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On the rank of CM-Type

Published online by Cambridge University Press:  22 January 2016

Hiromichi Yanai
Hiromichi Yanai*
Affiliation:
Department of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464, Japan
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In the present note, we prove that every simple CM-type is non-degenerate (i.e. the rank is maximal) if the dimension of corresponding abelian varieties is a prime. This follows directly from the argument of Tankeev [5], in which he has treated the 5-dimensional case.

Recently, S. G. Tankeev and K. A. Ribet have established similar results for more general types of abelian varieties (see [3], [4], [6]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 97 , March 1985 , pp. 169 - 172
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1985

References

[ 1 ] Kubota, T., On the field extension by complex multiplication, Trans. Amer. Math. Soc., 118 n°6 (1965), 113–122.Google Scholar
[ 2 ] Ribet, K. A., Division field of abelian varieties with complex multiplication, Soc. Math, de France 2é série Mémoire n°2 (1980), 75–94.Google Scholar
[ 3 ] Ribet, K. A., Generalization of a theorem of Tankeev, Sem. Bordeaux Année 1981–1982 exposé n°17 (1982).Google Scholar
[ 4 ] Ribet, K. A., Hodge classes on certain types of abelian varieties, Amer. J. Math., 105 (1983), 523–538.Google Scholar
[ 5 ] Tankeev, S. G., On algebraic cycles on simple 5-dimensional abelian varieties, Math. USSR-Izv. 19 No. 1 (1982), 95–123.CrossRefGoogle Scholar
[ 6 ] Tankeev, S. G., Cycles on simple abelian varieties of prime dimension, ibid., 20 No. 1, 157–171 (1983).Google Scholar