DOCSLIB.ORG

On the Modularity of Elliptic Curves

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Modularity of Elliptic Curves Uniwersytet Warszawski Wydziaª Matematyki, Informatyki i Mechaniki Przemysªaw Chojecki Nr albumu: 262926 On the modularity of elliptic curves Praca licencjacka na kierunku MATEMATYKA Praca wykonana pod kierunkiem dra hab. Adriana Langera Instytut Matematyki Czerwiec 2009 O±wiadczenie kieruj¡cego prac¡ Potwierdzam, »e niniejsza praca zostaªa przygotowana pod moim kierunkiem i kwa- lifikuje si¦ do przedstawienia jej w post¦powaniu o nadanie tytuªu zawodowego. Data Podpis kieruj¡cego prac¡ O±wiadczenie autora (autorów) pracy Świadom odpowiedzialno±ci prawnej o±wiadczam, »e niniejsza praca dyplomowa zostaªa napisana przeze mnie samodzielnie i nie zawiera tre±ci uzyskanych w sposób niezgodny z obowi¡zuj¡cymi przepisami. O±wiadczam równie», »e przedstawiona praca nie byªa wcze±niej przedmiotem pro- cedur zwi¡zanych z uzyskaniem tytuªu zawodowego w wy»szej uczelni. O±wiadczam ponadto, »e niniejsza wersja pracy jest identyczna z zaª¡czon¡ wersj¡ elektroniczn¡. Data Podpis autora (autorów) pracy Streszczenie Praca ta jest przegla,dem podstawowych definicji i twierdze«, które sa, potrzebne do zrozu- mienia wyniku znanego obecnie jako twierdzenie o modularno±ci. Szczególny przypadek tego twierdzenia, wystarczaja,cy do otrzymania Wielkiego Twierdzenia Fermata jako wniosku, zo- staª udowodniony w 1995 roku przez Andrew Wilesa (z pomoca, Richarda Taylora), a w caªej ogólności w 2000 roku przez Breuila, Conrada, Diamonda i Taylora. Dawniej twierdzenie o modularno±ci byªo znane jako hipoteza Taniyamy-Shimury-Weila. In this work I make a survey of basic definitions and theorems needed to understand a result called the Modularity Theorem. It was proved in '95 by Andrew Wiles (with a help of Richard Taylor) in the case when an elliptic curve in question is semistable and later in '00, it was proved in full generality by Breuil-Conrad-Diamond-Taylor. The statement of the theorem was known as the Taniyama-Shimura-Weil conjecture. Sªowa kluczowe modular form, Galois representation, elliptic curve Dziedzina pracy (kody wg programu Socrates-Erasmus) 11.1 Matematyka Klasyfikacja tematyczna 11F11 Modular forms, one variable Tytuª pracy w j¦zyku angielskim On the modularity of elliptic curves Table of contents Introduction • Modular forms • Modular curves • Elliptic curves • Geometric point of view • Hecke operators • Petersson inner product • Oldforms and Newforms • L-functions • Jacobians and abelian varieties • Galois representations • Modularity • Serres conjecture • Examples • Bibliography • 3 Introduction As a conjecture, the Modularity Theorem appeared around 1955 formulated by Yutaka Taniyama. Later it was reworked by Goro Shimura who gave it the correct wording, and popularized by Andre Weil who rediscovered it in 1967. It was Gerhard Frey who in the 80s made a remark that the Taniyama-Shimura-Weil con- jecture actually implies Fermat's Last Theorem. If FLT were false, it would give an elliptic curve which violate the Modularity Theorem. After years of struggle, Andrew Wiles managed to prove the Modularity Theorem when an elliptic case in question is semistable. The proof was announced in 1993, but it was found to be flawed. After a help of Richard Taylor, the correct proof appeared in 1994 and it was published in 1995 (see [W], [TW]). A few years later Breuil, Conrad, Diamond and Taylor proved the Modularity Theorem in full generality (see [BCDT]). The Taniyama-Shimura-Weil conjecture can be thought as a special case of Serre's conjec- ture which was probably formulated in the 70s. Extending results of Wiles and others, Khare with a help of Wintenberger proved Serre's conjecture in 2006 (see [KW1], [KW2]). The work which I present here, is a survey of basic definitions and theorems around the Modularity Theorem. In exposition, I have followed mostly the beautiful book of Diamond and Shurman A, first course in the modular forms". I do not include any proofs as they are to be found in the afore-mentioned book. I had rather tried to expose the crucial material in as short a form as possible and make a guide through [DS]. 4 Modular forms a b Let Γ(N) = SL (Z) : a d 1(modN); b c 0(modN) . c d 2 2 ≡ ≡ ≡ ≡ A subgroup Γ of SL (Z) is called a congruence subgroup if Γ(N) Γ for some N Z+, 2 ⊂ 2 in which case Γ is called a congruence subgroup of level N. The most important examples of conruence subgroups are: a b Γ (N) = SL (Z) : c 0(modN) 0 c d 2 2 ≡ and a b Γ (N) = SL (Z) : a d 1(modN); c 0(modN) : 1 c d 2 2 ≡ ≡ ≡ a b Let H = z C : Im(z) > 0 . Observe that SL (Z) acts on H by z = az+b . f 2 g 2 c d · cz+d a b For k Z, γ = SL (Z) and f : H C we define f[γ] by the formula 2 c d 2 2 ! k 1 az + b f[γ] (z) = f : k (cz + d)k cz + d Definition 1. Let Γ be a congruence subgroup of SL2(Z) and let k be a positive integer. A function f : H C is called a modular form of weight k with respect to Γ if the following conditions!are satisfied: 1. f is holomorphic, 2. f[γ] = f for all γ Γ, k 2 3. for every γ SL (Z) we have f[γ] (z) = 1 a(γ) enz=N(γ) for some a(γ) C and 2 2 k n=0 n n 2 N(γ) Z+. 2 P If in condition 3, a(γ) = 0 for every γ SL (Z) then f is called a cusp form. 0 2 2 Example 1. The basic example of a modular form of weight k for SL2(Z) is the Eisenstein 1 series 2 2 . Gk(z) = (m;n)2Z n(0;0) (mz+n) k P We will denote by Mk(Γ) modular forms of weight k with respect to Γ and by Sk(Γ) cusp forms of weight k with respect to Γ. 5 Modular curves Let Γ be a congruence subgroup for SL (Z) and define H∗ = H Q (where Q are 2 [ [ f1g rational numbers). A modular curve is a curve of the form X(Γ) = Γ H∗. The points Γ s n · in Γ Q are called cusps of X(Γ). It can be proved that each modular curve has only n [ f1g a finite number of cusps (2.4.1 in [DS]). To put a topology on the curve X(Γ), start by taking N = z H : Im(z) > M and take M f 2 g topology on H∗ with usual open sets in H plus the sets α(N ) (M > 0; α SL (Z)) M [ f1g 2 2 as a base of neighborhoods of the cusps. Now, give X(Γ) quotient topology. It can be proved that with this topology X(Γ) is a compact, connected Riemann surface (see 2.4.2 in [DS]). We use notation X0(N); X1(N) to denote X(Γ0(N)); X(Γ1(N)), respectively. 6 Elliptic curves A complex elliptic curve is a smooth, projective curve of genus 1 defined over complex numbers. It can be shown that an elliptic curve is a set of complex solutions of an equation of the form y2 = ax3 + bx + c where a; b; c C (see IV.4.6 in [Har] for details). 2 Each elliptic curve has a structure of an abelian group. A (1-dimensional) complex torus is a complex manifold of the form C=Λ where Λ is a lattice in C (a set of the form aZ + bZ for some a; b C linearly independent over Z). 2 With every complex torus we can associate Weierstrass function 1 1 1 } (z) = + + : Λ z2 (z w)2 !2 !2XΛ;!6=0 − 1 If Gk = !2Λ;!6=0 !2k is the Eisenstein series and g2(Λ) = 60G2(Λ), g3(Λ) = 140G3(Λ), then one can prove that P (}0 (z))2 = 4(} (z))3 g (Λ)} (z) g (Λ): Λ Λ − 2 Λ − 3 Hence we have an isomorphism between C=Λ and the curve E := (y2 = 4x3 g (Λ)x g (Λ)) − 2 − 3 given by z (} (z); }0 (z)). 7! Λ Λ This shows that complex elliptic curves and 1-dimensional complex tori are the same thing. By changing the coordinate system an elliptic curve can be put in the form y 2 = x3+Ax+B and then we can define its j-invariant as 1728 4A3 j(E) = · : 4A3 + 27B2 For the notion of a conductor of an elliptic curve we refer to IV.10 in [Silv2]. This will be used only once to state one of the versions of the Modularity Theorem. If E is an elliptic curve, let E[n] denote a subgroup of its n-torsion points that is such points P E that nP = 0. The l-adic Tate module of E is defined by setting T (E) = 2 l lim E[ln]. It can be proved that T (E) is isomorphic to Z Z , where Z denotes l-adic l l × l l integers (see III.7 of [Silv1]). When E is defined over Q (which is equivalent to j(E) Q) we have an action of 2 Gal(Q=Q) on each E[ln] and hence also on T (E). This gives a representation ρ : Gal(Q=Q) l E;l ! Aut(T (E)). After choosing a Z -basis for T (E) and using the inclusion Z Q we obtain a l l l l ⊂ l representation ρ : Gal(Q=Q) GL (Q ). E;l ! 2 l 7 Now, we define an L-function associated with an elliptic curve E over Q. Let ap(E) = e p+1 E(F ) where E is a reduction of E at p.
Load more

Recommended publications
  • Arxiv:2003.01675V1 [Math.NT] 3 Mar 2020 of Their Locations
    DIVISORS OF MODULAR PARAMETRIZATIONS OF ELLIPTIC CURVES MICHAEL GRIFFIN AND JONATHAN HALES Abstract. The modularity theorem implies that for every elliptic curve E=Q there exist rational maps from the modular curve X0(N) to E, where N is the conductor of E. These maps may be expressed in terms of pairs of modular functions X(z) and Y (z) where X(z) and Y (z) satisfy the Weierstrass equation for E as well as a certain differential equation. Using these two relations, a recursive algorithm can be used to calculate the q - expansions of these parametrizations at any cusp. Using these functions, we determine the divisor of the parametrization and the preimage of rational points on E. We give a sufficient condition for when these preimages correspond to CM points on X0(N). We also examine a connection between the al- gebras generated by these functions for related elliptic curves, and describe sufficient conditions to determine congruences in the q-expansions of these objects. 1. Introduction and statement of results The modularity theorem [2, 12] guarantees that for every elliptic curve E of con- ductor N there exists a weight 2 newform fE of level N with Fourier coefficients in Z. The Eichler integral of fE (see (3)) and the Weierstrass }-function together give a rational map from the modular curve X0(N) to the coordinates of some model of E: This parametrization has singularities wherever the value of the Eichler integral is in the period lattice. Kodgis [6] showed computationally that many of the zeros of the Eichler integral occur at CM points.
    [Show full text]
  • Supersingular Zeros of Divisor Polynomials of Elliptic Curves of Prime Conductor
    Kazalicki and Kohen Res Math Sci(2017) 4:10 DOI 10.1186/s40687-017-0099-8 R E S E A R C H Open Access Supersingular zeros of divisor polynomials of elliptic curves of prime conductor Matija Kazalicki1* and Daniel Kohen2,3 *Correspondence: [email protected] Abstract 1Department of Mathematics, p p University of Zagreb, Bijeniˇcka For a prime number , we study the zeros modulo of divisor polynomials of rational cesta 30, 10000 Zagreb, Croatia elliptic curves E of conductor p. Ono (CBMS regional conference series in mathematics, Full list of author information is 2003, vol 102, p. 118) made the observation that these zeros are often j-invariants of available at the end of the article supersingular elliptic curves over Fp. We show that these supersingular zeros are in bijection with zeros modulo p of an associated quaternionic modular form vE .This allows us to prove that if the root number of E is −1 then all supersingular j-invariants of elliptic curves defined over Fp are zeros of the corresponding divisor polynomial. If the root number is 1, we study the discrepancy between rank 0 and higher rank elliptic curves, as in the latter case the amount of supersingular zeros in Fp seems to be larger. In order to partially explain this phenomenon, we conjecture that when E has positive rank the values of the coefficients of vE corresponding to supersingular elliptic curves defined over Fp are even. We prove this conjecture in the case when the discriminant of E is positive, and obtain several other results that are of independent interest.
    [Show full text]
  • A Glimpse of the Laureate's Work
    A glimpse of the Laureate’s work Alex Bellos Fermat’s Last Theorem – the problem that captured planets moved along their elliptical paths. By the beginning Andrew Wiles’ imagination as a boy, and that he proved of the nineteenth century, however, they were of interest three decades later – states that: for their own properties, and the subject of work by Niels Henrik Abel among others. There are no whole number solutions to the Modular forms are a much more abstract kind of equation xn + yn = zn when n is greater than 2. mathematical object. They are a certain type of mapping on a certain type of graph that exhibit an extremely high The theorem got its name because the French amateur number of symmetries. mathematician Pierre de Fermat wrote these words in Elliptic curves and modular forms had no apparent the margin of a book around 1637, together with the connection with each other. They were different fields, words: “I have a truly marvelous demonstration of this arising from different questions, studied by different people proposition which this margin is too narrow to contain.” who used different terminology and techniques. Yet in the The tantalizing suggestion of a proof was fantastic bait to 1950s two Japanese mathematicians, Yutaka Taniyama the many generations of mathematicians who tried and and Goro Shimura, had an idea that seemed to come out failed to find one. By the time Wiles was a boy Fermat’s of the blue: that on a deep level the fields were equivalent. Last Theorem had become the most famous unsolved The Japanese suggested that every elliptic curve could be problem in mathematics, and proving it was considered, associated with its own modular form, a claim known as by consensus, well beyond the reaches of available the Taniyama-Shimura conjecture, a surprising and radical conceptual tools.
    [Show full text]
  • The Arbeitsgemeinschaft 2017: Higher Gross-Zagier Formulas Notes By
    The Arbeitsgemeinschaft 2017: Higher Gross-Zagier Formulas Notes by Tony Feng1 Oberwolfach April 2-8, 2017 1version of April 11, 2017 Contents Note to the reader 5 Part 1. Day One 7 1. An overview of the Gross-Zagier and Waldspurger formulas (Yunqing Tang) 8 2. The stacks Bunn and Hecke (Timo Richarz) 13 3. (Moduli of Shtukas I) (Doug Ulmer) 17 4. Moduli of Shtukas II (Brian Smithling) 21 Part 2. Day Two 25 5. Automorphic forms over function fields (Ye Tian) 26 6. The work of Drinfeld (Arthur Cesar le Bras) 30 7. Analytic RTF: Geometric Side (Jingwei Xiao) 36 8. Analytic RTF: Spectral Side (Ilya Khayutin) 41 Part 3. Day Three 47 9. Geometric Interpretation of Orbital Integrals (Yihang Zhu) 48 10. Definition and properties of Md (Jochen Heinloth) 52 Part 4. Day Four 57 11. Intersection theory on stacks (Michael Rapoport) 58 12. LTF for Cohomological Correspondences (Davesh Maulik) 64 µ 13. Definition and description of HkM;d: expressing Ir(hD) as a trace (Liang Xiao) 71 3 4 CONTENTS 14. Alternative calculation of Ir(hD) (Yakov Varshavsky) 77 Part 5. Day Five 83 15. Comparison of Md and Nd; the weight factors (Ana Caraiani) 84 16. Horocycles (Lizao Ye) 89 17. Cohomological spectral decomposition and finishing the proof (Chao Li) 93 NOTE TO THE READER 5 Note to the reader This document consists of notes I live-TEXed during the Arbeitsgemeinschaft at Oberwolfach in April 2017. They should not be taken as a faithful transcription of the actual lectures; they represent only my personal perception of the talks.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • The Role of the Ramanujan Conjecture in Analytic Number Theory
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, Number 2, April 2013, Pages 267–320 S 0273-0979(2013)01404-6 Article electronically published on January 14, 2013 THE ROLE OF THE RAMANUJAN CONJECTURE IN ANALYTIC NUMBER THEORY VALENTIN BLOMER AND FARRELL BRUMLEY Dedicated to the 125th birthday of Srinivasa Ramanujan Abstract. We discuss progress towards the Ramanujan conjecture for the group GLn and its relation to various other topics in analytic number theory. Contents 1. Introduction 267 2. Background on Maaß forms 270 3. The Ramanujan conjecture for Maaß forms 276 4. The Ramanujan conjecture for GLn 283 5. Numerical improvements towards the Ramanujan conjecture and applications 290 6. L-functions 294 7. Techniques over Q 298 8. Techniques over number fields 302 9. Perspectives 305 J.-P. Serre’s 1981 letter to J.-M. Deshouillers 307 Acknowledgments 313 About the authors 313 References 313 1. Introduction In a remarkable article [111], published in 1916, Ramanujan considered the func- tion ∞ ∞ Δ(z)=(2π)12e2πiz (1 − e2πinz)24 =(2π)12 τ(n)e2πinz, n=1 n=1 where z ∈ H = {z ∈ C |z>0} is in the upper half-plane. The right hand side is understood as a definition for the arithmetic function τ(n) that nowadays bears Received by the editors June 8, 2012. 2010 Mathematics Subject Classification. Primary 11F70. Key words and phrases. Ramanujan conjecture, L-functions, number fields, non-vanishing, functoriality. The first author was supported by the Volkswagen Foundation and a Starting Grant of the European Research Council. The second author is partially supported by the ANR grant ArShiFo ANR-BLANC-114-2010 and by the Advanced Research Grant 228304 from the European Research Council.
    [Show full text]
  • Andrew Wiles's Marvelous Proof
    Andrew Wiles’s Marvelous Proof Henri Darmon ermat famously claimed to have discovered “a truly marvelous proof” of his Last Theorem, which the margin of his copy of Diophantus’s Arithmetica was too narrow to contain. While this proof (if it ever existed) is lost to posterity, AndrewF Wiles’s marvelous proof has been public for over two decades and has now earned him the Abel Prize. According to the prize citation, Wiles merits this recogni- tion “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.” Few can remain insensitive to the allure of Fermat’s Last Theorem, a riddle with roots in the mathematics of ancient Greece, simple enough to be understood It is also a centerpiece and appreciated Wiles giving his first lecture in Princeton about his by a novice of the “Langlands approach to proving the Modularity Conjecture in (like the ten- early 1994. year-old Andrew program,” the Wiles browsing of Theorem 2 of Karl Rubin’s contribution in this volume). the shelves of imposing, ambitious It is also a centerpiece of the “Langlands program,” the his local pub- imposing, ambitious edifice of results and conjectures lic library), yet edifice of results and which has come to dominate the number theorist’s view eluding the con- conjectures which of the world. This program has been described as a “grand certed efforts of unified theory” of mathematics. Taking a Norwegian per- the most brilliant has come to dominate spective, it connects the objects that occur in the works minds for well of Niels Hendrik Abel, such as elliptic curves and their over three cen- the number theorist’s associated Abelian integrals and Galois representations, turies, becoming with (frequently infinite-dimensional) linear representa- over its long his- view of the world.
    [Show full text]
  • Modular Forms and Affine Lie Algebras
    Modular Forms Theta functions Comparison with the Weyl denominator Modular Forms and Affine Lie Algebras Daniel Bump F TF i 2πi=3 e SF Modular Forms Theta functions Comparison with the Weyl denominator The plan of this course This course will cover the representation theory of a class of Lie algebras called affine Lie algebras. But they are a special case of a more general class of infinite-dimensional Lie algebras called Kac-Moody Lie algebras. Both classes were discovered in the 1970’s, independently by Victor Kac and Robert Moody. Kac at least was motivated by mathematical physics. Most of the material we will cover is in Kac’ book Infinite-dimensional Lie algebras which you should be able to access on-line through the Stanford libraries. In this class we will develop general Kac-Moody theory before specializing to the affine case. Our goal in this first part will be Kac’ generalization of the Weyl character formula to certain infinite-dimensional representations of infinite-dimensional Lie algebras. Modular Forms Theta functions Comparison with the Weyl denominator Affine Lie algebras and modular forms The Kac-Moody theory includes finite-dimensional simple Lie algebras, and many infinite-dimensional classes. The best understood Kac-Moody Lie algebras are the affine Lie algebras and after we have developed the Kac-Moody theory in general we will specialize to the affine case. We will see that the characters of affine Lie algebras are modular forms. We will not reach this topic until later in the course so in today’s introductory lecture we will talk a little about modular forms, without giving complete proofs, to show where we are headed.
    [Show full text]
  • Fermat's Last Theorem
    Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have an infinite number of solutions.[1] The proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica; Fermat added that he had a proof that was too large to fit in the margin. However, there were first doubts about it since the publication was done by his son without his consent, after Fermat's death.[2] After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995; it was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016.[3] It also proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.[4] Contents The 1670 edition of Diophantus's Arithmetica includes Fermat's Overview commentary, referred to as his "Last Pythagorean origins Theorem" (Observatio Domini Petri Subsequent developments and solution de Fermat), posthumously published Equivalent statements of the theorem by his son.
    [Show full text]
  • Modular Dedekind Symbols Associated to Fuchsian Groups and Higher-Order Eisenstein Series
    Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series Jay Jorgenson,∗ Cormac O’Sullivan†and Lejla Smajlovi´c May 20, 2018 Abstract Let E(z,s) be the non-holomorphic Eisenstein series for the modular group SL(2, Z). The classical Kro- necker limit formula shows that the second term in the Laurent expansion at s = 1 of E(z,s) is essentially the logarithm of the Dedekind eta function. This eta function is a weight 1/2 modular form and Dedekind expressedits multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups Γ. The analogue of the eta function has a multiplier system that may be expressed in terms of a map S : Γ → R which we call a modular Dedekind symbol. We obtain detailed properties of these symbols by means of the limit formula. Twisting the usual Eisenstein series with powers of additive homomorphisms from Γ to C produces higher-order Eisenstein series. These series share many of the properties of E(z,s) though they have a more complicated automorphy condition. They satisfy a Kronecker limit formula and produce higher- order Dedekind symbols S∗ :Γ → R. As an application of our general results, we prove that higher-order Dedekind symbols associated to genus one congruence groups Γ0(N) are rational. 1 Introduction 1.1 Kronecker limit functions and Dedekind sums Write elements of the upper half plane H as z = x + iy with y > 0. The non-holomorphic Eisenstein series E (s,z), associated to the full modular group SL(2, Z), may be given as ∞ 1 ys E (z,s) := .
    [Show full text]