A finiteness theorem for abelian varieties with totally bad reduction

TL;DR

This paper proves a finiteness result for abelian varieties over number fields with specific reduction properties outside a finite set of places, extending understanding of their classification up to potential isogeny.

Contribution

It establishes a finiteness theorem for abelian varieties with prescribed reduction behavior, considering potential isogeny classes, which was previously unknown.

Findings

Finiteness of such abelian varieties up to potential isogeny

Classification based on reduction types at places outside a finite set

Extension of reduction theory to include totally bad reduction cases

Abstract

We show that up to potential isogeny, there are only finitely many abelian varieties of dimension $d$ defined over a number field $K$, such that for any finite place $v$ outside a fixed finite set $S$ of places of $K$ containing the archimedean places, it has either good reduction at $v$, or totally bad reduction at $v$ and good reduction over a quadratic extension of the completion of $K$ at $v$.