We state that a locally $(n-1)$-connected compactum with integral cohomological dimension $n$ has $n$-cohomological dimension modulo $p$ for some prime $p$. As a consequence, the integral cohomological dimension of the square of such a space is $2n$. In particular, the dimension of the square of an $n$-dimensional, locally $(n-1)$-connected compactum is $2n$.

The general Hodge conjecture for abelian varieties of CM-type is shown to be implied by the usual Hodge conjecture for those up to codimension two.

Let $\ell$ be a prime number, and $K$ a number field with $\zeta_{\ell} \in K^{\times}$. We give a simple necessary and sufficient condition for all tame Kummer extensions over $K$ of degree $\ell$ to have a relative normal integral basis. The result is given in terms of the class number and the group of units of $K$.

The present note studies the existence of unramified quaternion extensions over quadratic fields, and give an alternative proof of Lemmermeyer's result. Our method is based on the theory of embedding problems with restricted ramification.

We consider cohomology sets and exact sequences of groups with involutions. In particular, we study congruence subgroups of type $\Gamma_0(m)$ which are acted by the group generated either by the map $z \mapsto (-1/mz)$ of the upper half plane or by the map $x \mapsto (1/mx)$ of the set of irrational real numbers.

Extending the discussion in the previous note [6] of this series, the group $\mathrm{PSL}_2(\mathbf{C})$ will be dealt with in place of $\mathrm{PSL}_2(\mathbf{R})$. We shall indicate that the functional structure that supports the spectral theory of Kloosterman sums in the complex case is essentially the same as in the real case, though it is more involved as can be expected.

We seek a sufficient condition on initial data in order that the solution of Schrödinger equation has both analytically smoothing effect and exponential decay in time.

We prove a general optimal inequality for warped products in complex hyperbolic spaces and investigate warped products which satisfy the equality case of the inequality. As immediate applications, we obtain several non-immersion theorems for warped products in complex hyperbolic spaces.

In this paper, some new generalizations of Carleman's inequality and Hardy's inequalities on the constant $e$ are considered.