Kyushu Journal of Mathematics Vol. 71(2017) No. 1

In the present paper, we discuss the Grothendieck conjecture for hyperbolic curves over Kummer-faithful fields. In particular, we prove that every point-theoretic and Galois-preserving outer isomorphism between the étale/tame fundamental groups of affine hyperbolic curves over Kummer-faithful fields arises from a uniquely determined isomorphism between the original hyperbolic curves. This result generalizes results of Tamagawa and Mochizuki, i.e., our main result in the case where the basefields are either finite fields or mixed-characteristic local fields.

We consider a nonsingular transformation whose Perron-Frobenius operator is quasi-compact on an appropriate Banach algebra. We establish the central limit theorem of mixed type with a nice convergence rate for a real-valued observable in the Banach algebra. As an application, we show that generalized piecewise expanding maps on the unit interval with Hölder continuous derivatives and the Banach algebra of Lebesgue integrable functions with a version of bounded p-variation satisfy our conditions if p is not smaller than the reciprocal of the Hölder exponent of the derivatives.

A diffusion process associated with the real sub-Laplacian Δ_b, the real part of the complex Kohn-Spencer Laplacian □_b, on a strictly pseudoconvex CR manifold is constructed by H. Kondo and S. Taniguchi [A construction of diffusion processes associated with sub-Laplacian on CR manifolds and its applications. J. Math. Soc. Japan 69(1) (2017), 111-125]. In this paper, we investigate the diagonal short time asymptotics of the heat kernel corresponding to the diffusion process by using Watanabe's asymptotic expansion and give a representation for the asymptotic expansion of heat kernels which shows a relationship to the geometric structure.

In this paper, we study the Artin-Mazur zeta function of a generalization of the well-known β-transformation introduced by Góra [Invariant densities for generalized β-maps. Ergodic Theory Dynam. Systems 27 (2007), 1583-1598]. We show that the Artin-Mazur zeta function can be extended to a meromorphic function via an expansion of 1 defined by using the transformation. As an application, we relate its analytic properties to the algebraic properties of β.

Let ψ denote the Dedekind totient function by id ∗ μ², where μ is the Möbius function. We establish the asymptotic formula of the general kth Riesz mean of the arithmetical function n/ψ(n) for any positive integer k ≥ 2; namely

Assuming that the Riemann Hypothesis is true, then the error term of the above function can be estimated as O(k⁻¹x^−1/2+δ) for any small positive constant δ.

Let (X, L) be a polarized manifold of dimension n. In this paper, we study the dimension of H⁰(K_X + mL) with m ≥ n.

We study the fundamental properties of toroidal groups, complex line bundles and the meromorphic functions on them. As an application, we show that every toroidal group with zero Néron-Severi group has no non-constant meromorphic functions on it. Further, we shall give some examples of such toroidal groups.

For the fifth Painlevé equation, we present families of convergent series solutions near the origin and the corresponding monodromy data for the associated isomonodromy linear system. These solutions are of complex power type, of inverse logarithmic type and of Taylor series type. For generic parameters the total set of these critical behaviours is almost complete. For the complex power type of solutions in the generic case, we clarify the structure of the analytic continuation on the universal covering around the origin, and examine the distribution of zeros, poles and 1-points. It is shown that two kinds of spiral domains including a sector as a special case are alternately arrayed; the domains of one kind contain sequences both of zeros and of poles, and those of the other kind sequences of 1-points.

In this paper, it is shown that the set consisting of stable convex integrands Sⁿ → R₊ is open and dense in the set consisting of C^∞ convex integrands with respect to Whitney C^∞ topology. Moreover, examples are given representing well why stable convex integrands are preferred.

In this paper we obtain the residue modulo a prime power of cosine higher-order Euler numbers H ^(k) _2n(m) in terms of the linear combination of the Dirichlet L-function values L(s, χ) at positive integral arguments s or of generalized Bernoulli numbers. Our results are restricted to the equal parity case; i.e. s and χ are of the same parity. In the process, we employ Yamamoto's results on finite expressions in terms of Dirichlet L-function values for short interval character sums and in this sense our treatment is decisive, i.e. any ad-hoc transformation of short interval sums. The results obtained not only generalize the previous results pertaining to the congruences modulo a prime power of the class numbers as the special case of s = 1 in terms of Euler numbers but also closes the chapter on possible similar research.