Abelian varieties with prescribed embedding and full embedding degrees

Abstract

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 ≥4, prescribed integers m, n and a prescribed prime ℓ≡1(modm⋅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 ℓ and full embedding degree m⋅n with respect to ℓ. We also study a class of absolutely simple higher dimensional abelian varieties whose endomorphism algebras are central over imaginary quadratic fields.