In this thesis we look at simple abelian varieties defined over a finite field k = [special characters omitted]_pn with End_k( A) commutative. We derive a formula that connects the p -rank r(A) with the splitting behavior of p in E = [special characters omitted](π), where π is a root of the characteristic polynomial of the Frobenius endomorphism. We show how this formula can be used to explicitly list all possible splitting behaviors of p in [special characters omitted]_E, and we do so for abelian varieties of dimension less than or equal to four defined over [special characters omitted]_p. We then look for when p divides [[special characters omitted]_E : [special characters omitted][π, π]]. This allows us to prove that the endomorphism ring of an absolutely simple abelian surface is maximal at p when p ≥ 3. We also derive a condition that guarantees that p divides [[special characters omitted]_E: [special characters omitted][π, π]]. Last, we explicitly describe the structure of some intermediate subrings of p-power index between [special characters omitted][π, π] and [special characters omitted]_E when A is an abelian 3-fold with r(A) = 1.