In this paper, we announce a result on the relation of the analytic torsion with the Laurent expansion of the Ruelle zeta function at $s = 0$ for odd dimensional noncompact hyperbolic manifolds with cusps.

We obtain the Getzler’s relation in $H^{4}(\overline{M}_{2,2})$ by the localization theorem, which will give a constructive proof of Getzler’s conjecture, that is, the relation is actually algebraic.

We continue to studying value distribution of difference polynomials of meromorphic functions. In particular, we show that extending classical theorems of Tumura–Clunie type to difference polynomials needs additional assumptions.

For an imaginary quadratic field $K$ with class number $h$, we shall characterize $h$-dimensional CM abelian varieties over $K$ which descend to abelian varieties over $\mathbf{Q}$. These CM abelian varieties have minimal dimension $h$ both over $K$ and over $\mathbf{Q}$.

Let $P$ be a non constant polynomial. For $n\geq 1$, the $n$-th cyclic resultant of $P$ is the resultant of $P$ and of $x^{n}-1$. C.Hillar has proven a general result giving conditions on two polynomials to have the same set of non zero cyclic resultants. In this note, we give an alternative elementary proof of C.Hillar’s theorem.