Let ${\frak g}$ be a Kac-Moody-Borcherds algebra on a field ${\Bbb K}$ associated to a symetrizable matrix and with Cartan subalgebra ${\frak h}$. Let ${\frak L}$ be an ad$\,{\frak h}$-invariant subalgebra such that the restriction to ${\frak L}$ of the standard bilinear form is nondegenerate. We show that the root system $\Psi$ of $({\frak L}, {\frak h})$ is a subsystem according to [Ba] of $\Delta ({\frak g}, {\frak h})$. Moreover, if a subsystem $\Omega$ satisfies some conditions (i.e. $\Omega$ is "réduit et presque-clos") of $\Psi$, we construct inside of ${\frak L}$ a Kac-Moody-Borcherds algebra with root system $\Omega$.

Let $k$ be a subfield of ${\Bbb K}$. We prove similar results in the case of an action of a finite group of k-semi-automorphisms. In particular, we obtain a generalization to the Kac-Moody case of a result by Borel and Tits.

Let ${\frak g}$ be an almost-$k$-split form of a Kac-Moody algebra. We construct a Kac-Moody $k$-algebra with root system similar to the system of ${\frak g}$ (save on some multiples of certain roots).

Let $p$ be a prime number. Let $M$ be a finite Galois extension of a finite algebraic number field $k$. Suppose that $M$ contains a primitive $p$th root of unity and that the $p$-Sylow subgroup of the Galois group $G=Gal(M/k)$ is normal. Let $K$ be the intermediate field corresponding to the $p$-Sylow subgroup. Let ${\frak g}=Gal(K/k)$. The $p$-class group ${\cal C}$ of $M$ is a module over the group ring ${\bf Z}_p G$, where ${\bf Z}_p$ is the ring of $p$-adic integers. Let $J$ be the Jacobson radical of ${\bf Z}_p G$. ${\cal C}/J {\cal C}$ is a module over a semisimple artinian ring ${\bf F}_p {\frak g}$. We study multiplicity of an irreducible representation $\Phi$ apperaring in ${\cal C}/J{\cal C}$ and prove a formula giving this multiplicity partially. As application to this formula, we study a cyclotomic field $M$ such that the minus part of ${\cal C}$ is cyclic as a ${\bf Z}_p G$-module and a CM-field $M$ such that the plus part of ${\cal C}$ vanishes for odd $p$.

To show the formula, we apply theory of central extensions of algebraic number field andstudy global and local Kummer duality between the genus group and the Kummer radical for the genus field with respect to $M/K$.

We clarify the class of second and third order ordinary differential equations which can be tranformed to the simplest equations $Y''=0$ and $Y'''=0$. The coordinate changes employed to transform the equations are respectively area preserving maps for second order equations and contact form preserving maps for third order equations. A geometric explanation of the results is also given by using connections and associated covariant differentials both on tangent and cotangent spaces.

Explicit determinant formulas are presented for the $\tau$-functions of the generalized Painlevé equations of type $A$. This result allows an interpretation of the $\tau$-functions as the Plücker coordinates of the universal Grassmann manifold.

Let $\pi:Z\rightarrow Y$ be a Hermitian submersion. We study when the pull-back of an eigenform of the complex Laplacian on $Y$ is an eigenform of the complex Laplacian on $Z$.

We construct here a framework for a geometric variation of 1 dimensional geometric figures regarding them as sets of ordered points. In this framework, we can make full use of cut and paste technique to find a way to go down to the geometric smallest figure, including the topological change of the parameter space. Therefore we can discuss practical problems like switching of current flows and the minimal networks not only multiple closed geodesics.

We give two types of 3-dimensional CR-submanifolds of the 6-dimensional sphere. First we study whether there exists a 3-dimensinal CR-submanifold which is obtained as an orbit of a 3-dimensional simple Lie subgroup of $G_2$. There exists a unique (up to $G_2$) 3-dimensional CR-submanifold which is obtained as an orbit of reducible representations of $SU(2)$ on ${\bf R}^7$. As orbits of the subgroup which corresponds to the irreducible representation of $SU(2)$ on ${\bf R}^7$, we obtained 2-parameter family of 3-dimensional CR-submanifolds. Next we give a generalization of the example which was obtained by K. Sekigawa.

We determine explicitly the normalized potential, a Weierstrass-type representation, of a superconformal surface in an even-dimensional sphere $S^{2n}$ in terms of certain normal curvatures of the surface. When the Hopf differential is zero the potential embodies a system of first order equations governing the directrix curve of a superminimal surface in the twistor space of the sphere. We construct a birational map from the twistor space of $S^{2n}$ into ${\Bbb C}P^{n(n+1)/2}$. In general, birational geometry does not preserve the degree of an algebraic curve. However, we prove that the birational map preserves the degree, up to a factor 2, of the twistor lift of a superminimal surface in $S^{6}$ as long as the surface does not pass through the north pole. Our approach, which is algebro-geometric in nature, accounts in a rather simple way for the aforementioned first order equations, and as a consequence for the particularly interesting class of superminimal almost complex curves in $S^{6}$. It also yields, in a constructive way, that a generic superminimal surface in $S^{6}$ is not almost complex and can achieve, by the above degree property, arbitrarily large area.