Arithmetical Problems in Number Fields, Abelian Varieties and Modular Forms*
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(Elliptic Modular Curves) JS Milne
Modular Functions and Modular Forms (Elliptic Modular Curves) J.S. Milne Version 1.30 April 26, 2012 This is an introduction to the arithmetic theory of modular functions and modular forms, with a greater emphasis on the geometry than most accounts. BibTeX information: @misc{milneMF, author={Milne, James S.}, title={Modular Functions and Modular Forms (v1.30)}, year={2012}, note={Available at www.jmilne.org/math/}, pages={138} } v1.10 May 22, 1997; first version on the web; 128 pages. v1.20 November 23, 2009; new style; minor fixes and improvements; added list of symbols; 129 pages. v1.30 April 26, 2010. Corrected; many minor revisions. 138 pages. Please send comments and corrections to me at the address on my website http://www.jmilne.org/math/. The photograph is of Lake Manapouri, Fiordland, New Zealand. Copyright c 1997, 2009, 2012 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Contents Contents 3 Introduction ..................................... 5 I The Analytic Theory 13 1 Preliminaries ................................. 13 2 Elliptic Modular Curves as Riemann Surfaces . 25 3 Elliptic Functions ............................... 41 4 Modular Functions and Modular Forms ................... 48 5 Hecke Operators ............................... 68 II The Algebro-Geometric Theory 89 6 The Modular Equation for 0.N / ...................... 89 7 The Canonical Model of X0.N / over Q ................... 93 8 Modular Curves as Moduli Varieties ..................... 99 9 Modular Forms, Dirichlet Series, and Functional Equations . 104 10 Correspondences on Curves; the Theorem of Eichler-Shimura . 108 11 Curves and their Zeta Functions . 112 12 Complex Multiplication for Elliptic Curves Q . -
Canonical Heights and Division Polynomials 11
CANONICAL HEIGHTS AND DIVISION POLYNOMIALS ROBIN DE JONG AND J. STEFFEN MULLER¨ Abstract. We discuss a new method to compute the canonical height of an algebraic point on a hyperelliptic jacobian over a number field. The method does not require any geometrical models, neither p-adic nor complex analytic ones. In the case of genus 2 we also present a version that requires no factori- sation at all. The method is based on a recurrence relation for the `division polynomials' associated to hyperelliptic jacobians, and a diophantine approx- imation result due to Faltings. 1. Introduction In [EW] G. Everest and T. Ward show how to approximate to high precision the canonical height of an algebraic point on an elliptic curve E over a number field K with a limit formula using the (recurrence) sequence of division polynomials φn associated to E, and a diophantine approximation result. The φn have natural analogues for jacobians of hyperelliptic curves. In [Uc2] Y. Uchida shows how to obtain recurrence relations for the φn for hyperelliptic jaco- bians of dimension g ≥ 2. Further there exists a suitable analogue of the diophantine approximation result employed by Everest and Ward, proved by G. Faltings. In this paper we derive a limit formula for the canonical height of an algebraic point on a hyperelliptic jacobian from these inputs. We have implemented the resulting method for computing canonical heights in Magma for g = 2. The method does not require geometrical models, neither p-adic nor complex analytic ones. If the curve is defined over Q and the coordinates of the point are integral, then it also requires no factorisation. -
Elliptic Curves, Modularity, and Fermat's Last Theorem
Elliptic curves, modularity, and Fermat's Last Theorem For background and (most) proofs, we refer to [1]. 1 Weierstrass models Let K be any field. For any a1; a2; a3; a4; a6 2 K consider the plane projective curve C given by the equation 2 2 3 2 2 3 y z + a1xyz + a3yz = x + a2x z + a4xz + a6z : (1) An equation as above is called a Weierstrass equation. We also say that (1) is a Weierstrass model for C. L-Rational points For any field extension L=K we can consider the L-rational points on C, i.e. the points on C with coordinates in L: 2 C(L) := f(x : y : z) 2 PL : equation (1) is satisfied g: The point at infinity The point O := (0 : 1 : 0) 2 C(K) is the only K-rational point on C with z = 0. It is always smooth. Most of the time we shall instead of a homogeneous Weierstrass equation write an affine Weierstrass equation: 2 3 2 C : y + a1xy + a3y = x + a2x + a4x + a6 (2) which is understood to define a plane projective curve. The discriminant Define 2 b2 := a1 + 4a2; b4 := 2a4 + a1a3; 2 b6 := a3 + 4a6; 2 2 2 b8 := a1a6 + 4a2a6 − a1a3a4 + a2a3 − a4: For any Weierstrass equation we define its discriminant 2 3 2 ∆ := −b2b8 − 8b4 − 27b6 + 9b2b4b6: 1 Note that if char(K) 6= 2, then we can perform the coordinate transformation 2 3 2 y 7! (y − a1x − a3)=2 to arrive at an equation y = 4x + b2x + 2b4x + b6. -
The Klein Quartic in Number Theory
The Eightfold Way MSRI Publications Volume 35, 1998 The Klein Quartic in Number Theory NOAM D. ELKIES Abstract. We describe the Klein quartic X and highlight some of its re- markable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes divid- ing the order of the automorphism group of X; an explicit identification of X with the modular curve X(7); and applications to the class number 1 problem and the case n = 7 of Fermat. Introduction Overview. In this expository paper we describe some of the remarkable prop- erties of the Klein quartic that are of particular interest in number theory. The Klein quartic X is the unique curve of genus 3 over C with an automorphism group G of size 168, the maximum for its genus. Since G is central to the story, we begin with a detailed description of G and its representation on the 2 three-dimensional space V in whose projectivization P(V )=P the Klein quar- tic lives. The first section is devoted to this representation and its invariants, starting over C and then considering arithmetical questions of fields of definition and integral structures. There we also encounter a G-lattice that later occurs as both the period lattice and a Mordell–Weil lattice for X. In the second section we introduce X and investigate it as a Riemann surface with automorphisms by G. In the third section we consider the arithmetic of X: rational points, relations with the Fermat curve and Fermat’s “Last Theorem” for exponent 7, and some extremal properties of the reduction of X modulo the primes 2, 3, 7 dividing #G. -
The Modularity Theorem
R. van Dobben de Bruyn The Modularity Theorem Bachelor's thesis, June 21, 2011 Supervisors: Dr R.M. van Luijk, Dr C. Salgado Mathematisch Instituut, Universiteit Leiden Contents Introduction4 1 Elliptic Curves6 1.1 Definitions and Examples......................6 1.2 Minimal Weierstrass Form...................... 11 1.3 Reduction Modulo Primes...................... 14 1.4 The Frey Curve and Fermat's Last Theorem............ 15 2 Modular Forms 18 2.1 Definitions............................... 18 2.2 Eisenstein Series and the Discriminant............... 21 2.3 The Ring of Modular Forms..................... 25 2.4 Congruence Subgroups........................ 28 3 The Modularity Theorem 36 3.1 Statement of the Theorem...................... 36 3.2 Fermat's Last Theorem....................... 36 References 39 2 Introduction One of the longest standing open problems in mathematics was Fermat's Last Theorem, asserting that the equation an + bn = cn does not have any nontrivial (i.e. with abc 6= 0) integral solutions when n is larger than 2. The proof, which was completed in 1995 by Wiles and Taylor, relied heavily on the Modularity Theorem, relating elliptic curves over Q to modular forms. The Modularity Theorem has many different forms, some of which are stated in an analytic way using Riemann surfaces, while others are stated in a more algebraic way, using for instance L-series or Galois representations. This text will present an elegant, elementary formulation of the theorem, using nothing more than some basic vocabulary of both elliptic curves and modular forms. For elliptic curves E over Q, we will examine the reduction E~ of E modulo any prime p, thus introducing the quantity ~ ap(E) = p + 1 − #E(Fp): We will also give an almost complete description of the conductor NE associated to an elliptic curve E, and compute it for the curve used in the proof of Fermat's Last Theorem. -
"Rank Zero Quadratic Twists of Modular Elliptic Curves" by Ken
RANK ZERO QUADRATIC TWISTS OF MODULAR ELLIPTIC CURVES Ken Ono Abstract. In [11] L. Mai and M. R. Murty proved that if E is a modular elliptic curve with conductor N, then there exists infinitely many square-free integers D ≡ 1 mod 4N such that ED, the D−quadratic twist of E, has rank 0. Moreover assuming the Birch and Swinnerton-Dyer Conjecture, they obtain analytic estimates on the lower bounds for the orders of their Tate-Shafarevich groups. However regarding ranks, simply by the sign of functional equations, it is not expected that there will be infinitely many square-free D in every arithmetic progression r (mod t) where gcd(r, t) is square-free such that ED has rank zero. Given a square-free positive integer r, under mild conditions we show that there ×2 exists an integer tr and a positive integer N where tr ≡ r mod Qp for all p | N, such that 0 there are infinitely many positive square-free integers D ≡ tr mod N where ED has rank 0 0 Q zero. The modulus N is defined by N := 8 p|N p where the product is over odd primes p dividing N. As an example of this theorem, let E be defined by y2 = x3 + 15x2 + 36x. If r = 1, 3, 5, 9, 11, 13, 17, 19, 21, then there exists infinitely many positive square-free in- tegers D ≡ r mod 24 such that ED has rank 0. As another example, let E be defined by y2 = x3 − 1. If r = 1, 2, 5, 10, 13, 14, 17, or 22, then there exists infinitely many positive square-free integers D ≡ r mod 24 such that ED has rank 0. -
Lectures on Modular Forms 1St Edition Free Download
FREE LECTURES ON MODULAR FORMS 1ST EDITION PDF Joseph J Lehner | 9780486821405 | | | | | Lectures on Modular Forms - Robert C. Gunning - Google книги In mathematicsa modular form is a complex analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular groupand also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topologysphere packingand string theory. Instead, modular functions are meromorphic that is, they are almost holomorphic except for a set of isolated points. Modular form theory is a special case of the more general theory of automorphic formsand therefore can now be seen as just the most concrete part of a rich theory of discrete groups. Modular forms can also be interpreted as sections of a specific line bundles on modular varieties. The dimensions of these spaces of modular forms can be computed using the Riemann—Roch theorem [2]. A Lectures on Modular Forms 1st edition form of weight k for the modular group. A modular form can equivalently be defined as a function F from the set of lattices in C to the set of complex numbers which satisfies certain conditions:. The simplest examples from this point of view are the Eisenstein series. Then E k is a modular form of weight k. An even unimodular lattice L in R n is a lattice generated by n Lectures on Modular Forms 1st edition forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in L is an even integer. -
Elliptic Modular Curves)
Modular Functions and Modular Forms (Elliptic Modular Curves) J.S. Milne Version 1.31 March 22, 2017 This is an introduction to the arithmetic theory of modular functions and modular forms, with a greater emphasis on the geometry than most accounts. BibTeX information: @misc{milneMF, author={Milne, James S.}, title={Modular Functions and Modular Forms (v1.31)}, year={2017}, note={Available at www.jmilne.org/math/}, pages={134} } v1.10 May 22, 1997; first version on the web; 128 pages. v1.20 November 23, 2009; new style; minor fixes and improvements; added list of symbols; 129 pages. v1.30 April 26, 2010. Corrected; many minor revisions. 138 pages. v1.31 March 22, 2017. Corrected; minor revisions. 133 pages. Please send comments and corrections to me at the address on my website http://www.jmilne.org/math/. The picture shows a fundamental domain for 1.10/, as drawn by the fundamental domain drawer of H. Verrill. Copyright c 1997, 2009, 2012, 2017 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Contents Contents3 Introduction..................................... 5 I The Analytic Theory 13 1 Preliminaries ................................. 13 2 Elliptic Modular Curves as Riemann Surfaces................ 25 3 Elliptic Functions............................... 41 4 Modular Functions and Modular Forms ................... 48 5 Hecke Operators ............................... 67 II The Algebro-Geometric Theory 87 6 The Modular Equation for 0.N / ...................... 87 7 The Canonical Model of X0.N / over Q ................... 91 8 Modular Curves as Moduli Varieties..................... 97 9 Modular Forms, Dirichlet Series, and Functional Equations . 101 10 Correspondences on Curves; the Theorem of Eichler-Shimura . -
ON COMPUTING BELYI MAPS J. Sijsling & J. Voight
ON COMPUTING BELYI MAPS J. Sijsling & J. Voight Abstract.— We survey methods to compute three-point branched covers of the projective line, also known as Bely˘ımaps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and p-adic methods. Along the way, we pose several questions and provide numerous examples. Résumé.— Nous donnons un aperçu des méthodes actuelles pour le calcul des revêtements de la droite projective ramifiés en au plus trois points, connus sous le nom de morphismes de Bely˘ı.Ces méthodes comprennent une approche directe, se ramenant à la solution d’un système d’équations polynomiales ainsi que des méthodes analytiques complexes, de formes modulaires et p-adiques. Ce faisant, nous posons quelques questions et donnons de nombreux exemples. Contents Introduction. 6 1. Background and applications. 7 2. Gröbner techniques. 14 3. Complex analytic methods. 22 4. Modular forms. 28 5. p-adic methods. 38 6. Galois Bely˘ımaps. 42 7. Field of moduli and field of definition. 44 8. Simplification and verification. 50 9. Further topics and generalizations. 52 References. 55 2010 Mathematics Subject Classification.— 11G32, 11Y40. Key words and phrases.— Belyi maps, dessins d’enfants, covers, uniformization, computational algebra. 6 On computing Belyi maps Introduction Every compact Riemann surface X is an algebraic curve over C, and every meromorphic function on X is an algebraic function. This remarkable fact, generalized in the GAGA principle, links the analytic with the algebraic in a fundamental way. A natural problem is then to link this further with arithmetic; to characterize those Riemann surfaces that can be defined by equations over Q and to study the action of the absolute Galois group Gal(Q=Q) on these algebraic curves. -
Elliptic Curve Handbook
Elliptic Curve Handbook Ian Connell February, 1999 http://www.math.mcgill.ca/connell/ Foreword The first version of this handbook was a set of notes of about 100 pages handed out to the class of an introductory course on elliptic curves given in the 1990 fall semester at McGill University in Montreal. Since then I have added to the notes, holding to the principle: If I look up a certain topic a year from now I want all the details right at hand, not in an “exercise”, so if I’ve forgotten something I won’t waste time. Thus there is much that an ordinary text would either condense, or relegate to an exercise. But at the same time I have maintained a solid mathematical style with the thought of sharing the handbook. Montreal, August, 1996. 1 Contents 1 Introduction to Elliptic Curves 101 1.1 The a,b,c’s and ∆,j ...................... 101 1.2 Quartic to Weierstrass ..................... 105 1.3 Projective coordinates. ..................... 111 1.4 Cubic to Weierstrass: Nagell’s algorithm ........... 115 1.4.1 Example 1: Selmer curves ............... 117 1.4.2 Example 2: Desboves curves .............. 121 1.4.3 Example 3: Intersection of quadric surfaces ..... 123 1.5 Singular points. ......................... 125 − 1.5.1 Example: No E/Z has ∆ = 1 or 1 .......... 130 1.6 Affine coordinate ring, function field, generic points ..... 132 1.7 The group law: nonsingular case ............... 133 1.7.1 Halving points ..................... 140 1.7.2 The division polynomials ............... 145 1.7.3 Remarks on the group of division points ....... 151 1.8 The group law: singular case ................ -
Elliptic Curves in the Modern Age
Elliptic curves in the modern age (3rd millennium) Jens Berlips Elliptic curves in the modern age (3rd millennium) Jens Berlips August 21, 2006 Copyright c 2006 Jens Berlips. draft date: August, 2006. Typeset by the author with the LATEX 2" Documentation System. All rights reserved. No part of this work may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission. To my grandmother may she rest in peace CONTENTS 7 Contents 1 Introduction 10 2 Credits 12 3 Preliminiaries 13 3.1 Ane space ............................ 13 3.2 Projective space .......................... 14 3.3 From projective to Ane space and vice verse ......... 15 3.4 Notation .............................. 15 3.5 Cubic plane curve ......................... 16 3.5.1 Intersection of curves ................... 17 4 Elliptic Curves 18 4.1 Addition on E(k) ......................... 20 4.2 General theory .......................... 24 4.2.1 Order ........................... 24 4.2.2 Torsion points ....................... 25 4.2.3 Division polynomials ................... 26 5 Practical computational considerations 30 5.1 Binary ladder ........................... 30 6 Dierent parametrization 31 6.1 Ane coordinates ......................... 31 6.2 Projective coordinates ...................... 32 6.3 Montgomery coordinates ..................... 33 7 Finding the order 35 7.1 Schoof's method ......................... 36 7.1.1 The Frobenius endomorphism .............. 36 7.1.2 Division polynomials and Schoof's method ....... 37 7.1.3 Schoof's method explained ................ 37 8 CONTENTS 8 Factorization 40 8.1 Factorization methods ...................... 41 8.2 Pollard p ¡ 1 ........................... 41 8.3 Smooth numbers ......................... 43 8.4 Ideas of factorization ....................... 43 8.5 The general method ...................... -
MODULAR CURVES and MODULAR FORMS All of the Material Covered Here Can Be Found in the References, Though I Claim Responsibility
MODULAR CURVES AND MODULAR FORMS All of the material covered here can be found in the references, though I claim responsibility for mistakes. As these are very rough notes, I have not been diligent in adding citations for each specific results. The advanced reader should consult [1]. 1. Basic Theory of Elliptic Curves Elliptic curves admit numerous descriptions: projective curves of genus 1 with a distinguished point, varieties with equations y2 = x3 + ax + b, complete group varieties of dimension 1, etc. All are equivalent and it is instructive to see how one goes from one object to the next. Our goal is to classify elliptic curves up to isomorphism. Given a field K, we will denote by Ell(K) to be the set of elliptic curves over K (i.e. the equation defining the curve has coefficients in K) up to isomorphism. We consider curves E1;E2 over K to be isomorphic if there is a map ': E1 ! E2 defined over K (meaning the coefficients of ' are in K). This is an example of a moduli problem. We begin by studying Ell(C). Elliptic curves over C have a particularly nice form: Proposition 1. For every elliptic curve E=C, there is a lattice ΛE ⊂ C and a complex analytic group 0 isomorphism C=ΛE ! E(C). Further, maps E ! E correspond to complex numbers α such that αΛE ⊂ ΛE0 . 0 In particular two elliptic curves E; E are isomorphic if and only if αΛE = ΛE0 for some α 2 C, i.e. if the lattices ΛE; ΛE0 are homothetic.