Ono's number $p_D$ and the class number $h_D$, associated to an imaginary quadratic field with discriminant $-D$, are closely connected. For example, Frobenius-Rabinowitsch Theorem asserts that $p_D = 1$ if and only if $h_D = 1$. In 1986, T. Ono raised a problem whether the inequality $h_D \leq 2^{p_D}$ holds. However, in our previous paper [8], we saw that there are infinitely many $D$ such that the inequality does not hold. In this paper we give a modification to the inequality $h_D \leq 2^{p_D}$. We also discuss lower and upper bounds for Ono's number $p_D$.

We construct an elliptic curve over $\mathbf{Q}$ with non-trivial 2-torsion point and rank exactly equal to 15.

This paper gives examples of globally hypoelliptic operator on $S^3$, or on $S^7$, or on $S^{15}$ which is sum of squares of real vector fields. These operators fail to satisfy the infinitesimal transitivity condition (the Hörmander bracket condition) at every point and therefore they are not hypoelliptic in any subdomain.

We will prove the freeness of multi-Coxeter arrangements by constructing a basis of the module of vector fields which contact to each reflecting hyperplanes with some multiplicities using K. Saito's theory of primitive derivation.

We obtain an arithmetic expression of the Selberg zeta function for cocompact Fuchsian group defined via an indefinite division quaternion algebra over $\mathbf{Q}$. As application to the prime geodesic theorem, we prove certain uniformity of the distribution.

In the papers [KW2, KW3] we introduced and studied a new type of the Selberg zeta function called by a higher Selberg zeta function. We have established the analytic properties, especially the functional equation of the higher Selberg zeta functions in [KW3]. Motivated by this study of higher Selberg zeta functions we formulate the problem for general zeta functions which have the Euler products and discuss their general features.

We construct an action of the modular group $\varGamma(2)$ on a general $4$-parameter family of complex cubic surfaces and describe the nonlinear monodromy of the Painlevé VI equation in terms of this action.

We consider the equation $f_{xxt} + f f_{xxx} - a f_x f_{xx} = \nu f_{xxxx}$, $x \in (0,1)$, $t > 0 $, where $a \in \mathbf{R}$ is a constant, with the periodic boundary condition. We show that any solution exists globally in time if $-3 \le a \le 1$.

An isolation theorem of Weyl conformal tensor of positive Einstein manifolds is given, when its $L^{n/2}$-norm is small.

In this paper we characterize space forms by observing the extrinsic shape of circles on their geodesic spheres.

Let $\mathcal{A}$ be the class of functions $f(z)$ which are analytic in the open unit disk $\mathbf{U}$ with $f(0) = 0$ and $f'(0) = 1$. Using $g(z) \in \mathcal{A}$, the subclass $\mathcal{T}(\lambda, \mu, g)$ of $\mathcal{A}$ consisting of functions $f(z)$ is introduced. The object of the present paper is to consider some univalence conditions for functions $f(z)$ belonging to the class $\mathcal{T}(\lambda, \mu, g)$ applying the subordination properties of analytic functions.

In this paper, we deal with the problem of uniqueness of meromorphic functions sharing three values, and prove some results which are improvements and extensions of many known theorems. Examples are provided to show that the results in this paper are sharp.