On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field, by Everett W. Howe and Hui June Zhu

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 n the quantity S(Fq,n) approaches 1 as q approaches infinity over the prime powers.

Everett W. Howe and Hui June Zhu <however@alumni.caltech.edu, zhu@alum.calberkeley.org>