On some abelian varieties of type IV

TL;DR

This paper investigates simple abelian varieties of type IV over number fields, demonstrating their ordinary reduction outside a density-zero set of places and analyzing their reduction types and Newton polygons for small dimensions.

Contribution

It confirms a special case of Serre's conjecture for these abelian varieties and provides detailed analysis of their reduction properties.

Findings

Abelian varieties of type IV have ordinary reduction outside a density-zero set of places.

The paper characterizes splitting types and Newton polygons for small-dimensional cases.

It verifies a specific case of Serre's broader conjecture.

Abstract

We study a certain class of simple abelian varieties of type $\mathrm{IV}$ (in Albert's classification) over number fields with Mumford-Tate groups of type $A$. In particular, we show that such abelian varieties have ordinary reduction away from a set of places of Dirichlet density zero, thus confirming a special case of a broader conjecture of Serre's. We also study the splitting types and Newton polygons of the reductions of the abelian varieties of this type with small dimension (nine).