DOCSLIB.ORG

Algebra & Number Theory Vol. 8 (2014)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algebra & Number Theory Vol. 8 (2014) Algebra & Number Theory Volume 8 2014 No. 10 msp Algebra & Number Theory msp.org/ant EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Shigefumi Mori RIMS, Kyoto University, Japan Dave Benson University of Aberdeen, Scotland Raman Parimala Emory University, USA Richard E. Borcherds University of California, Berkeley, USA Jonathan Pila University of Oxford, UK John H. Coates University of Cambridge, UK Anand Pillay University of Notre Dame, USA J-L. Colliot-Thélène CNRS, Université Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Peter Sarnak Princeton University, USA Hélène Esnault Freie Universität Berlin, Germany Joseph H. Silverman Brown University, USA Hubert Flenner Ruhr-Universität, Germany Michael Singer North Carolina State University, USA Edward Frenkel University of California, Berkeley, USA Vasudevan Srinivas Tata Inst. of Fund. Research, India Andrew Granville Université de Montréal, Canada J. Toby Stafford University of Michigan, USA Joseph Gubeladze San Francisco State University, USA Bernd Sturmfels University of California, Berkeley, USA Roger Heath-Brown Oxford University, UK Richard Taylor Harvard University, USA Craig Huneke University of Virginia, USA Ravi Vakil Stanford University, USA János Kollár Princeton University, USA Michel van den Bergh Hasselt University, Belgium Yuri Manin Northwestern University, USA Marie-France Vignéras Université Paris VII, France Barry Mazur Harvard University, USA Kei-Ichi Watanabe Nihon University, Japan Philippe Michel École Polytechnique Fédérale de Lausanne Efim Zelmanov University of California, San Diego, USA Susan Montgomery University of Southern California, USA Shou-Wu Zhang Princeton University, USA PRODUCTION [email protected] Silvio Levy, Scientific Editor See inside back cover or msp.org/ant for submission instructions. The subscription price for 2014 is US $225/year for the electronic version, and $400/year (C$55, if shipping outside the US) for print and electronic. Subscriptions, requests for back issues and changes of subscribers address should be sent to MSP. Algebra & Number Theory (ISSN 1944-7833 electronic, 1937-0652 printed) at Mathematical Sciences Publishers, 798 Evans Hall #3840, c/o University of California, Berkeley, CA 94720-3840 is published continuously online. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. ANT peer review and production are managed by EditFLOW® from MSP. PUBLISHED BY mathematical sciences publishers nonprofit scientific publishing http://msp.org/ © 2014 Mathematical Sciences Publishers ALGEBRA AND NUMBER THEORY 8:10 (2014) msp dx.doi.org/10.2140/ant.2014.8.2297 K3 surfaces and equations for Hilbert modular surfaces Noam Elkies and Abhinav Kumar We outline a method to compute rational models for the Hilbert modular surfaces Y−.D/, which are coarse moduli spaces for principallyp polarized abelian surfaces with real multiplication by the ring of integers in Q. D/, via moduli spaces of elliptic K3 surfaces with a Shioda–Inose structure. In particular, we compute equations for all thirty fundamental discriminants D with 1 < D < 100, and analyze rational points and curves on these Hilbert modular surfaces, producing examples of genus-2 curves over Q whose Jacobians have real multiplication over Q. 1. Introduction Hilbert modular surfaces have been objects of extensive investigation in complex and algebraic geometry, and more recently in number theory. Since Hilbert modular varieties are moduli spaces for abelian varieties with real multiplication by an order in a totally real field, they have intrinsic arithmetic content. Their geometry is enriched by the presence of modular subvarieties. In[Hirzebruch 1973; Hirzebruch and van de Ven 1974; Hirzebruch and Zagier 1977] the geometric invariants of many of these surfaces were computed, and they were placed within the Enriques–Kodaira classification. A chief aim of the present work is to compute equations for birational models of some of these surfaces over the field of rational numbers. More precisely, let D be a positive fundamental discriminant,p i.e., the discrimi- nant of the ring of integers OD of the real quadratic field Q. D/. The quotient C − C − PSL2.OD/n.H × H / (where H and H are the complex upper and lower half-planes) parametrizes abelian surfaces with an action of OD. It has a natural Elkies was supported in part by NSF grants DMS-0501029 and DMS-1100511. Kumar was supported in part by NSF grants DMS-0757765 and DMS-0952486, and by a grant from the Solomon Buchsbaum Research Fund. The research was started when Kumar was a postdoctoral researcher at Microsoft Research. He also thanks Princeton University for its hospitality during Fall 2009. MSC2010: primary 11F41; secondary 14G35, 14J28, 14J27. Keywords: elliptic K3 surfaces, moduli spaces, Hilbert modular surfaces, abelian surfaces, real multiplication, genus-2 curves. 2297 2298 Noam Elkies and Abhinav Kumar compactification Y−.D/, obtained by adding finitely many points and desingulariz- ing these cusps. These surfaces Y−.D/ have models defined over Q, and the main goal of this paper is to describe a method to compute explicit equations for these models, as well as to carry out this method for all fundamental discriminants D with 1 < D < 100. This felt like a good place to stop for now, though these calculations may be extended to some higher D, as well as to non-fundamental discriminants. We briefly summarize the method, which we describe in more detail in later sections. The method relies on being able to explicitly parametrize K3 surfaces that are related by Shioda–Inose structure to abelian surfaces with real multiplication by some OD. The K3 surface corresponding to such an abelian surface has Néron– Severi lattice containing L D, a specific indefinite lattice of signature .1; 17/ and discriminant −D. In all our examples, we obtain the moduli space MD of L D- polarized K3 surfaces as a family of elliptic surfaces with a specific configuration of reducible fibers and sections. We then use the 2- and 3-neighbor method to transform to another elliptic fibration, with two reducible fibers of types II∗ and III∗ respectively. This lets us read off the map (generically one-to-one) of moduli spaces from MD into the 3-dimensional moduli space A2 of principally polarized abelian surfaces, using the formulae from[Kumar 2008]. The image of MD is the Humbert surface corresponding to discriminant D. The Hilbert modular surface Y−.D/ itself is a double cover of the Humbert surface, branched along a union of modular curves. We use simple lattice arguments to obtain the branch locus, and pin down the exact twist for the double cover by counting points on reductions of the related abelian surfaces modulo several primes. In all our examples, the Humbert surface happens to be a rational surface (i.e., birational to P2 over Q), and we display the equation of 2 Y−.D/ as a double cover of P branched over a curve of small degree. We analyze these equations in some detail, attempting to produce rational or elliptic curves on them, with the intent of producing several (possibly infinitely many) examples of genus-2 curves whose Jacobians have real multiplication. When Y−.D/ is a K3 surface, it often has very high Picard number (19 or 20), and we attempt to compute generators for the Picard group. When Y−.D/ is an honestly elliptic surface, we analyze the singular fibers and the Mordell–Weil group, and attempt to compute a basis for the sections. To our knowledge, this is the first algebraic description of most of these surfaces by explicit equations. We outline some related work in the literature. Wilson[1998; 2000] obtained equations for the Hilbert modular surface Y−.5/ corresponding to the smallest fundamental discriminant D > 1. Van der Geer[1988] gives a few examples of algebraic equations for Hilbert modular surfaces corresponding to a congruence subgroup of the full modular group (in other words, abelian surfaces K3 surfaces and equations for Hilbert modular surfaces 2299 with some level structure). Humbert surfaces have also been well-studied in the literature, and Runge[1999] and Gruenewald[2008] have obtained equations for some of these. However, these equations are quite complicated, and do not shed as much light on the geometry of Hilbert modular surfaces. While the methods are simpler, involving theta functions and q-expansions, the result is analogous to exhibiting the modular polynomial whose zero locus in A1 ×A1 is a singular model of the complement of the cusps in the modular curve X0.N/. The coefficients of these polynomials can quickly become enormous. We believe that our approach, giving simpler equations for these surfaces together with their maps to A2, is more conducive to an investigation of arithmetic properties. It is our hope that these equations will be of much help in subsequent arithmetic investigation of these surfaces. For instance, they should provide a testing ground for many conjectures in Diophantine geometry, because of the abundance of rational curves and points. Another direction of future investigation is to use these equations to investigate modularity of the corresponding abelian surfaces. Modularity of abelian varieties with real multiplication over Q is now proven, by combining results of Ribet[2004] with the recent proof of Serre’s conjecture by Khare and Wintenberger[2009a; 2009b]. However, unlike the case of dimension 1, where one has modular parametrizations and very good control of the moduli spaces, the situation in dimensions 2 and above is much less clear. For instance, it is not at all clear how to find a modular form corresponding to a given abelian surface with real multiplication.1 We hope that the abundance of examples provided by these equations will help pave the path for a better understanding of the 2-dimensional case.
Load more

Recommended publications
  • 18.06 Linear Algebra, Problem Set 2 Solutions
    18.06 Problem Set 2 Solution Total: 100 points Section 2.5. Problem 24: Use Gauss-Jordan elimination on [U I] to find the upper triangular −1 U : 2 3 2 3 2 3 1 a b 1 0 0 −1 4 5 4 5 4 5 UU = I 0 1 c x1 x2 x3 = 0 1 0 : 0 0 1 0 0 1 −1 Solution (4 points): Row reduce [U I] to get [I U ] as follows (here Ri stands for the ith row): 2 3 2 3 1 a b 1 0 0 (R1 = R1 − aR2) 1 0 b − ac 1 −a 0 4 5 4 5 0 1 c 0 1 0 −! (R2 = R2 − cR2) 0 1 0 0 1 −c 0 0 1 0 0 1 0 0 1 0 0 1 ( ) 2 3 R1 = R1 − (b − ac)R3 1 0 0 1 −a ac − b −! 40 1 0 0 1 −c 5 : 0 0 1 0 0 1 Section 2.5. Problem 40: (Recommended) A is a 4 by 4 matrix with 1's on the diagonal and −1 −a; −b; −c on the diagonal above. Find A for this bidiagonal matrix. −1 Solution (12 points): Row reduce [A I] to get [I A ] as follows (here Ri stands for the ith row): 2 3 1 −a 0 0 1 0 0 0 6 7 60 1 −b 0 0 1 0 07 4 5 0 0 1 −c 0 0 1 0 0 0 0 1 0 0 0 1 2 3 (R1 = R1 + aR2) 1 0 −ab 0 1 a 0 0 6 7 (R2 = R2 + bR2) 60 1 0 −bc 0 1 b 07 −! 4 5 (R3 = R3 + cR4) 0 0 1 0 0 0 1 c 0 0 0 1 0 0 0 1 2 3 (R1 = R1 + abR3) 1 0 0 0 1 a ab abc (R = R + bcR ) 60 1 0 0 0 1 b bc 7 −! 2 2 4 6 7 : 40 0 1 0 0 0 1 c 5 0 0 0 1 0 0 0 1 Alternatively, write A = I − N.
    [Show full text]
  • [Math.RA] 19 Jun 2003 Two Linear Transformations Each Tridiagonal with Respect to an Eigenbasis of the Ot
    Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array∗ Paul Terwilliger Abstract Let K denote a field. Let d denote a nonnegative integer and consider a sequence ∗ K p = (θi,θi , i = 0...d; ϕj , φj, j = 1...d) consisting of scalars taken from . We call p ∗ ∗ a parameter array whenever: (PA1) θi 6= θj, θi 6= θj if i 6= j, (0 ≤ i, j ≤ d); (PA2) i−1 θh−θd−h ∗ ∗ ϕi 6= 0, φi 6= 0 (1 ≤ i ≤ d); (PA3) ϕi = φ1 + (θ − θ )(θi−1 − θ ) h=0 θ0−θd i 0 d i−1 θh−θd−h ∗ ∗ (1 ≤ i ≤ d); (PA4) φi = ϕ1 + (Pθ − θ )(θd−i+1 − θ0) (1 ≤ i ≤ d); h=0 θ0−θd i 0 −1 ∗ ∗ ∗ ∗ −1 (PA5) (θi−2 − θi+1)(θi−1 − θi) P, (θi−2 − θi+1)(θi−1 − θi ) are equal and independent of i for 2 ≤ i ≤ d − 1. In [13] we showed the parameter arrays are in bijection with the isomorphism classes of Leonard systems. Using this bijection we obtain the following two characterizations of parameter arrays. Assume p satisfies PA1, PA2. Let ∗ ∗ A, B, A ,B denote the matrices in Matd+1(K) which have entries Aii = θi, Bii = θd−i, ∗ ∗ ∗ ∗ ∗ ∗ Aii = θi , Bii = θi (0 ≤ i ≤ d), Ai,i−1 = 1, Bi,i−1 = 1, Ai−1,i = ϕi, Bi−1,i = φi (1 ≤ i ≤ d), and all other entries 0. We show the following are equivalent: (i) p satisfies −1 PA3–PA5; (ii) there exists an invertible G ∈ Matd+1(K) such that G AG = B and G−1A∗G = B∗; (iii) for 0 ≤ i ≤ d the polynomial i ∗ ∗ ∗ ∗ ∗ ∗ (λ − θ0)(λ − θ1) · · · (λ − θn−1)(θi − θ0)(θi − θ1) · · · (θi − θn−1) ϕ1ϕ2 · · · ϕ nX=0 n is a scalar multiple of the polynomial i ∗ ∗ ∗ ∗ ∗ ∗ (λ − θd)(λ − θd−1) · · · (λ − θd−n+1)(θ − θ )(θ − θ ) · · · (θ − θ ) i 0 i 1 i n−1 .
    [Show full text]
  • Self-Interlacing Polynomials Ii: Matrices with Self-Interlacing Spectrum
    SELF-INTERLACING POLYNOMIALS II: MATRICES WITH SELF-INTERLACING SPECTRUM MIKHAIL TYAGLOV Abstract. An n × n matrix is said to have a self-interlacing spectrum if its eigenvalues λk, k = 1; : : : ; n, are distributed as follows n−1 λ1 > −λ2 > λ3 > ··· > (−1) λn > 0: A method for constructing sign definite matrices with self-interlacing spectra from totally nonnegative ones is presented. We apply this method to bidiagonal and tridiagonal matrices. In particular, we generalize a result by O. Holtz on the spectrum of real symmetric anti-bidiagonal matrices with positive nonzero entries. 1. Introduction In [5] there were introduced the so-called self-interlacing polynomials. A polynomial p(z) is called self- interlacing if all its roots are real, semple and interlacing the roots of the polynomial p(−z). It is easy to see that if λk, k = 1; : : : ; n, are the roots of a self-interlacing polynomial, then the are distributed as follows n−1 (1.1) λ1 > −λ2 > λ3 > ··· > (−1) λn > 0; or n (1.2) − λ1 > λ2 > −λ3 > ··· > (−1) λn > 0: The polynomials whose roots are distributed as in (1.1) (resp. in (1.2)) are called self-interlacing of kind I (resp. of kind II). It is clear that a polynomial p(z) is self-interlacing of kind I if, and only if, the polynomial p(−z) is self-interlacing of kind II. Thus, it is enough to study self-interlacing of kind I, since all the results for self-interlacing of kind II will be obtained automatically. Definition 1.1. An n × n matrix is said to possess a self-interlacing spectrum if its eigenvalues λk, k = 1; : : : ; n, are real, simple, are distributed as in (1.1).
    [Show full text]
  • Accurate Singular Values of Bidiagonal Matrices
    d d Accurate Singular Values of Bidiagonal Matrices (Appeared in the SIAM J. Sci. Stat. Comput., v. 11, n. 5, pp. 873-912, 1990) James Demmel W. Kahan Courant Institute Computer Science Division 251 Mercer Str. University of California New York, NY 10012 Berkeley, CA 94720 Abstract Computing the singular values of a bidiagonal matrix is the ®nal phase of the standard algo- rithm for the singular value decomposition of a general matrix. We present a new algorithm which computes all the singular values of a bidiagonal matrix to high relative accuracy indepen- dent of their magnitudes. In contrast, the standard algorithm for bidiagonal matrices may com- pute sm all singular values with no relative accuracy at all. Numerical experiments show that the new algorithm is comparable in speed to the standard algorithm, and frequently faster. Keywords: singular value decomposition, bidiagonal matrix, QR iteration AMS(MOS) subject classi®cations: 65F20, 65G05, 65F35 1. Introduction The standard algorithm for computing the singular value decomposition (SVD ) of a gen- eral real matrix A has two phases [7]: = T 1) Compute orthogonal matrices P 11and Q such that B PAQ 11is in bidiagonal form, i.e. has nonzero entries only on its diagonal and ®rst superdiagonal. Σ= T 2) Compute orthogonal matrices P 22and Q such that PBQ 22is diagonal and nonnega- σ Σ tive. The diagonal entries i of are the singular values of A. We will take them to be σ≥σ = sorted in decreasing order: ii+112. The columns of Q QQ are the right singular vec- = tors,andthecolumnsofP PP12are the left singular vectors.
    [Show full text]
  • Harvard University Department of Mathematics 2020 - 2021 Fall Directory
    Harvard University Department of Mathematics 2020 - 2021 Fall Directory Tel: (617) 495-2171 Fax: (617) 495-5132 www.math.harvard.edu DEPARTMENT OF MATHEMATICS DIRECTORY - FALL 2020 ASL: Associate Senior Lecturer PD: Postdoctoral Fellow BP: Benjamin Peirce Fellow S: Staff E: Emeritus SL: Senior Lecturer F: Faculty SP: Senior Preceptor G: Graduate Student V: Visitor L: Lecturer VP: Visiting Professor P: Preceptor GROUP EMAIL ADDRESSES Affiliates Lecturers Emeriti Mainoffice Everyone (Everyone at CMSA) Preceptors Everyone (Everyone at Math Dept) Snr_Fac (Senior Faculty) Grad (Grad students) Staff Jnr_Fac (Junior Faculty) Visitors (PostDocs, Research Associates and Fellows) Department email addresses followed by: @math.harvard.edu; CMSA email addresses followed by: @cmsa.fas.harvard.edu NAME EMAIL OFFICE PHONE # ADHIKARI, Arka (G) adhikari 321b ALAEE, Aghil (V - CMSA) aghil.alaee ARMSTRONG, Maureen (S) maureen 332 5-1980 AUROUX, Denis (F) auroux 539 5-5487 BALIBANU, Ana (BP) anab 236 6-4492 BALL, Andrew (S) ball 242h 6-1986 BAMBERG, Paul (SL) bamberg 322 5-9560 BARKLEY, Grant (G) gbarkley 333e BEJLERI, Dori (BP) bejleri 525 5-2334 BEN-ELIEZER, Omri (PD - CMSA) omribene BETTS, Alex (PD) 226h 5-2124 BOGAEVSKY, Tatyana (S - CMSA) bogaevsky 20 Garden 105 6-1778 BONGERS, Tyler (L) bongers 209.3 5-1365 BORETSKY, Jonathan (G) jboretsky 321c DEPARTMENT OF MATHEMATICS DIRECTORY - FALL 2020 BRALEY, Emily (P) braley 225 6-9122 BRENNECKE, Christian (BP) brennecke 239 5-8797 BRENTANA, Pam (S) pbrentan 325 5-5334 CAIN, Wes (SL) jcain2 515 5-1790 CASS,
    [Show full text]
  • Design and Evaluation of Tridiagonal Solvers for Vector and Parallel Computers Universitat Politècnica De Catalunya
    Design and Evaluation of Tridiagonal Solvers for Vector and Parallel Computers Author: Josep Lluis Larriba Pey Advisor: Juan José Navarro Guerrero Barcelona, January 1995 UPC Universitat Politècnica de Catalunya Departament d'Arquitectura de Computadors UNIVERSITAT POLITÈCNICA DE CATALU NYA B L I O T E C A X - L I B R I S Tesi doctoral presentada per Josep Lluis Larriba Pey per tal d'aconseguir el grau de Doctor en Informàtica per la Universitat Politècnica de Catalunya UNIVERSITAT POLITÈCNICA DE CATAl U>'YA ADA'UNÍIEÏRACÍÓ ;;//..;,3UMPf';S AO\n¿.-v i S -i i::-.« c:¿fCM or,re: iïhcc'a a la pàgina .rf.S# a;; 3 b el r;ú¡; Barcelona, 6 N L'ENCARREGAT DEL REGISTRE. Barcelona, de de 1995 To my wife Marta To my parents Elvira and José Luis "A journey of a thousand miles must begin with a single step" Lao-Tse Acknowledgements I would like to thank my parents, José Luis and Elvira, for their unconditional help and love to me. I will always be indebted to you. I also want to thank my wife, Marta, for being the sparkle in my life, for her support and for understanding my (good) moods. I thank Juanjo, my advisor, for his friendship, patience and good advice. Also, I want to thank Àngel Jorba for his unconditional collaboration, work and support in some of the contributions of this work. I thank the members of "Comissió de Doctorat del DAG" for their comments and specially Miguel Valero for his suggestions on the topics of chapter 4. I thank Mateo Valero for his friendship and always good advice.
    [Show full text]
  • Computing the Moore-Penrose Inverse for Bidiagonal Matrices
    УДК 519.65 Yu. Hakopian DOI: https://doi.org/10.18523/2617-70802201911-23 COMPUTING THE MOORE-PENROSE INVERSE FOR BIDIAGONAL MATRICES The Moore-Penrose inverse is the most popular type of matrix generalized inverses which has many applications both in matrix theory and numerical linear algebra. It is well known that the Moore-Penrose inverse can be found via singular value decomposition. In this regard, there is the most effective algo- rithm which consists of two stages. In the first stage, through the use of the Householder reflections, an initial matrix is reduced to the upper bidiagonal form (the Golub-Kahan bidiagonalization algorithm). The second stage is known in scientific literature as the Golub-Reinsch algorithm. This is an itera- tive procedure which with the help of the Givens rotations generates a sequence of bidiagonal matrices converging to a diagonal form. This allows to obtain an iterative approximation to the singular value decomposition of the bidiagonal matrix. The principal intention of the present paper is to develop a method which can be considered as an alternative to the Golub-Reinsch iterative algorithm. Realizing the approach proposed in the study, the following two main results have been achieved. First, we obtain explicit expressions for the entries of the Moore-Penrose inverse of bidigonal matrices. Secondly, based on the closed form formulas, we get a finite recursive numerical algorithm of optimal computational complexity. Thus, we can compute the Moore-Penrose inverse of bidiagonal matrices without using the singular value decomposition. Keywords: Moor-Penrose inverse, bidiagonal matrix, inversion formula, finite recursive algorithm.
    [Show full text]
  • Basic Matrices
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 373 (2003) 143–151 www.elsevier.com/locate/laa Basic matricesୋ Miroslav Fiedler Institute of Computer Science, Czech Academy of Sciences, 182 07 Prague 8, Czech Republic Received 30 April 2002; accepted 13 November 2002 Submitted by B. Shader Abstract We define a basic matrix as a square matrix which has both subdiagonal and superdiagonal rank at most one. We investigate, partly using additional restrictions, the relationship of basic matrices to factorizations. Special basic matrices are also mentioned. © 2003 Published by Elsevier Inc. AMS classification: 15A23; 15A33 Keywords: Structure rank; Tridiagonal matrix; Oscillatory matrix; Factorization; Orthogonal matrix 1. Introduction and preliminaries For m × n matrices, structure (cf. [2]) will mean any nonvoid subset of M × N, M ={1,...,m}, N ={1,...,n}. Given a matrix A (over a field, or a ring) and a structure, the structure rank of A is the maximum rank of any submatrix of A all entries of which are contained in the structure. The most important examples of structure ranks for square n × n matrices are the subdiagonal rank (S ={(i, k); n i>k 1}), the superdiagonal rank (S = {(i, k); 1 i<k n}), the off-diagonal rank (S ={(i, k); i/= k,1 i, k n}), the block-subdiagonal rank, etc. All these ranks enjoy the following property: Theorem 1.1 [2, Theorems 2 and 3]. If a nonsingular matrix has subdiagonal (su- perdiagonal, off-diagonal, block-subdiagonal, etc.) rank k, then the inverse also has subdiagonal (...)rank k.
    [Show full text]
  • Marshall Hall's Conjecture and Gaps Between Integer Points on Mordell
    On Marshall Hall's Conjecture and Gaps Between Integer Points on Mordell Elliptic Curves Ryan D'Mello Benet Academy, Lisle, IL Mailing address: Benet Academy, 2200 Maple Avenue, Lisle, IL 60532, USA e-mail: [email protected] October 2, 2014 Abstract 3 For a non-square integer x 2 N, let kx denote the distance between x and the perfect 3 p square closest to x . A conjecture of Marshall Hall states that the ratios rx = x=kx, are bounded above. (Elkies has shown that any such bound must exceed 46.6.) Let fxng be the sequence of "Hall numbers": positive non-square integers for which rxn exceeds 1. Extensive computer searches have identified approximately 50 Hall numbers. (It can be proved that infinitely many exist.) In this paper we study the minimum gap between consecutive Hall 1 1 6 numbers. We prove that for all n, xn+1 − xn > 5 xn , with stronger gaps applying when 1 1 3 4 xn is close to perfect even or odd squares (≈ xn or ≈ xn , respectively). This result has obvious implications for the minimum "horizontal gap" (and hence straight line and arc distance) between integer points (whose x-coordinates exceed k2) on the Mordell elliptic curves x3 − y2 = k , a question that does not appear to have been addressed. 1 Introduction [Appendix A includes a detailed expository coverage of interesting and relevant history and background on Hall's conjecture, its relationship to other math questions/conjectures, topics, and techniques; and some key applications of these techniques.] 3 For a non-square integer x 2 N, let kx denote the distance between x and the perfect square 3 p closest to x .
    [Show full text]
  • Notices of the AMS 595 Mathematics People NEWS
    NEWS Mathematics People contrast electrical impedance Takeda Awarded 2017–2018 tomography, as well as model Centennial Fellowship reduction techniques for para- bolic and hyperbolic partial The AMS has awarded its Cen- differential equations.” tennial Fellowship for 2017– Borcea received her PhD 2018 to Shuichiro Takeda. from Stanford University and Takeda’s research focuses on has since spent time at the Cal- automorphic forms and rep- ifornia Institute of Technology, resentations of p-adic groups, Rice University, the Mathemati- especially from the point of Liliana Borcea cal Sciences Research Institute, view of the Langlands program. Stanford University, and the He will use the Centennial Fel- École Normale Supérieure, Paris. Currently Peter Field lowship to visit the National Collegiate Professor of Mathematics at Michigan, she is Shuichiro Takeda University of Singapore and deeply involved in service to the applied and computa- work with Wee Teck Gan dur- tional mathematics community, in particular on editorial ing the academic year 2017–2018. boards and as an elected member of the SIAM Council. Takeda obtained a bachelor's degree in mechanical The Sonia Kovalevsky Lectureship honors significant engineering from Tokyo University of Science, master's de- contributions by women to applied or computational grees in philosophy and mathematics from San Francisco mathematics. State University, and a PhD in 2006 from the University —From an AWM announcement of Pennsylvania. After postdoctoral positions at the Uni- versity of California at San Diego, Ben-Gurion University in Israel, and Purdue University, since 2011 he has been Pardon Receives Waterman assistant and now associate professor at the University of Missouri at Columbia.
    [Show full text]
  • Moore–Penrose Inverse of Bidiagonal Matrices. I
    PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 2015, № 2, p. 11–20 Mathematics MOORE–PENROSE INVERSE OF BIDIAGONAL MATRICES.I 1 ∗ 2∗∗ Yu. R. HAKOPIAN , S. S. ALEKSANYAN ; 1Chair of Numerical Analysis and Mathematical Modeling YSU, Armenia 2Chair of Mathematics and Mathematical Modeling RAU, Armenia In the present paper we deduce closed form expressions for the entries of the Moore–Penrose inverse of a special type upper bidiagonal matrices. On the base of the formulae obtained, a finite algorithm with optimal order of computational complexity is constructed. MSC2010: 15A09; 65F20. Keywords: generalized inverse, Moore–Penrose inverse, bidiagonal matrix. Introduction. For a real m × n matrix A, the Moore–Penrose inverse A+ is the unique n × m matrix that satisfies the following four properties: AA+A = A; A+AA+ = A+ ; (A+A)T = A+A; (AA+)T = AA+ (see [1], for example). If A is a square nonsingular matrix, then A+ = A−1. Thus, the Moore–Penrose inversion generalizes ordinary matrix inversion. The idea of matrix generalized inverse was first introduced in 1920 by E. Moore [2], and was later redis- covered by R. Penrose [3]. The Moore–Penrose inverses have numerous applications in least-squares problems, regressive analysis, linear programming and etc. For more information on the generalized inverses see [1] and its extensive bibliography. In this work we consider the problem of the Moore–Penrose inversion of square upper bidiagonal matrices 2 3 d1 b1 6 d2 b2 0 7 6 7 6 .. .. 7 A = 6 . 7 : (1) 6 7 4 0 dn−1 bn−1 5 dn A natural question arises: what caused an interest in pseudoinversion of such matrices ? The reason is as follows.
    [Show full text]
  • Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods1
    Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods1 Richard Barrett2, Michael Berry3, Tony F. Chan4, James Demmel5, June M. Donato6, Jack Dongarra3,2, Victor Eijkhout7, Roldan Pozo8, Charles Romine9, and Henk Van der Vorst10 This document is the electronic version of the 2nd edition of the Templates book, which is available for purchase from the Society for Industrial and Applied Mathematics (http://www.siam.org/books). 1This work was supported in part by DARPA and ARO under contract number DAAL03-91-C-0047, the National Science Foundation Science and Technology Center Cooperative Agreement No. CCR-8809615, the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract DE-AC05-84OR21400, and the Stichting Nationale Computer Faciliteit (NCF) by Grant CRG 92.03. 2Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37830- 6173. 3Department of Computer Science, University of Tennessee, Knoxville, TN 37996. 4Applied Mathematics Department, University of California, Los Angeles, CA 90024-1555. 5Computer Science Division and Mathematics Department, University of California, Berkeley, CA 94720. 6Science Applications International Corporation, Oak Ridge, TN 37831 7Texas Advanced Computing Center, The University of Texas at Austin, Austin, TX 78758 8National Institute of Standards and Technology, Gaithersburg, MD 9Office of Science and Technology Policy, Executive Office of the President 10Department of Mathematics, Utrecht University, Utrecht, the Netherlands. ii How to Use This Book We have divided this book into five main chapters. Chapter 1 gives the motivation for this book and the use of templates. Chapter 2 describes stationary and nonstationary iterative methods.
    [Show full text]