早大学位記番号:新8412早稲田大

Aoki Takuya

早稲田大学博士(理学)早大学位記番号:新8412doctoral thesi

青木, 琢哉

104169

Aoki, Takuya

Waseda University Repository

  Graduate School of Fundamental Science and Engineering  Waseda University     博 士 論 文 概 要 Doctoral Thesis Synopsis     論  文  題  目  Thesis Theme On a class number problem of the cyclotomic 𝑍"-extension of Q(√5) Q(√5) の円分的 𝑍"-拡大の類数問題について      申  請  者 (Applicant Name)  Takuya  AOKI  青木  琢哉  Department of Pure and Applied Mathematics, Research on Number Theory      December,  2019        No. 1  The applicant studies on algebraic number theory. In algebraic number theory, the class number of an algebraic number field is one of the most important objects. An algebraic number field is an extension field of the rational number field 𝑄 with finite degree. Every algebraic number field 𝐾 has the special subring 𝑂(, which is called the ring of algebraic integer of 𝐾. The class number ℎ( of 𝐾 is the cardinal of the quotient group 𝐼(/𝑃(, where 𝐼( is the group of all the fractional ideals of 𝑂( and 𝑃(	is the group of all the principal fractional ideals of 𝑂(. It is well known that ℎ(  is finite for any algebraic number field 𝐾. Moreover, 𝑂(  is a unique factorization domain if and only if the class number of 𝐾 is equal to 1.  More than two hundred years ago, Gauss conjectured that there are infinitely many real quadratic fields with class number 1. This conjecture, however, is still open. Moreover, it is not known whether there are infinitely many algebraic number fields with class number 1.  In order to approach this problem, we study “Weber’s class number problem”. Let 𝑝 be a prime number and 𝑛 a non-negative integer. We denote by 𝜁1 a primitive 𝑛-th root of unity in the complex number field. Then the cyclotomic field 𝑄(𝜁"3456) has the unique real subfield 𝐵3,1  whose Galois group over 𝑄 is isomorphic to 𝑍/𝑝1𝑍. Since 𝐵3,1 ⊆ 𝐵3,1;<, the set 𝐵3,= =∪1@<= 𝐵3,1 is also a field. This field is called the cyclotomic 𝑍3-extension of 𝑄. 𝐵3,1 is called the 𝑛-th layer of the cyclotomic 𝑍3-extension of 𝑄. We denote by ℎ3,1 the class number of 𝐵3,1.  In terms of ℎ3,1 , Weber showed that ℎ",< = ℎ"," = ℎ",A = 1. Moreover, he proved that ℎ",1  is odd for all positive integer 𝑛. Later, ℎ",C = 1 was shown by Chon, Bauer and Masley. Van der Linden showed ℎ",D = 1. Recently, J. C. Miller proved that ℎ",E = 1. He also proved that ℎ",F = 1 holds under the assumption of the generalized Riemann hypothesis. Moreover, it is known that ℎ3,1 = 1 for (𝑝, 𝑛) = (3,1), (3,2), (3,3), (5,1), (5,2), (5,3), (7,1), (11,1), (13,1). Then we consider the following problem:  Weber’s class number problem. Is the class number ℎ3,1 equal to 1 for each prime number 𝑝 and positive integer 𝑛?  However, it is very difficult to calculate the class numbers directly. To approach Weber’s class number problem, we discuss on the indivisibility of ℎ3,1 by a prime number 𝑙:  Problem(𝑙-divisibility). Does there exist a prime number 𝑙 dividing some ℎ3,1?  For the case of 𝑝 = 𝑙, Weber and Iwasawa showed that 𝑝 does not divide ℎ3,1 for any positive integer 𝑛. For the case of p≠ 𝑙, on the other hand, Washington proved that the 𝑙-part of ℎ3,1 is bounded as 𝑛 tends to ∞. Regarding the 𝑙-indivisibility of ℎ3,1, K. Horie and K. Horie-M. Horie gave various results. The case of 𝑝 = 2 has been studied very deeply by Fukuda-Komatsu, Okazaki and Morisawa-Okazaki. The case of 𝑝 = 3 is also studied very deeply by Morisawa.  The applicant, on the other hand, studied the cyclotomic 𝑍"-extension of the quadratic field 𝐾M:= 𝑄(√5). For an arbitrary number field 𝐾, the cyclotomic 𝑍3-extension 𝐾3,= of 𝐾 is defined by 𝐾3,=:= 𝐾 ∙ 𝐵3,=. The reason why the applicant treated 𝐾M = 𝑄(√5) is because 𝐾M has the minimal discriminant among all the real quadratic fields. The 𝑛-th layer 𝐾1 of the cyclotomic 𝑍"-extension of 𝐾M is defined by 𝐾1 = 𝐵",1(√5).       No. 2  We denote by ℎ1  the class number of 𝐾1 . Applying the result of Iwasawa, we have that ℎ1  is odd for any positive integer 𝑛. Moreover, van der Linden showed that ℎ< = ℎ" = ℎA = 1. The aim of this thesis is to discuss the following problem:  Problem (divisibility for ℎ1). Does there exist an odd prime number dividing some ℎ1?  The above problem is a natural generalization of Weber’s class number problem for the cases of real quadratic fields. This problem is also related to Coates’ conjecture. An algebraic number field 𝐾 is called totally real if all the embeddings of 𝐾 into the complex number field are real. For a totally real field 𝐾, we denote by 𝐾PQP the composite of 𝐾3,= for all prime number 𝑝. By Galois theory, there is a unique intermediate field 𝐾(𝑚) of 𝐾PQP/𝐾 with degree 𝑚 over 𝐾 for all positive integer 𝑚. We denote by ℎ(𝑚) the class number of 𝐾(𝑚). Then Coates asked the following problem:  Problem (Coates). Does there exist a positive constant 𝐶(𝐾), which is not depending on 𝑚, such that ℎ(𝑚) is at most 𝐶(𝐾) for all positive integer 𝑚?   The applicant’s thesis consists of 6 chapters. In chapter 1, we recall fundamental facts of algebraic number fields, that is, the ideal class groups, the discriminants, the root discriminants and the cyclotomic 𝑍3-extensions. In this chapter, we explain how to calculate the discriminant of 𝐾1 because the explicit value of the discriminant plays an important role in chapter 5.  In chapter 2, we give an explicit upper bound of Washington’s theorem for the cyclotomic 𝑍"-extension of Q(√5). Our result in this chapter is as follows:  Theorem 1. For an odd prime number 𝑙, let 𝛿V be 0 or 1 according as 𝑙 ≡ 1	(𝑚𝑜𝑑	4) or not and 𝑐V the exact power of 2 dividing 𝑙\];< − 1. We denote by ⌊𝑥⌋ the maximal rational integer not exceeding a real number 𝑥. We put 𝑚V = 2𝑐V + ⌊𝑙𝑜𝑔"(5𝑙 − 1)⌋ − 𝛿V − 2 for l≠ 5 and  𝑚D = 2𝑐D + ⌊𝑙𝑜𝑔"(5 − 1)⌋ − 𝛿D − 2 = 4 for 𝑙 = 5. Then 𝑙 does not divide ℎ1/ℎd] for any n ≥ 𝑚V.  We show theorem 1 by calculating the 𝑙-parts of the generalized Bernoulli numbers and using the reflection theorem. We need to deal with the case of 𝑙 = 5 separately because the prime number 5 divides the discriminant of 𝐾M. For 𝑙 = 5, we cannot obtain the same group isomorphism as the case of 𝑙 ≠ 5, which plays an important role in our argument. To obtain a similar group isomorphism for 𝑙 = 5, we use the fact that the 5-part of ℎ",1 is trivial for all positive integer 𝑛. This is an original method of this thesis. In chapter 3, we give a numerical result as follows:  Theorem 2. Let 𝑙 be an odd prime number. If 𝑙 is less than 60000, then 𝑙 does not divide ℎ1 for any positive integer 𝑛.   Theorem 2 is obtained by using theorem 1 and a computer. In this chapter, we explain how to verify that 𝑙 does not divide ℎd]. For this purpose, we treat 4 cases and give 4 criteria. For 𝑙 with large 𝑐V, however, it takes too  much time to verify that 𝑙 does not divide ℎd]. In order to speed up the computer calculation, we also explain the logarithmic algorithm.         No. 3  In chapter 4, we explain how to establish an upper bound for the class number of a totally real number field 𝐾 by following to Miller’s method. This method is very important when we treat a totally real field with the root discriminant larger than 60.839. The root discriminants of 𝐾C and 𝐾D exceed this value. We define 𝐶=-function 𝐹P	 on the real number field by  𝐹P(𝑥) = 𝑒𝑥𝑝(−(𝑥/𝑐)")/𝑐𝑜𝑠ℎ(𝑥/2) for a positive constant 𝑐. 𝐹P is an important function to construct an upper bound of the class number. We denote by 𝑆(𝐾) the set of all prime numbers each of which splits completely into a product of principal prime ideals of 𝐾. By using Poitou version of Weil’s explicit formula and class field theory, we have that if we can choose a suitable 𝑐 and construct an enough subset 𝑇 of 𝑆(𝐾), then we can establish an upper bound of the class number of 𝐾.  In chapter 5, we establish an upper bound of ℎD, the class number of 𝐾D by using the method introduced in chapter 4. The upper bound we obtained is as follows:  Theorem 3. The class number of 𝐾D is at most 133.  To obtain theorem 3, we need to verify many small prime numbers are in 𝑆(𝐾D). A prime number 𝑞 is in 𝑆(𝐾D) if and only if there is an algebraic integer of 𝐾D whose absolute norm has absolute value 𝑞. For this purpose, we use a special integral basis of 𝐾D, some of which are Galois conjugate. Owing to this property, we can find algebraic integers whose absolute norms have small absolute values. Then we can construct a subset 𝑇M of 𝑆(𝐾D), which consists of 741,766 elements. This 𝑇M and 𝑐 = 210 are used to obtain theorem 3.  Theorems 2 and 3 imply that the class number of 𝐾D is 1. Moreover, since ℎ1n< divides ℎ1 for all positive integer 𝑛, we also have the following theorem in chapter 5:  Theorem 4. The class numbers of 𝐾C and 𝐾D are 1.  In chapter 5, we also show that the class number of 𝐾C is at most 518 without the knowledge of theorem 3.  In chapter 6, we describe perspectives of the research by comparing to known results on Weber’s class number problem. It is not known by this thesis whether there exist infinitely many prime numbers each of which does not divide ℎ1 for any positive integer 𝑛. To approach this question, it is natural to consider the following question:  Problem 1. We fix a prime number 𝑙. Can we provide an explicit lower bound for 𝑙-indivisibility of 𝐾=/𝐾?  As we mentioned, this thesis is related to Coates’ conjecture. To approach this conjecture, it is very important to discuss the following problem:  Problem 2. We fix a prime number 𝑙 . Let 𝑝<, 𝑝", . . . , 𝑝p  be distinct prime numbers. Does there exist an intermediate field of 𝑍36 × 𝑍3r × ⋯× 𝑍3t-extension of 𝐾M whose class number is divided by 𝑙 ?               No.1 早稲田大学 博士（理学） 学位申請 研究業績書 (List of research achievements for application of doctorate (Dr. of Science), Waseda University)  氏 名 Takuya AOKI          印 （As of December, 2019） 種 類 別(By Type)   題名、  発表・発行掲載誌名、  発表・発行年月、  連名者（申請者含む）(theme, journal name, date & year of publication, name of authors inc. yourself) Academic papers    Lectures (1) A Class Number Problem for the Cyclotomic 𝑍"-extension of Q(√5), Tokyo J. Math. 39,  No.1 (2016), 69-81, Takuya Aoki. (2) A Class Number Calculation of the 5pu layer of the Cyclotomic 𝑍"-extension of Q(√5), to appear in Funct. Approx. Commet. Math, Takuya Aoki.  (1) Q(√5) の円分的 𝑍"-拡大の類数問題, 早稲田整数論セミナー，早稲田大学， 2015年 1月 23日. (2) A Class Number Problem for the Cyclotomic 𝑍"-extension of Q(√5), 早稲田大学整数論 研究集会, 早稲田大学,2016年 3月 23日. (3) Odlyzko boundと総実代数体の類数評価について, 早稲田大学整数論セミナー, 早稲田大学, 2017年 12月 22 日. (4) Odlyzko bound と総実代数体の類数評価について, 野々市代数的整数論 2018, 金沢工業大学, 2018年 2月 24日. (5) Odlyzko bound と総実代数体の類数評価について, 早稲田大学整数論研究集会, 早稲田大学, 2018年 3月 15日. (6) Q(√5) の円分的 𝑍"-拡大の中間体の類数の計算について, 津田塾大学整数論ワークショップ 2019, 津田塾大学, 2019年 11月 24日. 

On a class number problem of the cyclotomic Z₂-extension of Q(√5)

https://waseda.repo.nii.ac.jp/?action=repository_action_common_download&item_id=57833&item_no=1&attribute_id=20&file_no=2

Q(√5)の円分的Z₂-拡大の類数問題について

Abstract

Similar works

Full text

Available Versions

Waseda University Repository