Non-isogenous abelian varieties sharing the same division fields

Two abelian varieties A 1 , 2 A_1, A_2 over a number field K"> K encoding="application/x-tex">K are strongly iso-Kummerian if the torsion fields K left-parenthesis left-bracket d right-bracket right-parenthesis"> ( stretchy="false">[ d stretchy="false">] stretchy="false">) encoding="application/x-tex">K(A_1[d]) and 2 encoding="application/x-tex">K(A_2[d]) coincide for all alttext="d greater-than-or-equal-to 1"> ≥ encoding="application/x-tex">d \geq 1 . For alttext="g 4"> g 4 encoding="application/x-tex">g 4 we construct geometrically simple, strongly iso-Kummerian alttext="g"> encoding="application/x-tex">g -dimensional that not isogenous. We also discuss related examples put significant constraints on any further pair.