DOCSLIB.ORG

The Local-Global Principle in Number Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Local-Global Principle in Number Theory The Local-Global Principle in Number Theory Arnélie Schinck A Thesis in The Department of Mathematics and Statistics Presented in Partial Fulfillment of the Requuements for the Degree of Master of Science at Concordia University Montréal, Québec, Canada Septernber 200 1 OAméiie Schinck, 200 1 Natiarial Library Bibiiïtheque nationale 1+1 ,calada du Canada Acquisitions and Acquisitions et BiMiographic Services senrices bibbgraphiques 395 W.liirg(on Street 395, IUO WeYington -ON KlAûN4 OüawaON KlAW CaMda Canada The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, 10- distribute or seli reproduire, prêter, disûibuer ou copies of this thesis in microfonn, vendre des copies de cette thèse sous paper or electronic formats. la forme de micrdche/film, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. ABSTRACT The Local-Global Principle in Number Theory Amélie Schinck "p-adic fields prmOZrlderernark~ble, easy and natural sol~utions to problems ,which apparently have no relation to p-udic fields and which othenrrise can be resolved, if at dl, only bt~deep and arduous methods". J .\\-.S. Cassels The first Local-Global Principle. as formulated by Rasse in 1921, relates the behaviour of rational quadratic forms in Q (global field) to their behaviour in the padic fields Q, (local fields). The notion of using local information as a stepping Stone towards understanding more dificult global properties has been generalized and applied to many problemsl making Local-Global met hods a powerful number t heoret ic tool. Even when the principle fails, m-e can sometimes saIvage some connection between the local and the global. This thesis aims to give a survey of the basic theory. Acknowledgements 1 wish to thank my thesis supervisor, Dr. Hershy Kisilevsky. His guidance and encouragement helped me stay on course. 1 am also grateful to Dr. Sebastien Pauli for his help in translating some of Hemel's German text to English, giving me a better understanding of the history. Finally, 1 would iike to thank Craig T. Achen for his loving support throughout the writing of this t hesis. Contents Preface 1 Introduction 1 1.1 Hensel's Analogy . - - . - . 1 1.2 The Transcendence of e . - . 3- 1.3 ThepadicAbsoluteValue. ..... ... ...... .... 3 1.4 Hensel.s Lemma . - - - . 6 2 The Hasse Principle 9 2.1 Helmut Hûsse . - . - . - . - . - - - - - - 9 2.2 The Hasse-Minkowski Theorem . 11 2.3 Hilbert's Eleventh Problem . - . - - 13 3 Diophantine Equations 15 3.1 Equations modula p . - . 1-5 3.2 Irrationality of fiand the Three Square Theorem . - - - - . 18 4 The Hilbert Symbol 20 4.1 The Hilbert Çymbol . - . - . - 20 4.2 Hilbert's Tenth ProbIem . 23 5 Proof of the Hasse-Minkowski Theorem 24 5.1 Proof for Ternary Quadratic Forms (n=3) . - . - - . - - 24 5.2 Proof for n >3 . - . 26 5.3 Artin's Conjecture . , . - . 30 5.4 Projecting a Conic ont0 a Line . - - - . - . - - . 31 6 Equivalence of Quadratic Forms 33 6.1 Preliminary Results ......................... 33 6.2 A LocaCGlobal Pt-inciple ...................... 34 7 Failure of the Hasse Principle 37 7.1 A Simple Counterexample ..................... 37 7.2 Seimer's Curve ............................ 38 7.3 Reichard's Equation ......................... 43 7.4 Representation of Integers by Quadratic Forms ........ 45 8 Finite Extensions of Qp 47 8.1 The p-adic Fields .......................... 47 8.2 Hasse-Minkowski for Algebraic Number FieIds ........ 49 9 The Product Formula and Quadratic Reciprocity 52 9.1 Euler's Criterion ........................... 53 9.2 Proof of the Product Fonnda ................... 56 9.3 Quadratic Reciprocity ........................ 57 9.1 Hasse's Product Formula ...................... 58 10 The Hasse Norm Theorem 60 10.1 The Hilbert Norm Tlieorern .................... 60 10.2 The Hasse Norm Theorem for Cyclic Extensions ....... 60 10.3 Abelian Extensions ......................... 62 10.4 Some-4pplications ......................... 63 11 Measuring the Failure 64 1 1.1 First Cohornology Group ..................... 64 11.2 Hasse Principle ........................... 66 11.3 Principal Homogeneous Spaces .................. 67 11.4 The Tate-Shafarevich Group ................... 68 Bibliography 69 Preface Chapter 1 is an overview of padic fields in their historical context. It contains important results on padic numbers, such as Hemel's Lemma, which will often be referred to throughout the course of this thesis. Chapter 2 contains a statement of the Local-Global Principle (Hasse Principle) and traces its historical development. For t his part of rny study, I am especially indebted to Dr. Giinther Frei, who kindly sent me a copy of the article he wrote on the subject for the Proceedings of the Class Field Theory Conference in Tokyo. We apply the Local-Global Principle to the study of Diophanthe equations in Chapter 3, easily obtaining seemingly difficult results. In Chapter 4, the Hilbert symbol and its properties are reviewed. The proofs of the Hasse-Minkowski Theorem for the repcesentation of a rational number by a quadratic form and the equivalence of quadratic forms are given in Chapter 5 and 6. Both chapters were heavily influenced by Borevich's and Shafarevic h's approach in their collaborative t ext Number Theory. Famous examples of the failure of the Hasse Principle are discussed at length in Chapter 7, among them, the very hst counterexample discovered by Reichard in 1942. The Hasse-Minkowski Theorem is extended to finite extensions of the padic field in Chapter 8, fully solving Hilbert's eleventh problem. Chapter 9 deduces quadratic reciprocity from a local-global result; The product formula. The Hasse Norm Theorem, of which the Hasse-Minkowski theorem is a particdm case, is the focus of Chapter 10. The failure of the Hasse Principle is again the topic of Chapter 11, but this time, put in the language of cohomology, allowing us to discuss the group which measures the obstruction caUed the Tate-Shafarevich group. vii Chapter 1 Introduction In the course of extending Kronecker's work on the factorization of prime ideals in number fields, one of his studeats was led to the creation of the padic numbers. This astute student was Kurt Hensel. Hensel camed over the method of cornplex analysis of expanding functions locaiiy in order to get information on their global properties to the land of Number Theory. He keenly obsenred that the linear factors (x - a), the prime ideals of the ring @[XI,play an andogous role in @(x) to the prime numbers p in Q. Hensel translateci the Laurent series expansion of a hctionflx) E @(x) about a point a E @ flx) =xai(x - a)' into a closely analogous Laurent series expansion of r E Q in progressive powers of a prime p Hensel called the latter the padic expmion of r. Any rational number r can be written padically with respect to every prime element p of Z. The padic expansion gives us local information about r near p, just as each Laurent expansion gives us / local information about flx) near a. Hensel showed that the collection of al1 such Laurent series in powers of p forms a field, which he dubbed the field of padic numbers and labeled K@). In modem shorthand this field is denoted Qp. An interesthg historicd side note is that, at the tirne, the dehition of a field forrnulated by Dedekind required that it be a subfield of the field of complex numbers, a requirement the padic fields fail to satisfy. The fields of padic numbers were the motivation behind the fundamental treatise on abstract Field Theory undertaken by Steinitz. The padic fields were the earliest examples of what are now known as Local Fields, with the exception of the fields of rdand complex numbers. Local Fields will be discussed further in Chapter 8. Hensel introduced padic numbers in 1897 in a short pape entitled &r eine neue BegrWldung der Theorie der algebraischen Zahlen (About a ~iewfmndation of the themg of algehaic nmbers). Hensel's treatment was extremely formal and failed to dazzle the mathematicd community. The padics were dismisseci at the time as imaginative play things, devoid of any real use because no substantial result for which they were indispensable had been found. Hilbert mentioried them in his famous Paris lecture in 1900 yet they went virtually unnoticed for decades. This attitude was in large part due to an error by Hensel. 1.2 The Transcendence of e From the beginning, Hensel was intent on applying padic methods to transcendency questions, a fashionable topic at the end of the 19th century. He gave a flawed proof of the transcendence of e in 1905 during an invited lecture given in Merano, Italy, to the Versnmmlung deutscher Nuturforscher und hze (Meeting of German Natural Scientists and Physicians). A discussion of Hensel's erroneous argument can be found in [29]. Hensel later had to admit that ''The proof of the transcendency of e given in ma( Meran lecture needs an essential campletion." This blunder did not help the perceiveci legitimacy of padic numbers in mathematical circles. Jean-Paul Bézivin and Philippe Robba obtained a padic proof of the Lindemann-Weierstrass Theorem in a 1987 article [4]. This theorem states that the values taken by the exponential function at different algebraic points are linearly independent over the field of algebraic numbers. The Lindemann- Weierstrass theorem impiies that e is transcendental. Referring to their result the authors wrote: "We are glad to vindicate Hensel in hzs idea that p-adic nwrnbers could be used to proue transcendental results althuugh the proof devised here has nothing to do urith the rnethod that Hensel intended to use." 1.3 The padic abdute value Nevertheless, Hensel continued to develop his theory more fully.
Load more

Recommended publications
  • Hasse Principles for Higher-Dimensional Fields 3
    Annals of Mathematics 183 (2016), 1{71 http://dx.doi.org/10.4007/annals.2016.183.1.1 Hasse principles for higher-dimensional fields By Uwe Jannsen dedicated to J¨urgen Neukirch Abstract For rather general excellent schemes X, K. Kato defined complexes of Gersten-Bloch-Ogus type involving the Galois cohomology groups of all residue fields of X. For arithmetically interesting schemes, he developed a fascinating web of conjectures on some of these complexes, which generalize the classical Hasse principle for Brauer groups over global fields, and proved these conjectures for low dimensions. We prove Kato's conjecture over number fields in any dimension. This gives a cohomological Hasse principle for function fields F over a number field K, involving the corresponding function fields Fv over the completions Kv of K. For global function fields K we prove the part on injectivity for coefficients invertible in K. Assuming resolution of singularities, we prove a similar conjecture of Kato over finite fields, and a generalization to arbitrary finitely generated fields. Contents 0. Introduction1 1. First reductions and a Hasse principle for global fields7 2. Injectivity of the global-local map for coefficients invertible in K 16 3. A crucial exact sequence, and a Hasse principle for unramified cohomology 26 4. A Hasse principle for Bloch-Ogus-Kato complexes 40 5. Weight complexes and weight cohomology 52 References 67 0. Introduction In this paper we prove some conjectures of K. Kato [Kat86] which were formulated to generalize the classical exact sequence of Brauer groups for a global field K, L (0.1) 0 −! Br(K) −! Br(Kv) −! Q=Z −! 0; v c 2016 Department of Mathematics, Princeton University.
    [Show full text]
  • 8 Complete Fields and Valuation Rings
    18.785 Number theory I Fall 2016 Lecture #8 10/04/2016 8 Complete fields and valuation rings In order to make further progress in our investigation of finite extensions L=K of the fraction field K of a Dedekind domain A, and in particular, to determine the primes p of K that ramify in L, we introduce a new tool that will allows us to \localize" fields. We have already seen how useful it can be to localize the ring A at a prime ideal p. This process transforms A into a discrete valuation ring Ap; the DVR Ap is a principal ideal domain and has exactly one nonzero prime ideal, which makes it much easier to study than A. By Proposition 2.7, the localizations of A at its prime ideals p collectively determine the ring A. Localizing A does not change its fraction field K. However, there is an operation we can perform on K that is analogous to localizing A: we can construct the completion of K with respect to one of its absolute values. When K is a global field, this process yields a local field (a term we will define in the next lecture), and we can recover essentially everything we might want to know about K by studying its completions. At first glance taking completions might seem to make things more complicated, but as with localization, it actually simplifies matters considerably. For those who have not seen this construction before, we briefly review some background material on completions, topological rings, and inverse limits.
    [Show full text]
  • Number Theoretic Symbols in K-Theory and Motivic Homotopy Theory
    Number Theoretic Symbols in K-theory and Motivic Homotopy Theory Håkon Kolderup Master’s Thesis, Spring 2016 Abstract We start out by reviewing the theory of symbols over number fields, emphasizing how this notion relates to classical reciprocity lawsp and algebraic pK-theory. Then we compute the second algebraic K-group of the fields pQ( −1) and Q( −3) based on Tate’s technique for K2(Q), and relate the result for Q( −1) to the law of biquadratic reciprocity. We then move into the realm of motivic homotopy theory, aiming to explain how symbols in number theory and relations in K-theory and Witt theory can be described as certain operations in stable motivic homotopy theory. We discuss Hu and Kriz’ proof of the fact that the Steinberg relation holds in the ring π∗α1 of stable motivic homotopy groups of the sphere spectrum 1. Based on this result, Morel identified the ring π∗α1 as MW the Milnor-Witt K-theory K∗ (F ) of the ground field F . Our last aim is to compute this ring in a few basic examples. i Contents Introduction iii 1 Results from Algebraic Number Theory 1 1.1 Reciprocity laws . 1 1.2 Preliminary results on quadratic fields . 4 1.3 The Gaussian integers . 6 1.3.1 Local structure . 8 1.4 The Eisenstein integers . 9 1.5 Class field theory . 11 1.5.1 On the higher unit groups . 12 1.5.2 Frobenius . 13 1.5.3 Local and global class field theory . 14 1.6 Symbols over number fields .
    [Show full text]
  • Little Survey on Large Fields – Old & New –
    Little survey on large fields { Old & New { Florian Pop ∗ Abstract. The large fields were introduced by the author in [59] and subsequently acquired several other names. This little survey includes earlier and new developments, and at the end of each section we mention a few open questions. 2010 Mathematics Subject Classification. Primary 12E, 12F, 12G, 12J. Secondary 12E30, 12F10, 12G99. Keywords. Large fields, ultraproducts, PAC, pseudo closed fields, Henselian pairs, elementary equivalence, algebraic varieties, rational points, function fields, (inverse) Galois theory, embedding problems, model theory, rational connectedness, extremal fields. Introduction The notion of large field was introduced in Pop [59] and proved to be the \right class" of fields over which one can do a lot of interesting mathematics, like (inverse) Galois theory, see Colliot-Th´el`ene[7], Moret-Bailly [45], Pop [59], [61], the survey article Harbater [30], study torsors of finite groups Moret- Bailly [46], study rationally connected varieties Koll´ar[37], study the elementary theory of function fields Koenigsmann [35], Poonen{Pop [55], characterize extremal valued fields as introduced by Ershov [12], see Azgin{Kuhlmann{Pop [1], etc. Maybe that is why the \large fields” acquired several other names |google it: ´epais,fertile, weite K¨orper, ample, anti-Mordellic. Last but not least, see Jarden's book [34] for more about large fields (which he calls \ample fields” in his book [34]), and Kuhlmann [41] for relations between large fields and local uniformization (`ala Zariski). Definition. A field k is called a large field, if for every irreducible k-curve C the following holds: If C has a k-rational smooth point, then C has infinitely many k-rational points.
    [Show full text]
  • Conics Over Function Fields and the Artin-Tate Conjecture José Felipe
    Conics over function fields and the Artin-Tate conjecture Jos´eFelipe Voloch Abstract: We prove that the Hasse principle for conics over function fields is a simple consequence of a provable case of the Artin-Tate conjecture for surfaces over finite fields. Hasse proved that a conic over a global field has a rational point if and only if it has points over all completions of the global field, an instance of the so-called local-global or Hasse principle. The case of the rational numbers is an old result of Legendre, who gave an elementary proof and a similar proof can be given in the case of rational functions over a finite field. Hasse’s proof, on the other hand, is a consequence of a more general result in Class Field Theory, the Hasse norm theorem. One purpose of this paper is to give a new proof of the local-global principle for conics over function fields, as a relatively simple consequence of a provable case of the Artin-Tate conjecture for surfaces over finite fields. This proof might be more complicated overall, once the work on the Artin-Tate conjecture is factored in, but it might be worthwhile recording since it suggests a new approach to local-global principles which might work in other contexts, such as number fields or higher dimensions. We also deduce from the Artin-Tate conjecture, together with a recent result [LLR] on Brauer groups, the Hilbert reciprocity law. We also do a careful study of conic bundles over curves over finite fields which might have independent interest.
    [Show full text]
  • Local-Global Methods in Algebraic Number Theory
    LOCAL-GLOBAL METHODS IN ALGEBRAIC NUMBER THEORY ZACHARY KIRSCHE Abstract. This paper seeks to develop the key ideas behind some local-global methods in algebraic number theory. To this end, we first develop the theory of local fields associated to an algebraic number field. We then describe the Hilbert reciprocity law and show how it can be used to develop a proof of the classical Hasse-Minkowski theorem about quadratic forms over algebraic number fields. We also discuss the ramification theory of places and develop the theory of quaternion algebras to show how local-global methods can also be applied in this case. Contents 1. Local fields 1 1.1. Absolute values and completions 2 1.2. Classifying absolute values 3 1.3. Global fields 4 2. The p-adic numbers 5 2.1. The Chevalley-Warning theorem 5 2.2. The p-adic integers 6 2.3. Hensel's lemma 7 3. The Hasse-Minkowski theorem 8 3.1. The Hilbert symbol 8 3.2. The Hasse-Minkowski theorem 9 3.3. Applications and further results 9 4. Other local-global principles 10 4.1. The ramification theory of places 10 4.2. Quaternion algebras 12 Acknowledgments 13 References 13 1. Local fields In this section, we will develop the theory of local fields. We will first introduce local fields in the special case of algebraic number fields. This special case will be the main focus of the remainder of the paper, though at the end of this section we will include some remarks about more general global fields and connections to algebraic geometry.
    [Show full text]
  • Counterexamples to the Hasse Principle
    COUNTEREXAMPLES TO THE HASSE PRINCIPLE W. AITKEN AND F. LEMMERMEYER Abstract. This article explains the Hasse principle and gives a self-contained development of certain counterexamples to this principle. The counterexam- ples considered are similar to the earliest counterexample discovered by Lind and Reichardt. This type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in Tate– Shafarevich groups. 1. Introduction In this article we develop counterexamplestotheHasseprincipleusingonlytech- niques from undergraduate number theory and algebra. By keeping the technical prerequisites to a minimum, we hope to provide a path for nonspecialists to this interesting area of number theory. The counterexamples considered here extend the classical counterexample of Lind and Reichardt. As discussed in an appendix, such counterexamples are important in the theory of elliptic curves, and today are interpreted as nontrivial elements in Tate–Shafarevich groups. 2. Background The problem of determining if the Diophantine equation aX2 + bY 2 + cZ2 =0 (1) has nontrivial solutions with values in Z has played a prominent role in the history of number theory. We assume that a, b, and c are nonzero integers and, using a simple argument, we reduce to the case where the product abc is square-free. Lagrange (1768) solved the problem by giving a descent procedure which determines in a finite number of steps whether or not (1) has a nontrivial Z-solution, but Legendre (1788) gave the definitive solution. Legendre proved that the following conditions, known by Euler to be necessary, are sufficient for the existence of a nontrivial Z- solution: (i) a, b,andc do not all have the same sign, and (ii) ab is a square modulo c , ca is a square modulo b ,and bc is a square modulo− a .
    [Show full text]
  • Algebraic Tori — Thirty Years After
    ALGEBRAIC TORI | THIRTY YEARS AFTER BORIS KUNYAVSKI˘I To my teacher Valentin Evgenyevich Voskresenski˘ı, with gratitude and admiration This article is an expanded version of my talk given at the Interna- tional Conference \Algebra and Number Theory" dedicated to the 80th anniversary of V. E. Voskresenski˘ı, which was held at the Samara State University in May 2007. The goal is to give an overview of results of V. E. Voskresenski˘ı on arithmetic and birational properties of algebraic tori which culminated in his monograph [Vo77] published 30 years ago. I shall try to put these results and ideas into somehow broader context and also to give a brief digest of the relevant activity related to the period after the English version of the monograph [Vo98] appeared. 1. Rationality and nonrationality problems A classical problem, going back to Pythagorean triples, of describing the set of solutions of a given system of polynomial equations by ratio- nal functions in a certain number of parameters (rationality problem) has been an attraction for many generations. Although a lot of various techniques have been used, one can notice that after all, to establish rationality, one usually has to exhibit some explicit parameterization such as that obtained by stereographic projection in the Pythagoras problem. The situation is drastically different if one wants to estab- lish non-existence of such a parameterization (nonrationality problem): here one usually has to use some known (or even invent some new) bi- rational invariant allowing one to detect nonrationality by comparing its value for the object under consideration with some \standard" one known to be zero; if the computation gives a nonzero value, we are done.
    [Show full text]
  • Branching Laws for Classical Groups: the Non-Tempered Case
    BRANCHING LAWS FOR CLASSICAL GROUPS: THE NON-TEMPERED CASE WEE TECK GAN, BENEDICT H. GROSS AND DIPENDRA PRASAD August 18, 2020 ABSTRACT. This paper generalizes the GGP conjectures which were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the nongeneric L-packets arising from Arthur parameters. The paper introduces the key no- tionof a relevant pair of A-parameters which governs the branching laws for GLn and all classical groups overboth local fields and global fields. It plays a role for all the branching problems studied in [GGP] including Bessel models and Fourier-Jacobi models. CONTENTS 1. Introduction 1 2. Notation and Preliminaries 8 3. Relevant Pair of A-Parameters 10 4. Correlator 12 5. Local Conjecture for GLn 15 6. Local Conjecture for Classical Groups 23 7. A Conjecture for A-packets 25 8. A Special Case of the Conjecture 31 9. Global Conjecture 34 10. Revisiting the Global Conjecture 43 11. Low Rank Examples 46 12. Automorphic Descent 55 13. L-functions: GL case 59 14. L-functions: Classical Groups 63 References 67 arXiv:1911.02783v2 [math.RT] 17 Aug 2020 1. INTRODUCTION This paper is a sequel to our earlier work [GP1, GP2, GGP, GGP2], which discussed several restriction (or branching) problems in the representation theory of classical groups. In the local case, the conjectural answer was given in terms of symplectic root numbers 1991 Mathematics Subject Classification. Primary 11F70; Secondary 22E55. WTG is partially supported by an MOE Tier 2 grant R146-000-233-112. DP thanks Science and En- gineering research board of the Department of Science and Technology, India for its support through the JC Bose National Fellowship of the Govt.
    [Show full text]
  • Cohomological Hasse Principle and Motivic Cohomology for Arithmetic Schemes
    COHOMOLOGICAL HASSE PRINCIPLE AND MOTIVIC COHOMOLOGY FOR ARITHMETIC SCHEMES by MORITZ KERZ and SHUJI SAITO ABSTRACT In 1985 Kazuya Kato formulated a fascinating framework of conjectures which generalizes the Hasse principle for the Brauer group of a global field to the so-called cohomological Hasse principle for an arithmetic scheme X. In this paper we prove the prime-to-characteristic part of the cohomological Hasse principle. We also explain its implications on finiteness of motivic cohomology and special values of zeta functions. CONTENTS Introduction........................................................ 123 1.Homologytheory................................................... 129 2.Log-pairsandconfigurationcomplexes....................................... 135 3.Lefschetzcondition.................................................. 140 4.Pullbackmap(firstconstruction)........................................... 146 5.Pullbackmap(secondconstruction)......................................... 159 6.Maintheorem..................................................... 162 7.Resultwithfinitecoefficients............................................. 166 8.Kato’sconjectures................................................... 169 9.Applicationtocyclemaps............................................... 174 10.Applicationtospecialvaluesofzetafunctions.................................... 176 Acknowledgements..................................................... 178 Appendix A: Galois cohomology with compact support . .............................. 179 References........................................................
    [Show full text]
  • Number Theory
    Number Theory Alexander Paulin October 25, 2010 Lecture 1 What is Number Theory Number Theory is one of the oldest and deepest Mathematical disciplines. In the broadest possible sense Number Theory is the study of the arithmetic properties of Z, the integers. Z is the canonical ring. It structure as a group under addition is very simple: it is the infinite cyclic group. The mystery of Z is its structure as a monoid under multiplication and the way these two structure coalesce. As a monoid we can reduce the study of Z to that of understanding prime numbers via the following 2000 year old theorem. Theorem. Every positive integer can be written as a product of prime numbers. Moreover this product is unique up to ordering. This is 2000 year old theorem is the Fundamental Theorem of Arithmetic. In modern language this is the statement that Z is a unique factorization domain (UFD). Another deep fact, due to Euclid, is that there are infinitely many primes. As a monoid therefore Z is fairly easy to understand - the free commutative monoid with countably infinitely many generators cross the cyclic group of order 2. The point is that in isolation addition and multiplication are easy, but together when have vast hidden depth. At this point we are faced with two potential avenues of study: analytic versus algebraic. By analytic I questions like trying to understand the distribution of the primes throughout Z. By algebraic I mean understanding the structure of Z as a monoid and as an abelian group and how they interact.
    [Show full text]
  • Counterexamples to the Local-Global Principle for Non-Singular Plane Curves and a Cubic Analogue of Ankeny-Artin-Chowla-Mordell Conjecture
    COUNTEREXAMPLES TO THE LOCAL-GLOBAL PRINCIPLE FOR NON-SINGULAR PLANE CURVES AND A CUBIC ANALOGUE OF ANKENY-ARTIN-CHOWLA-MORDELL CONJECTURE YOSHINOSUKE HIRAKAWA AND YOSUKE SHIMIZU Abstract. In this article, we introduce a systematic and uniform construction of non- singular plane curves of odd degrees n ≥ 5 which violate the local-global principle. Our construction works unconditionally for n divisible by p2 for some odd prime number p. Moreover, our construction also works for n divisible by some p ≥ 5 which satisfies 1=3 1=3 a conjecture on p-adic properties of the fundamental units of Q(p ) and Q((2p) ). This conjecture is a natural cubic analogue of the classical Ankeny-Artin-Chowla-Mordell 1=2 conjecture for Q(p ) and easily verified numerically. 1. Introduction In the theory of Diophantine equations, the local-global principle for quadratic forms established by Minkowski and Hasse is one of the major culminations (cf. [31, Theorem 8, Ch. IV]). In contrast, there exist many homogeneous forms of higher degrees which violate the local-global principle (i.e., counterexamples to the local-global principle). For example, Selmer [30] found that a non-singular plane cubic curve defined by (1) 3X3 + 4Y 3 = 5Z3 has rational points over R and Qp for every prime number p but not over Q. From eq. (1), we can easily construct reducible (especially singular) counterexamples of higher degrees. After that, Fujiwara [13] found that a non-singular plane quintic curve defined by (2) (X3 + 5Z3)(X2 + XY + Y 2) = 17Z5 violates the local-global principle. More recently, Cohen [9, Corollary 6.4.11] gave several p p p counterexamples of the form x + by + cz = 0 of degree p = 3; 5; 7; 11 with b; c 2 Z, and Nguyen [23, 24] gave recipes for counterexamples of even degrees and more complicated forms.
    [Show full text]