DOCSLIB.ORG

Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties Mathematical Surveys and Monographs Volume 198 Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties Jörg Jahnel American Mathematical Society Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties Mathematical Surveys and Monographs Volume 198 Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties Jörg Jahnel American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov Robert Guralnick MichaelI.Weinstein MichaelA.Singer 2010 Mathematics Subject Classification. Primary 11G35, 14F22, 16K50, 11-04, 14G25, 11G50. For additional information and updates on this book, visit www.ams.org/bookpages/surv-198 Library of Congress Cataloging-in-Publication Data Jahnel, J¨org, 1968– Brauer groups, Tamagawa measures, and rational points on algebraic varieties / J¨org Jahnel. pages cm. — (Mathematical surveys and monographs ; volume 198) Includes bibliographical references and index. ISBN 978-1-4704-1882-3 (alk. paper) 1. Algebraic varieties. 2. Geometry, Algebraic. 3. Brauer groups. 4. Rational points (Ge- ometry) I. Title. QA564.J325 2014 516.353—dc23 2014024341 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 191817161514 Contents Preface vii Introduction 1 Notation and conventions 11 Part A. Heights 13 Chapter I. The concept of a height 15 1. The naive height on the projective space over É 15 2. Generalization to number fields 17 3. Geometric interpretation 21 4. The adelic Picard group 25 Chapter II. Conjectures on the asymptotics of points of bounded height 35 1. A heuristic 35 2. The conjecture of Lang 38 3. The conjecture of Batyrev and Manin 40 4. The conjecture of Manin 44 5. Peyre’s constant I—the factor α 47 6. Peyre’s constant II—other factors 50 7. Peyre’s constant III—the actual definition 59 8. The conjecture of Manin and Peyre—proven cases 62 Part B. The Brauer group 81 Chapter III. On the Brauer group of a scheme 83 1. Central simple algebras and the Brauer group of a field 84 2. Azumaya algebras 89 3. The Brauer group 93 4. The cohomological Brauer group 94 5. The relation to the Brauer group of the function field 98 6. The Brauer group and the cohomological Brauer group 101 7. The theorem of Auslander and Goldman 103 8. Examples 107 Chapter IV. An application: the Brauer–Manin obstruction 119 1. Adelic points 119 2. The Brauer–Manin obstruction 122 3. Technical lemmata 126 4. Computing the Brauer–Manin obstruction—the general strategy 129 5. The examples of Mordell 132 v vi contents 6. The “first case” of diagonal cubic surfaces 146 7. Concluding remark 161 Part C. Numerical experiments 163 Chapter V. The Diophantine equation x4 +2y4 = z4 +4w4 165 Numerical experiments and the Manin conjecture 165 1. Introduction 166 2. Congruences 167 3. Naive methods 169 4. An algorithm to efficiently search for solutions 169 5. General formulation of the method 171 6. Improvements I—more congruences 172 7. Improvements II—adaptation to our hardware 176 8. The solution found 182 Chapter VI. Points of bounded height on cubic and quartic threefolds 185 1. Introduction—Manin’s conjecture 185 2. Computing the Tamagawa number 189 3. On the geometry of diagonal cubic threefolds 193 4. Accumulating subvarieties 195 5. Results 199 Chapter VII. On the smallest point on a diagonal cubic surface 205 1. Introduction 205 2. Peyre’s constant 208 3. The factors α and β 209 4. A technical lemma 211 5. Splitting the Picard group 212 6. The computation of the L-function at 1 216 7. Computing the Tamagawa numbers 219 8. Searching for the smallest solution 221 9. The fundamental finiteness property 222 10. A negative result 233 Appendix 239 1. A script in GAP 239 2. The list 241 Bibliography 247 Index 261 Preface In this book, we study existence and asymptotics of rational points on algebraic varieties of Fano and intermediate type. The book consists of three parts. In the first part, we discuss to some extent the concept of a height and formulate Manin’s conjecture on the asymptotics of rational points on Fano varieties. In the second part, we study the various versions of the Brauer group. We explain why a Brauer class may serve as an obstruction to weak approximation or even to the Hasse principle. This includes two sections devoted to explicit computations of the Brauer–Manin obstruction for particular types of cubic surfaces. The final part describes numerical experiments related to the Manin conjecture that were carried out by the author together with Andreas-Stephan Elsenhans. Prerequisites. We assume that the reader is familiar with some basic mathemat- ics, including measure theory and the content of a standard course in algebra. In addition, a certain background in some more advanced fields shall be necessary. This essentially concerns three areas. a) We will make use of standard results from algebraic number theory and class field theory, as well as such concerning the cohomology of groups. The content of articles [Cas67, Se67, Ta67, A/W, Gru] in the famous collection edited by J. W. S. Cassels and A. Fröhlich shall be more than sufficient. Here, the most important results that we shall use are the existence of the global Artin map and, related to this, the computation of the Brauer group of a number field [Ta67, 11.2]. b) We will use the language of modern algebraic geometry as described in the textbook of R. Hartshorne [Ha77, Chapter II]. Cohomology of coherent sheaves [Ha77, Chapter III] will be used occasionally. c) In Chapter III, we will make substantial use of étale cohomology. This is probably the deepest prerequisite that we expect from the reader. For this reason, we will formulate its main principles, as they appear to be of importance for us, at the beginning of the chapter. It seems to us that, in order to follow the arguments, an understanding of Chapters II and III of J. Milne’s textbook [Mi] should be sufficient. At a few points, some other background material may be helpful. This concerns, for example, Artin L-functions. Here, [Hei] may serve as a general reference. Pre- cise citations shall, of course, be given wherever the necessity occurs. A suggestion that might be helpful for the reader. Part C of this book describes experiments concerning the Manin conjecture. Clearly, the particular samples are vii viii preface chosen in such a way that not all the difficulties, which are theoretically possible, really occur. It therefore seems that Part C might be easier to read than the others, particularly for those readers who are very familiar with computers and the concept of an al- gorithm. Thus, such a reader could try to start with Part C to learn about the experiments and to get acquainted with the theory. It is possible then to continue, in a second step, with Parts A and B in order to get used to the theory in its full strength. References and citations. When we refer to a definition, proposition, theorem, etc., in the same chapter we simply rely on the corresponding numbering within the chapter. Otherwise, we add the number of the chapter. For the purpose of citation, the articles and books being used are encoded in the manner specified by the bibliography. In addition, we mostly give the number of the relevant section and subsection or the number of the definition, proposition, theorem, etc. Normally, we do not mention page numbers. Acknowledgments. I wish to acknowledge with gratitude my debt to Y. Tschinkel. Most of the work described in this book, which is a shortened version of my Habil- itation Thesis, was initiated by his numerous mathematical questions. During the years he spent at Göttingen, he always shared his ideas in an extraordinarily gen- erous manner. I further wish to express my deep gratitude to my friend and colleague Andreas- Stephan Elsenhans. He influenced this book in many ways, directly and indirectly. It is no exaggeration to say that most of what I know about computer program- ming I learned from him. The experiments, which are described in Part C, were carried out together with him as a joint work. I am also indebted to Stephan for proofreading. The computer part of the work described in this book was executed on the Sun Fire V20z Servers of the Gauß Laboratory for Scientific Computing at the Göttingen Mathematical Institute.The author is grateful to Y.
Load more

Recommended publications
  • A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature
    A survey of some arithmetic applications of ergodic theory in negative curvature Jouni Parkkonen Frédéric Paulin January 12, 2015 Abstract This paper is a survey of some arithmetic applications of techniques in the geometry and ergodic theory of negatively curved Riemannian manifolds, focusing on the joint works of the authors. We describe Diophantine approximation results of real numbers by quadratic irrational ones, and we discuss various results on the equidistribution in R, C and in the Heisenberg groups of arithmetically defined points. We explain how these results are consequences of equidistribution and counting properties of common perpendiculars between locally convex subsets in negatively curved orbifolds, proven using dynamical and ergodic properties of their geodesic flows. This exposition is based on lectures at the conference “Chaire Jean Morlet: Géométrie et systèmes dynamiques”, at the CIRM, Luminy, 2014. We thank B. Hasselblatt for his strong encouragements to write this survey. 1 1 Introduction For several decades, tools from dynamical systems, and in particular ergodic theory, have been used to derive arithmetic and number theoretic, in particular Diophantine approxi- mation results, see for instance the works of Furstenberg, Margulis, Sullivan, Dani, Klein- bock, Clozel, Oh, Ullmo, Lindenstrauss, Einsiedler, Michel, Venkatesh, Marklof, Green- Tao, Elkies-McMullen, Ratner, Mozes, Shah, Gorodnik, Ghosh, Weiss, Hersonsky-Paulin, Parkkonen-Paulin and many others, and the references [Kle2, Lin, Kle1, AMM, Ath, GorN, EiW, PaP5]. arXiv:1501.02072v1 [math.NT] 9 Jan 2015 In Subsection 2.2 of this survey, we introduce a general framework of Diophantine approximation in measured metric spaces, in which most of our arithmetic corollaries are inserted (see the end of Subsection 2.2 for references concerning this framework).
    [Show full text]
  • Algorithmic Factorization of Polynomials Over Number Fields
    Rose-Hulman Institute of Technology Rose-Hulman Scholar Mathematical Sciences Technical Reports (MSTR) Mathematics 5-18-2017 Algorithmic Factorization of Polynomials over Number Fields Christian Schulz Rose-Hulman Institute of Technology Follow this and additional works at: https://scholar.rose-hulman.edu/math_mstr Part of the Number Theory Commons, and the Theory and Algorithms Commons Recommended Citation Schulz, Christian, "Algorithmic Factorization of Polynomials over Number Fields" (2017). Mathematical Sciences Technical Reports (MSTR). 163. https://scholar.rose-hulman.edu/math_mstr/163 This Dissertation is brought to you for free and open access by the Mathematics at Rose-Hulman Scholar. It has been accepted for inclusion in Mathematical Sciences Technical Reports (MSTR) by an authorized administrator of Rose-Hulman Scholar. For more information, please contact [email protected]. Algorithmic Factorization of Polynomials over Number Fields Christian Schulz May 18, 2017 Abstract The problem of exact polynomial factorization, in other words expressing a poly- nomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of the adjoined elements used to form the algebraic number field in question, let s be the sum of the squares of these degrees, and let d be the degree of the polynomial to be factored; then the runtime of this algorithm is found to be O(d4sk2 + 2dd3).
    [Show full text]
  • Gaussian Integers1
    FORMALIZED MATHEMATICS Vol. 21, No. 2, Pages 115–125, 2013 DOI: 10.2478/forma-2013-0013 degruyter.com/view/j/forma Gaussian Integers1 Yuichi Futa Hiroyuki Okazaki Japan Advanced Institute Shinshu University of Science and Technology Nagano, Japan Ishikawa, Japan Daichi Mizushima2 Yasunari Shidama Shinshu University Shinshu University Nagano, Japan Nagano, Japan Summary. Gaussian integer is one of basic algebraic integers. In this artic- le we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaus- sian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic. MSC: 11R04 03B35 Keywords: formalization of Gaussian integers; algebraic integers MML identifier: GAUSSINT, version: 8.1.02 5.17.1179 The notation and terminology used in this paper have been introduced in the following articles: [5], [1], [2], [6], [12], [11], [7], [8], [18], [24], [23], [16], [19], [21], [3], [9], [20], [14], [4], [28], [25], [22], [26], [15], [17], [10], and [13]. 1. Gaussian Integer Ring Now we state the proposition: (1) Let us consider natural numbers x, y. If x + y = 1, then x = 1 and y = 0 or x = 0 and y = 1. Proof: x ¬ 1. 1This work was supported by JSPS KAKENHI 21240001 and 22300285. 2This research was presented during the 2012 International Symposium on Information Theory and its Applications (ISITA2012) in Honolulu, USA. c 2013 University of Białystok CC-BY-SA License ver.
    [Show full text]
  • Applications of Hyperbolic Geometry to Continued Fractions and Diophantine Approximation
    Applications of hyperbolic geometry to continued fractions and Diophantine approximation Robert Hines University of Colorado, Boulder April 1, 2019 A picture Summary Our goal is to generalize features of the preceding picture to some nearby settings: H2 H3 (H2)r × (H3)s Hn P 1(R) P 1(C) P 1(R)r × P 1(C)s Sn−1 SL2(R) SL2(C) SL2(F ⊗ R) SVn−1(R) SL2(Z) SL2(O) " SV (O) ideal right-angled triangles ideal polyhedra horoball bounded geodesic " neighborhoods trajectories quad. quad./Herm. " " forms forms closed geodesics closed surfaces aniso. subgroups " Ingredients Ingredients Upper half-space models in dimensions two and three Hyperbolic two-space: 2 H = fz = x + iy 2 C : y > 0g; 2 1 @H = P (R); 2 Isom(H ) = P GL2(R); az + b az¯ + b g · z = ; (det g = ±1); cz + d cz¯ + d + ∼ Stab (i) = SO2(R)={±1g = SO2(R): Hyperbolic three-space: 3 H = fζ = z + jt 2 H : t > 0; z 2 Cg; 3 1 @H = P (C); 3 Isom(H ) = P SL2(C) o hτi; g · ζ = (aζ + b)(cζ + d)−1; τ(ζ) =z ¯ + jt; + ∼ Stab (j) = SU2(C)={±1g = SO3(R): Binary quadratic and Hermitian forms Hyperbolic two- and three-space are the Riemannian symmetric spaces associated to G = SL2(R), SL2(C). The points can be identified with roots of binary forms: 2 SL2(R)=SO2(R) ! fdet. 1 pos. def. bin. quadratic formsg ! H p −b + b2 − 4ac g 7! ggt = ax2 + bxy + cy2 = Q 7! =: Z(Q) 2a 3 SL2(C)=SU2(C) ! fdet.
    [Show full text]
  • On a Certain Non-Split Cubic Surface
    ON A CERTAIN NON-SPLIT CUBIC SURFACE R. DE LA BRETECHE,` K. DESTAGNOL, J. LIU, J. WU & Y. ZHAO Abstract. In this note, we establish an asymptotic formula with a power-saving error term for the number of rational points of 3 bounded height on the singular cubic surface of PQ 2 2 3 x0(x1 + x2)= x3 in agreement with the Manin-Peyre conjectures. 1. Introduction and results 3 Let V ⊂ PQ be the cubic surface defined by 2 2 3 x0(x1 + x2) − x3 =0. The surface V has three singular points ξ1 = [1 : 0 : 0 : 0], ξ2 = [0 : 1 : i : 0] and ξ3 = [0 : 1 : −i : 0]. It is easy to see that the only three lines contained in VQ = V ×Spec(Q) Spec(Q) are ℓ1 := {x3 = x1 − ix2 =0}, ℓ2 := {x3 = x1 + ix2 =0}, and ℓ3 := {x3 = x0 =0}. Clearly both ℓ1 and ℓ2 pass through ξ1, which is actually the only rational point lying on these two lines. Let U = V r {ℓ1 ∪ ℓ2 ∪ ℓ3}, and B a parameter that can approach infinity. In this note we are concerned with the behavior of the counting arXiv:1709.09476v2 [math.NT] 12 Dec 2018 function NU (B) = #{x ∈ U(Q): H(x) 6 B}, where H is the anticanonical height function on V defined by 2 2 H(x) := max |x0|, x1 + x2, |x3| (1.1) n q o where each xj ∈ Z and gcd(x0, x1, x2, x3) = 1. The main result of this note is the following. Theorem 1.1. There exists a constant ϑ > 0 and a polynomial Q ∈ R[X] of degree 3 such that 1−ϑ NU (B)= BQ(log B)+ O(B ).
    [Show full text]
  • On a Certain Non-Split Cubic Surface Régis De La Bretèche, Kévin Destagnol, Jianya Liu, Jie Wu, Yongqiang Zhao
    On a certain non-split cubic surface Régis de la Bretèche, Kévin Destagnol, Jianya Liu, Jie Wu, Yongqiang Zhao To cite this version: Régis de la Bretèche, Kévin Destagnol, Jianya Liu, Jie Wu, Yongqiang Zhao. On a certain non- split cubic surface. Science China Mathematics, Science China Press, 2019, 62 (12), pp.2435-2446. 10.1007/s11425-018-9543-8. hal-02468799 HAL Id: hal-02468799 https://hal.sorbonne-universite.fr/hal-02468799 Submitted on 6 Feb 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ON A CERTAIN NON-SPLIT CUBIC SURFACE R. DE LA BRETECHE,` K. DESTAGNOL, J. LIU, J. WU & Y. ZHAO Abstract. In this note, we establish an asymptotic formula with a power-saving error term for the number of rational points of 3 bounded height on the singular cubic surface of PQ 2 2 3 x0(x1 + x2)= x3 in agreement with the Manin-Peyre conjectures. 1. Introduction and results 3 Let V ⊂ PQ be the cubic surface defined by 2 2 3 x0(x1 + x2) − x3 =0. The surface V has three singular points ξ1 = [1 : 0 : 0 : 0], ξ2 = [0 : 1 : i : 0] and ξ3 = [0 : 1 : −i : 0].
    [Show full text]
  • Algebraic Number Theory and Its Application to Rational Theory
    Journal of Shanghai Jiaotong University ISSN:1007-1172 Algebraic Number Theory And Its Application To Rational Theory Nitish Kumar Bharadwaj1 & Ajay Kumar2 1. Research Scholar, University Department of Mathematics, T. M. Bhagalpur University, Bhagalpur, Email: [email protected] 2. Research Scholar, University Department of Mathematics, T. M. Bhagalpur University, Bhagalpur, Email: [email protected] ABSTRACT In this paper we recall some nation connected with algebraic number theory and indicate some of its applications in the Gaussian field namely K(i) = √(–1). A Gaussian rational number is complex number of the form a+bi, where a and b both are rational numbers. The set of all Gaussian rationals forms the Gaussian rational field, denoted by K(i). Introduction Numbers have fascinated man from a very early period of human civilization. Pythagoreans studied many properties of natural numbers 1, 2, 3 ......... The famous theorem of Pythagoras (6th century B.C) through geometrical has a pronounced number theoretic content. The early Babylonians had noted many Pythagorean triads e.g. 3, 4, 5; 5, 12, 13 which are natural number a, b, c satisfying the equation. a2 + b2 = c2 ............. (1) In about 250 A.D Diophantus of Alexandria wrote treatise on polynomial equations which studied solution in fractions. Particular cases of these equations with natural number have been called Diophantine equations to this day. The study of algebra developed over centuries. The Hindu mathematicians deals with and introduced the nation of negative numbers and zero. In the 16th century and owned negative and imaginary numbers were used with increasing confidence and flexibility. Meanwhile the study of the theory of natural numbers went on side by side.
    [Show full text]
  • First Moment of Distances Between Centres of Ford Spheres
    First Moment of Distances Between Centres of Ford Spheres Kayleigh Measures (York) May 2018 Abstract This paper aims to develop the theory of Ford spheres in line with the current theory for Ford circles laid out in a recent paper by S. Chaubey, A. Malik and A. Zaharescu. As a first step towards this goal, we establish an asymptotic estimate for the first moment X 1 1 M (S) = + ; 1;I2 2jsj2 2js0j2 r r0 s ; 0 2GS consecs where the sum is taken over pairs of fractions associated with `consecutive' Ford 1 spheres of radius less than or equal to 2S2 . 1 Introduction and Motivation Ford spheres were first introduced by L. R. Ford in [3] alongside their two dimensional p analogues, Ford circles. For a Farey fraction q , its Ford circle is a circle in the upper 1 p half-plane of radius 2q2 which is tangent to the real line at q . Similarly, for a fraction r s with r and s Gaussian integers, its Ford sphere is a sphere in the upper half-space of 1 r radius 2jsj2 tangent to the complex plane at s . This paper studies moments of distances between centres of Ford spheres, in line with the moment calculations for Ford circles produced by S. Chaubey, A. Malik and A. Zaharescu. In [2], Chaubey et al. consider those Farey fractions in FQ which lie in a fixed interval I := [α; β] ⊆ [0; 1] for rationals α and β. They call the set of Ford circles corresponding to these fractions FI;Q, and its cardinality is denoted NI (Q).
    [Show full text]
  • Jugendtraum of a Mathematician
    F EATURE Principal Investigator Kyoji Saito Research Area：Mathematics Jugendtraum of a mathematician A number which appears when we count things §1 Kronecker’s Jugendtraum as one, two, three, ... is called a natural number. There is a phrase“ Kronecker’s Jugendtraum The collection of all natural numbers is denoted (dream of youth)” in mathematics. Leopold by N. When we want to prove a statement which Kronecker was a German mathematician who holds for all natural numbers, we use mathematical worked in the latter half of the 19th century. He induction as we learn in high school. It can be obtained his degree at the University of Berlin in proved by using induction that we can define 1845 when he was 22 years old, and after that, he addition and multiplication for elements of N (that successfully managed a bank and a farm left by his is, natural numbers) and obtain again an element of deceased uncle. When he was around 30, he came N as a result. But we cannot carry out subtraction back to mathematics with the study of algebraic in it. For example, 2－3 is not a natural number equations because he could not give up his love for anymore. Subtraction is defined for the system of mathematics. Kronecker’s Jugendtraum refers to a numbers ..., －3, －2, －1, 0, 1, 2, 3, .... We call such a series of conjectures in mathematics he had in those number an integer and denote by Z the collection days ̶ maybe more vague dreams of his, rather of integers. For Z, we have addition, subtraction, and than conjectures ̶ on subjects where the theories multiplication, but still cannot carry out division.
    [Show full text]
  • Gaussian Integers1
    FORMALIZED MATHEMATICS Vol. 21, No. 2, Pages 115–125, 2013 DOI: 10.2478/forma-2013-0013 degruyter.com/view/j/forma Gaussian Integers1 Yuichi Futa Hiroyuki Okazaki Japan Advanced Institute Shinshu University of Science and Technology Nagano, Japan Ishikawa, Japan Daichi Mizushima2 Yasunari Shidama Shinshu University Shinshu University Nagano, Japan Nagano, Japan Summary. Gaussian integer is one of basic algebraic integers. In this artic- le we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaus- sian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic. MSC: 11R04 03B35 Keywords: formalization of Gaussian integers; algebraic integers MML identifier: GAUSSINT, version: 8.1.02 5.17.1179 The notation and terminology used in this paper have been introduced in the following articles: [5], [1], [2], [6], [12], [11], [7], [8], [18], [24], [23], [16], [19], [21], [3], [9], [20], [14], [4], [28], [25], [22], [26], [15], [17], [10], and [13]. 1. Gaussian Integer Ring Now we state the proposition: (1) Let us consider natural numbers x, y. If x + y = 1, then x = 1 and y = 0 or x = 0 and y = 1. Proof: x ¬ 1. 1This work was supported by JSPS KAKENHI 21240001 and 22300285. 2This research was presented during the 2012 International Symposium on Information Theory and its Applications (ISITA2012) in Honolulu, USA. c 2013 University of Białystok CC-BY-SA License ver.
    [Show full text]
  • The Powers of Π Are Irrational
    The Powers of π are Irrational Tim Jones October 19, 2010 Abstract Transcendence of a number implies the irrationality of powers of a number, but in the case of π there are no separate proofs that powers of π are irrational. We investigate this curiosity. Transcendence proofs for e involve what we call Hermite’s technique; for π’s transcendence Lindemann’s adaptation of Hermite’s technique is used. Hermite’s technique is presented and its usage is demonstrated with irrational- ity proofs of e and powers of e. Applying Lindemann’s adaptation to a complex polynomial, π is shown to be irrational. This use of a complex polynomial generalizes and powers of π are shown to be irrational. The complex polynomials used involve roots of i and yield regular polygons in the complex plane. One can use graphs of these polygons to visualize various mechanisms used to proof π2, π3, and π4 are irrational. The transcendence of π and e are easy generalizations from these irrational cases. 1 Introduction There are curiosities in the treatment of the transcendence and irra- tionality of π and e. One such curiosity is the absence of proofs that powers of π are irrational. Certainly π2 is proven to be irrational fre- quently [4, 21, 37], but not higher powers.1 Proofs of the irrationality of powers of e [12, 28, 32]2 are given in text books, making it all the more odd that the irrationality of powers of π are not even mentioned. 1Ha¨ncl does give a proof of the irrationality of π4 [11].
    [Show full text]
  • Ford Circles and Spheres
    Ford Circles and Spheres Sam Northshield∗ Abstract Ford circles are parameterized by the rational numbers but are also the result of an iterative geometric procedure. We review this and introduce an apparently new parameterization by solutions of a certain quadratic Diophantine equation. We then generalize to Eisenstein and Gaussian rationals where the resulting “Ford spheres” are also the result of iterative geometric procedures and are also parameterized by solutions of certain quadratic Diophantine equations. We generalize still further to imaginary quadratic fields. 1 Introduction The set of Ford circles form an arrangement of circles each above but tangent to the x-axis at a rational number, with disjoint interiors, that is maximal in the sense that no additional such circles can be added. arXiv:1503.00813v1 [math.NT] 3 Mar 2015 Figure 1: Ford circles and their geometric construction They provide a natural way of visualizing the Diophantine approximation of real numbers by rationals, are used in the “circle method” of Ramanujan and Hardy, and give a geometric way of looking at continued fractions. They form part of an Apollonian circle packing, a topic of intense current study, as ∗SUNY-Plattsburgh 1 well as provide some connection to some older open problems (e.g., see [18] for connection to the Riemann hypothesis, [6] for Hausdorff dimension). Three dimensional analogues are then certainly of interest. In Section 2, we construct the set of Ford circles in three different ways. First, to each rational number a/b (in lowest terms), assign a circle above but tangent to the x-axis at a/b with radius1/2b2.
    [Show full text]