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The ABC-Conjecture and the Shimura Correspondence

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The ABC-Conjecture and the Shimura Correspondence NOTE TO USERS The original manuscript received by UMI contains pages with slanted print. Pages were microfilmed as received. This reproduction is the best copy available UMI The Wieferich Criterion, the ABC-conjecture and the Shimura correspondence by Satyagraha Mohit A thesis submitted to the Department of Mathematics and Statistics in conformity with the requirements for the degree of Master of Science Queen's University Kingston, Ontario, Canada Jdy, 1998 copyright @ Satya Mohit, 1998 National Library Bibliithèque nationale du Canada Acquisitions and Acquisitions et Bibliographie Services seMces bibliographiques 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, loan, distniute or seil reproduire, prêter, distri'buer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/lnlm, 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 substantid extracts fkom it Ni Ia thése 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 This thesis examines the Wieferich criterion, a criterion for the unsolvability of Fermat's Last Theorem for a given prime exponent. We explore the question of the existence of an infinitude of 'Wieferich primes' fkom a variety of perspectives, discussing its connection with the ABC-conjecture, fundamental units in rdquadratic fields and, most important ly, wit h twists of efiptic cwes. Acknowledgement s I would like to thank my supervisor, M. Ram Murty, for his guidance, support and generosity, both personally and professionally. His love of knowledge and his attitude toward research illustrate and exemplify all that is worthy and noble in the human endeavour that is mathematics. 1 will think of him as a guru and mentor always. 1 wodd like to thank Professor Ernst Kani, of Queen's University, for his generosity with his time and knowledge. In the time that 1 have spent at Queen's, he has given me many invaluable explanations and has always ben available to chat. I would also like to thank David Cardon, of Queen's University. The projects that we have worked on together have been both chdenging and fun; 1 have learned much hmthem. 1 would Like to thank Professors Hershy Kisilevsky and Chantal David, of Concordia University, for devoting many hours to teaching me the beautifid arithmetic t heory of elliptic cuves. Much of my appreciation of this subject has sprung from their insight and vision into it. Findy, 1 would like to thank my best buddy, Joanna Meadows, for saving my Life. Our once in a lifetime kiendship has taught me more than 1 could leam in any classroom. Contents Abstract Acknow ledgements Introduction Chapter 1. Fermat's Last Theorem 1. The Wieferich Criterion 2. Regular primes and the first case of FLT 3. Regular primes and the second case of FLT 4. Irreguiar primes and Bernoulli numbers Chapter 2. The Wieferich Criterion 1. The ABC-Conjecture and the Wieferich Criterion 2. Some numbers and conjectures 3. Some elementaq results 4. Wieferich primes and fundamental units Chapter 3. Wieferich Primes and Elliptic Cumes A Consequence of Hd's Conjecture Quadratic Twists of Elliptic Curves Modular Forms Modular Elliptic Curves Modular Forms of Half-Integer Weight Rank zero quadratic twists of modular eliiptic cuves Easy Consequences 1 CONTENTS Chapter 4. The ABC-Conjecture 1. Introduction 2. Diop hant ine Consequences 3. The Wieferich Conjecture in F, [t] 4. ABC and Elliptic Curves Bibiiography Introduction In the long and sinuous history of Fermat's last theorem, the foUowing criterion stands out. rf a prime p satisfies 2P-' f 1 (mod p2), then the first case of Fermat's Zast theorem is tnie for the exponent p. This criterion, proven by Wieferich in 1909, le& naturally to the question: Do them exist infiniteiy many primes p such that 2p-L $ 1 (mod or, more generally, For a given integer a, do there exist infinitely many primes p such that ap-' f 1 (mod p2) ? The answer to this question remai. unknown. The study of this and other, related problems is the central goal of this thesis. In Chapter 1, we prove Wieferich's criterion and Kiimmer's regularity criterion for the unsolvability of the Fermat equation for prime exponent. In Chapter 2, we discuss and summarize ail known results on the 'Wieferich Conjecture'. We also establish a connection between 'Wieferich primes' and the fundamentai units of real quadratic fields. In Chapter 3, we establish a comection between Wieferich primes and rank zero qua- dratic twists of modular elliptic curves. We state some results and give some examples of how the Shimura correspondance may be used to study such twists. In Chapter 4, we discuss the ABC-conjecture in Ml generality. We derive many impor- tant diophantine consequences of this conjecture and prove its equivalence with the degree 3 and height conjectures for elliptic cwes. We &O formulate and prove the analogue of the Wieferich Conjecture in the ring F, [t]. CHAPTER 1 Fermat's Last Theorem 1. The Wieferich Criterion In 1909, Wieferichpi] proved the following t heorem: THEOREM1.1. If the fist case of Fermat's Lûst Thwrem fails for the ezponent 1 (Z an odd prime), ie., if them ezist x,y, z E Z,1 ( zyr such that x' + y' + zf = O, then 2'-' 1 (mod 12). In this section, we will prove this remarkable theorem of Wieferich- In fxt, we will prove the following more general result, proven by Furtwihgler in 2912, from which Theorem 1 follows as an irnmediate corollq. THEQREM1.2. Let x,y, z be puiNiSe coprime integers such that x' + y' + zL= O with 1 t y+. Then for any pràme ply, pf-' 1 (mod 12). PROOF.(of Theorem 1.1) Since xL+yf + 2 = 0, one of x,y, z is even and, by syrnmetry, we may suppose that 21 y. By Theorem 1.2 then, 2'-' = 1 (mod 12). O We prove Theorem 1.2 using the Eisenstein Reciprocity Law. In order to state this relation, we first develop some background from dgebraic number theory. Let m be a positive integer and let Cm be a primitive mth root of unity. It is known that Z[C,j is the ring of integers in the field O(<,). Q(C,)/Q is easily seen to be a Galois extension with group G S (ZIrnZ)X. For p a rational prime, p 4. m, we can explicitly describe how p splits in the extension Q(k). For our purposes, it will sufEce to know that PZ [Cm] = nPi , L. THE WIEFERICH CRlTERION 6 where the Pi's are distinct prime ideah of Z[C,] and r f = q5(n), where f is the inertial degree of any Pi, that is, f = WPi)= #qCmIIPi. It is a fact that f is equd to the multiplicative order of p (mod m). PROPOSITION1.3. Let P be a prime idd of Z[Cm] not containing m. Then the wsets of 1, Cm, <;fr,. y <mm-' are al[ distinct in Z[Cm]/P. PROOF.Substituting x = 1 in the identity rn P =. (k f 1 (mod P) for ail 15 i 5 m- 1. Therefore f (mod P) for al1 i # j, PROPOSIT~ON1.4. Let a E Z[C,], a @ P. There is an anteger i, unique (mod m) such that q-l a m r Ck (mod P). PROOF.#Z[Cm]/P = q - 1, where q = 1 (mod m). Thus cP1 1 (mod P) so that Since P is a prime ideal, this implies that am = <& (mod P) for some i and by Proposition 3, this i is unique (mod m). O Definition. For a E Z[C,], P a prime ideal not containing ml we define the mth power residue symbot (P)mas follows: Define ($)m = O if a E P. If cr 6 P,define ($)m to be the unique mth root of unity such that 1. THE WTEFERICH CRITERION g-I We observe thah in particdar, (9)= Cmm and that, for any a E Z[C,j, !l$ a (p), (mod P). PROPOSITION1.5. 1) (F)~= 1 y and only if there is an z E Z[<,,,] such that zm cr (mod P). 2) (%lm = (%)m(F>, 3) If a = 0 (mod P), then ($)* = (g)m. PROOF. 1) Exm u (mod P), then Conversely, suppose that (F)m = 1 so that 0% = 1 (rnod P). The group (Z[Cm]/P)Xis cycbc, Say (Z[C,]/P) =< g > . Then a = gr (mod P) for some r so that and since g has order q - 1, we must have q - 119=+ & E Z or mir. Therefore, a is an mth power (mod P). 2) a aP 5 p (mod P) (+ = (4 ,a m 0 rn =(p)mP (-)m If one of a or 0 is in P, then and otherwise, ($)m, (g), are mth roots of unity, so the above congruence implies that (%, = (%,m($,,- 3) If u r p (mod P), then a a! G LI ,pm E (p)m (mod P) 1. THE WFERICH CRITERION We now extend the definition of the rnfi power residue symbol as follows: Definition. Let A C Z[<,] be an ideal coprime to m and let A = nzl P, be its factorization into prime ideals. For a E Z[Crn],we define If p E Z[Cm]is coprime to rn, we define (B)rn = (&)m PROPOSITION1.6.
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