Non-isogenous abelian varieties sharing the same division fields

Lombardo, Davide

doi:10.1090/tran/8767

Non-isogenous abelian varieties sharing the same division fields
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Trans. Amer. Math. Soc. 376 (2023), 2615-2640

Abstract:

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.

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