.1215/21562261-2019-0049