In this note, we define the Demjanenko matrices related to the cyclotomic $\mathbf{Z}_p$-extension, which can be regarded as generalization of the ordinary Demjanenko matrices. As a special case, we generalize the Maillet determinant defined by Girstmair.

Let $t \equiv 3 \mod 4$ ($t > 3$) be a prime and $\sigma_r\colon \zeta_t \rightarrow \zeta_t^r$ be a generator of $\operatorname{Gal}(\mathbf{Q}(\zeta_t) / \mathbf{Q}(\sqrt{-t}))$ for $r \in \{1,\dots,t-1\}$. If $p = tn + r$ is a prime, then $4p^h$ can be expressed as the form $4p^h = a^2 + tb^2$ where $h$ is the class number of $\mathbf{Q}(\sqrt{-t})$. Let $\alpha t$ be the sum of representatives of $\langle r \rangle $ in $(\mathbf{Z}/t\mathbf{Z})^{\times}$ and $\beta = \phi(t)/2 - \alpha$. If we choose the sign of $a$ then $a \equiv 2p^{\beta} \mod t$ and $a$ satisfies a certain congruence relation modulo $p$. We also treat the case of $t = 4k$ for a prime $k \equiv 1 \mod 4$.

Let $p$ be an odd prime number. We show that the Iwasawa invariants of a certain non-abelian $p$-extension fields of $\mathbf{Q}$ vanish. And we construct non-abelian $p$-extensions over some imaginary quadratic fields satisfying Leopoldt's conjecture on the $p$-adic regulator.

For an odd prime number $p$ and real abelian number fields $k$ with the degree $[k : \mathbf{Q}] = p$ in which $p$ splits completely, we give a criterion for vanishing of Iwasawa $\lambda$-invariants.

Let $G$ be the principal congruence subgroup of level $N \geq 3$ and $g$ be the group generated by the involution $z \mapsto -1/z$ of the upper half plane. We shall determine the cardinality of the (first) cohomology set $H(g,G)$ in terms of the binary form $x^2 + y^2 \mod N$.

This paper shows that the analogy of ``abc'' conjecture for non-Archimedean entire functions is true.