Counting Covers of Elliptic Curves

Orlando Marigliano

Geboren am 5. August 1994 in Rom 29. November 2017

Bachelorarbeit Mathematik Betreuer: Prof. Dr. Daniel Huybrechts Zweitgutachter: Dr. Vladimir Lazić Mathematisches Institut

Mathematisch-Naturwissenschaftliche Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn

Contents

1. Introduction 2

2. Quasimodular forms 4 2.1. The space of modular forms ...... 4 2.2. The space of quasimodular forms ...... 5

3. Covers of an elliptic curve 10

4. Classifying covers via the fundamental group 16 4.1. Covering spaces ...... 16 4.2. Marked covers and the monodromy map ...... 17

5. Conjugacy classes of the symmetric group 21 5.1. Conjugacy cycles ...... 21 5.2. Adjacency matrices ...... 22

6. The group algebra of the symmetric group 24 6.1. The centre of the group algebra ...... 24 6.2. Irreducible characters of the symmetric group ...... 25

7. Quasimodularity of the generating function 27 7.1. Subsets of the half integers ...... 27 7.2. The coefficients of the theta function ...... 28

8. Appendix – Modular curves 30 8.1. Congruence subgroups and modular curves ...... 30 8.2. Modular curves as Riemann surfaces ...... 32 8.3. Automorphic forms and modular forms ...... 35 8.4. The dimension formula ...... 37

1 1. Introduction

Let E be an elliptical curve over C and fix a genus д ≥ 1. The main interest of this thesis are the simply ramified morphisms p : C → E, where C is an arbitrary smooth irreducible complex curve of genus д. This thesis seeks to count such morphisms, in the way described below. For any d ≥ 1, the isomorphism classes of morphisms p of degree d may be classified by applying the theory of covering spaces in topology to the ramified covers (i. e. the morphisms) of interest. One finds that there are (up to isomorphism) finitely many such covers, so that one may ask about their number. Since a cover may have nontrivial automorphisms, a natural way to count covers would be to first apply the weighting p 7→ 1/| Aut(p)| to any cover p. The goal is to study the generating series Fд (q) over the weighted counts Nд,d of (connected) covers of genus д and degree d. The main theorem of this thesis states that for д ≥ 2, if q is viewed as the complex variable exp(2πiτ ) and the generating series Fд (q) as a function of τ , then the func- − tion Fд is a quasimodular form of weight 6д 6. This is one of the main theorems of [Dij95]. The concept of quasimodularity generalizes that of modularity; the definition here used comes from [KZ95]. The same article also contains a theorem that will provide the last step to the proof of the main theorem. As for getting to the last step, [Dij95] contains the sketch of an argument, expanded upon in [Rot09], for turning the gener- ating series Fд into something more useful, i. e. to which one may apply the theorem in [KZ95]. The main part of this thesis follows the outline of[Rot09].

Acknowledgements

I would like to thank Prof. Daniel Huybrechts for his valuable advice during the writing of this thesis. I also warmly thank my family for their support throughout my studies.

Einleitung

Sei E eine elliptische Kurve über C, sei д > 1 ein festes Geschlecht. Diese Arbeit befasst sich hauptsächlich mit den einfach verzweigten Morphismen p : C → E, wo- bei C eine beliebige glatte irreduzible komplexe Kurve vom Geschlecht д ist. Ziel der Arbeit ist es, solche Morphismen auf die unten beschriebene Weise zu zählen. Für d > 1 können die Isomorphieklassen von Morphismen p vom Grad d klassifiziert werden, indem man topologische Überlagerungstheorie auf die zu untersuchende verzweigte Überlagerungen (also auf die Morphismen) anwendet. Es stellt sich her- aus, dass es nur endlich viele solche Überlagerungen (bis auf Isomorphie) gibt, wes- halb die Frage nach deren Anzahl aufkommt. Überlagerungen können nichttriviale

2 Automorphismen haben; somit wäre eine natürliche Weise, die Überlagerungen p zu zählen, dadurch gegeben, dass man zunächst die Gewichtung p 7→ 1/| Aut(p)| anwendet. Von Interesse ist die erzeugende Funktion Fд (q) über die gewichteten An- zahlen Nд,d der (zusammenhängenden) Überlagerungen vom Geschlecht д und Grad d. Das Hauptresultat dieser Arbeit besagt, dass für д ≥ 2, falls man q als die komplexe Variable exp(2πiτ ) und die erzeugende Reihe Fд (q) als Funktion von τ auffasst, die − Funktion Fд eine quasimodulare Form vom Gewicht 6д 6 ist. Dieses Ergebnis ist eine der Hauptaussagen in [Dij95]. Der Begriff der Quasimodularität verallgemeinert den der Modularität, und istaus [KZ95] übernommen. Dieser Artikel liefert außerdem einen Satz, der den letzten Schritt des Beweises des Hauptresultats ermöglicht. Zum Erreichen dieses letztes Schrittes findet sich in[Dij95] die Skizze eines Arguments, welches in [Rot09] aus- geführt wird. Mit diesem kann die erzeugende Reihe Fд so umgeformt werden, dass sich der Satz aus [KZ95] anwenden lässt. Der Hauptteil dieser Arbeit orientiert sich an [Rot09].

3 2. Quasimodular forms

This section introduces quasimodular forms as described in[KZ95].

2.1. The space of modular forms

Let H = {τ ∈ C ; Im(τ ) > 0} denote the upper half-plane. For τ ∈ H , define = = Z ⊂ C q exp(2πiτ ) and Y 4π Im(τ ). Further, let SL2( ) SL2( ) denote the full Z H modular group. Then SL2( ) acts on by ( ) aτ + b a b γτ = , for γ = ∈ SL (Z). cτ + d c d 2

Definition 2.1. Let f : H → C be a function, let k ∈ Z. 1. The function f is Z-periodic, if it satisfies f (τ + 1) = f (τ ) for all τ ∈ H . In this case there exists a function f˜: B \{0} → C, defined on the open unit ball B ⊂ C with the origin removed, such that f (τ ) = f˜(q) for all τ . Now let f be holomorphic. Then so is f˜. We say that f is holomorphic at infinity, if f˜ has a holomorphic continuation to the whole of B. 2. The function f is said to satisfy the modular condition of weight k, if

f (γτ ) = (cτ + d)k f (τ ) H ∈ Z Z for all τ in ( and) all γ SL2( ). Such a function is -periodic, as can be seen by = 1 1 . setting γ 0 1 3. The function f is a modular form (of weight k) if it is holomorphic, satisfies the modular condition and is holomorphic at infinity.

Note that if k is odd, then any function satisfying the modular condition( ) of weight = −1 0 . k is zero. This follows by using the modular condition with γ 0 −1 There are several alternate conventions for handling the weights k. Some authors for instance replace k by 2k throughout, so that “modular forms of weight 2k” are considered. This is the convention used by[Ser12a].

The modular forms of weight k form a vector space, denoted¹by Mk . Multiplying two modular forms⊕ of weights k respectively l yields a modular form of weight k +l, giving the space k Mk the structure of a graded ring, denoted by M∗. Examples 2.2. For an even integer k ≥ 2, the Eisenstein series²of weight k is the function ∑ 2k n E (τ ) = 1 − σ − (n)q , k b k 1 k n≥1

¹ In [Ser12a], the space of modular forms of weight 2k is denoted by Mk .

4 ∑ = k−1 where bk is the k-th Bernoulli number, and σk−1(n) m|n m . By definition, these functions are holomorphic at infinity. For k ≥ 4, the Eisenstein series of weight k is a modular form of weight k. One ≥ proves this for example∑ by showing that for k 4, the series Ek is a multiple of the = + −k function Gk (τ ) (m,n)∈Z2∖(0,0) (mτ n) , which is indeed modular of weight k, see [Ser12a, Ch. VII, Prop. 8] and [Ser12a, Ch. VII, 2.3]. = −6 −3 3 − 2 The function ∆ 2 3 (E4 E6) is a modular form of weight 12. By a theorem of Jacobi [Ser12a, Ch. VII, Thm. 6], one has ∏∞ ∆(τ ) = q (1 − q)24. n=1

Proposition 2.3. We have the following dimension formula:

⌊k/12⌋ if k ≥ 0 and k ≡ 2 (mod 12)  dim(M ) = ⌊k/12⌋ + 1 if k ≥ 0 and k . 2 (mod 12) . k  0 if k < 0

Proof. An elementary approach is given in [Ser12a, Ch. VII, Corollary 1]. Alterna- tively, this formula arises as a corollary (8.39) to a more general formula proven in the Appendix, Theorem 8.38. □

Proposition 2.4. There is an isomorphism of graded rings ∼ C , −→ [X4 X6] M∗ mapping Xi to Ei, where the former ring is graded by assigning to Xi the degree i.

Proof. See [Ser12a, Ch. VII, Corollary 2]. □

2.2. The space of quasimodular forms

Let O(H ) denote the vector space of C-valued holomorphic functions on H . Recall the imaginary part function Y (τ ) = 4π Im(τ ). The following proposition shows that − one may compare coefficients of elements of O(H )[Y 1] as if Y was a formal variable. ∑ = M −m O H −1 = Proposition 2.5. Let F m=0 fmY be an element of ( )[Y ]. If F 0, then = fm 0 for all m.

²In [Ser12a], the Eisenstein series of weight k as defined below is denoted by Ek/2. A similar remark applies to the function Gk below.

5 d d −m = − −m−1 d = Proof. For the differential operator dτ¯ one has dτ¯Y 2πimY and dτ¯ fm 0, hence

∑M M∑−1 d −m−1 −2 −m 0 = F (τ ) = −2πi f (τ )Y = −2πiY ( f + τY ). dτ¯ m m 1 m=1 m=0 ≥ = □ By induction this implies that the fm are zero for m 1, hence also f0 0. ∑ = M −m O H −1 Corollary 2.6. Let F m=0 fmY be an element of ( )[Y ] satisfying the mod- Z ular condition of weight k. Then the fm are -periodic. Definition 2.7. An almost holomorphic modular form (of weight k) is an element

∑M = −m F fmY m=0 O H −1 H → C of ( )[Y ] such that F satisfies the modular condition and the fm : are holomorphic at infinity. ∑ = M −m Proposition 2.8. Let F (τ ) m=0 fm (τ )Y be an almost holomorphic modular form. − Then the leading coefficient fM is a modular form of weight k 2M. In particular, if , ≤ fM 0, then 2M k.

− Proof. This follows after comparing the coefficients of Y M in both sides of the mod- ularity condition F (γτ ) = (cτ + d)k F (τ ), using the equality

− − c(cτ + d) Y 1(γτ ) = (cτ + d)2Y (τ ) 1 + 2πi ( ) = a b ∈ Z . □ for γ c d SLn ( )

The almost holomorphic modular forms of weight k form a vector space, denoted by D D Mk . Let M∗ denote the associated graded ring. D → O H Definition 2.9. An element in the∑ image of the map Mk ( ) taking an almost = M −m holomorphic modular form F m=0 fmY of weight k to f0 is called a quasimod- ular form of weight k. Hence a quasimodular form is a holomorphic function on the upper plane appearing as the constant term of an almost holomorphic modular form.

H Again, denote the vector space of quasimodular forms of weight k by Mk and the H associated graded ring by M∗. The definition gives a surjective graded ring homo- D → H D ∩ H = morphism M∗ M∗ and one has Mk Mk Mk , the intersection being taken in the set of functions H → C. Example 2.10. Consider the second Eisenstein series ∑ = − n, E2(τ ) 1 24 σ1(n)q n≥1

6 ∑ ∏ = = ∞ − 24 where σ1(n) d|n d. For the weight 12 modular form ∆(τ ) q n=1(1 q) , = d one has the identity 2πiE2(τ ) dτ log(∆(τ )), which is proven by a straightforward computation. Using the modularity of ∆, one then computes

6c(cτ + d) E (γτ ) = (cτ + d)2E (τ ) + , 2 2 πi ( ) = a b ∈ Z for γ c d SLn ( ). −1 = + 2 −1 + c(cτ +d) ∗ = − / Now, since Y (γτ ) (cτ d) Y (τ ) 2πi , it follows that E2 E2 12 Y is an almost holomorphic modular form of weight 2. Hence, E2 is a quasimodular form of weight 2. H Proposition 2.11. The space M∗ of quasimodular forms satisfies the following proper- ties. D H 1. The canonical graded homomorphism M∗ → M∗ is an isomorphism. ⊗C ≃ C , , → H 2. There is an isomorphism of graded rings M∗ [X2] [X2 X4 X6] M∗ mapping Xi to Ei, where the former ring is graded by assigning to Xi the degree i. 3. Quasimodular forms are closed under taking derivatives. More precisely, the deriva- tive of a quasimodular form of weight k is a quasimodular form of weight k+2.

Proof. D → H 1. The map M∗ M∗ is surjective by definition. Injectivity follows∑ from Calculation = M −m 2.12 below. Given an almost holomorphic modular form F (τ ) m=1 fm (τ )Y with constant term zero, the strategy is to solve the modularity equation for the coefficients fm. This way, one finds for a fixed argument τ a polynomial equation , ∈ Z in the lower row components c d of any transformation γ SL2( ), involving the coefficients fm (τ ). By varying the transformation γ , one may force these coefficients to be zero. C , , → H 2. Express the map [X2 X4 X6] M∗ as the composition C ∗, , → D → H , [X2 X4 X6] M∗ M∗ ∗ ∗ where the first map takes X2 to E2 and Xi to Ei, and the second map is the canonical map, which is an isomorphism by the first point above. ∑ = M −m To prove the surjectivity of the first map, let F (τ ) m=0 fm (τ )Y be an almost ∗/ M holomorphic modular form of weight k. Then fM (E2 12) is an almost holomorphic − modular form of weight k, since fM is modular of weight k 2M, and the difference − ∗/ M F fM (E2 12) has degree smaller than M. Now use induction on M. ∑ = k/2 ∗ α To get injectivity, let F α=0(E2) fk−2α be an almost holomorphic modular form of weight k, in the image of the first map, where the fm are modular of weight m. If = −k/2 = F 0, then by comparing the coefficients of Y one obtains 0 f0. Now it follows

7 by induction on k that the other coefficients fm are zero. Hence F was the image of ⊗ C ∗ the zero element in M∗ [X2]. / ′ − 2 3. To prove the last statement, one verifies that (6 πi)E2 E2 is modular of weight / ′ − 4, and that if f is modular of weight k, then (6 πi)f kE2 f is modular of weight 2 + k. Now use the second point above. □ = ∑Calculation 2.12. This calculation follows the one found( in[BO00) ]. Let F (τ ) M −m = a b ∈ Z m=1 fm (τ )Y be an almost holomorphic modular form, γ c d SLn ( ), and − − τ ∈ H . Write j = cτ + d, and a = 6cj/2πi. Then Y 1(γτ ) = a + j2Y (τ ) 1. Hence,

∑M = + 2 −1 m F (γτ ) fm (γτ )(a j Y ) m=1 ( ) ∑M ∑m m − − = f (γτ )am l j2lY l l m m=1 l=0 ( ) ∑M ∑M ∑M m − − = f (γτ )am + f (γτ )am l j2ly l . m l m m=1 l=1 m=l On the other hand, ∑M = k −l , F (γτ ) fl (τ )j Y l=1 − by the modularity condition. By comparing the coefficients of Y l , one obtains the equalities ∑M m = fm (γτ )a 0 (1) m=1 and ( ) ∑M m − jk f (τ ) = f (γτ )am l j2l . l l m m=l Rewriting the second equality yields ( ) ∑M − m − f (γτ ) = f (τ )jk 2l − f (γτ )am l . (2) l l l m m=l+1

The latter may be solved recursively, starting by fM , to get equalities of the form = . fl (γτ ) (a polynomial in the f≥l (τ ) , j and c) (3)

The first two equalities are = k−2M fM (γτ ) fM (τ )j = k−2M+2 − · k−2M+1 . fM−1(γτ ) fM−1(τ )j const fM (τ )j c

8 In general, a straightforward inductive argument shows that in the summands of the expression (2) for fl (γτ ), the variable j appears with a power lower than or equal to − , k 2l. Now let r be the greatest index such that fr 0. Equation (1) finally gives, after substituting back the expressions for j and a and using (2) for l = r, the relation

∑M = + r r + + l l 0 κ1 fr (γτ )(cτ d) c κ3 fl (γτ )(cτ d) c l=r+1 = + k−r r − κ1 fr (τ )(cτ d) c ( ) ∑M ∑M m − − − κ f (γτ )(cτ + d)m rcm r + κ f (γτ )(cτ + d)lcl , 2 r m 3 l m=r+1 l=r+1 where the κi are some nonzero constants. To obtain a contradiction, choose a point τ in the upper half-plane and consider the last relation as a polynomial equation in c and d, letting P (c,d) denote the right-hand side of the equation. First look for ≥ the possible coefficients of monomials of the form crd 1. This excludes the third summand from the picture, since there c will always appear with a power greater − than r. Next look for the possible coefficients of the monomial crdk r . As seen when recursively solving the equations for fl (γτ ), the second summand will include only terms where (cτ + d) appears with a power lower than k − r. Hence the coefficient r k−r , of c d in P (c d) is κ1 fr (τ ). Now, if c ∈ Z, then there are infinitely many d ∈ Z such that P (c,d) = 0. Indeed, there are infinitely( many) d with gcd(c,d) = 1. For these d, find a,b ∈ Z such that − = a b ∈ Z , = ∈ Z ad bc 1. Since c d SL2( ), it follows that P (c d) 0. Similarly, for all d , there are infinitely many c such that P (c,d) = 0. It this follows that P (c,d) = 0 holds for all c,d ∈ C. These remarks may be summarized by the statement that the setof , Z C2 all c d belonging to the lower row of some matrix in SL2( ) is Zariski-dense in . Concluding, since P is zero as a function on C2, it is also zero as a polynomial, hence = the coefficient κ1 fr (τ ) is zero. Since τ was arbitrary, one finds fr 0, a contradiction.

9 3. Covers of an elliptic curve

In this section we define the central notions and objects of interest, i. e. finite coversof an elliptic curve with simple ramification type, the weighted counts of isomorphism classes thereof, and the generating functions associated to such weighted counts. In the following, let C be the ground field for all varieties considered. We begin by recalling some basic properties of complex curves.

Proposition 3.1. The assignment C 7→ K (C) defines a contravariant equivalence of categories between the category of irreducible smooth curves over C and the category of finitely generated field extensions of C of trascendence degree one. By definition, degree d maps of curves correspond to degree d field extensions.

Proof. See [Sil09, pp.20-22] □ → Proposition 3.2 (Riemann–Hurwitz formula). Let ϕ : C1 C2 be a finite map of smooth curves of genera д1 and д2, respectively. Let d be the degree of ϕ. Then ∑ − = − + − , 2д1 2 d(2д2 2) (eϕ (x) 1) ∈ x C1 where eϕ (x) is the ramification index of ϕ at x.

Proof. See [Sil09, Thm. 5.9] orLam09 [ , 7.2.1]. □

Definition 3.3. Let E be an elliptic curve. → = ∪k 1. A cover of E is a finite morphism p : C E of a disjoint union C i=1Ci of k irreducible smooth curves Ci. We shall denote the genus of C by д and the degree of p by d. Often a cover will be referred to by its source C. = { ,..., } − 2. Let S b1 b2д−2 be a set of 2д 2 distinct points of E. A cover C of genus д is simply branched over S, if it is simply branched over each point of S. This means − that for all points b of S there is exacly one point x in p 1(b) with ramification index = ep (x) 2, the others having ramification index one. It follows from the Riemann-Hurwitz formula of Proposition 3.2 that for a simply branched cover C → E, every point not in the pre-image of S has ramification index one. This justifies the choice of the number of points in S. , 3. Two covers C1 C2 are to be considered isomorphic, if there is an isomorphism → C1 C2 commuting with the respective structure maps into E. Accordingly, define the automorphism group Aut(C) of the cover C to be the group of cover isomor- phisms C → C. 4. A connected cover is a cover with connected sourceC, i. e. with only one irreducible component.

10 = ∪· · ·∪ Remark 3.4. LetC C1 Ck be a cover of genus д with structure map p of degree d. For all i, let p be the connected cover defined by the restriction p| . Denote the i Ci genus of Ci by дi and the degree of pi by di. By the Riemann-Hurwitz formula, the − maps pi have 2дi 2 ramification points on Ci. Hence, the following relations hold: ∑ ∑ = − = − . di d, and (2дi 2) 2д 2 i i Proposition 3.5. Let C be a connected cover of E. Then the automorphism group of C is finite.

Proof. By Proposition 3.1, if C is a connected cover of E, then the elements of the group Aut(C) correspond to the automorphisms of the finite field extension K (C)/K (E), of which only finitely many exist. □

Proposition 3.6. LetC = C ∪· · ·∪C be a cover, and p ·= p| . Then the automorphism 1 k i Ci group of C is given by the semidirect product ∏ = ⋊ , Aut(C) Aut(Ci ) Γ i ⊂ { ,..., } where Γ Sym C1 Ck is the subgroup generated by the automorphisms that permute isomorphic components. In particular, Aut(C) is finite.

Proof. The map Aut(C) → Γ given by looking at the action of an automorphism on { ,..., } the set C1 Ck is part of a short exact sequence ∏ 1 i Aut(Ci ) Aut(C) Γ 1 which admits a splitting Γ → Aut(C) given by the inclusion. □

Remark 3.7. If the cover C is simply branched over S, then no two components of genus greater than one are isomorphic as connected covers, since any isomorphism would have to preserve ramification indices (see for example [Sil09, II, Prop. 2.6]), but no two components share a branched point over E. In particular, if there are no components of genus one, then Γ = {1}. On the other hand, each component of genus one is unramified over E, and could be isomorphic to other components of genus one, in which case Γ is nontrivial.

Furthermore, note that theCk need not be connected for the statement of the previous proposition to hold. = { ,..., } − Definition 3.8. Let E be an elliptic curve, S b1 b2д−2 a set of 2д 2 distinct points of E. , 1. Let Cov(E S)д,d be the set of isomorphism classes of covers of E of genus д and degree d that are simply branched over S.

11 2. Any isomorphism of two equivalent covers defines a bijection of their automor- phism groups. This allows one to define the weight of the class [C] to be the number 1/| Aut(C)|. D 3. Define Nд,d to be the weighted count ∑ 1 D ·= Nд,d | | ∈ , Aut(C) C Cov(E S)д,d of the (classes of) covers of E. , ◦ ⊂ , 4. Let Cov(E S)д,d Cov(E S)д,d be the subset of classes [C] such that C is connected.

5. Similarly, define Nд,d to be the the weighted count ∑ · 1 N , ·= д d | Aut | ∈ , ◦ (C) C Cov(E S)д,d of the connected covers of E. To shorten the notation, the elliptic curve E and the D set of points S are omitted from the notation. It will turn outthat Nд,d and Nд,d are finite and do not depend on the choice of E and S. ≥ Definition 3.9. For any д 1, define Fд to be the generating series ∑ ·= d Fд (q) · Nд,d q d≥1 counting connected covers of genus д. ∑ = / Example 3.10. By the theory of elliptic curves we have N1,d j|d 1 j. To see this, we use the fact that the covers of degree d and genus 1 of an elliptic curve defined by a lattice Γ of C correspond to the subgroups of Γ of index d. If d is prime, then the number of such subgroups is d + 1. Furthermore, any such cover has exactly d automorphisms. Let R be the set of lattices of C and ZR the free abelian group generated by the set R. For n ∈ N, define the endomorphism T (n) : ZR → ZR by ∑ ′ T (n)Γ = Γ . ′ [Γ :Γ]=n

It is shown in [Ser12a, Ch. VII, Prop. 10] that the T (n) satisfy

T (m)T (n) = T (mn) for (m,n) = 1 and n = n+1 + n−1 ≥ , T (p )T (p) T (p ) pT (p )Rp for p prime, n 1 7→ where Rp is the homotety operator Γ pΓ. The number of index n subgroups of Γ is ZR R → c1(T (n)Γ), where c1 is the linear extension to of the constant function c1 :

12 Z, ′ 7→ R N → N = Γ 1 on . Now define∑ the function c : by c(n) c1(T (n)Γ) and define N → N = σ1 : by σ1(n) j|n j. It follows from the above equations that the function c satisfies c(m)c(n) = c(mn) for (m,n) = 1 and + − c(pn)c(p) = c(pn 1) + pc(pn 1) for p prime, n ≥ 1. = Since the function σ1 satisfies the same conditions and since c(p) σ1(p) holds for p prime, the two functions are equal. Finally, since any cover∑ of degree d and genus = / = / 1 has exactly d automorphisms, we have N1,d σ1(d) d j|d 1 j. By using the power series expansion for the logarithm ∑∞ − (−1)n 1 log(1 + z) = zn n n=1 ∑ ∑ ∑ | | < − − n = 1 d . for z 1, we find n≥1 log(1 q ) d≥1 j|d j q Hence, the first generating function is given by ∑ = − − n . F1(q) log(1 q ) n≥1

This thesis shall present a proof the following result. Theorem 3.11 ([Dij95]). Let д ≥ 2, and for τ ∈ C let q(τ ) = exp(2πiτ ). Then the − function Fд (q) is a quasimodular form of weight 6д 6.

The function F1 cannot be quasimodular. Indeed, if F1 was quasimodular of some ≥ ′ weight k 0, then by Proposition 2.11∑ the derivative F1 would be quasimodular of + ′ = d + / ′ = weight k 2. We have F1(q) 2πi d σ1(d)q , so 1 (24 2πi)F1 E2. This is a contradiction since the same proposition implies that the sum of two quasimodular forms of different weights cannot be quasimodular. The strategy to prove the main theorem will involve considering a more general generating function counting all covers of genus д and degree d. This generating function will be easier to compute. , D , D Definition 3.12. The generating functions Z (q λ) and Z (q λ) for Nд,d and Nд,d re- spectively, are defined as follows: ∑ ∑ ∑ Nд,d − Fд (q) − Z (q, λ) ·= qdλ2д 2 = λ2д 2, (2д − 2)! (2д − 2)! д≥1 d≥1 д≥1 ∑ ∑ D Nд,d − ZD(q, λ) ·= qdλ2д 2. (2д − 2)! д≥1 d≥1 Lemma 3.13. The above generating functions satisfy the relation

ZD(q, λ) = exp(Z (q, λ)) − 1.

13 Proof. The proof is subdivided into three parts. First, some notation and terminology − is introduced. Second, the coefficient of qdλ2д 2 in exp(Z (q, λ)) − 1 is expressed in terms of the new notation. Third, combinatorial arguments are used to prove that D / − this coefficient is equal to Nд,d (2д 2)!. Let C be a degree d, genus д cover. The combinatorial type of C is the tuple κ = , , r (kj дj dj )j=1 of natural numbers, such that for each j, the space C contains exactly kj → connected components Cj of genus дj such that the cover map Cj E is of degree − − dj. For simplicity, denote the Euler characteristics 2д 2 and 2дj 2 by χ and χj, respectively. Then ∑ ∑ = = . dj d, and χj χ j j D Further, define Nκ to be the weighted count of the covers of combinatorial type κ. Then ∑ D = D , Nд,d Nκ |κ|=(χ,d) ∑ ∑ | | , = , , where κ is defined as the tuple ( j kj χj j kjdj ), for κ (kj дj dj )j. For any j, we H also define the “unweighted” count N , of connected covers of degree of genus д дj dj j and degree dj by H · ′ ◦ N , ·= | Cov(E, S ) , |, дj dj дj dj ′ ⊂ where S S is any subset of cardinality χj. Finally, note that the relation

∏r qdλχ = qkjdj λkj χj j=1 = , , | | = , holds for each κ (kj дj dj )j with κ (χ d). The exponential of Z (q, λ) is given by

∏ ∏ ∑ k Nд,d exp(Z (q, λ)) = qkdλk χ . k д≥1 d≥1 k≥0 k!(χ!)

Expanding, one finds that the expression for exp(Z (q, λ)) is a sum over terms of the form <∞ k ∏ N , j * дj dj + 1 k d k χ q j j λ j j , χ ! k ! j=1 , j - j , , for some choices of parametersдj dj kj. Such choices may be collected to form combi- = , , natorial types κ (дj dj kj )j. Now, by collecting the summands arising from choices | | that induce combinatorial types of the same absolute value∑ κ , one obtains that the d χ , coefficient of q λ in exp(Z (q λ)) is equal to the sum |κ|=(χ,d) aκ, where

k ∏r N , j * дj dj + 1 a = . κ χ ! k ! j=1 , j - j

14 D / − It remains to prove that aκ = Nκ (2д 2)! for each combinatorial type κ. Here, we shall often make implicit use of Proposition 3.6 and the remark after it. ( ) ∏ χ r kj There are , ,..., = χ! = 1/(χ !) ways to subdivide the ramification locus S χ1 χ1 χr j 1 j into subsets that serve as the ramification loci of the connected components. Here, each χj appears in the binomial coefficient kj times. Fix one such decomposition, = ∪ · · · ∪ say S S1 Sr . Subdivide the decomposition further into subpartions such as (S + ,..., S + ), where all subsets belong to one type (д ,d ,k ) and have kj−1 1 kj−1 kj j j j cardinality χj. ≥ , , For дj 2, the number of covers of type (дj dj kj ) which ramify according to the ,..., / Hkj subpartition (Sk − +1 Sk − +k ) is exactly (1 kj!)N , . By tracking the automor- j 1 j 1 j дj dj phism group of the components involved in each choice, we obtain a weighted count / kj of (1 kj!)N , . дj dj ≥ For дj 1, there is no ramification locus. Instead, divide the kj components of genus 1 into isomorphism classes,( so that) there are, say, ℓ isomorphism classes with ∏ℓ ,..., kj = / cardinalities t1 tℓ. There are ,..., kj! i=1 1 tj! ways to perform such a t1 tℓ ∏ , , / ℓ Hkj subdivision, so that the number of of covers of type (дj dj kj ) is (1 kj!)( =1 tj!)N , . i дj dj Applying the weighting to an isomorphism class with ti elements gives an additional / factor of 1 ti! in the weighting, since the cardinality of its automorphism group as a cover has an additional factor of t !. Hence, the weighted count of covers of type ∏ ∏ i , , / ℓ ℓ / kj / kj (дj dj kj ) is (1 kj!)( =1 tj!)( =1 1 tj!)N , , which is equal to (1 kj!)N , . i i дj dj дj dj , , Since covers of different types (дj dj kj ) are not isomorphic, we deduce that the weighted count of covers with ramification type determined by the partition (S ,..., S ) ∏ 1 r r kj / of S is j=1(N , kj!). Finally, since covers belonging to different partitions are not дj dj ∏ r / kj isomorphic and since the number of such partitions is χ! j=1 1 (χj!) , we deduce D = □ that Nκ χaκ, as required.

15 4. Classifying covers via the fundamental group

The goal of this section is to use the theory of covering spaces to classify thecovers we are interested in, i. e. the covers of genus д and degree d that are simply branched D over S. This will get us to the first step to understanding their weighted count Nд,d .

4.1. Covering spaces

Definition 4.1. Let X be a topological space, F a set endowed with the discrete topol- × ogy, and G a group acting on both X and F. Define the fibred product X G F to be the topological space (X × F ) / ∼, where (x, f ) ∼ (дx,дf ) for all д in G. Proposition 4.2. Let X be a connected, locally pathwise connected, and semi-locally simply connected topological space. Let p : XH → X be a universal cover. Furthermore, H H H choose a point x0 of X, and let x0 be the image of x0 in X. Denote the fundamental , group π1(X x0) by π1. Then there is an eqivalence of categories { } { }, Unbranched covers of X π1-sets defined by the pair of quasi-inverse functors − (p : Y → X ) 7→ p 1(x ) and F 7→ XH × F . Y Y 0 π1

Proof. One verifies by hand that the given functors are mutually quasi-inverse, by using elementary covering theory. Nonetheless, the needed isomorphisms between objects are given below. Let F be a π -set and p : XH × F → X the associated covering. Define a map 1 F π1 → −1 H , ζF : F pF (x0) by sending an element f to the class of (x0 f ). → On the other hand, let pY : Y X be a cover of X. Define a map − η : XH × p 1 → Y Y π1 Y H, , → H H as follows. For a given class (x f ), let β : [0 1] X be a path starting in x0 and ending in xH. Consider the projection pβ of β to X and lift the path pβ to a path H H, = H H βf in Y, with starting point f . Finally, set ηY (x f ) βf (1). Note that since X is simply connected, this is independent of the choice of the path β. Also, the map is well-defined, since pβγH = pβ for any lift γH of a loop in X. □ Remark 4.3. In the above proposition, if X has the structure of a Riemann surface, then the first category may be taken to be the category of unbranched coversof Riemann surfaces over X. Indeed, every cover inherits a complex structure from X such that the structure map becomes holomorphic, and morphisms of covers of ′ X are automatically holomorphic. Indeed, if д : C → C is a continuous map and f : C → X is an open and holomorphic map such that f ◦ д is holomorphic, then д is holomorphic; see [Lam09, 1.3.7].

16 Furthermore, let X be a Riemann surface, let S ⊂ X be a finite set. Then putting − (C,p) 7→ (C \ p 1(S),p) defines an equivalence of categories between the category of finite covers of X with ramification locus contained in S and the category of finite unbranched covers of X \S. The reason is roughly that the local data of an unbranched cover around a “missing” branch point uniquely characterizes that of any extention of that cover to a ramified one, e. g. the local degree of the cover map will correspond to the ramification index. The topic of extending unbranched covers to branched ones is discussed in detail in [Lam09, 4.6]. ⊂ ∈ \ Corollary 4.4. Let X be a connected Riemann surface, S X a finite set, x0 X S \ , a point. Denote the fundamental group π1(X S x0) by π1. There is an equivalence of categories { } Finite ramified covers of X {π -sets}. with ramification locus contained in S 1

Remark 4.5. Let p : C → X be a ramified cover with ramification locus contained ∈ \ ∈ ∈ in S. Let x0 X S and s S. Let γ π1 be a loop of degree one about s and −1 → −1 ℓ θ : p (x0) p (x0) the permutation induced by γ . If θ decomposes in cycles, = ··· ,..., −1 say θ θ1 θℓ, having lengths k1 kℓ respectively, then the preimage p (s) ℓ ,..., ,..., consists of points s1 sℓ with ramification indices k1 kℓ respectively.

4.2. Marked covers and the monodromy map

= { ,..., } − Let E be an elliptic curve, S b1 b2д−2 a set of 2д 2 distinct points of E. Fix ∈ \ \ , a basis point b0 E S, and denote the fundamental group π1(E S b0) by π1. The equivalence of categories from the previous corollary is the equivalence { } Finite ramified covers of E {π -sets}. with ramification locus contained in S 1

Note that a simply branched cover of genus д is ramified over exactly 2д − 2 points = of E. It follows that if its ramification locus S0 is contained in S, then S0 S. , ∈ , Definition 4.6. A marking m on a cover (C p) Cov(E S)д,d is a bijective map −1 → { ,..., } , , , m : p (b0) 1 d .A marked cover is a triple (C p m), where (C p) is a cover and m is a marking on it. , , , , Two marked covers (C1 p1 m1) and (C2 p2 m2) are considered equivalent, if there is → = g , an isomorphism of covers ψ : C1 C2 such that m1 m2ψ . Let Cov(E S)д,d denote the set of equivalence classes of marked covers with respect to this relation. , Definition 4.7. Let (C p) be a cover of E. Denote the group action of π1 on the fibre −1 , 7→ . of p (b0) by (γ x) γ x. Define the monodromy map g , → , mon: Cov(E S)д,d Hom(π1 Sd ) − by mon(C,p,m)(γ )(i) = m(γ .m 1(i)).

17 . , , = , , Let the symmetric group Sd act on the first set by σ (C p m) (C p σm), and on the second by σ . ϕ = inn(σ )ϕ, i.e. by inner automorphisms. Then mon becomes a = , , morphism of Sd -sets. Furthermore, for ϕ mon(C p m) in the image of mon, the group action “forgetting the marking” −1 → −1 m ϕ( )m : π1 Aut(p (b0)) on the fiber of b0 is the same as the one defined by the above equivalence of cate- gories.

Definition 4.8. Let t denote the set of simple transpositions in Sd . Define the set

D = { ,..., , , ∈ S2д ∈ ··· = −1 −1}. Tд,d (τ1 τ2д−2 σ1 σ2) d ; τi t for all i, τ1 τ2д−2 σ1σ2σ1 σ2 D We define an Sd -action on Tд,d by conjugation in each component. This action is D well-defined, since conjugates of transpositions are transpositions, and makes Tд,d into an Sd -set. \ Remark 4.9. We can describe the fundamental group π1 of E S using generators and { , } , relations. Choose a generating set α1 α2 of π1(E b0) such that the images of the α in E do not intersect the ramification locus S. Furthermore, for i ∈ {1,..., 2д − 2} i ′ let γi be a simple loop about the point bi in S. For each i, there is a homotopy with \ ′ image in E S transforming γi into a loop γi about bi that contains the point b0. The picture below illustrates the situation.

α1

γ1 γ2д−2 ··· α2 α2

α b0 1

··· = −1 −1 The loops we defined satisfy the relationγ1 γ2д−2 α1α2α1 α2 . From the Seifert– Van Kampen theorem, it follows that the fundamental group π1 is described by the following generating set and relation: ⟨ ⟩ = ,..., , , ··· = −1 −1 . π1 γ1 γ2д−2 α1 α2 ; γ1 γ2д−2 α1α2α1 α2

In particular, any homomorphism of the group π1 into any group H is uniquely de- ,..., , termined by the images of the elements γ1 γ2д−2 α1, and α2. On the other hand, ,..., , , ··· = any tuple (τ1 τ2д−2 σ1 σ2) of elements of H satisfying the relation τ1 τ2д−2 −1 −1 σ1σ2σ1 σ2 defines a homomorphism of π1 into H.

18 D Proposition 4.10. The image of mon, as a Sd -set, is isomorphic to Tд,d .

D′ Proof. Define the set Tд,d by

′ − − D = { ,..., , , ∈ S2д ; ··· = 1 1}. Tд,d (τ1 τ2д−2 σ1 σ2) d τ1 τ2д−2 σ1σ2σ1 σ2 , → D′ 7→ By the previous remark there is a bijection ψ : Hom(π1 Sd ) Tд,d given by ϕ ,..., , , (ϕ(γ1) ϕ(γ2д−2) ϕ(α1) ϕ(α2)). The map ψ is also a morphism of Sd -sets, where D the Sd -action on Tд,d is defined by compenent-wise conjugation. The preimage of D → Tд,d under ψ consists of the homomorphisms ϕ : π1 Sd such that for all i, the permutation ϕ(γi ) is a simple transposition.

For covers that are simply branched over S, the image τi of each loop γi under the monodromy map is a simple transposition of the points in the fiber. Namely, there is over bi exactly one branch point of index 2, and τi interchanges the two fiber points corresponding to the two sheets of the branching, leaving the other fiber points un- −1 D changed. Hence, the image of mon is contained in ψ (Tд,d ). −1 D → It is left to show that Im(mon) contains ψ (Tд,d ). Let ϕ : π1 Sd be a homo- morphism such that ϕ(γi ) is a simple transposition for all i. By the equivalence of { ,..., } categories at the beginning of the section, the π1-action on 1 d defined by ϕ gives a finite, branched cover of Riemann surfaces p : C → E. The cover (C,p) also −1 → { ,..., } comes with a natural marking m : p (b0) 1 d , namely the inverse of the map ζ{1,...,d} defined in the proof of Proposition 4.2. By the same proposition, for all ∈ , ′ ∈ { ,..., } . −1 = −1 ′ ′ = θ π1 and all k k 1 d , if θ m (k) m (k ) then k ϕ(θ )(k). This , , { ,..., } shows that the π1-action defined by (C p m) on the set 1 d is the same as the action defined by ϕ. Finally, the condition that ϕ(γi ) be a simple transposition for all i implies by Remark 4.5 that the cover (C,p) is simply branched over S. Therefore, , , ∈ g , , , = □ we have (C p m) Cov(E S)д,d and mon(C p m) ϕ.

g , → D Define the morphism of Sd -sets ρ : Cov(E S)д,d Tд,d to be the composition of mon ∼ −→ D with the isomorphism Im(mon) Tд,d of the previous proposition. Proposition 4.11. Let (C,p,m) be a marked cover and θ its image under ρ. Then there ∼ is a group isomorphism Aut (C) −→ Stab (θ ). p Sd

→ Proof. Let ϕ : π1 Sd be the preimage of θ in the isomorphism of Proposition 4.10. By the equivalence of categories, the group Autp (C) is isomorphic to the group H { ,..., } of automorphisms of the π1-action on 1 d defined by ϕ. The group H consists ∈ = of all elements σ Sd commuting with ϕ, i. e. such that ϕ inn(σ )ϕ. Since the isomorphism of Proposition 4.10 is S -equivariant, we have H = Stab (θ ). □ d Sd Proposition 4.12. The morphism ρ induces a bijection on the sets of orbits g ∼ D \Cov(E, S) , −→ \T , . Sd д d Sd д d

19 Proof. The map ρ is surjective as the composition of surjective maps, hence the in- duced map on the sets of orbits is surjective too. , , = , , = . For the injectivity on the sets of orbits, let ρ(C1 p1 m1) θ and ρ(C2 p2 m2) σ θ, ∈ D ∈ , , −1 = for some θ Tд,d and σ Sd . Then ρ(C2 p2 σ m2) θ. It follows from the , ≃ , equivalence of categories that (C1 p1) (C2 p2), and hence the that the two marked □ covers only differ by the marking, so that they are in thesame Sd -orbit. , Corollary 4.13. The morphism ρ induces a bijection between Cov(E S)д,d and the set D , of Sd -orbits of Tд,d . In particular, the set Cov(E S)д,d is finite.

g , Proof. The statement follows from the last proposition, since theSd -orbits of Cov(E S)д,d , □ are in one-to-one correspondence with the elements of Cov(E S)д,d . D Lemma 4.14. The following equality for the weighted count Nд,d holds:

D = |D |/ . Nд,d Tд,d d!

D Proof. By Corollary 4.13 and Proposition 4.11, the weighted count Nд,d is equal to the D /| | weighted count of theSd -orbits ofTд,d , where each orbit is weighted by 1 StabS (θ ) , | . | = | |/|d | for any elementθ in the orbit. Now, it follows from the formula Sd θ Sd Stab(θ ) |D |/ □ that this weighted count equals Tд,d d!.

20 5. Conjugacy classes of the symmetric group

D In this section, we further the computation of Nд,d by using techniques inspired by graph theory. The rough picture is one of a graph with vertices the conjugacy classes of Sd and edges representing the passage from one class to another by multiplication with a simple transposition. We seek to count not cycles, but cycles starting and ending with the same representative, in the sense specified below. To do this, we make use of an analogue of the adjacency matrix for a graph. To abbreviate, we use the term “transposition” for simple transpositions. Again, we denote by t the set of simple transpositions in Sd .

5.1. Conjugacy cycles

Recall the definition

D = { ,..., , , ∈ S2д ∈ ··· = −1 −1}. Tд,d (τ1 τ2д−2 σ1 σ2) d ; τi t for all i, τ1 τ2д−2 σ1σ2σ1 σ2 Our aim is now to rewrite this definition using conjugacy classes. Note that the condition in the definition is equivalent to ··· = −1. (τ1 τ2д−2)σ2 σ1σ2σ1 ··· In particular, it is necessary for the product (τ1 τ2д−2)σ2 to be conjugated to σ2. ∈ For σ Sd , denote the conjugacy class of σ in Sd by c(σ ). ∈ Definition 5.1. For σ Sd , define − = { ,..., ∈ S2д 2 ; ∈ ··· = }. Pд,d (σ ) (τ1 τ2д−2) d τi t for all i, c(τ1 τ2д−2σ ) c(σ ) = {•} If д 1, define Pд,d (σ ) to be the singleton set .

Proposition 5.2. Let R = {σ (1),..., σ (r ) } be a system of (distinct) representatives of the conjugacy classes of Sd . Then ∑ |D | = | |. Tд,d d! Pд,d (σ ) σ ∈R

,..., ∈ ∈ Proof. We begin by noting that if (τ1 τ2д−2) Pд,d and σ Sd , then there is a bijection ′ ′ ′ −1 {σ ∈ S ; (τ ··· τ − )σ = σ σ (σ ) } → Stab (σ ). d 1 2д 2 Sd ′ Indeed, choose an arbitrary element σ0 of the first set and define the bijection by ′ ′ −1 ′ sending σ to (σ0) σ . D ,..., To count the elements inTд,d , we fix σ and we ask first first how many tuples (τ1 τ2д−2) ,..., satisfy the necessary condition of (τ1 τ2д−2)σ being conjugate to σ, next we

21 count the number of ways to realize such a conjugacy relation, which is exactly | Stab (σ )|. We obtain Sd ∑ ∑ D d! |T , | = | Stab ||P , (σ )| = |P , (σ )|. д d Sd д d | | д d ∈ ∈ c(σ ) σ Sd σ Sd | | → Z ′ ∈ The function Pд,d : Sd is constant on conjugacy classes. Indeed, for σ Sd ′ ′ −1 ′ there is a bijection of Pд,d (σ ) onto Pд,d (σ σ (σ ) ) given by conjugation with σ in each component. From this follows the required equality. □

The above proposition, together with Lemma 4.14, give the following reduction step: Corollary 5.3. We have the equality ∑ D = | |. Nд,d Pд,c (σ ) σ ∈R

From now on, let R = {σ (1),..., σ (r ) } be a fixed system of representatives of the = R conjugacy classes of Sd . Then the cardinality r part(d) of is the number of (unordered) partitions of {1,...,d}. This follows essentially from the fact that the conjugacy class of a permutation is uniquely determined by its cycle shape, that is the { ,..., } multiset a1 as whose elements are the respective sizes of the s cycles making up the (reduced) cycle decomposition of the permutation. Such sets are in a one-to- one correspondence with the partitions of d, since each cycle entry 1,...,d must appear exactly once in the cycle decomposition.

5.2. Adjacency matrices

Definition 5.4. Let d ≥ 1 and k ≥ 0. ≤ , ≤ k 1. For 1 i j r, define the sets Ed,i,j by k = { ,..., ∈ k ∈ ··· (i) = (j) }. Ed,i,j (τ1 τk ) Sd ; τi t for all i, c(τ1 τkσ ) c(σ ) = 0 , = For k 0, define Ed,i,j to be the empty set if i j and the singleton set if i j.

2. Define the square matrix Md by = | 1 |. (Md )i,j Ed,i,j

The matrix Md is a square matrix of size r, its definition does not depend on the choice of system of representatives R. Remark 5.5. If k is odd, applying the signum to the defining condition shows that − k = 2 − 2 2д 2 = (i) Ed,i,i is empty. If k д is even, then Ed,i,i Pд,d (σ ). k k = | k | Proposition 5.6. The entries of Md are given by (Md )i,j Ed,i,j .

22 Proof. The proof is by induction on k. For k = 0, 1, there is nothing to show. For the k induction step, note that if i (resp. j) are fixed, the sets Ed,i,j are pairwise disjoint for varying j (reps. i). We seek to define a bijection

⨿r k × 1 → k+1. φ : Ed,i,l Ed,l,j Ed,i,j l=1 ,..., k ∈ First, for each element (τ1 τk ) of Ed,i,l , choose an element σ Sd such that ··· (i) = (l) −1 ,..., , τ1 τkσ σσ σ . Second, define the image of ((τ1 τk ) τ0) under φ to be −1, ,..., (στ0σ τ1 τk ). By the definition of matrix multiplication, it suffices toprove that φ is a bijection. ′, ,..., Injectivity is clear. For surjectivity, given an element (τ0 τ1 τk ) in the target, ··· (i) (l) ··· (i) = (l) −1 choose an l such that τ1 τkσ is conjugate to σ , say τ1 τkσ σσ σ . −1 (l) (j) □ Then (σ τ0σ )σ is conjugate to σ . ≥ = ,..., Lemma 5.7. Let d 1 and r part(d). Let µ1,d µr,d be the eigenvalues of Md , listed according to their algebraic multiciplicities. Then

∑ ∑r D , = d . Z (q λ) exp(µi,dλ)q d≥1 i=1

Proof. Recall the definition of ZD: ∑ ∑ D Nд,d − ZD(q, λ) = qdλ2д 2. (2д − 2)! д≥1 d≥1

2д−2 (i) k The above proposition and remark give (M ) , = |P , (σ )| and (M ) , = 0 if k d i i д d ∑ d i i D = 2д−2 = r 2д−2 is odd, for all i. Hence, by 5.3 one has Nд,d Tr(Md ) i=1 µi,d , and since the terms for k odd vanish, ∑ ∑ 2д−2 Tr(M ) − ZD(q, λ) = d qdλ2д 2 (2д − 2)!) д≥1 d≥1 − ∑ ∑r ∑ 2д 2 µ , − = i d λ2д 2qd (2д − 2)! d≥1 i=1 д≥1 ∑ ∑r = d . exp(µi,dλ)q d≥1 i=1 □

23 6. The group algebra of the symmetric group

C Z Let [Sd ] be the group algebra of the symmetric group, let d be its centre. This is a commutative algebra, acting on itself linearly by multiplication. In this section, we relate this linear action to the matrix Md of the previous section, and we use the rep- resentation and character theory of the symmetric group to compute its eigenvalues.

6.1. The centre of the group algebra

Z ⊂ C Definition 6.1. Let d [Sd ] be the centre of the group algebra. If c is a conjugacy class of S , define the element z ∈ Z by d c d ∑ = . zc σ σ ∈c = Remark 6.2. The elements zc lie in the centre since αc cα for all conjugacy classes Z c and elements α of Sd . Further, the zc form a basis of d . Indeed, linear indepen- ∈ ⊂ C dence follows from the linear independence of the distinct elements σ Sd [Sd ]. ∈ Z −1 = C Furthermore, if z d , then the equalities αzα z show that the -coefficients Z of elements in the same conjugacy class are equal. Hence d is r-dimensional, with r = part (d).

Recall the definition of Md from the previous section. There, we fixed a system of representatives for the conjugacy classes of Sd . However, since the definition does not depend on the chosen representatives, we may also define Md to be a matrix indexed by the conjugacy classes of Sd , ordered in the same way as before. The new, equivalent definition is as follows. ′, Definition 6.3. Let c c be conjugacy classes of Sd . Define the matrix Md by ′ ′ = |{ ∈ }|, (Md )c ,c τ ; τ is a transposition such that τσ c where σ is any representative of c.

{ } From now on, we choose the ordering of the basis zc c and the ordering of the columns of Md to be compatible, i. e. coming from the same fixed ordering of the conjugacy classes {c}.

Proposition 6.4. Let t be the conjugacy class containing all transpositions. Let zt be the Z corresponding basis element of d . Let Mt be the size r square matrix representing the C · Z → Z -linear map (zt ) : d d given by multiplication with zt . Then the matrix Mt is the transpose of Md . ∑ ∑ , ′ = = Proof. Let c c be conjugacy classes. Note that if z σ ∈S λσ σ c˜ λc˜zc˜, then the ∈ d ∈ coefficient λc˜ is equal to the coefficient λσ , for any σ c˜. Now let σ c, and consider the product (∑ )( ∑ ) ∑ ∑ ′ ′ = = . ztzc τ σ σ ∈ ′ ′ ∈ ′ τ t σ ∈c σ Sd τσ ∈c

24 ∈ In this expansion, the coefficient λσ of any element σ c is the quantity ′ |{τ ∈ t ; τσ ∈ c }|.

′ = = = ′ □ It follows that (Mt )c ,c λc λσ (Md )c,c .

6.2. Irreducible characters of the symmetric group

We have reduced our problem of computing the eigenvalues of Md to the computation Z { } of the eigenvalues of Mt . More generally, we find that d actually has a basis wχ , indexed by the irreducible characters of Sd , such that each wχ is an eigenvector for Z all linear maps defined by multiplication with any element of d , and such that the corresponding eigenvalues are easy to compute. Definition 6.5. C 1. Let ρ be an irreducible representation of [Sd ], i. e. a group homomorphism → Cn ∈ ρ : Sd GL( ) such that for each σ Sd there are no ρ(σ )-invariant subspaces. → C, 7→ The irreducible character associated to ρ is defined as the map χρ : Sd σ Tr(ρ(σ )). → C = 2. An irreducible character of Sd is a map χ : Sd of the form χ χρ for some irreducible representation ρ. Its dimension dim(χ ) is defined as the dimension of the associated representation dim ρ = χ (1).

For brevity, we will refer to irreducible characters simply as characters. Remark 6.6. Characters are constant on conjugacy classes. It is therefore justified to write χ (c) ∈ C for a character χ and a conjugacy class c. Remark 6.7. The number of irreducible representations of a finite group, uptoiso- morphism, is equal to the number of its conjugacy classes (see for example [Ser12b, p. 19, Thm. 7]). In the case of the symmetric group, both the set of conjugacy classes and the set of irreducible representations are indexed by the set of Young diagrams, in a natural way. The irreducible representations are recovered from the Young dia- grams via Specht modules. , ′ ∈ Proposition 6.8. Let χ χ be characters, let σ1 Sd . Then ∑ ′  d! χ (σ ) if χ = χ ′ −1 =  dim(χ ) 1 χ (σ )χ (σ σ1)  ∈ 0 else. σ Sd

Further for conjugacy classes c and c1 we have ∑ ′  d! if c = c −1 =  |c| χ (c)χ (c1 )  χ 0 else, where the χ runs through the irreducible characters of Sd .

25 Proof. See [Isa13, Thm. 2.13 and Thm. 2.18]. □ ∈ Z Definition 6.9. Let χ be a character of Sd . Define the element wχ d by ∑ ∑ = dim(χ ) −1 = dim(χ ) −1 . wχ χ (c )zc χ (σ )σ d! d! ∈ c σ Sd ∑ Z = Proposition 6.10. The wχ form a basis of d . With respect to this basis, if z χ aχwχ Z · is any element of d , then the linear map (z ) is represented by the matrix Diag( )((aχ )χ ). · = d / Moreover, the matrix representing the map (zt ) has the diagonal entries aχ 2 χ (t) dim(χ ).

Proof. The two formulae in Proposition 6.8 lead to the formulae  ′ w if χ = χ ′ =  χ wχwχ  (1) 0 else and ∑ ( ) |c|χ (c) z = w (2) c dim( ) χ χ χ respectively. By (1), the wχ are linearly independent (multiply a linear relation with Z one of the wχ ), and by (2) they span d . The second statement( ) follows directly from = −1 | | = d □ (1). The last statement follows with (2) from t t and t 2 .

Lemma 6.11. With the notation from Lemma 5.7, the eigenvalues of Md are given by ( ) d 2 χ (t) µ , = , i d dim(χ ) where χ is the i-th character and t is the conjugation class of Sd containing all transpo- sitions.

Proof. By Proposition 6.4, the eigenvalues of Md are the same as the eigenvalues of · Mt . By Proposition 6.10, the wχ are eigenvectors for the map (zt ), represenented by □ the matrix Mt . The eigenvalues are given by the aχ from the same proposition.

26 7. Quasimodularity of the generating function

In this section, we use a formula of Frobenius to express the function ZD(q, λ) as the constant term of a certain product of Laurent series. This is exactly what is needed to ≥ prove that the generating function Fд is quasimodular for д 2, using the theorem about the generalized Jacobi function found in [KZ95]. The formula also exhibits a way to concretely compute the number of disconnected covers of given genus and degree.

7.1. Subsets of the half integers

Recall that the irreducible characters of Sd are parametrized by Young diagrams of Z + 1 size d. We will make use of the set of positive half integers ≥0 2 . Proposition 7.1. There is a bijection between the set of Young diagrams∑ ofsize d and , Z + 1 | | = | | = + ∑the set of pairs (U V ) of finite subsets of ≥0 2 such that U V and d u∈U u v∈V v.

Proof. Consider any a Young diagram of size d. Starting with the upper left corner, cut it diagonally in two pieces (see picture). This gives s “cut” columns in the lower ∈ Z + 1 piece and s “cut” rows in the upper piece. Let ui ≥0 2 denote the number of squares in the i-th cut row andvi the number of squares in the i-th cut∑ column.∑ Define = { ,..., } = { ,..., } | | = | | = + U u1 us and V v1 vs . Then U V and d u∈U u v∈V v. Conversely, let two suchU andV be given. The associated Young diagram is obtained by arranging both U and V in ascending order and then iteratively gluing the rows ∈ with ui squares to the columns with vi squares, for the appropriate elements ui U ∈ □ and vi V respectively. Proposition 7.2. Let χ be the character associated to the Young diagram corresponding , ∈ Z + 1 to the subsets U V ≥0 of equal cardinality s. Then (2 ) d χ (t) ∑s ∑s 2 1 * 2 2+ = u − v . dim(χ ) 2 i i , i=1 i=1 -

Proof. See [FH91], p. 52. □ Definition 7.3. Define the Laurent seriesθ (ζ ,q, λ) in ζ with coefficients formal power series in q and λ as follows: ∏ ( ) ∏ ( ) 2 / − − 2 / θ (ζ ,q, λ) = 1 + ζqueu λ 2 1 + ζ 1qve v λ 2 . ∈Z + 1 ∈Z + 1 u ≥0 2 v ≥0 2

27 Lemma 7.4. The counting function ZD(q, λ) is the coefficient of ζ 0 in the series θ (ζ ,q, λ)− 1. ∑ Proof. By expanding the product, one finds that θ (ζ ,q, λ) = , ⊂Z + 1 a , , where U V ≥0 2 U V = k d . aU ,V ζ q exp(µU ,V λ) Here, 1. k = |U | − |V | ∑ ∑ = + 2. d u∈U u v∈V v (∑ ∑ ) = 1 s 2 − s 2 3. µU ,V 2 i=1 ui i=1 vi .

Using the bijection in proposition 7.1, let the eigenvalues of the matrix∑ Md ∑be indexed , Z + 1 | | = | | = + by pairs (U V ) of subsets of ≥0 2 such that U V and d u∈U u v∈V v. By lemma 6.11 and proposition 7.2, the eigenvalue indexed by the pair (U ,V ) is equal to µU ,V . 0 , , − d ∑Now consider the coefficient of ζ in θ (ζ q λ) 1. There, the coefficient of q is U ,V exp(µU ,V λ), where the µU ,V are the eigenvalues of Md . By 5.7, this sum is equal to the coefficient of qd in ZD(q, λ). This proves the lemma. □

7.2. The coefficients of the theta function

Recall that Fд was defined as the series

∑ Fд (q) − Z (q, λ) = λ2д 2. (2д − 2)! д≥1

For an element τ of the upper half plane, set q = exp(2πiτ ). Sometimes q will be viewed as a formal variable. ∑ = k Proposition 7.5. Let a(x) k≥1 akx be a formal power series in x, with∑ holomorphic = k functions ak on the upper half plane as coefficients. Let exp(a(x)) k≥1 bkx be its formal exponential. Assume that each of the coefficients bk is quasimodular of weight kr, for some r. Then the ak are also quasimodular of weight kr.

□ Proof. This follows essentially by computing by hand the coefficients bi. Definition 7.6. Define the Laurent series Θ(ζ ,q, λ) in ζ with formal power series in q and λ as coefficients by ∏ Θ(ζ ,q, λ) = ( (1 − qn))θ (ζ ,q, λ). n≥1 , 0 , , Further, let Θ0(q λ) denote the coefficient of ζ in Θ(ζ q λ).

28 The following theorem about the quasimodularity of the coefficients of Θ0 is proved in [KZ95]. ∑ , = k Theorem 7.7. Let Θ0(q λ) k Ak (q)λ be the constant ζ -coefficient of Θ. Then the coefficient Ak (q) is a quasimodular form of weight 3k.

We may now prove the main result: ≥ Theorem 7.8 ([Dij95]). For д 2, the function Fд (q) is a quasimodular form of weight 6д − 6.

Proof. Lemma 7.4 gives the equality ∏ , = − n D , + . Θ0(q λ) ( (1 q ))(Z (q λ) 1) (1) n≥1

− By the previous theorem, the coefficient of λ2д 2 in this product is quasimodular of weight 6д − 6. By Lemma 3.13 one obtains, after taking the logarithm of both sides of (1), ∑ , = − n + , . log Θ0(q λ) log(1 q ) Z (q λ) n≥1 ∑ = − − n , As seen in Example 3.10, we have F1 n≥1 log(1 q ). Hence, in log Θ0(q λ) the coefficient of λ0 is zero. Thus, we may apply proposition 7.5 and the previous 2д−2 , / − theorem to find that the coefficient of λ in log Θ0(q λ), that is Fд (q) (2д 2)!, is a quasimodular form of weight 6д − 6. This concludes the proof. □

29 8. Appendix – Modular curves

One may weaken the definition of a modular form by requiring that the modular Z condition be met only for transformations lying in certain subgroups Γ of SL2( ). In this section we will calculate the dimension of the space Mk (Γ) of modular forms of even weight k associated to Γ, using the fact that modular forms can be seen as sections of a certain line bundle on a particular Riemann surface: the modular curve associated to the subgroup Γ. We will roughly follow [DS06, Ch. 1-3]. = Z Taking Γ SL2( ) will provide a proof to the dimension formula in Proposition 2.3 for the space Mk of modular forms defined in Section 2.

8.1. Congruence subgroups and modular curves

Definition 8.1. Let N ∈ Z. 1. The principal congruence subgroup of level N is the subgroup {( ) ( ) ( ) } a b a b 1 0 Γ(N ) = ∈ SL (Z) ; ≡ (mod N ) . c d 2 c d 0 1

⊆ Z ⊆ 2. A congruence subgroup is a subgroup Γ SL2( ) such that Γ(N ) Γ for some N ∈ Z. In that case, the group Γ is said to have level N .

Remark 8.2. The subgroup Γ(N ) is the kernel of the component-wise congruence Z → Z/ Z Z map SL2( ) SL2( N ). It is hence normal in SL2( ) and of finite index. Conse- Z quently, each congruence subgroup has finite index in SL2( ), while not necessarily being normal. ⊆ Remark 8.3. Since Γ(N ) ) Γ for some N , each congruence subgroup Γ contains an 1 ∗ element of the form 0 1 .

For the rest of the section, fix a congruence subgroup Γ.

Definition 8.4. > N ⊂ C ∪ {∞} 1. For M 0, define the set M by N = { ∈ H > } ∪ {∞}. M τ ; Im(τ ) M

∗ 2. Define the compact upper half-plane H to be the set H ∗ = H ∪ Q ∪ {∞}, endowed with the topology generated by the union of the topology of H with the set { } N ∈ Q , ∈ R . α ( M ) ; α SL2( ) M >0

30 Z H ∗ With this definition, the group SL2( ) acts on by continuous maps. This group action is transitive on the subset Q ∪ {∞}. Moreover, the space H is connected. ∗ 3. The modular curve X (Γ) is defined as the quotient space \H . Denote the canon- ∗ Γ ical projection map H → X (Γ) by π.

∈ H ∗ For τ , denote the stabilizer of τ under the action of Γ by Γτ . Remark 8.5. ( ) H → R + = √1 y x ∈ H 1. Define the function s : SL2( ) by s(x iy) y 0 1 . If τ , then . = R H s(τ ) i τ . Hence, the SL2( )-operation on is transitive. 2. The stabilizer of i with respect to the transitive group action of the topological R H R group SL2( ) on is the compact subgroup SO2( ). , ∈ H = ∈ R −1. 3. Let e1 e2 . We have γ (e1) e2 if and only if γ s(e2) SO2( )s(e1) , ∈ H ′ ′ Proposition 8.6. If τ1 τ2 , then there are open neighborhoods U1 of τ1 and U2 of ∈ Z ′ ′ τ2, such that for all but finitely many γ SL2( ), the sets γ (U1) and U2 do not meet.

′ ′ Proof. Choose U1 and U2 to be any open neighborhoods belonging to the topology H ∈ Z of , and with compact closure. Let γ SL2( ). The previous remark implies that ′ ∩ ′ , ∅ γ (U1) U2 is equivalent to ∩ ∈ Z ∩ R −1. γ SL2( ) s(e2) SO2( )s(e1) ∈ ′ e1 U1 ∈ ′ e2 U2 But this subgroup is compact and discrete, hence finite. □ H ∗. Proposition 8.7. Let τ1 and τ2 be elements of Then there exist open neighborhoods ∈ Z = U1 of τ1 and U2 of τ2 such that for all γ SL2( ), if γ (U1) meets U2 then γτ1 τ2.

Proof. We consider three cases separately. , ∈ H ′ ′ 1. Let τ1 τ2 . By Proposition 8.6, there are open neighborhoods U1 and U2 of τ1 and τ2 respectively, such that the set { ∈ Z ′ ∩ ′ , ∅, , } γ SL2( ) ; γ (U1) U2 γ (τ1) τ2 is finite. We denote this set by F. For eachγ ∈ F, choose disjoint open neighborhoods U1,γ and U2,γ of γτ1 and τ2, respectively, and put ∩ = ′ ∩ .* −1 /+ U1 U1 γ (U1,γ ) and ,γ ∈F - ∩ = ′ ∩ .* /+ . U2 U2 U2,γ ,γ ∈F -

31 The open neighborhoods U1 and U2 satisfy the required propertieties. ∈ Q ∪ {∞} ∈ H H 2. Let τ1 and τ2 . Choose U2 to be any open neighborhood in ≥ Z ¯ ∪ N = ∅ with compact closure. Now, there is some M 0 such that SL2( )U2 M . Let ∈ Z ∞ = = N ∈ Z α SL2( ) such that α ( ) τ1. Choose U1 α ( M ). Then for all γ SL2( ), the sets γ (U2) and U1 are disjoint. , ∈ Q ∪ {∞} , ∈ Z ∞ = ∞ = 3. Let τ1 τ2 . Let α1 α2 SL2( ) such that α1( ) τ1 and α2( ) τ2. = N = N Choose U1 α1( 2) and U2 α2( 2). The open sets U1 and U2 satisfy the required properties. □ ∗ Corollary 8.8. Let τ ∈ H . There is an open neighborhood U of τ such that for all ∈ H ∈ γ , if γ (U ) meets U then γ Γτ . Proposition 8.9. The modular curve X (Γ) is connected, compact, and Hausdorff.

∗ Proof. The connectedness of X (Γ) follows from the connectedness of H . For com- pactness, define the subsets

D = {τ ∈ H ; |τ | ≥ 1, Re(τ ) ≤ 1/2}

∗ ∗ ∗ and D = D ∪ {∞}. The subset D ⊂ H is compact and a fundamental domain Z H ∗ Z for the SL2( )-action on . Since Γ has finite index in SL2( ), it follows that one possible fundamental domain for the Γ-action is given by the union of finitely D∗ Z many images of under elements of SL2( ). Therefore, the modular curve X (Γ) is compact. Finally, the Hausdorff property follows from the previous proposition. □

8.2. Modular curves as Riemann surfaces

The modular curve X (Γ) may be given the structure of a Riemann surface. The needed local data is summarized below. We use the following convention: a subgroup G of Z − SL2( ) need not contain the matrix id; we denote the subroup generated by G and {− id} by G.

Proposition 8.10. ∈ H 1. For τ , the isotropy group Γτ is finite cyclic. ∈ Q ∪ {∞} Z 2. For s , the isotropy group Γs has finite index in the isotropy group SL2( )s.

R Proof. 1. By Remark 8.5, the isotropy group Γτ is conjugated via an element of SL2( ) R to some discrete subroup of SO2( ), but every such discrete subgroup is finite cyclic. ∈ Z ∞ = = −1 Z = 2. Let α SL2( ) such that α ( ) s. We have Γs( )αΓ∞α and SL2( )s Z −1 Z 1 1 □ α SL2( )∞α . Now use that SL2( )∞ is generated by 0 1 . Definition 8.11.

32 1. Let τ ∈ H . The period of τ is the number = | /{ }| hτ Γτ id of maps in the isotropy group of τ . The period of τ only depends on the class Γτ . If > hτ 1, then we call the point τ , or interchangeably the point π (τ ), an elliptic point H ∗ → CD for Γ. We further define the map δτ : to be the map represented by the matrix ( ) 1 −τ δ = ∈ GL (C) τ 1 −τ¯ 2

C → C = hτ . and the map ρτ : by ρτ (z) z 2. Let s ∈ Q ∪ {∞}. The width of s is the number = Z  . hs [SL2( )s : Γs]

This also only depends on the class Γs. We call the point s, or interchangeably the ∈ Z point π (s), a cusp for Γ. Furthermore, we define δs SL2( ) to be any map taking s ∞ H ∗ → C = / to . Finally, define the map ρs : by ρs (z) exp(2πiz hs ). Note that this is well-defined at ∞ since we restrict to the upper half-plane. Z Remark 8.12. There are only two elliptic points on X (SL2( )), namely i and µ3, = / where µ3 exp(2πi 3); see [DS06, 2.3]. Their periods are 2 and 3, respectively. ⊆ Z Since Γτ SL2( )τ , each elliptic point τ for Γ has period 2 or 3, and lies in one of the classes Γi or Γµ3, according to wether its period is 2 or 3. Since Γ has finite index in Z SL2( ), it follows that there are only finitely many elliptic points on X (Γ). Z ∞ Remark 8.13.(The) only cusp of SL2( ) is , whose isotropy subgroup is the group 1 1 ∞ generated by 0( 1 . Hence) the width of with respect to Γ is generally the smallest > 1 h ∈ h 0 such that( ) 0 1 Γ,( cf. Remark) 8.3. The only exception is when h is minimal − 1 h ∈ 1 h < such that ( 0 1) Γ but 0 1 Γ, in which case h is the width, but 2h is minimal 1 2h ∈ such that 0 1 Γ. As before, there are only finitely many cusps of Γ.

∈ H ∗ ∈ H For τ , let Uτ be an open neighborhood of τ such that for all γ , if γ (Uτ ) ∈ meets Uτ then γ Γτ , as per Corollary 8.8. If τ is an elliptic point or a cusp for Γ, we may assume thatUτ contains no further elliptic points or cusps. Ifτ is neither, we may assume that Uτ contains no ellliptic points or cusps altogether. These assumptions are justified, since there are only finitely many elliptic points orcups. = ◦ | C , ∈ Define the map ϕτ ρτ δτ U . This is a map with open image in . For τ1 τ2 Uτ , = τ = we have ϕτ (τ1) ϕτ (τ2) if and only if π (τ1) π (τ2); see [DS06, p. 50, 60-61]. Hence, → C , the map ϕτ induces an injective continuous map ψτ : Uτ . We take (π (Uτ ) ψτ ) to be our chosen chart around π (τ ).

, ∗ Proposition 8.14. The charts (π (Uτ ) ψτ )τ ∈H form a holomorphic atlas for the modular curve X (Γ).

33 Proof. See [DS06, 2.2, 2.4]. □ ⊆ − ∈ Proposition 8.15. Let Γ1 and Γ2 be two conjugation subgroups with Γ1 Γ2. If id \ → / Γ2 Γ1, then the induced morphism X (Γ1) X (Γ2) has degree [Γ2 : Γ1] 2, else it has ∈   degree [Γ2 : Γ1]. The ramification index of a point π1(τ ) X (Γ1) is [ Γ2,τ : Γ1,τ ].

Proof. See [DS06, p. 66-67]. □ → Z Corollary 8.16. Let f : X (Γ) X (SL2( )) be the morphism induced by the inclusion ∈ Z Z Γ SL2( ). If τ is a cusp for Γ or if τ is an elliptic point for SL2( ) but not for Γ, then the ramification index of π (τ ) is hτ , Else, the ramification index is 1.

∞ Let y2, y3, and y∞ be the images of i, µ3, and , respectively, under the projection H ∗ → Z ∈ { , } X (SL2( )). For h 2 3 , let εh be the number of elliptic points in X (Γ) of period h. Let ε∞ be the number of cusps in X (Γ). { , , } By Corollary 8.16, the ramification locus of f is contained in the set y1 y2 y∞ . Now let d be the degree of f . By applying the previous corollary and the formula ∑ = , d ef (x) ∈ −1 x f (yh ) we obtain the formulae ∑ h − 1 (e (x) − 1) = (d − ε ) f h h ∈ −1 x f (yh ) and ∑ − = − . (ef (x) 1) d ε∞ − x∈f 1 (∞) Proposition 8.17. The genus д of the modular curve X (Γ) is given by

d ε ε ε∞ д = 1 + − 2 − 3 − . 12 4 3 2

Proof. The statement follows from the Riemann–Hurwitz formula, the previous dis- Z cussion, and the fact that X (SL2( )) has genus 0, which will be proved later in Ex- ample 8.23. □

Corollary 8.18. We have the inequality 1 2 2д − 2 + ε + ε + ε∞ ≥ 0. 2 2 3 3

34 8.3. Automorphic forms and modular forms ( ) = a b C H → CD Definition 8.19. Let γ c d be an element of GL2( ) and let f : be a holomorphic function. For τ ∈ H define the factor of automorphy j(γ,τ ) ·= cτ + d ∈ Z H → C and for k the function f [γ ]k : by ·= k/2 , −k . f [γ ]k (τ ) · det(γ ) j(γ τ ) f (γτ ) , ′ ∈ Z ∈ H Remark 8.20. Let γ γ SL2( ) and τ . ′ ′ ′ 1. The factor of automorphy satisfies j(γγ ,τ ) = j(γ,γ (τ ))j(γ ,τ ). H → C ′ = ′ 2. For all holomorphic functions f : , we have f [γγ ]k (f [γ ]k )[γ ]k . Definition 8.21. Let f :(H) → C be a meromorphic function. Let h ∈ N be minimal 1 h ∈ ∈ H = / with the property that 0 1 Γ. For τ , set qh exp(2πiτ h). 1. The function f is h-periodic, if it satisfies f (τ + h) = f (τ ) for all τ ∈ H . N > 2. Assume that f is h-periodic. If f has no poles in the subset C for some C 0, ˜ \{ } → C = ˜ then there exists a meromorphic function f : B 0 such that f (τ ) f (qh) for N all τ . We say that f is meromorphic at infinity, if it has no poles in some C and if ˜ the associated function f has∑ a meromorphic continuation to the whole of B. In this = ∞ n ∈ Z , case, we may write f (τ ) n=m anqh, with m and am 0. We call m the order of f at infinity, and we denote it by ν∞(f ). The function f is holomorphic at infinity ≥ H if ν∞(f ) 0. We denote the order of f at a point τ of by ντ (f ). 3. The function f is said to satisfy the modular condition of weight( k) with respect to = ∈ 1 h ∈ Γ, if f [γ ]k f for all γ Γ. Such a function is h-periodic, since 0 1 Γ. ∈ Z −1 For all α SL2( ), the group α Γα is a conjugation subgroup. Now assume that the function f satisfies the modular condition of weight k with respect to Γ. For all α, the function f [α]k satisfies the modular condition of the same weight with respect −1 to the subgroup α Γα, and is hence hα -periodic for some hα . 4. Let s ∈ Q ∪ {∞} be a cusp for Γ. We define f to be meromorphic at s, if for ∈ Z ∞ = some α SL2( ) with α ( ) s, the function f [α]k is meromorphic at infinity. The order νs (f ) of f at s is the order of the function f [α]k at infinity. The function f is ≥ > holomorphic at s, if νs (f ) 0; it vanishes at s, if νs (f ) 0. These notions do not depend on the choice of α. 5. The function f is an automorphic form with respect to Γ, if it satisfies the modular condition of some weight k with respect to Γ and is meromorphic at all cusps of Γ. We call k the weight of f . 6. An automorphic form f is a modular form, if it is holomorphic and is holomorphic at all cusps of Γ. 7. A modular form f is a cusp form, if it vanishes at all cusps of Γ.

35 We denote the space of automorphic forms of weight k by Ak (Γ), the space of modular S forms of weight k by Mk (Γ), and the space of cusp forms of weight k by k (Γ). C Remark 8.22. The -algebra A0(Γ) of automorphic forms of weight 0 is isomorphic to the algebra C(X (Γ)) of meromorphic functions on the modular curve X (Γ). To see this, note that the two notions of meromorphy at a cusp are equivalent. Simi- ≃ C larly the space M0(Γ) is the space of holomorphic functions on X (Γ), so M0(Γ) . S = { } Accordingly, 0(Γ) 0 .

Example 8.23. Let ∆ and G4 be the cusp form of weight 12, respectively the modular form of weight 4 of Example 2.2. The j-invariant is the automorphic form of weight = 3/ H 0 defined as j 1728G4 ∆. The function ∆ has no zero on , but it is a cusp form with a nontrivial Fourier coefficient at place 1 in its expansion at infinity (see [DS06, p. 73, eq. 3.1]). Hence the automorphic form j has only one pole, which is simple. Z → CD Therefore the j-invariant may be seen as a holomorphic function j : X (SL2( )) Z of degree 1. It follows that the modular curve X (SL2( )) is isomorphic to the sphere CD, and has hence genus 0. ′ Remark 8.24. The derivative j of the j-invariant is a nonzero element of A2(Γ). ≥ Hence for k even with k 2, the algebra Ak (Γ) contains nonzero elements. Further- more, if f is a nonzero element of Ak (Γ), then division by f yields an isomorphism ≃ . Ak (Γ) A0(Γ)f

Since the factor of holomorphy j(γ,τ ) has no zeroes or poles at non-cusp points, the ∗ order of an automorphic form at a point τ of H does not depend on the Γ-class of τ . Hence we may define the order of an automorphic form at points of a modular curve, while taking the local coordinates into account. ∈ H ∗ Definition 8.25. Let f be an automorphic form of weight k, let τ . Let Uτ be a neighborhood of τ such that π (Uτ ) is a chart neighborhood of the point π (τ ) of X (Γ), | = ◦ ◦ → as per Proposition 8.14. Write f Uτ flocal ρ δ, with flocal : φ(Uτ ) V . The order νπ (τ ) (f ) of f at the point π (τ ) is defined as the order of flocal at 0. The order at π (τ ) does not depend on the choice of the representative τ . Remark 8.26. We have

ντ (f )  if τ ∈ H ,  h ν (f ) =  τ π (τ ) ν (f )h  τ τ if τ is a cusp,  pα Z ∞ = where in the second case α is some element of SL2( ) with α ( ) τ , and pα is the smallest period of the function f [α]k . Following Remark 8.13, the number pα is in most cases⟨ equal( to)⟩ the width hτ , the exception takes place when k is⟨ odd( and)⟩ − − 1 = − 1 hτ = 1 = − 1 hτ (αΓα )∞ 0 1 , in which case pα 2hτ . Note that if (αΓα )∞ 0 1 but k is even, then Definition 8.21 treats f [α]k as a 2hτ -periodic function, when f is in fact even hτ -periodic. For more detail, see [DS06, p. 74-75].

36 8.4. The dimension formula

∗ Definition 8.27. Let Y be a Riemann surface. Let T (Y ) denote the cotangent bundle ⊗ of Y. For n ∈ N, define the sheaf of n-fold meromorphic differentials Ω(Y ) n to be the ∗ ⊗ sheaf of sections into the bundle T (Y ) n.

If U is the domain of some chart for some Riemann surface, then the n-fold mero- morphic differentials are naturally identified with the meromorphic functions on U . In the following, we shall make implicit use of this fact. ′ Definition 8.28. Any morphism φ : Y → Y of Riemann surfaces gives rise to a pull- ∗ ′ ⊗ ⊗ ′ back map φ : Ω(Y ) n → Ω(Y ) n, given locally by f 7→ (φ )n (f ◦ φ).

Examples 8.29. The natural morphism π : H → X (Γ) gives rise to the pullback map

∗ ⊗ ⊗ π : Ω(X (Γ)) n → Ω(H ) n.

∈ C ′ = , −2 ∈ H ⊗n ∈ For δ GL2( ) we have δ det(δ )j(δ τ ) . Hence for f Ω( ) and α Z = ∗ SL2( ) we have the relation f [α]2n α f . → Remark 8.30. Let φ : Y1 Y2 be a morphism of Riemann surfaces with dense im- ∗ ⊗n → ⊗n age. Then the pullback map φ : Ω(Y2) Ω(Y1) is injective by the Morse-Sard theorem. → Proposition 8.31. Let φ : Y1 Y2 be a morphism of Riemann surfaces. Let f be a → CD meromorphic function on Y1. For each chart domain V in Y2, let ω : V be a ∗ V = | − { } holomorphic function such that φ ωV f φ 1 (V ). Then the collection ωV V forms an n-fold meromorphic differential on Y2.

′ Proof. Let V and V be two chart domains in Y2, let U be their intersection. Set − W = φ 1(U ). Since the map φ| : W → U is surjective, by Remark 8.30 it suffices ∗ ∗W to show that (φ| ) ω ′ = (φ| ) (д ′, ω ), where д ′, is the corresponding cocycle W V W V V V V V ∗ ∗ in the cotangent bundle. But a calculation shows that (φ| ) (д ′, ω ) = (φ| ) ω , ∗ ∗ W V V V W V | ′ = | □ and by assumption (φ W ) ωV (φ W ) ωV . ⊗ ∗ Proposition 8.32. Let ω ∈ Ω(X (Γ)) n, write π (ω) = f for some meromorphic function f : H → CD. Then f is an automorphic form of weight 2n.

∗ ∗ ∗ Proof. Let γ ∈ Γ. Then γ π ω = π ω, since πγ = γ . On the other hand, we have ∗ ∗ = ′ n ◦ = γ π ω (γ ) (f γ ) by the definition of the pullback map. Hence we have f f [γ ]2n, so that the f satisfies the modular condition. Furthermore, f is meromorphic at each cusp because ω is. □

By this proposition we get a map of C-vector spaces ∗ ⊗n → . π : Ω(X (Γ)) A2n (Γ)

37 ∗ ⊗ Proposition 8.33. The map π is bijective. Furthermore, if ω ∈ Ω(X (Γ)) n and f = ∗ π (ω), then the orders of f and ω at a point π (τ ) ∈ X (Γ) are related by the formula ( )  1  ( ) − 1 − ∈ H = νπ (τ ) f n if τ and τ has period h, . νπ (τ ) (ω)  h  − ∈ Q ∪ {∞} νπ (τ ) (f ) n if τ

∗ Proof. The map π is injective by Remark 8.30, since the map π has dense image. For surjectivity, let f be an automorphic form of weight 2n, viewed as an element of H ⊗n ∈ H ∗ ·= Ω( ) when convenient. Let τ0 , and let U · Uτ be the corresponding open 0 ∼ neighborhood as in 8.14, so that there is a chart of X (Γ) of the form ψ : π (U ) −→ V . For convenience, we are omitting the index τ0. Hence we have a diagram

U π π (U ) ≃

δ ≃ ψ ρ δ (U ) V , with δ ·= δ and ρ ·= ρ as in Definition 8.11. Define the n-fold differential τ0 τ0 = −1 ∗ | λ (δ ) (f U ∩H ) on δ (U ), which we also view as a meromorphic function. We distinguish between two cases, depending on whether τ0 is a cusp or not.  −1 /{ } Supposeτ0 is not a cusp and has periodh. The extended isotropy group (δΓδ )0 id is generated by the rotation rh represented by the matrix ( ) µ2h 0 , h−1 0 µ2h = / where µ2h exp(2πi h). Since the function f is Γ-invariant (under pullbacks), −1 ∈ = the function λ is δΓδ -invariant. In particular, for z δ (U ) we have λ(rhz) , 2n n = n j(rh z) λ(z), from which the relation (µhz) λ(µhz) z λ(z) follows. Hence the 7→ n 7→ function z z λ(z) is invariant under the transformation z µhz, so that there is a meromorphic function д on δ (U ) such that д(zh) = znλ(z) for all z. We have hν (д) = n + ν (f ), hence ν (д) = ν (f ) + n/h. 0 τ0 0 π (τ0) = / n = We define the n-fold differential θ on V by putting θ (z) д(z) (hz) , and set ωπ (U ) ∗ ∗ = ∗ = | ψ θ. A calculation shows that ρ θ λ, therefore we have π ωπ (U ) f U .

The case where τ0 is a cusp is treated analogously: here, V is the open unit( ball,)  −1 /{ } 1 h now the extended isotropy group (δΓδ )∞ id is generated by the matrix 0 1 , Z where h is the width of τ0, and the function λ is h -periodic. Hence there is a mero- / morphic function д on V \{0} such that д(e2πiz h) = λ(z) for all z. Since f is holo- morphic at the cusp τ0, the function д has a meromorphic continuation on all of V . Its order at 0 is given by ν (д) = ν (f ). 0 π (τ0) Define the n-fold differential θ on V by putting θ (z) = hnд(z)/(2πiz)n, and set again = ∗ ∗ = ωπ (U ) ψ θ, an n-fold differential on π (U ). Another calculation shows that ρ θ λ, ∗ = | hence π ωπ (U ) f U ∩H .

38 { } Now we put everything together: by Proposition 8.31, the collection ωπ (U ) U forms an n-fold differential on X (Γ), as required. By construction, we obtain the orders of the differential as in the statement. □

We now come to the Riemann–Roch theorem, stated here without proof, and its application to computing the dimension of Mk (Γ) for k even. Let X be a compact Riemann surface. To a divisor D ∈ Div(X ) we associate its linear space L(D), a finite- dimensional vector space over C whose dimension we denote by ℓ(D), defined as L(D) = {f ∈ C(X ) ; div(f ) + D ≥ 0} ∪ {0}. Theorem 8.34 (Riemann–Roch). Let X be a compact Riemann surface, д its genus. Let K be a canonical divisor on X. If D ∈ Div0(X ), then ℓ(D) = deg(D) − д + 1 + ℓ(K − D). Remark 8.35. From the Riemann–Roch theorem follow: 1. We have ℓ(K) = д and deg(K) = 2д − 2. 2. If deg(D) < 0, then ℓ(D) = 0. 3. If deg(D) > 2д − 2, then ℓ(D) = deg(D) − д + 1. Definition 8.36. Let X be a Riemann surface, k an even integer.

1. Define the group DivQ(X ) of rational divisors on X by DivQ(X ) = Div(X ) ⊗ Q. This group is equipped with a relation ≤, which extends the relation on Div(X ). ∈ 2. Let f Ak (Γ) be a nonzero∑ automorphic form. The rational divisor div(f ) of f is = the rational divisor div(f ) x∈X (Γ) νx (f )x.

3. The floor function ⌊⌋ : DivQ(X ) → Div(X ) is defined by applying the floor func- tion coefficient-wise. ⊗ Remark 8.37. Any differential in Ω(X (Γ)) n has degree 2n(д−1). To see this, choose ∈ ⊗1 a nonzero element f of A2(Γ), and let λ Ω(X (Γ)) be the preimage of f under ∗ π . The divisor div(λ) is a canonical divisor and has hence degree 2(д − 1) by the ⊗ previous remark. The element λn of Ω(X (Γ)) n has degree 2n(д − 1). Since we may ⊗ ⊗ write Ω(X (Γ)) n = C(X (Γ))λn, it follows that all elements of Ω(X (Γ)) n have the same degree as λn. Theorem 8.38. Let k be an even integer, set n = k/2. Let д be the genus of the modular ∈ { , } curve X (Γ); for h 2 3 , let εh denote the number of elliptic points in X (Γ) of period h. Let ε∞ be the number of cusps. The following formula for the dimension of the space Mk (Γ) of modular forms with respect to the congruence subgroup Γ holds: ⌊ ⌋ ⌊ ⌋  k k k (k − 1)(д − 1) + ε + ε + ε∞ if k ≥ 2 ,  4 2 3 3 2 dim(M (Γ)) =  . k 1 if k = 0,  0 if k ≤ 0

39 ≤ ≃ C S = { } < ∈ Proof. First let k 0. We have M0(Γ) and 0(Γ) 0 . If k 0 and f Mk (Γ), 12 −k S = then f ∆ lies in 0(Γ), where ∆ is the usual weight 12 cusp form. Hence f 0. ≥ ∈ Now let k 2. By Remark 8.24, there is a nonzero element f of Ak (Γ). Let ω ⊗n ∗ ≃ C Ω(X (Γ)) be the preimage of f under π . Using the isomorphism Ak (X (Γ))f , we obtain an isomorphism ≃ { ∈ C + ≥ } ∪ { }. Mk (Γ) f0 (X (Γ)) ; div(f0) div(f ) 0 0 + Since⌊ div⌋ (f0) is integral, the above condition is equivalent to⌊ the condition⌋ div(f0) ≥ ≃ div(f ) 0, hence⌊ there⌋ is an isomorphism Mk (Γ) L( div(f ) ), and we have = ℓ dim(Mk (Γ)) ( div(f ) ). Using the formula for the orders found in Proposition 8.33, we obtain ∑ ⌊ ⌋ ∑ ⌊ ⌋ ∑ ⌊ ⌋ * k k k + div(ω) = div(f ) − . x + x + x∞/ . 4 2 3 3 2 , x2 x3 x∞ - , Here the variables x2 x3 and x∞ run through the elliptic points of period 2, the elliptic points of period 3, and the cusps, respectively; all seen as points of X (Γ). It follows with Remark 8.37 that ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ k k k deg( div(f ) ) = k(д − 1) + ε + ε + ε∞. 4 2 3 3 2 Since k ≥ 2, we have ⌊k/4⌋ ≥ (k − 2)/4 and ⌊k/3⌋ ≥ (k − 2)/3. Together with Corollary 8.18, this implies ⌊ ⌋ deg( div(f ) ) > 2д − 2. Therefore, by Corollary 8.35, ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ k k k ℓ( div(f ) ) = (k − 1)(д − 1) + ε + ε + ε∞. 4 2 3 3 2 □ Corollary 8.39. Let k be an even integer. The following dimension formula holds: ⌊k/12⌋ if k ≥ 0 and k ≡ 2 (mod 12)  dim(M (SL (Z))) = ⌊k/12⌋ + 1 if k ≥ 0 and k . 2 (mod 12) . k 2  0 if k < 0

Proof. For k ≥ 2, Theorem 8.38 with Γ = SL (Z) gives 2 ⌊ ⌋ ⌊ ⌋ k k k dim(M (Γ)) = (1 − k) + + + . k 4 3 2 Now write k = 12q + r with 0 ≤ r ≤ 11 even, so that the formula becomes = + + ⌊ / ⌋ + ⌊ / ⌋ − / , dim(Mk (Γ)) 1 q r 4 r 3 r 2 and verify that the sum of the last three terms is 0 if r , 2 and −1 else. □

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