DOCSLIB.ORG

Classical Modular Forms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Classical Modular Forms Classical Modular Forms T.N. Venkataramana∗ School of Mathematics, Tata Institute of Fundamental Research, Colaba, Mumbai, India Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August 2000 LNS0821002 ∗[email protected] Contents 1 Introduction 43 2 A Fundamental Domain for SL(2, Z) 43 3 Modular Forms; Definition and Examples 49 4 Modular Forms and Representation Theory 54 5 Modular Forms and Hecke Operators 62 6 L-functions of Modular Forms 70 Classical Modular Forms 43 1 Introduction The simplest kind of automorphic forms (apart from Gr¨ossencharacters, which will also be discussed in this conference) are the “elliptic modular forms”. We will study modular forms and their connection with automor- phic forms on GL(2), in the sense of representation theory. Modular forms arise in many contexts in number theory, e.g. in questions involving representations of integers by quadratic forms, and in expressing elliptic curves over Q as quotients Jacobians of modular curves (this was the crucial step in the proof of Fermat’s Last Theorem by Andrew Wiles), etc. The simplest modular forms are those on the modular group SL(2, Z) and we will first define modular forms on SL(2, Z). To begin with, in section 2, we will describe a fundamental domain for the action of SL(2, Z) on the upper half plane h. The fundamental domain will be seen to parametrise isomorphism classes of elliptic curves. In section 3, we will define modular forms for SL(2, Z) and construct some modular forms, by using the functions E4 and E6 which we encounter already in the section on elliptic curves. In section 4, a representation theoretic interpretation of modular forms will be given, which will enable us to think of them as automorphic forms on GL(2, R). In section 5, we will give an adelic interpretation of modular forms. This will enable us to think of Hecke operators as convolution operators in the Hecke algebra; using this, we show the commutativity of the Hecke operators. We will also prove a special case of the Multiplicity One theorem. 2 A Fundamental Domain for SL(2, Z) Notation 2.1 Denote by h the “Poincar´eupper half-plane” i.e. the space of complex numbers whose imaginary part is positive: h = {z ∈ C; z = x + iy, x, y ∈ R, y> 0}. If z ∈ C, denote respectively by Re(z) and Im(z) the real and imaginary parts of z. On the upper half plane h, the group GL(2, R)+ of real 2 × 2 matrices a b with positive determinant operates as follows: let g = ∈ GL(2, R)+, c d and let z ∈ h. Set g(z) = (az + b)/(cz + d). Notice that if cz + d = 0 44 T.N. Venkataramana and c 6= 0 then, z = −d/c is real, which is impossible since z has positive imaginary part. Thus, the formula for g(z) makes sense. Observe that Im(g(z)) = Im(z)(det(g))/ | cz + d |2 . (1) The equation (1) shows that the map (g, z) 7→ g(z) takes GL(2, R)+ × h into h. One checks immediately that this map gives an action of GL(2, R)+ on the upper half plane h. Note also that | cz + d |2= c2y2 + (cx + d)2. (2) Therefore, | cz + d |2≥ y2 or 1 according as | c |= 0 or nonzero. Therefore, 2 Im(γ(z)) ≤ y/ min{1,y } ∀γ ∈ Γ0 ⊂ SL(2, Z), (3) 2 2 where min{1,y } denotes the minimum of 1 and y and Γ0 ⊂ SL(2, Z) is 1 1 0 1 the group generated by the elements T = ( 0 1 ) and S = ( −1 0 ). Later we will see that Γ0 is actually SL(2, Z). The element T acts on the upper half plane h by translation by 1: 1 1 T (z)= (z)= z + 1 ∀z ∈ h. (4) 0 1 Similarly, the element S acts by inversion: 0 1 S(z)= (z)= −1/z ∀z ∈ h. (5) −1 0 Consider the set F = {z ∈ h; −1/2 < Re(z) ≤ 1/2, | z |≥ 1, and 0 ≤ Re(x)} if | z |= 1. Theorem 2.2 Given z ∈ h there is a unique point z0 ∈ F and an element γ ∈ SL(2, Z) such that γ(z) = z0. Moreover, given γ ∈ SL(2, Z), we have γ(F ) ∩ F = φ unless γ lies in a finite set ( of elements of SL(2, Z) which fix the point ω = 1/2+ i31/2/2 ∈ h or i ∈ h). [one then says that F is a fundamental domain for the action of SL(2, Z) on the upper half plane h]. Proof We will first show that any point z on the upper half plane can be translated by an element of the subgroup Γ0 of SL(2, Z) (generated by 1 1 0 1 T ( 0 1 ) and S = ( −1 0 )) into a point in the “fundamental domain” F . Classical Modular Forms 45 Now, given a real number x, there exists an integer k such that −1/2 < x + k ≤ 1/2. Therefore, equation (4) shows that given z ∈ h there exists an integer k such that the real part x′ of T k(z) satisfies the inequalities −1/2 <x′ ≤ 1/2. Let y denote the imaginary part of z and denote by Sz the set Sz = {γ(z); γ ∈ Γ0, Im(γ(z)) ≥ y , −1/2 < Re(γ(z)) ≤ 1/2} . We will first show that Sz is nonempty and finite. Let k be as in the pre- vious paragraph. Then −1/2 < Re(T k(z)) ≤ 1/2 and Im(T k(z) = Im(z); k therefore, T (z) lies in Sz and Sz is nonempty. Now, equation (3) shows that the imaginary parts of elements of the set 2 Sz are all bounded from above by y/ min 1,y . By definition, the imaginary parts of points on Sz are bounded from below by y. The definition of Sz shows that Sz is a relatively compact subset of h. We get from (3) that 2 2 a b | cz + d | ≤ 1; now (2) shows that | c |≤ 1/y . Suppose γ ∈ γ0 = ( c d ) is 2 such that γ(z) ∈ Sz then, c is bounded by 1/y and is in a finite set. The fact that cz +d is bounded now shows that d also lies in a finite set. Since Sz is relatively compact in h, it follows that γ(z) = (az + b)/(cz + d) is bounded for all γ(z) ∈ Sz; therefore, az + b is bounded as well, and hence a and b run through a finite set. We have therefore proved that Sz is finite. Let y0 be the supremum of the imaginary parts of the elements of the ′ ′ finite set Sz; let S1 = {z ∈ Sz; Im(z )= y0} and let z0 ∈ S1 be an element whose real part is maximal among elements of S1. We claim that z0 ∈ F . ′ ′ ′ First observe that if z ∈ Sz then S(z ) = −1/z has imaginary part y0/ | 2 ′ ′ 2 ′ 2 z | = Im(z )/ | z | ≤ y0 whence | z | ≥ 1. If | z0 |> 1, then it is immediate from the definitions of F and Sz that z0 ∈ F . Suppose that | z0 |= 1. Then, S(z0)= −1/z0 also has absolute value 1, its imaginary part is y0 and its real part is the negative of Re(z0); hence S(z0) ∈ S1. The maximality of the real part of z0 among elements of S1 now implies that Re(z0) ≥ 0. Therefore, z0 ∈ F . We have proved that every element z0 may be translated by an element of Γ0 into a point in the fundamental domain F . −1 a b Suppose now that z ∈ γ (F )∩F for some γ ∈ SL(2, Z). Write γ = ( c d ) with a, b, c, d ∈ Z and ad − bc = 1. Suppose that Im(γ(z)) ≥ Im(z) = y (otherwise, replace z by γ(z)). Then, by (3) one gets (cx + d)2 + c2y2 ≤ 1. (6) Since z ∈ F , we have x2 + y2 ≥ 1 and 0 ≤ x ≤ 1/2. Therefore y2 ≥ 3/4 and (1 ≥)c2y2 ≥ c24/3. (7) 46 T.N. Venkataramana This shows that c2 ≤ 1 since c is an integer. Suppose c = 0. Then, ad = 1, a, d ∈ Z and we may assume (by multiplying by the matrix −Id [minus identity] if necessary) that d = 1. 1 b Hence γ = ( 0 1 ). Then, γ(z)= z + b ∈ D which means that 0 ≤ x + b ≤ 1/2 and 0 ≤ x ≤ 1/2. Thus, −1/2 ≤ b ≤ 1/2, i.e. b = 0 and γ is the identity matrix. The other possibility is c2 = 1, and by multiplying by the matrix −Id (minus identity) we may assume that c = 1. Suppose first that d = 0. Then, bc = −1 whence b = −1. Now, (7) shows that x2 + y2 ≤ 1. Moreover, γ(z)= az + b/z = a + bz/ | z |2= a − z whence its real part is a − x which lies between 0 and 1/2. Since 0 ≤ x ≤ 1, 0 −1 it follows that 0 ≤ a ≤ 1. If a = 0 then γ = ( 1 0 ) and lies in the isotropy 1 −1 of the point i ∈ h. If a = 1 then, γ = ( 1 0 ) which lies in the isotropy of the point ω = 1/2+ i31/2/2. We now examine the remaining case of c = 1 and d 6= 0. From (6) we get (x + d)2 + y2 ≤ 1. If d ≥ 1 then the inequality 0 ≤ x ≤ 1/2 shows that 1 ≤ d ≤ x + d which contradicts the inequality (x + d)2 + y2 ≤ 1, which is impossible.
Load more

Recommended publications
  • 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]
  • 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]
  • 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]
  • 18.785 Notes
    Contents 1 Introduction 4 1.1 What is an automorphic form? . 4 1.2 A rough definition of automorphic forms on Lie groups . 5 1.3 Specializing to G = SL(2; R)....................... 5 1.4 Goals for the course . 7 1.5 Recommended Reading . 7 2 Automorphic forms from elliptic functions 8 2.1 Elliptic Functions . 8 2.2 Constructing elliptic functions . 9 2.3 Examples of Automorphic Forms: Eisenstein Series . 14 2.4 The Fourier expansion of G2k ...................... 17 2.5 The j-function and elliptic curves . 19 3 The geometry of the upper half plane 19 3.1 The topological space ΓnH ........................ 20 3.2 Discrete subgroups of SL(2; R) ..................... 22 3.3 Arithmetic subgroups of SL(2; Q).................... 23 3.4 Linear fractional transformations . 24 3.5 Example: the structure of SL(2; Z)................... 27 3.6 Fundamental domains . 28 3.7 ΓnH∗ as a topological space . 31 3.8 ΓnH∗ as a Riemann surface . 34 3.9 A few basics about compact Riemann surfaces . 35 3.10 The genus of X(Γ) . 37 4 Automorphic Forms for Fuchsian Groups 40 4.1 A general definition of classical automorphic forms . 40 4.2 Dimensions of spaces of modular forms . 42 4.3 The Riemann-Roch theorem . 43 4.4 Proof of dimension formulas . 44 4.5 Modular forms as sections of line bundles . 46 4.6 Poincar´eSeries . 48 4.7 Fourier coefficients of Poincar´eseries . 50 4.8 The Hilbert space of cusp forms . 54 4.9 Basic estimates for Kloosterman sums . 56 4.10 The size of Fourier coefficients for general cusp forms .
    [Show full text]
  • 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.
    [Show full text]
  • Elliptic Curves and Modular Forms
    ALGANT Master Thesis - July 2018 Elliptic Curves and Modular Forms Candidate Francesco Bruzzesi Advisor Prof. Dr. Marc N. Levine Universit´adegli Studi di Milano Universit¨atDuisburg{Essen Introduction The thesis has the aim to study the Eichler-Shimura construction associating elliptic curves to weight-2 modular forms for Γ0(N): this is the perfect topic to combine and develop further results from three courses I took in the first semester of the academic year 2017-18 at Universit¨atDuisburg-Essen (namely the courses of modular forms, abelian varieties and complex multiplication). Chapter 1 gives a brief overview on the algebraic geometry results and tools we will need along the whole thesis. Chapter 2 introduces and develops the theory of elliptic curves, firstly as an algebraic curve over a generic field, and then focusing on the fields of complex and rational numbers, in particular for the latter case we will be able to define an L-function associated to an elliptic curve. As we move to Chapter 3 we shift our focus to the theory of modular forms. We first treat elementary results and their consequences, then we seehow it is possible to define the canonical model of the modular curve X0(N) over Q, and the integrality property of the j-invariant. Chapter 4 concerns Hecke operators: Shimura's book [Shi73 Chapter 3] introduces the Hecke ring and its properties in full generality, on the other hand the other two main references for the chapter [DS06 Chapter 5] and [Kna93 Chapters XIII & IX] do not introduce the Hecke ring at all, and its attributes (such as commutativity for the case of interests) are proved by explicit computation.
    [Show full text]
  • 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.
    [Show full text]
  • Modular Functions and Modular Forms
    Modular Functions and Modular Forms J. S. Milne Abstract. These are the notes for Math 678, University of Michigan, Fall 1990, exactly as they were handed out during the course except for some minor revisions andcorrections. Please sendcomments andcorrections to me at [email protected]. Contents Introduction 1 Riemann surfaces 1 The general problem 1 Riemann surfaces that are quotients of D.1 Modular functions. 2 Modular forms. 3 Plane affine algebraic curves 3 Plane projective curves. 4 Arithmetic of Modular Curves. 5 Elliptic curves. 5 Elliptic functions. 5 Elliptic curves and modular curves. 6 Books 7 1. Preliminaries 8 Continuous group actions. 8 Riemann surfaces: classical approach 10 Riemann surfaces as ringed spaces 11 Differential forms. 12 Analysis on compact Riemann surfaces. 13 Riemann-Roch Theorem. 15 The genus of X.17 Riemann surfaces as algebraic curves. 18 2. Elliptic Modular Curves as Riemann Surfaces 19 c 1997 J.S. Milne. You may make one copy of these notes for your own personal use. i ii J. S. MILNE The upper-half plane as a quotient of SL2(R)19 Quotients of H 21 Discrete subgroups of SL2(R)22 Classification of linear fractional transformations 24 Fundamental domains 26 Fundamental domains for congruence subgroups 29 Defining complex structures on quotients 30 The complex structure on Γ(1)\H∗ 30 The complex structure on Γ\H∗ 31 The genus of X(Γ) 32 3. Elliptic functions 35 Lattices and bases 35 Quotients of C by lattices 35 Doubly periodic functions 35 Endomorphisms of C/Λ36 The Weierstrass ℘-function 37 The addition formula 39 Eisenstein series 39 The field of doubly periodic functions 40 Elliptic curves 40 The elliptic curve E(Λ) 40 4.
    [Show full text]
  • On H 3 2.2. Fundamental Domain6 2.3
    ORTHOGONAL DECOMPOSITION OF MODULAR FORMS KARHAN K. KAYAN Abstract. The aim of this paper is to prove that the space of modular forms can be orthogonally decomposed into the space of Eisenstein series and the space of cusp forms. However, we assume no previous knowledge of modular forms and build them from ground up. Along the way, we develop the ma- chinery that is required to prove the decomposition theorem and delve into the theory of Hecke operators, Poincar´eseries, and Petersson inner product. Contents 1. Introduction1 2. Modular Forms2 2.1. Action of SL2(Z) on H 3 2.2. Fundamental Domain6 2.3. Modular Forms and Lattices6 2.4. Dimension of Mk 7 3. Fourier Expansion of Modular Forms8 4. Cusp Forms 10 5. Petersson Inner Product 11 6. Poincar´eSeries 12 7. Hecke Operators 15 7.1. Introduction to Hecke Operators 15 7.2. Eigenfunctions of Hecke operators 18 7.3. Hecke operators are Hermitian 19 8. Orthogonal Decomposition of Mk(Γ) 20 Acknowledgments 23 References 23 1. Introduction The study of modular forms dates back to the nineteenth century when they were developed to study elliptic functions. Today they are largely known for the central role they play in number theory, the proof of Fermat's last theorem being just an example. Granted, it would be unfair to overlook their connection with the rest of mathematics. In the words of Alain Connes: Whatever the origin of one's journey, one day, if one walks far enough, one is bound to stumble on a well-known town: for in- stance, elliptic functions, modular forms, or zeta functions.
    [Show full text]