We give a construction of positive integers with even period of minimal type.

In this paper, we announce some results on indivisibility of relative class numbers of CM quadratic extensions $K/F$ of a fixed totally real number field $F$ which is Galois over $\mathbf{Q}$ and on vanishing of these relative Iwasawa $\lambda_{p}$-, $\mu_{p}$-invariants. In particular, we give a lower bound of the number of such CM extensions $K/F$ with bounded (norm of) relative discriminants. To prove them, we use Hilbert modular forms of half-integral weight.

In~[1], Wilson defined holomorphic 1-cochains and combinatrial period matrices of triangulated Riemann surfaces by using the combinatorial Hodge star operator, introduced in~[2]. In this paper, we define a matrix and call this matrix the associate matrix. Then, we prove that among the three matrices, which are a period matrix, a combinatorial period matrix which is introduced by Wilson~[2] and an associate matrix, there exists a matrix equation. Then we also show that an associate matrix is an element of the Siegel upper half space, so this means that a trianguted Riemann surface gives three elements of the Siegel upper half space.

The aim of this survey article is to bring together recent advances concerning the fundamental groups of join-type curves. Though the paper is of purely expository nature, we do also announce a new result.