We give a sufficient condition for the injectivity of the global-to-local map of the relative Chow group of zero-cycles on a quadric fibration of dimension $\leq 3$ defined over a number field.

We study the rational torsion subgroup of the modular Jacobian variety $J_0(N)$ when $N$ is square-free. We prove that the $p$-primary part of this group coincides with that of the cuspidal divisor class group for $p\geq 3$ when $3 \nmid N$, and for $p\geq 5$ when $3 \mid N$. We further determine the structure of each eigenspace of such $p$-primary part with respect to the Atkin-Lehner involutions. This is based on our study of the Eisenstein ideals in the Hecke algebras.

Let $X$ be a complex $n$-dimensional Fano manifold. Let $s(X)$ be the sum of $l(R)-1$ for all the extremal rays $R$ of $X$, the edges of the cone $\operatorname{NE}(X)$ of curves of $X$, where $l(R)$ denotes the minimum of $(-K_X \cdot C)$ for all rational curves $C$ whose classes $[C]$ belong to $R$. We show that $s(X)\leq n$ if $n\leq 4$. And for $n\leq 4$, we completely classify the case the equality holds. This is a refinement of the Mukai conjecture on Fano fourfolds.

Let $S$ be a connected, compact and orientable surface of genus two having exactly one boundary component. We describe any automorphism of the Torelli complex of $S$, and describe any automorphism of the Torelli group of $S$. More generally, we study superinjective maps from the Torelli complex of $S$ into itself, and show that any finite index subgroup of the Torelli group of $S$ is co-Hopfian.

Let $L$ be a nonnegative self-adjoint operator on $L^2(\mathbb{R}^n)$ satisfying the full off-diagonal estimates $L^{q_0}-L^2$ for some $q_0\in [1,2)$. In this paper, we study the sharp weighted $L^p$ estimates for the spectral multipliers of the operator $L$ and their commutators with BMO functions $b$. As an application, we study the weighted norm inequalities for spectral multipliers of Schr\"odinger operators with negative potentials.

In this paper, we construct {\it round fold maps} or {\it stable fold maps} with singular value sets of concentric spheres introduced by the author [11] on smooth bundles over spheres and bundles over more general manifolds. The class of round fold maps includes some {\it special generic maps} on spheres (see [20] for example) and such maps have been constructed on smooth bundles over the standard sphere $S^k$ with $k \geq 2$ and connected sums of smooth bundles over $S^k$ with $k \geq 2$ with fibers diffeomorphic to standard spheres, for example, in previous studies by the author ([10], [12]). In this paper, we obtain round fold maps which do not appear in these studies with information on the diffeomorphism types of their source manifolds in new manners.

We study the Iwasawa modules related to certain tamely ramified extensions (tamely ramified Iwasawa modules). Let $p$ be an odd prime number, and $k$ an imaginary quadratic field. In the present paper, we shall give some results concerning the $\mu$-invariant of tamely ramified Iwasawa modules for $\mathbb{Z}_p$-extensions of $k$.

In this paper, we will study Earle slices of quasi-fuchsian space for once punctured torus associated with involutions of its fundamental group induced by orientation reversing diffeomorphism of this surface. First we classify Earle slices into two types: rhombic Earle slices and rectangular Earle slices. The main purpose of this paper is to study the configuration of Earle slices. Especially, we obtain a necessary and sufficient condition for two Earle slices to intersect each other. We also show that the union of all Earle slices is connected. In the end, we describe Earle slices by using trace coordinates of quasi-fuchsian space.

A Lefschetz trace formula is proved for $p^n$-torsion \'etale cohomology of open algebraic varieties defined over a finite field of characteristic $p$. These are analogues of Deligne's conjecture on Lefschetz trace formula for $\ell$-adic cohomology, which was proved by Fujiwara.

The aim of this paper is to determine the structure of the cut locus for a class of surfaces of revolution homeomorphic to a cylinder. Let $M$ denote a cylinder of revolution which admits a reflective symmetry fixing a parallel called the equator of $M$. It will be proved that the cut locus of a point $p$ of $M$ is a subset of the union of the meridian and the parallel opposite to $p$ respectively, if the Gaussian curvature of $M$ is decreasing on each upper half meridian.

In this paper, we study the best constant of $L^p$ Sobolev inequality including $j$-th derivative: \begin{align*} \sup_{0\leq y\leq 1}\big\vert u^{(j)}(y)\big\vert \leq C \Bigg(\int_0^1 \big|\,u^{(M)}(x)\,\big|^p dx \Bigg)^{1/p}\,, \end{align*} where $u$ is an element of Sobolev space with periodic or Neumann boundary condition. The best constant can be expressed by $L^q$ norm of Bernoulli polynomial. In [1, 4], the best constant of the above inequality was obtained for the case of $1<p<\infty$ and $j=0$. This paper extends the results of [1, 4] to $j=1,2,3,\ldots,M-1$.

We prove Minding's Theorem for $C^2$-immersions with constant negative Gaussian curvature. As a Corollary we also prove Minding's Theorem for $C^{1M}$-immersions in the sense of [1].

Kobayashi [13] introduced complex contact manifolds as a variant of real contact manifolds. Later, Ishihara and Konishi [11] defined normality of complex contact manifolds as for Sasakian manifolds in real contact geometry. In this paper, we construct normal complex contact manifolds via reduction from hyperk\"{a}hler manifolds, and give a new example of normal complex contact manifolds. To check the normality for the new examples, we give a useful identity about sectional curvatures of normal complex contact manifolds. We also give an explicit example of a non-normal complex almost contact metric structure on $S^{4m+3} \times S^{4n+3}$.

In this paper, we classify real hypersurfaces in a non-flat complex space with $\eta$-recurrent shape operator.

A submanifold of a Riemannian manifold is called reflective, if it is a connected component of an involutive isometry. If every shortest geodesic arc of a complete submanifold is still shortest in the ambient space, we say that the submanifold is convex. In this note we show that reflective submanifolds in special unitary groups are convex.