DOCSLIB.ORG

Elliptic Modular Forms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Elliptic Modular Forms Elliptic modular forms Hironori Shiga Version May 10, 2008 Contents 1 SL2(Z) and elliptic curves 2 1.1 SL2(Z) and the moduli of complex tori . 2 1.2 The Fundamental region and a system of generators . 3 1.3 The Weierstrass } function . 5 1.4 Nonsingular cubics and the invariant j ............................. 9 1.5 Elliptic modular function j(¿).................................. 11 2 Modular forms for SL2(Z) 14 2.1 Cusps . 14 2.2 Concept of modular forms . 15 2.3 Eisenstein series . 16 2.4 Discriminant form . 19 2.5 Eisenstein series E2(z)...................................... 19 2.6 Algebra M(¡)........................................... 21 2.7 The Dedekind ´ function . 24 3 Modular form for congruence subgroups 26 3.1 Geometry of congruence subgroups . 27 3.2 Principal congruence subgroup ¡(N).............................. 27 3.3 Recalling the Riemann-Roch theorem . 30 3.4 Dimension formula for congruence subgroups . 30 4 Hecke operators and Hecke eigen forms 34 4.1 Preparatory consideration . 34 4.2 Hecke operator T (n)....................................... 35 4.3 Hecke eigen form . 37 4.4 Examples . 39 5 Theta functions 39 6 An example: how it works the theory applied to number theoretic problems 40 1 1 SL2(Z) and elliptic curves 1.1 SL2(Z) and the moduli of complex tori De¯nition 1.1. For !1;!2 2 C ¡ f0g with ¿ = !2=!1 2= R, we de¯ne a lattice in C by ¤ = ¤(!1;!2) = fm!1 + n!2 : m; n 2 Zg: Proposition 1.1. We have 0 0 () 9 2 0 0 ¤(!1;!2) = ¤(!1;!2) M GL(2; Z); (!1;!2) = (!1;!2)M: Proposition 1.2. For ¿; ¿ 0 2 H = fz 2 C : Im (z) > 0g , we have ( ¶ a b a¿ + b ¤(1;¿ 0) = k¤(1;¿); (k 2 C ¡ f0g) () 9M = 2 SL (Z); ¿ 0 = : c d 2 c¿ + d De¯nition 1.2. We de¯ne a complex torus T (!1;!2) = C=¤(!1;!2);T (¿) = C=¤(1;¿): Theorem 1.1. ( ¶ a b a¿ + b T (¿) » T (¿ 0) () 9M = 2 SL (Z); ¿ 0 = : biholo c d 2 c¿ + d [proof]. ((=) is apparaent from the above Proposition. We show (=)). Let ¼(resp. ¼0) be the canonical projection from C to T (¿)( resp.T (¿ 0)). And let f : T (¿) ! T (¿ 0) be a biholomorphic map. We suppose f(O) = O0, with ¼(0) = O; ¼0(0) = O0. f 0 ~ has a lifting f1 : C ! T (¿ ) in a natural way. It induces a analytic function f : U ! C de¯ned in a neighborhood U of 0. And f~ has an analytic continuation on the whole plane C as a possibly multivalued analytic function. Here, by the Monodromy theorem f~ is single valued. f~ C - C ¼ ¼0 ? f ? T (¿) - T (¿ 0) Diagram 1：Lifting f~ of f If we consider f~¡1, by the Monodromy theorem again, we can show f~ is injective. So f~ : C ! f~(C) is a biholomorphic map. By the Riemann uniformization theorem, the image f~(C) cannot be a proper subdomain. So f~ : C ! C is a bijective map, and it is biholomorphic. By the Weierstrass singularity theorem we can show that Aut(H) is a group of nontrivial linear functions. Hence we may put f~(z) = kz; note that we supposed f(O) = O0. So we have ¤(1;¿ 0) = k¤(1;¿). q.e.d. 2 1.2 The Fundamental region and a system of generators Notations: P SL2(R) = SL2(R)= § I = Aut(H); ¡ := SL2(Z); ¡ := SL2(Z)= § I; ½ ¡ \ in general,( for a¶ subgroup G ¡; G := G=(< I > G); a b ¡(N) := f 2 ¡) : a ´ d ´ 1 (mod N ); b ´ c ´ 0 (mod N )g C ¡(N 2 Z+): c d De¯nition 1.3. ¡(N) is called the principal congruence subgroup of level N. Remark 1.1. Note ¡I2 = ¡(N) N > 2 . So ¡(2) = ¡(2)= § I; ¡(N) = ¡(N)(N > 2): De¯nition 1.4. ( ¶ a b ¡ (N) := f 2 ¡: c ´ 0 (mod N )g; 0 c d ( ¶ a b ¡ (N) := f 2 ¡ (N): a ´ d ´ 1 (mod N )g: 1 c d 0 De¯nition 1.5. Let G be a subgroup of ¡. G acts on H. Two points z1 and z2 are said to be G-equivalent, if we have z2 = g(z1) for some g 2 G. A closed region in H is said to be a fundamental region of G, if (1) Every point z 2 H is G-equivalent to a point in F . (2) Any two di®erent points in the interior of F are not G-equivalent. Theorem 1.2. The closed region 1 F = fz 2 H : jRe zj · and jzj ¸ 1g (1.1) 2 is a fundamental region for ¡. F i e2 Πi3 eΠi3 0 Re - 1 1 2 2 Fundamental region for ¡ [proof]. (i) F contains all representatives. Set ( ¶ ( ¶ 1 1 0 ¡1 T = ;S = : 0 1 1 0 3 ( ¶ a b G := hS; T i ½ ¡. For a ¯xed z 2 H and g = 2 G we have c d Im z Im g(z) = : jcz + dj2 2 2 fjcd + dj : c; d 2 Z ¡ f(0; 0)g; coprimeg has the minimum, so Im g(z) has the maximum. Let g0 realizes k § ¢ ¢ ¢ k0 2 j j · 1 the maximum, so do T g0 (k = 0; 1; ). And we can ¯nd g = T g0 G such that g(z) 2 . We have g(z) 2 F . In fact, if we have g(z) 2= F , it holds jg(z)j < 1. Take S(g(z)). We have Im g(z) Im S(g(z)) = > Im g(z): jg(z)j2 This is contradicts the maximality. So we could ¯nd a g 2 G such that g(z) 2 F . (ii) Equivalent points. ( ¶ a b Suppose z1; z2 are equivalent, and we have z2 = g(z1); g = 2 ¡ ¡ f§Ig. Assume Im z2 = p c d Im z1 ¸ j j ¢ 3 · j j · j j j j · 2 Im z1 . So we have c c Im z1 (c Re z1 + d) + ic Im z1 = cz1 + d 1. We can jcz1+dj 2 see that it must hold jcj · 1. By observing jcz1j · 1=2 we have jdj · 1. So we have only restricted possibilities: (a) c = 0; d = §1 =) g : az + b § ) ¡ 1 (b) c = 1; d = 0 = g : a z § ) ¡1 2¼i=3 (c) c = d = 1 = g : a + z+1 and z1 = e ¡ § ) ¡1 ¼i=3 (d) c = d = 1 = g : a + z¡1 and z1 = e . In any case z1 and z2 cannot stay inside F at the same time. -1 -1 F2 F z + 1 z - 1 1 i i e2 Πi3 eΠi3 ¬ ® - 1 1 - 1 1 2 0 2 2 0 2 Figures of case (c)(d) q.e.d. Proposition 1.3. Let z1; z2 2 @F . z1 and z2 are ¡-equivalent () (1) z2 ¡ z1 = §1; jRe z1j = jRe z2j = 1=2 or 1 (2) z2 = ¡ ; jz1j = jz2j = 1. z1 We use the notation Gz = fg 2 G : g(z) = zg, the isotropy group for z. Set ( ¶ ( ¶ 1 1 0 ¡1 T = ;S = : 0 1 1 0 4 Proposition 1.4. We have (i) ¡i = f§I; §Sg 2 2¼i=3 (ii) ¡! = f§I; §ST; §(ST ) g;! = e 2 2¼i=6 (iii) ¡¡! = f§I; §T S; §(TS) g; ¡! = e (iv) ¡z = §I; otherwise: [proof]. We can solve cz2 + (d ¡ a)z ¡ b = 0; ad ¡ bc = 1 (a; b; c; d 2 Z) for z = i; !; ¡!2. Corollary 1.1. ¡ acts on H as a discrete group. Theorem 1.3. We have ¡ = hS; T: § idi: [proof]. Take an element γ 2 ¡. According to the part (i) of the proof of Theorem 1.2 we can ¯nd g 2 G = hS; T i so that we have gγ(2i) 2 F . Namely it holds gγ(2i) = 2i. So gγ 2 I2i = f§idg. Hence we obtain the required conclusion. q.e.d. 1.3 The Weierstrass } function De¯nition 1.6. Let ¤ = ¤(!1;!2) be a lattice in C. The Weierstrass } function is de¯ned by ( ¶ 1 X 1 1 }(z) = + ¡ : (1.2) z2 (z ¡ !)2 !2 !2¤¡f0g Remark 1.2. For a lattice ¤, 0 0 X 1 X X ( means the sum ) !k !2¤¡f0g is absolute convergent for k ¸ 3, and is conditional convergent for k = 2. Theorem 1.4. (i) }(z) is meromorphic on C and doubly periodic, i.e. }(z + !) = }(z)(! 2 ¤): (ii) }(z) has double poles at z = ! 2 ¤, and is an even function of order 2. 0 (iii) } is an odd function of order 3. It has zeros at half lattice points z = !1=2;!2=2; (!1 + !2)=2, and has a triple poles at z = ! 2 ¤. 0 0 [proof]. (i) We have } (z + !i) = } (z)(i = 1; 2). So }(z + !i) ¡ }(z) = ci (c : constant). By putting z = ¡!i=2 we get ci = 0. (ii)(iii) follows from the following general argument. Lemma 1.1. (i) Let f be a doubly periodic meromorphic function (=6 0) for a lattice ¤. f takes every complex value ® same times (counting multiplicities) in a period parallelogram Pa = f¸!1 + ¹!2 + a : 0 < ¸ < 1; 0 < ¹ < 1g provided f =6 ®; 1 on @Pa. 5 (ii) Let f be a doubly periodic meromorphic function. a1; : : : ; ar be its representatives of zeros, and b1; : : : ; br be the representatives of poles. Then we have a1 + ::: + ar ¡ (b1 + ::: + br) 2 ¤: [proof]. (i) By the argument principle Z 0 1 f f 2 g ¡ f 2 1g ¡ dz = ] z Pa : f(z) = ® ] z Pa : f(z) = : 2¼i @Pa f ® By the periodicity the left hand side is equal to 0.
Load more

Recommended publications
  • 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.
    [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]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [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]
  • 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.
    [Show full text]
  • ELLIPTIC CURVES and MODULAR FORMS Contents 1. Curves Over A
    ELLIPTIC CURVES AND MODULAR FORMS HARUZO HIDA Contents 1. Curves over a field 2 1.1. Plane curves 2 1.2. Tangent space and local rings 5 1.3. Projective space 8 1.4. Projective plane curve 9 1.5. Divisors 11 1.6. The theorem of Riemann–Roch 13 1.7. Regular maps from a curve into projective space 14 2. Elliptic curves 14 2.1. Abel’s theorem 15 2.2. Weierstrass Equations of Elliptic Curves 16 2.3. Moduli of Weierstrass Type 18 3. Modular forms 20 3.1. Elliptic curves over general rings 20 3.2. Geometric modular forms 22 3.3. Topological Fundamental Groups 23 3.4. Classical Weierstrass Theory 25 3.5. Complex Modular Forms 26 3.6. Hurwitz’s theorem, an application 28 4. Elliptic curves over p–adic fields 30 4.1. Power series identities 31 4.2. Tate curves 33 References 36 In this notes, we hope to go through basics of elliptic curves and modular curves in three steps: (1) As plane curves over a field (first 3 to 4 weeks); (2) As scheme/group functor over a ring (next 3 weeks); (3) Modular forms on modular curves. Date: August 24, 2015. The author is partially supported by the following NSF grant: DMS 0753991 and DMS 0854949. 1 ELLIPTIC CURVES AND MODULAR FORMS 2 Elliptic curves and modular curves are one of the most important objects studied in number theory. As everybody knows, the theory is a base of the proof by Wiles (through Ribet’s work) of Fermat’s last theorem, it supplies a fast prime factorization algorithm (cf.
    [Show full text]
  • Computing Modular Polynomials with Theta Functions
    ERASMUS MUNDUS MASTER ALGANT UNIVERSITA` DEGLI STUDI DI PADOVA UNIVERSITE´ BORDEAUX 1 SCIENCES ET TECHNOLOGIES MASTER THESIS COMPUTING MODULAR POLYNOMIALS WITH THETA FUNCTIONS ILARIA LOVATO ADVISOR: PROF. ANDREAS ENGE CO-ADVISOR: DAMIEN ROBERT ACADEMIC YEAR 2011/2012 2 Contents Introduction 5 1 Basic facts about elliptic curves 9 1.1 From tori to elliptic curves . .9 1.2 From elliptic curves to tori . 13 1.2.1 The modular group . 13 1.2.2 Proving the isomorphism . 18 1.3 Isogenies . 19 2 Modular polynomials 23 2.1 Modular functions for Γ0(m)................... 23 2.2 Properties of the classical modular polynomial . 30 2.3 Relations with isogenies . 34 2.4 Other modular polynomials . 36 3 Theta functions 39 3.1 Definitions and basic properties . 39 3.2 Embeddings by theta functions . 45 3.3 The functional equation of theta . 48 3.4 Theta as a modular form . 51 3.5 Building modular polynomials . 54 4 Algorithms and computations 57 4.1 Modular polynomials for j-invariants . 58 4.1.1 Substitution in q-expansion . 58 4.1.2 Looking for linear relations . 60 4.1.3 Evaluation-interpolation . 63 4.2 Modular polynomials via theta functions . 65 4.2.1 Computation of theta constants . 66 4.2.2 Looking for linear relations . 69 4.2.3 Evaluation-interpolation . 74 4 CONTENTS 4.3 Considerations . 77 5 Perspectives: genus 2 81 5.1 Theta functions . 81 5.1.1 Theta constants with rational characteristic . 81 5.1.2 Theta as a modular form . 83 5.2 Igusa invariants . 86 5.3 Computations .
    [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]
  • 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]