Non-isogenous abelian varieties sharing the same division fields

Abstract

Two abelian varieties A 1 , A 2 A_1, A_2 over a number field K K are strongly iso-Kummerian if the torsion fields K ( A 1 [ d ] ) K(A_1[d]) and K ( A 2 [ d ] ) K(A_2[d]) coincide for all d ≥ 1 d \geq 1 . For all g ≥ 4 g \geq 4 we construct geometrically simple, strongly iso-Kummerian g 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.