Extending finite subgroup schemes of semi-stable abelian varieties via log abelian varieties

TL;DR

This paper proves that finite subgroup schemes of semi-stable abelian varieties over discrete valuation fields extend to log finite flat group schemes, and characterizes isogenies of log abelian varieties.

Contribution

It establishes the extension of finite subgroup schemes to log finite flat group schemes and provides new criteria for isogenies of log abelian varieties.

Findings

Finite subgroup schemes extend to log finite flat group schemes.

Weak log abelian varieties are sheaves for the Kummer flat topology.

Equivalent conditions for isogenies of log abelian varieties.

Abstract

For a semi-stable abelian variety A_K over a complete discrete valuation field K, we show 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 prove that every weak log abelian variety over an fs log scheme with its underlying scheme locally noetherian, is a sheaf for the Kummer flat topology, which answers a question of Chikara Nakayama. We also give several equivalent conditions defining isogenies of log abelian varieties.