Elliptic Functions and Modular Forms

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1 Elliptic Functions and Modular Forms DRAFT, Release 0.8 March 22, 2018

2 2 I have prepared these notes for the students of my lecture. The lecture took place during the winter term 2017/18 at the mathematical department of LMU (Ludwig- Maximilians-Universität) at Munich. Compared to the oral lecture in class these written notes contain some additional material. Please report any errors or typos to wehler@math.lmu.de Release notes: Release 0.8: Chapter 5 revised, Chapter 6 added. Release 0.7: Introduction and preliminary version of Chapter 5 added. Release 0.6: Chapter 4 added. Update list of references. Release 0.5: Chapter 3 added. Update list of references. Minor revisions passim. Release 0.4: Chapter 2 added. Release 0.3: Update list of references. Release 0.2: Completion of Chapter 1. Update list of references. Release 0.1: Document created, Chapter 1.

3 Contents Introduction The elliptic functions of a lattice Doubly periodic meromorphic functions The -function of Weierstrass Abel s Theorem Complex tori Compact Riemann surfaces Isomorphism classes of complex tori and the modular group The modular curve of the modular group Elliptic curves Plane cubics The group structure Reduction mod p Modular forms of the modular group Eisenstein series The algebra of modular forms Hecke operators Application to Number Theory The τ-function of Ramanujan Complex multiplication and imaginary quadratic number fields Applications of the modular polynomial Outlook to Congruence Subgroups Modular forms of congruence subgroups The four squares theorem Modularity of elliptic curves v

4 vi Contents References Index

5 Introduction Modular Forms and the theory of Elliptic Functions are a classical domain from mathematics. At least, since the proof of Fermat s last conjecture the domain attracts widespread attention. The domain of Modular Forms integrates the three mathematical disciplines Complex Analysis, Algebraic Geometry, and Algebraic Number Theory. 1. Modular Forms Modular group Proper group operation Fundamental domain Congruence subgroup Modular curve Modular form Fourier explansion Eisenstein series and and j M k (Γ ) = H 0 (P 1,Ω Γ,k ) Weight formula Dimension formula Hecke operator Uniformization of elliptic curves over C Congruence subgroups Modularity of elliptic curves over Q 2. Complex Analysis Laurent series Residue theorem Elliptic functions Weierstrass -function Abel s theorem Lemma of Schwarz 1

6 2 Introduction Compact Riemann surface Complex torus Line bundle Divisor Riemann-Roch Theorem Serre duality Very ample line bundle 3. Algebraic Geometry Elliptic curve Embedding elliptic curves into P 2 Weierstrass equation Discriminant Group structure Class group Reduction mod prime ζ -function L-series 4. Algebraic Number Theory Fractional ideal Ideal class group Imaginary quadratic number field Ramanujan τ-function Class number of quadratic imaginary number fields Four squares theorem Legendre symbol Law of quadratic reciprocity

7 Chapter 1 The elliptic functions of a lattice The present chapter deals with complex analysis, i.e. we study holomorphic and meromorphic functions on domains in the complex plane. 1.1 Doubly periodic meromorphic functions A meromorphic function on a domain G C is a holomorphic function f : G \ S C with a discrete subset S G of poles of f - but no essential singularities. Definition 1.1 (Period of a meromorphic function). Consider a meromorphic function f on a domain G. A number ω C is a period of f if for all z G also z ± ω G and for all z G \ P f (z + ω) = f (z). It looks like Definition 1.1 excludes the poles from the definition of periodicity. A pole z 0 of f is an isolated singularity of f. Therefore in a neighbourhood of z 0 the function f expands into a convergent Laurent series around z 0 with coefficients a n = 1 2πi z z 0 =ε f (z) = a n (z z 0 ) n n n 0 f (z) dz with suitable ε > 0. (z z 0 ) n+1 3

8 4 1 The elliptic functions of a lattice If f has the period ω then the Laurent expansions of f around z 0 and z 0 + ω have the same coefficients because the integrands with z and z + ω give the same value. Hence periodicity of f includes also the poles. Example 1.2 (Periodic meromorphic functions). 1. The functions sin, cos : C C are holomorphic with period ω = 2π. 2. The tangent-function is a meromorphic function on C with the singularities π/2 + kπ, k Z and period ω = π. 3. The function C C,z e 2πiz, is holomorphic with period ω = 1. If a meromorphic function f has a period ω 0 one can ask for the set of all periods. Apparently elements of the form kω, k Z are also periods of f. How many independent periods of f exist? Definition 1.3 (Discrete subgroup). A subset Γ C is a discrete subgroup, if it satisfies the following conditions 1. Γ is a subgroup of the additive group (C,+). 2. Γ is discrete, i.e. a discrete topological space when equipped with the subspace topology of C: A neighbourhood U of 0 in C exists such that U Γ = {0}. Proposition 1.4 (Period group). Consider a non-constant meromorphic function f on a domain G C. The set Γ f of all periods of f is a discrete subgroup of C. It is named the period group of f. Proof. i) Apparently Γ f is a subgroup. ii) In case Γ f were not discrete, a sequence (ω n ) n N of periods ω n Γ f \ {0} would exist with lim n ω n = 0. Then for any arbitrary but fixed z 0 G \ P and for all n N f (z 0 ) = f (z 0 + ω n ) = f (ω n ). The identity theorem implies that f is constant, a contradiction, q.e.d.

9 1.1 Doubly periodic meromorphic functions 5 Proposition 1.5 (Discrete subgroups of (C, +)). Each discrete subgroup Γ (C, +) belongs to one of the following types: 1. Rank = 0: 2. Rank = 1: with a suitable element ω C. 3. Rank = 2: Γ = {0} Γ = Zω = {m ω : m Z} Γ = Zω 1 + Zω 2 = {m 1 ω 1 + m 2 ω 2 : m 1,m 2 Z} with ω 1,ω 2 C linearly independent over the base field R. The subgroup Γ is named a lattice. For a proof see [1, Chap. 7, Sect. 2.1]. Note. If we specify a lattice in the form Λ = Zω 1 + Zω 2 we will assume that the basis (ω 1,ω 2 ) is positively oriented, i.e. det (ω 1 ω 2 ) > 0. Definition 1.6 (Elliptic function). Consider a non-constant meromorphic function f on a domain G C with period group Γ f. If rank Γ f 1 then f is named periodic. If rank Γ f = 2 then f is named doubly periodic. Its period group Γ f = Zω 1 + Zω 2 is also named the period lattice of f. The subset P = {λ 1 ω 1 + λ 2 ω 2 : 0 λ 1,λ 2 < 1} is the fundamental period parallelogram of f. A doubly periodic meromorphic function defined on C is an elliptic function with respect to a lattice Λ if its period group Γ f contains Λ, i.e. all lattice points ω Λ are periods of f. Proposition 1.7 (Constant elliptic functions). Each holomorphic elliptic function is constant. f : C C

10 6 1 The elliptic functions of a lattice Proof. An entire function f is holomorphic and attends the maximum of its modulus on the closure of its period parallelogram which is compact set. Being a bounded entire function, f is constant according to Liouville s theorem, q.e.d. Proposition 1.8 (Residue theorem for elliptic functions). An elliptic function f with period parallelogram P satisfies res ζ ( f ) = 0. ζ P Proof. The boundary P comprises only finitely many poles of f. Therefore we can choose a number a C such the boundary of the translated period parallelogram P a := a + P contains no poles of f. The residue theorem implies 1 f (z) dz = 2πi P a res ζ ( f ) = res ζ ( f ). ζ P a ζ P The integrand on the left-hand side is doubly periodic, therefore the integral vanishes, q.e.d. Corollary 1.9 (Poles of elliptic functions). There are no elliptic functions f with one pole of order = 1 modulo the period lattice Γ f, and no other pole modulo Γ f. Proof. A pole of order 1 at a C has res a ( f ) 0, which contradicts Proposition 1.8, q.e.d. Corollary 1.10 (Counting poles and zeros of elliptic functions). Any non-constant elliptic function f assumes modulo its period lattice Γ f each value a C { } with the same multiplicity. In particular, f has modulo Γ f the same number of poles and zeros. Proof. The case a reduces to the second claim by considering the function f a. It has the same poles as f. Therefore it suffices to show that f has the same number of zeros and poles, taken with multiplicity. We apply Proposition 1.8 to the meromorphic function f / f and obtain ( ) f res ζ = 0. f ζ P Each residue evaluates to ( ) f res ζ = f { k k f has a zero of order k at ζ f has a pole of order k at ζ q.e.d.

11 1.2 The -function of Weierstrass 7 Corollary 1.11 (Zeros of an elliptic function). There are no elliptic functions f with only one zero of order = 1 modulo the period lattice Γ f and no other zero modulo Γ f. Proof. The proof follows from Corollary 1.9 and 1.10, q.e.d. 1.2 The -function of Weierstrass In the present section we consider an arbitrary but fixed lattice Λ := Z ω 1 + Z ω 2. We use the notation Λ := Λ \ {0}. Moreover P denotes the fundamental period parallelogram of Λ. Lemma 1.12 (Lattice constants). The infinite series G Λ,k := 1 ω Λ ω k, with k N, k > 2, is absolutely convergent. Its values for k > 2 are named the lattice constants of Λ. Proof. We choose the exhaustion of Λ by the sequence (Λ n ) n N of subsets Λ n := {µ ω 1 + ν ω 2 Λ : µ,ν Z; µ, ν n and ( µ = n or ν = n)}, i.e. Λ = n N Λ n. Then card Λ n = 8n. A suitable constant c exists with ω c n for all n N and for all ω Λ n. Therefore 1 ω Λ n ω k 8 n ( ) 8 c k n k 1 c k n k 1. For k > 2 Therefore the claim follows from n=1 1 ω Λ ω k = n=1 1 <. nk 1 ( ) 1 ω Λ n ω k, q.e.d.

12 8 1 The elliptic functions of a lattice We will study elliptic functions with period lattice Λ. According to Corollary 1.11 a candidate f for the most simple example would have only one pole mod Λ, the pole having order = 2 and residue = 0. The Laurent expansion of f around z 0 = 0 would start f (z) = 1 z 2 + a 1 z + O(2). Being doubly periodic, f would have poles exactly at the points ω Λ. The following Theorem 1.13 shows that such a function actually exists. Theorem 1.13 (Weierstrass -function). The series (z) := 1 ( 1 z 2 + ω Λ (z ω) 2 1 ) ω 2 is absolutely and compactly convergent for all z C \Λ. It defines an even elliptic function with respect to Λ, named the Weierstrass -function of Λ. The pole set of is Λ, each pole has order = 2. If we consider only the first summand with fixed z then 1 (z ω) 2 1 ω 2. Therefore the series with these summands alone does not converge. We will show that the difference 1 (z ω) 2 1 ω 2 is proportional to Therefore the series (z) converges. 1 ω 3. Note: TeX reserves the separate symbol backslash wp to denote the Weierstrass function. Proof. i) Meromorphic with pole set Λ: For each compact set K C only finitely many summands of have a pole in K. We show that the remaining series converges absolutely and uniform on K: Choose an arbitrary but fixed R > 0. For z R and 2R ω holds ω z 0. 2

13 1.2 The -function of Weierstrass 9 The triangle inequality ω ω z ω = ω z ω z = z ω 2 implies 1 (z ω) 2 1 ω 2 = ω 2 (z ω) 2 ω 2 (z ω) 2 ) z2 + 2zω ( ) 2 4 z z ω ω ω 4 ω z 2 ω z ω 3 According to Lemma 1.12 the series 4 R2 2 R ω R ω 3 = 10 1 ω Λ ω 3 R ω 3. converges absolutely. For z R we decompose the series with respect to the summation over ω Λ [ 1 z 2 + ω Λ, ω <R ( 1 (z ω) 2 1 )] [ ω 2 + ω Λ, ω 2R ( 1 (z ω) 2 1 )] ω 2 The first summand is meromorphic with poles exactly at the points z Λ (R). The second summand converges absolutely and compactly on (R) \ Λ due to Lemma 1.12, and defines a holomorphic function on (R). Because R can be choosen arbitrary, the series is compact convergent on C\Λ and defines a meromorphic function on C with pole set Λ. ii) Even function with periods from Λ: is even: After rearranging the summation, which is admissible due to the absolute convergence, ( z) = 1 ( z ω Λ ( z ω) 2 1 ) ω 2 = 1 ( z ω Λ (z ( ω)) 2 1 ) ( ω) 2 = (z). has periods from Λ: The derivative is (z) = 2 z ω Λ (z ω) 3 = 2 ω Λ 1 (z ω) 3

14 10 1 The elliptic functions of a lattice Apparently for all ω Λ (z + ω) = (z) after rearranging the summation. As a consequence (z + ω j ) = (z) + c j with two suitable constants c j C, j = 1,2. For j = 1,2 and the specific arguments z := ω j /2: On one hand (( ω j /2) + ω j ) = (ω j /2) + c j, on the other hand due to periodicity ( (ω j /2) + ω j ) = (ω j /2), which implies c j = 0, q.e.d. Lemma 1.14 (Half-lattice points). Denote by { ω1 Z := 2, ω 2 2, ω } 1 + ω 2 2 the set of half-lattice points from the fundamental period parallelogram P Λ. 1. Consider an even elliptic function f with period lattice Λ. Any u Z has even order ord u ( f ). 2. Modulo Λ, the odd elliptic function has a pole of order = 3 at 0 C and no other poles, and exactly three zeros mod Λ, namely the points u Z. Proof. 1. Set u := ω/2 Z,ω Λ. We compute the Laurent expansion of f around u: On one hand, because ω is a period On the other hand, because f is even As a consequence f (u + z) = f (u + z ω) = f ( u + z). f ( u + z) = f (u z). f (u + z) = f (u z). Therefore the Laurent expansion around u

15 1.2 The -function of Weierstrass 11 f (w) = c n (w u) n n=n 0 satisfies c n = 0 for odd n. In particular ord u ( f ) is even. 2. Set u := ω/2 Z,ω Λ. According to Theorem 1.13 the function has no pole at u. The derivative has the same pole set Λ as. Because is even, the function is an odd elliptic function Therefore (u) = ( u) = ( u + 2u) = (u). (u) = 0. Because has a single pole at 0 C of order = 3 mod Λ and no other pole mod Λ. Corollary 1.10 implies that Z is the zero set of mod Λ, and each zero has order = 1, q.e.d. Note that the determination of the two zeros modulo Λ of is much more delicate [9]. Corollary 1.15 (Period lattice of and ). The Weierstrass function of the lattice Λ and its derivative have the period lattice Λ, i.e. Λ = Λ = Λ, the only periods of and of are the points from Λ. Proof. We know Λ Λ Λ Here the first inclusion has been shown in Theorem 1.13, while the second inclusion is obvious. We show Λ Λ : Let ω Λ be a period of. Hence (ω/2) = ((ω/2) ω) = ( ω/2) = (ω/2) which implies (ω/2) = 0. Lemma 1.14 implies ω/2 Λ/2 or ω Λ, q.e.d. See the PARI-file Weierstrass_p_function_08 as motivation for the following Proposition 1.16.

16 12 1 The elliptic functions of a lattice Proposition 1.16 (Laurent expansion of ). The Weierstrass function of Λ has the Laurent expansion around zero (z) = 1 z 2 + a 2k z 2k k=1 with coefficients a 2k = (2k + 1) G Λ,2k+2. Proof. We determine the Taylor series of 1 (z ω) 2 1 ω 2 around 0 C: Taking the derivative 1 (z ω) 2 = d ( ) 1 dz z ω and expanding into a geometric series shows and 1 z ω = 1 ω z = 1/ω 1 (z/ω) = (1/ω) (1/ω) ν z ν ν=0 1 (z ω) 2 = (1/ω) ν (1/ω) ν z ν 1 = ν=1 ν=0 (ν + 1) 1 (z ω) 2 1 ω 2 = 1 (ν + 1) ν=1 ω ν+2 zν. 1 zν ων+2 Because is even and the series converges absolutely, see Theorem 1.13, we obtain after rearrangement (z) = 1 ( ) z ) ν=1(ν ω Λ ω ν+2 z ν = 1 ( ) z k=1(2k+1) ω Λ ω 2k+2 z 2k, q.e.d. Theorem 1.17 (Differential equation of ). The Weierstrass function of the lattice Λ satisfies the differential equation with the constants 2 = 4 3 g 2 g 3

17 1.2 The -function of Weierstrass 13 derived from the lattice constants of Λ. g 2 := 60 G Λ,4, g 3 := 140 G Λ,6 Proof. To simplify the notation we omit the index Λ from the lattice constants G Λ,k. Due to Proposition 1.16 the Laurent expansions start (z) = 1 z 2 + (2k + 1) G 2k+2 z 2k = 1 k 1 z G 4 z G 6 z 4 + O(6) (z) = 2 z G 4 z + 20 G 6 z 3 + O(5) (z) 2 = 4 z 6 24 G 4 z 2 80 G 6 + O(2) (z) 2 = 1 z G G 6 z 2 + O(4) (z) 3 = 1 z G 4 z G G 4 z 2 +5 G 6 +O(2) = 1 z G 4 z G 6 +O(2) Therefore g 2 + g 3 = O(2). The left-hand side is an elliptic function. The right-hand side shows that the function is even holomorphic and vanishes at 0 C. According to Proposition 1.7 the lefthand side is constant, i.e. zero, q.e.d. Theorem 1.18 (Field of elliptic functions). The field M (Λ) of elliptic functions with respect to the lattice Λ is a field extension of C of transcendence degree = 1, more precisely M (Λ) = C( )[ ] with algebraic over the field C( ) with quadratic minimal polynomial with the lattice constants f (T ) = T g 2 + g 3 C( )[T ] g 2 := 60 G Λ,4, g 3 := 140 G Λ,6. Note the equality C( )[ ] = C( )( ) because is algebraic over C( ). Proof. i) Due to Theorem 1.17 we have f ( ) = 0 C( ).

18 14 1 The elliptic functions of a lattice The derivative is odd while is even. Therefore / C( ), and the minimal polynomial must have degree 2. ii) Consider an elliptic function f M (Λ), f 0. We show f C(, ). We first consider the case that f has pole set Λ. The unique pole of f mod Λ has order k 2 due to Corollary 1.9. Induction on the pole order k. If k = 2 then for suitable c C the function f c has at most a single pole mod Λ of order < 2. Therefore the function is holomorphic, and due to Corollar 1.9 even constant. For the induction step assume k > 2. On one hand, for even k = 2m a suitable constant c C exists such that f c m has only poles of order < k. Therefore f c m C(, ) by induction assumption. On the other hand, for odd k = 2m + 1 a suitable constant c C exists such that f c m 1 has only poles of order < k. By induction assumption In both cases f c m 1 C(, ). f C(, ). Eventually, we consider the case of an arbitrary pole set of f. Then f has only finitely many poles z ν P\Λ,ν = 1,...,n. For suitable constants k 1,...,k n N the function f (z) n ν=1 ( (z) (z ν )) k ν has poles at most in Λ. According to the first part f (z) n ν=1 ( (z) (z ν )) k ν C(, ). Therefore f C(, ), q.e.d.

19 1.3 Abel s Theorem Abel s Theorem We continue with fixing an arbitrary lattice Λ := Z ω 1 + Z ω 2, keeping the notations Λ := Λ \{0} and P for the fundamental period parallelogram of Λ. Proposition 1.19 (Zero and pole divisor). Consider an elliptic function and denote by A = (a i ) i=1,...,n its family of zeros in P and by B = (b i ) i=1,...,n its family of poles in P. Then n 2 and i=1,...,n a i i=1,...,n b i Λ. Note that each point p of A and B appears as often as its multiplicity as zero or pole of f indicates. By Corollary 1.9 holds n 2, and by Corollary 1.10 both families A and B have the same cardinality. Proof. We consider a translation P a := P+a of the fundamental parallelogram such that f has neither zeros nor poles on the boundary P a. With a counter-clockwise orientation of P a the residue theorem gives 1 z f ( (z) 2πi P a f (z) dz = res p z f ) (z). p P a f (z) Right-hand side: Assume ord p ( f ) = k Z. Then { p is a zero of f of order k if k > 0 p is a pole of f of order k if k < 0 In a neighbourhood of p with f 1 holomorph. As a consequence f (z) f (z) = k z p + f 1(z) z f (z) f (z) = (p + (z p)) f (z) f (z) = p k z p + f 2(z) with f 2 holomorph. Therefore ( res p z f ) (z) = p k f (z) and

20 16 1 The elliptic functions of a lattice Left-hand side: AB+ CD z f (z) f (z) = AB res p (z f (z) n p P a f (z) ) = i=1 a i (z (z + ω 2 )) f (z) f (z) = ω 2 AB n i=1 b i. f (z) f (z) = ω 2 [log f (z)] B A Here log denotes a branch of the logarithm in a simply-connected neighbourhood of the line AB. AB+ CD z f (z) f (z) dz = ω 2 (log f (A) log(b)). Because f (A) = f (B) an integer m 2 Z exists with log f (A) log f (B) = 2πi m 2. A similar argument for the lines BC and DA provides a second constant m 1 Z such that f (z) z f (z) dz = m 1 ω 1 + m 2 ω 2 Λ, q.e.d. P a Theorem 1.20 (Abel s theorem). Consider n 2 and two finite families of complex points A := (a i ) i=1,...,n, B := (b i ) i=1,...,n P with A disjoint to B. Then an elliptic function f M (Λ) exist with its zeros A and its poles B mod Λ if and only if n i=1 a i n i=1 b i Λ. Proof. Proposition 1.19 proves that the zero set and the pole set of f satisfy the condition stated above. In order to prove the opposite direction of the theorem assume two sets A and B with the property stated above. Without restriction we may assume n j=1 (a j b j ) = 0, otherwise replace a 1 by a suitable point which is congruent mod Λ. The proof for the existence of a suitable elliptic function f is by induction, starting with n = 2. If a 1 b 1 = b 2 a 2

21 1.3 Abel s Theorem 17 then consider z 0 := a 1 + a 2 + b 1 + b 2. 4 Geometrically the point z 0 is the center of the parallelogram with vertices the points a 1,a 2,b 1,b 2. We may assume z 0 = 0, otherwise we translate all coordinates by z 0. As a consequence The even function has a a 1 = a 2 and b 1 = b 2. g(z) := (z) (a 1 ) zero of order = 1 at a 1 and at a 2 if a 1 a 2 and a pole of order = 2 at z = 0. An analogous result holds for the function h(z) := (z) (b 1 ). mod Λ, and of order = 2 otherwise, Hence the quotient f := g solves the problem. h Induction step n n + 1: If mod Λ then choose ã n C with Then (a 1 b 1 ) (a n b n ) + (a n+1 b n+1 ) 0 ã n (a 1 b 1 ) (a n 1 b n 1 ) + a n. (a 1 b 1 ) (a n 1 b n 1 ) + (a n a n ) 0. By induction assumption an elliptic function f 1 M (Λ) exists with zeros (a 1,...,a n ) and poles (b 1,...,b n 1,ã n ). A second elliptic function f 2 M (Λ) exists with zeros (ã n,a n+1 ) and poles (b n,b n+1 ). Hence the function solves the problem, q.e.d. f := f 1 f 2 Abel s theorem generalizes to all compact Riemann surfaces, it is not restricted to tori, see [10, 20].

22 18 1 The elliptic functions of a lattice Corollary 1.21 (Elliptic functions with pole of order = 3). An elliptic function f M (Λ) with a pole mod Λ at 0 Λ of order = 3 and no other poles mod Λ, has exactly three zeros a 1,a 2,a 3 mod Λ. They satisfy a 1 + a 2 + a 3 = 0 mod Λ.

23 Chapter 2 Complex tori 2.1 Compact Riemann surfaces This section recalls some necessary prerequisites from the theory of Riemann surfaces. A reference is the textbook [10]. Fig. 2.1 Riemann surface X 19

24 20 2 Complex tori Definition 2.1 (Riemann surface). manifold of complex dimension 1. A Riemann surface X a connected complex We assume all manifolds to be Hausdorff spaces. By a theorem of Radó every Riemann surface has a countable basis of its topology. Example 2.2 (Riemann surfaces). 1. Any domain G C is a Riemann surface. 2. The complex projective space P 1 = P 1 (C) (Riemann sphere) is a compact Riemann surface. 3. According to Riemann s mapping theorem any simply-connected Riemann surface is biholomorphically equivalent to either the unit disc := {z C : z < 1}, or to C, or to P 1. The upper half-plane, also called Poincaré upper half-plane, H := {τ C : Im τ > 0} of complex numbers with positive imaginary part is a Riemann surface, biholomorphically equivalent to. 4. The complex torus C/Λ, with Λ (C,+) a lattice, is a compact Riemann surface. Note that the torus is also an Abelian group with respect to addition. (C,+)/(Λ,+) Remark 2.3 (Meromorphic function). 1. Consider a Riemann surface X and an open subset U X. A meromorphic function f defined on U is determined by a discrete set S U and a holomorphic function f : U \ S C, such that for all p S lim f (z) =. z p The set S is the pole set of f : With respect to a local coordinate φ around a point p S the function f φ 1 has a pole at φ(p). The field of all meromorphic functions defined on U is denoted M (U).

25 2.1 Compact Riemann surfaces 21 The attachment U M (U), U X open, defines on X the sheaf M X of meromorphic functions. It is sheaf of rings. The meromorphic functions, which are non-zero on any connected component of their domain of definition, define the subsheaf MX. The latter is a sheaf of multiplicative Abelian groups. Analogously one defines the sheaves O X and OX of respectively holomorphic functions and holomorphic functions without zeros. 2. Consider the canonical embedding and the point := (1 : 0) P 1. C P 1,z (z : 1), Any meromorphic function f defined on a Riemann surface X with pole set S can be considered a holomorphic map by defining for all poles z S f : X P 1 = C { } f (z) := P 1. Vice versa, any holomorphic map f : X P 1 with f (X) { } defines a meromorphic function f on X. In the following we will not distinguish between a meromorphic function f defined on X and the corresponding holomorphic map f : X P Consider a lattice Λ C and the corresponding torus T := C/Λ. Holomorphic maps f : T P 1 correspond bijectively to elliptic functions f M (Λ): f ([z]) = f (z),z C. If there is no risk of confusion we will denote both functions with the same letter f. In particular, the Weierstrass function of the lattice Λ will be considered also a meromorphic function on T. Proposition 2.4 (Non-constant holomorphic maps). Any non-constant holomorphic map f : X Y between two compact Riemann surfaces is surjective. It attains each value y Y with the same multiplicity. The value is named deg f, the degree of f : For all y Y the fibre X y := f 1 (y) has cardinality

26 22 2 Complex tori card X y = deg f. Proof. A non-constant holomorphic map between two Riemann surfaces is an open map. Locally f is the map z z k for a suitable k N. If X is compact then f is also closed. The open and closed subset f (X) Y equals Y because Y is connected. The set of S X of ramification points of f, i.e. with k 2 in a neighbourhood of the point, is a discrete subset, hence a finite set. Denote by f (S) Y the set of branch points. Then f X \ f 1 ( f (S)) : X \ f 1 ( f (S)) Y \ f (S) is a covering projection. Because Y \ f (S) is connected, each fibre of f (X \ f 1 ( f (S)) has the same finite cardinality, and a posteriori also each fibre of f, q.e.d. Note: The degree of a non-constant map f according to the topological definition from Proposition 2.4 equals the degree of the field extension which is an algebraic concept. [M (X) : f M (Y )] Definition 2.5 (Divisor). Consider a compact Riemann surface X. 1. A divisor on X is a formal sum D := n x (x) x X with integers n x Z, and n x = 0 for all but finitely many x X. The degree of D is deg D := n x. x X For each x X set D(x) := n x. The divisors of meromorphic functions f M (X) ( f ) := n x (x), n x := ord x ( f ), x X are named principal divisors. Note that f is non-zero, hence has only finitely many zeros and poles. Here

27 2.1 Compact Riemann surfaces 23 { n if f has zero at x of order n ord x ( f ) = n if f has pole at x of order n Principal divisors satisfy deg ( f ) = 0 according to Proposition Consider two divisors Their sum is defined as One defines D i = n x.i (x), i = 1,2. x X D 1 + D 2 := n x (x), n x := n x.1 + n x,2. x X D 1 D 2 n x,1 n x,2 f or all x X. If D 1 D 2 then one names D 1 a multiple of D 2. Two divisors D 1,D 2 Div(X) are equivalent if their difference D 2 D 1 is a principal divisor. 3. The Abelian group of all divisors on X, the free Abelian group on the set X, is denoted Div(X). The divisors of degree = 0 form the subgroup Div 0 (X) Div(X). The quotient of Div(X) with respect to the subgroup of pricipal divisors is the group of divisor classes Cl(X). The notation Cl 0 (X) denotes the subgroup of Cl(X) generated by the classes from Div 0 (X). Example 2.6 (Abel s Theorem: Version with divisors). The families A and B from Abel s Theorem 1.20 define on the torus T two divisors D A := n a (a) and D B := n b (b) a A b B with respectively n a and n b the multiplicities of the points a in the family A and b in B. The theorem proves the existence of a meromorphic function f M (T ) with divisor ( f ) = D := D A D B Div(T ) if and only if n a a = n b b T a A b B with respect to addition and multiples from the group structure on T - not with respect to the group structure due to Div(T ).

28 24 2 Complex tori Remark 2.7 (Exact sequence of divisor class groups). On a compact Riemann surface X the following sequence of group morphisms is exact Here The map 1 O (X) M (X) Div 0 (X) Cl 0 (X) 0 O (X) := H 0 (X,O X) = C. M (X) Div 0 (X), f ( f ), is well-defined due to Proposition 2.4. For a torus T Abel s theorem shows Cl 0 (T ) 0. Later Theorem 3.17 will compute this group. Proposition 2.8 (Divisors and line bundles). Consider a compact Riemann surface X. 1. Consider a divisor D = n x (x) Div(X). x X For each open set U X define the Abelian group i.e. for all x U O D (U) := { f M (U) : ( f ) D U} {0} f has in x Then the attachment { a pole of order at most n x i f n x 0 a zero of order at least n x i f n x 0 U O D (U), U X open, defines a subsheaf O D M X, named the sheaf of meromorphic multiples of D. 2. The sheaf O D is isomorphic to the sheaf L D of sections of the holomorphic line bundle L D on X, which is defined by the following cocycle g := (g i j ) i, j I Z 1 (U,O X) : Choose an open covering U = (U i ) i I of X by open subsets U i X with meromorphic functions ψ i M (U i ) such that (ψ i ) = D U i, i I, and set g i j := ψ i ψ j O X(U i U j ).

29 2.1 Compact Riemann surfaces Two divisors D 1, D 2 Div(X) are equivalent iff The map O D1 O D2. F : (Cl(X),+) (Pic(X), ),[D] [L D ], is a group isomorphism from the additive group of divisor classes to the multiplicative Picard group of isomorphism classes of holomorphic line bundles. Proof. 1. Apparently, O D M is a subsheaf of the sheaf of meromorphic functions. 2. For i, j I the quotient g i j := ψ i ψ j O X(U i U j ) is holomorphic without zeros because ψ i and ψ j define the same divisor on U i U j. Apparently the family g := (g i j ) i, j I satisfies the cocycle relation g i j g jk = g ik f or i, j,k I. Consider an open subset U X and a meromorphic function f O D (U). Then for all i I: ( f (U U i ) ψ i (U U i )) 0, i.e. the function f i := f (U U i ) ψ i (U U i ) O X (U U i ) is holomorphic. Apparently on the intersections U (U i U j ): g i j f j = ψ i ψ j ( f ψ j ) = ψ i f = f i, i.e. the family ( f i ) i I is a section of the line bundle L D on the open set U X. 3. In order to obtain the inverse G : Pic(X) Cl(X) of the morphism F we use the following deep fact: Any line bundle on a compact Riemann surface X has a non-constant meromorphic section [10, Satz 29.16]. For a line bundle L on X choose a non-constant meromorphic section ψ of L and define G(L ) := [(ψ)] Cl(X).

30 26 2 Complex tori One checks: The definition of G is independent from the choice of ψ, and G = F 1, q.e.d. The sheaf L D of sections of a line bundle L D is a locally free sheaf of rank = 1. These sheaves are also named invertible sheaves. We shall often identify line bundles with their sheaf of sections, i.e. we shall not distinguish between L D and L D. Corollary 2.9 (Divisors of negative degree). Consider a compact Riemann surface and a divisor D Div(E) with deg D < 0. Then H 0 (X,O D ) = 0. Proof. A non-zero section f H 0 (X,O D ) is a meromorphic function f with divisor ( f ) D, in particular deg ( f ) deg D > 0, a contradiction to Proposition 2.4, q.e.d. A deep result from the theory of compact Riemann surfaces is the following theorem on the finiteness of the cohomology of coherent sheaves, in particular the cohomology of invertible sheaves like O D, Ω 1, Θ, but in general not the cohomology of M. Remark 2.10 (Finiteness of cohomology groups). Consider a compact Riemann surface X and a coherent sheaf F on X. Then for all i N the complex vector space H i (X,F ) has finite dimension. Definition 2.11 (Genus of a compact Riemann surface). The (geometric) genus of a compact Riemann surface X is defined as g(x) := dim H 0 (X,Ω 1 X), the dimension of the complex vector space of holomorphic differential forms. Example 2.12 (Genus of projective space). The projective space P 1 is a compact Riemann surface of genus g = 0. Proof. To compute H 0 (P 1,Ω 1 P 1 ) we use the standard covering (U 0,U 1 ) of P 1 and the corresponding charts U i := {(z 0 : z 1 ) P 1 : z i 0}, i = 0,1,

31 2.1 Compact Riemann surfaces 27 w 0 = φ 0 (z 0 : z 1 ) := z 1 z 0 and w 1 = φ 1 (z 0 : z 1 ) := z 0 z 1. A differential form ω H 0 (P 1,Ω 1 P 1 ) is a pair ( f i dw i ) i=0,1 with holomorphic functions f i, which on satisfy Because w 1 = 1/w 0 we have Therefore C U 0 U 1 = P 1 \ {(0 : 1), (1 : 0)}, f 0 (w 0 ) dw 0 = f 1 (w 1 ) dw 1. dw 1 = dw 0 w 2. 0 w 2 0 f (w 0 ) dw 0 = f 1 (1/w 0 ) dw 0 or w 2 0 f (w 0 ) = f 1 (1/w 0 ). The Taylor expansions of f 0 and f 1 shows Therefore ω = 0, q.e.d. f 0 = f 1 = 0. We recall the main result to compute the dimension of cohomology groups on compact Riemann surfaces. Theorem 2.13 (Theorem of Riemann-Roch). Consider a compact Riemann surface X of genus g and a line bundle on X with sheaf of sections L. Then: dimh 0 (X,L ) dimh 1 (X,L ) = 1 g + c 1 (L ). Here c 1 (L ) Z denotes the first Chern number of L. If L = O D for the divisor D Div(X), then c 1 (L ) = deg D. For the proof of Theorem 2.13 see [10, Satz 16.9]. A second basic result is Serre s theorem on duality: Theorem 2.14 (Serre duality). Consider a compact Riemann surface X. Then the sheaf ω X := ΩX 1 is the dualizing sheaf of X, i.e. for any line bundle L on X and its dual line bundle L with invertible sheaves of sections L and L 1 a canonical pairing of vector spaces exists In particular H 0 (X,L 1 ω X ) H 1 (X,L ) C.

32 28 2 Complex tori dim H 0 (X,L 1 ω X ) = dim H 1 (X,L ). For a proof of Theorem 2.14 see [10, Satz 17.9]. Due to Theorem 2.14 the genus of a compact Riemann surface X satisfies the equality g(x) := dim H 0 (X,Ω 1 X) = dim H 1 (X,O X ). Moreover from the viewpoint of algebraic topology 2 g(x) = dim H 1 (X,C). Combining Theorem 2.13 and 2.14 the theorem of Riemann-Roch takes on the following form: Corollary 2.15 (Riemann-Roch with Serre duality). Consider a compact Riemann surface X of genus g. Then for any line bundle on X with sheaf L of sections: dim H 0 (X,L ) dim H 0 (X,L 1 Ω 1 X) = 1 g + c 1 (L ). Corollary 2.16 (Degree of differential forms). Consider a compact Riemann surface X of genus g N. Then Proof. According to Corollary 2.15 c 1 (Ω 1 X) = 2g 2. dim H 0 (X,Ω 1 X) dim H 0 (X,O X ) = 1 g + c 1 (L ). Due to the maximum principle all holomorphic functions on X are constant, hence According to the definition As a consequence dim H 0 (X,O X ) = 1. g := dim H 0 (X,Ω 1 X). 1 g + c 1 (L ) = dim H 0 (X,Ω 1 X) dim H 0 (X,O X ) = g 1 i.e c 1 (L ) = 2g 2, q.e.d. A divisor K Div(X) with O K ΩX 1 is named canonical divisor of X.

33 2.1 Compact Riemann surfaces 29 Theorem 2.17 (Theorem of Riemann-Hurwitz). Consider a non-constant holomorphic map f : X Y between two compact Riemann surfaces and denote by S X the set of its ramification points. Then 2g(X) 2 = (deg f )(2g(Y ) 2) + r x f. x S Recall from the proof of Proposition 2.4 the local representation of f around an arbitrary point x X z z n with n 2. The number r x f := n 1 N is the ramification order of f at x. The point x X is a ramification point of f if r x ( f ) 1. Recall the holomorphic Euler characteristic χ(o X ) := dim H 0 (X,O X ) dim H 1 (X,O X ) and the topological Euler characteristic χ(x) := They relate as χ(x) = 2 χ(o X ). Then 2 i=0 dim H i (X,C). 2g(x) 2 = (χ(x)) = 2 χ(o X ). Theorem 2.17 shows that a non-constant holomorphic map f : X Y cannot increase the genus. It is either an isomorphism or it decreases the genus, i.e. g(y ) < g(x). The following theorem 2.18 describes how to obtain holomorphic maps f : X P n into the projective space from suitable line bundles L on X. Actually, any holomorphic map f : X P n arises in a similar vein. Theorem 2.18 (Maps into a projective space). Consider a compact Riemann surface X and a line bundle L on X with sheaf of sections L.

34 30 2 Complex tori 1. If L has sufficiently many sections, then L defines an embedding φ L : X P n (C) of X as a closed submanifold of P n (C), more precisely: Choose a basis (s i ) i=0,...,n of H 0 (X,L ) and consider the following conditions: The sheaf L has no base points, i.e. for all p X a section s H 0 (X,L ) with s(p) 0 exists: Then define φ L : X P n (C), p (s 0 (p) :... : s n (p)). The sheaf L separates points, i.e. for any two distinct points p q X a section s H 0 (X,L ) exists with s(p) = 0 but s(q) 0: Then φ L is injective. The sheaf L separates tangent vectors, i.e. for all points p X the map {s H 0 (X,L ) : s(p) = 0} m p L p /m 2 pl p, s [s p ], is surjective: Then φ L is an immersion. If L satisfies all three conditions, then φ L : X P n (C) is defined and embeds X into P n (C) as a closed submanifold. 2. Assume L = O D for the divisor D Div(X). Then: L has no base points iff for all points p X dim H 0 (X,O D (p) ) = dim H 0 (X,O D ) 1. L separates points and tangent lines iff for all points p,q X, not necessarily distinct, dim H 0 (X,O D (p) (q) ) = dim H 0 (X,O D ) 2. The definition of the map φ L uses explicitly a basis of H 0 (X,L ). This makes the definition easy to understand but dependent on the choice of the basis. To obtain a map which is independent from any basis, one uses the more refined definition with the dual space: X P(H 0 (X,L ) ),x [λ x ], with λ x : H 0 (X,L ) L (x) := L x /m x L x C,σ [σ x ]. Here L x denotes the stalk of the sheaf L at the point x X. The class of λ x does not depend on the choice of a trivialization of L around x. For a proof of Theorem 2.18 in the context of Algebraic Geometry see [15, Prop. 7.3].

35 2.1 Compact Riemann surfaces 31 Remark 2.19 (Very ample line bundle). A line bundle L on a compact Riemann surface X is named very ample if is defined and is a closed embedding. φ L : X P 1 (H 0 (X,L ) ) Lemma 2.20 (Genus of a torus). Any complex torus T = C/Λ is a compact Riemann surface of genus g = 1. Proof. The holomorphic differential form dz H 0 (C,ΩC 1 ) on C has no zeros. It is invariant with respect to translation by arbitrary but fixed constants a C: d(z + a) = dz. Therefore dz induces a holomorphic differential form ω H 0 (T,Ω 1 T ) on T with deg (ω) = 0. According to Corollary 2.16 the genus g of T satisfies 2g 2 = deg (ω) = 0, q.e.d. Note: In the proof of Lemma 2.20 the reference to Corollary 2.16 can be avoided. Instead one argues that the line bundle ΩT 1 is trivial because it has a holomorphic section without zeros. From ΩT 1 O T follows H 0 (T,ΩT 1 ) H 0 (T,O T ) = C. We now apply the theorem of Riemann-Roch to the question on the existence of meromorphic functions on a torus. Example 2.21 (Meromorphic functions on tori). Consider a complex torus X. In order to prove the existence or non-existence of meromorphic functions f M (X) with a single pole at a given point x X we consider the divisors They satisfy D(n) := n(x) Div(X), n N. deg D(n) = n. According to Corollary 2.15 with genus g = 1 and according to Lemma 2.20 dim H 0 (X,O D(n) ) dim H 0 (X,O D( n) ) = n.

36 32 2 Complex tori For all n 1 H 0 (X,O D( n) ) = 0 according to Corollary 2.9. Hence for all n 1 dim H 0 (X,O D(n) ) = n, and in addition dim H 0 (X,O D(0) ) = dim H 0 (X,O X ) = 1 because H 0 (X,O X ) C. From these dimension formulas we now rediscover the result from Corollary 1.9 and the result about the existence of non-constant elliptic functions: i) According to Proposition 2.8 the 1-dimensional vector space H 0 (X,O D(1) ) is the vector space of all meromorphic functions with at most one pole at x of order = 1 and no other pole. The vector space comprises the 1-dimensional subspace of constant functions. Therefore H 0 (X,O D(1) ) = C, in particular there do not exist meromorphic functions with only one pole at x of order = 1 and no other pole. ii) Similarly, the 2-dimensional vector space H 0 (X,O D(2) ) of meromorphic functions with at most one pole at x of order = 2 and no other pole comprises H 0 (X,O D(1) ) = H 0 (X,O D(0) ) as a proper subspace. Hence at least one meromorphic function f H 0 (X,M (X)) exists with a pole at x of order = 2 and no other pole. 2.2 Isomorphism classes of complex tori and the modular group The basis of a lattice Λ = Zω 1 + Zω 2 is not uniquely determined. Two positively oriented bases generate the same lattice if and only if they are equivalent modulo SL(2,Z), i.e. iff they belong to the same orbit of the following SL(2,Z)-operation: Lemma 2.22 (Positively oriented bases of a lattice). Denote by { ( ) } M := (ω 1,ω 2 ) basis o f R 2 ω1 ω with det 2 > 0

37 2.2 Isomorphism classes of complex tori and the modular group 33 the set of positively oriented bases of R 2. The set R of all lattices Λ C is the orbit set SL(2,Z)\M of the left SL(2,Z)-operation SL(2,Z) M M, ( γ, ( ω1 ω 2 )) ( ) ω 1 ω 2 := γ ( ω1 Proof. The invertible matrices γ M(2 2,Z) have determinant ±1. Because both bases are positively oriented we have det γ = 1, q.e.d. ω 2 ). Two tori belonging to different lattices may be biholomorphically equivalent. The following Definition 2.23 introduces a name for the relation between lattices with isomorphic tori. Definition 2.23 (Similar lattices). Two lattices Λ, Λ C are similar if a complex number µ C exists with Λ = µ Λ. Proposition 2.24 (Lattices of biholomorphically equivalent tori). Two tori T = C/Λ and T = C/Λ are biholomorphically equivalent iff their lattices Λ and Λ are similar. Proof. i) Assume the existence of a biholomorphic map f : T T satisfying without restriction f (0) = 0. Because C is simply connected and the canonical map p : C C/Λ is a covering projection, the map f extends to a commutative diagram C F C p p f C/Λ C/Λ with a unique holomorphic map F such that F(Λ) Λ and F(0) = 0. For arbitrary but fixed ω Λ consider the holomorphic map F ω : C C, z F(z + ω) F(z).

38 34 2 Complex tori By assumption F ω (C) Λ. Because the lattice Λ is a discrete space and the domain of definition C is connected, the map F ω is constant. Hence the derivative satisfies F ω(z) = df df (z + ω) dz dz (z) = 0 for all z C. As a consequence, the holomorphic map F is elliptic with respect to Λ. Proposition 1.7 implies that F is constant, i.e. a number µ C exists with F(z) = µ z for all z C. Analogously the unique lift F of f 1 with F (0) = 0 satisfies for a suitable µ C. Then F (z) = µ z, z C, C C,z (µ µ) z, is the unique lift of id = f 1 f which maps 0 0. Hence µ µ = 1, in particular µ Λ = Λ. ii) Assume Λ = µ Λ with µ C. Then the biholomorphic map F : C C, z µ z, induces a biholomorphic map f : T T, q.e.d. Theorem 2.25 (Biholomorphic equivalence classes of complex tori). 1. Any complex torus is biholomorphic equivalent to a torus T = C/Λ with a normalized lattice Λ = Z 1 + Z τ C with τ H in the upper half-plane. 2. The period τ of a normalized lattice is determined up to a transformation of SL(2,Z), i.e. two tori C/Λ τ and C/Λ τ with normalized lattices are biholomorphic equivalent iff Λ τ = Z 1 + Z τ and Λ τ = Z 1 + Z τ τ = a τ + b c τ + d

39 2.2 Isomorphism classes of complex tori and the modular group 35 with integers a,b,c,d Z satisfying ad bc = 1. Proof. 1. Consider an arbitrary torus T = C/Λ with a lattice satisfying Λ = Zω 1 + Zω 2 and det (ω 1 ω 2 ) > 0. Then τ := ω 2 ω 1 satisfies Im τ > 0 and the lattice Λ is similar to the normalized lattice Λ τ := Z 1 + Z τ due to 1 ω 1 Λ = Λ τ. According to Proposition 2.24 the two tori T and T := C/Λ τ are biholomorphic equivalent. 2. i) Assume Set Then As a consequence because τ = a τ + b with γ := c τ + d ( ω2 ω 1 ( ) a b SL(2,Z). c d ) ( ) τ := γ. 1 Z ω 1 + Z ω 2 = Z 1 + Z τ = Λ τ. 1 ω 1 Λ τ = Λ τ τ = ω 2 ω 1. The lattices Λ and Λ are similar. Hence the tori C/Λ τ and C/Λ τ are biholomorphic equivalent according to Proposition ii) For the opposite direction assume that two tori with normalized lattices C/Λ τ and C/Λ τ,τ,τ H, are biholomorphic equivalent. According to Proposition 2.24 a number µ C exists with Λ τ = µ Λ τ. As a consequence Therefore a matrix Λ τ = Z µ + Z µτ.

40 36 2 Complex tori ( ) a b γ = SL(2,Z) c d exists with ( τ i. e. 1 τ = a µτ + b µ c µτ + d ) ( ) µτ = γ, µ = a τ + b c τ + d, q.e.d. Theorem 2.25 shows the consequences when two points of the upper half-plane are related by an element of the group SL(2,Z). We formalize this relation by the concept of a group operation of SL(2,Z). Definition 2.26 (Operation of the modular group). 1. The group is named the modular group. SL(2,Z) := {γ M(2 2,Z) : det γ = 1} 2. The SL(2,Z)-left operation on the upper half-plane is the map Φ : SL(2,Z) H H,(γ,τ) γ(τ) := aτ + b cτ + d, γ := 3. The isotropy group of a point τ H is the subgroup SL(2,Z) τ := {γ SL(2,Z) : γ(τ) = τ}. The point τ H is an elliptic point of Φ if its isotropy group is not trivial, i.e. if Γ τ := {γ Γ : γ(τ) τ} Γ τ {±id}. ( ) a b. c d The period of an arbitrary, not necessary elliptic τ H is the number 4. The orbit of a point τ H is the set h τ := (1/2) ord Γ τ. {γ(τ) H : γ SL(2,Z)}. 5. To points τ 1,τ 2 H are equivalent with respect to Φ iff they belong to the same orbit. The orbit set of Φ is the set of all equivalence classes

41 2.3 The modular curve of the modular group 37 Y := SL(2,Z)\H. Note that Φ is well-defined because γ(τ) H for τ H: One checks Im γ(τ) = Im τ cτ + d 2. The operation Φ is continuous. The element id SL(2, Z) operates trivially on H, i.e. id SL(2,Z) τ for all τ H. Therefore some authors apply the name modular group to the quotient PSL(2,Z) := SL(2,Z)/{±id}. We will not follow this convention. Later we will study also the operation of certain subgroups of SL(2,Z) which do not contain the element id. 2.3 The modular curve of the modular group In the present section we denote by Φ : SL(2,Z) H H,(γ,τ) γ(τ) := aτ + b cτ + d, γ := ( ) a b c d the continuous SL(2, Z)-left operation introduced in Definition We will often use the shorthand notation for the modular group Γ := SL(2,Z). Our aim is to provide the orbit space Y := Γ \H with an analytic structure and to compactify the resulting Riemann surface to obtain a compact Riemann surface X. The latter will be named the modular curve of the operation Φ. We proceed along the following steps: Constructing a Hausdorf topology on Y, see Corollary Constructing an analytic structure on Y, see Proposition Compactifying Y to a compact Riemann surface X, see Theorem To provide the orbit space of a group operation with a suitable topological or differentiable structure is always a subtle task. In the present context the problem is intensified by the fact that the modular group does not operate freely, see Theorem 2.31: There are elliptic points. The difficulty consist in dealing with the elliptic points with respect to the Hausdorff property and with respect to complex charts on the quotient.

42 38 2 Complex tori The following Lemma 2.27 covers the main step for proving the Hausdorff property for the quotient topoplogy on Y. Lemma 2.27 (Separating orbits). For any two points τ 1,τ 2 H exist neighbourhoods U i H of τ i, i = 1,2, such that for all γ Γ : γ(u 1 ) U 2 /0 = γ(τ 1 ) = τ 2. In particular: Any point τ H has a neighbourhood U H such that for all γ Γ : γ(u) U /0 = γ Γ τ. The set U has no elliptic points except possibly τ. Lemma 2.27 can be paraphrased as follows: If an element γ Γ maps a point of the distinguished neighbourhood U 1 of τ 1 to the distinguished neighbourhhood U 2 of τ 2, then γ maps τ 1 to τ 2. The add-on states: If γ Γ does not fix τ, then γ moves all points from U away from U. Proof. 1. Finiteness: We choose two relatively compact neighbourhoods V i H of τ i, i = 1,2, and show: Only finitely many matrices γ Γ exists with γ(v 1 ) V 2 /0. The idea is to extend the operation of the modular group defined over Z to an operation of SL(2, R) defined over R, hereby embedding each isotropy group of Γ into a compact isotropy group of SL(2,R). Working out this idea we show that H is a homogeneous space for the SL(2,R)- left operation φ R : SL(2,R) H H,(γ,τ) γ(τ) := aτ + b cτ + d, γ := The point i has the isotropy group because one checks SL(2,R) i = SO(2,R) γ(i) = i a = d and b = c. ( ) a b. c d

43 2.3 The modular curve of the modular group 39 The operation φ R is transitive because any point τ H belongs to the orbit of i: If τ = x + iy then γ τ (i) = τ for the matrix γ τ := 1 y Note that γ τ depends continuously on τ. ( ) y x SL(2,R). 0 1 As a consequence for any two points e 1,e 2 H and any γ SL(2,R): γ(e 1 ) = e 2 γ γ e2 SO(2,R) γ 1 e 1. γ e 1 e 2 γ 1 e 1 γ e2 i i If a matrix γ Γ satisfies γ(v 1 ) V 2 /0 then also and γ(v 1 ) V 2 /0 G := {γ SL(2,R) : γ(v 1 V 2 /0} = The latter set is the continuous image of the compact set V 2 SO(2,R) V 1 e 1. e 1 V 1,e 2 V 2 γ e2 SO(2,R) γ 1 and therefore compact. Its intersection G Γ with the discrete group Γ is finite. 2. Shrinking V 1 and V 2 : According to the first part only finitely many γ Γ exist with γ(v 1 ) V 2 /0. In particular, the set F := {γ Γ : γ(v 1 ) V 2 /0 and γ(τ 1 ) τ 2 } is finite. For each γ F we have γ(τ 1 ) τ 2. Because H is a Hausdorff space we can choose two disjoint neighbourhoods U 1,γ H of γ(τ 1 ) and U 2,γ H of τ 2.

44 40 2 Complex tori We define two neighbourhoods of respectively τ 1 and τ 2 in H ( ) ( U 1 := V 1 γ 1 (U 1,γ ), U 2 := V 2 U 2,γ ). γ F γ F Then for all γ F γ(u 1 ) U 2 = /0. Eventually, consider an arbitrary element γ Γ with γ(u 1 ) U 2 /0. We show γ(τ 1 ) = τ 2 by an indirect proof: Otherwise γ F and by construction and Then a contradiction to the choice of γ. U 1 γ 1 (U 1,γ ) and U 2 U 2,γ U 1,γ U 2,γ = /0. (γ(u 1 ) U 2 ) (U 1,γ U 2,γ ) = /0, 3. τ 1 = τ 2 : The add-on referring to the isotropy group Γ τ is the specific case τ 1 = τ 2 and U := U 1 U 2, q.e.d. Definition 2.28 (Proper group operation). Consider a continuous operation F : G X X, (g,x) g(x), of a locally compact group G on a compact space X. The operation is a proper group operation if σ : G X X X X, (g,x) (g(x),x), is a proper map. Corollary 2.29 (Proper operation of the modular group). The operation is a proper group operation. φ : Γ H H Proof. Consider a compact subset K = K 1 K 2 X X. We have to show the compactness of σ 1 (K) G X.

45 2.3 The modular curve of the modular group 41 It suffices to show that σ 1 (K) is sequentially compact. We claim: Any sequence (γ ν,τ ν ) ν N in σ 1 (K) has a convergent subsequence. Due to compactness of K we may assume that the image sequence is convergent. Set σ((γ ν,τ ν )) ν N = (γ ν (τ ν ),τ ν ) ν N x 1 := lim ν τ ν, x 2 := lim ν γ ν (τ ν ). Choose open neighbourhoods U i X of x i,i = 1,2 with the properties from Lemma For all but finitely many ν N we have τ ν U 1 and γ ν (τ ν ) U 2. The Lemma implies for all but finitely many ν N γ ν (x 1 ) = x 2. The set {γ Γ : γ(x 1 ) = x 2 } has the same cardinality like the isotropy group Γ x1. Hence it is a finite set. As a consequence: A suitable subsequence (γ τν k ) k N is constant, hence a posteriori convergent, q.e.d. Corollary 2.30 (Hausdorff topology of the orbit space). The quotient topology on the orbit space Y := Γ \H is a connected Hausdorff space and the canonical projection p : H Y is an open map. Proof. i) Open map: We provide Y with the quotient topology with respect to p, i.e. a subset U Y is open iff p 1 (U) H is open. Connectedness of H implies connectedness of Y. Eventually, p is open: For an open set U H the inverse image p 1 (p(u)) = γ(u) is open as union of open sets. By definition of the quotient topology p(u) Y is open. ii) Hausdorff topology: We use the following general result: Consider a topological space X and an equivalence relation R X X. If the canonical projection γ Γ

46 42 2 Complex tori p : X X/R is an open map with respect to the quotient topology, then X/R is Hausdorff iff R X X is closed. According to Corollary 2.29 the map σ : Γ H H H is proper. It satisfies σ(γ H) = R H H with R the equivalence relation on H induced from the operation of Γ, and the quotient space Y = H/R. Therefore R H H is closed, and Y is Hausdorff, q.e.d. The modular group contains the two distinguished elements Reflection at the y-axis and dividing by squared absolute value: S := ( ) 0 1 Γ, S(τ) = 1/τ = τ 1 0 τ 2. Translation by one: T := ( ) 1 1 Γ, T (τ) = τ Theorem 2.31 (Fundamental domain and elliptic points). 1. The modular group Γ is generated by the two elements S and T. Their relations are generated by the two relations S 4 = id, (ST ) 6 = id. 2. The orbit space Y = Γ \H maps bijectively to the set D := {τ H : τ > 1 and 1/2 Re τ < 1/2} {τ H : τ = 1 and 1/2 Re τ 0}, the fundamental domain of Φ. 3. The operation Φ has exactly two elliptic points z 0 D. Their isotropy groups are cyclic: z 0 = i with Γ i =< S > Γ, ord Γ i = 4,

47 2.3 The modular curve of the modular group 43 z 0 = ρ := e 2πi/3 with Γ ρ =< ST > Γ,ord Γ ρ = 6.

48 44 2 Complex tori Fig. 2.2 Fundamental domain of the modular group [15] Figure 2.2 shows the fundamental domain D of the modular group. In order to exclude pairs of congruent points within D one has to exclude part of the boundary of D. This convention is followed by [15] but not by all references. Most references name fundamental domain the closure D. Proof. Note that the three parts of the subsequent proof do not correspond one to one to the three parts of the theorem. i) The domain D: Denote by G :=< S,T > Γ the subgroup generated by the two distinguished elements. We show: Any given point τ H is equivalent to a point τ D mod G. We choose an element γ 0 G with Im γ 0 (τ) maximal. Such choice is possible because Im γ(τ) = Im τ ( ) a b c τ + d 2 for γ = G. c d

49 2.3 The modular curve of the modular group 45 The translation T Γ does not change the imaginary part of its argument. Therefore we may assume 1/2 Re γ 0 (τ) < 1/2. In addition, γ 0 (τ) 1 because otherwise ( ) γ0 (τ) Im (S γ 0 )(τ) = Im γ 0 (τ) 2 > Im γ 0 (τ). If γ 0 (τ) > 1 then τ := γ 0 (τ). If γ 0 (τ) = 1 but 0 < Re γ 0 (τ) 1/2 then In both cases τ D. τ := (S γ 0 )(τ). ii) Isotropy groups and fundamental domain: Consider two points τ,τ D with ( ) τ a b = γ(τ),γ = Γ. c d We show τ = τ, and the latter equation implies γ Γ τ. Because Im τ = Im τ c τ + d 2. we may assume without restriction Im τ Im τ, i.e. c τ + d 1. Because Im τ 3/2 only the cases c = 0,±1 are possible: Case 1, c = 0: Then d = ±1 and a = d. Without restriction a = d = 1, hence τ = τ + b, b Z. Because Re τ = (Re τ) + b we obtain b = 1; i.e. τ = τ and ( ) 1 0 γ = ± = ± id Γ 0 1 τ. Case 2, c = 1: Case 2a, d = 0: If d = 0 then b = 1 and τ = a τ 1 = a (1/τ), a Z. τ

50 46 2 Complex tori Due to τ D the equation implies τ = 1. Hence c τ + d = τ 1 1/τ = τ and τ + τ = 2 Re τ = a Z. Therefore: Either a = 0 and τ = i which implies and τ = (1/τ) = i = τ γ = ± Or a = 1 and τ = ρ which implies and γ = ± Case 2b, d = ±1: If d = ±1 then ( ) 0 1 = ±S Γ 1 0 i. τ = 1 (1/τ) = ρ = τ ( ) 1 1 = ±(ST ) 2 Γ 1 0 ρ. c τ + d = τ ± 1 1. Because of τ,τ D the case d = 1 with τ 1 1 is excluded. Therefore d = 1 leaving τ + 1 = τ ( 1) 1. As a consequence, τ = ρ and hence a = 0, τ = ρ = τ and ad bc = a b = 1 and τ = a ρ + b ρ + 1 = a + ρ, γ = ( ) 0 1 = ST Γ 1 1 ρ. Case 3, c = 1: This case reduces to case 2, c = 1, by considering γ 0. iii) G = Γ : Consider an arbitrary element γ Γ and the point τ 0 := 2 i D. Due to part i) an element γ 0 G exists with (γ 0 γ)(τ 0 ) D. Due to part ii)

51 2.3 The modular curve of the modular group 47 τ 0 = (γ 0 γ)(τ 0 ) D, i.e. Because S 2 = id we obtain which implies γ G, q.e.d. γ 0 γ Γ τ0 = {±id}. γ 0 γ = ±S 2, It are the elliptic points in particular which complicate the introduction of a complex structure on the orbit space Y. To overcome this difficulty we make a detailed study of the operation in the neighbourhood of an elliptic point. Fortunately, here the operation restricts to the operation of the isotropy group and the latter turns out as a group of rotation. The statement of the following Lemma 2.32 is trivial for non-elliptic points. For the elliptic points it relies on the Lemma of Schwarz. Lemma 2.32 (Isotropy groups operate as rotations). For any point τ H exist a neighbourhood U τ H of τ such that for all γ Γ γ(u τ ) U τ /0 = γ Γ τ, and a fractional linear transformation λ τ GL(2,C) satisfying λ τ : U τ τ with a disc τ = {z : z < r τ } for a suitable radius r τ > 0 such that: For all γ Γ τ the conjugate map from the commutative diagram γ τ := λ τ γ λτ 1 : τ τ γ U τ γ τ (U τ ) λ τ τ γ τ τ λ τ is a rotation around 0 τ. The angle of rotation is an integer multiple of 2π/h τ. Proof. Denote by λ τ : P 1 P 1, λ τ (z) := z τ z τ, the fractional linear automorphism of P 1 with λ(h) = and

52 48 2 Complex tori λ τ (τ) = 0, λ τ (τ) =. We choose a radius r τ > 0 and a neighbourhood U τ of τ with the properties from Lemma 2.27 such that the restriction λ τ U τ : U τ τ is biholomorphic. The conjugate γ τ := λ τ γ λ 1 τ : τ has the fixed point 0. Moreover γ τ (0) = γ (τ) = 1 because in particular S (i) = (ST ) (ρ) = 1. Therefore the Lemma of Schwarz implies that γ τ is a rotation-dilation around 0 τ. Because γ τ (0) = 1 we obtain that γ τ reduces to a rotation. The angle of rotation is a multiple of 2π/h τ, because the isotropy group Γ τ is cyclic with order 2h τ. Note that the whole statement is trivial for all non-elliptic points τ H because here Γ τ = {±id}, q.e.d. Proposition 2.33 (Analytic structure of the orbit space). On the orbit space Y = SL(2,Z)\H exists an analytic structure such that Y becomes a Riemann surface with holomorphic canonical projection p : H Y. Considered from a general point of view the complex manifold Y is the coarse moduli space of complex tori. Proof. i) Homeomorphisms: For each τ H choose a neighbourhood U τ H of τ with the properties from Lemma Using the notation from the Lemma define pow τ : τ C,z z h τ, and set W τ := pow τ ( τ ). Eventually, define a chart of Y around y = p(τ) Y φ τ : p(u τ ) W τ as the commutative completion of the following diagram

53 2.3 The modular curve of the modular group 49 p U τ U τ p(u τ ) λ τ φ τ τ powτ W τ The map φ τ is well-defined and bijective according to Lemma 2.32, continuous, because p is open, open, because λ τ and pow τ are open. As a consequence, φ τ is a homeomorphism. ii) Holomorphic transition functions: We show that the family is a complex atlas of Y. Assume two charts A := (φ τ : p(u τ ) W τ ) τ H φ τi : p(u τi ) W τi around points y i = p(τ i ) Y, i = 1,2, and assume To simplify the subscripts we set Consider a point We show that the map p(u τ1 ) p(u τ2 ) /0. U i := U τi,φ i := φ τi,λ i := λ τi,h i := h τi,w τi := W i, i = 1,2. x p(u 1 ) p(u 2 ). φ 2,1 := φ 2 φ 1 1 is holomorphic in a neighbourhood of φ 1 (x) W 1 C: By definition of φ i two points τ i U i, i = 1,2, exist with λ i ( τ i ) h i = φ i (x), i = 1,2. They satisfy τ 2 = γ( τ 1 ) for a suitable γ Γ. For q C in a neighbourhood of φ 1 (x) ( ) φ 2,1 (q) = (λ 2 γ λ1 1 )(q 1/h h2 1 )

54 50 2 Complex tori Therefore we have to prove: Taking the h 1 -root does not prevent the holomorphy of the map. We prove the non-trivial case h 1 > 1, i.e. the case of an elliptic point τ 1 H:

55 2.3 The modular curve of the modular group 51 Case φ 1 (x) = 0 W 1, i.e. τ 1 = τ 1 : The point τ 1 = τ 1 is elliptic. Therefore also τ 2 = γ( τ 1 ) U 2 is elliptic. Because τ 2 is the only elliptic point in U 2 we obtain τ 2 = τ 2, and τ 2 = γ(τ 1 ) and h := h 2 = h 1. By construction of λ i, i = 1,2, the composition λ 2 γ λ 1 1 is a fractional linear transformation of P 1 mapping 0 to 0 and to. Therefore a matrix ( ) a b A = GL(2,C) c d exists with and As a consequence and φ 2,1 (q) = λ 2 γ λ 1 1 = A A (0 1) = (0 1), A (1 0) = (1 0). A = ( ) a 0 0 d ( ) h ( h (λ 2 γ λ1 1 )(q 1/h ) = A(q )) 1/h = ((a q 1/h )/d) h = (a/d) h q. Therefore φ 2,1 is holomorphic in a neighbourhood of 0 = φ 1 (x). Case φ 2 (x) = 0 W 2, i.e. τ 2 = τ 2 : The previous case shows the holomorphy of φ 1,2 in a suitable neighbourhood. Because the map is bijective, its inverse φ 2,1 is holomorphic. Case φ 1 (x) W 1 arbitrary, i.e. τ 1 U 1 arbitrary: We reduce this case to the other two cases. Choose τ 3 D with p(τ 3 ) = x and consider the chart φ τ3 : U τ3 W τ3 with the shorthand U 3 := U τ3,φ 3 := φ τ3,w 3 := W τ3. In a suitable neighbourhood W of y with W (p(u 1 ) p(u 2 ) p(u 3 ))

56 52 2 Complex tori Fig. 2.3 Transition functions of Y

57 2.3 The modular curve of the modular group 53 we have φ 1,2 = φ 1,3 φ 3,2. The right-hand side is holomorphic due to part i) and ii). Therefore φ 1,2 is holomorphic, q.e.d. The important steps in the proof of Proposition 2.33 and its prerequisites: The modular group operates properly. Isotropy groups operate as rotations (Lemma of Schwarz). Existence of fractional linear maps λ : H. Theorem 2.34 (The modular curve of the modular group). 1. The operation Φ of the modular group Γ extends to a continuous Γ -left operation on the extended upper half-plane 2. On the orbit space Φ : Γ H H H := H Q { }. X := Γ \H exists the structure of a compact Riemann surface such that the following diagram commutes with canonical projections p and p p H Y H X p 3. The modular curve X is biholomorphic equivalent to P 1. The orbit set Y compactifies because the operation of the modular group extends to the point. The orbit of encloses all points of Q. Therefore one has to consider the extended operation on the extended half-plane H. Apparently, Q is not discrete as a subspace of the Euclidean space. This fact hinders the quotient topology from being Hausdorff. Therefore one introduces a coarser topology on the orbit of. The new topology derives from the usual topology around the point.

58 54 2 Complex tori Proof. i) Extending φ to H : The complex plane embedds canonically into the projective space P 1 via C P 1, z (z : 1). The complement P 1 \ C is the singleton with the point := (1 : 0). The Γ -left operation φ : Γ H H extends via the canonical Γ -left operation by matrix multiplication to the Γ -left operation on P 1 In particular Γ C 2 C 2,(γ,z) γ z Γ P 1 P 1, (γ,(z 1 : z 2 )) (a z 1 + b z 2 : c z 1 + d z 2 ). (γ, ) = (γ,(1 : 0)) (a : c) = (a/c : 1). The last expression makes precise the intuitive expression a + b c + d = a + (b/ ) c + (d/ ) = a c. For any rational number q = a/c Q with coprime a,c Z exist integers b,d Z with da bc = 1. As a consequence, the matrix maps γ := ( ) a b Γ c d (γ, ) γ( ) = (a/c : 1) = q Q. If we compactify H by adding the point we have to add also the whole orbit of, i.e. the set Q { }. This additional orbit is named the cusp of the extended operation on the extended upper half-plane with Φ : Γ H H,(γ,τ) γ(τ) := a τ + b c τ + d,

59 2.3 The modular curve of the modular group 55 ( ) a b γ :=. c d ii) Topology on H : Take as neighbourhood basis of in the new topology on H the sets U N := {τ H : Im τ > N} { }, N N, and as neighbourhood basis of a/b Q, gcd(a,c) = 1, the sets ( ) a γ(u N ), N N, γ = Γ. c These neighbourhood bases together with the open sets of H generate a unique topology on H. Apparently H H is an open subspace. Note. Because fractional linear transformations from Γ take circles to circles or lines the neighbourhoods of γ(u N ) are circles tangent to the real axis at the distinguished point a/b Q. The topology in the neighbourhood of is not the usual topology with the complement of compact sets (with respect to the Euclidean topology) as neighbourhoods of. iii) Hausdorff topology on Γ \H : We provide X := Γ \H with the quotient topology with respect to the canonical projection p : H X. Then p is an open map: The proof is the same as the corresponding proof for Corollary The proof uses only the fact that the quotient arises from a group operation. In order to show that X is Hausdorff we consider two points and distinguish the following cases: τ 1 H, τ 2 = : For each the formula x i = p (τ i ) X, τ i H,i = 1,2, and x 1 x 2, Im γ(τ) = 1 Im τ d 2 c = 0 1 c 2 Im 2 c 0 τ shows γ = ( ) a b Γ c d Im τ c τ + d 2 = Im τ (c Re τ + d) 2 + c 2 Im 2 τ Im τ c = 0 1 Im τ c 0 max{im τ,1/im τ}

60 56 2 Complex tori Im γ(τ) max{im τ,1/(im τ)}. Therefore a relatively compact neighbourhood U 1 H of τ 1 and a neighbourhood U 2 H of τ 2 exist such that for all γ Γ As a consequence γ(u 1 ) U 2 = /0. p (U 1 ) p (U 2 ) = /0. τ 1,τ 2 H: This case has been considered already in the proof of Corollary iv) Compactness of X: Denote by D H the closure of D with respect to H. Note that D { } H is compact with respect to H : Consider an open covering (U i ) i I of D { }. If U i0 then for a number N R: The remaining set {τ D { } : Im τ > N} U i0. {τ D { } : Im τ N} = {τ D : Im τ N} H is compact in H, and therefore covered by a finite subcovering of (U i ) i I. As a consequence X is compact as continuous image of the compact set D { }. One has Y = X \ {p ( )}. Therefore Y X is an open subset. v) Charts of the modular curve: In order to define a chart φ : p (U ) W around the point x = p ( ) X we consider the neighbourhood of in H We define and set For two points τ i U,i = 1,2, U := {τ H : Im τ > 2} { }. pow : U, τ { e 2πi τ if τ 0 if τ = W := pow (U ) C.

61 2.3 The modular curve of the modular group 57 τ 2 = p (τ 1 ) τ 2 = γ(τ 1 ) for a suitable γ Γ τ 2 = τ 1 + m for a suitable m Z pow (τ 2 ) = pow (τ 1 ). We define the chart φ as the commutative completion of the diagram p U U p (U ) pow φ W One shows analogously to the proof of Proposition 2.33 that the map φ is a homeomorphism. Eventually we have to show that the charts of Y and the chart φ are compatible, It suffices to prove the compatibility of the chart around p ( ) with the chart φ : p (U ) W φ τ : p(u τ ) = p (U τ ) W τ around an arbitrary point p(τ), τ H. The fact that pow : U W is locally biholomorphic, proves that the transition function of the two charts is holomorphic. vi) Modular curve X P 1 : The image of the fundamental domain Y = p(d) is homeomorphic to the plane C: Identify points on the left and right boundary of D and equivalent points of D on the unit circle. Therefore the compactification of Y is homeomorphic to the 1-point compactification of C, i.e. X is homeomorphic to S 2. By the Riemann mapping theorem the only complex structure on S 2 is the complex projective space P 1 (C), q.e.d. For the fact X P 1 (C) a second proof will be given in Corollary This second proof uses the modular invariant from the theory of modular forms, which provides an explicit identification.

62

63 Chapter 3 Elliptic curves The first two sections of the present chapter deal with Algebraic Geometry. We leave the analytical domain of complex manifolds and investigate non-singular projectivealgebraic varieties. The focus switches from holomorphic and meromorphic functions to polynomials and rational functions. A torus, being a Riemann surface, corresponds to a projective-algebraic curve, named an elliptic curve. Complex Analysis with several variables and Algebraic Geometry are theories with many concepts in common and distinguished concepts in parallel. We compile the following dictionary. Proposition 3.1 (Dictionary Complex Analysis - Algebraic Geometry). Complex Analysis Algebraic Geometry Complex Manifold Smooth variety Euclidean topology Zariski topology Compact complex manifold Smooth projective-algebraic variety Compact Riemann surface Smooth projectiv-algebraic curve Stein manifold Smooth affine variety Structure sheaf O X Holomorphic function Regular function Meromorphic function Rational function Local ring O X,x Line bundle Cohomology Riemann-Roch Theorem Serre Duality Note that not any compact manifold, and not even any compact Kaehler manifold has a counter part in Algebraic Geometry. 59

64 60 3 Elliptic curves In Algebraic Geometry a variety is assumed to be irreducible. Irreducibility corresponds to connectedness in the case of manifolds. If one skips the requirement of smoothness on the right-hand side one obtains the irreducible reduced complex space as the analogue of a variety. Remark 3.2 (Algebraic and analytic geometry). 1. Any smooth projective-algebraic variety X P n (C) can be equipped with the structure of a connected compact complex manifold. In particular, a projectivealgebraic curve C P n (C) carries also the structure of a compact Riemann surface. The idea is to cover X by affine subsets (U i ) i I and to consider the regular transition functions of their charts with respect to the Zariski topology as holomorphic functions with respect to the Euclidean topology of U i, i I. Much deeper is the result in the opposite direction: By a theorem of Chow any connected submanifold of X P n (C) carries the structure of a projective-algebraic variety, i.e. X can be defined by homogeneous polynomials. It is a deep result that many invariants X from the algebraic context translate to invariants in the analytical context. E.g., the genus of a projective-algebraic curve is the same as the genus of its corresponding compact Riemann surface. Relations of this type are subsumed under the keyword GAGA. The latter is a shorthand for Géométrie Algébrique et Géométrie Analytique, the title of Serre s seminal paper [31]. 2. In both contexts, in the algebraic and in the analytic context the same form of the Riemann-Roch theorem (Theorem 2.13) and Serre duality (Theorem 2.14) are valid. For the phrasing of these theorems and their corollaries we will always point to the statements from the analytic category. But in general the proof does not translate from the algebraic category to the analytic category. This is due to the fact that a general compact complex manifold does not embed into a projective space. The case is different for compact Riemann surfaces: All compact Riemann surfaces X embed into P 3 (C). But to prove this result one first needs the Riemann-Roch theorem on X. 3. We will use the analytic theory to deal with modular forms. They turn out to be meromorphic differential forms on the modular curves. The letter are compact Riemann surfaces. The algebraic theory is the context for elliptic curves defined over Q and their reductions to curves over the finite fields F p. Elliptic curves are the bridge to Number Theory. The base field for elliptic curves is no longer restricted to C. Instead we will make a refined study if the curve is defined over the prime field Q or a number field Q k C. If the curve is defined by a polynomial with coefficients from Q

65 3.1 Plane cubics 61 one can clear the denominators and obtain polynomials with even integer coefficients. They can be reduced modulo arbitrary primes. As a consequence, the original curve defined in characteristics 0 is accompanied by a family of curves in prime characteristics. They arise from reducing the coefficients of the defining polynomials. Apparently a rational projective-algebraic curve encodes more information than the set of its complex points. This fact suggests that the basic objects of algebraic geometry are not the point-sets but instead the defining polynomials, or more precisely the corresponding ideal of polynomials. The analogy between Riemann manifolds and projective-algebraic curves is far reaching. But nevertheless, both live in different categories: As a topological space a Riemann manifold carries locally the Euclidean topology of C R 2. On the contrary, a projective-algebraic variety carries the Zariski topology: Its closed subsets are the zero sets of polynomials, i.e. the the algebraic subsets. Both topologies are different in a qualitative aspect: E.g., in dimension = 1 proper closed subsets in the Zariski topology are finite sets. And in general, the Zariski topology is not Hausdoff, in particular the concept of compacteness is not defined. Nevertheless the Zariski has available analogoue concepts: Any projective-algebraic variety X is separared, i.e. the diagonal morphism : X X X is a closed immersion. In addition, any morphisms f : X Y between projective-algebraic varieties is closed, i.e. it maps Zariski-closed subsets of X to Zariski-closed subsets of Y. We use the notation k for a perfect field, i.e. a field with all finite field extensions separable. The algebraic closure of k is denoted k. Typical perfect fields in our context are: Fields of characteristic = 0 like Q or C, more general number fields Q K C. Finite fields like F p or F p n. A typical imperfect field is the field F p (T ) of rational functions with coefficients from F p. As a textbook for the present chapter we recommend [35]. A more general reference to algebraic geometry is the classic [15]. 3.1 Plane cubics Many properties of elliptic curves derive from the Riemann-Roch Theorem. Its numerical results rely on the fact that elliptic curves have genus g = 1. As a first consequence any elliptic curve is a plane curve. It is the zero set of one polynomial

66 62 3 Elliptic curves of degree = 3. Such polynomials are named Weierstrass polynomials. They are the main data structure to represent an elliptic curve. Elliptic curves are distinguished from other projective-algebraic curves by the existence of a group structure on its set of points. We introduce this group structure in two different ways. The first uses the Weierstrass polynomial, the second applies directly the Riemann-Roch Theorem. Theorem 3.17 shows that both methods lead to the same result. Definition 3.3 (Variety with structure sheaf). Consider a field k. 1. A projective-algebraic variety X is the zero set X = V (a) P n (k) of a prime ideal a k[x 0,...,X n ] generated by homogeneous polynomials, i.e. X := {(x = (x 0 :... : x n ) P n (k) : f (x) = 0 for all homogeneous f a}. If the ideal a can be generated by polynomials with coefficients from k then X is defined over k, denoted X/k. 2. The Zariski topology on X is the subspace topology induced from the Zariski topology of P n (k). Closed sets of P n (k) are the projective-algebraic subvarieties. 3. The sheaf O X of regular functions on X is the functor U O X (U) := { f : U k f is regular }, U X affine, with the restriction maps. A function f : U k is regular if locally it has the form f = g h with polynomials g,h k[x 1,...,X n ],h without zeros. 4. The local ring O X,x at a point x X is the ring of germs of regular functions defined in a neighbourhood of x X. 5. The projective-algebraic variety X P n (k) is non-singular or smooth at a point x X if the local ring O X,x is a regular local ring, i.e. if dim O X,x = dim k(x) (m x /m x 2 ) (Ring dimension equals tangent dimension). Here k(x) := O X,x /m x denotes the residue field at the point x. 6. A projective-algebraic curve C/k is a 1-dimensional, non-singular projectivealgebraic variety defined over k. If C P 2 (k)

67 3.1 Plane cubics 63 then C/k is a plane projective-algebraic curve. Definition 3.3 for X/k involves two fields: First, the field of definition k is not necessarily algebraically closed. It is the field where the coefficients of a set of generators of a may be taken from. Secondly, the algebraically closed field k is used to visualize the point set of the ideal a k[x 0,...,X n ]. Also the condition on smoothness refers to k. But the set X P n (k) is not the only point set attached to a. Definition 3.4 will define point sets X(K) attached to any intermediate field K between k and k. We have included the requirement of non-singularity into our definition of a projective-algebraic curve because all curves will be assumed smooth if not stated otherwise. Note that any plane projective algebraic curve is defined by a single homogeneous polynomial. Its degree is the degree of C/k. Definition 3.4 (K-valued points). Consider a projective-algebraic variety X/k with X P n (k). Then for any intermediate field k K k the set of K-valued points of X is defined as X(K) := X P n (K) = {(x 0 :... : x n ) P n (k) : x X and x i K f or all i = 0,...,n}. Definition 3.5 (Elliptic curve). An elliptic curve defined over k is a pair (E,O) with a projective-algebraic curve E/k of genus g = 1 and a k-valued point O E In Definition 3.5 O denotes the upper-case letter O, not the digit 0. The genus of a projective-algebraic curve C is defined as g = dim k H 0 (C,ΩC), 1 the dimension of the k-vector space of regular differential forms, compare Definition 2.11.

68 64 3 Elliptic curves Note that there exist smooth projective-algebraic curves C/Q of genus g = 1 without rational points, i.e. with C(Q) = /0. An example is the Selmer curve in P 2 (C) defined by the inhomogeneous equation 3X 3 + 4Y 3 + 5Z 3 = 0. The fact g = 1 for an elliptic curve E allows a simple application of the Riemann- Roch theorem to compute the vector spaces of multiples of a divisor on E. Lemma 3.6 (Divisors on an elliptic curve). Consider an elliptic curve E and a divisor D Div(E). Then 0 if deg D < 0 dim H 0 0 if deg D = 0, D not principal (E,O D ) = 1 if deg D = 0, D principal deg D if deg D > 0 Proof. An elliptic curve E has genus g = 1. Corollary 2.16 implies for the canonical divisor deg K = deg Ω 1 E = 2g 2 = 0. Hence Corollary 2.15 implies for any divisor D Div(E) dim H 0 (E,O D ) dim H 1 (E,O D ) = dim H 0 (E,O D ) dim H 0 (E,O D+K ) = = 1 g + deg D = deg D If deg D > 0 then deg ( D) < 0. As a consequence { dim H 0 0 if deg D < 0 (E,O D ) = deg D if deg D > 0 The statement for deg D = 0 follows from the fact that a non-trivial line bundle has no holomophic sections, q.e.d. Proposition 3.7 (Rational functions and function field). Consider a projective algebraic variety X/k. 1. Any non-empty open subset U X is dense in X with respect to the Zariski topology. 2. Each ring O X (U),U X affine, not empty, is an integral domain. All these integral domains have the same quotient field. It is named the field k(x) of rational functions of X or the function field of X.

69 3.1 Plane cubics 65 According to Proposition 3.7 for any rational function f k(x) exist a nonempty affine subset U X and two regular functions g,h O X (U), h 0, such that f = g h. The rational functions with all coefficients from k form the subfield k(x) k(x). The following Lemma 3.8 will be used in the proof of Theorem 3.9. It indicates why the existence of a k-valued point is requested for an elliptic curve defined over k. Lemma 3.8 (Divisors defined over a subfield). Consider a curve X/k and a divisor D = n x (x) Div(X) x X which is fixed under the canonical action of the Galois group G k/k, i.e. D = D σ := n x (x σ ) f or all σ G k/k. x X Then the finite dimensional k-vector space O D (X) = { f k (X) : ( f ) D} {0} has a basis of rational functions from k(x). For the proof see [35, II, Prop. 5.8]. Elliptic curves are by definition subvarieties of a projective space P n (k). Theorem 3.9 proves that they are even plane curves, i.e. that they embed as hypersurfaces into P 2 (k). Theorem 3.9 (Elliptic curves are plane cubics). For an elliptic curve(e, O) defined over k exist two rational functions x,y k(e) with poles only at O such that the map { φ : E P 2 (1 : x(p) : y(p)) p O (k), p (0 : 0 : 1) p = O induces an isomorphism onto the plane cubic C P 2 (k), which is defined by a homogeneous polynomial F hom k[x 0,X 1,X 2 ] of the form F hom (X 0,X 1,X 2 ) = X 0 X a 1 X 0 X 1 X 2 + a 3 X 2 0 X 2 (X a 2X 0 X a 4 X 2 0 X 1 + a 6 X 3 0 ) with five suitable constants a i k,i = 1,2,3,4,6. In particular, C is defined over k. Proof. i) We construct a very ample line bundle L = O D on E and define φ := φ L. Therefore consider the divisor

70 66 3 Elliptic curves We claim: The line bundle D := 3(O) Div(E). L := O D Pic(E) satisfies the ampleness conditions from Theorem The proof follows from the dimension formula in Lemma 3.6 because for all points p,q X dim H 0 (E,O D (p) ) = dim H 0 (E,O D ) 1 and dim H 0 (E,O D (p) (q) ) = dim H 0 (E,O D ) 2. For n 0 H 0 (E,O n(o) ) = { f k (E) : f has pole at O of order at most = n} {0} and the dimension formula implies the proper inclusion H 0 (E,O 3(O) ) H 0 (E,O 2(O) ) H 0 (E,O (O) ) = H 0 (E,O E ). Hence a basis (1,x,y) of H 0 (E,L ) exists with y k(e) has a pole of order = 3 at O and no other pole, x k(e) has a pole of order = 2 at O and no other pole. By assumption O E(k). Therefore the divisor D is defined over k. According to Lemma 3.8 we may choose even x,y k(e). With respect to the basis (1,x,y) the line bundle L defines the closed embedding onto a curve C/k in P 2 (k). The family φ := φ L : E P 2 (k), p (1 : x(p) : y(p)) (1, x, y, x 2, xy, x 3, y 2 ) of seven rational functions from the 6-dimensional vector space H 0 (E,O 6(O) ) is linearly dependent. Therefore the functions satisfy a linear relation with coefficients from k. Among them the two functions x 3 and y 2 are the only ones with a pole at O E of order = 6. Hence in the linear relation their coefficients are not zero. After division we may assume that their coefficients are = ±1 and y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 with coefficients a i k. This relation defines the inhomogeneous polynomial (Weierstrass polynomial) F(X,Y ) := Y 2 + a 1 XY + a 3 Y (X 3 + a 2 X 2 + a 4 X + a 6 ) k[x,y ] and its homogenization, the polynomial F hom k[x 0,X 1,X 2 ] with

71 3.1 Plane cubics 67 F hom (X 0,X 1,X 2 ) := X 0 X 2 1 +a 1 X 0 X 1 X 2 +a 3 X 2 X 2 0 (X 3 1 +a 2X 2 1 X 0 +a 4 X 1 X 2 0 +a 6 X 3 0 ). The inhomogeneous variables (X,Y ) relate to the homogeneous variables (X 0,X 1,X 2 ) according to X = X 1 X 0 and Y = X 2 X 0. The curve φ(e) is a subvariety of the 1-dimensional cubic hypersurface C := V (F hom ) {p P 2 (k) : F hom (p) = 0}. The cubic C intersects the line at infinity P 1 (k) {(x 0 : x 1 : x 2 ) P 2 (k) : x 0 = 0} in the single point (0 : 0 : 1). Its affine part is the variety of the Weierstrass polynomial F(X,Y ) k[x][y ] C aff = {(x 0 : x 1 : x 2 ) P 2 (k) : F hom ((x 0 : x 1 : x 2 )) = 0 and x 0 0} = = {(x,y) A 2 (k) : F(x,y) = 0} = V (F). The Weierstrass polynomial F k[x,y ], considered as polynomial in Y with coefficients from the ring k[x], is irreducible: It does not split as F(X,Y ) = (Y + b 1 )(Y + b 2 ) with polynomials b 1,b 2 k[x]. Therefore C aff and also C are irreducible. Hence the non-constant regular map φ : E C is surjective, cf. [15, Chap. II, Prop. 6.8] as an analogue of Proposition 2.4. As a consequence φ(e) = C and E C over k. ii) In order to show that φ is a morphism also in a neighbourhood of O choose an affine neighbourhood U E of O such that x = f x g x and y = f y g y with regular functions satisfying f x,g x, f y,g y O E (U) f x (O), f y (O) 0, g x has a zero of order 2 at O, g y has a zero of order 3 at O. Multiplying for p U \ {O} the homogenous components of φ(p) by g y (p) 0 implies

72 68 3 Elliptic curves φ(p) = (1 : x(p) : y(p)) = (g y (p) : g y (p)/g x (p) : 1). The singularity of φ at O is removable by defining φ(o) := (0 : 0 : 1), q.e.d. In the following we will identify the elliptic curve (E,O) defined over k with the plane cubic from Theorem 3.9. We say that (E,O), more precisely its affine part, is defined over k by the cubic Weierstrass equation F(X,Y ) = 0. Definition 3.10 (Discriminant of a Weierstrass polynomial). Consider a Weierstrass polynomial F(X,Y ) = Y 2 + a 1 XY + a 3 Y (X 3 + a 2 X 2 + a 4 X + a 6 ) k[x,y ] Its discriminant F k is defined as with F := b 2 2b 8 8b b b 2b 4 b 6 b 2 = a a 2, b 4 = 2a 4 +a 1 a 3, b 6 = a a 6, b 8 = a 2 1a 6 +4a 2 a 6 a 1 a 3 a 4 +a 2 a 2 3 a 2 4. Proposition 3.11 (Weierstrass equation in short form). Assume char k 2, 3 and consider a plane elliptic curve (E/k,(0 : 0 : 1)) defined by a homogeneous cubic equation. 1. After applying a projective-linear coordinate transformation from PGL(3,k) the elliptic curve E is defined by a homogeneous polynomial of the form F hom (X 0,X 1,X 2 ) = X 0 X 2 (X AX 2 0 X 1 + BX 3 0 ) k[x 0,X 1,X 2 ], A, B k. Its affine part E aff is defined by the Weierstrass polynomial in short form Its discriminant is F(X,Y ) = Y 2 (X 3 + AX + B) k[x,y ]. F = 16 (4A B 2 ) k. 2. Any two Weierstrass polynomials F(X,Y ) and F (X,Y ) in short form of (E/k,(0 : 0 : 1)) relate to each other by a linear coordinate change ( ) ( ) ( X X u Y = 2 X ) + r Y u 3 Y + s u 2 X +t with constants

73 3.1 Plane cubics 69 u,r,s,t k,u 0 The corresponding transformation of the discriminant is F = u 12 F. 3. In particular, any two Weierstrass polynomials F(X,Y ) and F (X,Y ), which define (E/k,(0 : 0 : 1)) in short form, relate to each other by a linear coordinate transformation If ( ) X Y ( X Y ) := ( u 2 ) 0 0 u 3 ( ) X. Y F(X,Y ) = Y 2 (X 3 + AX + B) and F (X,Y ) = Y 2 (X 3 + A X + B ) then and the j-invariant A = u 4 A, B = u 6 B, F := u 12 F. j F := 1728 (4A)3 = 1728 (4A ) 3 =: j F F F is independent from the choice of a Weierstrass polynomial in short form. For the proof of Proposition 3.11 see [35]: For part 1) cf. Chap. III, 1, for part 2) cf. Chap. III, Prop. 3.1(b) Proposition 3.11 shows that a coordinate transformation allows to eliminate all 12th-power from the discriminant of a Weierstrass polynomial in short form. Example 3.22 will show why it is necessary to represent elliptic curves by a minimal Weierstrass polynomial in short form. Definition 3.12 (Minimal Weierstrass polynomial). For an elliptic curve E/Q = (E/Q,(0 : 0 : 1)) defined over Q a Weierstrass polynomial F Z[X,Y ] in short form is minimal if the discriminant F Z is minimal, i.e. for all Weierstrass polynomials F of E in short form and for all primes p Z ord p ( F ) ord p ( F ). Every elliptic curve defined over Q has a minimal Weierstrass polynomial F. In general, F itself is not uniquely determined, but its discriminant F is unique. It is named min (E).

74 70 3 Elliptic curves Because minimality of the discriminant is a property referring to all primes of Z that property is named global minimality in a more general context. For a number field k with class number = 1, in particular for k = Q, any elliptic curve defined over k has a global minimal model [35, Chap. VIII, Cor. 8.3]. See PARI-file Elliptic_curve_weierstrass_equation_12. The discriminant F k of a Weierstrass polynomial F(X,Y ) k[x,y ] indicates whether the projective-algebraic variety X = V (F hom ) P 2 (k) is non-singular or not. If F 0 then (C/k,(0 : 0 : 1)) is non-singular, i.e. an elliptic curve. For an elliptic curve (E/Q,(0 : 0 : 1)) the prime factors of min (E) are the primes p N with singular reduction mod p of E/Q, see Section 3.3. Proposition 3.13 (Singularities of cubics). Assume k 2, 3 and consider a plane cubic curve C := V (F hom ) = P 2 (k) defined by homogenization of the Weierstrass polynomial in short form with A, B k and discriminant F(X,Y ) = Y 2 (X 3 + AX + B) k[x,y ] F = 16 (4A B 2 ). The curve C has at most one singular point. The curve is non-singular iff F 0 has a node iff F = 0 and A 0 has a cusp iff F = A = 0. For a proof of Proposition 3.13 see [35, Chap. III, Prop. 1.4]. See PARI-file Elliptic_curve_plot_06 for a graphical illustration of Proposition The illustrations show the elliptic curves Y 2 = X 3 3X + 3,Y 2 = X 3 + X,Y 2 = X 3 X and the singular cubics Y 2 = X 3 (cusp),y 2 = X 3 + X 2 (node).

75 3.2 The group structure The group structure In the present section we provide an elliptic curves (E,O) with the structure of an Abelian group with the point O as neutral element This properties distinguishes curves of genus = 1 from curves of any other genus. Due to Theorem 3.9 and Proposition 3.11 we may assume that a given elliptic curve defined over the field k with char k 2,3 is a plane cubic in P 2 (k) defined by the homogenization of a Weierstrass polynomial in short form F(X,Y ) = Y 2 (X 3 + AX + B) k[x,y ]. We introduce the following notation: For two points P Q E we denote by L (P,Q) the line in P n (k) passing through P and Q. The symbol L (P,P) denotes the tangent to E at the point P. Definition 3.14 (Composition Law). Consider an elliptic curve (E, O) defined over k. Then a law of composition is defined by the following construction: : E E E,(P,Q) P Q, Step 1: For arbitrary points P, Q E denote by R E the third point of intersection from L (P,Q) E. Step 2: Set P Q E as the third point of intersection from L (R,O) E.

76 72 3 Elliptic curves Fig. 3.1 Group law on an elliptic curve Theorem 3.15 (Group structure: Geometric version). Assume chark 2, 3 and consider an elliptic curve (E,O) defined over k. The composition : E E E

77 3.2 The group structure 73 from Definition 3.14 is well-defined and has the following properties, in particular it defines an Abelian group on E: i) For all P,Q E: ii) Neutral element: For all P E: iii) Abelian: For all P,Q E: (P Q) R = O P O = P P Q = Q P iv) Inverse: For all P E exists a point P E satisfying If P = (x 0 : x 1 : x 2 ) E then v) Associativity: For all P,Q,S E: vi) For any intermediate field P ( P) = O P = (x 0 : x 1 : x 2 ). (P Q) S = P (Q S). k K k the group law restricts to a group structure on E(K). Proof. We use a simple version of Bezout s theorem ([15, Chap. I, Cor. 7.8]), namely: Any line in P 2 (k) intersects the cubic E in three points. Apparently the statement is equivalent to the fact that any cubic polynomial has three zeros over an algebraically closed field. Therefore a line passing through two points P,Q E intersects E in a third point R. In general R is not the sum P Q but the construction of R is an intermediate step in the construction of the point P Q. To obtain three points of intersection requires to consider points with coordinates from an algebraically closed field. Some remarks to fix the notation: Homogeneous coordinates (X 0 : X 1 : X 2 ) on P 2 (k) Plane cubic E = V (F hom ) P 2 (k) with F hom (X 0,X 1,X 2 ) = X 0 X 2 (X AX 1X BX 3 0 ) k[x 0,X 1,X 2 ] homogenization of a Weierstrass polynomial F(X,Y ) k[x,y ] in short form.

78 74 3 Elliptic curves Inhomogeneous coordinates around (1 : 0 : 0) P 2 (k) on the affine space A 2 (k) = P 2 (k) \ {X 0 = 0} : X := X 1 X 0 and Y := X 2 X 0. Inhomogeneous coordinates around O = (0 : 0 : 1) =: P 2 (k) on the affine space P 2 (k) \ {X 2 = 0} A 2 (k) : Then Ẽ := E {X 2 0} = V ( F) with Tangent to Ẽ at is L = V (G) with i.e. Hence V := X 0 X 2 and U := X 1 X 2. F(U,V ) = V (U 3 + AUV 2 + BV 3 ) G(U,V ) = F U (0,0) U + F (0,0) V, V G(U,V ) = V. L Ẽ = {(u,0) : u 3 = 0} = {(0,0)} with multiplicity = 3. The tangent to E at the point has no further point of intersection with E. i) We compute The third point of intersection from (P Q) R. L (P Q,R) E is R 1 = O. Then L (R1,O) = L (O,O) is the tangent to E at. The preliminary remarks imply L (O,O) E = {O}. Hence by construction (P Q) R = O. ii) Denote by R the third point of intersection from L (P,O) E.

79 3.2 The group structure 75 The third point of intersection from L (R,O) E is P, because the two lines L P,O and L R,O are the same. Hence by construction iii) The proof is obvious by construction. P O = P. iv) Due to part ii) holds O = O. Assume P = (1 : x 1 : x 2 ) O. Then L (P,Q) intersect E in the third point R = O. The line L (O,O) is tangent at E at O, hence it does not intersect E in a point different from O. Therefore P Q = O. v) To verify the law of associativity is a tedious task. One proof is to calculate using coordinates on E as done in [35, Chap. III, Algorithm 2.3]. A second proof employs a calculation with intersection numbers as in [12, Chap. III, Sect. 6, Prop. 4]. A third proof follows from Theorem vi) For two points P,Q E(K) the line L (P,Q) is defined over K. Hence R, its point of intersection with E has coordinates from K. Analogously the line L (R,O) is defined over K and intersects E in the point P Q with coordinates from K. In addition, part iv) shows that P E(K) implies P E(K), q.e.d. Definition 3.14 uses a geometric construction to introduce a group law on an elliptic curve. The construction relies on the fact, that any line intersects the cubic curve E in three points, counted with multiplicity. We now give a second construction of the group structure. It links the points of E to the divisors of degree 0. The second construction relies on the Riemann-Roch theorem and the fact, that an elliptic curve has genus g = 1. The second construction does not make use of any geometric argument. Lemma 3.16 prepares the proof of Theorem The lemma shows that a pair (P,Q) of distinct points from E never defines the divisor D := (P) (Q) of a rational function, though deg D = 0. Note that the lemma gives also a second proof of Corollary 1.9. Lemma 3.16 (No single pole of order one). Consider an elliptic curve (E, 0) defined over a field k. If two points P,Q E satisfy (P) (Q) = ( f ) for a rational function f k(e), then P = Q.

80 76 3 Elliptic curves Proof. Consider the divisor D := (Q) Div(E) with degree deg D = 1. Lemma 3.6 implies dim H 0 (E,O D ) = 1. The inclusion implies k H 0 (E,O D ) k = H 0 (E,O D ). Hence f is a constant function and ( f ) = 0 Div(E), which implies P = Q, q.e.d. Lemma 3.16 shows: The obstructions for a divisor D Div 0 (E) to be principal are divisors of type (P) (Q) or even of type (P) (O). A divisor D Div 0 (E) is the divisor of a rational function up to a divisor of type (P) (O). Because the point P E is uniquely determined by D, the obstruction is expressed by P alone. Theorem 3.17 (Group structure: Version with divisor class group). Consider an elliptic curve (E,O) defined over a field k. i) For any divisor D Div 0 (E) a unique point P E exists with Denote the corresponding map by ii) The map σ is surjective. D (P) (O). σ : Div 0 (E) E, D P. iii) For two divisors D 1,D 2 Div 0 (E) σ(d 1 ) = σ(d 2 ) D 1 D 2. Therefore σ induces a bijective map - also denoted by σ - Cl 0 (E) E. The inverse of σ is the map ρ : E Cl 0 (E), P [(P) (O)]. iv) The maps

81 3.2 The group structure 77 σ : Cl 0 (E) E and ρ : E Cl 0 (E) are isomorphisms between the additive group of classes of divisors of degree = 0 on E and the group (E, ) from Definition Proof. i) The divisor D + (O) has degree = 1. Lemma 3.6 implies dim H 0 (E,O D+(O) ) = 1. We choose a non-zero rational function f k(e), f 0, with divisor satisfying Because ( f ) D (O). deg ( f ) = 0 but deg( D (O)) = 1, a point P E exists with ( f ) = D (O) + (P), i.e. D (P) (O). If also D (Q) (O) for a point Q E, then 0 = D D (P) (Q). Hence for a suitable rational function g k (E) Lemma 3.16 implies P = Q. (P) (Q) = (g). ii) Consider an arbitrary point P E. Set D := (P) (O) Div 0 (E). Then σ(d) is the unique point Q E with D (Q) (O). Therefore σ(d) = P, which proves the surjectivity of σ. iii) Set P j := σ(d j ), j = 1,2, i.e. D j P j (O). Then D 1 D 2 (P 1 ) (P 2 ). On one hand, if σ(d 1 ) = σ(d 2 ) then P 1 = P 2. Hence D 1 D 2 0 i.e. On the other hand, if D 1 D 2 then D 1 D 2.

82 78 3 Elliptic curves (P 1 ) (P 2 ) 0 which implies P 1 = P 2 according to Lemma iv) Consider two arbitrary but fixed points P,Q E. We claim: ρ(p Q) = ρ(p) + ρ(q) Cl (E). The idea is now to translate classical geometry into algebrai geometry. Starting at the geometric construction from Definition 3.14 we represent the distinguished points of E as divisors of suitable rational functions on E. The translation proceeds along the following steps: Convert the information about the points into information about lines: Consider the line L passing through P and Q and its third point of intersection R L E and the line L passing through R and 0 and its third point of intersection P Q L E. Convert the lines into linear homogeneous polynomials: Denote by G hom (X 0,X 1,X 2 ) k[x 0,X 1,X 2 ] the linear homogeneous polynomial with L = V (G hom ) and by G hom (X 0,X 1,X 2 ) k[x 0,X 1,X 2 ] the linear homogeneous polynomial with V (G hom ) = L. Eventually the line at infinite distance L := {(x 0 : x 1 : x 2 ) P 2 (k) : x 0 = 0} is L = V (X 0 ). These three lines incorporate the gemometrically constructed group structure of E. Construct suitable rational functions from these linear polynomials: Consider on E the rational functions G hom, G hom k(e). X 0 X 0 Determine the divisors of these rational functions: The rational function G hom X 0 has zeros of order = 1 at the points P, Q, R of its affine part and at infinite distance a pole of order = 3 at O. Hence its divisor is ) ( Ghom X 0 = (P) + (Q) + (R) 3(O).

83 3.2 The group structure 79 Similar the rational function G hom has the divisor X 0 ( ) G hom = (R) + (O) + (P Q) 3(O). X 0 Hence ( ) ( ) G 0 hom G = hom G hom X 0 ( Ghom X 0 ) = As a consequence = {(P Q) (O)} {(P) (O)} {(Q) (O)} Div 0 (E). 0 = ρ(p Q) ρ(p) ρ(q) Cl 0 (E) i.e. ρ(p Q) = ρ(p) + ρ(q) Cl 0 (E), q.e.d. Corollary 3.18 (Abel s Theorem for elliptic curves). Consider an elliptic curve (E, O) defined over k. A divisor is a principal divisor iff D = n P (P) Div (E) P E [n P ]P = 0 E. P E Here the last equation refers to the group structure on E: [k]p denotes taking k- times the sum of a point P E if k 0 and taking ( k)-times the sum of P E if k 0. Proof. Because deg D = 0 we have Due to Theorem 3.17, part v) D = D n P (O) = n p ((P) (O)) Div 0 (E) P E p E D principal D 0 Div 0 (E) σ(d) = O E σ( n p ((P) (O))) = O E p E [n P ](σ((p) (O)) = O P ]P = 0 (E, ), q.e.d. p E p E[n

84 80 3 Elliptic curves Remark 3.19 (The theorems of Mordell and Faltings). 1. Consider an elliptic curve (E, O) defined over Q. Mordell proved: The Abelian group E(Q) of Q-valued points of E is finitely generated, see [35, Chap. VIII, Theor. 4.1]. As a consequence, E(Q) decomposes as the product of an Abelian torsion group and a free Abelian group of finite rank E(Q) = E tors Z r. Many open questions are related to the rank r. 2. Elliptic curves have genus g = 1. Curves of genus g > 1 do no longer carry a group structure. Mordell conjectured: Curves C/Q with genus g > 1 have only finitely many points with rational coordinates, i.e. card C(Q) is finite. The Mordell-conjecture has been proved by Faltings, see [37]. Any elliptic curve E/k P 2 (C) is not only a projective algebraic curve. When equipped with the Euclidean topology it defines also a compact Riemann manifold. To indicate the different point of view we denote the latter by E an. Theorem 3.20 (Isomorphism between tori and elliptic curves). Consider a torus T := C/Λ with lattice Λ = Zω 1 + Zω 2. Set g 2 := 60 G Λ,4, g 3 := 140 G Λ,6, (Lemma 1.12), and denote by = Λ the Weierstrass function of Λ. The map { φ : T P 2 (1 : (p) : (p)) if p 0 an(c), p (0 : 0 : 1) if p = 0 maps the torus T biholomorphically onto the elliptic curve E an P 2 an(c) defined by the cubic polynomial F hom (X 0,X 1,X 2 ) = X 2 2 X 0 (4X 3 1 g 2 X 1 X 2 0 g 3 X 3 0 ) C[X 0,X 1,X 2 ], the homogenization of F(X,Y ) = Y 2 (4X 3 g 2 X g 3 ) C[X,Y ]. Moreover, φ is an isomorphism of groups between (T,+) and (E an, ). Note. The coordinate transformation X := X, Y := 2 Y

85 3.2 The group structure 81 transforms the inhomogeneous polynomial F(X,Y ) to the Weierstrass polynomial F (X,Y ) = Y 2 (X 3 (1/4) g2 X (1/4) g 3 ). Proof. The proof of Theorem 3.20 makes use of the properties of and from Section 1.2. i) Biholomorphic map: The proof relies on the Riemann-Roch theorem. Its statement is literally the same for both complex analysis and algebraic geometry, cf. Proposition 3.1. Therefore we can copy the corresponding part of the proof of Theorem 3.9 with the substitutions from Proposition 3.1. Hereby, the two rational functions x,,y k(e) can be made explicit by the two meromorphic functions, M (T ). ii) Isomorphism of groups: For the sake of clarity we distinguish the group composition / on the elliptic curve E an from the group composition +/ on the torus T. As a shorthand we denote the image of a point p T by the corresponding upper-case letter We prove: For all p, q T P := φ(p) E an. P Q = φ(p + q). Assume first p q and denote by L (P,Q) P 1 (C) the line passing through P and Q. The line L (P,Q) has the equation c 0 X 0 + c 1 X 1 + c 2 X 2 = 0 with constants c 0,c 1,c 2 C. We attach to the pair (p,q) the meromophic function on T h (P,Q) : T P 1 (C), z c 0 + c 1 (z) + c 2 (z). The zeros of h (P,Q) correspond to the points of intersection from L P,Q E an : {z T : h (P,Q) (z) = 0} L (P,Q) E an,z φ(z). With respect to L (P,Q) we distinguish the following cases : If c 2 0, i.e. the line L (P,Q) does not pass through O = (0 : 0 : 1), then the meromorphic function h (P,Q) has a pole of order 3 at z = 0 T. On one hand, the function has three zeros {p,q,r}, see Proposition 1.19, which satisfy according to Theorem p + q + r = 0 i.e. p + q = r

86 82 3 Elliptic curves On the other hand, according to Theorem 3.15 the three intersection points P, Q, R from L E an satisfy (P Q) R = 0 i.e. P Q = R. Using that the function is even and is odd we obtain P Q = R = (1 : (r) : (r)) = (1 : (r) : (r)) = = (1 : ( r) : ( r)) = φ( r) = φ(p + q). If c 2 = 0 but c 1 0, then the third intersection point of L (P,Q) E an is the point at infinite distance. Therefore The function O = (0 : 0 : 1) = φ(0) P Q O = O i.e. P Q = O. h (P,Q) = c 0 + c 1 has two zeros p i 0,i = 1,2. They satisfy p 2 = p 1 i.e. p 1 + p 2 = 0 because is an even function. We obtain φ(p 1 + p 2 ) = φ(0) = O = P Q. If c 1 = c 2 = 0 then L (P,Q) is the line at infinite distance. This case is excluded: p q implies P := φ(p) Q := φ(q), but the line at infinite distance contains only one single point from E an, namely the point O. So far we have proved the claim for the case of two points p q. From this follows the claim also for the case p = q by a limit argument because all involved maps are continuous, q.e.d. Conversely, we will show in Theorem 4.18 that any elliptic curve can be obtained from a torus with a normalized period lattice via a biholomorphic map Φ according to Theorem 3.20.

87 3.3 Reduction mod p Reduction mod p Reduction mod p is a powerful tool to investigate the information encoded in an elliptic curve defined over Q. The method initiates the arithmetic theory of elliptic curves. Consider an elliptic curve (E,O) defined over Q by a homogeneous polynomial F hom Q[X 0,X 1,X 2 ] being the homogenization of a Weierstrass polynomial F(X,Y ) Q[X,Y ]. As noted subsequently to Definition 3.12 we may assume that F is even a minimal Weierstrass polynomial in short form Then F(X,Y ) Z[X,Y ]. F hom Z[X 0,X 1,X 2 ]. Because the coefficients of F hom are integers, for any prime p reducing mod p these coefficients provides a homogeneous polynomial It defines the projective variety F hom,p F p [X 0,X 1,X 2 ]. E p := {(x 0 : x 1 : x 2 ) P 2 (F p ) : F hom,p (x 0,x 1,x 2 ) = 0} P 2 (F p ). Definition 3.21 (Stable, semistable, unstable). Consider an elliptic curve (E, O) over Q. At a prime p Z the elliptic curve is 1. stable (good reduction) if (E p,o) is non-singular, i.e. an elliptic curve over F p. 2. semistable (bad reduction of multiplicative type) if (E p,o) is singular with a node. If both tangents at the singularity have slope in F p, then the reduction is a split reduction, otherwise a non-split reduction. 3. unstable (bad reduction of additive type) if (E p,o) is singular with a cusp. Note. According to Proposition 3.13 the distinction between stable and not-stable at p depends on the reduction of the minimal discriminant.

88 84 3 Elliptic curves The following example 3.22 shows why it is necessary to study elliptic curves by considering minimal Weierstrass equations. Example 3.22 (Minimal Weierstrass polynomial and reduction mod p). Consider the elliptic curve (E,O) defined over Q by the inhomogeneous Weierstrass polynomial F(X,Y ) = Y 2 (X ) with discriminant Note = 5 6. F = Z. i) The Weierstrass polynomial F Z[X,Y ] is in short form but it is not minimal. Reducing its coefficients mod p = 5 gives the Weierstrass polynomial with discriminant F p (X,Y ) = Y 2 X 3 F p [X,Y ] Fp = 0 F p. Hence the Weierstrass polynomial F p defines a singular cubic with a cusp. Note that the discriminant F contains the 12th-power factor u = 5. ii) The coordinate transformation with the matrix ( 1/u 2 0 ) 0 1/u 3 transforms F(X,Y ) to the minimal Weierstrass polynomial with discriminant F (X,Y ) = Y 2 (X 3 + 1) F = Z. Reducing the coefficients of F mod p = 5 creates the Weierstrass polynomial with discriminant F 5 (X,Y ) = Y 2 (X 3 + 1) F 5 = 3 F 5. The reduction of E mod p = 5 defines an elliptic curve E 5 over F 5. Remark 3.23 (Hasse estimate).

89 3.3 Reduction mod p One of the most interesting properties of an elliptic curve E/Q is the function, which attaches to each prime p of good reduction the order of the groups E p (F p n),n N. These groups collect those points from E p P 2 (F p ) with all coordinates in the finite fields F p n. We recall the following result of Hasse for the case n = 1: 1 + p card E p (F p ) 2 p. For a proof see [18, Theor. 10.5]. See the PARI-file Elliptic_curve_Hasse_estimate_05 for a numerical example. Hasses result had been conjectured by Artin. The value 1 + p for comparison can be motivated as follows: The field F p has p elements. A Weierstrass equation Y 2 = X 3 + AX + B is quadratic in Y. Hence for each x F p there exist at most two solutions (x,y) A 2 (F p ). Taking into account the point at infinity (0 : 0 : 1) E p (F p ) gives the estimate card Ep (F p ) (1 + 2p). In the average, i.e. by choosing the Weierstrass polynomial at random, there is a 50% chance to find a solution (x,y) at all. Therefore 1+ p is a plausible average for card E p (F p ). 2. One defines for any prime p the difference a p (E) := 1 + p card E p (F p ) between the average number and the actual number of F p -valued points of E. The sequence (a p (E)) p prime will turn out to be a fundamental characteristic of the elliptic curve E, see Remark These numbers determine also the cardinalities card E(F p n) for n 2. The local information about an elliptic curve E/Q, which is stable at a prime p, is encoded by the ζ -function Z(E p,t ) of the reduction E p. Therefore we first consider elliptic curves defined over the finite field F p. Definition 3.24 (Zeta-function). Consider an elliptic curve E/F p defined over the prime field F p. The ζ -function of E is the generating formal power series of the sequence (card E(F p n)) n N

90 86 3 Elliptic curves Z(E,T ) := n=1 card E(F p n) T n Q[[T ]]. n Remark 3.25 (Properties of the Zeta-function). The ζ -function of an elliptic curve E/F p, p prime, has the following properties: 1. Rationality: The ζ -function is a rational function, more precisely: Z(E,T ) = with the quadratic polynomial L(E,T ) (1 T ) (1 p T ) Z(T ) L(E,T ) = 1 a p (E) T + p T 2 Z[T ]. 2. Functional equation: The ζ -function satisfies the functional equation Z(E,1/(pT )) = Z(E,T ). 3. Riemann hypothesis: Both complex zeros α, β C of L(E,T ) have modulus α = β = p. The proof of these properties is due to Weil, [35, Chap. V, Theor. 2.2]. He also proved an analogue for projective-algebraic curves of arbitrary genus. Morevoer, Weil made a conjecture for the general case of higher-dimensional smooth projectivealgebraic varieties. Deligne proved the Weil conjecture, cf. [15, Appendix C]. The L-series L(E) of an elliptic curve E/Q collects for each prime p Z the local information about the reduction E p. For stable primes of E this information is the quadratic polynomial L(E p,t ) from the ζ -function of E p. For non-stable primes of E information is added which characterizes the type of singularity. Definition 3.26 (L-series of an elliptic curve). Consider an elliptic curve E/Q defined over Q. For each prime p set L(E p,t ) E stable at p 1 T E semistable at p with split multiplicative reduction L p (T ) := 1 + T E semistable at p with non-split multiplicative reduction 1 E unstable at p The L-series of E is the function

91 3.3 Reduction mod p 87 L(E) : {s C : Re s > 3/2} C, L(E,s) := p prime 1 L p (1/p s ). Note: The absolute convergence of the L-series follows from Hasses estimation in Remark 3.23 and the theory of Euler products, see [18, Chap. X, Cor. 10.6]. Hence L(E) is a holomorphic function. Current research deals with the task of extending the domain of definition for E/K with K a number field, see [35, App. C, 16], [13].

92

93 Chapter 4 Modular forms of the modular group Chapter 1 investigated elliptic functions. Elliptic functions are those meromorphic functions on C which are invariant under the left operation on C of the discrete Abelian group Λ = Zω 1 + Zω 2. We proved that the field M (Λ) of elliptic functions is determined by the Weierstrass -function of Λ, namely M (Λ) = C( Λ )[ Λ ]. Similar to the groups Λ we now consider the modular group Γ := SL(2,Z) which is also discrete but no longer Abelian. We proved in Chapter 2 that Γ acts proper on the upper half-plane H. Moreover this actions extends to H. We showed that the orbit space X := Γ \H is a compact Riemann surface, more precisely X P 1 (C). An obvious question asks for the meromorphic functions on the upper half-plane, invariant under the left operation of the modular group. The complete answer is given by the modular invariant j. The modular invariant is the most prominent example of the larger class of meromorphic functions on H which satisfy a certain symmetry with respect to the modular group and extend meromorphically to. The homolomorphic functions from this class are named modular forms. In this chapter we denote the modular group by Γ := SL(2,Z). 89

94 90 4 Modular forms of the modular group 4.1 Eisenstein series Modular forms are holomorphic functions on H which transform in a simple manner under the operation of Γ. These functions have a discrete non-abelian group of symmetries. In particular, they are invariant under the translation ( ) 1 1 T := Γ. 0 1 Holomorphic functions with period = 1 develop into a convergent Fourier series f (τ) = similar to the trigonometric function n= a n q n, q = e 2πi τ, The holomorphic map surjects to the punctured unit disk cos(2πz) = 1 2 (e2πi z + e 2πi z ). H,τ e 2πiτ, := {z C : 0 < z < 1}. A meromorphic function on H with period = 1 induces a corresponding meromorphic function f on by the definition f (q) := f (τ), q = e 2πi τ. Definition 4.1 (Fourier expansion around ). Consider a meromorphic function f on H invariant with respect to the operation of T and the induced meromorphic function f on with f (e 2πiτ ) := f (τ). 1. The function f is named meromorphic at if f has an isolated singularity at 0 and if this singularity is at most a pole. 2. If f is meromorphic at then the induced function f has a Laurent expansion around 0 f (q) = a n q n, q = e 2πiτ, n=n 0 with a suitable n 0 Z. The elements (a n ) n n0 are the Fourier coefficients of f. 3. A function f, meromorphic at, is named holomorphic at if n 0 N, i.e. if 0 is a removable singularity of f. In this case one defines

95 4.1 Eisenstein series 91 f ( ) := f (0). Note: For a meromorphic function f on H, invariant with respect to T : f holomorphic at lim f (τ) < R > 0 : f (τ) bounded for Im τ R. Im τ Modular forms are not necessarily invariant under the operation of the modular group, but they multiply with an automorphy factor which is a holomorphic function without zeros. Definition 4.2 (Automorphic and modular forms). Consider an integer k Z and a meromorphic function f on H. 1. The function f is weakly modular of weight k if for all γ Γ and for all τ H 1 f (γ(τ)) = f (τ), γ = (cτ + d) k 2. The function f is an automorphic form of weight k if f is weakly modular of weight k and is meromorphic at. 3. The function f is a modular form of weight k if f is weakly modular of weight k, is holomorphic on H, and is holomorphic at. ( ) a b. c d 4. A modular form f of weight k satisfying f ( ) = 0 is a cusp form of weight k. 5. An automorphic function is an automorphic form of weight = 0, i.e. satisfying f γ = f. Similarly, a modular function is a modular form of weight = 0. On denotes by A k (Γ ) M k (Γ ) S k (Γ ) the vector spaces of automorphic forms of weight k, the subspace of modular forms, and the subspace of cusp forms. Note that Definition 4.2 uses the weight convention from [7, Chap. 1.1]. It differs from Serre s convention in [33, Chap. VII, 2] by the factor 2. In the literature also the notation to discriminate between modular functions and modular forms is not uniform. According to Definition 4.2, a modular form of weight k > 0, when considered a function f on H, is not invariant under the operation of the modular group. But

96 92 4 Modular forms of the modular group for even k > 0 one easily finds a related object which is actually invariant: These invariant objects are holomorphic differential forms on H. Proposition 4.3 (Modular forms are invariant differential forms). 1. The Γ -left operation on H induces the Γ -left operation on H 0 (H,Ω 1 H ) with A differential form Φ Ω : Γ H 0 (H,Ω 1 H ) H0 (H,Ω 1 H ), (γ,ω) γ (ω), γ ( f dτ) := ( f γ) dγ. ω = f dτ H 0 (H,Ω 1 H ) is invariant under Φ Ω if and only if f is weakly modular of weight = 2 and holomorphic. 2. More general: For any even k N denote by Ω k/2 H := (Ω 1 H ) k/2 the (k/2)-th tensor power. A differential form is invariant under ω = f dτ k/2 H 0 (H,Ω k/2 H ) Φ Ω k/2 := (Φ Ω ) k/2, the (k/2)-th tensor power of Φ Ω, if and only if f is weakly modular of weight k and holomorphic. Proof. 1) For we compute Therefore (dγ)(τ) = d γ = ( ) a b Γ c d ( ) aτ + b a(cτ + d) (aτ + b)c 1 = cτ + d (cτ + d) 2 dτ = (cτ + d) 2 dτ Φ Ω (γ, f dτ) = ( f (γ(τ)) d(γ(τ)) = f (τ) (cτ + d) 2 = f (τ) dτ. 2) For even k > 2 the proof is analogous, q.e.d. 1 (cτ + d) 2 dτ

97 4.1 Eisenstein series 93 Remark 4.4 (Factor of automorphy). The following definition is slightly more general than needed in the present context. We consider the action of on the upper half-plane GL(2,R) + := {A GL(2,R) : det A > 0} Φ GL + : GL(2,R) + H H,(A = ( ) a b,τ) A(τ) := aτ + b c d cτ + d, which extends the operation Φ of the modular group from Definition The map h : GL(2,R) + H H, h(a,τ) := cτ + d, is named the automorphy factor of Φ GL We denote by M (H) the complex vector space of meromorphic functions on H. For a matrix A GL(2,R) + and an integer k Z we define the linear operator with [A] k : M (H) M (H), f f [A] k, f [A] k (τ) := (det A) k/2 h(a,τ) k f (A(τ)). Employing the new notation, the condition of weak modularity of weight k from Definition 4.2 reads: f [γ] k = f, γ Γ. 3. The factors of automorphy satisfy the composition rules h(b A,τ) = h(b,a(τ)) h(a,τ) [B A] k = [B] k [A] k. Note. The latter operators apply to a function written on their left-hand side. Proof. i) For A,B GL(2,R) + and τ H: ( ) ( ) τ A(τ) A = h(a,τ). 1 1 Here we distinguish between the matrix product of A with a vector on the lefthand side and the fractional linear transformation A(τ) on the right-hand side. Applying this identity for the product B A ( ) ( ) τ (B A)(τ) (B A) = h(b A,τ). 1 1

98 94 4 Modular forms of the modular group Left-hand side: ( ( )) (( ) ) τ A(τ) B A = B h(a,τ) = 1 1 ( (B(A(τ)) Equating with the right-hand side gives for the lower row hence for the upper row h(b A,τ) = h(b,a(τ)) h(a,τ), (B A)(τ) = B(A(τ)). ii) Starting with the right-hand side we compute ( f [B] k )[A] k (τ) = ( f [B] k )(A(τ)) h(a,τ) k det A = 1 ) h(b,a(τ)) h(a,τ). = f (B(A(τ))) h(b,a(τ)) k h(a,τ) k det B det A = = f ((B A)(τ)) h(b A,τ) k det (B A) = f [B A] k (τ), using the result from part i) to obtain the penultimate equality, q.e.d. As a consequence of Theorem 2.31 and Remark 4.4 it is sufficient to require in Definition 4.2 the transformation law only for the two generators S,T of Γ. Modular forms f are distinguished by their two properties 1. On one hand, they have the discrete group SL(2,Z) as group of symmetries. 2. On the other hand, they expand into a Fourier series around the point. The interplay of these two properties is coined The magic of modular forms [39]. Therefrom a series of remarkable identities between the Fourier coefficients of modular forms arises. In many cases these identities imply interesting statements about arithmetic functions. We will see examples in Chapter 5 and Chapter 6. Remark 4.5 (Modular functions of odd weight). A modular function of odd weight k Z satisfies for all τ H and for the particular element ( ) 1 0 γ = Γ 0 1 the equation f (γ(τ)) = f (τ) = ( 1) k f (τ). Therefore f = 0 is the only modular function of odd weight.

99 4.1 Eisenstein series 95 We will now construct modular forms G k of even weight k 4 and a distinguished holomorphic function G 2, which behaves in a different way under symmetries. To achieve the construction we need the convergence of certain infinite series. Lemma 4.6 provides the necessary prerequisites. Lemma 4.6 (Some convergent series). Assume k N. For τ C set q := e 2πi τ. 1. For k 2 and Im τ 0 n Z r 1 k 1 q r Im τ > 0 (τ + n) k = ( 2πi)k r=1 (k 1)! ( 1) k r k 1 q r Im τ < 0 2. For Im τ > 0 the double series is absolutely convergent. 3. For even k 2, Im τ > 0, ( m Z n Z r=1 ) d q m=1( md d=1 ) 1 (mτ + n) k = 2 (2πi)k (k 1)! n=1 σ k 1 (n) q n. Here for any k 2 the arithmetic function has the values σ k 1 : N N σ k 1 (n) := d k 1, n N, d n the sum of the (k 1)-powers of the non-negative integer divisors of n. Proof. 1. i) We first consider the case Im τ > 0. We expand the cotangent function into a partial fraction π cot(πτ) = 1 ( 1 τ + m=1 τ + m + 1 ) τ m and also into a Fourier series π cot(πτ) = π cos(πτ) 2πi = πi sin(πτ) 1 q = πi 2πi q m. m=1

100 96 4 Modular forms of the modular group Here the second equality results from the Euler formulas which imply The third equality uses cos πτ = 1 2 (eiπτ + e iπτ ) and sin πτ = 1 2i (eiπτ e iπτ ) cot(πτ) = i eiπτ + e iπτ e iπτ e iπτ = i q + 1 q 1 = ( i) 1 + q 1 q 1 + q 1 q = (1 + q) q m = m=0 m=1 The geometric series is absolutely convergent because q < 1 due to τ H. Equating both expansions gives ( 1 1 τ + m=1 τ + m + 1 ) = πi 2πi τ m q m m=1 q m, and successive (k 1)-times differentiation with respect to τ shows m Z ii) For Im τ < 0 we first use 1 (τ + m) k = ( 2πi)k (k 1)! m=1 m k 1 q m. cot(πτ) = i eiπτ + e iπτ 1 + (1/q) e iπτ = i e iπτ 1 (1/q) = i(1+(1/q)) m=1 q m = i+2i Equating the partial fraction expansion and the Fourier expansion gives ( 1 1 τ + m=1 τ + m + 1 ) = πi + 2πi τ m q m, m=1 and successive (k 1)-times differentiation with respect to τ shows m Z 1 ( 2πi)k = ( 1)1+(k 1) (τ + m) k (k 1)! m=1 m k 1 q m. q m m=1 2. For any complex number x C with x < 1 the geometric series is absolutely convergent satisfying d=0 x d = 1 1 x

101 4.1 Eisenstein series 97 Taking the derivative and multiplying by x shows d x d x = d=1 (1 x) 2. For m 1 set x = q m with q = e 2π Im τ < 1 gives m=1( d=1 d q md ) = m=1( d=1 d (q m ) d ) = m=1 The last series is absolutely convergent because for fixed q and. Part 1) justifies the second equal- m>0 ) 3. We split the summation into m Z m<0 ity sign: ( m<0 (2πi) k (k 1)! ( n Z lim 1 m qm 2 = 1. q m (1 q m ) 2. 1 m Z n Z (mτ + n) k = ) ( ) 1 1 (mτ + n) k + m>0 (mτ + n) k = n Z ( ) )) r m<0( k 1 q rm + r r=1 m>0( k 1 q rm = r=1 = 2 (2πi)k (k 1)! m=1 ( r=1 r k 1 q rm ). For k 4 any rearrangement is permitted. For k = 2 the last series is absolutely convergent according to part 2). We rearrange the series by changing the indices of summation from (m,r) to (n = rm,d n): m=1( r=1 r k 1 q rm ) = n=1 ( d n d k 1 q n ) = n=1 σ k 1 (n) q n, q.e.d. Theorem 4.7 (Eisenstein series, k 4). Consider an even integer k > 2. Then: 1. The Eisenstein series G k (τ) := 1 (c,d) Z 2 (cτ + d) k is absolutely and compactly convergent for τ H. The series defines a modular form G k of weight k with

102 98 4 Modular forms of the modular group Here ζ (s) := denotes the Riemann ζ -function. G k ( ) = 2ζ (k). n=1 (1/n s ), Re s > 1, 2. The Eisenstein series G k M k (Γ ) has the Fourier expansion G k (τ) = 2ζ (k) + 2 ( 2πi)k (k 1)! σ k 1 (n) q n, q = e 2πiτ. n=1 The normalized Eisenstein series of weight k is the function E k := G k 2ζ (k). The identity between the Riemann ζ -function and the Bernoullis numbers shows E k (τ) = 1 2k B k 2 ζ (k) = (2πi)k B k k! n=1 σ k 1 (n) q n M k (Γ ). In particular, all Fourier coefficients of E k are rational. The first values (k,ζ (k),b k, 2k B k ) are (4,(π 4 / ), (1/30),240) (6,(π 6 / ),(1/42), 504) (8,(π 8 / ), (1/30),480) (10,(π 10 / ),(5/66), 264) (12,(691 π 12 / ), (691/2730),(65520/691)) Proof. i) Modularity: For τ H consider the normalized lattice Λ τ := Z + Zτ and for k 2 its lattice constants G Λτ,k. Apparently G k (τ) := G Λτ,k

103 4.1 Eisenstein series 99 The series G Λ,k is absolute convergent due to Lemma 1.12, hence allows arbitrary rearrangement. With respect to the translation T : H H, τ T (τ) := τ + 1, we obtain With respect to the reflection G k (τ) = G k (τ + 1). we have S : H H, τ S(τ) := 1 τ 1 τ k G k( (1/τ)) = 1 τ k (c,d) Z 2 1 ( c(1/τ) + d) k = 1 τ k (c,d) Z 2 τ k ( c + dτ) k = G k(τ). Because S and T generate the modular group we have for all γ Γ and for all τ H G k (τ) = h(γ,τ) k G k (γ(τ)). Note that h(γ,τ) k depends holomorphically on τ. ii) Holomorphy: The lattice constant G Λρ,k, ρ = e 2πi/3, dominates the Eisenstein series and shows its compact convergence on D: We obtain for τ D and c, d Z using τ 1 and Re τ 1/2 Therefore c τ + d 2 = c 2 τ 2 + 2cd Re τ + d 2 c 2 cd + d 2 cd 0 c ρ + d 2 cd 0 = c 2 + cd + d 2 cd < 0 c ρ d 2 cd < 0 1 (cτ + d) k (c,d) Z 2 (c,d) Z 2 1 (cρ + d) k + (c,d) Z 2 1 = 2 (cρ + d) k < (c,d) Z 2 1 (cρ d) k = Hence the restriction G k D is compact convergent. Each summand of G k is holomorphic in H. Therefore G k D is holomorphic. For any γ Γ and τ D we have

104 100 4 Modular forms of the modular group G k (γ(τ)) = h(γ,τ) G k (τ) with h(γ, τ) 0 and holomorphic with respect to τ. As a consequence, the restriction G k γ(d) is absolutely and compactly convergent, hence holomorphic. The representation H = γ(d) γ Γ shows that G k : H C is absolutely and compactly convergent, hence also holomorphic. iii) Fourier expansion: We split the summation over (c,d) Z 2,(c,d) (0,0), into the simple sum for (c,d) with c = 0 and d Z and the double sum with c Z and d Z. G k (z) = (c,d) Z 2 1 (cτ + d) k = 1 d Z = 2ζ (k) + 2 (2πi)k (k 1)! n=1 d k + c Z,d Z σ k 1 (n) q n, q = e 2πi τ, 1 (cτ + d) k = due to Lemma 4.6, part 3). The Fourier expansion proves the holomorphy of G k at the point, q.e.d. Remark 4.8 (Eisenstein series and lattice constants). For even k > 2 the Eisenstein series G k relate to the lattice constants G Λ,k of an arbitrary lattice with positively oriented basis Λ = Zω 1 + Zω 2 as follows: Set τ := ω 2 /ω 1 H. Then G Λ,k = 1 ω1 k G k (τ). Similar to the series n=1 (1/n) in one dimension, in two dimensions the series 1 (mτ + n) 2 (m,n) Z 2

105 4.1 Eisenstein series 101 is not convergent with respecto to any order of summation. Anyhow, in both cases a special order of summation exists which provides a finite value. In the 2-dimensional case we obtain the Eisenstein series G 2. Remark 4.9 (Eisenstein series, k = 2). 1. The Eisenstein series is holomorphic on H. G 2 (τ) := d=1 ( 1 d 2 + c Z d Z 2. The Eisenstein series has the Fourier expansion G 2 (τ) = 2ζ (2) 8π 2 n=1 ) 1 (cτ + d) 2 σ 1 (n) q n, q = e 2πi τ, and the normalized Eisenstein series of weight k = 2 is E 2 (τ) := G 2(τ) 2ζ (2) = 1 24 σ 1 (n) q n. n=1 3. The Eisenstein series G 2 is invariant with respect to the translation T Γ. With respect to the reflection S Γ it transforms as In particular, G 2 is not weakly modular. G 2 (S(τ)) = τ 2 G 2 (τ) 2πiτ. Proof. ad 1) and 2): Lemma 4.6, part 3) applied to the second summand shows ( ) ( ) 1 1 c Z d Z (cτ + d) 2 = d Z d c Z d Z (cτ + d) 2 = = 2ζ (2) 8π 2 n=1 σ 1 (n) q n,q = e 2πi τ. The Fourier representation shows: G 2 is holomorphic on H { } and invariant with respect to translation. ad 3) Cf. [20, Kap. III, 6, Abschn. 1]. The lack of modularity is due to the conditional convergence G 2 of G 2. Definition 4.10 (Discriminant form).

106 102 4 Modular forms of the modular group The discriminant form is the modular form of weight k = 12 defined as := g g2 3 : H C derived from the Eisenstein series g 2 := 60 G 4, g 3 := 140 G 6. Proposition 4.11 (Fourier coefficients of the dicriminant form). All Fourier coefficients of the normalized discriminant modular form are integers: (2π) 12 (τ) (2π) 12 = τ n q n Z{q}, q = e 2πi τ, n=0 with τ 0 = 0 and τ 1 = 1. In particular S 12 (Γ ) is a cusp form of weight 12. Proof. We refer to the Fourier series expansion from Theorem 4.7 for the normalized Eisenstein series In addition, E 4 = G4 π 4 = σ 3 (n) q n n=1 E 6 = G6 2 π 6 = σ 5 (n) q n. n=1 Hence g 2 = 60 G 4 = 22 3 π4 E 4 and g 3 = 140 G 6 = π6 E 6. and = g g 2 3 = π (E3 4 E2 6 ) = (2π)12 E3 4 E = (2π)12 E3 4 E We set and obtain (τ) (2π) 12 = E3 4 E = E3 4 E X k (q) := n=1 σ k (n) q n

107 4.1 Eisenstein series 103 (2π) 12 = ( X 3) 3 (1 504 X 5 ) Denotbe by Q = Q(q) the numerator series Q of the latter fraction. The explicit form of Q(q) proves τ 0 = 0. Moreover Q = X X X X X 2 5 In lowest order with X 3 (q) = q + O(2) and X 5 (q) = q + O(2) we get which proves τ 1 = 1. Q(q) = q q + O(2) = 1728 q + O(2) The remaining part of the lemma reduces to the claim that also all other coefficients of the series Q are divisible by We compute Q = X X X X X 2 5 Q 12 2 (5 X X 5 ) mod 12 3 To complete the proof we have to show that all coefficients of the power series 5 X X 5 are divisible by 12. We show even more: For all d Z 12 divides 5 d d 5. According to the Chinese remainder theorem the claim reduces to the two congruences 5 d d 5 0 mod 3 and 5 d d 5 0 mod 4 or The latter two congruences reduce to d 5 d 3 mod 3, mod 4. d 2 1 mod 3, mod 4 if d 3 0 mod 3, mod 4 The latter congruences are obvious, q.e.d. Definition 4.12 (Modular invariant). The modular invariant is the automorphic function defined on H as

108 104 4 Modular forms of the modular group j := 1728 g3 2 = 1728 g 3 2 g g2 3 Proposition 4.11 shows: j is a well-defined meromorphic function because its denominator is not constant. The proposition shows in addition, that j has a pole of order = 1 at the point with residue = The algebra of modular forms Continuing with the investigation started in Proposition 4.3 we show: Modular forms of weight k are those meromorphic differential forms on the modular curve which are multiples of a certain divisor which depends on k. Applying once more the Riemann-Roch theorem we draw several conclusions from this result: We derive a formula for the dimension of the vector spaces M k (Γ ). We show: The modular invariant j is a biholomorphic map between the modular curve and the Riemann surface P 1 (C). We show: Any complex elliptic curve arises from a complex torus. More precisely, the elliptic curve is parametrized by the two elliptic functions Λ and Λ defined on a suitable torus C/Λ, see Theorem We continue with the notation Γ = SL(2,Z). By denoting the vector space of modular forms of weight k, we will often skip Γ and set M k := M k (Γ ). We denote by p : H X the canonical projection to the modular curve, skipping the -superscript used in the notation from Theorem We introduce the notation [τ] = p(τ). In addition we use the floor function: For x R x := max{n Z : n x}. Theorem 4.13 (Modular forms are differential forms on the modular curve). Denote by Γ := SL(2,Z) the modular group and by X := H \Γ its modular curve. Consider an even integer k Z.

109 4.2 The algebra of modular forms Differential forms: The map of vector spaces ( ) α : M k (Γ ) H 0 X,M X Ω k/2 X, f ω f := p ( f dτ k/2 ), is injective. 2. Divisor: Consider the divisor The attachment with Ω Γ,k (U) := D k := k 2 ([ ]) + k 4 ([i]) + k ([ρ]) Div(X). 3 U Ω Γ,k (U), U X open, { } ω H 0 (U,M X Ω k/2 X ) : ω 0 and (ω) D k U {0} is a line bundle Ω Γ,k on X. The image of the map α from part 1) is α(m k (Γ )) = H 0 (X,Ω Γ,k ). If K Div(X) denotes the canonical divisor of X, then Ω Γ,k O (k/2)k+dk. Proof. 1. If f M k (Γ ) then Proposition 4.3 implies that the modular group leaves invariant the differential form f dτ k/2 on H. Hence the latter defines a meromorphic form ( ) α( f ) := ω f := p ( f dτ k/2 ) H 0 X,M X Ω k/2 X. 2. The sheaf Ω Γ,k is well-defined. The definition of the divisor D k results from the following relation between the order of f at τ H and the order of ω f at [τ] X: ord [ ] ω f + (k/2) ord τ f = h τ ord [τ] ω f + (k/2) (h τ 1) τ = τ H Here h τ denotes the period of τ, see Definition To prove the relation for τ {, i, ρ} we use the local form of according to Theorem 2.34: p : H X

110 106 4 Modular forms of the modular group For t H { } set Then q(t) := { e 2πit t 0 t = q := e 2πit is a local parameter around [ ] X satisfying dq = 2πiq dt or dt = dq 2πiq. Hence with dq k/2 ω f = p ( f (t) dt k/2 ) = f (q) (2πiq) k/2 f (q(t) := f (t). Determining the order of the right-hand side at q = 0 and using the definition ord f := ord 0 f, shows ord [ ] ω f = ord f (k/2). For τ {i, ρ} choose a local parameter t on H around τ such that the isotropy group Γ τ operates by multiplication by e-th root of unity, e = h τ. Then q = t e is a local parameter on X around [τ]. Denote by N the order of ω f at [τ], such that ω f (q) = u(q) q N dq k/2, u([τ]) 0. One has dq = e t e 1 dt dq k/2 = t (k/2)(e 1) e k/2 dt k/2 and q N = t e N. Hence f (t) dτ k/2 = p ( ω f (q) ) = p ( u(q) q N dq k/2) = = u(q(t)) e k/2 t N e+k/2(e 1) dt k/2, and In particular: ord τ f = N e + (k/2) (e 1) = h τ ord [τ] ω f + (k/2) (h τ 1).

111 4.2 The algebra of modular forms 107 ord [τ] ω f = ord τ f, ord τ f = ord [τ] ω f, τ D \ {i, ρ}, ord [ ] ω f = ord f (1/2)k, ord f = ord [ ] ω f + (k/2) ord [i] ω f = (1/2) ord i f (1/4)k, ord i f = 2 ord [i] ω f + (k/2) ord [ρ] ω f = (1/3) ord ρ f (1/3)k, ord ρ f = 3 ord [ρ] ω f + k. As a consequence { α(m k (Γ )) = ω H 0 ( X,M X Ω k/2 X ) : (ω) k 2 ([ ]) k ([i]) 4 } k ([ρ]). 3 According to Proposition 2.8 the sheaf Ω 1 X is isomorphic to the subsheaf O K M X. Therefore Ω k/2 O (k/2)k M X and Ω Γ,k O (k/2)k+dk M X. The map α is injective because p is a branched covering. The map α is surjective: For ω H 0 (X,Ω Γ,k ) the pull back p (ω) = f dτ k/2 defines a modular form f M k (Γ ) with α( f ) = ω, q.e.d. Theorem 4.13 explains the name modular form: Modular forms are differential forms. The formulas in the proof of part 2) show: While a modular form f on H is holomorphic when considered as a function, its representing differential form ω f on the modular curve X may have poles. Actually, it is the divisor of ω f which encodes the information about f. Corollary 4.14 (Dimension of M k (Γ )). The vector spaces M k (Γ ) of modular forms and S k (Γ ) of cusp forms of weight k have the following dimensions: For even k 4 holds dim M 0 (Γ ) = 1, and for even k < 4 : dim M k (Γ ) = 0. dim M k (Γ ) = Proof. We compute for X P 1 the dimension { k/12 k 2 mod 12 k/ otherwise H 0 (X,Ω Γ,k )

112 108 4 Modular forms of the modular group by the Theorem of Riemann-Roch according to Corollary We claim deg( D Γ,k + K) < 0 : The divisor has degree D Γ,k := (k/2)k + D k Div(X) deg D Γ,k = k + (k/2) + k/4 + k/3 = (k/2) + k/4 + k/3. Hence deg( D Γ,k + K) = (k/2) 2 k/4 k/3 1 using k/4 < (k/4) + 1 and k/3 < (k/3) + 1. Therefore Corollary 2.15 implies for all even k Z dim M k (Γ ) = dim H 0 (X,Ω Γ,k ) = deg D Γ,k = 1 (k/2) + k/4 + k/3. We consider the different values of k mod12 separately: i) k 0 mod 12: 1 (k/2) + (k/4) + (k/3) = 1 + (k/12) = 1 + k. 12 For the remaining values of k mod 12 we make the ansatz k = α 12 + β ii) β = 2: 1 (k/2) + k/4 + k/3 = 1 6α 1 + 3α + 4α = α = α = k/12 iii) β = 4: 1 (k/2) + k/4 + k/3 = 1 6α 2 + 3α α + 1 = 1 + α = 1 + k/12 iv) β = 6: 1 6α α α2 = 1 + α = 1 + k/12 and similarly for the two remaining cases β = 8 and β = 10, q.e.d. In particular

113 4.2 The algebra of modular forms 109 M k = 0,k < 0, dim M 0 = 1 dim M 2 = 0, dim M 4 = dim M 6 = dim M 8 = dim M 10 = 1, dim M 12 = 2. The non-zero vector spaces in the last line are spanned by the Eisenstein series G 4, G 6 and their products. Only M 12 contains a non-zero cusp form, namely the discriminant modular form. Corollary 4.15 (Weight formula for modular forms). Consider an even integer k 4 and a modular form f M k (Γ ). Then the sum of the orders of f satisfies the following equation: ord f ord i f ord ρ f + ordτ f = k τ 12 Here the symbol denotes the summation over all points τ D \ {i, ρ}. Proof. We recall the relation between the order of f M k (Γ ) at a point τ and the order of the differential form ω f H 0 (X,M X Ω k/2 ) at the point p (τ) established during the proof of Theorem 4.13: ord f = ord [ ] ω f ord i f = 2 ord [i] ω f + k 2 ord ρ f = 3 ord [ρ] ω f + k ord τ f = ord [τ] ω f, τ D \ {i, ρ}. The modular curve X has genus g = 0 according to Theorem The line bundle Ω X k/2 has degree (2g 2)(k/2) = k according to Corollary Therefore = ( ord [ ] ω f + k 2 deg(ω f ) = k. ord f ord i f ord ρ f + ) + ( ord [i] ω f + k 4 ) + ( τ ordτ f = ord [ρ] ω f + k 3 = k + k 2 + k 4 + k 3 = k 12, q.e.d. ) + ord[τ] ω f = τ Note: A non-zero modular form f M 4 has zeros exactly along the orbit of ρ. While a non-zero modular form f M 6 has zeros exactly along the orbit of i. In both cases, each zero has order = 1.

114 110 4 Modular forms of the modular group Corollary 4.16 (Modular curve and modular invariant). 1. The discriminant form has no zeros in H. 2. The modular invariant is a bihomorphic map j : X P(C). It attains the distinguished values j(i) = 1728 and j(ρ) = All Fourier coefficents (b n ) n 1 of the modular invariant are integers, and b 1 = 1. j(q) = n= 1 b n q n Proof. i) According to Proposition 4.11 the discrriminant modular form has a zero of order 1 at the point, i.e. ord = 1. According to Corollary 4.15 the modular form of weight k = 12 has no further zeros. ii) Part i) implies that the modular invariant has a single pole at, and this pole has order 1. We consider j as holomorphic map on the quotient X = Γ \H j : X P(C). According to Proposition 2.4 the modular invariant j is surjective and assumes all values with the same multiplicity. Therefore j is biholomorphic. Corollary 4.15 implies G 4 (ρ) = 0 and G 6 (i) = 0. The definition j = 1728 g 2 3 g g 3 2 implies the values of j at the distinguished points. iii) To compute the Fourier expansion of j = g3 2 we combine from Proposition 4.11 and from its proof two formulas. Firstly, the formula g 2 = 22 3 π4 E 4

115 4.2 The algebra of modular forms 111 which implies and secondly the formula g 3 2 = π12 E 3 4, (2π) 12 = a n q n Z{q}. n=1 We obtain j = g3 2 = π12 = 1 n=1 a n q n = 1 q a n q n 1 n=2 and Z{q}, q.e.d. a n q n 1 n=2 Corollary 4.17 (The field of automorphic functions). 1. If an automorphic form f A 0 (Γ ) is holomorphic on H with Fourier series f (τ) = c n q n n k for a suitable k N, then a poylnomial P(X) C[X] of degree k exists with f = P( j), and all coefficients of P(X) are Z-linear combinations of the Fourier coefficients c k,...,c 1,c The field of automorphic functions is A 0 (Γ ) = C( j). Proof. 1. The proof goes by induction on k and uses the weight formula from Corollary 4.15: If k = 0 than f is a modular function, hence equals the constant c 0. For the induction step k 1 k consider the function g(τ) := f (τ) c k j(τ) k. The function g has at most a pole of order k 1 at. Therefore the induction assumption applies to g.

116 112 4 Modular forms of the modular group 2. Denote by A := {x 1,...,x n } H { } and B = {y 1,...,y n } H { } the respective zero set and pole set of f, counted with multiplicity. Then g := n ν=1 j j(x ν ) j j(y ν ) A 0(Γ ) has the same zero set and pole set as f. The quotient f /g has no poles and zeros on H { }. According to part 1) a constant a C exists with a = f g or f = a g C( j), q.e.d. We now prove the converse of Theorem Theorem 4.18 (Uniformization of elliptic curves E/C by tori). For any elliptic curve E defined over C and equipped with the analytic structure from E an P n (C) exists a torus T = C/Λ and a biholomorphic map Φ : T E an. Proof. Assume that E is defined by the homogenization of the Weierstrass polynomial F(X,Y ) := Y 2 (4 X 3 A X B) C[X,Y ]. Making the substitution Y = 2 Y and applying Proposition 3.11 shows for the discriminant F = A 3 27B 2 0. Note the signature of the different coefficients of F when applying the proposition. We search for a lattice Λ = Zω 1 + Zω 2 with lattice constants G Λ,4 and G Λ,6 satisfiying The restriction of the modular invariant A = 60 G Λ, 4 and B = 140 G Λ, 6. g 3 2 j H : H C, j(τ) := 1728 (τ) g 3 2 (τ) 27g2 3 (τ), is surjective according to Corollary Hence a number τ H exists with

117 4.2 The algebra of modular forms 113 j(τ) = 1728 A3 F. If g 2 (τ) = 0 then τ = ρ according to Corollary We choose Λ := Λ ρ. Similarly, if g 3 (τ) = 0 then τ = i, and we choose Λ := Λ i. Otherwise, clearing the denominators implies We choose ω 1 C with ( ) 3 ( ) 2 A B = g 2 (τ) g 3 (τ) ( ) 4 ( ) 6 A 1 g 2 (τ) = B 1 and ω 1 g 3 (τ) =. ω 1 This choice is possible because we may replace ω 1 by iω 1. In addition, we set ω 2 := τ ω 1, i.e. τ = ω 2 /ω 1. Then (ω 1,ω 2 ) is a positively oriented basis of the lattice Λ := Zω 1 + Zω 2. Its lattice constants are ( ) 1 4 ( ) 1 4 G Λ,4 = G 4 (τ) = (1/60) g 2 (τ) = (1/60) A ω 1 ω 1 and similarly Therefore, according to Theorem 3.20 G Λ,6 = (1/140) B. Φ(C/Λ) = E an, q.e.d. Lemma 4.19 (Exact sequence for cusp forms). Denote by ε : M k C, f f ( ), the evaluation with kernel S k, and for k 4 by

118 114 4 Modular forms of the modular group µ : M k 12 M k, f f, the multiplication with the discriminant modular form. Then for k 4 the sequence of vector space morphisms is exact In particular for k 4: µ 0 M ε k 12 Mk C 0. M k = S k C G k, and multiplication by the discriminant modular form defines a vector space isomorphism µ : M k 12 Sk. Proof. The Eisenstein series G k M k satisfies G k ( ) = 2 ζ (k) 0. Therefore ε is surjective. If f M k with ε( f ) = 0, then the quotient g := f is holomorphic on H, because has no zero on H due to Corollary In addition g is holomorphic at the point because has a zero of order 1 at according to Proposition Hence g M k 12. Apparently the map µ is injective, q.e.d. Theorem 4.20 (Algebra of modular forms). The associative algebra ( ) M (Γ ) := M 2k (Γ ),+, k 0 of modular forms of the module group Γ is freely generated by the two Eisenstein series G 4 and G 6 : M = C[G 4,G 6 ] C[T 1,T 2 ] (Algebra of polynomials in two indeterminates). The subalgebra S (Γ ) := k 0 S 2k (Γ ) M (Γ ) is an ideal. Proof. i) Generated by G 4 and G 6 : We give a proof by induction on k 4. Any integer k 4 decomposes as k = 4 α + 6 β with non-negative integers α,β Z. Hence G α 4 Gβ 6 M k

119 4.3 Hecke operators 115 and ε(g α 4 Gβ 6 ) 0. Consider an arbitrary element f M k. For a suitable λ C ε( f λ G α 4 Gβ 6 ) = ( f λ Gα 4 Gβ 6 )( ) = 0. According to Lemma 4.19 an element g M k 12 exists with f λ G α 4 Gβ 6 = g. The claim follows by applying the induction assumption to g. ii) Freely generated: The generators G 4 and G 6 are algebraically independent. Assume the existence of a polynomial P C[T 1,T 2 ] with P(G 4,G 6 ) = 0. We may assume that all monomials of P(G 4,G 6 ) have the same weight. The case P(G 4,G 6 ) = a G m 4 + G 6 P (G 4,G 6 ), a C, is excluded because G 4 (i) 0, but G 6 (i) = 0. Analogously the case P(G 4,G 6 ) = b G m 6 + G 4 P (G 4,G 6 ) b C, is excluded because G 6 (ρ) 0, but G 4 (ρ) = 0. Hence G 4 G 6 divides P(G 4,G 6 ), i.e. P(G 4,G 6 ) = G 4 G 6 P (G 4,G 6 ) = 0. Iteration of the argument for P (G 4,G 6 ) = 0 proves P = 0 by descending induction. iii) Ideal of cusp forms: According to Lemma 4.19 any cusp form is a multiple of, q.e.d. 4.3 Hecke operators Hecke operators are linear maps on modular forms. Hecke operators average the behaviour of modular forms with respect to certain sets of matrices Γ m GL(2,R) +. These sets generalize Γ 1 := Γ when considered a set. Definition 4.21 (Integral matrices with positive determinant). For any m N consider the set of matrices and its subset Γ m := {M M(2 2,Z) : det M = m} GL(2,R) +

120 116 4 Modular forms of the modular group {( ) } a b Γ m,prim := Γ c d m : (a,b,c,d) = 1 of primitive matrices. We denote by the canonical Γ -left operation and by Φ m : Γ Γ m Γ m,(a,m) A M Φ m,prim := Φ m Γ m,prim : Γ Γ m,prim Γ m,prim its restriction. The respective orbits sets are denoted Γ \Γ m and Γ \Γ m,prim. The subset Γ m,prim Γ m is stable under the Γ -left operation Φ m. For the proof consider any ( ) ( ) α β a b A = Γ and M = Γ γ δ c d m,prim. The ideal contains the element I := (αa + βc,αa + βd,γa + δc,γb + δd) Z δ (αa + βc) + ( β) (γa + δc) = (αδ βγ)a + (δβ βδ)c = a and similar for the other generators of (a,b,c,d). Therefore and I = (a,b,c,d) = (1) Φ m (A,M) = A M Γ m,prim. Lemma 4.22 (Orbit sets of left-operation). Consider a positive integer m N. 1. A bijective map exists Γ \Γ m V (m) := {( a b 0 d ) } Γ m : 0 b < d to a subset of upper triangular matrices. In particular: The orbit set of Φ m has cardinality σ 1 (m). 2. A bijective map exists Γ \Γ m,prim V (m, prim) := {( a b 0 d In particular: The orbit set of Φ m,prim has cardinality ) } V (m) : (a,b,d) = 1.

121 4.3 Hecke operators 117 Proof. 1. We define a map ψ(m) := m (1 + (1/p)) p m p prime p : Γ m V (m), which attaches to any M Γ m a matrix p(m) belonging to the same orbit: i) Consider a fixed but arbitrary element ( ) a b M = Γ c d m. We construct a matrix A Γ such that A M V (m): First, in order to remove the entry c of M we choose coprime integers γ,δ Z with γ a + δ c = 0. In order to obtain a matrix with det = 1 we choose integers α,β Z with α δ β γ = 1. Such a choice is possible because γ and δ are coprime. Set ( ) α β A 1 := Γ. γ δ Then and ( a A 1 M = b ) 0 d det (A 1 M) = det M = a d = m. In particular d 0 and possibly after replacing A 1 by A 1 even a, d > 0. Secondly, because T Γ, for any k Z also Then We choose k 0 Z such that and set A 2 (k) := T k A 1 Γ. ( ) ( 1 k a A 2 (k) M = b ) ( a b d = + k d ) 0 d. 0 b + k 0 d < d

122 118 4 Modular forms of the modular group A := A 2 (k 0 ). ii) The construction from part i) determines uniquely the element Assume two matrices A i Γ such that Then Set From follows: A M V (m) : A i M =: M i V (m), i = 1,2. M = A 1 1 M 1 = A 1 2 M 2 or A 21 M 1 = M 2, A 21 := A 2 A 1 1. A 21 =: ( ) α β and M γ δ i := ( ) ai b i, i = 1,2. 0 d i ( ) ( ) ( ) α β a1 b 1 a2 b = 2 γ δ 0 d 1 0 d 2 γ a 1 = 0, hence γ = 0. From d 2 = δ d 1 follows δ > 0. Together with α δ = 1 it implies α = δ = 1. As a consequence a 1 = a 2, d 1 = d 2, and b 1 + β d 1 = d 2. Because 0 b 1,b 2 < d 1 = d 2 we get b 1 = b 2 and β = 0. As a consequence iii) Part i) and ii) show that is well-defined. A 21 = ( ) 1 0 or A = A 2. p : Γ m V (m), M A M, iv) The map p is surjectice because V (m) Γ m. In addition, p(m 1 ) = p(m 2 ) A Γ with A M 1 = M 2 M 1 and M 2 on the same orbit of Φ m. As a consequence p induces the bijective map Γ \Γ m V (m), M p(m). v) The value card V (m) follows at once from the explicit representation of the elements of V (m).

123 4.3 Hecke operators The representation of the orbit set Γ \Γ m,prim follows from part 1) by restriction. For the cardinality of V (m, prim) see [23, Chap. 5, 1]. Corollary 4.23 (Representatives of the orbit set). For any γ Γ : 1. The set V (m) γ Γ m is a complete set of representatives of the orbit set Γ \Γ m, i.e. with respect to the canonical projection Γ m Γ \Γ m, M [M], we have the equality of sets 2. The set Γ \Γ m = {[M] : M V (m)} = {[M γ] : M V (m)}. V (m, prim) γ Γ m,prim is a complete set of representatives of the orbit set Γ \Γ m,prim. Proof. Consider a fixed, but arbitrary γ Γ. We show: For arbitrary but fixed M V (m) exist an element M 1 V (m) and a matrix A Γ such that A M = M 1 γ. According to Lemma 4.22 a unique element A Γ exists with We define Then A (M γ 1 ) V (m). M 1 := A M γ 1 V (m). M 1 γ := A M. Hence M 1 γ lies on the same orbit as M, q.e.d. Definition 4.24 (Hecke operators of the modular group). For any positive integer m 1 the m-th Hecke operator is the C-linear map on the algebra of modular forms which is defined on M k (Γ ), k N, as T m : M (Γ ) M (Γ )

124 120 4 Modular forms of the modular group M k (Γ ) M k (Γ ), f T m ( f ) := m k 1 Γ \Γ m f [M] k. Here the summation is over an arbitrary complete set of representatives M Γ m of Γ \Γ m. Apparently, for m = 1: T 1 = id : M k (Γ ) M k (Γ ). Note that the value T m ( f ) of the m-th Hecke operator in Definition 4.24 does not depend on the choice of a fixed representative M: For any γ Γ due to Remark 4.4. f [γ M] k = ( f [γ] k )[M] k = f [M] k Theorem 4.25 (Fourier coefficients of Hecke transforms). 1. The Hecke operators from Definition 4.24 are well-defined, i.e. f M k (Γ ) = T m ( f ) M k (Γ ). 2. For a modular form f M k (Γ ) with Fourier expansion f (τ) = n=0 a n q n, q = e 2πi τ, its Hecke transform T m ( f ) M k (Γ ) has the Fourier series T m ( f )(τ) = n=0 r>0 r (m,n) r k 1 a (mn/r 2 ) q n. 3. The ideal of cusp forms is stable under all Hecke operators, i.e. T m (S (Γ )) S (Γ ). Proof. 1. To prove that T m ( f ) is a modular form of the same weight as f we have to verify the defining properties from Definition 4.2: Appararently T m ( f ) is holomorphic on H. Weakly modularity: Consider a fixed but arbitrary matrix γ Γ. We replace the summation over the orbit set V (m) by the summation over the set of representatives V (m) γ, see Corollary 4.23:

125 4.3 Hecke operators 121 T m ( f )[γ] k = m k 1 M V (m) ( f [M] k )[γ] k = m k 1 M V (m) = m k 1 f [N] k = T m ( f ). N V (m) γ ( f [M γ] k ) = Holomorphy of T m ( f ) at the point : We have to compute the Fourier series of T m f. During the computation we will use the following results: i) If then exp(2πin M(τ)) = exp M = ( ) a b V (m) 0 d ( 2πin aτ + b ) d = e (2πin (aτ/d) e 2πin (b/d) ii) { e 2πib (n/d) d d n = 0 b<d 0 otherwise The first case follows from e 2πib (n/d) = 1 for each of the d summands with 0 b < d. The second case follows from the formula of the finite geometric series. We now use the representation of the orbit set Γ \Γ m from Lemma 4.22 and insert the Fourier series of f. We obtain T m ( f )(τ) = m k 1 = m k 1 M V (m) a,d>0 ad=m m k 1 1 d k = m k 1 1 d k d>0 d m = m k 1 d>0 d m 1 f [M] k (τ) = m k 1 a,d>0 ad=m d 1 1 d k b=0 ( n=0 d 1 b=0 ( n=0 d k ( n=0 d 1 b=0 M V (m) f (M(τ)) = h(m,τ) k f (M(τ)) = a n e 2πin (aτ/d) e 2πin (b/d)) ) a n e 2πin (mτ/d2) e 2πib (n/d)) ) ) a n q (nm/d2) d 1 e 2πib (n/d)) b=0

126 122 4 Modular forms of the modular group Now we make the substitution obtaining Hence ( )k 1 ( m ) = a n q (nm/d2 ) d>0 d n=0 d (m,n) (d,n) (t = (mn)/d 2,r = m/d) T m ( f )(τ) = n = d2 t m = t=0 ( ) 2 d tm = tm m r 2. r k 1 a (tm)/r 2 q t r>0 r (m,t) with the inner sum being finite. Apparently, the last Fourier series has no coefficients with negative index. Hence T m ( f ) is holomorphic at For the proof see the Fourier series from the previous part. 3. According to the previous part of the theorem, for a modular form f M k (Γ ) the Fourier expansion of T m ( f ) is A cusp form satisfies a 0 = 0, hence T m ( f )(τ) = a 0 σ k 1 (m) + a m q + O(2). T m ( f )(τ) = a m q + O(2), q.e.d. The following corollary specializes the general formula from Theorem It gives the formula for the Fourier coefficients of the Hecke transform for prime indices. Corollary 4.26 (Fourier coefficients with prime index of Hecke transforms). Consider a modular form f M k (Γ ) with Fourier coefficients For arbitrary but fixed m 1 denote by (a(n) := a n ) n 0. (b(n) := b n ) n 0 the Fourier coefficients of its Hecke transform T m ( f ). Then for any prime p

127 4.3 Hecke operators 123 { a(mp) p m b(p) = a(mp) + p k 1 a(m/p) p m Proof. The claim follows directly from the general formula in Theorem 4.25: If p m then r (p,m) implies r = 1 or r = p; note n = p. If p m then r (p,m) implies r = 1, q.e.d. Corollary 4.27 (Simultaneous eigenform of all Hecke operators). The discriminant modular form S 12 (Γ ) is a simultaneous eigenform, i.e. eigenvector, of all Hecke operators (T m ) m 1. The Fourier coefficients of the normalized discriminant modular form (2π) 12 m=1 τ m q m are the corresponding eigenvalues, i.e. for all m N T m ( ) = τ m. Proof. According to Corollary 4.14 and Lemma 4.19 the vector space S 12 (Γ ) of cusp forms of weight k = 12 is 1-dimensional. It has the discriminant modular form from Definition 4.10 as a basis. Hence is an eigenvector of each Hecke operator T m, m 1. To calculate the corresponding eigenvalue we recall the Fourier expansion of the normalized discriminant modular form from Proposition 4.11: (τ) (2π) 12 = τ n q n, τ 1 = 1. n=1 ( ) 1 According to Theorem 4.25 the Fourier expansion of T m (2π) 12 is ( ) 1 T m (2π) 12 (τ) = τ m q + O(2). We obtain T m ( ) = τ m, q.e.d. Remark 4.28 (Algebra of Hecke operators).

128 124 4 Modular forms of the modular group 1. The algebra of all Hecke operators is Abelian. H (Γ ) := {T m : M (Γ ) M (Γ ) m 1},+, ) 2. On the vector spaces S k (Γ ),k 12 even, exists a Hermitian scalar product, the Petersson scalar product, <, >: S k (Γ ) S k (Γ ) C such that all Hecke operators become Hermitian operators. In particular, for each arbitrary but fixed k N the vector space S k (Γ ) of cusps has a basis, such that each element from the basis is a simultaneous eigenform for all Hecke operators (T m ) m N. For a proof consider [20, Kap. IV, 2, 3].

129 Chapter 5 Application to Number Theory 5.1 The τ-function of Ramanujan Definition 5.1 (Ramanujan function). The Fourier coefficients (τ n ) n N of the normalized discriminant form define the Ramanujan function (τ) (2π) 12 = τ n q n, q = e 2πi τ, n=1 τ : N Z,n τ(n) := τ n. According to Proposition 4.11 all Fourier coefficients are integers τ(n) Z. Computing the first coefficients τ(n) up to n = 6 gives (τ) (2π) 12 = q 24q q q q q 6 + O(7) From these values one verifies at once the product formula τ(2) τ(3) = = = τ(6), and as a particular case of the recursion formula the formula τ(p 2 ) = τ(p) 2 p 11 τ(p) τ(2 2 ) = = τ(2) = ( 24)

130 126 5 Application to Number Theory From explicit computation of the first 30 values of τ(n) Ramanujan [29] conjectured in 1916 several generalizations. Figure 5.1 shows with equations (103) and (104) two of Ramanujan s conjectures from his original paper.

131 5.1 The τ-function of Ramanujan 127 Fig. 5.1 Ramanujan s conjecture on τ(n) from [29] The conjecture was proved inter alia one year later by Mordell [26]. Figure 5.2 shows Mordell s introduction from his original paper.

132 128 5 Application to Number Theory Fig. 5.2 Mordell s introduction to [26] Today Ramanujan s product formula and recursion formula are implied by Hecke theory: Proposition 5.2 (Product and recursion formula of the Ramanujan function). The Ramanujan function satisfies: 1. Product formula: For all integers m,n 1 τ(m) τ(n) = r (m,n) r 11 τ(mn/r 2 ).

133 5.2 Complex multiplication and imaginary quadratic number fields 129 In particular, for coprime m and n: τ(m) τ(n) = τ(m n). 2. Recursion formula: For all primes p and integers k > 1 τ(p) τ(p k ) = τ(p k+1 ) + p 11 τ(p k 1 ). Proof. 1. Consider an arbitrary, but fixed m N. Corollary 4.27 shows: ( ) T m (2π) 12 = τ(m) (2π) 12 The explicit formula for the Fourier coefficients of the Hecke transform from Theorem 4.25 implies for all n N: r 11 τ(mn/r 2 ) = τ(m) τ(n). r (m,n) For coprime integers m,n we have (m,n) = 1 and the summation reduces to the single term for r = 1: τ(m n) = τ(m) τ(n). 2. We set m = p k and n = p in the product formula from part i). Due to (p k, p) = p we now have two summands, referring to the indices r = 1 and r = p. We obtain τ(p k+1 ) + p 11 τ(p k 1 ) = τ(p k ) τ(p), q.e.d. Remark 5.3 (Growth estimation of the Ramanujan function). The final conjecture of Ramanujan on the τ-function is the growth estimation (104) from 5.1. This conjecture lies much deeper than his other conjectures. In the form τ(p) 2 p 11/2, p prime, Deligne proved the growth condition of the τ-function as an implication of his proof of the Weil conjectures in Note the typo of the factor 2 in Ramanujan s original equation (104). 5.2 Complex multiplication and imaginary quadratic number fields The present section introduces the subclass of tori T with complex multiplication, i.e. with a non-trivial endomorphism ring End(T ). Complex multiplication is a sin-

134 130 5 Application to Number Theory gular phenomenon in the theory of tori. Tori with complex multplication provide a bridge from complex analysis to number theory because their lattice satisfies an arithmetic condition: The generator τ of the normalized lattice Λ τ of T generates an imaginary quadratic number field K = Q[τ], and End(T ) becomes an order in O K, see Proposition 5.9. In the opposite direction, for a given imaginary quadratic number field K the number of isomorphism classes of tori T with End(R) = O K equals the class number of K, see Theorem Proposition 5.4 (Holomorphic maps between tori). Consider a holomorphic map F : C/Λ 1 C/Λ 2 between two tori with F(0) = 0. Then a number α C exists such that the following diagram commutes C µ α C p 1 p 2 F C/Λ 1 C/Λ 2 Here µ α denotes the multiplication by α and p i, i = 1,2, denote the canonical projections. In particular, any holomorphic map between tori which respects the neutral element is a group homomorphism. Proof. The holomorphic map F lifts to a holomorphic map F between the universal coverings, satisfying F(0) = 0 and for all z C,ω Λ 1, F(z + ω) F(z) Λ 2. The lattice Λ 2 is discrete and F is continous. Therefore F(z + ω) F(z) does not depend on z. As a consequence for all z C F (z + ω) F (z) = 0. Hence F is a holomorphic and doubly periodic function. Therefore F is a constant a C and for all z C F(z) = a z, q.e.d. Apparently α has to satisfy α Γ 1 Γ 2. Note that the multiplier α in Proposition 5.4 satisfies

135 5.2 Complex multiplication and imaginary quadratic number fields 131 α Γ 1 Γ 2. Definition 5.5 (Endomorphisms of a torus). Any holomorphic map F : (T,0) (T,0) of a torus (T,0) with F(0) = 0 is an endomorphism of (T,0). Proposition 5.6 (Endomorphism ring). The set of endomorphisms of a torus T = C/Γ constitutes in a canonical way a ring End(T ), the endomorphism ring of T. The map from Proposition 5.4 is bijective. End(T ) {α C : α Γ Γ },F µ α, Because each torus T is an Abelian group its endomorphism ring End(T ) comprises at least the group Z: Z End(T ) C. Which intermediate rings are possible? The case End(T ) = Z is the general case, justifying a distinguished name for tori with additional endomorphisms: Definition 5.7 (Complex multiplication). A torus T = C/Γ has complex multiplication if Z End(T ). Complex multiplication is an exceptional phenomenon. Its existence depends on an arithmetic property of the lattice of the corresponding torus. Remark 5.8 (Number field). A number field K is an extension field Q K C of finite degree [K : Q] =: n <. 1. Any number field K is an algebraic extension of Q generated by a single element K = Q(ω).

136 132 5 Application to Number Theory 2. The Galois group Gal(K/Q) of K has order n. It operates on K as a group of automorphisms. An element x K has the trace and the norm Tr(x) := N(x) := σ Gal(K/Q) σ Gal(K/Q) 3. The subring O K K of algebraic integers σ(x) Q, σ(x) Q. O K := {x K : f (x) = 0 for a normed polynomial f Z[X]} is a lattice of rank = n Its quotient field is n O K = Zω i. i=1 Q(O K ) = K. For all x O K Tr(x) Z and N(x) Z. 4. An integral basis of K is a Q-basis (ω i ) i=1,...,n of K such that O K = Zω Zω n. In particular, all elements of an integral basis are algebraic integers. The discriminant δ K of K is the determinant of an integral basis (ω i ) i=1,...,n, i.e. δ K := det(tr(ω i ω j ) 1 i, j n ) Z. 5. An order of K is a subring O O K which is an n-dimensional lattice. 6. A quadratic number field is a field K = Q( d) with a squarefree integer d Z. If d < 0 then K is a quadratic imaginary number field.. Consider an imaginary quadratic number field K = Q( d) with d > 0. Its ring of integers is O K = Z + Zω with Its discriminant is d d 3 mod 4 ω = 1 + d 2 d 3 mod 4

137 5.2 Complex multiplication and imaginary quadratic number fields 133 4d d 3 mod 4 δ K = d d 3 mod 4 Examples are d = 1: Number field K = Q(i); ω = i, δ K = 4 d = 3: Number field K = Q(ρ),ρ := e 2πi/3 ; ω = 1 + ρ, δ K = 3. Proposition 5.9 (Complex multiplication and imaginary quadratic number fields). A torus T = C/Λ τ with a normalized lattice Λ τ = Z 1 + Z τ, τ H, has complex multiplication if and only τ belongs to an imaginary quadratic number field K. In this case the ring of endomorphisms End(T ) is an order of K. Proof. i) Assume that T has complex multiplication: Then exists α C \ Z with We obtain a matrix with We obtain γ = α Λ τ Λ τ. ( ) a b M(2 2,Z) c d α which implies b 0, because α / Z, and ( ) ( ) 1 1 = γ. τ τ α = a + b τ, α τ = c + d τ, b τ 2 + (a d) τ c = 0. Therefore τ H belongs to an imaginary quadratic number field. ii) Assume that τ H belongs to an imaginary quadratic number field. Then integers a,b,c Z exist such that Because τ / H we have a 0. From ( 1 (aτ) τ a τ 2 + b τ + c = 0. ) = ( ) 0 a c b ( ) 1. τ

138 134 5 Application to Number Theory follows aτ End(T ), and T has complex multiplication, iii) Assume that T has complex multiplication and consider α End(T ). Alike to the proof of part i) we obtain a matrix ( ) a b γ = M(2 2,Z) c d with α ( ) ( ) 1 1 = γ. τ τ Therefore α is an eigenvalue of γ and satisfies the integral equation As a consequence End(T ) O K. α 2 (d + a) α cd = 0. As submodule of the lattice O K also End(T ) is a lattice. Because End(T ) Z the lattice End(T ) has rank = 2. Hence End(T ) is an order, q.e.d. Example 5.10 (Tori with complex multiplication). The modular group has exactly two elliptic points modulo its fundamental domain, see Theorem 2.31: 1. Elliptic point i: The point i H belongs to the imaginary quadratic number field K := Q( d),d = 1. The torus T = C/Λ i with the normalized lattice Λ i = Z + Z i has as endomorphisms the multiplication with arbitrary elements Hence its endomorphism ring is α Λ i. End(T ) = Λ i, which equals the ring O K of algebraic integers from K. Recall from Corollary 4.16 j(i) = 1728 = Z. 2. Elliptic point ρ = e 2πi/3 : The point ρ H belongs to the imaginary quadratic number field

139 5.2 Complex multiplication and imaginary quadratic number fields 135 K = Q( d), d = 3. The torus T = C/Λ ρ with the normalized lattice Λ ρ = Z + Z ρ has as endomorphisms the multiplication with arbitrary elements Hence its endomorphism ring is α Λ ρ. End(T ) = Λ ρ. The result is similar to the first example: End(T ) is the ring O K of algebraic integers from K. Recall from Corollary 4.16 j(ρ) = 0 Z. Also the element 2ρ H belongs to K. The torus with the normalized lattice T = C/Λ 2ρ Λ 2ρ = Z + Z 2ρ has as endomorphisms the multiplication with arbitrary elements For the proof one uses the equation α Λ 2ρ. ρ 2 + ρ + 1 = 0. The endomorphism ring is with an order of K. Note End(T ) = O O = Λ 2ρ O K j(2ρ) = j(ρ) = 0 Z. Remark 5.11 (Fractional ideals in number fields). Consider a number field K of degree [K : Q] = n. 1. A fractional ideal of K is an O K -submodule a K for which there exists an element d O K,a 0, with

140 136 5 Application to Number Theory d a O K. Hence fractional ideals are the subsets of the form a = ã d K with an ideal ã O K. The element d is named a common denominator of a. Fractional ideals generated by a single element α K are named principal fractional ideal. a = (α) := O K α 2. The fractional ideals of K are the finitely generated O K -submodules of K. As a consequence: Any fractional ideal of K is a lattice of rank = n. 3. For two fractionals ideals a, b K the product a b is the fractional ideal generated by all products a b, a a, b b. Any fractional ideal a {0} has a well-defined fractional ideal a 1 such that a a 1 = (1) = O K. The fractional ideal a 1 is named the inverse of a. 4. With respect to multiplication the set of fractional ideals is an Abelian group J(K). Its quotient by the subgroup P(K) of principal fractional ideals is named the ideal class group of K Cl(K) := J(K)/P(K). We recall from Remark 2.7 the exact sequence of divisor classes on a compact Riemann surface X 1 C M (X) Div 0 (X) Cl 0 (X) 0. For a number field K we have the analogue 1 K P(K) J(K) Cl(K) 1. With respect to this analogue fractional ideals correspond to divisors or line bundles of degree = 0, and principal fractional ideals correspond to principal divisors, i.e. divisors of meromorphic functions. A fundamental theorem from algebraic number theory states the finiteness of the class number h K := ord Cl(K). Examples:

141 5.2 Complex multiplication and imaginary quadratic number fields 137 For K = Q(i) we have O K = Z[i] = Z + Z i, which is a principal ideal domain. Hence h K = 1. For K = Q(ρ) we have O K = Z[ρ] = Z + Z ρ with h K = 1. There are exactly 9 imaginary quadratic number fields K = Q( d) with h K = 1. They belong to the values d {1,2,3,7,11,19,43,67,163}. We now investigate the consequences of Proposition 5.9 for an imaginary quadratic number field K, in particular for its class group Cl(K). Theorem 5.12 (Ideal classes and endomorphisms of elliptic curves). Consider a quadratic imaginary number field K and denote by O K its ring of integers. Then a bijection exists between the ideal class group Cl(K) and the set of isomorphism classes of complex elliptic curves E with End(E) O K. The bijection is induced by the attachement Its inverse is induced by the attachement a J(K) E = C/a with End(E) = O K. E = C/Λ with End(E) = O K Λ J(K). Proof. i) Consider a fractional ideal a J(K). Due to Remark 5.11 the ideal a C is a lattice and defines the torus E = C/a. According to Proposition 2.24 the torus depends only on the ideal class [a] Cl(K). Proposition 5.6 implies End(E) = {α C : α a a}. As a consequence O K End(E) because a is a fractional ideal. For the opposite inclusion consider a number α End(E). Then one checks: The representation as lattice a = Zω 1 + Zω 2 K implies α K. Hence End(E) K. We are left to prove, that all elements α End(E) are integral. Consider a Z-basis (ω 1,ω 2 ) of a. The equation

142 138 5 Application to Number Theory ( ) ( ) ( ) ω1 a b ω1 a = ω 2 c d ω 2 implies that α is an eigenvalue of the matrix ( ) a b GL(2,Z). c d Therefore α K satisfies a quadratic equation with integer coefficients, i.e. α O K. ii) Consider an elliptic curve E = C/Λ. We may assume that the lattice Λ is normalized Λ = Z + Zτ. By assumption End(E) = O K. We claim Λ K: If O K = Z + Zω then ( ) ( ) ( ) 1 a b 1 ω =. τ c d τ Hence ω is an eigenvalue of ( a b c d We obtain an equation It implies The assumption ) M(2 2, Z) with eigenvector ω = a + b τ. τ Q(ω) = K. End(E) = {α C : αλ Λ} = O K implies that Λ is even a fractional ideal of K. ( ) 1. τ iii) The two constructions from part i) and ii) are inverse, q.e.d. Remark 5.13 (Ideal class group of orders). Theorem 5.20 is a statement about fractional ideals in K with respect to the ring O K. The ideal class group Cl(K) refers to the maximal order O K of K. The theorem generalizes literally to the case of fractional ideals with respect to an arbitrary order O O K K. The general case sets in a bijective relation the ideal classes with respect to an order O of K and the isomorphism classes of elliptic curves E with End(E) = O.

143 5.3 Applications of the modular polynomial Applications of the modular polynomial The present section takes up the study of imaginary quadratic number fields K from the viewpoint of modular forms. The modular invariant j links both domains: Theorem 5.18 proves that the value j(z) C is integral over Z for any non-rational z K H. If O K = Z + Z ω K then [Q( j(ω)) : Q] equals the class number of K, see Theorem 5.20 for a lower bound. The main tool to derive the necessary properties of the modular invariant is the modular polynomial from Definition We take up the investigation of the integral matrices Γ m from Section 4.3, in particular Definition 4.21 of the set Γ m,prim of primitive matrices and its set V (m, prim) of representatives with respect to the canonical left Γ -operation. Definition 5.14 (Class invariants and modular polynomial). Consider a positive integer m N and the complete set of representatives of Γ \Γ m,prim V (m, prim) = {M 1,...,M ψ(m) }. Each matrix M µ V (m, prim) induces an automorphism of the upper half-plane The functions α µ : H H, τ M µ (τ). j µ := j α µ M (H), µ = 1,...,ψ(m), are the class invariants of Γ \Γ m. The modular polynomial of order m is the polynomial ψ(m) F m (X) := (X j µ ) M (H)[X]. µ=1 Note. The modular invariant is constant along the orbits of the operation Φ. Therefore the value of j µ in Definition 5.14 does not depend on representing a coset from Γ \Γ m,prim by the matrix M V (m, prim) or by the matrix γ M,γ Γ. Apparently, the roots of the modular polynomial are exactly the class invariants j µ, µ = 1,...,ψ(m), corresponding to the Γ -left operation on Γ m,prim. If

144 140 5 Application to Number Theory σ µ, µ = 1,...,ψ(m), denotes the µ-th elementary symmetric function in ψ(m) variables, then up to sign the functions σ µ ( j 1,..., j ψ(m) ) M (H) are the coefficients of the modular polynomial F m (X). E.g., is the constant term and is the coefficient of the term X ψ(m) 1. ( 1) ψ(m) σ ψ (m) = j 1... j ψ(m) ( 1) σ 1 ( j 1,..., j ψ(m) ) = ( j j ψ (m)) The modular polynomial F m (X) has coefficients from the field of meromorphic functions M (H). The next Theorem 5.15 investigates these meromorphic functions. The theorem shows that the coefficients are Z-linear combinations of the modular invariant j, i.e. the coefficients of the modular polynomial belong to the smaller subring Z[ j] M (H). Theorem 5.15 (The modular polynomial). For all positive integers m N ψ(m) F m (X) := µ=1 (X j µ ) Z[ j][x]. Proof. We choose an arbitrary but fixed index µ = 1,...,ψ(m) and consider the meromorphic function c = c µ := σ µ ( j 1,..., j ψ(m) ) M (H). i) Weakly modular: According to Corollary 4.23 right multiplication with an element γ Γ permutes the orbits of the Γ -left operation Φ m {[M 1 ],...,[M ψ(m) ]} = {[M 1 γ],...,[m ψ(m) γ]}. Hence right multiplication does not change the value of the elementary symmetric functions: For all j = 1,...,ψ(m) Therefore c is weakly modular. σ µ ( j 1,..., j ψ(m) ) = σ µ ( j 1 γ),..., j ψ(m) γ)). ii) Holomorphic on H: The modular invariant is holomorphic on H according to Corollary Therefore also the coefficient c is holomorphic.

145 5.3 Applications of the modular polynomial 141 iii) Meromorphic at : The coefficient c is weakly modular according to part i). Hence c has a Fourier expansion c(τ) = a ν q ν, q = e 2πiτ. ν Z We have to show that the singularity q = 0 is a pole of c. For the proof we estimate the growth of c: According to Corollary 4.16 the modular invariant has the Fourier expansion j(τ) = 1 q + P(q) with P Z{q} a power series with integer coefficients. For a matrix ( ) a b α = V (m, prim) 0 d the Fourier series of the class invariant j α derives from the Fourier series of j by substituting the variable q = e 2πiτ by Here e 2πiα(τ) = e 2πi(b/d) q (a/d) = ζ m ab (q 1/m ) a2. ζ m := e 2πi/m, denotes a primitive m-th root of unity. Therefore ( j α)(τ) 1 + O(1) q a/d for a single class invariant, and a constant K exists with for small q \ {0}. c(τ) K 1 q a/d iv) Integrality c O K [ j]: Due to the part already proved, Corollary 4.17 implies that c is a polynomial with complex coefficients in the modular invariant, i.e. We show that even with c C[ j]. c O K [ j] K := Q(ζ m ) the m-th cyclotomic number field. For the proof we recall the substitution of q by

146 142 5 Application to Number Theory ζ m ab (q 1/m ) a2. As a consequence the class invariant j α has a Fourier expansion with respect to q 1/m, namely j(α(τ)) = 1 (q 1/m ) a2 ζ m ab + P(qa/d ζ m ab ), which shows that the Fourier coefficents of c belong to K. According to Corollary 4.17 the coefficients of the polynomial c C[ j] are Z-linear combinations of the Fourier coefficients of c. Hence the coefficients are algebraic integers from Z + Z ζ m = O K. v) c Z[ j]: The operation of the Galois group Gal(K/Q) (Z/mZ). extends to an operation on the set { j 1,..., j ψ (m)} of class invariants Gal(K/Q) { j 1,..., j ψ(m) } { j 1,..., j ψ(m) }, (χ, j µ ) j µ χ : Assume χ Gal(K/Q) maps χ(ζ m ) = ζ e m with (e,m) = 1. If j µ is the class invariant of the matrix ( ) a b M = 0 d then we define j µ χ as the class invariant of the matrix with ( ) M χ a b χ = 0 d b χ b e mod d. Because the function c is fixed under the operation of Gal(K/Q), its Fourier coefficients must belong to the fixed field Q. Due to part iv) the Fourier coefficients even belong to Q O K = Z, q.e.d. We now evaluate the modular polynomial F m (X) Z[ j][x] at X = j, i.e. we consider the polynomial in j ψ(m) Φ m ( j) := F m ( j, j) = µ=1 ( j j µ ) Z[ j]. Lemma 5.16 (Leading coefficient of the modular polynomial). For squarefree m 2 the polynomial

147 5.3 Applications of the modular polynomial 143 has leading coefficient ±1. Φ m ( j) Z[ j] Proof. The function Φ m ( j) : H C is an automorphic function. Consider its Fourier expansion Φ m ( j)(τ) = c k q k + c k 1 + higher terms qk 1 Because j has a pole at τ = of order = 1, the coefficient c k of the Fourier series is also the leading coefficient of F m as polynomial in j. The coefficient is the product of the highest negative coefficients of the factors j j µ, µ = 1,...,ψ(m). For arbitrary but fixed µ the representation j(τ) = 1 q + P(q) implies j µ (τ) = j ( ) aτ + b d = e 2πi(b/d) q a/d + P (q a/d e 2πi(b/d)) with j µ corresponding to the matrix ( ) a b V (m, prim). 0 d Hence the highest negative coefficient in the Fourier expansion of j j µ is { 1 a < d e 2πi(b/d) a > d Note that the case a = d is excluded because m = a d is squarefree. In any case the highest negative coefficent of j j µ is a root of unity, and the same holds for the coefficient c k of the product ψ(m) Φ = µ=1 Because c k Z we have c k = ±1, q.e.d. ( j j µ ).

148 144 5 Application to Number Theory According to Theorem 5.15 the modular polynomial ψ(m) F m ( j,x) = N n=0 µ=1 (X j µ ) Z[ j,x] is a polynomial in the two variables j and X with integer coefficients. We indicate its polynomial dependency on the two variables by the notation ( F m ( j,x) = c rs j r X ), s c rs Z. r+s=n The following Proposition 5.17 shows that F m (t,x) is symmetric in both variables, i.e. the coefficients satisfy c rs = c sr. As a consequence, the highest exponent of j in F m ( j,x) is ψ(m). Proposition 5.17 (Symmetry of the modular polynomial). The modular polynomial F m ( j,x) Z[ j,x] is symmetric with respect to j and X, i.e. F m ( j,x) = F m (X, j). Proof. We set R = Z[ j] with quotient field K := Q(R) = Q( j). i) Irreducibility of F m (X) R[X] over the base field K: The polynomial has the roots ψ(m) F m (X) = µ=1 (X j µ (τ)) R[X] j µ, µ = 1,...,ψ(m). To prove that F m is irreducible, we show that the Galois group Gal(L/K) of the splitting field L = K( j 1,..., j ψ(m) ) of F m (X) acts transitively on the roots. Then no root belongs to the fixed field K. For the proof set with matrices j µ = j α µ

149 5.3 Applications of the modular polynomial 145 auch that α µ V (m, prim), µ = 1,...,ψ(m), {α µ : µ = 1,...,ψ(m)} V (m, prim) is a complete set of representatives of Γ \Γ m. According to Corollary 4.23 for any γ Γ the same holds for the set {α µ γ : µ = 1,...,ψ(m)} Γ m. Hence the Γ -orbit of any matrix (α µ γ) equals the Γ -orbit of a suitable matrix α µ. Therefore the functions ( j (α µ γ)) µ=1,...,ψ(m) are a permutation of the functions As a consequence, for any γ Γ the map ( j α) µ=1,...,ψ(m). L L, f f γ is an element of the Galois group Gal(L/K). For each pair (ν, µ) exist elements γ,γ Γ with α µ = γ α ν γ. Therefore Gal(L/K) operates transitively on the roots of F m (X), and the latter polynomial is irreducible over K. ii) Common root of F m ( j,x) and F m (X, j): We set f (X) := F m ( j,x) and g ( X) := F m (X, j). For any matrix α V (m, prim) the function x α := j α is a root of the polynomial f (X). In particular, ( ) 1 0 α := V (m, prim) 0 m satisfies for all τ H f (x α )(τ) = F m ( j(τ),( j α)(τ)) = F m ( j(τ), j(τ/m)) = 0. The latter equation can be also interpreted: For all τ H 0 = F m ( j(m τ), j(τ)) = F m (( j β)(τ), j(τ)) = g(x β )(τ)

150 146 5 Application to Number Theory with the matrix β := ( ) m 0 V (m, prim). 0 1 But j β is also a zero of f (X) because β = α µ for a suitable µ {1,...,ψ(m)}. We now apply the basic fact about the minimal polynomial of algebraic field extensions: Both polynomials have the common zero f (X), g(x) K[X] x β := j β. The polynomial f (X) is irreducible, see part i), with leading coefficient = 1. Therefore f (X) is the minimal polynomial of the field extension k K[x β ], and a polynomial h(x) K[X] exists with g(x) = h(x) f (X). According to Theorem 5.15 both polynomials f (X) and g(x) have their coefficients already in the ring R. The ring R is factorial as a ring of polynomials over the factorial ring Z. The Lemma of Gauss implies that also the polynomial h[x] has all coefficients from R. Set Then H( j,x) := h(x) R[X] = Z[ j][x]. F m (X, j) = H( j,x) F m ( j,x) Z[ j,x]. iii) Invariance: The last equation from part ii), considered as an equation of polynomials in two variables, is invariant when interchanging the role of j and X. Hence also F m ( j,x) = H(X, j) F m (X, j). Therefore and F m ( j,x) = H(X, j) H( j,x) F m ( j,x), H(X, j) H( j,x) = 1, H(X, j) = H( j,x) = ±1. The case H(X, j) = 1 would imply H( j, j) = H( j, j) and H( j, j) = 0, contradicting the fact that Φ m ( j) = F m ( j, j) has leading coefficient ±1 according to Lemma Therefore H(X, j) = 1 and

151 5.3 Applications of the modular polynomial 147 F m ( j,x) = F m (X, j), q.e.d. E.g. the modular polynomial F 2 ( j,x) Z[ j,x] with ψ(2) = 3 is F 2 ( j,x) = (X 3 + j 3 ) (X 2 j + X j 2 ) (X 2 + j 2 ) X 2 j (X + j) (X + j) Theorem 5.18 (Integrality of the modular invariant). For any imaginary quadratic irrational number τ H the value j(τ) C is an algebraic integer. Proof. Consider an imaginary quadratic field with d Z, d > 0 squarefree. Set K = Q( d) O K = Z + Z z. i) j(z) is integral over Z: Any algebraic integer ξ O K has norm N(ξ ) Z. Choose d d > 1 ξ := 1 + i d = 1 Then N(ξ ) Z is squarefree. Multiplication with ξ defines an endomorphism Hence a matrix exists with The last equation shows: α = O K O K, x ξ x. ( ) a b M(2 2,Z) c d ξ ( ) ( ) z z = α 1 1 The matrix α has the two complex eigenvalues ξ and ξ. Hence m := N(ξ ) = det α Z is squarefree, which implies in particular that α is primitive: α Γ m,prim.

152 148 5 Application to Number Theory The generator z O K satisfies Because a matrix z = az + b cz + d = α(z). α µ V (m, prim), µ {1,...,ψ(m)}, exists with the same Γ -orbit as α. W.l.o.g. Then which implies: The function has the zero z H: We know µ = 1. j(z) = j(α(z)) = j(α 1 (z)) = j 1 (z), ψ(m) Φ m ( j) = Φ m ( j) = µ=1 ( j j µ ) Z[ j] Φ m ( j)(z) = 0. N ν=0 a ν j ν Z[ j] with integer coefficients a ν. Due to Lemma 5.16 the leading coefficient satsfies a N = ±1. Hence proves that j(z) is integral over Z. 0 = Φ m ( j)(z) = N ν=0 ii) j(τ) is integral over Z[ j(z)]: The element τ Q + Q z a ν j(z) ν has a representation with a primitive matrix β = τ = β(z) := az + b d ( ) a b Γ 0 d m,prim, m := a d. Alike to the final argument from part i) the Γ -orbit of β passes through a matrix α µ V (p, prim), which proves j(τ) = j(β(z)) = j(α µ (z)) = j µ (z).

153 5.3 Applications of the modular polynomial 149 Theorem 5.15 implies that j µ is integral over the ring Z[ j]. Hence j(τ) = j µ (z) is integral over Z[ j(z)]. iii) j(τ) is integral over Z: When combining part i) and part ii) the transitivity of integral dependence proves the integrality of j(τ) over Z, q.e.d. Lemma 5.19 (Operation of the Galois group). Consider an elliptic curve E P 2 (C) defined as the variety of a homogeneous polynomial F(X 0,X 1,X 2 ) C[X 0,X 1,X 2 ]. 1. Any element σ Gal(C,Q) defines the conjugate elliptic curve E σ := V (F σ ) with the conjugate polynomial. F σ C[X 0,X 1,X 1 ] 2. The assignment E E σ extends in a functorial way to morphisms: If φ : E 1 E 2 is a regular map between two elliptic curves in P 2 (C), then σ defines in a canonical way a regular map φ σ : E 1 σ E 2 σ such that 3. The j-invariant transforms according to Proof. If then id σ = id and (φ 2 φ 1 ) σ = φ 2 σ φ 1 σ. j (F σ ) = σ( j F ). F(X 0,X 1,X 2 ) = a ν X ν ν F σ (X 0,X 1,X 2 ) := σ(a ν ) X ν. ν The discriminant of the defining Weierstrass polynomial transforms as (F σ ) = σ( F ) 0,

154 150 5 Application to Number Theory and analogously transforms the j-invariant. Also regular maps transform with σ because they are locally the quotient of two polynomials, q.e.d. The class number of an imaginary quadratic number field K relates to the value of the modular function j at the generator z H of its ring O K of integers. Theorem 5.20 (Lower bound of the class number). Consider an imaginary quadratic number field K and denote by O K = Z + Z z, z H, its ring of integers. Then the class number h K satisfies the estimate [Q( j(z)) : Q] h K. Note that j(ω) C is an algebraic integer due to Theorem Theorem 5.20 holds even in the strong form [Q( j(z)) : Q] = h K. Proof. During the proof we will often switch between tori and elliptic curves, using implicitly Theorem 3.20 and Theorem If the extension field Q( j(ω)) has degree then m automorphisms exist such that the conjugates are pairwise distinct. m := [Q( j(z)) : Q] id = σ 1,σ 2,...,σ m Gal(C/Q) σ 1 ( j(z)),...,σ m ( j(z)) C We consider the normalized lattice Λ z := O K and the torus T = C/Λ z. We denote by E the elliptic curve corresponding to T. Due to Lemma 5.19 for any σ Gal(C/Q) the conjugate elliptic curve E σ has the j-invariant σ( j(z)) and the same ring of endomorphisms End(E σ ) End(E). Because the elements of the family (σ µ ( j(z)) µ=1,...,m are pairwise distinct, the elliptic curves (E σ µ ) µ=1,...,m

155 5.3 Applications of the modular polynomial 151 with these j-invariants are pairwise non-isomorphic. Theorem 5.12 implies that the corresponding ideal classes are pairwise distinct. Their number is bounded by the class number h K = card Cl(K). Hence [Q( j(z) : Q] = m h K, q.e.d. Corollary 5.21 (Integer values of the modular invariant). Consider an imaginary quadratic number field K with class number h K = 1. If then j(z) Z. O K = Z + Zz Proof. The proof follows from Theorem 5.18 and Theorem 5.20.

156

157 Chapter 6 Outlook to Congruence Subgroups 6.1 Modular forms of congruence subgroups The theory of modular forms M k (Γ ) of the modular group Γ = SL(2,Z) has been started in Chapter 4. The whole theory generalizes to the congruence subgroups of Γ. These groups are defined by arithmetic properties. In the present section we consider one type of congruence subgroups. Definition 6.1 (Operation of the conguence subgroups Γ 0 (N)). Consider a positive integer N N and the residue map Z Z/NZ. 1. The congruence subgroup of upper triangular matrices of level N is {( ) } a b Γ 0 (N) := SL(2,Z) : c 0 mod N c d 2. The canonical left-action of the modular group from Definition 2.26 on the extended half-plane H restricts to a left-operation Φ N : Γ 0 (N) H H. The orbits of the points from Q { } are the cusps of Γ 0 (N). Due to the possibility of several cusps Definition 4.2 for modular forms generalizes as follows: Definition 6.2 (Modular forms of the congruence subgroup Γ 0 (N)). Consider a congruence subgroup Γ 0 (N) with a positive integer N N. 153

158 154 6 Outlook to Congruence Subgroups 1. A modular form of weight k Z, k Z, with respect to Γ 0 (N) is a function with the following properties: The function f is holomorphic. For all γ Γ 0 (N) holds f [γ] k = f. f : H C For all γ Γ the function f [γ] k is holomorphic at, i.e. it expands into a Fourier series with non-negative indices. If in addition for all γ Γ the Fourier series of f [γ] k has no constant term, then f is a cusp form of weight k with respect to Γ 0 (N). The symbols M k (Γ 0 (N)) and S k (Γ 0 (N)) M k (Γ 0 (N)) denote the complex vector space of modular forms of weight k and level N with respect to the congruence subgroup Γ 0 (N) and its subspace of cusp forms. 2. For each pair of positive integers (d,m) with (dm) N on has an injective map The image j d,m : S k (Γ 0 (M)) S k (Γ 0 (N)), f (τ) f (d τ). (d,m) (dm) N, M N im j d,m S k(γ 0 (N)) is named the subspace of old forms, and its orthogonal complement with respect to the Petersson scalar product the subspace Sk new (Γ 0 (N)) of newforms of S k (Γ 0 (N)). Remark 6.3 (Modular curve). Consider a congruence subgroup Γ (N) with a positive integer N N. The orbit space of the operation Φ N is a compact Riemann surface, named the modular curve X 0 (N) of Γ 0 (N). For even k Z a suitable line bundle and a canonical isomorphism exist Ω Γ0 (N),k Pic(X 0 (N)) M k (Γ 0 (N)) H 0 (X 0 (N),Ω Γ0 (N),k).

159 6.1 Modular forms of congruence subgroups 155 The injection Γ 0 (N) Γ of the groups induces a branched covering of the corresponding modular curves X 0 (N) X. Example 6.4 (Congruence subgroups Γ 0 (2) and Γ 0 (4)). 1. The congruence subgroup Γ 0 (4) Γ has index [Γ : Γ 0 (4)] = 6. The right nebenclasses Γ 0 (4) γ are represented by the elements ( ) ( ) ( ) ( ) ( ) ( ) ,,,,, The congruence subgroup Γ 0 (4) has three cusps, namely { } p Orbit of H : Q : (p,q) = 1 and (4,q) = 4 { }, q divisible q by 4. Orbit of 1/2 H : by 4, Orbit of 0 H : { } p Q : (p,q) = 1 and (4,q) = 2, q even, not divisible q { } p Q : (p,q) = 1 and (4,q) = 1, q odd. q 3. The modular curve X 0 (4) has genus g = 0, hence X 0 (4) P 1. The modular curve is a 6-fold branched covering X 0 (4) X P 1 4. For even k the dimension of the space of modular forms is (3/2) k k 2 dim M k (Γ 0 (4)) = 1 k = 0 0 k < 0 and the subspace of cusp forms has dimension ( ) k 3 dim S k (Γ 0 (4)) = 2 1 k 4 0 k 2 All formulas stated above are the particular case N = 4 of general formulas which invoke the Euler function

160 156 6 Outlook to Congruence Subgroups Φ(N) = card (Z/NZ) and the number of orbits with non-trivial isotropy group, in particular the number of cusps. cf. [7]. See PARI-file Congruence_subgroup_20: Fig. 6.1 Modular forms of Γ 0 (2) Figure 6.1 shows: S 12 (Γ 0 (2)) contains 2 two old forms. Both result from the cusp form S 12 (Γ ) embedded via j 1,1 ( )(τ) := (τ) and via j 2,1 ( )(τ) := (2 τ).

161 6.1 Modular forms of congruence subgroups 157 Fig. 6.2 Modular forms of Γ 0 (4) Analogously Figure 6.2 shows: S 8 (Γ 0 (4)) contains 2 two old forms. Both result from the cusp form 8 S 8 (Γ 0 (2)) embedded via j 1,2 ( 8 )(τ) := 8 (τ) and via j 2,2 ( 8 )(τ) := 8 (2 τ). S 10 (Γ 0 (4)) contains 2 two old forms. Both result from the cusp form 10 S 10 (Γ 0 (2)) embedded via j 1,2 ( 10 )(τ) := 10 (τ) and via j 2,2 ( 10 )(τ) := 10 (2 τ). S 12 (Γ 0 (4)) contains 3 two old forms. They result from the cusp form S 12 (Γ ) embedded via j 1,1 ( )(τ) := (τ), via and via j 2,1 ( )(τ) := (2 τ) j 4,1 ( )(τ) := (4 τ).

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