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Transactions of the American Mathematical Society

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The structure of Galois groups of $ {\rm CM}$-fields

Author: B. Dodson
Journal: Trans. Amer. Math. Soc. 283 (1984), 1-32
MSC: Primary 11R32; Secondary 11G10, 20B25
MathSciNet review: 735406
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Abstract: A $ CM$-field $ K$ defines a triple $ (G,H,\rho )$, where $ G$ is the Galois group of the Galois closure of $ K$, $ H$ is the subgroup of $ G$ fixing $ K$, and $ \rho \in G$ is induced by complex conjugation. A "$ \rho $-structure" identifies $ CM$-fields when their triples are identified under the action of the group of automorphisms of $ G$. A classification of the $ \rho $-structures is given, and a general formula for the degree of the reflex field is obtained. Complete lists of $ \rho $-structues and reflex fields are provided for $ [K:\mathbb{Q}] = 2n$, with $ n = 3,4,5$ and $ 7$. In addition, simple degenerate Abelian varieties of $ CM$-type are constructed in every composite dimension. The collection of reflex fields is also determined for the dihedral group $ G = {D_{2n}}$, with $ n$ odd and $ H$ of order $ 2$, and a relative class number formula is found.

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Keywords: Complex multiplication of Abelian varieties, reflex field, nondegenerate Abelian variety of $ CM$-type, imprimitive permutation groups, relative class number
Article copyright: © Copyright 1984 American Mathematical Society