  Let $K$ be an imaginary quadratic field. For an order $\mathcal{O}$ in $K$
and a positive integer $N$, let $K_{\mathcal{O},\,N}$ be the ray class field of
$\mathcal{O}$ modulo $N\mathcal{O}$. We deal with various subjects related to
$K_{\mathcal{O},\,N}$, mainly about Galois representations attached to elliptic
curves with complex multiplication, form class groups and $L$-functions for
orders.
