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On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field. (English) Zbl 0998.11031
The main result of the paper under review is the proof that for any field \(k\) and any positive integer \(n\), there exists an absolutely simple abelian variety of dimension \(n\) over \(k\). The proof is easily reduced to the case where \(k\) is a prime field and then, the case \(k = {\mathbb Q}\) already being solved [S. Mori, Jap. J. Math., New Ser. 3, 105-109 (1977; Zbl 0379.14011) or Yu. G. Zarhin, Math. Res. Lett. 7, 123-132 (2000; Zbl 0959.14013)], to the case that \(k\) is a finite prime field.
For finite prime fields, the authors then even prove that there exist absolutely simple ordinary abelian varieties of every dimension. In addition, an asymptotic result about the proportion of absolutely simple ordinary isogeny classes of abelian varieties over general finite fields is proved. This result states that for fixed dimension \(n\), this proportion tends to \(1\) as the size of the finite field tends to infinity.
The basic tool used in the proofs is Honda-Tate theory, which sets up a bijection between isogeny classes of simple abelian varieties over \({\mathbb F}_q\) and Galois conjugacy classes of Weil \(q\)-numbers or irreducible Weil \(q\)-polynomials.

11G10 Abelian varieties of dimension \(> 1\)
14K99 Abelian varieties and schemes
11G25 Varieties over finite and local fields
Full Text: DOI arXiv
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