Non-isogenous abelian varieties sharing the same division fields

Davide Lombardo

paper

Non-isogenous abelian varieties sharing the same division fields

arXiv:2108.04270

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.

22 pages, includes some unpublished parts of arXiv:1402.1478. Comments are very welcome!