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.