Interdisciplinary Information Sciences

Special Section: Japan-Korea Joint Seminar on Number Theory and Related Topics 2008

A Note on Galois Cohomology of Algebraic Integers

Takashi ONO

2010 年 16 巻 1 号 p. 1-4
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.1

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We consider an admissible system ({\\mathfrak g},(G,M)) where {\\mathfrak g} is the group of a Galois extension k⁄Q of number fields, G the ring {\\mathcal O}_k of integers of k and M=({\\mathcal O}_k)². The cohomology group H¹({\\mathfrak g},{\\mathcal O}_k) is nicely described by a naive submodule Ξ_k of {\\mathcal O}_k and the Poincaré index i_γ({\\mathfrak g},M), γ∈H¹({\\mathfrak g},{\\mathcal O}_k), for the system ({\\mathfrak g},(G,M)), can be expressed in terms of Ξ_k.

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2-Adic Arithmetic-Geometric Mean and Elliptic Curves

Kensaku KINJO, Yuken MIYASAKA

2010 年 16 巻 1 号 p. 5-15
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.5

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We prove that the ratio of the sequences of the arithmetic geometric mean over a finite extension of {\\mathbb Q}₂ is periodically convergent, and that the limit is related to the canonical lift of an elliptic curve defined over a finite field of characteristic two.

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A Kummer Theoretic Construction of an S₃-Polynomial with Given Quadratic Subfield

Masanari KIDA

2010 年 16 巻 1 号 p. 17-20
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.17

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An example of an S₃-polynomial arising from a Kummer theory of certain algebraic torus is computed.

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Greenberg Conjecture for the Cyclotomic {\\mathbb Z}₂-Extension of {\\mathbb Q}(√p)

Takashi FUKUDA

2010 年 16 巻 1 号 p. 21-32
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.21

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We summarize the present state of Greenberg conjecture for the cyclotomic {\\mathbb Z}₂-extension of {\\mathbb Q}(√p) and suggest a coming direction of studying Greenberg conjecture.

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Flat Modules and Gröbner Bases over Truncated Discrete Valuation Rings

Toshiro HIRANOUCHI, Yuichiro TAGUCHI

2010 年 16 巻 1 号 p. 33-37
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.33

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We present basic properties of Gröbner bases of submodules of a free module of finite rank over a polynomial ring R with coefficients in a graded truncated discrete valuations ring A. As an application, we give a criterion for a finitely generated R-module to be flat over A. Its non-graded version is also given.

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Construction of Number Fields of Odd Degree with Class Numbers Divisible by Three, Five or by Seven

Atsushi SATO

2010 年 16 巻 1 号 p. 39-43
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.39

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We introduce a simple way to construct a family of number fields of given degree with class numbers divisible by a given integer, by using the arithmetic theory of elliptic curves. In particular, we start with an elliptic curve defined over the rational number field with a rational torsion point of order l ∈ {3,5,7}, and show a way to construct infinitely many number fields of given odd degree d ≥ 3 with class numbers divisible by l.

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A Note on the Field Isomorphism Problem of X³+sX+s and Related Cubic Thue Equations

Akinari HOSHI, Katsuya MIYAKE

2010 年 16 巻 1 号 p. 45-54
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.45

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We study the field isomorphism problem of cubic generic polynomial X³+sX+s over the field of rational numbers with the specialization of the parameter s to nonzero rational integers m via primitive solutions to the family of cubic Thue equations x³−2mx²y−9mxy²−m(2m+27)y³=λ where λ² is a divisor of m³(4m+27)⁵.

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Pseudo-Null Submodules of the Unramified Iwasawa Module for {\\mathbb Z}_p²-Extensions

Satoshi FUJII

2010 年 16 巻 1 号 p. 55-66
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.55

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In this article, we study generalized Greenberg’s conjecture for imaginary quadratic fields. In particular, for the prime number p=3, we show some examples.

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Remark on the Iwasawa Invariants of p-Extensions of a Totally Real Number Field

Manabu OZAKI

2010 年 16 巻 1 号 p. 67-70
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.67

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We shall investigate vanishing problem of Iwasawa λ_p- and μ_p-invariants especially for p-extensions of a fixed ground totally real number field with λ_p=μ_p=0.

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Congruence Relations Connecting Tate-Shafarevich Groups with Hurwitz Numbers

Yoshihiro ÔNISHI

2010 年 16 巻 1 号 p. 71-86
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.71

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Let p > 3 be a rational prime congruent to 3 modulo 4, and h(−p) be the class number of the imaginary quadratic field Q(\\sqrt{-p}). Then h(−p)≡−2B_{\\frac{p+1}{2}}modp, where B_n is the n-th Bernoulli number. This is a quite classical congruence. Under the full BSD conjecture, we provide an easy method to obtain the natural explicit generalization of this, which is a congruence between the conjectural order of the Tate-Shafarevich group for certain elliptic curve with Mordell-Weil rank 0 and a coefficient of power series expansion of an elliptic function associating the elliptic curve.

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On the Density of Real Quadratic Fields with λ₂=μ₂=ν₂=0

Hisao TAYA, Gen YAMAMOTO

2010 年 16 巻 1 号 p. 87-92
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.87

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According to the previous works by Nakagawa-Horie and the first author, the density of real quadratic fields with λ₃=μ₃=ν₃=0 is greater than or equal to \\frac{17}{24}. In this note, however, we will report the density of real quadratic fields with λ₂=μ₂=ν₂=0 to be zero.

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Regular Papers

Affine Weyl Groups and the Boundary Value Eigenvalue Problems of the Laplacian

Yoshinori HOSHIKAWA, Hajime URAKAWA

2010 年 16 巻 1 号 p. 93-109
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.93

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In this paper, we give the explicit formulas for the eigenvalues, and the eigenfunctions, of the Dirichlet and also the Neumann eigenvalue problems of the Laplacian on the crystallographic Euclidean domains. We show the Poisson summation formulas of the Dirichlet and Neumann heat kernels.

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Combinatorial Proof of the Identity for Cover Times on Finite Graphs

Takuya OHWA

2010 年 16 巻 1 号 p. 111-118
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.111

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We give a combinatorial proof and an extension of the identity for the probability density of the cover time of a random walk on a finite graph obtained in [10] by considering it as the Möbius inversion formula. In addition, we obtain a similar identity for the cover time of multiple random walks.

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Distance Trisector of a Segment and a Point

Jinhee CHUN, Yuji OKADA, Takeshi TOKUYAMA

2010 年 16 巻 1 号 p. 119-125
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.119

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Motivated by the work of Asano et al. [1], we consider the distance trisector problem and zone diagram considering segments in the plane as the input geometric objects. As the most basic case, we first consider the pair of curves (distance trisector curves) trisecting the distance between a point and a line, as shown in Figure 1. This is a natural extension of the bisector curve (that is a parabola) of a point and a line. In this paper, we show that these trisector curves C₁ and C₂ exist and are unique.

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Neumann-Series Solution of Fractional Differential Equation

Tohru MORITA, Ken-ichi SATO

2010 年 16 巻 1 号 p. 127-137
発行日: 2010年
公開日: 2010/03/25

DOIhttps://doi.org/10.4036/iis.2010.127

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For the initial-value problem of a fractional differential equation with constant coefficients, the solution is given by the Neumann series for the corresponding Volterra integral equation, and is expressed by a generalized Mittag-Leffler function. The derivation is presented in the style of operational calculus, which is formulated by distribution theory.

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