We study the distribution of points on the $(n+1)$-dimensional modular hyperbola $a_1\cdots a_{n+1} \equiv c \pmod q$, where $q$ and $c$ are relatively prime integers. In particular, we show that an elementary argument leads to a straight-forward proof of a recent result of T.~Zhang and W.~Zhang, with a stronger error term. We also use character sums to obtain an asymptotic formula for the number of points in a given box that lie on such hyperbolas.

We calculate the number of Galois extensions, up to isomorphism, of a local field whose Galois groups are isomorphic to the semidihedral (resp. modular) group of order $2^m$ ($m\ge 4$).

The present note studies a sufficient condition that the length of the $3$-class field tower of a cyclic cubic field is greater than $1$.

Let $h$ be a separately harmonic function on an open neighborhood of a $(m-1)$-dimensional compact submanifold $\Sigma$ in R$^m$ with $m\geq 2$. We show that $h$ can be extended to a separately harmonic function on the bounded component of R$^m-\Sigma$.