DOCSLIB.ORG

Modular Forms: a Computational Approach William A. Stein

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Modular Forms: A Computational Approach William A. Stein (with an appendix by Paul E. Gunnells) Department of Mathematics, University of Washington E-mail address: [email protected] Department of Mathematics and Statistics, University of Massachusetts E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary 11; Secondary 11-04 Key words and phrases. abelian varieties, cohomology of arithmetic groups, computation, elliptic curves, Hecke operators, modular curves, modular forms, modular symbols, Manin symbols, number theory Abstract. This is a textbook about algorithms for computing with modular forms. It is nontraditional in that the primary focus is not on underlying theory; instead, it answers the question “how do you explicitly compute spaces of modular forms?” v To my grandmother, Annette Maurer. Contents Preface xi Chapter 1. Modular Forms 1 1.1. Basic Definitions 1 § 1.2. Modular Forms of Level 1 3 § 1.3. Modular Forms of Any Level 4 § 1.4. Remarks on Congruence Subgroups 7 § 1.5. Applications of Modular Forms 9 § 1.6. Exercises 11 § Chapter 2. Modular Forms of Level 1 13 2.1. Examples of Modular Forms of Level 1 13 § 2.2. Structure Theorem for Level 1 Modular Forms 17 § 2.3. The Miller Basis 20 § 2.4. Hecke Operators 22 § 2.5. Computing Hecke Operators 26 § 2.6. Fast Computation of Fourier Coefficients 29 § 2.7. Fast Computation of Bernoulli Numbers 29 § 2.8. Exercises 33 § Chapter 3. Modular Forms of Weight 2 35 3.1. Hecke Operators 36 § 3.2. Modular Symbols 39 § 3.3. Computing with Modular Symbols 41 § vii viii Contents 3.4. Hecke Operators 47 § 3.5. Computing the Boundary Map 51 § 3.6. Computing a Basis for S (Γ (N)) 53 § 2 0 3.7. Computing S (Γ (N)) Using Eigenvectors 58 § 2 0 3.8. Exercises 60 § Chapter 4. Dirichlet Characters 63 4.1. The Definition 64 § 4.2. Representing Dirichlet Characters 64 § 4.3. Evaluation of Dirichlet Characters 67 § 4.4. Conductors of Dirichlet Characters 70 § 4.5. The Kronecker Symbol 72 § 4.6. Restriction, Extension, and Galois Orbits 75 § 4.7. Alternative Representations of Characters 77 § 4.8. Dirichlet Characters in SAGE 78 § 4.9. Exercises 81 § Chapter 5. Eisenstein Series and Bernoulli Numbers 83 5.1. The Eisenstein Subspace 83 § 5.2. Generalized Bernoulli Numbers 83 § 5.3. Explicit Basis for the Eisenstein Subspace 88 § 5.4. Exercises 90 § Chapter 6. Dimension Formulas 91 6.1. Modular Forms for Γ (N) 92 § 0 6.2. Modular Forms for Γ (N) 95 § 1 6.3. Modular Forms with Character 98 § 6.4. Exercises 102 § Chapter 7. Linear Algebra 103 7.1. Echelon Forms of Matrices 103 § 7.2. Rational Reconstruction 105 § 7.3. Echelon Forms over Q 107 § 7.4. Echelon Forms via Matrix Multiplication 110 § 7.5. Decomposing Spaces under the Action of Matrix 114 § 7.6. Exercises 119 § Chapter 8. General Modular Symbols 121 Contents ix 8.1. Modular Symbols 122 § 8.2. Manin Symbols 124 § 8.3. Hecke Operators 128 § 8.4. Cuspidal Modular Symbols 133 § 8.5. Pairing Modular Symbols and Modular Forms 137 § 8.6. Degeneracy Maps 142 § 8.7. Explicitly Computing M (Γ (N)) 144 § k 0 8.8. Explicit Examples 147 § 8.9. Refined Algorithm for the Presentation 154 § 8.10. Applications 155 § 8.11. Exercises 156 § Chapter 9. Computing with Newforms 159 9.1. Dirichlet Character Decomposition 159 § 9.2. Atkin-Lehner-Li Theory 161 § 9.3. Computing Cusp Forms 165 § 9.4. Congruences between Newforms 170 § 9.5. Exercises 176 § Chapter 10. Computing Periods 177 10.1. The Period Map 178 § 10.2. Abelian Varieties Attached to Newforms 178 § 10.3. Extended Modular Symbols 179 § 10.4. Approximating Period Integrals 180 § 10.5. Speeding Convergence Using Atkin-Lehner 183 § 10.6. Computing the Period Mapping 185 § 10.7. All Elliptic Curves of Given Conductor 187 § 10.8. Exercises 190 § Chapter 11. Solutions to Selected Exercises 191 11.1. Chapter 1 191 § 11.2. Chapter 2 193 § 11.3. Chapter 3 194 § 11.4. Chapter 4 196 § 11.5. Chapter 5 197 § 11.6. Chapter 6 197 § 11.7. Chapter 7 198 § x Contents 11.8. Chapter 8 199 § 11.9. Chapter 9 201 § 11.10. Chapter 10 201 § Appendix A. Computing in Higher Rank 203 A.1. Introduction 203 § A.2. Automorphic Forms and Arithmetic Groups 205 § A.3. Combinatorial Models for Group Cohomology 213 § A.4. Hecke Operators and Modular Symbols 225 § A.5. Other Cohomology Groups 232 § A.6. Complements and Open Problems 244 § Bibliography 253 Index 265 Preface This is a graduate-level textbook about algorithms for computing with mod- ular forms. It is nontraditional in that the primary focus is not on underly- ing theory; instead, it answers the question “how do you use a computer to explicitly compute spaces of modular forms?” This book emerged from notes for a course the author taught at Harvard University in 2004, a course at UC San Diego in 2005, and a course at the University of Washington in 2006. The author has spent years trying to find good practical ways to compute with classical modular forms for congruence subgroups of SL2(Z) and has implemented most of these algorithms several times, first in C++ [Ste99b], then in MAGMA [BCP97], and as part of the free open source computer algebra system SAGE (see [Ste06]). Much of this work has involved turning formulas and constructions buried in obscure research papers into precise computational recipes then testing these and eliminating inaccuracies. The author is aware of no other textbooks on computing with modular forms, the closest work being Cremona’s book [Cre97a], which is about computing with elliptic curves, and Cohen’s book [Coh93] about algebraic number theory. In this book we focus on how to compute in practice the spaces Mk(N, ε) of modular forms, where k 2 is an integer and ε is a Dirichlet character ≥ of modulus N (the appendix treats modular forms for higher rank groups). We spend the most effort explaining the general algorithms that appear so far to be the best (in practice!) for such computations. We will not dis- cuss in any detail computing with quaternion algebras, half-integral weight forms, weight 1 forms, forms for noncongruence subgroups or groups other xi xii Preface than GL2, Hilbert and Siegel modular forms, trace formulas, p-adic modular forms, and modular abelian varieties, all of which are topics for additional books. We also rarely analyze the complexity of the algorithms, but instead settle for occasional remarks about their practical efficiency. For most of this book we assume the reader has some prior exposure to modular forms (e.g., [DS05]), though we recall many of the basic defini- tions. We cite standard books for proofs of the fundamental results about modular forms that we will use. The reader should also be familiar with basic algebraic number theory, linear algebra, complex analysis (at the level of [Ahl78]), and algorithms (e.g., know what an algorithm is and what big oh notation means). In some of the examples and applications we assume that the reader knows about elliptic curves at the level of [Sil92]. Chapter 1 is foundational for the rest of this book. It introduces congru- ence subgroups of SL2(Z) and modular forms as functions on the complex upper half plane. We discuss q-expansions, which provide an important computational handle on modular forms. We also study an algorithm for computing with congruence subgroups. The chapter ends with a list of ap- plications of modular forms throughout mathematics. In Chapter 2 we discuss level 1 modular forms in much more detail. In particular, we introduce Eisenstein series and the cusp form ∆ and describe their q-expansions and basic properties. Then we prove a structure theorem for level 1 modular forms and use it to deduce dimension formulas and give an algorithm for explicitly computing a basis. We next introduce Hecke operators on level 1 modular forms, prove several results about them, and deduce multiplicativity of the Ramanujan τ function as an application. We also discuss explicit computation of Hecke operators. In Section 2.6 we make some brief remarks on recent work on asymptotically fast computation of values of τ. Finally, we describe computation of constant terms of Eisenstein series using an analytic algorithm. We generalize many of the constructions in this chapter to higher level in subsequent chapters. In Chapter 3 we turn to modular forms of higher level but restrict for simplicity to weight 2 since much is clearer in this case. (We remove the weight restriction later in Chapter 8.) We describe a geometric way of view- ing cuspidal modular forms as differentials on modular curves, which leads to modular symbols, which are an explicit way to present a certain homol- ogy group. This chapter closes with methods for explicitly computing cusp forms of weight 2 using modular symbols, which we generalize in Chapter 9. In Chapter 4 we introduce Dirichlet characters, which are important both in explicit construction of Eisenstein series (in Chapter 5) and in de- composing spaces of modular forms as direct sums of simpler spaces. The Preface xiii main focus of this chapter is a detailed study of how to explicitly represent and compute with Dirichlet characters. Chapter 5 is about how to explicitly construct the Eisenstein subspace of modular forms. First we define generalized Bernoulli numbers attached to a Dirichlet character and an integer then explain a new analytic algorithm for computing them (which generalizes the algorithm in Chapter 2). Finally we give without proof an explicit description of a basis of Eisenstein series, explain how to compute it, and give some examples.
Recommended publications
  • Determining Large Groups and Varieties from Small Subgroups and Subvarieties

    Determining Large Groups and Varieties from Small Subgroups and Subvarieties

    Determining large groups and varieties from small subgroups and subvarieties This course will consist of two parts with the common theme of analyzing a geometric object via geometric sub-objects which are `small' in an appropriate sense. In the first half of the course, we will begin by describing the work on H. W. Lenstra, Jr., on finding generators for the unit groups of subrings of number fields. Lenstra discovered that from the point of view of computational complexity, it is advantageous to first consider the units of the ring of S-integers of L for some moderately large finite set of places S. This amounts to allowing denominators which only involve a prescribed finite set of primes. We will de- velop a generalization of Lenstra's method which applies to find generators of small height for the S-integral points of certain algebraic groups G defined over number fields. We will focus on G which are compact forms of GLd for d ≥ 1. In the second half of the course, we will turn to smooth projective surfaces defined by arithmetic lattices. These are Shimura varieties of complex dimension 2. We will try to generate subgroups of finite index in the fundamental groups of these surfaces by the fundamental groups of finite unions of totally geodesic projective curves on them, that is, via immersed Shimura curves. In both settings, the groups we will study are S-arithmetic lattices in a prod- uct of Lie groups over local fields. We will use the structure of our generating sets to consider several open problems about the geometry, group theory, and arithmetic of these lattices.
  • Zombies in Western Culture: a Twenty-First Century Crisis

    Zombies in Western Culture: a Twenty-First Century Crisis

    JOHN VERVAEKE, CHRISTOPHER MASTROPIETRO AND FILIP MISCEVIC Zombies in Western Culture A Twenty-First Century Crisis To access digital resources including: blog posts videos online appendices and to purchase copies of this book in: hardback paperback ebook editions Go to: https://www.openbookpublishers.com/product/602 Open Book Publishers is a non-profit independent initiative. We rely on sales and donations to continue publishing high-quality academic works. Zombies in Western Culture A Twenty-First Century Crisis John Vervaeke, Christopher Mastropietro, and Filip Miscevic https://www.openbookpublishers.com © 2017 John Vervaeke, Christopher Mastropietro and Filip Miscevic. This work is licensed under a Creative Commons Attribution 4.0 International license (CC BY 4.0). This license allows you to share, copy, distribute and transmit the work; to adapt the work and to make commercial use of the work providing attribution is made to the authors (but not in any way that suggests that they endorse you or your use of the work). Attribution should include the following information: John Vervaeke, Christopher Mastropietro and Filip Miscevic, Zombies in Western Culture: A Twenty-First Century Crisis. Cambridge, UK: Open Book Publishers, 2017, http://dx.doi. org/10.11647/OBP.0113 In order to access detailed and updated information on the license, please visit https:// www.openbookpublishers.com/product/602#copyright Further details about CC BY licenses are available at http://creativecommons.org/licenses/ by/4.0/ All external links were active at the time of publication unless otherwise stated and have been archived via the Internet Archive Wayback Machine at https://archive.org/web Digital material and resources associated with this volume are available at https://www.
  • SYMBOLS Te Holy Apostles & Evangelists Peter

    SYMBOLS Te Holy Apostles & Evangelists Peter

    SYMBOLS Te Holy Apostles & Evangelists Peter Te most common symbol for St. Peter is that of two keys crossed and pointing up. Tey recall Peter’s confession and Jesus’ statement regarding the Office of the Keys in Matthew, chapter 16. A rooster is also sometimes used, recalling Peter’s denial of his Lord. Another popular symbol is that of an inverted cross. Peter is said to have been crucifed in Rome, requesting to be crucifed upside down because he did not consider himself worthy to die in the same position as that of his Lord. James the Greater Tree scallop shells are used for St. James, with two above and one below. Another shows a scallop shell with a vertical sword, signifying his death at the hands of Herod, as recorded in Acts 12:2. Tradition states that his remains were carried from Jerusalem to northern Spain were he was buried in the city of Santiago de Compostela, the capital of Galicia. As one of the most desired pilgrimages for Christians since medieval times together with Jerusalem and Rome, scallop shells are often associated with James as they are often found on the shores in Galicia. For this reason the scallop shell has been a symbol of pilgrimage. John When shown as an apostle, rather than one of the four evangelists, St. John’s symbol shows a chalice and a serpent coming out from it. Early historians and writers state that an attempt was made to poison him, but he was spared before being sent to Patmos. A vertical sword and snake are also used in some churches.
  • Generalization of a Theorem of Hurwitz

    Generalization of a Theorem of Hurwitz

    J. Ramanujan Math. Soc. 31, No.3 (2016) 215–226 Generalization of a theorem of Hurwitz Jung-Jo Lee1,∗ ,M.RamMurty2,† and Donghoon Park3,‡ 1Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea e-mail: [email protected] 2Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada e-mail: [email protected] 3Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 120-749, South Korea e-mail: [email protected] Communicated by: R. Sujatha Received: February 10, 2015 Abstract. This paper is an exposition of several classical results formulated and unified using more modern terminology. We generalize a classical theorem of Hurwitz and prove the following: let 1 G (z) = k (mz + n)k m,n be the Eisenstein series of weight k attached to the full modular group. Let z be a CM point in the upper half-plane. Then there is a transcendental number z such that ( ) = 2k · ( ). G2k z z an algebraic number Moreover, z can be viewed as a fundamental period of a CM elliptic curve defined over the field of algebraic numbers. More generally, given any modular form f of weight k for the full modular group, and with ( )π k /k algebraic Fourier coefficients, we prove that f z z is algebraic for any CM point z lying in the upper half-plane. We also prove that for any σ Q Q ( ( )π k /k)σ = σ ( )π k /k automorphism of Gal( / ), f z z f z z . 2010 Mathematics Subject Classification: 11J81, 11G15, 11R42.
  • Bian Que's Viewpoint on Medicine and the Preventative Treatment of Diseases

    Bian Que's Viewpoint on Medicine and the Preventative Treatment of Diseases

    Northeast Asia Traditional Chinese Medicine Communication and Development Base of traditional medicine in Northeast Asia, conduct academic seminars and collaborative innovation, and form annual report on development of traditional medicine in Northeast Asia. 2. Special Belt & Road scholarships for Northeast Asia: are ready to provide yearly funding support such as full & partial scholarships and grants for overseas TCM talents with medical background. 3. Exhibitions on traditional medicine in Northeast Asia: to held exhibitions on traditional medicine of northeast Asian countries, health-care foods, welfare In November 2016, the Northeast Asia Traditional equipments and service trade negotiations, and promote Chinese Medicine Communication and Development multilateral p roject cooperation. Base was established in Changchun University of Chinese Medicine being approved by the State 4. The forum on Traditional Medicine in Northeast Administration of Traditional Chinese Medicine of Asia (Planning): to invite principals of traditional China. This foundation will serve as an important medicine departments from northeast Asian or platform for the spread of TCM in northeast China. relevant countries to make keynote speeches, and distinguished specialists and experts to participate in Distinct Regional Advantages Historically, this conference discussion. area is the core of the Northern Silk Road that extends to Russia, Japan, Mongolia, Republic of Korea and 5. One journal and one bulletin: to issue restrictedly Democratic People’s Republic of Korean. The city of Northeast Asia Traditional Chinese Medicine (quarterly) Changchun aims to be a regional center in Northeast and Bulletin on Traditional Chinese Medicine Asia. The China-Northeast Asia Expo in Changchun Information in Northeast Asia (monthly). serves as an important window to open to the north.
  • Arxiv:2009.05223V1 [Math.NT] 11 Sep 2020 Fdegree of Eaeitrse Nfidn Function a finding in Interested Are We Hoe 1.1

    Arxiv:2009.05223V1 [Math.NT] 11 Sep 2020 Fdegree of Eaeitrse Nfidn Function a finding in Interested Are We Hoe 1.1

    COUNTING ELLIPTIC CURVES WITH A RATIONAL N-ISOGENY FOR SMALL N BRANDON BOGGESS AND SOUMYA SANKAR Abstract. We count the number of rational elliptic curves of bounded naive height that have a rational N-isogeny, for N ∈ {2, 3, 4, 5, 6, 8, 9, 12, 16, 18}. For some N, this is done by generalizing a method of Harron and Snowden. For the remaining cases, we use the framework of Ellenberg, Satriano and Zureick-Brown, in which the naive height of an elliptic curve is the height of the corresponding point on a moduli stack. 1. Introduction ′ Let E be an elliptic curve over Q. An isogeny φ : E E between two elliptic curves is said to be cyclic ¯ → of degree N if Ker(φ)(Q) ∼= Z/NZ. Further, it is said to be rational if Ker(φ) is stable under the action of the absolute Galois group, GQ. A natural question one can ask is, how many elliptic curves over Q have a rational cyclic N-isogeny? Henceforth, we will omit the adjective ‘cyclic’, since these are the only types of isogenies we will consider. It is classically known that for N 10 and N = 12, 13, 16, 18, 25, there are infinitely many such elliptic curves. Thus we order them by naive≤ height. An elliptic curve E over Q has a unique minimal Weierstrass equation y2 = x3 + Ax + B where A, B Z and gcd(A3,B2) is not divisible by any 12th power. Define the naive height of E to be ht(E) = max A∈3, B 2 .
  • Modular Forms, the Ramanujan Conjecture and the Jacquet-Langlands Correspondence

    Modular Forms, the Ramanujan Conjecture and the Jacquet-Langlands Correspondence

    Appendix: Modular forms, the Ramanujan conjecture and the Jacquet-Langlands correspondence Jonathan D. Rogawski1) The theory developed in Chapter 7 relies on a fundamental result (Theorem 7 .1.1) asserting that the space L2(f\50(3) x PGLz(Op)) decomposes as a direct sum of tempered, irreducible representations (see definition below). Here 50(3) is the compact Lie group of 3 x 3 orthogonal matrices of determinant one, and r is a discrete group defined by a definite quaternion algebra D over 0 which is split at p. The embedding of r in 50(3) X PGLz(Op) is defined by identifying 50(3) and PGLz(Op) with the groups of real and p-adic points of the projective group D*/0*. Although this temperedness result can be viewed as a combinatorial state­ ment about the action of the Heckeoperators on the Bruhat-Tits tree associated to PGLz(Op). it is not possible at present to prove it directly. Instead, it is deduced as a corollary of two other results. The first is the Ramanujan-Petersson conjec­ ture for holomorphic modular forms, proved by P. Deligne [D]. The second is the Jacquet-Langlands correspondence for cuspidal representations of GL(2) and multiplicative groups of quaternion algebras [JL]. The proofs of these two results involve essentially disjoint sets of techniques. Deligne's theorem is proved using the Riemann hypothesis for varieties over finite fields (also proved by Deligne) and thus relies on characteristic p algebraic geometry. By contrast, the Jacquet­ Langlands Theorem is analytic in nature. The main tool in its proof is the Seiberg trace formula.
  • Batman in the 50S Free

    Batman in the 50S Free

    FREE BATMAN IN THE 50S PDF Joe Samachson,Various,Edmond Hamilton,Bill Finger,Bob Kane,Dick Sprang,Stan Kaye,Sheldon Moldoff | 191 pages | 01 May 2002 | DC Comics | 9781563898105 | English | United States Batman in the Fifties (Collected) | DC Database | Fandom An instantly recognizable theme song, outrageous death traps, ingenious gadgets, an army of dastardly villains and femme fatales, and a pop-culture phenomenon unmatched for generations. James Bondright? When it first premiered inBatman was the most faithful adaptation of a bona fide comic book superhero ever seen on the screen. It was a nearly perfect blend of the Saturday matinee movie serials where most comic book characters had their first Hollywood break and the comics of its time. But the TV series, particularly during its genesis, was both a product of its own time, and that of an earlier era. Both Flash Gordon and Dick Tracy had made the leap to the big screen before Superman had even hit newsstands, and both saw their serial adventures get two sequels. While Flash Gordonparticularly the first one, was a faithful within the limitations of its budget translation of the Alex Raymond comic strips, Dick Tracy was less so. Years before Richard Donner and Christopher Reevethis one made audiences believe a man could fly, and featured a perfectly cast Tom Tyler in the title role, but was still rather beholden to serial storytelling conventions and the aforementioned budgetary limitations. Ad — content continues below. Batman and Robin are portrayed much as they are in the comics, despite some unfortunately cheap costumes, and less than physically convincing actors in the title roles.
  • Portrayals of Stuttering in Film, Television, and Comic Books

    Portrayals of Stuttering in Film, Television, and Comic Books

    The Visualization of the Twisted Tongue: Portrayals of Stuttering in Film, Television, and Comic Books JEFFREY K. JOHNSON HERE IS A WELL-ESTABLISHED TRADITION WITHIN THE ENTERTAINMENT and publishing industries of depicting mentally and physically challenged characters. While many of the early renderings were sideshowesque amusements or one-dimensional melodramas, numerous contemporary works have utilized characters with disabilities in well- rounded and nonstereotypical ways. Although it would appear that many in society have begun to demand more realistic portrayals of characters with physical and mental challenges, one impediment that is still often typified by coarse caricatures is that of stuttering. The speech impediment labeled stuttering is often used as a crude formulaic storytelling device that adheres to basic misconceptions about the condition. Stuttering is frequently used as visual shorthand to communicate humor, nervousness, weakness, or unheroic/villainous characters. Because almost all the monographs written about the por- trayals of disabilities in film and television fail to mention stuttering, the purpose of this article is to examine the basic categorical formulas used in depicting stuttering in the mainstream popular culture areas of film, television, and comic books.' Though the subject may seem minor or unimportant, it does in fact provide an outlet to observe the relationship between a physical condition and the popular conception of the mental and personality traits that accompany it. One widely accepted definition of stuttering is, "the interruption of the flow of speech by hesitations, prolongation of sounds and blockages sufficient to cause anxiety and impair verbal communication" (Carlisle 4). The Journal of Popular Culture, Vol. 41, No.
  • The Reflection of Sancho Panza in the Comic Book Sidekick De Don

    The Reflection of Sancho Panza in the Comic Book Sidekick De Don

    UNIVERSIDAD DE OVIEDO FACULTAD DE FILOSOFÍA Y LETRAS MEMORIA DE LICENCIATURA From Don Quixote to The Tick: The Reflection of Sancho Panza in the Comic Book Sidekick ____________ De Don Quijote a The Tick: El Reflejo de Sancho Panza en el sidekick del Cómic Autor: José Manuel Annacondia López Directora: Dra. María José Álvarez Faedo VºBº: Oviedo, 2012 To comic book creators of yesterday, today and tomorrow. The comics medium is a very specialized area of the Arts, home to many rare and talented blooms and flowering imaginations and it breaks my heart to see so many of our best and brightest bowing down to the same market pressures which drive lowest-common-denominator blockbuster movies and television cop shows. Let's see if we can call time on this trend by demanding and creating big, wild comics which stretch our imaginations. Let's make living breathing, sprawling adventures filled with mind-blowing images of things unseen on Earth. Let's make artefacts that are not faux-games or movies but something other, something so rare and strange it might as well be a window into another universe because that's what it is. [Grant Morrison, “Grant Morrison: Master & Commander” (2004: 2)] TABLE OF CONTENTS 1. Acknowledgements v 2. Introduction 1 3. Chapter I: Theoretical Background 6 4. Chapter II: The Nature of Comic Books 11 5. Chapter III: Heroes Defined 18 6. Chapter IV: Enter the Sidekick 30 7. Chapter V: Dark Knights of Sad Countenances 35 8. Chapter VI: Under Scrutiny 53 9. Chapter VII: Evolve or Die 67 10.
  • Congruences Between Modular Forms

    Congruences Between Modular Forms

    CONGRUENCES BETWEEN MODULAR FORMS FRANK CALEGARI Contents 1. Basics 1 1.1. Introduction 1 1.2. What is a modular form? 4 1.3. The q-expansion priniciple 14 1.4. Hecke operators 14 1.5. The Frobenius morphism 18 1.6. The Hasse invariant 18 1.7. The Cartier operator on curves 19 1.8. Lifting the Hasse invariant 20 2. p-adic modular forms 20 2.1. p-adic modular forms: The Serre approach 20 2.2. The ordinary projection 24 2.3. Why p-adic modular forms are not good enough 25 3. The canonical subgroup 26 3.1. Canonical subgroups for general p 28 3.2. The curves Xrig[r] 29 3.3. The reason everything works 31 3.4. Overconvergent p-adic modular forms 33 3.5. Compact operators and spectral expansions 33 3.6. Classical Forms 35 3.7. The characteristic power series 36 3.8. The Spectral conjecture 36 3.9. The invariant pairing 38 3.10. A special case of the spectral conjecture 39 3.11. Some heuristics 40 4. Examples 41 4.1. An example: N = 1 and p = 2; the Watson approach 41 4.2. An example: N = 1 and p = 2; the Coleman approach 42 4.3. An example: the coefficients of c(n) modulo powers of p 43 4.4. An example: convergence slower than O(pn) 44 4.5. Forms of half integral weight 45 4.6. An example: congruences for p(n) modulo powers of p 45 4.7. An example: congruences for the partition function modulo powers of 5 47 4.8.
  • 25 Modular Forms and L-Series

    25 Modular Forms and L-Series

    18.783 Elliptic Curves Spring 2015 Lecture #25 05/12/2015 25 Modular forms and L-series As we will show in the next lecture, Fermat's Last Theorem is a direct consequence of the following theorem [11, 12]. Theorem 25.1 (Taylor-Wiles). Every semistable elliptic curve E=Q is modular. In fact, as a result of subsequent work [3], we now have the stronger result, proving what was previously known as the modularity conjecture (or Taniyama-Shimura-Weil conjecture). Theorem 25.2 (Breuil-Conrad-Diamond-Taylor). Every elliptic curve E=Q is modular. Our goal in this lecture is to explain what it means for an elliptic curve over Q to be modular (we will also define the term semistable). This requires us to delve briefly into the theory of modular forms. Our goal in doing so is simply to understand the definitions and the terminology; we will omit all but the most straight-forward proofs. 25.1 Modular forms Definition 25.3. A holomorphic function f : H ! C is a weak modular form of weight k for a congruence subgroup Γ if f(γτ) = (cτ + d)kf(τ) a b for all γ = c d 2 Γ. The j-function j(τ) is a weak modular form of weight 0 for SL2(Z), and j(Nτ) is a weak modular form of weight 0 for Γ0(N). For an example of a weak modular form of positive weight, recall the Eisenstein series X0 1 X0 1 G (τ) := G ([1; τ]) := = ; k k !k (m + nτ)k !2[1,τ] m;n2Z 1 which, for k ≥ 3, is a weak modular form of weight k for SL2(Z).