On the Existence of Absolutely Simple Abelian Varieties of a Given Dimension over an Arbitrary Field Original Research Article

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 we have S(Fq, n)→1 as q→∞ over the prime powers.