This paper is the first part of our project towards classifications of smooth projective $3$-folds $X$ with $\kappa(X) = -\infty$ admitting a non-isomorphic étale endomorphism. We can prove that for any extremal ray $R$ of divisorial type, the contraction morphism $\pi_R\colon X\to X'$ associated to $R$ is the blowing-up of a smooth $3$-fold $X'$ along an elliptic curve. The difficulty is that there may exist infinitely many extremal rays on $X$. Thus we introduce the notion of an `ESP' which is an infinite sequence of non-isomorphic finite étale coverings of $3$-folds with constant Picard number. We can run the minimal model program (`MMP') with respect to an ESP and obtain the `FESP' $Y_{\bullet}$ of $(X, f)$ which is a distinguished ESP with \textit{extremal rays of fiber type} (cf. Definition 3.6). We first classify $Y_{\bullet}$ and then blow-up $Y_{\bullet}$ along elliptic curves to recover the original $X$. The finiteness of extremal rays of $\overline{\rm NE}(X)$ is verified in certain cases (cf. Theorem 1.4). We encounter a new phenomenon showing that our \'{e}taleness assumption is related with torsion line bundles on an elliptic curve (cf. Theorem 1.5).

In this article we prove a local Riemman-Hurwitz formula which compares the dimensions of the spaces of vanishing cycles in a finite Galois cover of type $(p,p,\cdots,p)$ between formal germs of $p$-adic curves and which generalises the formula proven in [6] in the case of Galois covers of degree $p$. We also investigate the problem of the existence of a torsor structure for a finite Galois cover of type $(p,p,\cdots,p)$ between $p$-adic schemes.

There are many studies about twisted Alexander invariants for knots and links, but calculations of twisted Alexander invariants for spatial graphs, handlebody-knots, and surface-links have not been demonstrated well. In this paper, we give some remarks to calculate the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links, and observe their behaviors. For spatial graphs, we calculate the invariants of Suzuki's theta-curves and show that the invariants are nontrivial for Suzuki's theta-curves whose Alexander ideals are trivial. For handlebody-knots, we give a remark on abelianizations and calculate the invariant of the handlebody-knots up to six crossings. For surface-links, we correct Yoshikawa's table and calculate the invariants of the surface-links in the table.

Let $b \ge 2$ be an integer. Not much is known on the representation in base $b$ of prime numbers or of numbers whose prime factors belong to a given, finite set. Among other results, we establish that any sufficiently large integer which is not a multiple of $b$ and has only small (in a suitable sense) prime factors has at least four nonzero digits in its representation in base $b$.

In this paper, we solve affirmatively B.-Y. Chen's conjecture for hypersurfaces in the Euclidean space, under a generic condition. More precisely, every biharmonic hypersurface of the Euclidean space must be minimal if their principal curvatures are simple, and the associated frame field is irreducible.

In this paper, we study the theory of geodesics with respect to the Tanaka-Webster connection in a pseudo-Hermitian manifold, aiming to generalize some comparison results in Riemannian geometry to the case of pseudo-Hermitian geometry. Some Hopf-Rinow type, Cartan-Hadamard type and Bonnet-Myers type results are established.

The word \textit{type number} of an algebra means classically the number of isomorphism classes of maximal orders in the algebra, but here we consider quaternion hermitian lattices in a fixed genus and their right orders. Instead of inner isomorphism classes of right orders, we consider isomorphism classes realized by similitudes of the quaternion hermitian forms.The number $T$ of such isomorphism classes are called \textit{type number} or \textit{$G$-type number}, where $G$ is the group of quaternion hermitian similitudes. We express $T$ in terms of traces of some special Hecke operators. This is a generalization of the result announced in [5] (I) from the principal genus to general lattices. We also apply our result to the number of isomorphism classes of any polarized superspecial abelian varieties which have a model over ${\Bbb F}_p$ such that the polarizations are in a "fixed genus of lattices". This is a generalization of [8] and has an application to the number of components in the supersingular locus which are defined over ${\Bbb F}_p$.

We consider a diffusion processes $\{ X_t \}$ on an interval in the natural scale. Some results are known under which $\{ X_t \}$ is a martingale, and we give simple and analytic proofs for them.