MODULAR FORMS and APPLICATIONS in NUMBER THEORY Contents 1. Modular Forms 1 1.1. Definitions 1 1.2. Connection to Number Theory
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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. -
Construction of Free Subgroups in the Group of Units of Modular Group Algebras
CONSTRUCTION OF FREE SUBGROUPS IN THE GROUP OF UNITS OF MODULAR GROUP ALGEBRAS Jairo Z. Gon¸calves1 Donald S. Passman2 Department of Mathematics Department of Mathematics University of S~ao Paulo University of Wisconsin-Madison 66.281-Ag Cidade de S. Paulo Van Vleck Hall 05389-970 S. Paulo 480 Lincoln Drive S~ao Paulo, Brazil Madison, WI 53706, U.S.A [email protected] [email protected] Abstract. Let KG be the group algebra of a p0-group G over a field K of characteristic p > 0; and let U(KG) be its group of units. If KG contains a nontrivial bicyclic unit and if K is not algebraic over its prime field, then we prove that the free product Zp ∗ Zp ∗ Zp can be embedded in U(KG): 1. Introduction Let KG be the group algebra of the group G over the field K; and let U(KG) be its group of units. Motivated by the work of Pickel and Hartley [4], and Sehgal ([7, pg. 200]) on the existence of free subgroups in the inte- gral group ring ZG; analogous conditions for U(KG) have been intensively investigated in [1], [2] and [3]. Recently Marciniak and Sehgal [5] gave a constructive method for produc- ing free subgroups in U(ZG); provided ZG contains a nontrivial bicyclic unit. In this paper we prove an analogous result for the modular group algebra KG; whenever K is not algebraic over its prime field GF (p): Specifically, if Zp denotes the cyclic group of order p, then we prove: 1- Research partially supported by CNPq - Brazil. -
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
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). -
Elliptic Curves, Modular Forms, and L-Functions Allison F
Claremont Colleges Scholarship @ Claremont HMC Senior Theses HMC Student Scholarship 2014 There and Back Again: Elliptic Curves, Modular Forms, and L-Functions Allison F. Arnold-Roksandich Harvey Mudd College Recommended Citation Arnold-Roksandich, Allison F., "There and Back Again: Elliptic Curves, Modular Forms, and L-Functions" (2014). HMC Senior Theses. 61. https://scholarship.claremont.edu/hmc_theses/61 This Open Access Senior Thesis is brought to you for free and open access by the HMC Student Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in HMC Senior Theses by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. There and Back Again: Elliptic Curves, Modular Forms, and L-Functions Allison Arnold-Roksandich Christopher Towse, Advisor Michael E. Orrison, Reader Department of Mathematics May, 2014 Copyright c 2014 Allison Arnold-Roksandich. The author grants Harvey Mudd College and the Claremont Colleges Library the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced ma- terials and notice is given that the copying is by permission of the author. To dis- seminate otherwise or to republish requires written permission from the author. Abstract L-functions form a connection between elliptic curves and modular forms. The goals of this thesis will be to discuss this connection, and to see similar connections for arithmetic functions. Contents Abstract iii Acknowledgments xi Introduction 1 1 Elliptic Curves 3 1.1 The Operation . .4 1.2 Counting Points . .5 1.3 The p-Defect . .8 2 Dirichlet Series 11 2.1 Euler Products . -
Modular Forms and the Hilbert Class Field
Modular forms and the Hilbert class field Vladislav Vladilenov Petkov VIGRE 2009, Department of Mathematics University of Chicago Abstract The current article studies the relation between the j−invariant function of elliptic curves with complex multiplication and the Maximal unramified abelian extensions of imaginary quadratic fields related to these curves. In the second section we prove that the j−invariant is a modular form of weight 0 and takes algebraic values at special points in the upper halfplane related to the curves we study. In the third section we use this function to construct the Hilbert class field of an imaginary quadratic number field and we prove that the Ga- lois group of that extension is isomorphic to the Class group of the base field, giving the particular isomorphism, which is closely related to the j−invariant. Finally we give an unexpected application of those results to construct a curious approximation of π. 1 Introduction We say that an elliptic curve E has complex multiplication by an order O of a finite imaginary extension K/Q, if there exists an isomorphism between O and the ring of endomorphisms of E, which we denote by End(E). In such case E has other endomorphisms beside the ordinary ”multiplication by n”- [n], n ∈ Z. Although the theory of modular functions, which we will define in the next section, is related to general elliptic curves over C, throughout the current paper we will be interested solely in elliptic curves with complex multiplication. Further, if E is an elliptic curve over an imaginary field K we would usually assume that E has complex multiplication by the ring of integers in K. -
Special Unitary Group - Wikipedia
Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Special unitary group In mathematics, the special unitary group of degree n, denoted SU( n), is the Lie group of n×n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U( n), consisting of all n×n unitary matrices. As a compact classical group, U( n) is the group that preserves the standard inner product on Cn.[nb 1] It is itself a subgroup of the general linear group, SU( n) ⊂ U( n) ⊂ GL( n, C). The SU( n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1] The simplest case, SU(1) , is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+ I, − I}. [nb 2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. Contents Properties Lie algebra Fundamental representation Adjoint representation The group SU(2) Diffeomorphism with S 3 Isomorphism with unit quaternions Lie Algebra The group SU(3) Topology Representation theory Lie algebra Lie algebra structure Generalized special unitary group Example Important subgroups See also 1 of 10 2/22/2018, 8:54 PM Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Remarks Notes References Properties The special unitary group SU( n) is a real Lie group (though not a complex Lie group). -
Foundations of Algebraic Geometry Class 41
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41 RAVI VAKIL CONTENTS 1. Normalization 1 2. Extending maps to projective schemes over smooth codimension one points: the “clear denominators” theorem 5 Welcome back! Let's now use what we have developed to study something explicit — curves. Our motivating question is a loose one: what are the curves, by which I mean nonsingular irreducible separated curves, finite type over a field k? In other words, we'll be dealing with geometry, although possibly over a non-algebraically closed field. Here is an explicit question: are all curves (say reduced, even non-singular, finite type over given k) isomorphic? Obviously not: some are affine, and some (such as P1) are not. So to simplify things — and we'll further motivate this simplification in Class 42 — are all projective curves isomorphic? Perhaps all nonsingular projective curves are isomorphic to P1? Once again the answer is no, but the proof is a bit subtle: we've defined an invariant, the genus, and shown that P1 has genus 0, and that there exist nonsingular projective curves of non-zero genus. Are all (nonsingular) genus 0 curves isomorphic to P1? We know there exist nonsingular genus 1 curves (plane cubics) — is there only one? If not, “how many” are there? In order to discuss interesting questions like these, we'll develop some theory. We first show a useful result that will help us focus our attention on the projective case. 1. NORMALIZATION I now want to tell you how to normalize a reduced Noetherian scheme, which is roughly how best to turn a scheme into a normal scheme. -
1 Second Facts About Spaces of Modular Forms
April 30, 2:00 pm 1 Second Facts About Spaces of Modular Forms We have repeatedly used facts about the dimensions of the space of modular forms, mostly to give specific examples of and relations between modular forms of a given weight. We now prove these results in this section. Recall that a modular form f of weight 0 is a holomorphic function that is invari- ant under the action of SL(2, Z) and whose domain can be extended to include the point at infinity. Hence, we may regard f as a holomorphic function on the compact Riemann surface G\H∗, where H∗ = H ∪ {∞}. If you haven’t had a course in com- plex analysis, you shouldn’t worry too much about this statement. We’re really just saying that our fundamental domain can be understood as a compact manifold with complex structure. So locally, we look like C and there are compatibility conditions on overlapping local maps. By the maximum modulus principle from complex analysis (see section 3.4 of Ahlfors for example), any such holomorphic function must be constant. We can use this fact, together with the fact that given f1, f2 ∈ M2k, then f1/f2 is a meromorphic function invariant under G (i.e. a weakly modular function of weight 0). Usually, one proves facts about the dimensions of spaces of modular forms using foundational algebraic geometry (the Riemann-Roch theorem, in particular) or the Selberg trace formula (an even more technical result relating differential geometry to harmonic analysis in a beautiful way). These apply to spaces of modular forms for other groups, but as we’re only going to say a few words about such spaces, we won’t delve into either, but rather prove a much simpler result in the spirit of the Riemann-Roch theorem. -
Chapter 1 GENERAL STRUCTURE and PROPERTIES
Chapter 1 GENERAL STRUCTURE AND PROPERTIES 1.1 Introduction In this Chapter we would like to introduce the main de¯nitions and describe the main properties of groups, providing examples to illustrate them. The detailed discussion of representations is however demanded to later Chapters, and so is the treatment of Lie groups based on their relation with Lie algebras. We would also like to introduce several explicit groups, or classes of groups, which are often encountered in Physics (and not only). On the one hand, these \applications" should motivate the more abstract study of the general properties of groups; on the other hand, the knowledge of the more important and common explicit instances of groups is essential for developing an e®ective understanding of the subject beyond the purely formal level. 1.2 Some basic de¯nitions In this Section we give some essential de¯nitions, illustrating them with simple examples. 1.2.1 De¯nition of a group A group G is a set equipped with a binary operation , the group product, such that1 ¢ (i) the group product is associative, namely a; b; c G ; a (b c) = (a b) c ; (1.2.1) 8 2 ¢ ¢ ¢ ¢ (ii) there is in G an identity element e: e G such that a e = e a = a a G ; (1.2.2) 9 2 ¢ ¢ 8 2 (iii) each element a admits an inverse, which is usually denoted as a¡1: a G a¡1 G such that a a¡1 = a¡1 a = e : (1.2.3) 8 2 9 2 ¢ ¢ 1 Notice that the axioms (ii) and (iii) above are in fact redundant. -
Lecture 16 : Algebraic Modular Forms Cont'd
Math 726: L-functions and modular forms Fall 2011 Lecture 16 : Algebraic Modular forms cont'd Instructor: Henri Darmon Notes written by: Maxime Turgeon In the last lecture, we gave two important examples of marked elliptic curves over a ring: 1. The universal elliptic curve 2 3 dx (C=h1; τi) = (y = 4x − g (τ)x − g (τ); )O ; OH 4 6 y H 2. The Tate elliptic curve × dt 2 3 dx C =hqi; = (y = x − a(q)x + b(q); )O ; t y D× OD× 1 1 n 2 1 n where a(q) = 3 (1 + 240 n=1 σ3(n)q ) and b(q) = 27 (1 − 504 n=1 σ5(n)q ). P P Z 1 Note that the Tate elliptic curve can be viewed as an elliptic curve over the ring [ 6 ]((q)). But with more care (for example, by introducing new parameters), one can show that it can also be viewed as an elliptic curve over Z((q)): dt dt ET ate; = Gm=hqi; : t Z((q)) t Z((q)) We also defined a weakly holomorphic algebraic modular form of weight k and level 1 to be a rule (E; !)R −! f(E; !)R 2 R; subject to two conditions, namely, that it should be compatible with base change (see Lecture 15) and that it should satisfy the following homogeneity condition f(E; λω) = λ−kf(E; !); for all λ 2 R×: C In particular, f ( =h1; τi; dz)OH 2 OH is the classical view of (weakly holomorphic) dt Z modular forms as holomorphic functions on H. -
MODULAR GROUP IMAGES ARISING from DRINFELD DOUBLES of DIHEDRAL GROUPS Deepak Naidu 1. Introduction the Modular Group SL(2, Z) Is
International Electronic Journal of Algebra Volume 28 (2020) 156-174 DOI: 10.24330/ieja.768210 MODULAR GROUP IMAGES ARISING FROM DRINFELD DOUBLES OF DIHEDRAL GROUPS Deepak Naidu Received: 28 October 2019; Revised: 30 May 2020; Accepted: 31 May 2020 Communicated by A. C¸i˘gdem Ozcan¨ Abstract. We show that the image of the representation of the modular group SL(2; Z) arising from the representation category Rep(D(G)) of the Drinfeld double D(G) is isomorphic to the group PSL(2; Z=nZ) × S3, when G is either the dihedral group of order 2n or the dihedral group of order 4n for some odd integer n ≥ 3. Mathematics Subject Classification (2020): 18M20 Keywords: Drinfeld double, modular tensor category, modular group, con- gruence subgroup 1. Introduction The modular group SL(2; Z) is the group of all 2 × 2 matrices of determinant 1 whose entries belong to the ring Z of integers. The modular group is known to play a significant role in conformal field theory [3]. Every two-dimensional rational con- formal field theory gives rise to a finite-dimensional representation of the modular group, and the kernel of this representation has been of much interest. In particu- lar, the question whether the kernel is a congruence subgroup of SL(2; Z) has been investigated by several authors. For example, A. Coste and T. Gannon in their paper [4] showed that under certain assumptions the kernel is indeed a congruence subgroup. In the present paper, we consider the kernel of the representation of the modular group arising from Drinfeld doubles of dihedral groups.