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An Introduction to the Birch and Swinnerton-Dyer Conjecture
Rose-Hulman Undergraduate Mathematics Journal Volume 16 Issue 1 Article 15 An Introduction to the Birch and Swinnerton-Dyer Conjecture Brent Johnson Villanova University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Johnson, Brent (2015) "An Introduction to the Birch and Swinnerton-Dyer Conjecture," Rose-Hulman Undergraduate Mathematics Journal: Vol. 16 : Iss. 1 , Article 15. Available at: https://scholar.rose-hulman.edu/rhumj/vol16/iss1/15 Rose- Hulman Undergraduate Mathematics Journal An Introduction to the Birch and Swinnerton-Dyer Conjecture Brent A. Johnson a Volume 16, No. 1, Spring 2015 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal Villanova University Rose-Hulman Undergraduate Mathematics Journal Volume 16, No. 1, Spring 2015 An Introduction to the Birch and Swinnerton-Dyer Conjecture Brent A. Johnson Abstract. This article explores the Birch and Swinnerton-Dyer Conjecture, one of the famous Millennium Prize Problems. In addition to providing the basic theoretic understanding necessary to understand the simplest form of the conjecture, some of the original numerical evidence used to formulate the conjecture is recreated. Recent results and current problems related to the conjecture are given at the end. Acknowledgements: I would like to thank Professor Robert Styer and Professor Alice Deanin for their incredible mentorship, patience, and friendship. RHIT Undergrad. Math. J., Vol. 16, No. 1 Page 271 1 Introduction An elliptic curve is a projective, nonsingular curve given by the general Weierstrass equation 2 3 2 E : y + a1xy + a3y = x + a2x + a4x + a6: There is no doubt that elliptic curves are amongst the most closely and widely studied objects in mathematics today. -
EMS Newsletter September 2012 1 EMS Agenda EMS Executive Committee EMS Agenda
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. Matemática, Faculdade Facultad de Matematicas de Ciências, Edifício C6, Universidad Complutense Piso 2 Campo Grande Mathematical de Madrid 1749-006 Lisboa, Portugal e-mail: [email protected] Plaza de Ciencias 3, 28040 Madrid, Spain Eva-Maria Feichtner e-mail: [email protected] (2012–2015) Society Department of Mathematics Lucia Di Vizio (2012–2016) Université de Versailles- University of Bremen St Quentin 28359 Bremen, Germany e-mail: [email protected] Laboratoire de Mathématiques Newsletter No. 85, September 2012 45 avenue des États-Unis Eva Miranda (2010–2013) 78035 Versailles cedex, France Departament de Matemàtica e-mail: [email protected] Aplicada I EMS Agenda .......................................................................................................................................................... 2 EPSEB, Edifici P Editorial – S. Jackowski ........................................................................................................................... 3 Associate Editors Universitat Politècnica de Catalunya Opening Ceremony of the 6ECM – M. -
Iasinstitute for Advanced Study
G13-15849_FacultyMembersCOV-12_Layout 1 10/28/13 12:26 PM Page 1 IASInstitute for Advanced Study INSTITUTE FOR ADVANCED STUDY EINSTEIN DRIVE PRINCETON, NEW JERSEY 08540 (609) 734-8000 www.ias.edu Faculty and Members 2013–2014 G13-15849_FacultyMembersCOV-12_Layout 1 10/28/13 12:26 PM Page 2 It is fundamental in our purpose, and our express desire, that in the appointments to the staff and faculty as well as in the admission of workers and students, no account shall be taken, directly or indirectly, of race, religion, or sex. We feel strongly that the spirit characteristic of America at its noblest, above all the pursuit of higher learning, cannot admit of any conditions as to personnel other than those designed to promote the objects for which this institution is established, and particularly with no regard whatever to accidents of race, creed, or sex. —Louis Bamberger and Caroline Bamberger Fuld, in a letter dated June 4, 1930, to the Institute’s first Board of Trustees Cover: Kazuya Yonekura (kneeling), Member in the School of Natural Sciences, with Yuji Tachikawa (Member, 2006–11) Photo: Andrea Kane Contents Mission and History . 2 School of Historical Studies . 4 School of Mathematics . 21 School of Natural Sciences . 41 School of Social Science . 58 Program in Interdisciplinary Studies . 68 Director’s Visitors . 70 Artist-in-Residence Program . 71 Trustees and Officers of the Board and of the Corporation . 72 Administration . 74 Past Directors and Faculty . 76 Index . 77 Information contained herein is current as of September 23, 2013. Mission and History The Institute for Advanced Study is one of the world’s leading centers for theoretical research and intellectual inquiry. -
Elliptic Curves Over Q∞ Are Modular
Elliptic curves over Q1 are modular Jack A. Thorne∗ July 11, 2016 Abstract We show that if p is a prime, then all elliptic curves defined over the cyclotomic Zp-extension of Q are modular. 1 Introduction Our goal in this paper is to prove the following theorem: Theorem 1. Let p be a prime, and let F be a number field which is contained in the cyclotomic Zp-extension of Q. Let E be an elliptic curve over F . Then E is modular. By definition, the cyclotomic Zp-extension of Q is the unique subfield Q1 of Q(ζp1 ) with Galois ∼ group Gal(Q1=Q) = Zp. It is totally real. An elliptic curve E over a number field F is said to be modular if there is a regular algebraic automorphic representation π of GL2(AF ) which has the same L-function as E. This is one of several equivalent formulations; if F is totally real, then π will be generated by vectors which can be interpreted as Hilbert modular forms of parallel weight 2. The modularity of a given elliptic curve E has many useful consequences. It implies that the L-function of E has an analytic continuation to the whole complex plane, allowing one to formulate the Birch{Swinnerton-Dyer conjecture for E unconditionally. When the order of vanishing of the L-function at the point s = 1 is at most 1, this conjecture is known, in its weak form, in many cases [Zha01]. The modularity of all elliptic curves over Q has been known since work of Wiles and Breuil, Conrad, Diamond, and Taylor [Wil95], [TW95], [CDT99], [BCDT01]. -
Arxiv:1806.01691V2 [Math.NT] 22 Aug 2018 Ubro Primes
INDIVISIBILITY OF HEEGNER POINTS AND ARITHMETIC APPLICATIONS ASHAY A. BURUNGALE, FRANCESC CASTELLA, AND CHAN-HO KIM Abstract. We upgrade Howard’s divisibility towards Perrin-Riou’s Heegner point main con- jecture to the predicted equality. Contrary to previous works in this direction, our main result allows for the classical Heegner hypothesis and non-squarefree conductors. The main ingredi- ents we exploit are W. Zhang’s proof of Kolyvagin’s conjecture, Kolyvagin’s structure theorem for Shafarevich–Tate groups, and the explicit reciprocity law for Heegner points. Contents 1. Introduction 1 2. Selmer structures 4 3. Heegner point Kolyvagin systems 5 4. Equivalent main conjectures 7 5. Equivalent special value formulas 7 6. Skinner–Urban lifting lemma 9 7. Proof of the main results 9 References 10 1. Introduction Let E/Q be an elliptic curve of conductor N and let K be an imaginary quadratic field of discriminant DK with (DK ,N) = 1. Then K determines a factorization N = N +N − with N + (resp. N −) divisible only by primes which are split (resp. inert) in K. Throughout this paper, the following hypothesis will be in force: arXiv:1806.01691v2 [math.NT] 22 Aug 2018 Assumption 1.1 (Generalized Heegner hypothesis). N − is the square-free product of an even number of primes. Let p > 3 be a good ordinary prime for E with (p, DK ) = 1, and let K∞ be the anti- cyclotomic Zp-extension of K. Under Assumption 1.1, the theory of complex multiplication + provides a collection of CM points on a Shimura curve with “Γ0(N )-level structure” attached to the quaternion algebra B/Q of discriminant N − defined over ring class extensions of K. -
On the P-Part of the Birch-Swinnerton
ON THE p-PART OF THE BIRCH{SWINNERTON-DYER FORMULA FOR MULTIPLICATIVE PRIMES FRANCESC CASTELLA Abstract. Let E=Q be a semistable elliptic curve of analytic rank one, and let p > 3 be a prime for which E[p] is irreducible. In this note, following a slight modification of the methods of [JSW15], we use Iwasawa theory to establish the p-part of the Birch and Swinnerton-Dyer formula for E. In particular, we extend the main result of loc.cit. to primes of multiplicative reduction. Contents 1. Introduction 1 2. Selmer groups 3 3. A p-adic Waldspurger formula 8 4. Main Conjectures 10 5. Proof of Theorem A 12 References 14 1. Introduction Let E=Q be a semistable elliptic curve of conductor N, and let L(E; s) be the Hasse{Weil L-function of E. By the celebrated work of Wiles [Wil95] and Taylor{Wiles [TW95], L(E; s) is known to admit analytic continuation to the entire complex plane, and to satisfy a functional equation relating its values at s and 2 − s. The purpose of this note is to prove the following result towards the Birch and Swinnerton-Dyer formula for E. Theorem A. Let E=Q be a semistable elliptic curve of conductor N with ords=1L(E; s) = 1, and let p > 3 be a prime such that the mod p Galois representation ρ¯E;p : Gal(Q=Q) −! AutFp (E[p]) is irreducible. If p j N, assume in addition that E[p] is ramified at some prime q 6= p. -
2018-06-108.Pdf
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Feature S E European Tensor Product and Semi-Stability M M Mathematical Interviews E S Society Peter Sarnak Gigliola Staffilani June 2018 Obituary Robert A. Minlos Issue 108 ISSN 1027-488X Prague, venue of the EMS Council Meeting, 23–24 June 2018 New books published by the Individual members of the EMS, member S societies or societies with a reciprocity agree- E European ment (such as the American, Australian and M M Mathematical Canadian Mathematical Societies) are entitled to a discount of 20% on any book purchases, if E S Society ordered directly at the EMS Publishing House. Bogdan Nica (McGill University, Montreal, Canada) A Brief Introduction to Spectral Graph Theory (EMS Textbooks in Mathematics) ISBN 978-3-03719-188-0. 2018. 168 pages. Hardcover. 16.5 x 23.5 cm. 38.00 Euro Spectral graph theory starts by associating matrices to graphs – notably, the adjacency matrix and the Laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph theoretic or combinatorial. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. The first half is devoted to graphs, finite fields, and how they come together. This part provides an appealing motivation and context of the second, spectral, half. The text is enriched by many exercises and their solutions. -
Arxiv:2008.01607V2 [Math.NT] 23 Feb 2021
ELLIPTIC CURVES AND THOMPSON’S SPORADIC SIMPLE GROUP MARYAM KHAQAN ABSTRACT. We characterize all infinite-dimensional graded virtual modules for 3 Thompson’s sporadic simple group whose graded traces are weight 2 weakly holo- morphic modular forms satisfying certain special properties. We then use these mod- ules to detect the non-triviality of Mordell–Weil, Selmer, and Tate-Shafarevich groups of quadratic twists of certain elliptic curves. 1. INTRODUCTION In 1978, McKay and Thompson observed [Thompson 1979b] that the first few 1 coefficients of the normalized elliptic modular invariant J(τ) = q− + 196884q + 21493760q2+864299970q3+O(q4), a central object in the theory of modular forms, can be written as sums involving the first few dimensions of irreducible representations of the monster group M, e.g., 196884 = 2 11+ 2 1196883 · · 21493760 = 2 11+ 2 1196883 + 21296876 (1.1) · · 864299970 = 12 1+ 12 196883 + 21296876 + 842609326. · · This coincidence inspired Thompson’s conjecture [Thompson 1979a] that there is an infinite-dimensional M-module V = Vn whose graded dimension is J(τ) and n 1 whose McKay–Thompson series M≥− T (τ) := tr(g V )qn (1.2) g | n n 1 X≥− are distinguished functions on the upper half-plane. Conway and Norton [Conway and Norton 1979] explicitly described the relevant McKay-Thompson se- ries, and also christened this phenomenon “monstrous moonshine.” Their conjecture was proven by Borcherds [Borcherds 1992] (building on work by Frenkel, Lepowsky and Meurman [Frenkel et al. 1988]) in 1992. In the few decades since the first ob- arXiv:2008.01607v2 [math.NT] 23 Feb 2021 servations of McKay and Thompson, it has become clear that monstrous moonshine is just the first of a series of similar phenomena encompassing several finite groups and their counterparts in the world of modular forms. -
Ternary Cubic Forms Having Bounded Invariants, and the Existence of a Positive Proportion of Elliptic Curves Having Rank 0
Annals of Mathematics 181 (2015), 587{621 http://dx.doi.org/10.4007/annals.2015.181.2.4 Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0 By Manjul Bhargava and Arul Shankar Abstract We prove an asymptotic formula for the number of SL3(Z)-equivalence classes of integral ternary cubic forms having bounded invariants. We use this result to show that the average size of the 3-Selmer group of all elliptic curves, when ordered by height, is equal to 4. This implies that the average rank of all elliptic curves, when ordered by height, is less than 1.17. Combining our counting techniques with a recent result of Dokchitser and Dokchitser, we prove that a positive proportion of all elliptic curves have rank 0. Assuming the finiteness of the Tate{Shafarevich group, we also show that a positive proportion of elliptic curves have rank 1. Finally, combining our counting results with the recent work of Skinner and Urban, we show that a positive proportion of elliptic curves have analytic rank 0; i.e., a positive proportion of elliptic curves have nonvanishing L-function at s = 1. It follows that a positive proportion of all elliptic curves satisfy BSD. 1. Introduction Any elliptic curve E over Q is isomorphic to a unique curve of the form 2 3 6 EA;B : y = x + Ax + B, where A; B 2 Z and for all primes p: p - B 4 whenever p j A. The (naive) height H(EA;B) of the elliptic curve E = EA;B is then defined by 3 2 H(EA;B) := maxf4jA j; 27B g: In a previous paper [8], we showed that the average rank of all elliptic curves, when ordered by height, is finite. -
Lectures on the Iwasawa Theory of Elliptic Curves
LECTURES ON THE IWASAWA THEORY OF ELLIPTIC CURVES CHRISTOPHER SKINNER Abstract. These are a preliminary set ot notes for the author's lectures for the 2018 Arizona Winter School on Iwasawa Theory. Contents 1. Introduction 2 2. Selmer groups 3 2.1. Selmer groups of elliptic curves 3 2.2. Bloch{Kato Selmer groups 9 2.3. Selmer structures 12 3. Iwasawa modules for elliptic curves 13 3.1. The extension F1=F 13 3.2. Selmer groups over F1 15 3.3. S?(E=F1) as a Λ-module 17 3.4. Control theorems 19 4. Main Conjectures 22 4.1. p-adic L-functions 23 4.2. The Main Conjectures 27 4.3. Main Conjectures without L-functions 28 5. Theorems and ideas of their proofs 30 5.1. Cyclotomic Main Conjectures: the ordinary case 30 ac 5.2. The Main Conjectures for SGr(E=K1) and SBDP(E=K1) 32 5.3. Cyclotomic Main Conjectures: the supersingular case 34 5.4. Perrin-Riou's Heegner point Main Conjecture 35 6. Arithmetic consequences 36 1 2 CHRISTOPHER SKINNER 6.1. Results when L(E; 1) 6= 0 36 6.2. Results when L(E; 1) = 0 36 6.3. Results when ords=1L(E; s) = 1 37 6.4. Converses to Gross{Zagier/Kolyvagin 39 References 40 1. Introduction Iwasawa theory was introduced around 1960 in the context of class groups of cyclotomic and other Zp-extensions of number fields. The Main Conjecture of Iwasawa theory proposed a re- markable connection between the p-adic L-functions of Kubota and Leopoldt and these class groups [19, x1], [12, x5], including among its consequences certain refined class number formulas for values of Dirichlet L-functions. -
Of the Annual
Institute for Advanced Study IASInstitute for Advanced Study Report for 2013–2014 INSTITUTE FOR ADVANCED STUDY EINSTEIN DRIVE PRINCETON, NEW JERSEY 08540 (609) 734-8000 www.ias.edu Report for the Academic Year 2013–2014 Table of Contents DAN DAN KING Reports of the Chairman and the Director 4 The Institute for Advanced Study 6 School of Historical Studies 10 School of Mathematics 20 School of Natural Sciences 30 School of Social Science 42 Special Programs and Outreach 50 Record of Events 58 79 Acknowledgments 87 Founders, Trustees, and Officers of the Board and of the Corporation 88 Administration 89 Present and Past Directors and Faculty 91 Independent Auditors’ Report CLIFF COMPTON REPORT OF THE CHAIRMAN I feel incredibly fortunate to directly experience the Institute’s original Faculty members, retired from the Board and were excitement and wonder and to encourage broad-based support elected Trustees Emeriti. We have been profoundly enriched by for this most vital of institutions. Since 1930, the Institute for their dedication and astute guidance. Advanced Study has been committed to providing scholars with The Institute’s mission depends crucially on our financial the freedom and independence to pursue curiosity-driven research independence, particularly our endowment, which provides in the sciences and humanities, the original, often speculative 70 percent of the Institute’s income; we provide stipends to thinking that leads to the highest levels of understanding. our Members and do not receive tuition or fees. We are The Board of Trustees is privileged to support this vital immensely grateful for generous financial contributions from work. -
Arxiv:1411.6352V2 [Math.NT] 23 Jun 2015 Aiiso H Leri at Ftecrepnigspecia Corresponding the of Parts Algebraic the of Families Ealtefruaino 3] Let [31]
IWASAWA Main Conjecture for Supersingular Elliptic curves Xin Wan Abstract In this paper we prove the -main conjecture formulated by Kobayashi for elliptic curves ± with supersingular reduction at p such that ap = 0, using a completely new idea of reducing it to another Iwasawa-Greenberg main conjecture which is more accessible. We also prove as a corollary the p-part of the BSD formula at supersingular primes when the analytic rank is 0. The argument uses in an essential way the recent study on explicit reciprocity law for Beilinson-Flach elements by Kings-Loeffler-Zerbes. 1 Introduction Let p be an odd prime. Iwasawa theory studies relations between special values of L-functions and arithmetic objects such as class numbers of number fields or more generally p-adic Selmer groups. The central problem for this study is the Iwasawa main conjecture, which roughly speaking, says that the size (or more precisely the characteristic ideal) of certain module parameterizing the p-adic families of Selmer groups is generated by the so called p-adic L-function, which interpolates p-adic families of the algebraic parts of the corresponding special L-values. Iwasawa main conjecture is also a useful tool in proving the refined Birch-Swinnerton-Dyer (BSD) formula for elliptic curves. Earlier work on Iwasawa main conjecture includes the work of Mazur-Wiles [19], Wiles [38] for p-adic families of Hecke characters of totally real fields using the Eisenstein congruence on GL2, Rubin [27] for characters for quadratic imaginary fields using Euler systems of elliptic units, the work of Hida-Tilouine for anticyclotomic characters of general CM fields [9], the work of E.Urban [32] on symmetric square functions, the work of Bertolini-Darmon [2] for anticyclotomic main conjecture L for modular forms, and the recent work of Kato [10] and Skinner-Urban [31] which proves the arXiv:1411.6352v2 [math.NT] 23 Jun 2015 Iwasawa main conjecture for ordinary elliptic curves E/Q (and this list is not complete).