This is an announcement of our result showing that the Bando-Calabi-Futaki character for an integral Kähler class, if any, of a compact complex manifold is regarded as an obstruction to semistability of the manifold. This develops a pioneering work of Fujiki [3].

We show an elliptic curve over $\mathbf{Q}(t)$ of rank $\geq 14$.

Fix an odd prime number $p$. For an even Dirichlet character $\chi$, it is conjectured that the Iwasawa $\lambda$-invariant $\lambda_{p,\chi}$ related to the $\chi$-part of ideal class group is zero ([5], [2]). In this note, we show (under some assumptions) that there exist infinitely many characters $\chi$ of order divisible by $p$ for which the conjecture is true by using Kida's formula ([6]).

We give a sufficient condition for an imaginary cyclic field of degree $p - 1$ containing $\mathbf{Q}(\zeta + \zeta^{-1})$ to have the relative class number divisible by $p$. As a consequence, we see that there exist infinitely many imaginary cyclic fields of degree $p - 1$ with the relative class number divisible by $p$.

Model independent splittings of metaplectic representations of unitary groups are given.

Character formulas of metaplectic representations of unitary groups are given.

Let $K(r)$ be the 3-manifold obtained by a Dehn surgery on a hyperbolic knot $K$ in the 3-sphere a along a slope $r \ne \infty$. We show that if $|r| > 3 \cdot 2^{7/4} g$, then $K(r)$ is an irreducible 3-manifold with infinite and word-hyperbolic fundamental group, where $g$ denotes the genus of $K$.

We say that an entire function is structurally finite if it is constructed from a finite number of quadratic polynomials and exponential functions by Maskit surgeries. In this note, we show that every structurally finite entire function has an explicit representation.