On nondegenerate CM-types

Abstract

We study bounds of the rank of CM-types of CM-fields, and using the bounds, we show that all CM-types of Galois CM-fields with cyclic or dicyclic Galois groups of order $2^{k}p$ with a prime $p$ are nondegenerate. We also show the nondegeneracy of CM-types for certain non-Galois CM-fields. As a consequence, the Hodge conjecture is true for abelian varieties with complex multiplication by such CM-fields.