On Parities of Relative Class Numbers of certain CM-Extensions

Abstract

Let $k/F$ and $k'/F$ be CM-extensions, $K = kk'$ and $K_+$ the maximal totally real subfield of $K$. It holds $h^-(k)\;|\; h^-(K)$ if $k/F$ is unramified at all finite primes or $K_+/F$ is unramified. Hence, $h^-(k)$ is an obstacle that prevents $h^-(K)$ from being 1. On the other hand, analytic class number formula implies the class number relation $h^-(K) = h^-(k)h^-(k')/c(K/F)$, where $c(K/F)$ is an integer determined by units. Consistency of the first assertion and the class number relation is guaranteed by $h^-(k')$. As a reason of the consistency, the parity equality $h^-(k) \equiv h^-(k')$ (mod 2) is formulated under the situation of the first assertion. (See Theorems 1 and 2.) Non-trivial Examples (§§3.2) and a proof of the parity equality are given. The tool behind the first assertion and indices related with the class number relation are discussed in detail.