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

2000/02/24 by Everett W. Howe, Hui June Zhu · 1 citation
Mathematics · #math.AG #math.NT #msc:14G15 #msc:11G10 #msc:14K15

paper · pdf

published as J. Number Theory 92 (2002) 139--163. · 17 pages, AMS-LaTeX

arxiv created 2000/02/24 · arxiv updated 2009/11/30

Abstract

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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• Existence of simple non-cyclic abelian varieties over arbitrary finite fields and of a given dimension g>1

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