In this note we give a new approach to the rationality problem of some Cremona transformation. Let $k$ be any field, $k(x,y)$ be the rational function field of two variables over $k$. Let $\sigma$ be a $k$-automorphism of $k(x,y)$ defined by \begin{align*} &\sigma(x) = \frac{-x(3x-9y-y^{2})^{3}}{(27x+2x^{2}+9xy+2xy^{2}-y^{3})^{2}},\quad & \qquad \sigma(y) = \frac{-(3x+y^{2})(3x-9y-y^{2})}{27x+2x^{2}+9xy+2xy^{2}-y^{3}}. \end{align*} Theorem. The fixed field $k(x,y)^{\langle\sigma\rangle}$ is rational (= purely transcendental) over $k$. Embodied in a new proof of the above theorem are several general guidelines for solving the rationality problem of Cremona transformations, which may be applied elsewhere.

Two new concepts of zeta functions for schemes over the field of one element are proposed. A localization formula and an explicit formula in the affine case are given. This allows for a computation for every scheme.

This article is a geometric application of polarized logarithmic Hodge theory of Kazuya Kato and Sampei Usui. We prove generic Torelli theorem for the well-known quintic-mirror family in two ways by using different logarithmic points at the boundary of the fine moduli of polarized logarithmic Hodge structures.

We show that two analytic function germs $(\mathbf{C}^{2},0) \to (\mathbf{C},0)$ are topologically right equivalent if and only if there is a one-to-one correspondence between the irreducible components of their zero sets that preserves the multiplicites of these components, their Puiseux pairs, and the intersection numbers of any pair of distinct components.

The number of Galois extensions, up to isomorphism, of a local field whose Galois groups are isomorphic to the modular group $M_{p^{m}}=\langle x,y\mid x^{p^{m-1}}=y^{p}=1,y^{-1}xy=x^{p^{m-2}+1}\rangle$, where $p$ is an odd prime, is counted.

We shall locate Katsurada’s results, in our framework of modular relations, on two series involving the values of the Riemann zeta-function, which are decisive generalizations of earleir results of Chowla and Hawkins and of Buschman and Srivastava \textit{et al.} We shall elucidate these results as an improper or a proper modular relation according as the involved parameter $\nu$ exerts effects on the series or not, eventually indicating that they are disguised form of modular relations as given by Theorem 4 in 3.

In this note we show that Bollobás’ theorem on the trace of sets has polynomial-size Frege proofs.