Spaces of modular forms. Modular curves and dimensions

FACULTAD DE MATEM´ ATICAS DEPARTAMENTOS DE ´ ALGEBRA Y AN´ ALISIS SPACES OF MODULAR FORMS, MODULAR CURVES AND DIMENSIONS Jes´us L´opez S´anchez Supervised by: Sara Arias de Reyna Dom´ınguez Juan Arias de Reyna Mart´ınez FACULTAD DE MATEM´ ATICAS DEPARTAMENTOS DE ´ ALGEBRA Y AN´ ALISIS Spaces of modular forms, modular curves and dimensions Jes´us L´opez S´anchez Memory presented as part of the requirements to obtain a Master’s degree in Mathematics from the University of Seville. Supervisors: Sara Arias de Reyna Dom´ınguez Juan Arias de Reyna Mart´ınez June 20, 2019 Contents Abstract vii Symbols ix Introduction 11 1 Modular curves 17 1.1 The modular group ........................ 17 1.2 Congruence subgroups ...................... 19 1.3 The Riemann surfaces Y(Γ) = Γ\H............... 23 1.3.1 Elliptic points ....................... 25 1.3.2 Complex charts ...................... 29 1.4 The Riemann surfaces X(Γ) = Γ\H∗.............. 33 1.4.1 Cusps ........................... 34 1.4.2 Complex charts ...................... 37 1.4.3 Genus ........................... 40 2 Automorphic, modular and cusp forms 43 2.1 Basic definitions ......................... 43 2.1.1 Eisenstein series for SL2(Z)............... 53 2.1.2 More examples of modular forms ............ 60 2.2 Some applications ........................ 64 2.2.1 Transformation law of the Dedekind eta function . . . 64 2.2.2 Four squares problem .................. 66 3 Modularity Theorem 73 3.1 Weil pairing ............................ 73 3.2 Moduli spaces for modular curves ................ 76 3.3 Complex analytic version of the Modularity Theorem ..... 79 v vi Contents A Riemann surfaces 81 A.1 Basic definitions ......................... 81 A.2 Morphisms ............................ 84 A.2.1 Theorems on morphisms ................. 86 A.2.2 Meromorphic functions ................. 89 A.3 Meromorphic differentials .................... 91 A.4 Divisors and Riemann-Roch Theorem ............. 94 B Complex elliptic curves 99 B.1 Complex Tori ........................... 99 B.1.1 Analytic group structure .................102 B.2 Morphisms between complex tori ................103 B.3 Weierstrass ℘-function ......................104 B.3.1 Algebraic models .....................106 Bibliography 112 Abstract The Modularity Theorem states that all rational elliptic curve arise from modular forms. In 1995, Andrew Wiles proved a special case of this theorem (then known as the Taniyama–Shimura conjecture) for semistable elliptic curves, completing the proof of Fermat’s Last Theorem after some 350 years. Later, Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor extended Wiles’s techniques to prove completely the Modularity Theorem. In this work we explain a complex analytic version of this notable theorem. Translation to Spanish El Teorema de Modularidad afirma que todas las curvas el´ıpticas racionales surgen de formas modulares. En 1995, Andrew Wiles prob´o un caso especial de este teorema (entonces conocido como la conjetura de Taniyama–Shimura) para curvas el´ıpticas semiestables, completando as´ı la prueba del ´ Ultimo Teorema de Fermat despu´es de unos 350 a˜nos. M´as tarde, Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor extendieron las t´ecnicas de Wiles para probar completamente el Teorema de Modularidad. En este trabajo nosotros explicamos una versi´on anal´ıtica compleja de este notable teorema. vii Symbols Zset of integer numbers Z+set of positive integer numbers Qset of rational numbers Rset of real numbers Cset of complex numbers C∗set of nonzero complex numbers b CRiemann sphere Hupper half plane Dunit disc Re zreal part of a complex number z Im zimaginary part of a complex number z |z|absolute value of a complex number z D(z, ε){w∈C||w−z|< ε} ˙ D(z, ε){w∈C|0<|w−z|< ε} D(z, ε){w∈C||w−z| ≤ ε} ∅empty set int(A) interior of a subset A A0derived of a subset A Aclousure of a subset A Y(Γ) noncompact modular curve Γ\H X(Γ) compact modular curve Γ\H∗(H∗=H∪Q∪{∞}) Ak(Γ) set of automorphic forms of weight kwith respect to Γ Mk(Γ) set of modular forms of weight kwith respect to Γ Sk(Γ) set of cusp forms of weight kwith respect to Γ OX(U) set of holomorphic functions on U MX(U) set of meromorphic functions on U M(n)(X) set of meromorphic differentials on Xof degree n Div(X) set of divisors on X ix 16 Modular forms Outline of the work In Chapter 1 we introduce the modular group, its congruence subgroups and the modular curves, which are quotient spaces of the upper half plane by the action of a congruence subgroup of the modular group. Furthermore, we show these curves are Riemann surfaces that can be compactified. In Chapter 2 we introduce the automorphic, modular and cusp forms. They are (meromorphic) holomorphic functions on the upper half plane that satisfy certain transformation and (meromorphy) holomorphy conditions. We comment on the dimension formulas of the vector spaces of modular and cusp forms, and we conclude with two interesting applications. In Chapter 3 we introduce moduli spaces (isomorphism classes of complex elliptic curves enhanced by associated torsion data) for some modular curves and we explain the complex analytic version of the Modularity Theorem we have mentioned above. In Appendix A we recall all the theory of Riemann surfaces that we need in this work: Holomorphic maps, meromorphic differentials, divisors and the Riemann-Roch Theorem. In Appendix B we explain the principal results over compact Riemann surfaces of genus equal to 1. These Riemann surfaces are called complex elliptic curves for reasons to be explained in this chapter. Comments on Bibliography Most of the content of this work has been extracted from the book [DS05]. This book explains (equivalent) distinct versions of the Modularity Theorem. Other books about this topic we have used are [Apo90,Miy06,Ser73,Shi71]. A good reference book about Riemann surfaces we have used in Appendix is [Mir95]. Also, we have used this magnificent book [Sil09] about elliptic curves in Appendix B. The results of complex analysis we utilize in this work are really basic. The reader can consult them in any of these two books [Ahl78,SS03]. Chapter 1 Modular curves In this first chapter we introduce the modular curves which are quotient spaces of the upper half plane by the action of a congruence subgroup Γ of SL2(Z). We show these curves are Riemann surfaces (see Appendix A) that can be compactified. The theory of compact Riemann surfaces allows us to calculate the topological genus of these compactified curves. 1.1 The modular group The modular group is SL2(Z) = a b c d ∈M2(Z)|ad −bc = 1. Lemma 1.1. The modular group SL2(Z)is generated by the two matrices T=1 1 0 1 and S=0−1 1 0 . Proof: Let Γ be the subgroup of SL2(Z) generated by these matrices and α=a b c d ∈SL2(Z). Below, we describe an algorithm to compute γ∈Γ such that αγ ∈Γ (and therefore we will be able to conclude that α∈Γ). First observe that we can suppose without loss of generality that c6= 0, since otherwise α=±1b 0±1∈Γ. Indeed, Tn=1n 0 1 and S2Tn=−Tn=−1n 0 1 ,∀n∈Z. The identity αTn=a b c d 1n 0 1 =a0b0 c nc +d, n ∈Z, 17 18 Modular forms shows that there exists a matrix γ1∈Γ such that αγ1has bottom row (c0, d0)=(c, nc +d),with |d0|≤|c0|/2. On the other hand, the identity αγ1S=a0b0 c0d00−1 1 0 =b0−a0 d0−c0, n ∈Z, shows that this process can be iterated (a finite number of times) to find a matrix γ∈Γ such that αγ has bottom row (0,±1), and therefore αγ ∈Γ. Each element of the modular group induces naturally an automorphism of the Riemann sphere, the fractional linear transformation α(z) = az +b cz +d,∀z∈b C,for α=a b c d ∈SL2(Z). The identity matrix Iand its negative −Iboth induce the identity map. It is not difficult to prove that two matrices α, α0∈SL2(Z) induce the same transformation if and only if α0=±α. Furthermore, note that (αα0)(τ) = α(α0(τ)),∀z∈b C,∀α, α0∈SL2(Z). Therefore, the modular group acts on the Riemann sphere. The subgroup of transformations defined by the modular group is generated by the two maps induced by the two matrix generators, T(τ) = τ+ 1 and S(τ) = −1/τ. The upper half plane is H={τ∈C|Im (τ)>0}. The formula Im (α(τ)) = Im (τ) |cτ +d|2,∀τ∈H,∀α=a b c d ∈SL2(Z) (1.1) shows that each element of the modular group maps the upper half plane back to itself. Hence the modular group acts also on the upper half plane. Definition 1.2. A matrix α=a b c d ∈SL2(Z),α6=±I, is called elliptic, parabolic, or hyperbolic if the absolute value of its trace, |a+d|, is less than 2, equal to 2, or greater than 2, respectively. 1. Modular curves 19 To see the geometrical meaning of this classification, we have to study the fixed points of the induced transformation on the Riemann sphere. If c= 0, then αis of the form ±1m 0 1 ,for some m∈Z\{0}, so αis parabolic and its unique fixed point is ∞ ∈ b C. If c6= 0, then ∞ ∈ b C can not be a fixed point of α, since α(∞) = a/c. In this case, observe that the fixed points of αsatisfy the quadratic equation cz2+ (d−a)z−b= 0. As the discriminant of this equation is (d−a)2+ 4cb = (d+a)2−4, the fixed points of αare two conjugate complex numbers, a real number, or two distinct real numbers if αis elliptic, parabolic or hyperbolic, respectively. Theorem 1.3. A matrix α=a b c d ∈SL2(Z),α6=±I, is characterized by its fixed points in b Cas follows: . α is elliptic if and only if αhas two fixed points τand τ, with τ∈H. . α is parabolic if and only if αhas only one fixed point in R∪{∞}. . α is hyperbolic if and only if αhas two distinct fixed points in R. 1.2 Congruence subgroups The principal congruence subgroups are Γ(N) = a b c d ∈SL2(Z)|a b c d ≡1 0 0 1 (mod N), N ∈Z+. The matrix congruence is interpreted by entries, i.e., a≡1 (mod N), b ≡0 (mod N), c≡0 (mod N), d ≡1 (mod N). Let us consider the canonical homomorphism from SL2(Z) to SL2(Z/NZ), a b c d ∈SL2(Z)7→ a b c d ∈SL2(Z/NZ). By definition, its kernel is Γ(N). So Γ(N) is a normal subgroup of SL2(Z). Let a b c d be a matrix of SL2(Z/NZ). As ad −bc ≡1 (mod N), 20 Modular forms we deduce that gcd(c, d, N) = 1. Using the Chinese remainder theorem it is possible to prove that there exist integers c1and d1such that c1≡c(mod N), d1≡d(mod N) and gcd(c1, d1)=1. Let g= gcd(c, d). If c6= 0, consider the system t≡1 (mod p),with pprime, p|g, t≡0 (mod p),with pprime, p|c, p -g, and define c1=cand d1=d+tN. If c= 0, then necessarily d6= 0 (unless N= 1, but this case is trivial). Therefore, consider the system s≡1 (mod p),with pprime, p|g, s≡0 (mod p),with pprime, p|d, p -g, and define c1=c+sN and d=d1. These two cases prove the statement. Let kbe the unique integer such that ad1−bc1= 1 + kN. As gcd(c1, d1) = 1, there exist integers a1and b1such that a1d1−b1c1=−k. Letting a2=a+a1N, b2=b+b1N, c2=c1, d2=d1, we obtain that a2b2 c2d2∈SL2(Z) and a2b2 c2d2≡a b c d (mod N). Thus we can conclude that this homomorphism is surjective. Therefore, it induces an isomorphism SL2(Z)/Γ(N)∼ −→ SL2(Z/NZ). As a consequence, note that Γ(N) has finite index in SL2(Z). In fact, it is possible to compute that [SL2(Z) : Γ(N)] = N3Y p|N1−1 p2, where the product is taken over all prime divisors of N[Miy06, p.105]. Definition 1.4. A subgroup Γof SL2(Z)is a congruence subgroup if there exists N∈Z+such that Γ(N)⊂Γ, in which case Γis called congruence subgroup of level N. 1. Modular curves 21 Remarks 1.5. .Each congruence subgroup Γ has finite index in SL2(Z), since Γ(N)⊂Γ,for some N∈Z+. .Each congruence subgroup Γ contains a translation matrix of the form 1h 0 1 :τ→τ+h, h ∈Z+. Let hΓ= min{h∈Z+|1h 0 1 ∈Γ} and let us fix a positive integer Nsuch that Γ(N)⊂Γ. As 1N 0 1 =1hΓ 0 1 q1r 0 1 ,whenever N=qhΓ+r, q, r ∈Z, we deduce that hΓnecessarily divides N. The reason why these subgroups are called congruence subgroups is justified in the following lemma which describes the congruence subgroups of level N. Lemma 1.6. Let Γbe a subgroup of SL2(Z)and N∈Z+. The following conditions are equivalent: 1. Γis a congruence subgroup of level N. 2. There exist γ1, . . . , γd∈SL2(Z)such that Γ = Sd j=1 Γ(N)γj, i.e., Γ = d [ j=1{γ∈SL2(Z)|γ≡γj(mod N)}. Proof: If Γ(N)⊂Γ, then Γ(N) has finite index in Γ. Reciprocally, if there exist γ1, . . . , γd∈SL2(Z) such that Γ = Sd j=1 Γ(N)γj, then I≡γj(mod N),for some j= 1, . . . , d, where Iis the identity matrix. So Γ(N) = Γ(N)γj⊂Γ. An immediate consequence of the fact that Γ(N) is normal in SL2(Z) is that any conjugated subgroup of a congruence subgroup of level Nis also a congruence subgroup of level N. 22 Modular forms Lemma 1.7. Let Γbe a congruence subgroup of SL2(Z),N∈Z+and α∈SL2(Z). If Γis a congruence subgroup of level N, then α−1Γα={α−1γα |γ∈Γ} is also a congruence subgroup of level N. Proof: As Γ(N)⊆Γ, we deduce that Γ(N) = α−1Γ(N)α⊆α−1Γα, since Γ(N) is a normal subgroup of SL2(Z). In addition to the principal congruence subgroups, the most important congruence subgroups are Γ0(N) = a b c d ∈SL2(Z)|a b c d ≡∗ ∗ 0∗(mod N), N ∈Z+, and Γ1(N) = a b c d ∈SL2(Z)|a b c d ≡1∗ 0 1 (mod N), N ∈Z+, where ∗means “unspecified”. Note that Γ(N)⊂Γ1(N)⊂Γ0(N)⊂SL2(Z),∀N∈Z+. In the special case N= 1, Γ(1) = Γ1(1) = Γ0(1) = SL2(Z). The map Γ1(N)→Z/NZ, a b c d ∈Γ1(N)7→ b∈Z/NZ, is surjective, since 1k 0 1 ∈Γ1(N),∀k∈Z. Its kernel is Γ(N). Therefore, Γ(N) is also a normal subgroup of Γ1(N) and Γ1(N)/Γ(N)∼ =Z/NZ,with [Γ1(N) : Γ(N)] = N. Analogously, the map Γ0(N)→(Z/NZ)∗, a b c d ∈Γ0(N)7→ d∈(Z/NZ)∗. 1. Modular curves 23 is surjective as well, since for each d∈(Z/NZ)∗there exist integers edand kd such that edd= 1 + kdN. Therefore edkd N d ∈Γ0(N),∀d∈(Z/NZ)∗. Further, its kernel is Γ1(N). So Γ1(N) is a normal subgroup of Γ0(N) and Γ0(N)/Γ1(N)∼ =(Z/NZ)∗,with [Γ0(N):Γ1(N)] = φ(N), where φis the Euler’s totient function from number theory, φ(N) = |(Z/NZ)∗|=NY p|N1−1 p. Finally, taking into account all the above, we can deduce that [SL2(Z) : Γ0(N)] = NY p|N1 + 1 p, where the product is taken over all prime divisors of N. 1.3 The Riemann surfaces Y(Γ) = Γ\H Let Γ be a congruence subgroup of SL2(Z) acting on the upper half plane. The modular curve Y(Γ) is defined as the quotient space of orbits under Γ, Y(Γ) = Γ\H={Γτ|τ∈H}. The topology of this space is induced by the natural projection π:H→Y(Γ), π(τ)=Γτ, ∀τ∈H, that is, a subset of Y(Γ) is open if its inverse image under πis open in H. This makes πan open map, since π−1(π(U)) = [ γ∈Γ γ(U),∀U⊂H. Observe that πis also a continuous map by definition. As a consequence, the modular curve Y(Γ) is a connected topological space, since π(H) = Y(Γ). The modular curves for Γ(N), Γ0(N) and Γ1(N) are denoted Y(N) = Γ(N)\H, Y0(N) = Γ0(N)\Hand Y1(N)=Γ1(N)\H. 24 Modular forms In this section we show that Y(Γ) can be made into a Riemann surface. The reader can consult the theory of Riemann surfaces in Appendix A. One of the main results we need to carry out this task is that the action of the modular group on the upper half plane is properly discontinuous, i.e., any two points in Hhave neighbourhoods small enough so that each transformation of the modular group taking one point away from the other also takes its neighbourhood away from the other’s. Proposition 1.8. Let τ1, τ2∈H. Then there exist open neighbourhoods U1of τ1and U2of τ2in H, with the following property: For all α∈SL2(Z), if α(U1)∩U26=∅, then α(τ1) = τ2. Proof: Let U0 1and U0 2be any open neighbourhoods of τ1and τ2, respectively, with compact closure in H. Consider the intersection α(U0 1)∩U0 2, α ∈SL2(Z). We claim that this intersection is empty for all but finitely many α∈SL2(Z). Note that for all but finitely many pairs (c, d)∈Z2, with gcd(c, d) = 1, sup{Im (α(τ)) |α=∗ ∗ c d ∈SL2(Z), τ ∈U0 1}<inf{Im (τ)|τ∈U0 2} holds (making the intersection empty), since Im (τ) |cτ +d|2≤min{1 c2y1 ,Y1 (cRe (τ) + d)2},∀τ∈U0 1,∀α=a b c d ∈SL2(Z), where y1= inf{Im (τ)|τ∈U0 1}and Y1= sup{Im (τ)|τ∈U0 1}. Furthermore, for each pair (c, d)∈Z2, with gcd(c, d) = 1, the matrices α∈SL2(Z) with bottom row (c, d) are 1k 0 1 a b c d , k ∈Z, where (a, b)∈Z2is any particular pair such that ad −bc = 1.As α(U0 1)∩U0 2=a b c d (U0 1) + k∩U0 2, we deduce that the intersection is empty for all but finitely many α∈SL2(Z) with bottom row (c, d). Therefore, combining these two remarks we can conclude that there are only finitely many α∈SL2(Z) such that α(U0 1)∩U0 26=∅, 1. Modular curves 25 as claimed. Finally, let Fbe the finite set {α∈SL2(Z)|α(U0 1)∩U0 26=∅, α(τ1)6=τ2}. For each α∈Fthere exist disjoint open neighbourhoods U1,α of α(τ1) and U2,α of τ2in H, since α(τ1)6=τ2. Define U1=U0 1∩ \ α∈F α−1(U1,α)!,an open neighbourhood of τ1in H, and U2=U0 2∩ \ α∈F U2,α!,an open neighbourhood of τ2in H. Observe that these open neighbourhoods satisfy the desired property, i.e., if there exists α∈SL2(Z) such that α(U1)∩U26=∅, then necessarily α /∈F, since otherwise α(U1)∩U2⊆U1,α ∩U2,α =∅. The first immediate consequence of this result is the following corollary. Corollary 1.9. The modular curve Y(Γ) is a Hausdorff topological space. Proof: Let π(τ1), π(τ2) be distinct points in Y(Γ). By Proposition 1.8 there exist open neighbourhoods U1of τ1and U2of τ2in Hsuch that γ(U1)∩U2=∅,∀γ∈Γ. As a consequence, π(U1) and π(U2) are disjoint open subsets of Y(Γ) containing π(τ1) and π(τ2), respectively. 1.3.1 Elliptic points To define charts on the curve Y(Γ) we have to prove that the isotropy subgroups of Γ are finite cyclic. This requirement will bee more clearly seen in the next subsection where we study the action of an isotropy subgroup of Γ on the upper half plane. Definition 1.10. Let Γτdenote the isotropy subgroup of a point τ∈H, i.e., the τ-fixing subgroup of Γ, Γτ={γ∈Γ|γ(τ) = τ}. The point τis an elliptic point of Γif there exists γ∈Γτsuch that γ6=±I, i.e., if Γτdefines a nontrivial subgroup of transformations. 32 Modular forms i 1 −1 δi0 δi(−1) δi(1) ρ −1 21 −2 δρ0 δρ(−2) δρ(1) Figure 1.2: Γ = SL2(Z) and τ=i, ρ - For all γ∈Γ, if γ(U)∩U6=∅, then γ∈Γτ. -Uhas no elliptic points of Γ except possibly τ. Define ψτ:U→Cas ψτ(τ0) = λτ(δτ(τ0)),∀τ0∈U, and let V=ψτ(U). Then for any points τ1, τ2∈U, π(τ1) = π(τ2)⇔τ1∈Γτ2⇔τ1∈Γττ2⇔ψτ(τ1) = ψτ(τ2), i.e., the projection πand the wrapping ψτidentify the same points of U. As a consequence, there exists a bijection φτ:π(U)→Vmaking the diagram U π }} ψτ  π(U)φτ//V commutative. Further, as πand ψτare both continuous and open maps, we deduce that this bijection φτ:π(U)→Vis in fact a homeomorphism. Caution: This complex chart depend on the open neighbourhood Uof τ. Below, we show that all these charts determine an atlas on the curve Y(Γ). Let φ1:π(U1)→V1and φ2:π(U2)→V2be two charts such that φ1=φτ1, φ2=φτ2and π(U1)∩π(U2)6=∅. 1. Modular curves 33 Consider the commutative diagram π(U1)∩π(U2) φ2 && V1,2 φ−1 1 88 φ2,1//V2,1 where φ2,1=φ2◦φ−1 1,V1,2=φ1(π(U1)∩π(U2)) and V2,1=φ2(π(U1)∩π(U2)). We have to prove that φ2,1is holomorphic at φ1(x) for all x∈π(U1)∩π(U2). Put x=π( ˜τ1) = π( ˜τ2) with ˜τ1∈U1, ˜τ2∈U2and ˜τ2=γ˜τ1for some γ∈Γ. Let U1,2=U1∩γ−1(U2), an open neighbourhood of ˜τ1in H. Since πis open, its projection π(U1,2) is an open neighbourhood of xin π(U1)∩π(U2). An input point q=φ1(x0) to φ2,1in φ1(π(U1,2)) is of the form q=φ1(π(τ0)) = ψ1(τ0)=(δ1(τ0))h1,for some τ0∈U1,2, where δ1=δτ1and h1is the period of τ1. So the corresponding output is φ2(x0) = φ2(π(γ(τ0))) = ψ2(γ(τ0)) (since γ(τ0)∈U2) = (δ2(γ(τ0)))h2= ((δ2γδ−1 1)(δ1(τ0)))h2, where δ2=δτ2and h2is the period of τ2. This calculation shows that the only case possible where the transition function might not be holomorphic at φ1(x) is when δ1(˜τ1) = 0 and h1>1, i.e., when τ1= ˜τ1and τ1is an elliptic point of Γ. In this case, ˜τ2=γ(τ1) would also be an elliptic point of Γ with the same period, implying τ2= ˜τ2 (recall that U2has no elliptic points except possibly τ2by construction). Therefore, δ2γδ−1 1=α0 0β,for some α, β ∈C∗, since 0δ−1 1 7−→ τ1 γ 7−→ τ2δ2 7−→ 0 and ∞δ−1 1 7−→ τ1 γ 7−→ τ2δ2 7−→ ∞. As a consequence, the formula for φ2,1becomes φ2,1(q) = α0 0β(q1/h1)h1 = (α/β)h1q, ∀q∈φ1(π(U1,2)), Observe that this proves that transition function is holomorphic at φ1(x) = 0. The modular curve Y(Γ) is now a (noncompact) Riemann surface. 1.4 The Riemann surfaces X(Γ) = Γ\H∗ In this last section of the chapter we show that the Riemann surface Y(Γ) can be compactified. The resulting compact Riemann surface is denoted X(Γ). The Riemann-Hurwitz formula A.28 will allow us to calculate its genus. 34 Modular forms 1.4.1 Cusps To compactify the modular curve Y(Γ) = Γ\H, define H∗=H∪b Q,where b Q=Q∪{∞}, and consider the following topology on H∗: - A fundamental system of neighbourhoods at τ∈His formed by the usual open neighbourhoods of τin H. - A fundamental system of neighbourhoods at s∈b Qis formed by the subsets α(NM∪{∞}) : M > 0, α ∈SL2(Z), α(∞) = s, where NM={τ∈H|Im (τ)> M}for any real number M > 0 (N:= N1). As fractional linear transformations are conformal and take circles to circles, if α(∞)∈Q, then α(NM∪{∞}) is a disc tangent to the real line at α(∞). The formula (1.1) allow us to compute that the radius of this disc is 1/2c2M, where cis the lower left entry of the matrix α. Therefore, α(NM∪{∞})∩(N ∪{∞}) = ∅,∀M≥1. If α(∞) = ∞, then αis a translation matrix and α(NM∪{∞})=NM∪{∞}, since the isotropy subgroup of ∞in SL2(Z) is SL2(Z)∞=±1m 0 1 |m∈Z. Lemma 1.21. Let α∈SL2(Z). Then the following conditions are equivalent: 1. α∈SL2(Z)∞ 2. α(NM) = NM, for any M > 0. 3. α(NM)∩NM6=∅, for some M≥1. As a consequence, observe that Ndoes not contains elliptic points of SL2(Z). The proof of this lemma is easy and is left as an exercise to the reader. Below, the Figure 1.3 shows N ∪{∞} and some of its SL2(Z)-translates. Note that H∗is a Hausdorff topological space with respect to this topology. The modular group acts on this space via fractional linear transformations. Let Γ be a congruence subgroup of SL2(Z) acting on H∗. The compact modular curve X(Γ) is defined as the quotient space of orbits under Γ, X(Γ) = Γ\H∗=Y(Γ) ∪Γ\b Q. 1. Modular curves 35 −1 2 1 2 2 3 −1 3 1 3 −2 30 1 −1 Figure 1.3: Neighbourhoods of ∞and of some rational points As in Section 1.3, the topology of this quotient space is induced by the natural projection π:H∗→X(Γ) which is an open continuous map. The Γ-equivalence classes of points in b Qare also called the cusps of X(Γ). The (compact) modular curves for Γ(N), Γ0(N) and Γ1(N) are denoted X(N) = Γ(N)\H, X0(N) = Γ0(N)\Hand X1(N)=Γ1(N)\H. Lemma 1.22. The modular curve SL2(Z)\H∗has only one cusp, SL2(Z)∞. Proof: Let s=a/c be a rational number in reduced form, gcd(a, c) = 1. By the B´ezout’s identity, there exist integers band dsuch that ad −bc = 1. Then a b c d ∈SL2(Z) and a b c d (∞) = s. Corollary 1.23. The modular curve X(Γ) has only finitely many cusps. Proof: Let SL2(Z) = Sd j=1 Γαj. For each s∈b Qthere exists α∈SL2(Z) such that α(∞) = s. So sis Γ-equivalent to αj(∞), for some j= 1, . . . , d. Theorem 1.24. The modular curve X(Γ) is Hausdorff, connected and compact. Proof: Let x1, x2∈X(Γ) be distinct points. To prove that X(Γ) is Hausdorff, we have to find disjoint open neighbourhoods of these two points. 36 Modular forms Note that the case x1= Γτ1,x2= Γτ2, with τ1, τ2∈H, was already proved in Corollary 1.9, since the natural inclusion Y(Γ) ,→X(Γ) is an open map. Suppose that x1= Γs1,x2= Γτ2, with s1∈Q∪{∞} and τ2∈H. Then s1=α1(∞) for some α1∈SL2(Z). Let U2be any open neighbourhood of τ2 with compact closure in H. The inequality Im (α(τ)) ≤max{Im (τ),1/Im (τ)},∀τ∈H,∀α∈SL2(Z), shows that there exists M > 0 such that α(U2)∩ NM=∅,∀α∈SL2(Z). Let U1=α1(NM∪{∞}). Then π(U1) and π(U2) are disjoint open subsets of the curve X(Γ) containing x1and x2, respectively. Suppose now that x1= Γs1,x2= Γs2, with s1, s2∈Q∪ {∞}. Then s1=α1(∞), s2=α2(∞) for some α1, α2∈SL2(Z).Let U1=α1(N ∪{∞}) and U2=α2(N ∪{∞}). Then π(U1) and π(U2) be must disjoint open subsets of X(Γ), since otherwise there exists γ∈Γ such that α−1 2γα1(N ∪ {∞})∩(N ∪ {∞})6=∅, i.e., α−1 2γα1∈SL2(Z)∞, but this is not possible since the points x1and x2 are distinct. These three cases prove that the curve X(Γ) is Hausdorff. Let H∗=G1∪G2be a disjoint union of two open subsets. Then H= (G1∩H)∪(G2∩H). But His a connected subset and the subsets G1∩Hand G2∩Hare open, so this implies that either H⊂G1and G2=∅or G1=∅and H⊂G2. Thus we conclude that H∗is connected and therefore so is the curve X(Γ). For compactness, let SL2(Z) = Sd j=1 Γαj. As F∗=F∪{∞} is compact subset of H∗and H∗= SL2(Z)F∗= d [ j=1 Γαj(F∗), we deduce that the curve modular X(Γ) is a finite union of compact subsets, X(Γ) = d [ j=1 π(γj(F∗)). 1. Modular curves 37 1.4.2 Complex charts Each point s∈b Qhas an associated positive integer hs=hs,Γ=|SL2(Z)∞/(δs{±I}Γδ−1 s)∞|, where δs∈SL2(Z) takes sto ∞. This hsis called the width of swith respect to Γ for reasons to be explained. As the subgroup SL2(Z)∞={±I}1 1 0 1  is infinite cyclic as a group of transformations, the width of sis characterized by the conditions {±I}(δsΓδ−1 s)∞={±I}1h 0 1 ,with h > 0. Observe that the width of sis independent of the matrix δstaking sto ∞, since hs=|SL2(Z)s/{±I}Γs|. Let α∈SL2(Z). Then the isotropy subgroups SL2(Z)sand SL2(Z)α(s) are conjugated subgroups, SL2(Z)α(s)=αSL2(Z)sα−1. Further, {±I}(αΓα−1)α(s)=α{±I}Γsα−1. Hence the width of α(s) under αΓα−1is equal to the width of sunder Γ. This proves in particular that the width of π(s)∈X(Γ) is also well defined. Examples 1.25. 1. The width of a point s∈b Qwith respect to SL2(Z) is hs= 1. 2. More generally, the width of a point s∈b Qwith respect to Γ(N) is hs=N. Let us construct a chart on X(Γ) about the point π(s). Define U=δ−1 s(N ∪{∞}). Note that this open neighbourhood of sin H∗has the following property: For all γ∈Γ, if γ(U)∩U6=∅, then γ∈Γs. As a consequence, z1, z2∈Uare Γ-equivalent ⇔z1, z2∈Uare Γs-equivalent. Consider the subgroup {±I}(δsΓsδ−1 s) = {±I}(δsΓδ−1 s)∞={±I}1hs 0 1 . 38 Modular forms δs i 0 Figure 1.4: Γ = SL2(Z), s= 0 and δs=0−1 1 0  Then for any points τ1, τ2∈U, π(τ1) = π(τ2)⇔τ1∈Γsτ2 ⇔δs(τ1)∈δsΓsδ−1 s(δs(τ2)) ⇔δs(τ1) = δs(τ2) + mhs,for some m∈Z. This shows that the width of a cusp is the number of unit vertical strips in N that are distinct under isotropy. The Figure 1.4 illustrates the situation. Observe that each unit vertical strip in Ncorresponds in Uwith a “triangle” formed by three circular arcs (including lines). Hence the width of a cusp is also the number of such triangles that are not identified under isotropy. This time δsis straightening neighbourhoods of sto neighbourhoods of ∞. Define ψs:U→Cas ψs(τ) = λs(δs(τ)),∀τ∈U, where λsis the hs-periodic wrapping map λs(z) = e2πiz/hs(λs(∞) := 0). Taking into account the above, it is immediate to check that this map and the projection πidentify the same points of U. Let V=ψs(U) (=(e−2π/hs)D). Then there exists a bijection φs:π(U)→Vmaking the following diagram U π }} ψs  π(U)φs//V commutative. Further, as πand ψsare both continuous and open maps, we deduce that this bijection φs:π(U)→Vis in fact a homeomorphism. Caution: This chart complex depend on the matrix δstaking sto ∞. Below, we show that all these complex charts together with the ones constructed in the previous section determine an atlas on the curve X(Γ). 1. Modular curves 39 Let φ1:π(U1)→V1and φ2:π(U2)→V2be two charts such that π(U1)∩π(U2)6=∅. Consider the commutative diagram π(U1)∩π(U2) φ2 && V1,2 φ−1 1 88 φ2,1//V2,1 where φ2,1=φ2◦φ−1 1,V1,2=φ1(π(U1)∩π(U2)) and V2,1=φ2(π(U1)∩π(U2)). We have to prove that φ2,1is holomorphic at φ1(x) for all x∈π(U1)∩π(U2). Put x=π( ˜τ1) = π( ˜τ2) with ˜τ1∈U1, ˜τ2∈U2and ˜τ2=γ˜τ1for some γ∈Γ. Let U1,2=U1∩γ−1(U2), an open neighbourhood of ˜τ1in H∗. Since πis open, its projection π(U1,2) is an open neighbourhood of xin π(U1)∩π(U2). Note that the case φ1=φτ1and φ2=φτ2, with τ1, τ2∈H, was already proved in the previous section. Suppose that φ1=φτ1, with τ1∈H, and φ2=φs2, with s2∈Q∪ {∞}. As before, an input point q=φ1(x0) to φ2,1in φ1(π(U1,2)) is of the form q=φ1(π(τ0)) = ψ1(τ0)=(δ1(τ0))h1,for some τ0∈U1,2, where δ1=δτ1and h1is the period of τ1. So the corresponding output is φ2(x0) = φ2(π(γ(τ0))) = ψ2(γ(τ0)) (since γ(τ0)∈U2) = exp(2πiδ2(γ(τ0))/h2) = exp(2πiδ2γδ−1 1(δ1(τ0))/h2), where δ2:s27→ ∞ is the straightening map of φ2and h2is the width of s2. As a consequence, observe that the only case possible where the transition function might not be holomorphic at φ1(x) is when δ1(˜τ1) = 0 and h1>0, i.e., τ1= ˜τ1and τ1is an elliptic point of Γ. But this case is not possible, since otherwise δ2(γ(τ1)) ∈ N would also be an elliptic point of Γ, which is an contradiction. This argument also covers the case φ1=φs1, with s1∈Q∪ {∞}, and φ2=φτ2, with τ2∈H, since the inverse of a holomorphic biyection is also holomorphic. Suppose now that φ1=φs1and φ2=φs2, with s1, s2∈Q∪ {∞}. Let δ1:s17→ ∞ and δ2:s27→ ∞ be the corresponding straightening maps. As π(U1)∩π(U2)6=∅, there exists γ∈Γ such that δ2γδ−1 1is a translation, δ2γδ−1 1=±1m 0 1 ,for some m∈Z. As a consequence, γ(s1) = γδ−1 1(∞) = ±δ−1 21m 0 1 (∞) = s2. 40 Modular forms In this case, an input point q=φ1(x0) to φ2,1in φ1(π(U1,2)) is of the form q=φ1(π(τ0)) = ψ1(τ0) = exp(2πiδ1(τ)/h1),for some τ0∈U1,2, where h1is the width of s1. So the corresponding output is φ2(x0) = φ2(π(γ(τ0))) = ψ2(γ(τ0)) (since γ(τ0)∈U2) = exp(2πiδ2γδ−1 1(δ1(τ0))/h1) = exp(2πi(δ1(τ0) + m)/h1) = exp(2πim/h1)q Observe that this proves that transition function is holomorphic at φ1(x). The modular curve X(Γ) is now a compact Riemann surface. Figure 1.5 summarizes the complex charts of X(Γ) for future references. π:H∗→X(Γ) is the natural projection. U⊂H∗is a neighbourhood containing at most one elliptic point or cusp. The complex chart φ:π(U)→Vsatisfies φ◦π=ψ, where ψ:U→Vis a composition ψ=λ◦δ. About τ∈H: About s∈Q∪{∞}: The straightening map is z=δ(τ0), The straightening map is z=δ(τ0), where δ=1−τ 1−¯τ,δ(τ) = 0. where δ∈SL2(Z), δ(s) = ∞. δ(U) is a neighbourhood of 0. δ(U) is a neighbourhood of ∞. The wrapping map is q=λ(z) The wrapping map is q=λ(z) where λ(z) = zh,λ(0) = 0, where λ(z) = e2πiz/h,λ(∞) = 0, with period h=|{±I}Γτ/{±I}|. with width h=|SL2(Z)s/{±I}Γs|. V=λ(δ(U)) is a neighbourhood of 0. V=λ(δ(U)) is a neighbourhood of 0. Figure 1.5: Complex charts on X(Γ) 1.4.3 Genus Let Γ1,Γ2be congruence subgroups of SL2(Z) such that Γ1⊂Γ2. Theorem 1.26. The natural projection of the corresponding modular curves F:X(Γ1)→X(Γ2),Γ1τ7→ Γ2τ, is a surjective morphism of Riemann surfaces. Its degree is deg(F) = |{±I}Γ2:{±I}Γ1|=[Γ2: Γ1]/2if −I∈Γ2and −I /∈Γ1, [Γ2: Γ1]otherwise. 1. Modular curves 41 Proof: Consider the commutative diagram U δ zz π1  id //U π2  δ $$ δ(U) λ1## π1(U) φ1  F//π2(U) φ2  δ(U) λ2 {{ V1 Flocal //V2 where φj:πj(U)→Vjis a chart on X(Γj), for j= 1,2. Observe that Flocal ◦ψ1=ψ2,where ψj=λj◦δ. Let δ=δτ, with τ∈H. Then λ1(z) = zh1,λ2(z) = zh2and the local map is q∈V17→ qh2/h1∈V2, where hj=|{±I}Γj,τ |/2, for j= 1,2. Further, as hj∈{1,2,3}and h2/h1∈Z, we deduce that the ramification index of Fat π1(τ) is eπ1(τ)(F) = h2/h1=h2if τis an elliptic point of Γ2and not of Γ1, 1 otherwise, =|{±I}Γ2,τ :{±I}Γ1,τ |. Let δ:s7→ ∞, with s∈b Q. Then λ1(z) = e2πiz/h1,λ2(z) = e2πiz/h2and the local map is q∈V17→ qh1/h2∈V2, where hj= [SL2(Z) : {±I}Γj,s], for j= 1,2. Further, as h1/h2∈Z, we deduce that the ramification index of Fat π1(s) is eπ1(s)(F) = h1/h2=|{±I}Γ2,s :{±I}Γ1,s|. As a consequence, π1(s) is a ramification point of Fif and only if h1> h2. This proves that the natural projection is a morphism of Riemann surfaces. To compute its degree, let {±I}Γ2=Sd j=1{±I}Γ1γj, where γjare coset representatives. Then the inverse image of a nonelliptic point π2(τ)∈X(Γ2) is F−1(π2(τ)) = {π1(γ1(τ)), . . . , π1(γd(τ))}. This shows that deg(F) = |{±I}Γ2:{±I}Γ1|, since deg(F) = X x∈F−1(y) ex(F),∀y∈X(Γ2). To calculate the genus of X(Γ), specialize to Γ1= Γ and Γ2= SL2(Z). 48 Modular forms where µhis the complex h-th root of the unity e2πi/h (e2πi(τ+j)/h =µj hqh). Thus we can conclude that the definition of the order of fat the cusp s is independent of the chosen matrix α∈SL2(Z) with α(∞) = s. Note that the order of fis also well-defined on the modular curve X(Γ), i.e., if s∈b Qand γ∈Γ, then νs(f) = νγ(s)(f). The modular forms are defined the same way as automorphic forms except with holomorphy in place of meromorphy. Definition 2.6. Let Γbe a congruence subgroup of SL2(Z)and k∈Z. A holomorphic function f:H→b Cis a modular form of weight kwith respect to Γif . f is weight-kinvariant under Γ, and . f[α]kis holomorphic at ∞, for all α∈SL2(Z). If in addition, . a0= 0 in the Fourier series of f[α]k, for all α∈SL2(Z), then fis a cusp form of weight kwith respect to Γ. The sets of modular and cusp forms of weight kwith respect to Γ are denoted Mk(Γ) and Sk(Γ), respectively. Note that both sets are vector subspaces of Ak(Γ), since Mk(Γ) = {f∈ Ak(Γ) |fis holomorphic and ν∞(f[α]k)≥0,∀α∈SL2(Z)} and Sk(Γ) = {f∈ Ak(Γ) |fis holomorphic and ν∞(f[α]k)≥1,∀α∈SL2(Z)}. As a consequence, Sk(Γ) ⊂ Mk(Γ) ⊂ Ak(Γ) ⊂ M(H). Remarks 2.7. .Let Γ1, Γ2be congruence subgroups of SL2(Z) such that Γ1⊂Γ2. Then Sk(Γ2)⊂ Sk(Γ1),Mk(Γ2)⊂ Mk(Γ1) and Ak(Γ2)⊂ Ak(Γ1). 2. Automorphic, modular and cusp forms 49 .Let α∈SL2(Z). The weight-koperator associated to the matrix α induces the following isomorphisms of C-vector spaces: Ak(Γ) ∼ =Ak(α−1Γα)Mk(Γ) ∼ =Mk(α−1Γα)Sk(Γ) ∼ =Sk(α−1Γα) .If f∈Ak(Γ) (resp. Mk(Γ) or Sk(Γ)) and g∈Al(Γ) (resp. Ml(Γ) or Sl(Γ)), then fg ∈ Ak+l(Γ) (resp. Mk+l(Γ) or Sk+l(Γ)). Indeed, -fg is weight-(k+l) invariant under Γ, and - the Fourier series of (fg)[α](k+l), with α∈SL2(Z), is (fg)[α](k+l)=X n∈Z (X s+j=n asbj)qn h, qh=e2πiτ/h, where f[α]k=X s∈Z asqs h, g[α]l=X j∈Z bjqj hand h=h(α−1Γα). Thus the direct sums A(Γ) = M k∈ZAk(Γ),M(Γ) = M k∈ZMk(Γ) and S(Γ) = M k∈ZSk(Γ) forms graded rings. .Let R=A,Mor S. For each α∈SL2(Z), the map X k∈Z fk∈ R(Γ) 7→ X k∈Z fk[α]k∈ R(α−1Γα) is an isomorphism of graded rings. .If f∈ Mk(Γ) and g∈ Sl(Γ), then fg ∈ Sk+l(Γ). As a consequence, observe that S(Γ) is a graded ideal of M(Γ), since M(Γ)S(Γ) ⊂ S(Γ) and S(Γ) = M k∈Z (Mk(Γ) ∩S(Γ)). .If f∈ Ak(Γ) and g∈ Al(Γ) (g6= 0), then f/g ∈ Ak−l(Γ). Indeed, 1/g ∈ A−l(Γ) and f/g =f(1/g)∈ Ak−l(Γ). In particular, note that Ak(Γ) = fA0(Γ) = {ff0|f0∈ A0(Γ)},whenever f6= 0. 50 Modular forms .If f, g ∈ A0(Γ) and λ, β ∈C, then λf +βg,fg, 1/f (f6= 0) ∈ A0(Γ). This implies that A0(Γ) is a C-algebra that presents structure of field. Moreover, the map f∈ A0(Γ) 7→ F∈ M(X(Γ)), F(x) = f(τ) if x= Γτ, f(s) if x= Γs, defines an isomorphism of C-algebras. .If −I∈Γ and kis odd, then Sk(Γ) = Mk(Γ) = Ak(Γ) = {0}, since the only function that is weigh-kinvariant under Γ is the zero function. Theorem 2.8. Let Γbe a congruence subgroup of SL2(Z)of level N,k∈Z and f:H→Ca holomorphic function. If fis weight-kinvariant under Γ, fis holomorphic at ∞, f(τ) = ∞ X n=0 anqn N, qN=e2πiτ/N ,∀τ∈H, and in addition, there exist positive constants Cand rsuch that |an| ≤ Cnr,∀n≥1, then f[α]kis holomorphic at ∞,∀α∈SL2(Z). As a consequence, f∈ Mk(Γ). Proof: Let α∈SL2(Z). The function (f[α]k)(τ) = j(α, τ)−kf(α(τ)),∀τ∈H, is holomorphic and weight-kinvariant under α−1Γα, so it has an expansion (f[α]k)(τ) = X n∈Z a0 nqn N,∀τ∈H. Therefore, to prove that f[α]kis holomorphic at ∞, it suffices to see that lim qN→0((f[α]k)(τ)·qN)=0.(2.1) Let us suppose that we have proved that there exist constants C0, C1>0 satisfying the following property: |f(τ)| ≤ C0+C1/yr,as y→ ∞ (τ=x+iy ∈H).(2.2) If α(∞) = ∞, then (2.1) is immediate, since α=±1m 0 1 , with m∈Z, and (f[α]k(τ)) = (±1)kf(τ+m) = (±1)kµm N ∞ X n=0 anqn N,∀τ∈H, 2. Automorphic, modular and cusp forms 51 where µN=e2πi/N . Otherwise, we have c6= 0 and |(f[α]k)(τ)|=|f(α(τ))||cτ +d|−k ≤C0+C1Im (α(τ))−r|cτ +d|−k =C0+C1|cτ +d|2ry−r|cτ +d|−k,as y→ ∞, where cand dare the lower entries of the matrix α. If we assume that 0≤x≤N((f[α]k)(τ)=(f[α]k)(τ+N)), then |cτ +d|grows as y, and as a consequence, there exists a constant C2>0 such that |(f[α]k)(τ)| ≤ C2yr−k,as y→ ∞. Using this estimation we obtain (2.1), since y= (N/2π) log(1/|qN|) and |(f[α]k)(τ)·qN| ≤ C2(N/2π)r−klog(1/|qN|)r−k|qN| → 0,as qN→0. To conclude this proof we need to prove that there exist constants C0, C1>0 satisfying the property (2.2). By hypothesis, |f(τ)|≤|a0|+C ∞ X n=1 nre−2πny/N ,∀τ=x+iy ∈H. Let gy:R→Rbe the function gy(t) = tre−2πty/N ,∀t∈R. As its derivative is g0 y(t) = rtr−1−2πy Ntre−2πty/N ,∀t∈R, we deduce that this function decreases monotonically on the interval [ rN 2πy ,∞). Therefore, |f(τ)| ≤ |a0|+Cgy(1) + ∞ X n=2 gy(n) ≤ |a0|+C1 + ∞ X n=2 Zn n−1 gy(t)dt[rN/2π < y] ≤ |a0|+C+CZ∞ 0 tre−2πty/N dt [t= 2πty/N] =|a0|+C+C(N/2π)r+11/yr+1 Z∞ 0 tre−tdt, as y→ ∞. 52 Modular forms Letting C0=|a0|+Cand C1=C(N/2π)r+1 Z∞ 0 tre−tdt, we obtain that |f(τ)| ≤ C0+C1/yr,as y→ ∞ (τ=x+iy ∈H). The condition of holomorphy at the cusps, f[α]kis holomorphic at ∞, keeps the vector spaces of modular and cusp forms finite-dimensional. Theorem 2.9. Let kbe an even integer. Let Γbe a congruence subgroup of SL2(Z),gthe genus of X(Γ),ε2the number of elliptic points with period 2, ε3the number of elliptic points with period 3and ε∞the number of cusps. Then dim(Mk(Γ)) =    (k−1)(g−1) + k 4ε2+k 3ε3+ε∞,if k≥2, 1,if k= 0, 0,if k < 0, and dim(Sk(Γ)) =    (k−1)(g−1) + k 4ε2+k 3ε3+ (k 2−1)ε∞,if k≥4, g, if k= 2, 0,if k≤0. Proof: [DS05, p.86-88] Let s∈b Qand α∈SL2(Z) taking ∞to s. Recall that the width h∈Z+ of swith respect to a congruence subgroup Γ satisfies the condition {±I}(Γα)∞={±I}Dh1h 0 1 iE,where Γα=α−1Γα. Therefore, this implies that (Γα)∞={±I}Dh1h 0 1 iE,(Γα)∞=Dh1h 0 1 iE or (Γα)∞=D−h1h 0 1 iE, since the negative identity matrix −Imight not be in the subgroup α−1Γα. The cusp π(s)∈X(Γ) is called a regular cusp of Γ if (Γα)∞={±I}Dh1h 0 1 iE or (Γα)∞=Dh1h 0 1 iE. Otherwise, π(s) is called irregular cusp of Γ, (Γα)∞=D−h1h 0 1 iE. Note that this definition is independent of the chosen matrix α∈SL2(Z). 2. Automorphic, modular and cusp forms 53 Theorem 2.10. Let kbe an odd integer. Let Γbe a congruence subgroup of SL2(Z),gthe genus of X(Γ),ε3the number of elliptic points with period 3, εreg ∞the number of regular cusps and εirr ∞the number of irregular cusps. If −I /∈Γ, then dim(Mk(Γ)) = (k−1)(g−1) + k 3ε3+k 2εreg ∞+k−1 2εirr ∞,if k≥3, 0,if k < 0, and dim(Sk(Γ)) = (k−1)(g−1) + k 3ε3+k−2 2εreg ∞+k−1 2εirr ∞,if k≥3, 0,if k < 0. If εreg ∞>2g−2, then dim(M1(Γ)) = εreg ∞/2and dim(S1(Γ)) = 0. If εreg ∞≤ 2g−2, then dim(M1(Γ)) ≥εreg ∞/2and dim(S1(Γ)) = dim(M1(Γ)) −εreg ∞/2. Proof: [DS05, p.90-91] The demonstration of these dimension formulas uses the Riemann-Roch Theorem A.44. Specifically, the following consequence: Let Xbe a compact Riemann surface of genus g. If Dis a divisor on X such that deg(D)>2g−2, then dim L(D) = deg(D)−g+ 1. We introduce some applications of these dimension formulas in Section 2.2. Remark 2.11. Observe that we have not given dimension formulas for the vector spaces M1(Γ) and S1(Γ),when εreg ∞≤2g−2. Determining dimension formulas for these spaces is an open problem. 2.1.1 Eisenstein series for SL2(Z) Let k≥3 be an even integer. The Eisenstein series of weight kfor SL2(Z) is defined as Gk(τ) = X0 (c,d) 1 (cτ +d)k,∀τ∈H, where the primed summation means to sum over the pairs (c, d)∈Z2\{(0,0)}. If we let Λτdenote the lattice τZ+Z={cτ +d|c, d ∈Z}, τ ∈H, 54 Modular forms then another expression for the Eisenstein series of weight kis Gk(τ) = X0 w∈Λτ 1 wk,∀τ∈H, where the primed summation means now to sum over the points w∈Λτ\{0}. The following lemma is an important technical result. We will use it to prove that the Eisenstein series define holomorphic functions on the upper half plane. Lemma 2.12. Let Λbe a lattice in Cand r > 0. Then the series X0 w∈Λ 1 |w|r<∞if and only if r > 2. Proof: Let (w1, w2) be a basis of Λ. For each k∈Z+, define Ak={mw1+nw2|(m, n)∈Z2,|m|+|n|=k}and Sk={xw1+yw2|(x, y)∈R2,|x|+|y|=k}. Then Ak⊂Sk, Sk=kS1and |Ak|= 4k, ∀k∈Z+. Using this, we obtain that X0 w∈Λ 1 |w|r= ∞ X k=1 X w∈Ak 1 |w|r,(2.3) 4 Cr ∞ X k=1 1 kr−1≤ ∞ X k=1 X w∈Ak 1 |w|r≤4 cr ∞ X k=1 1 kr−1,(2.4) where C= max z∈S1|z|and c= min z∈S1|z|. As the series of positive terms ∞ X k=1 1 kr−1<∞if and only if r > 2, the proof of the lemma is an immediate consequence of (2.3) and (2.4). Theorem 2.13. The Eisenstein series Gk(τ) = X (c,d) 01 (cτ +d)k,with k≥3, defines a holomorphic function on the upper half plane. 2. Automorphic, modular and cusp forms 55 Proof: We show that the series converges uniformly on the subset Ω(a, b) = {τ∈H||Re τ| ≤ a, Im τ≥b}, a, b > 0. Let τ∈H,τ=x+iy, and c, d ∈Z. Then |cτ +d|2=c2x2+d2+ 2cdx +c2y2. Choose δ > 0 such that a2 a2+b2< δ2<1 and rewrite the anterior equality as |cτ +d|2=y2+1−1 δ2x2c2+δd +cx δ2+ (1 −δ2)d2. If τ∈Ω(a, b) (i.e., |x| ≤ aand y≥b), we obtain that |cτ +d|2≥y2+1−1 δ2x2c2+ (1 −δ2)d2 ≥b2+1−1 δ2a2c2+ (1 −δ2)d2. As the coefficients of c2and d2are both positive, there exists ε > 0 such that |cτ +d|2≥ε2(c2+d2),∀c, d ∈Z, or equivalently, 1 |cτ +d|≤1 ε 1 |ci +d|,∀(c, d)∈Z2,(c, d)6= (0,0).(2.5) Furthermore, by Lemma 2.12 X0 (c,d) 1 |ci +d|k<∞, so it suffices to apply the Weierstrass M-test to deduce that the series X0 (c,d) 1 (cτ +d)k converges uniformly on the subset Ω(a, b). We only have defined Eisenstein series of weight k≥3 even. If k≥3 is odd, then Gkis also a holomorphic function, but in this case it is the zero function, since the terms corresponding to (c, d),−(c, d)∈Z2\{(0,0)}, 1 (cτ +d)kand 1 (−1)k(cτ +d)k, 56 Modular forms cancel out - are cancelled. Below, we show that Gkis a modular form of weight kwith respect to SL2(Z). Let α=a b c d ∈SL2(Z). Then Gk(α(τ)) = X0 (c0,d0) 1 (c0(α(τ)) + d0)k = (cτ +d)kX0 (c0,d0) 1 ((c0a+d0c)(τ)+(c0b+d0d))k,∀τ∈H. Furthermore, as the map (c0, d0)7→ (c0, d0)a b c d = (c0a+d0c, c0b+d0d) is a bijection from Z2\{(0,0)}to itself, we deduce that the right side of this equality is (cτ +d)kGk(τ). Therefore, Gkis weight-kinvariant under SL2(Z), Gk[α]k=Gk,∀α∈SL2(Z). To compute the Fourier series of Gk, we use the following two identities of the cotangent function: πcot(πz) = 1 z+ ∞ X d=1 1 z−d+1 z+d,∀z∈C\Z(2.6) πcot(πτ) = πiq+ 1 q−1=πi −2πi ∞ X m=0 qm, q =e2πiτ ,∀τ∈H(2.7) The first identity is a partial fraction decomposition of the meromorphic function πcot(πz). We omit its proof since it is not trivial. It can be consulted in [SS03, p.142]. The second identity follows from the expansion 1 q−1=− ∞ X m=0 qm,∀q∈C,|q|<1. Equating these two expressions for πcot(πτ) and differentiating k−1 times with respect to τ, we obtain that X d∈Z 1 (τ+d)k=(−2πi)k (k−1)! ∞ X m=1 mk−1qm,∀τ∈H, k ≥2.(2.8) For k≥3 even, X0 (c,d) 1 (cτ +d)k=X d6=0 1 dk+ 2 ∞ X c=1 X d∈Z 1 (cτ +d)k! = 2ζ(k)+2 (2πi)k (k−1)! ∞ X c=1 ∞ X m=1 mk−1qcm,∀τ∈H, 2. Automorphic, modular and cusp forms 57 where ζdenotes the Riemann zeta function, ζ(s) = ∞ X n=1 1 ns,∀s∈C,Re (s)>1. Rearranging the terms of the last double series gives the Fourier series, Gk(τ) = 2ζ(k)+2 (2πi)k (k−1)! ∞ X n=1 σk−1(n)qn,∀τ∈H,(2.9) where the coefficients of the series are the values of the arithmetic function σk−1(n) = X m|n m>0 mk−1,∀n≥1. This proves that Gkis a modular form of weight kwith respect to SL2(Z). Remark 2.14. For each 0 < r < 1, observe that ∞ X c=1 ∞ X m=1 mk−1|q|cm = ∞ X m=1 mk−1|q|m 1−|q|m ≤1 1−r ∞ X m=1 mk−1rm<∞,∀q∈rD. Therefore, the doubles series ∞ X c=1 ∞ X m=1 mk−1qcm, q ∈D, k ∈Z, converges uniformly on compact subsets of the unit disc. Remark 2.15. There is another way of proving that Gkis holomorphic at ∞ without having to compute its Fourier series. Let a= 1/2 and b= 1. By the proof of Theorem 2.13 there exists a positive constant Csuch that |Gk(τ)| ≤ X0 (c,d) 1 |cτ +d|k≤CX0 (c,d) 1 |ci +d|k,∀τ∈Ω(a, b), and since Gkis Z-periodic |Gk(τ)| ≤ CX0 (c,d) 1 |ci +d|k,∀τ=x+iy ∈H, y ≥1. Hence Gkis bounded as Im (τ)→∞, and the function gassociated to Gk, g(q) = Gk(τ), q =e2πiτ ,∀τ∈H, has a removable singularity at q= 0. 64 Modular forms Then this holomorphic function is a example of modular form of weight 2 with respect to Γ0(N), since (G2,N [α]2)(τ) = G2(τ)−2πic0N j(α, τ)−N j(α, τ)2G2(Nα(τ)) =G2(τ)−N2πic0 j(α0, Nτ)+G2[α0]2(Nτ)α0=a N b c0d =G2(τ)−NG2(Nτ),∀τ∈H,∀α=a b c0N d ∈Γ0(N), and its Fourier series is G2,N (τ)=2ζ(2)(1 −N)−8π2 ∞ X n=1 σ1,N (n)qn, q =e2πiτ ,∀τ∈H, where the coefficients of the series are the values of the arithmetic function σ1,N (n) = X 0<d|n N-d d, ∀n≥1. 2.2 Some applications 2.2.1 Transformation law of the Dedekind eta function The Dedekind eta function η:H→Cis defined as the infinite product η(τ) = q24 ∞ Y n=1 (1 −qn), q24 =e2πiτ/24, q =e2πiτ ,∀τ∈H. Lemma 2.22. The series ∞ X n=1 log(1 −qn), q ∈D, converges uniformly on compact subsets of the unit disc. As a consequence, the function ηdefines a holomorphic function on the upper half plane. Proof: Let 0 < r < 1. Recall that log(1 + z) = ∞ X n=0 (−1)nzn+1 n+ 1 ,∀z∈C,|z|<1. Using this Taylor series, we obtain the estimation |log(1 + z)| ≤ ∞ X n=0 |z|n+1 ≤|z| 1−r,∀z∈C,|z|< r. 2. Automorphic, modular and cusp forms 65 Therefore, ∞ X n=1 |log(1 −qn)| ≤ 1 1−r ∞ X n=1 rn=r (1 −r)2,∀q∈D,|q|< r. Note that the function ηis nonvanishing on H, since q24 6= 0 and ∞ Y n=1 (1 −qn) = eP∞ n=1 log(1−qn)6= 0,∀τ∈H. Theorem 2.23. The Dedekind eta function satisfies the transformation law η(−1/τ) = √−iτη(τ),∀τ∈H, where √is the principal branch of the multivalued function z1/2. Proof: The logarithmic derivative of ηis d dτ log(η(τ)) = πi 12 −2πi ∞ X d=1 dqd 1−qd=πi 12 −2πi ∞ X d=1 d ∞ X m=1 qdm =πi 12 −2πi ∞ X m=1 ∞ X d=1 dqdm =πi 12 −2πi ∞ X n=1  X 0<d|n d qn =πi 12E2(τ),∀τ∈H, where E2:H→Cis the normalized Eisenstein series of weight 2, E2(τ) = G2(τ) 2ζ(2) = 1 −24 ∞ X n=1 σ(n)qn, q =e2πiτ ,∀τ∈H. Therefore, d dτ log(η(−1/τ)) = πi 12τ−2E2(−1/τ),∀τ∈H, and d dτ log(√−iτη(τ)) = 1 2τ+πi 12E2(τ) = πi 12 E2(τ) + 12 2πiτ ,∀τ∈H. Using Theorem 2.21 with α=0−1 1 0 , we obtain that τ−2E2(−1/τ) = E2(τ) + 12 2πiτ ,∀τ∈H, 66 Modular forms so d dτ log(η(−1/τ)) −d dτ log(√−iτη(τ)) = 0,∀τ∈H. As a consequence, observe that there exists a constant c∈Csuch that η(−1/τ) = c√−iτη(τ),∀τ∈H. Letting τ=i, we conclude that this constant cmust be equal to 1, since η(i) = η(−1/i) = cp−i2η(i) = cη(i). The function η24 :H→C, η24(τ) = q ∞ Y n=1 (1 −qn)24, q =e2πiτ ,∀τ∈H, is a cusp form of weight 12 with respect to SL2(Z), since η24(τ+ 1) = η24(τ), η24(−1/τ) = τ12η24(τ),∀τ∈H, and lim Im (τ)→∞ η24(τ)=0. As the function ∆ is also a cusp form of weight 12 with respect to SL2(Z) and dim(S12(SL2(Z))) = 1 by Theorem 2.9, we deduce that ∆ = cη24,for some constant c∈C. Equating the coefficients of their Fourier series, we obtain the identity ∆ = (2π)12η24. 2.2.2 Four squares problem Let us consider the following questions: Can each positive integer nbe written as a sum of four squares, n=z2 1+z2 2+z2 3+z2 4, zi∈Z? In which case, in how many ways can nbe written as a sum of four squares? Is it possible to determine an explicit formula? 2. Automorphic, modular and cusp forms 67 The answers to these questions are affirmative, i.e, each positive integer n can be written as a sum of four squares, and in addition, there exists an explicit formula that counts the representation number of nby four squares. The proof that we present here uses basically the fact of that the functions G2,2(τ) = G2(τ)−2G2(2τ) and G2,4(τ) = G2(τ)−4G2(Nτ) (see Subsection 2.1.2) form a basis of the space of modular forms M2(Γ0(4)). Define r(n, k) = #{z∈Zk|n=z2 1+···+z2 k}, k ≥1, and consider the generating function of the representation numbers, θ(τ, k) = ∞ X n=0 r(n, k)qn, q =e2πiτ , τ ∈H. This series defines a holomorphic function on the upper half plane, since it converges uniformly on the subsets Ω(b) = {τ∈H|Im τ≥b},b > 0. Indeed, ∞ X n=0 r(n, k)|q|n= ∞ X n=0 r(n, k)(e−2πy)n ≤ ∞ X n=0 (1 + 2√n)k(e−2πb)n<∞,∀τ=x+iy ∈Ω(b). Furthermore, as r(n, k) = X s+j=n r(s, k1)r(j, k2), k1+k2=k, we deduce that θ(τ, k1)θ(τ, k2) = θ(τ, k1+k2),∀τ∈H. Let θ:H→Cbe the function θ(τ) = θ(τ, 1),∀τ∈H. The following lemma summarizes the principal properties of this holomorphic function. Its proof is immediate and it is left as an exercise to the reader. Lemma 2.24. Let kbe a positive integer. Then: 1. θis Z-periodic 2. θ(τ)k=θ(τ, k),∀τ∈H 68 Modular forms 3. θ(τ) = P∞ n=0 r(n, 1)qn=Pn∈Ze2πin2τ,q=e2πiτ ,∀τ∈H. This last expression for θallows us to determine another transformation law. Theorem 2.25. The function θsatisfies the transformation law θ(−1/2τ) = √−iτθ(τ/2),∀τ∈H, where √is the principal branch of the multivalued function z1/2. Proof: Observe that it suffices to prove this transformation law for τ=it, with t > 0, since the two sides of the equation are holomorphic functions on H. Let f, ft:R→Rbe the functions f(x) = e−πx2and ft(x) = e−πtx2. Recall that the Fourier transform of the function fis itself, i.e., Z+∞ −∞ e−πx2e−2πixξdx =e−πξ2,∀ξ∈R. The change of variables x=t1/2xin the integral shows that the Fourier transform of the function ftis b ft(ξ) = t−1/2e−πξ2/t,∀ξ∈R. As a consequence, using the Poisson summation formula, we obtain that X n∈Z e−πtn2=X n∈Z t−1/2e−πn2/t,∀t > 0, or equivalently, θ(τ/2) = 1 √−iτ θ(−1/2τ), τ =it, ∀t > 0. Note that the transformation law of the function θcan also be written as θ(−1/4τ) = √−2iτθ(τ),∀τ∈H.(2.14) The matrix 0−1 4 0 taking τto −1/4τis not in SL2(Z), but the product 0 1/4 −1 0 1−1 0 1 0−1 4 0 =1 0 4 1  2. Automorphic, modular and cusp forms 69 taking τto τ/(4τ+ 1), is in SL2(Z). Applying the corresponding succession of transformations and using (2.14) twice, we obtain that θτ 4τ+ 1=θ−1 4(−1/4τ−1) =s2i1 4τ+ 1θ−1 4τθ−1 4τ=θ−1 4τ−1 =s2i1 4τ+ 1(−2iτ)θ(τ) = √4τ+ 1θ(τ),∀τ∈H, where in the penultimate equality we have used tacitly that p2i(1/4τ+ 1)√−2iτ =p2i(1/4τ+ 1)(−2iτ),∀τ∈H. As a consequence, observe that θ(γ(τ),4) = (cτ +d)2θ(τ, 4),for γ=±1 1 0 1 and ±1 0 4 1 . Lemma 2.26. The subgroup Γθ,4of SL2(Z)generated by the matrices ±1 1 0 1 and ±1 0 4 1  is Γ0(4) = a b c d ∈SL2(Z)|a b c d ≡∗ ∗ 0∗(mod 4). Proof: Let α=a b c d ∈Γ0(4). Below, we describe an algorithm to compute γ∈Γθ,4such that αγ ∈Γθ,4(and therefore α∈Γθ,4). We can suppose without loss of generality that c6= 0, since otherwise α=±1b 0±1∈Γθ,4. The identity a b c d 1n 0 1 =a0b0 c nc +d, n ∈Z, shows that there exists a matrix γ1∈Γθ,4such that αγ1∈Γ0(4) has bottom row (c0, d0) = (c, nc +d),with |d0|<|c0|/2 (the inequality is clearly strict because of properties of cand dmodulo 4). On the other hand, the identity a0b0 c0d01 0 4n1=a00 b00 c0+ 4nd0d0, n ∈Z, 70 Modular forms shows that there exists a matrix γ2∈Γθ,4such that αγ1γ2∈Γ0(4) has bottom row (c00, d00)=(c0+ 4nd0, d0),with |c00|<2|d00| (the inequality is again strict because of properties of c0and d0modulo 4). As a consequence, observe that these two multiplications allow us to reduce strictly the absolute value of lower left entry of the matrix α, since |c00|<2|d00|= 2|d0|<|c0|=|c|. Repeating this argument whenever the lower left entry of the resulting matrix is nonzero (a finite number of times), we deduce that there exists a matrix γ∈Γθ,4such that αγ ∈Γ0(4) has bottom row (0,±1), and therefore αγ ∈Γθ,4. Using Theorem 2.8 we can conclude that θ(·,4) ∈ M2(Γ0(4)), since θ(τ, 4) = ∞ X n=0 r(n, 4)q4n 4, q4=e2πiτ/4,∀τ∈H, and r(n, 4) ≤(1 + 2√n)4,∀n≥1. Theorem 2.27. The representation number of a positive integer nby four squares is r(n, 4) = 8 X 0<d|n 4-d d. As a consequence, observe that ncan be written as a sum of four squares. Proof: Consider the modular forms G2,2, G2,4∈ M2(Γ0(4)), G2,2(τ) = −π2 3 1 + 24 ∞ X n=1 σ1,2(n)qn!,∀τ∈H and G2,4(τ) = −π2 1+8 ∞ X n=1 σ1,4(n)qn!,∀τ∈H The subset {G2,2, G2,4}forms a basis of the vector space M2(Γ0(4)), since it is linearly independent and dim(M2(Γ0(4))) = 2 (compute the dimension 2. Automorphic, modular and cusp forms 71 of this space using [DS05, p.107]). Therefore, the function θ(·,4) can be expressed in a unique way as a linear combination of these modular forms, θ(·,4) = aG2,2+bG2,4, a, b ∈C. The expansions θ(τ, 4) = 1 + 8q+··· , −3 π2G2,2(τ) = 1 + 24q+··· , −1 π2G2,4(τ) = 1 + 8q+··· , show that θ(·,4) = −(1/π2)G2,4. Equating the Fourier coefficients, we obtain that the representation number of a positive integer nas a sum of four squares is r(n, 4) = 8σ1,4(n)=8 X 0<d|n 4-d d. 72 Modular forms Chapter 3 Modularity Theorem In this third chapter we introduce moduli spaces for the modular curves Y0(N), Y1(N),and Y(N), N ∈Z+, and we explain a complex analytic version of the Modularity Theorem which is equivalent to the original version that was proved about twenty years ago. 3.1 Weil pairing Let Λ be a lattice in Cwith basis (w1, w2), Λ = w1Z+w2Z(assume w1/w2∈H), and Na positive integer. Consider the multiply-by-Nmap [N] : C/Λ→C/Λ, z + Λ 7→ Nz + Λ. This map is a holomorphic group homomorphism (Remark B.6). Its kernel is the set of N-torsion points of C/Λ, ker[N] = {P∈C/Λ|[N]P= 0}=hw1/N + Λi+hw2/N + Λi, a subgroup isomorphic to Z/NZ×Z/NZ. 0w1 w2 Figure 3.1: ker[5]: the 5-torsion points of C/Λ 73 80 Modular forms Appendix A Riemann surfaces A Riemann surface is simply a nonempty connected Hausdorff topological space endowed with an equivalence class of complex atlases. In this chapter we introduce all the theory of Riemann surfaces we need in this work: - Holomorphic maps - Meromorphic differentials - Divisors and Riemann-Roch Theorem A good reference book about Riemann surfaces is for example [Mir95]. A.1 Basic definitions Let Xbe a topological space. We want Xto behave locally as an open subset of the complex plane, thus it will allow us to define complex coordinates at each point of X. Definition A.1. A complex chart, or simply chart, on Xis a homeomorphism φ:U→V, where U⊂Xis an open subset of Xand V⊂Cis an open subset of C. A complex chart φ:U→Vis denoted by (U, φ). The open subset Uis called domain of the chart φ. Furthermore, the chart φis centered at p∈U if φ(p) = 0. Examples A.2. 1. In the euclidean plane, consider any open subset U⊂R2. The map φU:U→U,φU(x, y) = x+iy, is a chart on R2. 81 82 Appendix A 2. In the Riemann sphere, consider the open subsets U1=C,U2=b C\{0}. The maps φi:Ui→C, φ1(z) = z, φ2(z) = 1 z(where 1 ∞:= 0), are charts on b C. Two complex charts φ1:U1→V1and φ2:U2→V2on Xare compatible if either U1∩U2=∅or U1∩U26=∅and φ2◦φ−1 1:φ1(U1∩U2) | {z } V1,2 →φ2(U1∩U2) | {z } V2,1 is holomorphic. Note that the definition is symmetric, i.e., if U1∩U26=∅ and φ2◦φ−1 1:φ1(U1∩U2)→φ2(U1∩U2) is holomorphic, then φ1◦φ−1 2:φ2(U1∩U2)→φ1(U1∩U2) is holomorphic. As a consequence, (φ2◦φ−1 1)0(z)6= 0,∀z∈φ1(U1∩U2). The function φ2◦φ−1 1is called transition function between the two charts. U1∩U2 U1U2 V1,2V2,1 V1V2 φ1φ2 φ2◦φ−1 1 Figure A.1: Transition function Definition A.3. A complex atlas, or simply atlas, Aon Xis a collection {φα:Uα→Vα} of pairwise compatible complex charts whose domains cover X, i.e., X=SUα. Riemann surfaces 83 Examples A.4. 1. In the euclidean plane, the collection of charts A={φU:U→U|U⊂R2is an open subset} is an atlas on R2. 2. In the Riemann sphere, the collection of charts A={φi:Ui→C|i= 1,2} is an atlas on b C. Indeed, U1∩U2=C∗and φ2◦φ−1 1(z)=1/z,∀z∈C∗. Two complex atlases on Xare equivalent if every chart of one is compatible with every chart of the other. Note that two complex atlases are equivalent if and only if their union is also a complex atlas. Moreover, every complex atlas Aon Xis contained in a unique maximal complex atlas A∗on X which consists of all charts on Xthat are compatible with every chart of A. Therefore, two complex atlases are equivalent if and only if they are both contained in the same maximal complex atlas. Definition A.5. A complex structure on Xis an equivalence class of complex atlases on X, or equivalently, a maximal complex atlas on X. As any complex atlas on Xdetermines a unique complex structure on X, this will be the usual way to define a complex structure on X. Definition A.6. A Riemann surface is a nonempty connected Hausdorff topological space endowed with a complex structure. Convention. If Xis a Riemann surface, then by a chart on Xwe always mean a chart belonging to the maximal atlas of the complex structure on X. Remarks A.7. .A domain in a Riemann surface is a nonempty connected open subset. Note that these subsets inherit naturally the structure of Riemann surface. .Each point of a Riemann surface has an open neighbourhood which is homeomorphic to an open disc of the complex plane. As a consequence, the topological local properties of the euclidean plane are preserved. For example, any Riemann surface is locally path-connected, locally compact, locally contractible, locally metrizable. . . .According to Rad´o’s Theorem [NR11, p.71], every Riemann surface is second-countable, i.e., there exists a countable base for its topology. 84 Appendix A .Classically, a compact Riemann surface is called closed while a noncompact surface is called open. It is important to note that there exist notable differences between the theory of compact Riemann surfaces and that of noncompact Riemann surfaces. .Every Riemann surface is an orientable connected 2-dimensional smooth manifold, since any complex atlas is an oriented smooth atlas. Therefore, every compact Riemann surface is diffeomorphic to a torus with gholes, for some unique integer g≥0. This integer gis called the genus of the Riemann surface and is a topological invariant. Examples A.8. 1. The euclidean plane endowed with the complex structure determined by the atlas A={φR2:R2→R2}is a noncompact Riemann surface called Complex Plane. It is denoted by C. 2. The Riemann sphere endowed with the complex structure determined by the atlas A={φi:Ui→C|i= 1,2}is a compact Riemann surface called Riemann Sphere. It is denoted by b C. A.2 Morphisms Let X, Y, Z be Riemann surfaces. Definition A.9. Let Ube a nonempty open subset of X. .A map F:U→Yis holomorphic at p∈Uif there exist charts φ1:U1→V1on X, with p∈U1, and φ2:U2→V2on Y, with F(p)∈U2, such that the composition φ2◦F◦φ−1 1 is holomorphic at φ1(p). .A map F:U→Yis holomorphic on a nonempty open subset V⊆Uif Fis holomorphic at each point of V. .A map F:X→Yis a morphism if Fis holomorphic on X. Examples A.10. 1. The holomorphic functions on a nonempty open subset of C. 2. The fractional linear transformations T:b C→b C, T(z) = az +b cz +d,with ad −bc 6= 0,are bijective morphisms. Riemann surfaces 85 3. Let C/Λ1,C/Λ2be complex tori and αa complex number such that αΛ1⊂Λ2. For each β∈C, the map Fα,β :C/Λ1→C/Λ2, Fα,β(z+ Λ1) = (αz +β)+Λ2,∀z+ Λ ∈C/Λ1, is a morphism. 4. Let Γ1,Γ2be congruence subgroups of SL2(Z) such that Γ1⊂Γ2. The natural projection of the corresponding compact modular curves F:X(Γ1)→X(Γ2),Γ1τ7→ Γ2τ, is a surjective morphism. In the special case Y=C, it is important to note that a function f:U→Y is holomorphic at p∈Uif and only if there exists a chart φ1:U1→V1on X, with p∈U1, such that the composition f◦φ−1 1is holomorphic at φ1(p). The set of holomorphic functions on Uis denoted by OX(U), OX(U) = {f:U→C|fis holomorphic on U}. It is a C-algebra, since the constant functions are holomorhic functions and the sum and product of holomorhic functions are also holomorphic functions. Moreover, if Uis connected, then O(U) is an integral domain. The following proposition states the main results concerning holomorphic maps between Riemann surfaces. Proposition A.11. Let Ube an open subset of X,F:U→Ya map and p∈U. 1. If Fis holomorphic at p, then Fis continuous at p. 2. If Fis holomorphic at p, then Fis holomorphic on an open subset Up⊂U with p∈Up. 3. Fis holomorphic at pif and only if for any pair of charts φ1:U1→V1on X, with p∈U1, and φ2:U2→V2on Y, with F(p)∈U2, the composition φ2◦F◦φ−1 1 is holomorphic at φ1(p). 4. If Fis continuous at pand there exists an open subset Up⊂U, with p∈Up, such that Fis holomorphic on Up\{p}, then Fis holomorphic at p. 5. Let Vbe an open subset of Yand G:V→Za map such that F(U)⊂V. If Fis holomorphic at pand Gis holomorphic at F(p), then G◦Fis holomorphic at p. 86 Appendix A It is not difficult to prove that if F:X→Yis a bijective morphism, then F−1:Y→Xis also a morphism. A bijective morphism F:X→Yis called isomorphism. The Riemann surfaces Xand Yare isomorphic if there exists an isomorphism F:X→Y. Note that the relation of being isomorphic is an equivalence relation on Riemann surfaces, since the composition of morphisms is a morphism. As usual, a self-isomorphism F:X→Xis also called automorphism. The automorphisms of Xform a group under the operation of composition. A.2.1 Theorems on morphisms Now we present several results about morphisms between Riemann surfaces. In the majority of cases these results are immediate consequences of the corresponding results of complex analysis about holomorphic functions. Theorem A.12 (Identity Theorem).Let F, G :X→Ybe morphisms. If there exists a subset S⊂Xsuch that S06=∅and F(x) = G(x),∀x∈S, then F=G. Theorem A.13 (Open Mapping Theorem).Let F:X→Ybe a nonconstant morphism. Then Fis an open map. Corollary A.14. Let F:X→Ybe a nonconstant morphism. If Xis compact, then Fis surjective and Yis compact. As a consequence, observe that if Xis compact, then O(X) = C. Corollary A.15 (Discreteness of Preimages).Let F:X→Ybe a nonconstant morphism. Then F−1(q)is a discrete subset of Xfor all q∈Y. In particular, if Xis compact, then F−1(q)is a nonempty finite subset of X for all q∈Y. Theorem A.16. Let F:X→Ybe an injective morphism. Then its restriction on its image F:X→F(X)is an isomorphism. Theorem A.17 (Local normal form).Let F:X→Ybe a nonconstant morphism and p∈X. There exists a unique integer m≥1which satisfies the following property: For each chart φ2:U2→V2on Y, centered at F(p), there exists a chart φ1:U1→V1on X, centered at p, with F(U1)⊂U2, such that φ2◦F◦φ−1 1(z) = zm,∀z∈V1. Proof: [Mir95, p.44] Riemann surfaces 87 Definition A.18. Let F:X→Ybe a nonconstant morphism. The ramification index of Fat p∈Xis the unique integer m≥1which satisfies the property cited in the previous theorem. It is denoted by ep(F). Examples A.19. 1. Let f:C→Cbe the holomorphic function f(z) = zm, with m≥1. The index ramification of fat z∈C∗is ez(f) = 1 and at 0 is e0(f) = m. 2. Let f:b C→b Cbe the holomorphic function f(z) = zm+c1zm−1···+cm(where f(∞) := ∞), with m≥1. The index ramification of fat ∞is e∞(f) = m. There exists an easy way to compute the ramification index without having to find centered charts which put the morphism into local normal form. Let us fix two charts φ1:U1→V1on X, with p∈U1, and φ2:U2→V2on Y, with F(p)∈U2. As the composition φ2◦F◦φ−1 1 is holomorphic at z0=φ1(p), φ2◦F◦φ−1 1= ∞ X n=1 an(z−z0)n. Then ep(F) = min{n≥1|an6= 0}. Proposition A.20. Let F:X→Ybe a nonconstant morphism and p∈X. The following conditions are equivalent: 1. ep(F)=1 2. Fis a local isomorphism at p, i.e., there exist connected open subsets Up⊂X, with p∈Up, and VF(p)⊂Y, with F(p)∈VF(p), such that F|Up:Up→VF(p)is an isomorphism. Definition A.21. Let F:X→Ybe a nonconstant morphism. A point p∈Xis a ramification point of Fif ep(F)>1. A point y∈Yis a branch point of Fif it is the image of a ramification point of F. As a consequence of the above, the ramification points of a nonconstant morphism F:X→Yform a discrete closed subset of X. 88 Appendix A Definition A.22. A morphism F:X→Yis a branched covering if for each q∈Yholds that . F−1(q)is a nonempty finite subset of X,F−1(q) = {p1, . . . , pr}, and .there exist charts φi:Ui→Vion X, centered at pi,φ:U→Von Y, centered at q, with F−1(U) = r G i=1 Ui, and integers ei≥1, such that for all i= 1, . . . , r, φ◦F◦φ−1 i(z) = zei,∀z∈Vi. Note that the integers eiare the ramification indexes of the points pi. Examples A.23. 1. The holomorphic function f:C→C,f(z) = zm, with m≥1, is a branched covering. 2. The holomorphic function f:b C→b C,f(z) = zm+c1zm−1··· +cm, with m≥1, is a branched covering. The following theorem is really useful to prove that a morphism is a branched covering since it characterizes the branched coverings in a simple way. Theorem A.24 (Characterization of Branched Coverings).Let F:X→Ybe a nonconstant morphism. The following conditions are equivalent: 1. Fis a branched covering. 2. For each q∈Y,F−1(q)is a nonempty finite subset of X, and in addition, the map q∈Y7→ X p∈F−1(q) ep(F)∈Z is constant. 3. Fis a proper map. In particular, observe that if Xis compact, then Fis a branched covering. Proof: [Gir70, p.7] Since any proper map between locally compact topological spaces is closed, we deduce from this result that any branched covering is a closed map. Riemann surfaces 89 Definition A.25. Let F:X→Ybe a branched covering. The number deg(F) = X p∈F−1(q) ep(F) is called the degree of the branched covering F. Proposition A.26. Let F:X→Y,G:Y→Zbe branched coverings. Then the composition G◦Fis a branched covering and deg(G◦F) = deg(G) deg(F). Proposition A.27. Let F:X→Ybe a branched covering and R={p∈X|ep(F)>1}. Then F(R)is a closed discrete subset of Y. Moreover, Ris empty if and only if Fis an ´etale covering. Finally, we present the Riemann-Hurwitz formula which relates the genus of two compact Riemann surfaces through a nonconstant morphism between each other. Theorem A.28 (Riemann-Hurwitz formula).Let F:X→Ybe a nonconstant morphism. If Xis compact, then 2g(X)−2 = deg(F)(2g(Y)−2) + X p∈X (ep(F)−1). Proof: [Mir95, p.52] A.2.2 Meromorphic functions Definition A.29. Let Ube a nonempty open subset of X. .A function f:U→b Cis meromorphic at p∈Uif there exists a chart φ1:U1→V1on X, with p∈U1, such that the composition f◦φ−1 1is meromorphic at φ1(p). In this case, for any chart ψ1:U0 1→V0 1on X, with p∈U0 1, the composition f◦ψ−1 1is meromorphic at ψ1(p). .A function f:U→b Cis meromorphic on a nonempty open subset V⊂U if fis meromorphic at each point of V. Note that a function f:U→b Cis meromorphic at p∈Uif and only if fis holomorphic at p(as Riemann surfaces) and there does not exist a neighbourhood Upof p, with Up⊂U, such that fis identically infinity on Up. 96 Appendix A Proof: Let {φα:Uα→Vα}be a complex atlas on X. There exist local representations λ1={(φα, f1 α)}and λ2={(φα, f2 α)}. Then the meromorphic function f:X→b C, f(p) = f2 α(φα(p)) f1 α(φα(p)),∀p∈Uα, is well-defined and is the unique nonzero meromorphic function on Xthat satisfies the desired property. Theorem A.44. [Riemann-Roch] Let Xbe a compact Riemann surface of genus g. If div(λ)is a canonical divisor on X, then for any divisor D∈Div(X), `(D) = deg(D)−g+1+`(div(λ)−D). Proof: [Mir95, p.192] Observe that if div(λ0) is another canonical divisor on X, then the vector spaces Lλ=L(div(λ)−D) and Lλ0=L(div(λ0)−D) are isomorphic, since the map g∈Lλ7→ g/f ∈Lλ0,where f∈ M(X)∗, λ0=fλ, is an isomorphism of C-vector spaces. As a consequence, `(div(λ)−D) = `(div(λ0)−D). Corollary A.45. Let X,g,div(λ)and Dbe as above. Then 1. `(div(λ)) = g 2. deg(div(λ)) = 2g−2 3. If deg(D)<0, then `(D) = 0 4. If deg(D)>2g−2, then `(D) = deg(D)−g+ 1 Proof: To prove the first equality let D= 0 in the Riemann-Roch Theorem and recall that `(0) = 1. To prove the second equality it suffices to apply the Riemann-Roch Theorem with D= div(λ), since g(1) =`(div(λ)) = deg(div(λ)) −g+ 2. Riemann surfaces 97 We now argue the third assertion by contradiction. Let us suppose that there exists a nonzero meromorphic function f∈L(D). Then div(f)≥ −D, and taking degrees follows that deg(D)≥0, which is a contradiction. Finally, the fourth assertion is consequence of (2) and (3) since if deg(D)>2g−2, then deg(div(λ)−D)(2) = 2g−2−deg(D)<0. 98 Appendix A Appendix B Complex elliptic curves The compact Riemann surfaces of genus equal to 1 are called complex elliptic curves for reasons to be explained in this chapter. It is possible to prove that any complex elliptic curve is isomorphic to a complex torus (as Riemann surfaces). As a consequence, these curves can be endowed with an analytic group structure which is uniquely determined by the choice of identity element. In this chapter we detail all these results over complex elliptic curves. B.1 Complex Tori Let us begin by defining the Riemann surfaces called complex torus. It is well-known that the discrete subgroups of Care -{0}, -Zw, with w∈C∗, and -Zw1+Zw2, with w1, w2∈Clinearly independent over R. A lattice Λ in Cis a discrete subgroup of Cof the third kind, that is, Λ = Zw1+Zw2,with w1, w2∈Clinearly independent over R. Note that the pair (w1, w2) has the property that any w∈Λ has a unique representation w=m1w1+m2w2,with m1, m2∈Z. Any pair with this property is called a basis of the lattice Λ. It is usual to make the normalizing convention w1/w2∈H, but this still does not determine a basis given a lattice. In fact, two pairs (w1, w2) and (w0 1, w0 2) are bases of a same lattice if and only if w0 1 w0 2=a b c d w1 w2,for some α=a b c d ∈GL2(Z). 99 100 Appendix B 0 w2 w1 Figure B.1: Lattice with basic (w1, w2) If we make the normalizing convention w1/w2, w0 1/w0 2∈H, then α∈SL2(Z). Let Λ be a lattice in C, with basis (w1, w2), Λ = Zw1+Zw2. Let us consider the quotient group C/Λ endowed with the topology induced by the natural projection π:C→C/Λ, that is, a subset U⊂C/Λ is open if and only if π−1(U) is open in C. Note that C/Λ is a compact connected topological space, since πis a continuous map with respect to this topology and π(Pa) = C/Λ,for any period parallelogram Pa=Pa(w1, w2) = {a+λw1+βw2|0≤λ, β < 1}, a ∈C. Pa(w1, w2) aa+w1 a+w2 Figure B.2: Period parallelogram Furthermore, πis an open map, since π−1(π(V)) = [ w∈Λ w+V, ∀V⊂C. Let us now prove that C/Λ is a Hausdorff topological space. Let z1+ Λ, z2+ Λ be points in C/Λ, with z2−z1/∈Λ. Then η= min w∈Λ|(z2−z1)−w|>0. Complex elliptic curves 101 As a consequence, π(D(z1, η/2)) and π(D(z2, η/2)) are disjoint open subsets of C/Λ containing z1+ Λ and z2+ Λ, respectively. Finally, let us define an atlas on C/Λ. For each z∈C, consider the disc Dz=D(z, δ/2), where δ= min w∈Λ\{0}{|w|}. Note that π|Dz:Dz→π(Dz) is a homeomorphism. We denote by φzits inverse homeomorphism. Below, we show that the collection of charts A={φz:π(Dz)→Dz|z∈C} is an atlas on C/Λ: - Evidently, C/Λ = Sz∈Cπ(Dz). - Furthermore, if π(Dz1)∩π(Dz2)6=∅, with z1, z2∈C, then there exists w∈Λ such that φz2◦φ−1 z1(z) = z+w, ∀z∈V1,2, where V1,2=φz1(π(Dz1)∩π(Dz2)). Indeed, ϕ(z) := φz2◦φ−1 z1(z)−z∈Λ,∀z∈V1,2, and |ϕ(s1)−ϕ(s2)|< δ, ∀s1, s2∈V1,2. The topological space C/Λ endowed with the complex structure determined by the atlas Ais a compact Riemann surface called Complex Torus. It is important to note that the group structure on C/Λ is analytic, i.e., in terms of local charts about any two given points in the complex torus, addition is a holomorphic function of two complex variables. Let z1+ Λ, z2+ Λ ∈C/Λ. Consider the charts φz1:π(Dz1)→Dz1, φz2:π(Dz2)→Dz2, φz3:π(Dz3)→Dz3, where z3=z1+z2, and fix an open neighbourhood Vof (z1, z2) in Dz1×Dz2 such that s1+s2∈Dz3,∀(s1, s2)∈V. Then φz3φ−1 z1(s1) + φ−1 z2(s2)=s1+s2,∀(s1, s2)∈V. 102 Appendix B B.1.1 Analytic group structure As mentioned at the beginning of this chapter, if Eis a complex elliptic curve, then there exists a lattice Λ in Csuch that Eis isomorphic to C/Λ as Riemann surfaces [Mir95, p.265]. Therefore, there exists an isomorphism F:C/Λ→E. Now, through this isomorphism the complex elliptic curve Einherits the analytic group structure on C/Λ. To see this, it suffices to define F(z1+ Λ) + F(z2+ Λ) := F((z1+z2) + Λ),∀z1+ Λ, z2+ Λ ∈C/Λ. Note that the isomorphism becomes a group isomorphism from C/Λ to E with respect to this group structure. The following theorem states that an analytic group structure on a complex elliptic curve is uniquely determined by the identity element. Theorem B.1. Let Ebe a complex elliptic curve and p0∈E. Then there exists a unique analytic group structure on Esuch that p0is the identity element. Proof: Let Λ be a lattice in Csuch that Eis isomorphic to C/Λ and F:E→C/Λ an isomorphism. The map G:E→C/Λ, G(p) = F(p)−F(p0),∀p∈E, is an isomorphism, taking p0to 0 + Λ. So the inverse isomorphism defines an analytic group structure on Eas above, such that p0=G−1(0 + Λ) is the identity element. Let us now prove the uniqueness. First, we consider the case E=C/Λ, with Λ a lattice in C, and p0= 0 + Λ. Let ⊕:E×E→Ebe another addition which defines an analytic group structure with identity element 0 + Λ. Below, we show that ez1+ez2=ez1⊕ez2,∀˜z1=z1+ Λ,˜z2=z2+ Λ ∈E. - For ez2= 0 + Λ, it is obvious. - For ez26= 0 + Λ, the map Fez2:E→E, Fez2(ez) = ez⊕ez2,∀ez=z+ Λ ∈E, is an automorphism without fixed points. Then Fez2(ez) = ez+ez3,∀ez=z+ Λ ∈E, Complex elliptic curves 103 for some ez3=z3+Λ ∈E, with z3/∈Λ (Corollary B.5). As a consequence, ez2=Fez2(0 + Λ) = ez3. The arbitrary case of an elliptic curve E, with identity element p0∈E, follows from the case just considered since if there exists another analytic group structure on E, with identity element p0, then the isomorphism G:E→C/Λ defined above carries the two analytic group structures on Eto two distinct analytic group structures on C/Λ, with identity element 0 + Λ. From the uniqueness stated in the previous theorem, we obtain the following corollary. Corollary B.2. Let E1,E2be two complex elliptic curves with identity elements p1 0and p2 0, respectively. If F:E1→E2is an isomorphism of Riemann surfaces such that F(p1 0) = p2 0, then F(p) + F(q) = F(p+q),∀p, q ∈E1. Therefore, the isomorphism Fis also a group isomorphism from E1to E2. B.2 Morphisms between complex tori Let C/Λ1,C/Λ2be two complex tori. Theorem B.3. If F:C/Λ1→C/Λ2is a morphism of Riemann surfaces, then there exist α, β ∈C, with αΛ1⊂Λ2, such that F(z+ Λ1)=(αz +β)+Λ2,∀z+ Λ1∈C/Λ1. Reciprocally, if there exists α∈C, with αΛ1⊂Λ2, then for each β∈C the map Fα,β :C/Λ1→C/Λ2, Fα,β(z+ Λ1)=(αz +β)+Λ2,∀z+ Λ1∈C/Λ1. is a morphism of Riemann surfaces. Its degree is the index of αΛ1in Λ2, deg(Fα,β) = [Λ2:αΛ1]. Proof: [Mir95, p.63] Note that the map Fα,β is bijective if and only if αΛ1= Λ2. In such case, its inverse is F−1 α,β(z+ Λ2) = (1/α)z−β/α + Λ1,∀z+ Λ2∈C/Λ2. 104 Appendix B Corollary B.4. The isomorphisms from C/Λ1to C/Λ2are F(z+ Λ1) = (αz +β)+Λ2,∀z+ Λ1∈C/Λ1, where α, β ∈C,αΛ1= Λ2. Therefore, the complex tori C/Λ1and C/Λ2are isomorphic (as Riemann surfaces) if and only if there exists α∈Csuch that αΛ1= Λ2, i.e., if the lattices Λ1and Λ2are homothetic. Corollary B.5. Let C/Λbe a complex torus. 1. The morphisms from C/Λto itself are F(z+ Λ) = (αz +β)+Λ,∀z+ Λ ∈C/Λ, where α, β ∈C,αΛ⊂Λ. 2. The automorphisms of C/Λare F(z+ Λ) = (αz +β)+Λ,∀z+ Λ ∈C/Λ, where α, β ∈C,αΛ=Λ. 3. The automorphisms of C/Λwithout fixed points are F(z+ Λ) = (z+β)+Λ,∀z+ Λ ∈C/Λ, where β∈C\Λ. Remark B.6. The maps Fα,β :C/Λ1→C/Λ2, Fα,β(z+ Λ1) = (αz +β)+Λ2,∀z+ Λ1∈C/Λ1. are group homomorphisms if and only if Fα,β(0 + Λ1) = 0 + Λ2, i.e., β∈Λ2. B.3 Weierstrass ℘-function The Weierstrass ℘-function for a lattice Λ in Cis defined as ℘Λ(z) = 1 z2+X0 w∈Λ1 (z−w)2−1 w2,∀z∈C, z /∈Λ, where the primed summation means to sum over the points w∈Λ\ {0}. It is the most important specific example of elliptic function with periods Λ, ℘Λ(z) = ℘Λ(z+w),∀z∈C,∀w∈Λ. Complex elliptic curves 105 As the series converges uniformly on compact subsets which do not meet Λ, the function is holomorphic on C/Λ and has double poles at lattice points. So its order is 2, i.e, the number of poles in any period parallelogram Pa(w1, w2) = {a+λw1+βw2|0≤λ, β < 1}, a ∈C, where (w1, w2) is a basic of Λ, is equal to 2. Its derivative is also an elliptic function with periods Λ, ℘0 Λ(z) = −2X0 w∈Λ 1 (z−w)3,∀z∈C, z /∈Λ. In fact, it turns out that ℘Λand ℘0 Λare the only specific examples we need since the field of elliptic functions with periods Λ is generated by ℘Λand ℘0 Λ. It is important to note that ℘Λis an even function while ℘0 Λis an odd function. The following theorem states the main results of the Weierstrass ℘-function. Theorem B.7. Let ℘Λbe the Weierstrass function with respect to a lattice Λ. 1. The Laurent series of ℘Λis ℘Λ(z) = 1 z2+ ∞ X n=2 neven (n+ 1)Gn+2(Λ)zn,∀z∈˙ D(0, δ), where Gk(Λ) = X0 w∈Λ 1 wk, k > 2even, and δ= min w∈Λ\{0}{|w|}. 2. The functions ℘Λand ℘0 Λsatisfy the cubic equation (℘0 Λ)2= 4℘3 Λ−g2(Λ)℘Λ−g3(Λ), where g2(Λ) = 60G4(Λ) and g3(Λ) = 140G6(Λ). 3. Let (w1, w2)be a basis of Λand w3=w1+w2. Then the cubic equation satisfied by ℘Λand ℘0 Λis y2= 4(x−e1)(x−e2)(x−e3),where ei=℘Λ(wi/2),∀i= 1,2,3. This equation is nonsingular, meaning its right side has distinct roots. In particular, its discriminant up to constant multiple ∆(Λ) = g2(Λ)3−27g3(Λ)26= 0. 112 Bibliography [Ser73] J.-P. Serre. A course in arithmetic. Springer-Verlag, New York- Heidelberg, 1973. Translated from the French, Graduate Texts in Mathematics, No. 7. [Shi71] Goro Shimura. Introduction to the arithmetic theory of automorphic functions. Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanˆo Memorial Lectures, No. 1. [Sil09] Joseph H. Silverman. The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics. Springer, Dordrecht, second edition, 2009. [SS03] Elias M. Stein and Rami Shakarchi. Complex analysis, volume 2 of Princeton Lectures in Analysis. Princeton University Press, Princeton, NJ, 2003. [TW95] Richard Taylor and Andrew Wiles. Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2), 141(3):553–572, 1995. [vdP96] Alf van der Poorten. Notes on Fermat’s last theorem. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, Inc., New York, 1996. A Wiley-Interscience Publication. [Wil95] Andrew Wiles. Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2), 141(3):443–551, 1995.