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

Abstract

We show—for a semistable abelian variety $A_K$ over a complete discrete valuation field $K$—that every finite-subgroup scheme of $A_K$ extends to a log finite-flat group scheme over the valuation ring of $K$ 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.