Let $F$ be a real quadratic field and $\mathfrak{p}$ a prime ideal of degree 2. We construct a quadratic extension of the Hilbert class field in the ray class field of conductor $\mathfrak{p}$.

For some algebraic number fields $F$ with only one fundamental unit, we give a lower bound of the extension degree of the ray class field of conductor a rational prime $p$ over the Hilbert class field of $F$.

For each positive integer $p$, there exists a holomorphic curve of order $p$ mean type with an infinite number of deficiencies, the sum of which to the a power is divergent, where $0 < a < 1/3$.

Our aim in this paper is to discuss continuity of Sobolev functions of variable exponent on metric spaces in the limiting case of Sobolev's imbedding theorem.

We announce results of [F1] on automorphic forms on $\mathrm{SO}(4)$. An initial result is the proof by means of the trace formula that the functorial product of two automorphic representations $\pi_1$ and $\pi_2$ of the adèle group $\mathrm{GL}(2, \mathbf{A}_F)$ whose central characters $\omega_1$, $\omega_2$ satisfy $\omega_1 \omega_2 = 1$, exists as an automorphic representation $\pi_1 \boxtimes \pi_2$ of $\mathrm{PGL}(4, \mathbf{A}_F)$. The product is in the discrete spectrum if $\pi_1$ is inequivalent to a twist of the contragredient $\check{\pi}_2$ of $\pi_2$, and $\pi_1$, $\pi_2$ are not monomial from the same quadratic extension. If $\pi_2 = \check{\pi}_1$ then $\pi_1 \boxtimes \pi_2$ is the $\mathrm{PGL}(4, \mathbf{A}_F)$-module normalizedly parabolically induced from the $\mathrm{PGL}(3, \mathbf{A}_F)$-module $\operatorname{Sym}^2(\pi_1)$ on the Levi factor of the parabolic subgroup of type $(3,1)$. Finer results include the definition of a local product $\pi_{1v} \boxtimes \pi_{2v}$ by means of characters, injectivity of the global product, and a description of its image. Thus the product $(\pi_1, \pi_2) \mapsto \pi_1 \boxtimes \pi_2$ is injective in the following sense. If $\pi_1$, $\pi_2$, $\pi_1^0$, $\pi_2^0$ are discrete spectrum representations of $\mathrm{GL}(2, \mathbf{A})$ with central characters $\omega_1$, $\omega_2$, $\omega_1^0$, $\omega_2^0$ satisfying $\omega_1 \omega_2 = 1 = \omega_1^0 \omega_2^0$, and for each place $v$ outside a fixed finite set of places of the global field $F$ there is a character $\chi_v$ of $F_v^{\times}$ such that $\{ \pi_{1v}\chi_v, \pi_{2v}\chi_v^{-1} \} = \{ \pi_{1v}^0, \pi_{1v}^0 \}$, then there exists a character $\chi$ of $\mathbf{A}^{\times} / F^{\times}$ with $\{ \pi_1\chi, \pi_2\chi^{-1} \} = \{ \pi_1^0, \pi_2^0 \}$. In particular, starting with a pair $\pi_1$, $\pi_2$ of discrete spectrum representations of $\mathrm{GL}(2, \mathbf{A})$ with $\omega_1 \omega_2 = 1$, we cannot get another such pair by interchanging a set of their components $\pi_{1v}$, $\pi_{2v}$ and multiplying $\pi_{1v}$ by a local character and $\pi_{2v}$ by its inverse, unless we interchange $\pi_1$, $\pi_2$ and multiply $\pi_1$ by a global character and $\pi_2$ by its inverse. The injectivity of $(\pi_1, \pi_2) \mapsto \pi_1 \boxtimes \pi_2$ is a strong rigidity theorem for $\mathrm{SO}(4)$. The self contragredient discrete spectrum representations of $\mathrm{PGL}(4, \mathbf{A})$ of the form $\pi_1 \boxtimes \pi_2$ are those not obtained from the lifting from the symplectic group $\mathrm{PGSp}(2, \mathbf{A})$.

We show that each equation in the first Painlevé hierarchy is equivalent to a system of nonlinear equations determined by a kind of generating function, and that it admits the Painlevé property. Our results are derived from the fact that the first Painlevé hierarchy follows from isomonodromic deformation of certain linear systems with an irregular singular point.

We formulate Poincaré formulas of complex submanifolds in almost Hermitian homogeneous spaces, using Howard's formulation of Poincaré formulas in Riemannian homogeneous spaces. This formula is an extension of Santaló's one in complex space forms.

The main purpose of this paper is using the properties of primitive characters and the mean value theorems of Dirichlet $L$-functions to study the hybrid mean value of Gauss sums and generalized Bernoulli numbers, and give a sharper asymptotic formula.

Consider an o-minimal expansion of the real field. We deal with the real spectrums of the ring $C_{\mathrm{df}}^r$ of definable $C^r$ functions on an affine definable $C^r$ manifold $M$ in the present paper. Here $r$ denotes a nonnegative integer. We show that the natural map $\operatorname{Sper}(C_{\mathrm{df}}^r(M)) \rightarrow \operatorname{Spec}(C_{\mathrm{df}}^r(M))$ is a homeomorphism when the o-minimal structure is polynomially bounded. If the o-minimal structure is not polynomially bounded, it is not known whether the natural map $\operatorname{Sper}(C_{\mathrm{df}}^r(M)) \rightarrow \operatorname{Spec}(C_{\mathrm{df}}^r(M))$ is a homeomorphism or not. However, the natural map $\operatorname{Sper}(C_{\mathrm{df}}^0(M)) \rightarrow \operatorname{Spec}(C_{\mathrm{df}}^0(M))$ is bijective even in this case.