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

摘要

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(F_q, n) → 1 as q → ∞ over the prime powers.