In this paper, we will give determinant formulas of zeta functions for real abelian extensions over a rational functions field with one variable. By a class number formula, our formula can be regard as a generalization of determinant formulas of class numbers.

We show that a bounded linear operator from the Sobolev space $W^{-m}_{r}(\Omega)$ to $W^{m}_{r}(\Omega)$ is a bounded operator from $L_{p}(\Omega)$ to $L_{q}(\Omega)$, and estimate the operator norm, if $p,q,r\in [1,\infty]$ and a positive integer $m$ satisfy certain conditions, where $\Omega$ is a domain in $\mathbf{R}^{n}$. We also deal with a bounded linear operator from $W^{-m}_{p'}(\Omega)$ to $W^{m}_{p}(\Omega)$ with $p'=p/(p-1)$, which has a bounded and continuous integral kernel. The results for these operators are applied to strongly elliptic operators.

We show for a minimal counterexample ($G,B$) to the Alperin-McKay conjecture, the Fitting subgroup of $G$ is central and $G$ has a unique $G$-conjugacy class of components.

We consider another application of the Ogg-Shafarevich-Grothendieck formula (abbr. OSG) to pencils of plane curves of degree $d$ with certain extremal properties of reducibility. Besides some new results for extremal pencils we treat also various aspects of the topics, e.g. pencils with small number of special fibres, families with $(2IV_{d})$, etc.

We show that a Fano manifold $(X,-K_{X})$ is \textit{not} slope stable with respect to a smooth curve $Z$ if and only if $(X,Z)$ is isomorphic to one of (projective space, line), (product of projective line and projective space, fiber of second projection) or (blow up of projective space along linear subspace of codimension two, nontrivial fiber of blow up).

We consider some function field analogs of some main cohomological results of Kottwitz theory used in stable trace formula and Colliot-Thélène (and Sansuc) theory and give some applications.

In this article, we study discrete curvature and torsion for spatial polygonal lines of unit sides. We express geometric conditions on polygons by using inner products of oriented sides. As a application, we prove a generalization of van der Waerden’s theorem. The theorem given in this paper and its proof clarify how the conditions on the sides affect the polygon being planar from the discrete torsion point of view.