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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. We try to take an intermediate approach to the subject and rephrase everything just in terms of SL2(Z) and congruence subgroups. At the end of the chapter we will be able to associate an L-function to a cusp form for Γ0(N). Lastly in Chapter 5 we are ready to illustrate how to obtain an elliptic curve from a weight-2 cusp form for Γ0(N), making use of the theory of abelian varieties. i ii Notes and References Chapter 1: The organization of Sections 1.1 and 1.2 is based on [Sil09, Chapter 1 & Chapter 2]. The first statement of Proposition 1.10 is taken from [DS06, Proposition 7.2.6]; Proposition 1.14 is from [Kna93, Proposition 11.43]; Proposition 1.22 comes from [Was03, Proposition C.2]. The main references for these two sections are [Har77], [Mir95] and [Sha77]. Section 1.3 treats some standard results on the Jacobian variety associated to a Riemann surface. The main references are [Kir92] and [Mir95]. Chapter 2: Section 2.1 is organized as [Sil09, Chapter 3]. Proposition 2.1 is taken from [Sil09, III.1.4(i)]; Proposition 2.3 and Proposition 2.5 are based on filling the detailsKna93 of[ , Theorem 11.57 and Theorem 11.58] (respectively); Proposition 2.12 comes from [Sil09, Theorem III.4.10]; Lemma 2.15 is from [Kna93, Lemma 11.63]; Theorem 2.16 and Theorem 2.17 try to fill the details of [Kna93, Theorem 11.64 and Theorem 11.66] (respectively). The main refer- ence for Section 2.2 is [Shi73, Chapter 4]. For Section 2.3 we followed [Kna93, Chapter X]. Chapter 3: Some of the results arise as homeworks and/or are taken from the course of modular forms mentioned in the Introduction above and most of the results are standards: main references are [DS06], [Miy89] and [Lan76]. However Section 3.2.2 is taken from [Kna93, Chapter XI, Section 8]. Chapter 4: The structure of the Chapter is as in [Shi73, Chapter 3]. Propo- sition 4.5 is from [Shi73, Proposition 3.8]; Lemma 4.9 is from[Shi73, Lemma 3.12]; Proposition 4.10 is from [Shi73, Proposition 3.14]; Lemma 4.40 and 4.41 are respectively from [Shi73, Lemma 3.61 and 3.62]. Results from Theorem 4.33 to Proposition 4.39 are taken from [Kna93, Chapter XI, Section 5]. Chapter 5: The organization of the whole chapter is as in [Kna93, Chap- ter XI, Sections 10 & 11]. For Section 5.1 we used as references [Lan59] and [Swi74]. Section 5.2 is from the last part of [Kna93, Chapter XI, Section 10]; Proposition 5.15 is taken from [Kna93, Theorem 11.74]. iii Notation Z; Q; R; C integers, rationals, reals, complex numbers Fp the field with p elements H complex upper half plane Re(z); Im(z) real, imaginary part of z K algebraic closure of the field K Aut(K=L) automorphism group of K fixing the subfield L ⊂ K Mn(K) n × n matrices with coefficients in K GLn(K) invertible n × n matrices with coefficients in K SLn(K) n × n matrices with coefficients in K and determinant 1 [A : B] index of B in A or degree of A in B W _ dual space of the vector space W A× group of invertible elements of the ring A #S cardinality of the set S iv Acknowledgements First and foremost, I would like to express my gratitude to Professor Dr. Marc N. Levine, who accepted to supervise my study in this topic and patiently spent time to enlight me with his deep mathematical insights. He motivated me to do always better. I want to thank all the people I met in the last two year in the Algant Master, both Professors and students. In particular Bob, John and Francesco: we shared every moment, all the laughs and all the dissapointments. We constantly helped each other, both in life and in university. We all know how frustating it feels to study math sometimes, and I am thankful that we overcome the difficulties of these two years together. A special thanks goes to Federica, who had to bear me every single day of this stressful period. Even though she had to listen to my complaints, she made me smile in every situation. Thank you for having coped with all my struggles and problems. Also I would like to mention my dear friend Antonio, the person who helped me to develop a taste for number theory, I couldn't ask for a better mentor and friend. No matter how long we don't see each other, we always have a great time and connection. I am thankful to all my hometown friends and relatives. You are the reason why I keep coming back home and on all occasions it feels like time didn't pass. Last, but certainly most important, I want to thank my parents Lanfranco and Laura for providing me with unfailing support and continuous encourage- ment throughout my years of study. This accomplishment would not have been possible without them: words cannot really describe how grateful I am. Francesco Contents Introduction i Notes and References . ii Notation . iii Ackowledgements . iv 1 Algebraic Geometry 1 1.1 Algebraic Varieties . .1 1.2 Algebraic Curves . .6 1.2.1 Fields of positive characteristic . 10 1.2.2 Riemann-Roch theorem . 13 1.3 Riemann Surfaces . 16 2 Elliptic Curves 19 2.1 Elliptic curves over an arbitrary field . 19 2.1.1 Weierstrass form & abstract elliptic curves . 19 2.1.2 Isogenies . 24 2.2 Elliptic curves over C ........................ 27 2.2.1 The Weierstrass }-function . 29 2.2.2 Isogenies over C ....................... 33 2.2.3 Automorphisms of an elliptic curve . 35 2.3 Elliptic curves over Q ........................ 36 2.3.1 L-function associated to an elliptic curve . 37 2.3.2 Hasse theorem . 38 3 Modular Forms 40 3.1 Modular forms for SL2(Z)...................... 40 3.1.1 Functions of lattices . 40 3.1.2 The action of SL2(Z) on H ................. 42 3.1.3 Divisors of modular functions . 45 3.1.4 The space of modular forms . 49 3.1.5 The modular curve X0(1) . 52 3.2 Congruence subgroups . 58 3.2.1 Modular functions of higher level . 63 3.2.2 The canonical model of X0(N) over Q ........... 67 3.3 Integrality of the j-invariant . 69 3.3.1 j(z) is an algebraic number . 70 3.3.2 j(z) is integral . 71 v CONTENTS vi 4 Hecke Operators 74 4.1 The Hecke ring . 74 4.1.1 The structure of R(Γ; ∆) . 76 4.2 Action on modular functions . 83 4.2.1 Hecke operators on SL2(Z)................. 84 4.2.2 Hecke operators on congruence subgroups . 86 4.3 L-function of a cusp form . 93 5 Eichler-Shimura Theory 97 5.1 Complex abelian varieties and Jacobian varieties . 97 5.2 Technical results . 101 5.3 Elliptic curves associated to weight-2 cusp forms . 104 5.3.1 Perspective . 108 Bibliography 109 Chapter 1 Algebraic Geometry Throughout this whole chapter, let K0 denote a field and K be an alge- braically closed field containing0 K 1.1 Algebraic Varieties Definition. Define the affine n-dimensional space over K as n n A = A (K) = fP = (x1; :::; xn) j xi 2 Kg n Similarly, define the set of K0-points (or K0-rational points) of A as the set n n A (K0) = fP = (x1; :::; xn) 2 A j xi 2 K0g n Remark 1.1. We have an action of the Galois group Gal(K=K0) on A : let n σ σ σ σ 2 Gal(K=K) and P 2 A , then P = (x1 ; :::; xn). It follows that n n σ A (K0) = fP 2 A j P = P 8σ 2 Gal(K=K0)g Recall that by the Hilbert basis theorem any polynomial ring over a field is a Noetherian ring, thus every ideal is finitely generated.
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