For a dominant rational self-map on a smooth projective variety defined over a number field, Shu Kawaguchi and Joseph H. Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at an algebraic point whose forward orbit is well-defined and Zariski dense. We give some examples of self-maps on product varieties and rational points on them for which the Kawaguchi-Silverman conjecture holds.

We consider the Cauchy problem for the higher-order nonlinear Schrödinger equation in two dimensional case

\[ \left\{\!\!\! \begin{array}{c} i\partial _{t}u+\frac{b}{2}\Delta u-\frac{1}{4}\Delta ^{2}u=\lambda \left\vert u\right\vert u,\text{ }t>0,\text{\ }x\in \mathbb{R}^{2}\,\mathbf{,} \\

u\left ( 0,x\right) =u_{0}\left ( x\right) ,\text{\ }x\in \mathbb{R}^{2} \,\mathbf{,} \end{array} \right. \]

where $\lambda \in \mathbb{R}$, $b>0$. We develop the factorization techniques for studying the large time asymptotics of solutions to the above Cauchy problem. We prove that the asymptotics has a modified character.

On the basis of fractional calculus, the author’s previous study [9] introduced an approach to the integral of controlled paths against Hölder rough paths. The integral in [9] is defined by the Lebesgue integrals for fractional derivatives without using any arguments based on discrete approximation. In this paper, we revisit the approach of [9] and show that, for a suitable class of Hölder rough paths including geometric Hölder rough paths, the integral in [9] is consistent with that obtained by the usual integration theory of rough path analysis, given by the limit of the compensated Riemann–Stieltjes sums.

We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) étale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric étale fundamental group of a (geometrically connected) formal fibre of a $p$-adic curve is (pro-) prime-to-$p$ free of finite computable rank.

We show that a toric Fano contraction associated to an extremal ray whose length is greater than the dimension of its fiber is a projective space bundle.

We consider a certain linear combination of zeta-functions of root systems for a root system. Showing two different expressions of this linear combination, we find that a certain signed sum of zeta-functions of root systems is equal to a sum involving Bernoulli functions of root systems. This identity gives a non-trivial functional relation among zeta-functions of root systems, if the signed sum does not identically vanish. This is a generalization of the authors’ previous result (Proc. London Math. Soc. 100 (2010), 303–347). We present several explicit examples of such functional relations. We give a criterion of the non-vanishing of the signed sum, in terms of Poincaré polynomials of associated Weyl groups. Moreover we prove a certain converse theorem.

We give a general method for constructing compact Kähler manifolds $X_1$ and $X_2$ whose intermediate Jacobians $J^k (X_1)$ and $J^k (X_2)$ are isogenous for each $k$, and we exhibit some examples. The method is based upon the algebraic transplantation formalism arising from Sunada’s technique for constructing pairs of compact Riemannian manifolds whose Laplace spectra are the same. We also show that the method produces compact Riemannian manifolds whose Lazzeri Jacobians are isogenous.

We give a finite analogue of the well-known formula $\mathrm{Li}_{\underbrace{1, \ldots, 1}_n}(t)=\frac{1}{n!}\mathrm{Li}_1(t)^n$ of multiple polylogarithms for any positive integer $n$ by using the shuffle relation of finite multiple polylogarithms of Ono–Yamamoto type. Unlike the usual case, the terms regarded as error terms appear in this formula. As a corollary, we obtain “$t \leftrightarrow 1-t$” type new functional equations of finite multiple polylogarithms of Ono–Yamamoto type and Sakugawa-Seki type.