Let $L$ be a finite extension of a number field $K$ with ring of integers ${\cal O}_L$ and ${\cal O}_K$ respectively. One can consider ${\cal O}_L$ as a projective module over ${\cal O}_K$. The highest exterior power of ${\cal O}_L$ as an ${\cal O}_K$ module gives an element of the class group of ${\cal O}_K$, called the Steinitz module. These considerations work also for algebraic curves where we prove that for a finite unramified cover $Y$ of an algebraic curve $X$, the Steinitz module as an element of the Picard group of $X$ is the sum of the line bundles on $X$ which become trivial when pulled back to $Y$. We give some examples to show that this kind of result is not true for number fields. We also make some remarks on the capitulation problem for both number field and function fields. (An ideal in ${\cal O}_K$ is said to capitulate in $L$ if its extension to ${\cal O}_L$ is a principal ideal.)

As a generalization of the conformal structure of type $(2, 2)$, we study Grassmannian structures of type $(n, m)$ for $n, m \geq 2$. We develop their twistor theory by considerin the complete integrability of the associated null distributions. The integrability corresponds to global solutions of the geometric structures.

A Grassmannian structure of type $(n, m)$ on a manifold $M$ is, by definition, an isomorphism from the tangent bundle $TM$ of $M$ to the tensor product $V \otimes W$ of two vector bundles $V$ and $W$ with rank $n$ and $m$ over $M$ respectively. Because of the tensor product structure, we have two null plane bundles with fibres $P^{m-1}(\mathbb{R})$ and $P^{n-1}(\mathbb{R})$ over $M$. The tautological distribution is defined on each two bundles by a connection. We relate the integrability condition to the half flatness of the Grassmannian structures. Tanaka's normal Cartan connections are fully used and the Spencer cohomology groups of graded Lie algebras play a fundamental role.

Besides the integrability conditions corr[e]sponding to the twistor theory, the lifting theorems and the reduction theorems are derived. We also study twistor diagrams under Weyl connections.

Let ${\cal G}$ be an infinitesimal group scheme, defined over an algebraically closed field of characteristic $p$. We employ rank varieties of ${\cal G}$-modules to study the stable Auslander-Reiten quiver of the distribution algebra of ${\cal G}$. As in case of finite groups, the tree classes of the AR-components are finite or infinite Dynkin diagrams, or Euclidean diagrams. We classify the components of finite and Euclidean type in case ${\cal G}$ is supersolvable or a Frobenius kernel of a smooth, reductive group.

In this paper we consider the following nonlinear parabolic variational inequality; $u(t) \in D(\Phi)$ for all $t \in I$, $(u_t(t), u(t) - v) + \langle \Delta_p u(t), u(t) - v \rangle + \Phi(u(t)) - \Phi(v) \leqq (f(t), u(t) -v )$ for all $v \in D(\Phi)$ a.e. $t\in I$, $u(x,0) = u_0(x)$, where $\Delta_p$ is the so-called $p$-Laplace operator and $\Phi$ is a proper, lower semicontinuous functional. We have obtained two results concerning to solutions of this problem. Firstly, we prove a few regularity properties of solutions. Secondly, we show the continuous dependence of solutions on given data $u_0$ and $f$.

We denote by (A) Artin's reciprocity law for a general abelian extension of a finite degree over an algebraic number field of a finite degree, and denote two special cases of (A) as follows: by (AC) the assertion (A) where $K/F$ is a cyclotomic extension; by (AK) the assertion (A) where $K/F$ is a Kummer extension. We will show that (A) is derived from (AC) and (AK) only by routine, elementarily algebraic arguments provided that $n = (K : F)$ is odd. If $n$ is even, then some more advanced tools like Proposition 2 are necessary. This proposition is a consequence of Hasse's norm theorem for a quadratic extension of an algebraic number field, but weaker than the latter.

In this paper a certain type of Dirichlet series, attached to a pair of Jacobi forms and Siegel modular forms is studied. It is shown that this series can be analyzed by a new variant of the Rankin-Selberg method. We prove that for eigenforms the Dirichlet series have an Euler product and we calculate all the local $L$-factors. Globally this Euler product is essentially the quotient of the standard $L$-functions of the involved Jacobi- and Siegel modular form.

Let $K$ be the simplest cubic field defined by the irreducible polynomial $$f(x) = x^{3}+mx^{2}-(m+3)x+1,$$ where $m$ is a nonnegative rational integer such that $m^{2}+3m+9$ is square-free. We estimate the value of the Dedekind zeta function $\zeta_{K}(s)$ at $s = -1$ and get class number $1$ criterion for the simplest cubic fields.

Let $F$ be an algebraic number field of a finite degree, and $K$ be a normal extension over $F$ of a finite degree $n$. Let $\mathfrak{p}$ be a prime ideal of $F$ which is unramified in $K/F$, $\mathfrak{P}$ be a prime ideal of $K$ dividing $\mathfrak{p}$ such that $N_{K/F}\mathfrak{P} = \mathfrak{p}^f$, $n=fg$. Denote by $\delta(K/F)$ the greatest common divisor of discriminants of integers of $K$ with respect to $K/F$. Then, $\mathfrak{p}$divides $\delta(K/F)$ if and only if $\Sigma_{d|f} \mu(\frac fd)N\mathfrak{p}^d < n$.