Using the localization property, we construct triangulated categories of motives over quasi-projective $T$-schemes for any coefficient where $T$ is a noetherian separated scheme, and we prove the Grothendieck six operations formalism. We also construct integral étale realization of motives.

In this paper, we give a sufficient condition for a positive constant scalar curvature metric on a manifold with boundary to be a relative Yamabe metric, which is a natural relative version of the classical Yamabe metric. We also give examples of non-Einstein relative Yamabe metrics with positive scalar curvature.

In this paper, we study the univalency and quasiconformal extension of sense-preserving harmonic mappings $f$ in the unit disk. For $f$, we introduce a quantity similar to Ahlfors's criteria and obtain a criterion of univalency and quasiconformal extension of $f$, which can be regarded as generalizations of the results obtained by Ahlfors [Sufficient conditions for quasiconformal extension, Ann. of Math. Stud. 79 (1974), 23-29], Hernández and Martín [Quasiconformal extensions of harmonic mappings in the plane, Ann. Acad. Sci. Fenn. Math. 38 (2013), 617-630], and Chen and Que [Quasiconformal extension of harmonic mappings with a complex parameter, J. Aust. Math. Soc. 102 (2017), 307-315]. By Schwarzian derivatives of harmonic mappings, we also obtain a criterion for univalency and quasiconformal extension for harmonic Techmüller mappings.

We provide an equivalent condition for the monogenity of the ring of integers of any cyclic cubic field. Although a large part of the main results is covered by the classical one of Gras, we write the condition more explicitly. First, we show that if a cyclic cubic field is monogenic then it is a simplest cubic field $K_t$ defined by Shanks' cubic polynomial $f_t(x):=x^3-tx^2-(t + 3)x-1$ with $t \in \mathbf{Z}$. Then we give an equivalent condition for when $K_t$ is monogenic, which is explicitly written in terms of $t$.

In this paper, we characterize conformally flat Riemannian manifolds with constant scalar curvature by the standard contact metric structure of their unit tangent sphere bundles.

In this note, we show that the sets of all stable commutator lengths in the braided Ptolemy-Thompson groups are equal to non-negative rational numbers.

To a dynamical system is attached a non-negative real number called entropy. In 1990, Lind, Schmidt and Ward proved that the entropy for the dynamical system induced by the Laurent polynomial algebra over the ring of the rational integers is described by the Mahler measure. In 2009, Deninger introduced the $p$-adic entropy and obtained a $p$-adic analogue of Lind-Schmidt-Ward's theorem by using the $p$-adic Mahler measures. In this paper, we prove the existence and the explicit formula about $p$-adic entropies for two dynamical systems; one is induced by the Laurent polynomial algebra over the ring of the integers of a number field $K$, and the other is defined by the solenoid.

We study complex projective plane curves with a given group of automorphisms. Let $G$ be a simple primitive subgroup of $\mathrm{PGL}(3, \mathbf{C})$, which is isomorphic to $\mathfrak{A}_{6}$, $\mathfrak{A}_{5}$ or $\mathrm{PSL}(2, \mathbf{F}_{7})$. We obtain a necessary and sufficient condition on $d$ for the existence of a nonsingular projective plane curve of degree $d$ invariant under $G$. We also study an analogous problem on integral curves.

Let $f^1,f^2,f^3$ are three holomorphic curves from a complex disc $\Delta (R)$ into $\mathbf{P}^n(\mathbf{C})\ (n\ge 2)$ with finite growth indexes $c_{f^1},c_{f^2},c_{f^3}$ and sharing $q (q \ge 2n+2)$ hyperplanes in general position regardless of multiplicity. In this paper, we will show the above bounds for the sum $c_{f^1}+c_{f^2}+c_{f^3}$ to ensure that $f^1\wedge f^2\wedge f^3=0$ or there are two curves among $\{f^1,f^2,f^3\}$ coincide to each other. Our results are generalizations of the previous degeneracy and finiteness results for linearly non-degenerate meromorphic mappings from $\mathbf{C}^m$ into $\mathbf{P}^n(\mathbf{C})$ sharing $(2n+2)$ hyperplanes regardless of multiplicities.