Updated on 2025/03/30

写真a

 
KOMATSU, Keiichi
 
Affiliation
Faculty of Science and Engineering
Job title
Professor Emeritus
Degree
(BLANK)
理学博士 ( 東京工業大学 )

Education Background

  • 　
    -
    1973

    Tokyo Institute of Technology   Graduate School, Division of Science and Engineering  

  • 　
    -
    1971

    Waseda University   Faculty of Science and Engineering  

Professional Memberships

  • 　
    　
    　

    日本数学会

Research Areas

  • Algebra
 

Papers

  • Weber’s class number problem in the cyclotomic ℤ2-extension of ℚ, II

    Takashi Fukuda, Keiichi Komatsu

    Journal de Theorie des Nombres de Bordeaux   22 ( 2 ) 359 - 368  2010

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    Let hn denote the class number of n-th layer of the cyclotomic ℤ2-extension of ℚ. Weber proved that hn (n ≥ 1) is odd and Horie proved that hn (n ≥ 1) is not divisible by a prime number ℓ satisfying ℓ ≡ 3, 5 (mod 8). In a previous paper, the authors showed that hn (n ≥ 1) is not divisible by a prime number ℓ less than 107. In this paper, by investigating properties of a special unit more precisely, we show that hn (n ≥ 1) is not divisible by a prime number ℓ less than 1.2 • 108. Our argument also leads to the conclusion that hn (n ≥ 1) is not divisible by a prime number ℓ satisfying ℓ = ± 1 (mod 16).

    DOI

  • Weber's Class Number Problem in the Cyclotomic Z(2)-Extension of Q

    Takashi Fukuda, Keiichi Komatsu

    EXPERIMENTAL MATHEMATICS   18 ( 2 ) 213 - 222  2009

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    Let h(n) denote the class number of Q(2cos(2 pi/2(n+2))). Weber proved that h(n) is odd for all n >= 1. We claim that if l is a prime number less than 10(7), then for all n >= 1, l does not divide h(n).

  • On the Iwasawa lamda-invariant of the cyclotomic Z_2-extension of Q(sqrt{p})

    Keiichi Komatsu, Takashi Fukuda

    Math. Comp.   78   1797 - 1808  2009

  • Iwasawa lambda-invariants and Mordell-Weil ranks of abelian varieties with complex multiplication

    Takasih Fukuda, Keiichi Komatsu, Shuji Yamagata

    ACTA ARITHMETICA   127 ( 4 ) 305 - 307  2007

  • On the Iwasawa λ-invariant of the cyclotomic Z?-extension of a real quadratic field

    Tokyo Journal of Mathematics   Vol.28,No.1,pp.259-264  2005

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Research Projects

  • On Weber's class number one problem

    Project Year :

    2012.04
    -
    2015.03
     

    　View Summary

    Let p be a prime number, B_p,∞ the cyclotomic Z_p-extension of the rational number field Q, B_p,n the n-th layer of B_p,∞/Q and h_p,n the class number of B_p,n. We obtained the following:Let p be a prime number which is not congruent to 1 or -1 modulo 16. Then the p-part of the class number h_p,m,n of B_2,mB_p,n is bounded as n tends to infinity for the fixed m. We can see the result in [④]

  • Noether's Problem for Cremona Groups over algebraic number fields and its application to Number theory

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2007
    -
    2009
     

    HASHIMOTO Kiichiro, KOMATSU Keiichi, MURAKAMI Jun, MIYAKE Katsuya, KIDA Masanari, TSUNOGAI Hiroshi

    　View Summary

    We studied the Noether's Problem, which asks the rationality of the fixed field of the rational function field of several variables over a given field, with respect to a given finite subgroup G of the Cremona group. We solved this problem affirmatively in the case where G is one of the transitive permutation groups of degree six, and obtained the explicit description of the fixed field as expected. The results are now being collected and prepared in some papers, although it will take some time before the completion. During the period of the research, we had in each year a workshop entitled as "Galois theory and related topics", and discussed the various related problems.. They were held in the university of Yamagata (2007), Tokushima (2008), and Kanazawa (2009). We also had a conference on number theory each year at Waseda university and communicated with many experts of this subject, including those from foreign countries.

  • Special values of Siegel modular functions and Jacobian variety

    Project Year :

    2004
    -
    2006
     

    　View Summary

    We construct certain algebraic integers αm as special values of two variable theta functions in the ray class field of a certain quartic field modulo 2m, and study a property of prime i deals which appear in αm in connection to the relationships between cyclotomic units and exponential functions and between elliptic units and elliptic functions, Moreover, we study a relationship between the Mordell-Weil rank of an abelian variety with complex multiplication and the Iwasawa λ-invariant of a certain ZLP-extension.

  • Construction of Generic Polynomials in Galois Theory and application to Number Theory

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2003
    -
    2005
     

    HASHIMOTO Kiichiro, KOMATSU Keiichi, MURAKAMI Jun, MIYAKE Katsuya, FUKUDA Takashi, TSUNOGAI Hiroshi

    　View Summary

    Thanks to the current Grant-in-Aid, we were able to organize seven research workshops inviting the most active mathematicians on this field, through which we had many discussions on our subjects.
    This enabled us to make a considerable developments along our reseach project on Galois theory.
    As for the main theme of constructing generic polynomials with given finite groups over Q, our first result is the construction of concrete and simple families of quintic polynomials with two parameters for each of the five transitive permutation groups of degree 5. As a remarkable application we have established the proof of the genericity of the famous family of A_5 polynomials of degree 6 found by A.Bumer, in connection with algebraic curves of genus two whose Jacobian have real multiplication of discriminant 5.
    Our second result is concerned with the Noethers' Problem for the meta abelian groups of exponent 8 which are subgroups of the affine transformation group over Z/8Z. We have proved the affirmative answer for the linear representation of degree 4 for each of them, in contrast with the negative answer for cyclic group of order 8. As a biproduct of this result, we obtained a simple crite

  • Units groups generated by special values of Siegel modular functions

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2001
    -
    2003
     

    KOMATSU Keiichi, HASHIMOTO Kiichiro

    　View Summary

    In 1976, Coates and Wiles gave large improvement to Birch-Swinnerton-Dyer conjecture for some elliptic curves with complex multiplication by using elliptic units in abelian extensions of imaginary quadratic fields. Main purpose of car investigation is to consider Birch-Swinnerton-Dyer conjecture of the Jacobian variety of some genus-2-curves with complex multiplications.
    In our investigation, we obtained the following :
    We put ζ=e^(2πi)/(13) and α=5+5^3+5^9. Then the field k=Q(α) is the CM-field
    corresponding to the Jacobian variety J(C) of the curve C :
    y^2=x^5-156x^4+10816x^3-421824x^2+8998912x-8042776.
    We construct unit groups in abelian extensions of k by special values of Siegel modular functions at a CM-point corresponding to J(C). moreover we write the values of Hecke L-functions associated to the above abelian fields using units given by Siegel modular functions.

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