Abelian varieties with prescribed embedding and full embedding degrees

We study the problem of the embedding degree of an abelian variety over a finite field and extend some of the results of [1]. In particular, we show that for a prescribed CM field L of degree & GE; 4, prescribed integers m, n and a prescribed prime .e & EQUIV; 1 (mod m & BULL; n) that splits completely in L, there exists an ordinary abelian variety over a prime finite field with endomorphism algebra L, embedding degree n with respect to .@ and full embedding degree m & BULL; n with respect to @. We also study a class of absolutely simple higher dimensional abelian varieties whose endomorphism algebras are central over imaginary quadratic fields.& COPY; 2023 Elsevier Inc. All rights reserved.