Non-isogenous abelian varieties sharing the same division fields.

Two abelian varieties A_1, A_2 over a number field K are 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. [ABSTRACT FROM AUTHOR]