Isogeny and overconvergence

In this paper, we apply Tsuzuki’s main theorem in [12] to establish a criterion for when two abelian varieties over a function field K of characteristic p are isogenous. Specifically, assuming that their endomorphism algebras tensored with Qp are division algebras, we prove that if the maximal quotients of minimal slope (i.e., the unique maximal isoclinic quotient corresponding to the minimal slope, defined up to isogeny) of their associated p-divisible groups are isogenous, then the abelian varieties themselves are isogenous over K . We also extend this result to certain p-divisible groups, highlighting the deep connection between isogenies of abelian varieties and the structure of overconvergent F-isocrystals.