Fields with no everywhere good abelian varieties

TL;DR

This paper develops criteria to identify number fields lacking non-zero abelian varieties with everywhere good reduction, and applies these criteria under GRH to find thousands of such fields up to degree 16.

Contribution

It extends existing methods to determine fields with no everywhere good abelian varieties and provides a large list of such fields under GRH.

Findings

Identified 24,744 number fields with no everywhere good abelian varieties up to degree 16.

Extended Fontaine, Abrashkin, and Schoof's methods for this purpose.

Provided criteria applicable under GRH for these determinations.

Abstract

We extend methods of Fontaine, Abrashkin and Schoof to obtain criteria determining number fields K over which no non-zero abelian variety with everywhere good reduction exists. As an application, under the GRH, we find 24744 such fields of various degrees up to 16.