Everett W. Howe and Hui June Zhu: On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field, J. Number Theory 92 (2002) 139–163, MR 2003g:11063, Zbl 0998.11031.
(An official and an unoffical electronic version are available.)
We prove that for every field k and every positive integer n, there exists an absolutely simple n-dimensional abelian variety over k. We also prove an asymptotic result for finite fields: For every finite field k and positive integer n, we let S(k,n) denote the fraction of the isogeny classes of n-dimensional abelian varieties over k that consist of absolutely simple ordinary abelian varieties. Then for every integer n, as q approaches infinity over the prime powers, the ratio S(Fq,n) approaches 1.