Non-isogenous abelian varieties sharing the same division fields

TL;DR

This paper constructs examples of high-dimensional abelian varieties over number fields that share all torsion division fields but are not isogenous, revealing new phenomena in the arithmetic of abelian varieties.

Contribution

It provides the first known examples of non-isogenous, strongly iso-Kummerian abelian varieties of dimension at least 4 over number fields.

Findings

Existence of non-isogenous, strongly iso-Kummerian abelian varieties for all dimensions ≥ 4

Construction of geometrically simple examples over number fields

Constraints on possible iso-Kummerian pairs

Abstract

Two abelian varieties $A_1, A_2$ over a number field $K$ are called strongly iso-Kummerian if the torsion fields $K(A_1[d])$ and $K(A_2[d])$ coincide for all $d \geq 1$. For all $g \geq 4$ we construct geometrically simple, strongly iso-Kummerian $g$-dimensional abelian varieties over number fields that are not geometrically isogenous. We also discuss related examples and put significant constraints on any further iso-Kummerian pair.