On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field

Details On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field

2000-02-24

arxiv.org/abs/math/0002205v1

J. Number Theory 92 (2002) 139--163.

Mathematics

Algebraic Geometry

17 pages, AMS-LaTeX

Scientific paper

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(F_q,n) approaches 1 as q approaches infinity over the prime powers.

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