×

zbMATH — the first resource for mathematics

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.

MSC:
11G10 Abelian varieties of dimension \(> 1\)
14K99 Abelian varieties and schemes
11G25 Varieties over finite and local fields
Software:
Magma
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bosma, W.; Cannon, J.; Playoust, C., The magma algebra system. I. the user language, J. symbolic comput, 24, 235-265, (1997) · Zbl 0898.68039
[2] C.-L. Chai, and, F. Oort, A note on the existence of absolutely simple Jacobians, electronic preprint available online at, http://arXiv.org/abs/math.AG/9905063, 1999.
[3] DiPippo, S.A.; Howe, E.W., Real polynomials with all roots on the unit circle and abelian varieties over finite fields, J. number theory, 73, 426-450, (1998) · Zbl 0931.11023
[4] DiPippo, S.A.; Howe, E.W., Corrigendum to “real polynomials with all roots on the unit circle and abelian varieties over finite fields”, J. number theory, 83, 182, (2000)
[5] Howe, E.W., Principally polarized ordinary abelian varieties over finite fields, Trans. amer. math. soc, 347, 2361-2401, (1995) · Zbl 0859.14016
[6] Howe, E.W., The Weil pairing and the Hilbert symbol, Math. ann, 305, 387-392, (1996) · Zbl 0854.11031
[7] Hsu, C.-N., The distribution of irreducible polynomials in fq[t], J. number theory, 61, 85-96, (1996)
[8] Katz, N.M.; Sarnak, P., Random matrices, Frobenius eigenvalues, and monodromy, American mathematical society colloquium publications, 45, (1999), American Mathematical Society Providence
[9] Lenstra, H.W.; Oort, F., Simple abelian varieties having a prescribed formal isogeny type, J. pure appl. algebra, 4, 47-53, (1974) · Zbl 0279.14009
[10] Mori, S., The endomorphism rings of some abelian varieties, II, Japan. J. math (N.S.), 3, 105-109, (1977) · Zbl 0379.14011
[11] Oort, F., CM-liftings of abelian varieties, J. algebraic geom, 1, 131-146, (1992) · Zbl 0803.14025
[12] Oort, F.; Ueno, K., Principally polarized abelian varieties of dimension two or three are Jacobian varieties, J. fac. sci. univ. Tokyo sect. IA math, 20, 377-381, (1973) · Zbl 0272.14008
[13] Robinson, R.M., Intervals containing infinitely many sets of conjugate algebraic integers, Studies in mathematical analysis and related topics, (1962), Stanford Univ. Press Stanford, p. 305-315 · Zbl 0116.25402
[14] Rück, H.-G., Abelian surfaces and Jacobian varieties over finite fields, Compositio math, 76, 351-366, (1990) · Zbl 0742.14037
[15] Silverberg, A., Fields of definition for homomorphisms of abelian varieties, J. pure appl. algebra, 77, 253-262, (1992) · Zbl 0808.14037
[16] Tate, J., Classes d’isogénie des variétés abéliennes sur un corps fini, exposé 352, Séminaire bourbaki 1968/1969, Lecture notes in math, 179, (1971), Springer-Verlag Berlin, p. 95-110
[17] Zarhin, Y.G., Hyperelliptic Jacobians without complex multiplication, Math. res. lett, 7, 123-132, (2000) · Zbl 0959.14013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.