Rademacher Sums, Hecke Operators and Moonshine

Home , Hecke operator

RADEMACHER SUMS, HECKE OPERATORS AND MOONSHINE

PAUL BRUNO

Submitted in partial fulﬁllment of the requirements For the degree of Doctor of Philosophy

Thesis Advisor: Dr. John F. R. Duncan

Department of Mathematics, Applied Mathematics and Statistics

CASE WESTERN RESERVE UNIVERSITY

May 2016 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the thesis/dissertation of

Paul Bruno

candidate for the Doctor of Philosophy degree *

Committee Chair: Dr. John F. R. Duncan Dissertation Advisor Assistant Professor, Department of Mathematics, Applied Mathematics and Statistics Department of Mathematics and Computer Science, Emory University

Committee: Dr. David A. Singer Professor and Interim Chair, Department of Mathematics, Applied Mathematics and Statistics

Committee: Dr. Elisabeth M. Werner Professor, Department of Mathematics, Applied Mathematics and Statistics

Committee: Dr. Colin McLarty Truman P. Handy Professor of Philosophy, Department of Philosophy

May 2016

*We also certify that written approval has been obtained for any proprietary material contained therein. DEDICATION

For Margaret

iii Table of Contents

Dedication ...... iii Table of Contents ...... iv Acknowledgements ...... v Abstract ...... vi

1 Introduction and Background 1 1.1 Introduction and Overview of Results ...... 1 1.2 Modular Group ...... 3 1.2.1 Modular Functions and Lattice Functions ...... 5 1.2.2 Eisenstein Series ...... 9 1.2.3 Space of Modular Forms ...... 11 1.3 Hecke Operators ...... 13 1.4 Rademacher Sums ...... 17 1.5 Monstrous Moonshine ...... 20

2 Rademacher Sums 22 2.0 Matrices and Cosets ...... 22 2.1 Generalization of Rademacher Sums ...... 25 2.2 Coeﬃcients ...... 30

3 Rademacher Sums and Hecke Operators 43

4 Rademacher Sums, Hecke Operators and Moonshine 56

5 Conclusions 70 Bibliography ...... 72

iv ACKNOWLEDGEMENTS

I would like to thank my advisor, Dr. John F. R. Duncan, for his mentoring over the past four years. He has been a constant role model and a true friend. I would also like to thank the Department of Mathematics, Applied Mathematics and Statistics at Case Western Reserve University for their support and guidance. I am particularly thankful for Dr. David A. Singer, Dr. Elisabeth M. Werner and Dr. Colin McLarty for their service on my thesis defense committee. Thank you to my fellow graduate students for making our department a family. Thanks to Richard Lartey for putting up with my music in the oﬃce. Thank you to my friends and family for having faith in me throughout the years. Thank you to Margaret for being the perfect partner.

v Rademacher Sums, Hecke Operators and Moonshine

Abstract

PAUL BRUNO

In this thesis, we analyze the actions of Hecke operators on Rademacher sums. To investigate the problem, we consider the coeﬃcients of the Fourier expansion of a Rademacher sum. We derive explicit formulae for these coeﬃcients and prove a number of relationships between them. Using this analysis, we identify distinguished families of Hecke operators for a large class of discrete groups of isometries of the hyperbolic plane. These groups coincide with the groups appearing in monstrous moonshine.

vi Chapter 1

Introduction and Background

1.1 Introduction and Overview of Results

The theory of modular forms has been an important area of study for mathematicians for the past two centuries. One of the most important modular functions is the elliptic modular invariant, which defines a correspondence between isomorphism classes of complex elliptic curves and points on the complex plane. In 1938, Hans Rademacher derived an explicit formula for the coefficients of the Fourier series expansion of the modular invariant. Rademacher’s method was applied to certain groups other than the modular group by Knopp in the 1960s. Now, functions constructed in this way are called Rademacher sums. The primary goal of this thesis is to find the Hecke operators for certain subgroups of the modular group that generalize the following result from Duncan and Frenkel.

(−k) Let T (k) denote the kth Hecke operator for the modular group and let RΓ (z) denote the order k Rademacher sum associated to the modular group. (For full deﬁnitions,

1 see Sections 1.3 and 1.4.) Then,

1 T (k)R(−1) (z) = R(−k)(z). (1.1.1) Γ k Γ

We are able to deﬁne modiﬁed Hecke operators for positive integers k and N, denoted

Tb(k, N), for certain subgroups of the modular group that give the result

(−1) 1 (−k) Tb(k, N)R (z) = R (z) (1.1.2) Γ0(N) k Γ0(N) where R(−k) (z) is the order k Rademacher sum associated to a particular subgroup Γ0(N) of the modular group, denoted Γ0(N). A full description of these Hecke operators is found in Chapter 3. We apply these Hecke operators to more general groups in Chapter 4. The monstrous moonshine conjecture associates with each conjugacy class of the monster group a modular function. Thanks to the work of Duncan and Frenkel, we know that these modular functions are equal to normalized Rademacher sums. For instance, the modular function associated to the conjugacy class of the identity element of the monster group is equal to the normalized Rademacher sum associated to the modular group. Note that this particular Rademacher sum coincides with the elliptic modular invariant with constant term set equal to zero. This example illustrates some of the remarkable relationships between the monster group and modular functions. In particular, the coeﬃcients of the aforementioned normalized Rademacher sums turn out to be sums of the dimensions of irreducible representations of the monster group. Our analysis of Hecke operators and Rademacher sums has signiﬁcance for monstrous moonshine, as well. We show that the Hecke operators in equation (1.1.2) act

2 on the modular functions associated to certain conjugcay classes of the monster in the following way.

(−1) 1 (−k) Tb(k, N)R (z) = R (z) (1.1.3) Γg k Γg

For the remainder of Chapter 1, we give a survey of basic results in the theory of modular forms. Then, we briefly describe Hecke operators, Rademacher sums and monstrous moonshine. In Chapter 2, we generalize the Rademacher sums of [DF11] and derive formulae for their Fourier series coefficients. We will prove a number of relationships between these coefficients. In Chapter 3, we apply Hecke operators to Rademacher sums. We will see that for particular Hecke operators acting on Rademacher sums associated to certain discrete subgroups of the modular group, the result is another Rademacher sum. In Chapter 4, we investigate the implications of the results of the previous chapters with respect to monstrous moonshine.

1.2 Modular Group     a b    ˆ The group SL2 (R) =   | a, b, c, d ∈ R, ad − bc = 1 acts on C = C∪{∞}  c d    a b in the following way. For γ =   ∈ SL ( ) and z ∈ , set γ · z = az+b , and for   2 R C cz+d c d a z = ∞, set γ · z = c when c 6= 0, and γ · z = ∞ otherwise. Denote the upper half-plane by H = {z ∈ C|=(z) > 0} . We can verify that H is   a b   stable under the above action of SL2 (R) . Take γ =   ∈ SL2 (R) and z ∈ H. c d

3 Then,

az + b =(γ · z) = = cz + d 1 az + b az¯ + b = − 2i cz + d cz¯ + d (1.2.1) 1 z(ad − bc) − z¯(ad − bc) = |cz + d|2 2i =(z) = . |cz + d|2

Hence, if z ∈ H, then γ · z ∈ H. Denote

    a b    SL2 (Z) =   | a, b, c, d ∈ Z, ad − bc = 1 . (1.2.2)  c d 

We call this group the modular group. We now investigate properties of the modular group found in [Ser73]. Denote   0 −1   D = {z ∈ H | |z| ≥ 1, |<(z)| ≤ 1/2} . Denote S =   ∈ SL2 (Z) and 1 0   1 1   T =   ∈ SL2 (Z) . Then we have S · z = −1/z and T · z = z + 1. One can show 0 1 that SL2 (Z) = hS,T i and that D is a fundamental domain for the action of SL2 (Z) on H. Further, we can conclude that the stabilizer of any point in D is the identity except in the following cases. The stabilizer of z = i is the group of order 2 generated by S. The stabilizer of z = e(1/3) is the group of order 3 generated by ST and the stabilizer of z = e(1/6) is the group of order 3 generated by TS, where e(z) represent the quantity e2πiz.

4 1.2.1 Modular Functions and Lattice Functions

A complex-valued function is meromorphic on an open set if it is holomorphic everywhere in that set except for a set of isolated points, the poles of the function. That is, the function has no essential singularities. Every meromorphic function is expressible as a fraction of two holomorphic functions. Hence, the isolated points are the roots of the holomorphic function of the denominator. For example, the quotient of two polynomials is a meromorphic function so long as the denominator is not identically zero.

A function, f(z), is weakly modular of weight 2k if f(z) is meromorphic on H   a b   and satisﬁes the following condition for any γ ∈ SL2 (Z) . For γ =   , set c d jac(γ, z) = (cz + d)−2.

f(z) = jac (γ, z)k f (γ · z) (1.2.3)

Since SL2 (Z) is generated by S and T, we can rewrite the relationship of (1.2.3) with the following pair of equations.

f(z) = f(z + 1) (1.2.4) f(z) = z−2kf(−1/z)

The ﬁrst condition of (1.2.4) allows us to write weakly modular functions as functions of e(z). In particular, we can rewrite f(z) as a sum over n as follows.

∞ X f(z) = cne(nz) (1.2.5) −∞

5 A weakly modular function is simply a modular function if it is meromorphic at inﬁnity. In which case (1.2.5) becomes the following. For some n0 ∈ Z,

∞ X f(z) = cne(nz). (1.2.6)

n=n0

A modular function is a modular form if it is holomorphic everywhere, including at inﬁnity. Then, we can rewrite (1.2.6) with n0 = 0. In particular,

∞ X f(z) = cne(nz). (1.2.7) n=0

A modular form that vanishes at inﬁnity is called a cusp form. In this case we have c0 = 0 in (1.2.7). If f(z) is a modular form of weight 2k and f 0(z) is a modular form of weight 2k0, then the product ff 0(z) is a modular form of weight 2(k + k0). A holomorphic function on the upper half-plane h(z) is a weight zero mock modular form for Γ0(N) if it satisﬁes the following conditions [DMZ].

1. There exists a weight two modular form g(z) for Γ0(N), called the shadow of

∗ ∗ h(z), such that h(z) + g (z) is invariant under the action of Γ0(N) where g (z) solves

∂g∗(z) = −2πig(z). (1.2.8) ∂z¯

C=(z) 2. For γ ∈ SL2(Z), there exists a C > 0 such that h(γz) = O e as =(z) → ∞.

Note that a modular function is a mock modular form with shadow equal to zero.

6 Suppose ω1 and ω2 are nonzero elements of C such that = (ω1/ω2) > 0. Denote the lattice associated to ω1 and ω2 by

L (ω1, ω2) = {α1ω1 + α2ω2 | α1, α2 ∈ Z} . (1.2.9)

Denote the nonzero elements of L (ω1, ω2) by

0 L (ω1, ω2) = {z ∈ L (ω1, ω2) | z 6= 0} . (1.2.10)

0 0 In order that ω1 and ω2 deﬁne the same lattice as ω1 and ω2, it is necessary and suﬃcient that the pairs are congruent modulo SL2 (Z) .

Note that for a nonzero λ ∈ C, we have (λω1) / (λω2) = ω1/ω2. Then the map

(ω1, ω2) → ω1/ω2 is a bijection between the set of distinct lattices of C up to scale and the quotient H/SL2 (Z) . But for each lattice L, there is an associated elliptic curve, C/L. Moreover, two elliptic curves are isomorphic if and only if they are deﬁned by the same lattices up to scale. So we can think of H/SL2 (Z) as a parameter space for isomorphism classes of elliptic curves. Now consider F (L) to be a complex-valued function of lattices. Then, F is of weight 2k if for any nonzero λ ∈ C,

F (λL) = λ−2kF (L). (1.2.11)

For L = L (ω1, ω2) , we can write F (L) = F (ω1, ω2) , in which case (1.2.11) becomes

−2k F (λω1, λω2) = λ F (ω1, ω2) (1.2.12)

7 −1 Setting λ = ω2 , (1.2.12) becomes

−2k F (ω1, ω2) = ω2 F (ω1/ω2, 1) . (1.2.13)

Thus, the value of F up to scale depends only on ω1/ω2. Then, we can set f (ω1/ω2) =

F (ω1/ω2, 1) , and so we can write (1.2.13) as

−2k F (ω1, ω2) = ω2 f (ω1/ω2) . (1.2.14)

As we saw above, the pair (ω1, ω2) deﬁnes the same lattice under the action of the modular group. So we can conclude that F is invariant under the action of the   a b   modular group. Let γ =   ∈ SL2 (Z) . For z ∈ H, we can rewrite (1.2.14) as c d

f(z) = F (z, 1)

= F (az + b, cz + d) az + b (1.2.15) = (cz + d)−2kf cz + d = jac (γ, z)k f (γ · z) .

Therefore, we can identify modular functions with certain lattice functions of the same weight.

8 1.2.2 Eisenstein Series

For a lattice L and an integer k > 1, denote the Eisenstein series of weight 2k by

X 2k Gk(L) = 1/z . (1.2.16) z∈L0

This series converges absolutely. Since L is determined by some pair of ω1 and ω2, we can consider the series as a function of these complex values. That is,

X 1 Gk (ω1, ω2) = 2k . (mω1 + nω2) (1.2.17) m∈Z n∈Z (m,n)6=(0,0)

As we see in (1.2.15), we can equivalently write the series as a function of a single complex variable.

X 1 Gk(z) = (mz + n)2k (1.2.18) m∈Z n∈Z (m,n)6=(0,0)

Note that Gk(z) is a modular form of weight 2k. One can also compute the limit

∞ X 2k lim Gk(z) = 2 1/n =(z)→∞ n=1 (1.2.19) = 2ζ(2k)

where ζ denotes the Riemann zeta function. It will be useful to consider the following

power series representation for Gk(z), which we state without proof. Let σk(n) denote the divisor function, which is the sum of the kth powers of the positive divisors of a

P k positive integer n. That is, σk(n) = ` . `|n

9 ∞ X 2(2πi)2k G (z) = 2ζ(2k) + σ (n) e (nz) (1.2.20) k (2k − 1)! 2k−1 n=1

Eisenstein series can be used to construct cusp forms. For example, set g2(z) =

60G2(z) and g3 = 140G3(z). Using known values of the Riemann zeta function, we

4 6 3 2 have that g2 (∞) = 4π /3 and g3 (∞) = 8π /27. Thus, (g2 − 27g3)(∞) = 0. We also

3 2 know from properies of modular forms that g2(z) and g3(z) are both modular forms

3 2 of weight 12. Therefore, (g2 − 27g3)(z) is a cusp form of weight 12. This cusp form is denoted

3 2 ∆ = g2 − 27g3 . (1.2.21)

The function ∆ is called the modular discriminant. It also can be written using the following product formula.

∞ Y ∆ = (2π)12e(z) (1 − e(nz))24 (1.2.22) n=1

One can also see a connection between Eisenstein series and elliptic curves. Let

℘L(z) denote the Weierstrass elliptic function over the lattice L (ω1, ω2) . Explicitly, it is deﬁned by

1 X 1 1 ℘L(z) = + − . (1.2.23) z2 (z + mω + nω )2 (mω + nω )2 n2+m26=0 1 2 1 2

0 Set x = ℘L(z) and y = ℘L(z). Then, one can show that we have the cubic equation

2 3 y = 4x −g2x−g3, which is isomorphic to the elliptic curve C/SL2 (Z) . Additionally,

10 the discriminant of the curve equals the modular discriminant.

1.2.3 Space of Modular Forms

Denote the vector space of modular forms of weight 2k over the ﬁeld C by Mk.

0 Similarly, denote the space of cusp forms of weight 2k by Mk . Deﬁne the map φ :

0 Mk → C by φ(f) → lim f(z). Then, ker φ = Mk . Hence, the ﬁrst isomorphism =(z)→∞ 0 0 0 theorem implies that either Mk/Mk '{0}, or Mk/Mk ' C. Thus, Mk/Mk has dimension 0, or 1, respectively. As we saw in (1.2.19), Gk(z) does not vanish at

0 inﬁnity for k ≥ 2. Hence, Gk(z) ∈ Mk, but Gk(z) 6∈ Mk . Combining with the above dimension argument, when k ≥ 2,

0 (1.2.24) Mk = Mk ⊕ CGk(z).

Note that Mk is empty when k < 0 or k = 1.

Consider the linear map φ : f(z) → ∆f(z), where f(z) ∈ Mk. Clearly, ∆f(z) is

0 a cusp form of weight 2k + 12. So φ : Mk → Mk+6. Further, φ is bijective. That is,

0 0 Mk ' Mk+6. Thus, Mk is empty when k = 0, 2, 3, 4, 5.

0 0 As we have seen, Mk = Mk ⊕ CGk(z), which implies that dim Mk = dim Mk +

1. Applying the above bijection, dim Mk = dim Mk−6 + 1. One can calculate that dim Mk = 1 when k = 0, 2, 3, 4, 5, and dim M1 = 0. The following formula summarizes the discussion about dimension. For k ≥ 0,

  bk/6c if k ≡ 1 (mod 6) dim Mk = (1.2.25)  bk/6c + 1 if k 6≡ 1 (mod 6).

11 Now that we know the dimension of the space of modular forms for a particular

weight, we can give a basis for each. We need only to ﬁnd dim Mk linearly independent modular forms for a particular weight. Consider the set,

n α β o Bk = G2 G3 | α, β ∈ Z, α, β ≥ 0, 2α + 3β = k . (1.2.26)

An elementary counting argument shows that there are exactly dim Mk choices for

the pair (α, β) to satisfy the conditions of the deﬁnition of Bk, so it must be that

|Bk| = dim Mk. Further, the elements of Bk are linearly independent. Hence, Bk

forms a basis for Mk. We conclude our discussion of the space of modular forms with the modular invariant. It is deﬁned by

1728g3 j = 2 . (1.2.27) ∆

Note that the coeﬃcient 1728 = 123 is introduced so that j(z) has a residue equal to 1 as =(z) → ∞.

3 3 Since g2 = (60G2) and ∆ are both modular forms of weight 12, we can conclude that j must be a modular function of weight 0. In particular, one can show that ∆ is

nonzero in H, therefore we have that j is holomorphic in H. Further, we know that

3 g2 is nonzero at infinity, while ∆ has a simple zero at infinity. Hence, j has a simple pole at infinity.

3 For any λ ∈ C, set fλ = 1728g2 − λ∆. Then, we have that fλ ∈ M6, and that

fλ must have a unique zero in H/SL2 (Z) . On the other hand, for any point p ∈

H/SL2 (Z) , setting λ = j(p) gives fλ(p) = 0. Hence, we have a bijection between

H/SL2 (Z) and C. Considering compactiﬁcation of these spaces, the isomorphism

12 2 2 extends to H/SL\2 (Z) ' C ∪ {∞} . In particular, H/SL\2 (Z) ' S , where S denotes the unit sphere in three dimensions.

Let f(z) be a meromorphic function on H. Then, the following conditions are equivalent.

1. f is a rational function of j.

2. f is the quotient of two modular forms of the same weight.

3. f is a modular function of weight 0.

1.3 Hecke Operators

Let L denote a lattice in C. We say that L¯ is a sublattice of L when L¯ 6= ∅ and L¯ ⊆ L. Denote the index of L¯ in L, by L : L.¯ Then, for any positive integer k, we have the kth Hecke operator for the modular group given as a summation over sublattices by

X T (k)L = L.¯ (1.3.1) L:L¯=k

Note that we realize a sum of lattices in the context of the free abelian group generated

by the lattices in C. First, we check that the sum is finite. Since each L¯ contains kL, then it suffices to find the number of subgroups of L/kL of order k. But, that is

2 equivalent to ﬁnding the number of subgroups of (Zk) of order k, which is clearly ﬁnite. Hecke operators satisfy a number of convenient properties. For example, they

commute with lattice scaling. That is, for any nonzero λ ∈ C, denoting the scaling

13 operator by Rλ. Explicitly, we write RλL = λL. Then,

RλT (k)L = T (k)(λL) . (1.3.2)

Further, if k and k0 are positive integers with (k, k0) = 1, then

T (k)T (k0) L = T (kk0) L. (1.3.3) where (k, k0) represents the greatest common divisor of k and k0. The following relationship holds for a prime p.

T (pn) T (p)L = T pn+1 L + pT pn−1 (pL) (1.3.4)

It follows directly from equation (1.3.4) that T (pn) is a polynomial in T (p) and

Rλ, where λ can take the value of a power of p. Therefore, the T (p) and Rpn , generate a commutative algebra that contains T (k) for every positive integer k, when p is prime. Let F (L) be a complex-valued function of lattices of weight 2κ. Then, T (k) acts on F (L) by T (k)F (L) = F (T (k)L) . Combining (1.2.11) and (1.3.2), we have

RλT (k)F (L) = T (k)F (λL) (1.3.5) = λ−2κT (k)F (L).

Thus, T (k)F is also of weight 2κ. Equations (1.3.3) and (1.3.4) also extend to

14 functions of lattices. That is,

T (k)T (k0) F (L) = T (kk0) F (L), (1.3.6) when (k, k0) = 1, and if p is prime,

T (pn) T (p)F (L) = T pn+1 F (L) + pT pn−1 F (pL) . (1.3.7)

It is helpful to write Hecke operators as sums over sets of matrices instead of sums over sublattices. To do that, we state a bijection between a set of matrices of determinant k and the set of sublattices of a particular lattice of index k. Let Sk denote the following set of matrices.

    a b    Sk =   | ad = k, 0 ≤ b < d (1.3.8)  0 d 

  a b   For a lattice L = L (ω1, ω2) and for γ =   ∈ Sk, the map φ (γ) → 0 d

L (aω1 + bω2, dω2) is a bijection between Sk and the set of sublattices of L of index k. Now we can give the formula for a Hecke operator acting on a modular function f(z). In particular, we consider the formula when f(z) is of weight zero.

1 X az + b (T (k)f)(z) = f k d (1.3.9) ad=k 0≤b

As a result of the established bijection, equations (1.3.3) and (1.3.4) can be reformulated in terms of the action of Hecke operators on modular functions. That

15 is, when (k, k0) = 1,

(T (k)T (k0) f)(z) = (T (kk0) f)(z), (1.3.10)

and for a prime p, we have

1 (T (p)T (pn) f)(z) = T pn+1 f (z) + T pn−1 f (z). (1.3.11) p

Since the summation in (1.3.9) is ﬁnite, then (T (k)f)(z) must also be a modular function of weight zero. Also, if f(z) is a modular form or a cusp form, then so is (T (k)f)(z). So far, we have only considered Hecke operators for the modular group. We want to consider Hecke operators for certain discrete subgroups of the modular group. In

particular, we say that a Hecke operator for Γ0(N) is a Hecke operator of level N where Γ0(N) denotes the subgroup of SL2(Z) for which the lower left entries of the matrices are integer multiples of N. That is,

    a b    Γ0(N) =   ∈ SL2(Z) | c ≡ 0 (mod N) . (1.3.12)  c d 

Hecke operators generalize to level N. For a modular function f(z) for Γ0(N), the kth Hecke operator of level N acts on f(z) by

1 X az + b (T (k, N)f)(z) = f . k d (a,N)=1 (1.3.13) ad=k 0≤b

In Chapter 3, we deﬁne a modiﬁed Hecke operator for a positive integer `, which

16 divides (k, N). For a modular function f(z) for Γ0(N), we write,

1 X az + b (T (k, N)f)(z) = f . ` k d (a,N)=` (1.3.14) ad=k 0≤b

Then, we consider the action of the modiﬁed Hecke operators on particular Rademacher sums.

1.4 Rademacher Sums

In 1932, Hans Petersson proved an explicit formula for the coeﬃcients of the modular invariant,

X j(z) = e(−z) + 744 + cne(nz). (1.4.1) n≥1

In 1938, Hans Rademacher published an alternate proof in [Rad38] deriving the same formula for the coeﬃcients,

∞ k−1 ∞ √ 2m+1 2π X X X 1 −nh −h0 (−1)m 2π n c = √ e e , n n k k k m!(m + 1)! k (1.4.2) k=1 h=1 m=0 (h,k)=1 where hh0 ≡ −1 (mod k). One might recognize the right-most factors of the summands as the order 1 Bessel function. In [Kno90], Knopp analyzes the proof found in [Rad38], commenting that Rademacher writes j(z) as a modiﬁed Poincar´eseries.

17 Let Γ denote the modular group and set

    ±1 b    Γ∞ =   | b ∈ Z . (1.4.3)  0 ±1 

We denote the set of complete and irredundant representatives for the coset space

∗ ∗ Γ∞\Γ0(N) as Γ∞\Γ0(N)

    a b  Γ \Γ (N)∗ =   ∈ Γ (N) | 0 < c < L, −L2 < d < L2 . (1.4.4) ∞ 0

That is, we can write j(z) in terms of coset representation for the modular group.

X j(z) = e (−z) + lim e (−γ · z) − e (−γ · ∞) L→∞ (1.4.5) ∗ γ∈Γ∞\Γ

The modiﬁcation of the Poincar´eseries is the subtraction of the term e (−γ · ∞) , which is essential to guarantee convergence of the series. This representation has been generalized in the following way. For a positive integer m, the Rademacher sum of weight zero and order m associated to the modular group is given by

(−m) X RΓ (z) = e (−mz) + lim e (−mγ · z) − e (−mγ · ∞) . L→∞ (1.4.6) ∗ γ∈Γ∞\Γ

Clearly, the Rademacher sum of weight zero and order 1 associated to the modular group coincides with the modular invariant. In a series of papers published between 1961 and 1962, Knopp generalized the

18 Rademacher sum further by considering certain discrete subgroups of the modular group in place of Γ in (1.4.6). Much is known about Rademacher sums thanks to the analysis found in [DF11]. In particular, we know that any Rademacher sum associated to a discrete subgroup of

SL2(R) that is commensurable with the modular group converges locally uniformly in H, and we have explicit formulae for the coeﬃcients of their Fourier series expansions. Additionally, while they are modular forms under certain conditions, they are more generally mock modular forms. Mock modular forms of weight zero are classically known as abelian integrals. Duncan and Frenkel also give the relationship between the nth Hecke operator and the Rademacher sum of weight zero and order 1 associated to the modular group,

1 T (k)R(−1)(z) (z) = R(−k)(z). (1.4.7) Γ k Γ

In Chapter 3, we improve this result by generalizing it to the action of certain Hecke operators on Rademacher sums associated to certain discrete subgroups of the modular group. In Section 2.1, we consider Rademacher sums associated to more general sets of matrices. It will become apparent that these Rademacher sums are generally also mock modular forms. In Section 2.2, we state a formula for the coeﬃcients of the Fourier series expansion of the generalized Rademacher sums and derive some useful relationships between them.

19 1.5 Monstrous Moonshine

The monster group, denoted M, is the largest sporadic simple group. It has order approximately equal to 8 × 1053. Although the monster was conjectured to exist in the 1970s, it was not explicitly constructed, and thus proved to exist, until 1980. In the 1970s, a number of coincidences were observed as a result of the conjectured monster group. The ﬁrst such coincidence, now referred to the Jack Daniels problem,

was that the primes occuring in the factorization of the order of M coincide with the primes that Ogg found while considering the automorphism groups of certain algebraic curves [DGO]. Perhaps the most well-known of these coincidences is the observation that the coeﬃcients of j(z) − 744 are equal to sums of the dimensions of

the 194 irreducible representations of M. In [CN79], Conway and Norton gave the monstrous moonshine conjecture, which

associates a modular function and group to each conjugacy class of M that satisﬁes

the following condition. For g ∈ M, there is a subgroup of SL2 (R) , denoted Γg, and

modular function, denoted Tg(z), such that Tg is the unique Γg-invariant function on

H with Tg(z) = e(−z)+O(e(z)) as =(z) → ∞, and remains bounded as z approaches any non-inﬁnite cusp of Γg.

Thompson conjectured further that there is a graded infinite-dimensional M- module which can be written as an infinite direct sum of finite modules, each with dimension equal to one of the coefficients of j(z) − 744. Explicitly,

∞ M V = Vn, (1.5.1) n=−1

∞ P where dim (Vn) = cn for j(z) − 744 = cne(nz). This suggests that the coeﬃcients n=−1

20 of the modular functions arising from the monstrous moonshine conjecture are equal to the graded-traces of elements of the monster in the conjectured M-module. That is,

∞ X Tg(z) = tr (g|Vn) e(nz). (1.5.2) n=−1

This M-module was constructed by Frenkel et al. in the 1980s, thus conﬁrming Thompson’s theory. The M-module is generalized to form the moonshine tower, (see Section 7 of [DGO]) which inspires the results of Chapter 4. Duncan proved in [DF11] that the modular functions associated to the conjugacy classes of the monster are equal to order 1 Rademacher sums associated to the group

Γg. Therefore, we now turn our attention to Rademacher sums.

21 Chapter 2

Rademacher Sums

2.0 Matrices and Cosets

In this section we give descriptions of the sets of matrices and cosets that appear in this chapter. Let M(k, N) denote the matrices of determinant k for which the lower left entries of the matrices are integer multiples of N and the upper left entries are relatively prime to N. That is,

    a b    M(k, N) =   | ad − bc = k, (a, N) = 1, c ≡ 0 (mod N) . (2.0.1)  c d 

Similarly, set

    a b    M(k, N)∞ =   | ad = k, (a, N) = 1, b ∈ Z . (2.0.2)  0 d 

22 More generally, we denote

    a b    M`(k, N) =   | ad − bc = k, (a, N) = `, c ≡ 0 (mod N) . (2.0.3)  c d 

Note that M1(k, N) = M(k, N). As above, denote

    a b    M`(k, N)∞ =   | ad = k, (a, N) = `, b ∈ Z . (2.0.4)  0 d 

  a b For γ =  , we set γ · z = az+b and c (γ) = c. When c 6= 0, we set γ · ∞ = a .   cz+d c c d

It will be useful to consider M(k, N) as a ﬁnite union of cosets for Γ0(N).

Lemma 2.0.1. Let k and N be positive integers. Then,

    a b a 0 [   [   M(k, N) = Γ0(N)   = Γ0(N)   Γ0(N). (2.0.5) ad=k 0 d ad=k 0 d 0≤b

We also consider the relationship between M(k, N) and M`(k, N).

Lemma 2.0.2. Let `, k and N be positive integers such that `|(k, N). Then, there is a one-to-one correspondence between M(k/`, N/`) and M`(k, N). In particular, f : M`(k, N) 7→ M(k/`, N/`) deﬁned by

    a b a/` b     f   =   (2.0.6) c d c/` d is a bijective map.

23 The proofs of Lemmas 1.2.1 and 1.2.2 are simple computations, which we omit. We now provide a description of the coset spaces found in the remainder of this document. set

    ±1 b    Γ∞ =   | b ∈ Z . (2.0.7)  0 ±1 

A complete and irredundant set of representatives for the coset space Γ∞\M(k, N)∞ is given by

    a b      ∈ M(k, N) | 0 ≤ a, d ≤ k, 0 ≤ b ≤ d − 1 . (2.0.8)  0 d 

We denote the set of complete and irredundant representatives for the coset space

∗ ∗ Γ∞\Γ0(N) as Γ∞\Γ0(N)

    a b  Γ \Γ (N)∗ =   ∈ Γ (N) | 0 < c < L, −L2 < d < L2 . (2.0.9) ∞ 0

More generally, we have

    a b  Γ \M(k, N)∗ =   ∈ M(k, N) | 0 < c < L, −L2 < d < L2 . (2.0.10) ∞

We denote the set of complete and irredundant representatives for the coset space

Γ∞\Γ0(N) as Γ∞\Γ0(N)

24 Explicitly, we write the set of representatives for the coset space as

    a b    2 2 Γ∞\Γ0(N)

We denote the set of complete and irredundant representatives for the double

∗ ∗ coset space Γ∞\Γ0(N)/Γ∞ as [Γ∞\Γ0(N)/Γ∞]≤L . For each γ ∈ [Γ∞\Γ0(N)/Γ∞]≤L , we have 0 < c (γ) ≤ L. Explicitly, we write the set of representatives for the double coset space as

    a b  [Γ \Γ (N)/Γ ]∗ =   ∈ Γ (N) | 0 < c ≤ L, 0 ≤ a, d ≤ c − 1 . ∞ 0 ∞ ≤L   0  c d  (2.0.12)

More generally, we have

    a b  [Γ \M(k, N)/Γ ]∗ =   ∈ M(k, N) | 0 < c ≤ L, 0 ≤ a, d ≤ c − 1 . ∞ ∞ ≤L    c d  (2.0.13)

2.1 Generalization of Rademacher Sums

In this section we will generalize Rademacher sums found in [DF11]. Let m, k and N, be positive integers. Then, the Rademacher sum of weight zero and order m

25 associated to M`(k, N) is given by

(−m) X R (z) = e (−mγ · z) M`(k,N) γ∈Γ∞\M (k,N)∞ ` (2.1.1) X + lim e (−mγ · z) − e (−mγ · ∞) L→∞ ∗ γ∈Γ∞\M`(k,N)

Rademacher sums associated to M`(k, N).

Proposition 2.1.1. Let `, m, k and N be positive integers such that `|(k, N). Then,

R(−m) (z) = R(−m) (`z). (2.1.2) M`(k,N) M(k/`,N/`)

  a b   Proof. Denote γ =   . c d

R(−m) (z) = M`(k,N) X X e (−mγ · z) + lim e (−mγ · z) − e (−mγ · ∞) L→∞ ∗ γ∈Γ∞\M`(k,N)∞ γ∈Γ∞\M`(k,N)

26 By Lemma 2.0.2, M(k, N) and M`(k/`, N/`) are in one-to-one correspondence. Thus,

(−m) X a(`z) + b R (z) = e −m M`(k,N) d Γ∞\M(k/`,N/`)∞ X a(`z) + b a + lim e −m − e −m L→∞ ∗ c(`z) + d c Γ∞\M(k/`,N/`)

(−m) = RM(k/`,N/`)(`z). (2.1.4)

With Proposition 2.1.1 in mind, we direct our attention toward Rademacher sums associated to M(k, N).

Proposition 2.1.2. The Rademacher sum associated to M(k, N) converges locally uniformly to a holomorphic function on the upper half-plane.

Using Lemma 2.0.1, we can rewrite M(k, N) as a ﬁnite union of cosets for Γ0(N). Then, Proposition 2.1.2 follows directly from Theorem 3.3.2 of [DF11]. Hence, we omit the proof.

Lemma 2.1.3. Let m, k and N be positive integers. The Rademacher sum of weight

27 zero and order m associated to M(k, N) has the representation

(−m) X az X X az + b a R (z) = de −m + lim e −m − e −m M(k,N) d L→∞ cz + d c (a,N)=1 0

  P P az+b a e (−kz) + lim e − − e − (k, N) = 1  L→∞ cz+d c  0

Proof. It suﬃces to evaluate the ﬁrst sum that appears in equation (2.1.1).

X X az + b e (−mγ · z) = e −m d γ∈Γ∞\M(k,N)∞ (a,N)=1 ad=k 0≤b≤d−1 (2.1.7) X −maz X −mb = e e d d (a,N)=1 0≤b≤d−1 ad=k

We notice the summation over values of b evaluates to 0 unless d|m, in which case

28 the sum evaluates to d. Then,

X X az e (−mγ · z) = de −m . d (2.1.8) γ∈Γ∞\M(k,N)∞ (a,N)=1 d|(m,k) ad=k

Setting m = 1, we must have d = 1. Then we have a = k unless (k, N) 6= 1, in which case the sum has no terms. Hence,

  X e(−kz)(k, N) = 1 e (−γ · z) = (2.1.9)  γ∈Γ∞\M(k,N)∞ 0 (k, N) 6= 1.

We now consider the action of Γ (N) on R(−m) (z). That is, we investigate the 0 Γ0(N) diﬀerence between R(−m) (z) and R(−m) (σz) for σ ∈ Γ (N). It will develop that Γ0(N) Γ0(N) 0

Rademacher sums associated to M(k, N) are mock modular forms for Γ0(N). Let m and N be positive integers. Recall the Poincar´eseries of weight two and order m associated to Γ0(N), given by

(m) X P (z) = lim e(mγ · z)jac(γ, z) Γ0(N) L→∞ (2.1.10) γ∈Γ∞\Γ0(N)

Proposition 2.1.4. Let m and N be positive integers. The Poincaréseries of weight two and order m associated to Γ0(N) converges to a holomorphic function on the upper half-plane. Further, the Poincaréseries defines a weight two cusp form for

29 Γ0(N).

Proposition 2.1.5. Let m and N be positive integers. For σ ∈ Γ0(N),

∞ Z R(−m) (σz) − R(−m) (z) = −2πi (−m)P (m) (z)dz. (2.1.11) Γ0(N) Γ0(N) Γ0(N) σ−1·∞

Thus, R(−m) (z) is a weight zero mock modular form for Γ (N) with shadow equal Γ0(N) 0 to (−m)P (m) (z). Γ0(N)

Proof. Equation (2.1.12) follows directly from equation (3.4.14) of [DF11]. By Propo- sition 2.1.4, (−m)P (m) (z) is a weight two modular form for Γ (N). By Proposition Γ0(N) 0 2.1.2, R(−m) (z) is holomorphic on the upper half-plane and its construction allows Γ0(N) for, at most, exponential growth as =(z) → ∞.

2.2 Coeﬃcients

The coefficient functions in this section will allow for a Fourier series expansion of Rademacher sums. We begin with a description of the coefficient functions, which results in the convenient representation of Lemma 2.2.2. We generalize the coefficient functions of [DF11] for positive integers k and N as follows. Let m be an integer. We introduce the weight zero coefficient function associated to M(k, N) as

cM(k,N)(m, j) = s X X −4π2k ms+1 js (2.2.1) lim e (mγ · ∞) e −jγ−1 · ∞ L→∞ 2 (s + 1)! s! ∗ s≥0 c (γ) γ∈[Γ∞\M(k,N)/Γ∞]≤L

30 where the sum is over complete and irredundant sets of coset representatives for the

double coset space Γ∞\M(k, N)/Γ∞.

As before, we can rewrite M(k, N) as a ﬁnite union of cosets for Γ0(N). Then, Proposition 2.2.1 follows directly from Proposition 3.2.8 of [DF11]. Hence, it is stated here without proof.

Proposition 2.2.1. The weight zero coeﬃcient function associated to M(k, N) converges.

We now derive explicit formulae for these coeﬃcient functions. Recall that

∗ [Γ∞\M(k, N)/Γ∞]≤L is given explicitly by

    a b      ∈ M(k, N) | 0 < c ≤ L, 0 ≤ a, d ≤ c − 1 . (2.2.2)  c d 

For ﬁxed integers a and c such that c > 0, c ≡ 0(mod N) and 0 ≤ a ≤ c − 1, set

a c 0 −1 a¯ = (a,c) , c¯ = (a,c) anda ¯ =a ¯ (modc ¯). From the condition ad − bc = k above on

the representatives for Γ∞\M(k, N)/Γ∞, we have ad ≡ k (mod c). So we must have

ka¯0 d ≡ (a,c) (modc ¯). In particular, for some integer z0, if we restrict d to be in the set

ka¯0 + zc¯ | z = z , z + 1, ..., z + (a, c) − 1 , (2.2.3) (a, c) 0 0 0

then we have every value of d for which 0 ≤ d ≤ c − 1 and for which there is a unique b ∈ Z so that ad − bc = k.

Lemma 2.2.2. The weight zero coeﬃcient function associated to M(k, N) has the

31 representation

cM(k,N)(m, j) =

0 L c−1 (a,c)−1 ka¯ ! 2 s+1 s+1 s X X X X a (a,c) + zc¯ −4π k m j lim e m e j L→∞ c c c2 (s + 1)! s! c=1 a=0 s≥0 z=0 c≡0(N) (a,c)|k (a,N)=1 (2.2.4) where a¯0 and c¯ are as above. In particular, if k = 1, then

L c−1 s+1 X X X a a0 −4π2 ms+1 js cΓ (N)(m, j) = lim e m e j 0 L→∞ c c c2 (s + 1)! s! (2.2.5) c=1 a=0 s≥0 c≡0(N) (a,c)=1

Proof. The construction of a complete and irredundant set of representatives for the corresponding double coset space follows Proposition 2.2.1.

For integers m, j and c, with c > 0, recall the Kloosterman sum

c−1 X a a0 K(m, j; c) = e m e j c c (2.2.6) a=0 (a,c)=1 where a0 ≡ a−1 (mod c). Kloosterman sums will allow for a more concise representation of our coeﬃcient functions. The following properties of Kloosterman sums will be useful in later proofs. First, we can rearrange our inputs using K(m, j; c) = K(j, m; c) and K(m, j; c) = K(−m, −j; c). We also have the relationship

X mj c K(m, j; c) = `K , 1; `2 ` (2.2.7) `|(m,j,c)

32 which is known as the Selberg Identity [Ma]. Now we establish the relationship between the coeﬃcient function associated to

M(k, N) and the coeﬃcient function associated to Γ0(N).

Proposition 2.2.3. Let j ∈ Z. For positive integers k and N,

X k kj c (m, j) = c m, . M(k,N) ` Γ0(N) `2 (2.2.8) `|(k,j) (`,N)=1

In particular, if j = 0, m = −1 and N = 1, then

(2.2.9) cM(k,1)(−1, 0) = σ1(k)cΓ0(1)(−1, 0).

Proof. For integers a and c, we denotea, ¯ c¯ anda ¯0 as in Lemma 2.2.2.

cM(k,N)(m, j) =

0 L c−1 (a,c)−1 ka¯ ! 2 s+1 s+1 s X X X X a (a,c) + zc¯ −4π k m j lim e m e j L→∞ c c c2 (s + 1)! s! c=1 a=0 s≥0 z=0 c≡0(N) (a,c)|k (a,N)=1 (2.2.10)

When c ≡ 0(mod N), we see that (a, N) = 1 and ((a, c),N) = 1 are equivalent conditions. Furthermore, this givesc ¯ ≡ 0(mod N). Then, since (a, c) must divide k, we introduce a sum over the divisors of k and set ` = (a, c).

33 cM(k,N)(m, j) =

L c¯−1 `−1 X X X X X a¯ ka¯0 z lim e m e j e j L→∞ c¯ `2c¯ ` (2.2.11) `|k c¯=1 a¯=0 s≥0 z=0 (`,N)=1 c¯≡0(N) (¯a,c¯)=1 s 2 s+1 s+1 jk −4π k m 2 · ` c¯2 `2 (s + 1)! s!

Now we notice we can compute the sum over z.

 `−1  X z ` `|j e j = (2.2.12) `  z=0 0 `6 |j

So we may replace this sum with a factor of ` so long as we restrict ` to divide (k, j).

cM(k,N)(m, j) = s L c¯−1 0 2 s+1 s+1 jk X k X X X a¯ jk a¯ −4π m 2 lim e m e ` L→∞ ` c¯ `2 c¯ c¯2 (s + 1)! s! `|(k,j) c¯=1 a¯=0 s≥0 (2.2.13) (`,N)=1 c¯≡0(N) (¯a,c¯)=1 X k kj = c m, ` Γ0(N) `2 `|(k,j) (`,N)=1

34 Setting j = 0, m = −1 and N = 1, we have

X k c (−1, 0) = c (−1, 0) M(k,1) ` Γ0(1) `|k (2.2.14)

= σ1(k)cΓ0(1)(−1, 0).

The next proposition considers the eﬀect of rearranging the inputs of a coeﬃcient

function associated to Γ0(N).

Proposition 2.2.4. Let j ∈ Z. For positive integers k and N,

X X X k k cΓ0(N)(−k, j)e(jz) = cΓ N −1, j e(`jz). (2.2.15) ` 0( (`,N) ) ` j>0 j>0 `|k

Proof. We can use Kloosterman sums to write our coeﬃcient function as follows.

L c−1 s+1 X X X a a0 −4π2 (−k)s+1 js cΓ (N)(−k, j) = lim e −k e j 0 L→∞ c c c2 (s + 1)! s! c=1 a=0 s≥0 c≡0(N) (a,c)=1 L s+1 s+1 X X −4π2 (−1)s+1 (k) js = lim K(−k, j; c) L→∞ c2 (s + 1)! s! c=1 s≥0 c≡0(N) (2.2.16)

Now we apply the Selberg Identity, and the two other properties of Kloosterman sums preceding equation (2.2.7).

35 L s+1 s+1 X X X −kj c −4π2 (−1)s+1 (k) js cΓ (N)(−k, j) = lim `K , 1; 0 L→∞ `2 ` c2 (s + 1)! s! c=1 s≥0 `|(k,j,c) c≡0(N) L s+1 s+1 X X X kj c −4π2 (−1)s+1 (k) js = lim `K −1, ; L→∞ `2 ` c2 (s + 1)! s! c=1 s≥0 `|(k,j,c) c≡0(N) (2.2.17)

Rearranging terms,

L s+1 X X X kj c −4π2 (−1)s+1 (kj)s cΓ (N)(−k, j) = lim k`K −1, ; 0 L→∞ `2 ` c2 (s + 1)! s! c=1 s≥0 `|(k,j,c) c≡0(N) c L −1 s+1 X X X` X −a kj a0 −4π2 (−1)s+1 (kj)s = lim k`e e L→∞ c `2 c c2 (s + 1)! s! c=1 `|(k,j,c) a=0 s≥0 ` ` c≡0(N) c (a, ` )=1 (2.2.18)

0 −1 c where a ≡ a (mod ` ). Then, we introduce the sum over values of j and rearrange terms.

36 X cΓ0(N)(−k, j)e(jz) j>0 c L −1 X X X X` X −a kj a0 = lim k`e e L→∞ c `2 c j>0 c=1 `|(k,j,c) a=0 s≥0 ` ` c≡0(N) c (a, ` )=1 −4π2 s+1 (−1)s+1 (kj)s · e(jz) (2.2.19) c2 (s + 1)! s! c L −1 X X X X` X k −a k j a0 = lim e e L→∞ ` c ` ` c j>0 c=1 `|(k,j,c) a=0 s≥0 ` ` c≡0(N) c (a, ` )=1 !s+1 s −4π2 (−1)s+1 k j · ` ` e(jz) c 2 (s + 1)! s! `

`N For each divisor ` of k, we restrict c ≡ 0(mod (`,N) ) so that `|c and replace j with `j in the summation. Then we can rewrite the sum over the divisors of (k, j, c) as a sum over the divisors of k.

X cΓ0(N)(−k, j)e(jz) j>0 c L −1 X X X X` X k −a k a0 = lim e e j L→∞ ` c ` c (2.2.20) j>0 `|k c=1 a=0 s≥0 ` ` `N c c≡0( (`,N) ) (a, ` )=1 !s+1 s −4π2 (−1)s+1 k j · ` e(`jz) c 2 (s + 1)! s! `

c N Now we replace ` with c in the summation and restrict c ≡ 0(mod (`,N) ) to adjust accordingly.

37 X cΓ0(N)(−k, j)e(jz) j>0 L c−1 X X X X X k −a k a0 = lim e e j L→∞ ` c ` c j>0 `|k c=1 a=0 s≥0 N (a,c)=1 c≡0( (`,N) ) (2.2.21) s −4π2 s+1 (−1)s+1 k j · ` e(`jz) c2 (s + 1)! s! X X k k = cΓ N −1, j e(`jz) ` 0( (`,N) ) ` j>0 `|k

In the remaining propositions of this section, we present relationships between the coeﬃcient function associated to M(k, N) and the coeﬃcient function associated to

Γ0(N).

Proposition 2.2.5. Let m, k and N be positive integers such that (k, N) = 1 and (m, k) = 1. Then,

X X cM(k,N)(−m, j)e(jz) = cΓ0(N)(−mk, j)e(jz). (2.2.22) j>0 j>0

Proof. Applying Proposition 2.2.4 to the right-hand side of the equation we have,

X X X mk mkj cΓ0(N)(−mk, j)e(jz) = cΓ N −1, e(`jz). (2.2.23) ` 0( (`,N) ) ` j>0 j>0 `|mk

38 Applying Proposition 2.2.3 to the left-hand side of the equation we have,

X X X k kj c (−m, j)e(jz) = c −m, e(jz). M(k,N) ` Γ0(N) `2 (2.2.24) j>0 j>0 `|(k,j) (`,N)=1

Since we assume (k, N) = 1, we can eliminate the condition (`, N) = 1 on the sum over the divisors of (k, j).

X X X k k c (−m, j)e(jz) = c −m, j e(jz) M(k,N) ` Γ0(N) `2 (2.2.25) j>0 j>0 `|(k,j)

j For each divisor `, replace j with ` so we can rearrange the sum again to consider only the divisors of k.

X X X k k c (−m, j)e(jz) = c −m, j e(`jz) M(k,N) ` Γ0(N) ` (2.2.26) j>0 j>0 `|k

Now we apply Proposition 2.2.4 to the resulting sum.

X X X X k m k m cM(k,N)(−m, j)e(jz) = cΓ N −1, j e(d`jz) (2.2.27) ` d 0( (d,N) ) ` d j>0 j>0 `|k d|m

Since we assume (k, N) = 1 and (m, k) = 1, we may combine the sums over divisors of m and k into a single sum over the divisors of mk.

39 X X X mk mk cM(k,N)(−m, j)e(jz) = cΓ N −1, j e(`jz) (2.2.28) ` 0( (`,N) ) ` j>0 j>0 `|mk

Therefore,

X X cM(k,N)(−m, j)e(jz) = cΓ0(N)(−mk, j)e(jz). (2.2.29) j>0 j>0

Proposition 2.2.6. Let m, N and r be positive integers and let p be a prime such that p|N and p6 |m. Then,

X X r X r−1 r cM(p ,N)(−m, j)e(jz) = cΓ0(N) (−mp , j) e(jz) − c N −mp , j e(pjz). Γ0( p ) j>0 j>0 j>0 (2.2.30)

Proof. Applying Proposition 2.2.4 to the sums of the right-hand side of the equation we have the following relationships.

r r X r X X mp mp j cΓ0(N) (−mp , j) e(jz) = cΓ N −1, e(`jz) ` 0( (`,N) ) ` j>0 j>0 `|mpr r−1 r−1 X r−1 X X mp mp j cΓ N −mp , j e(pjz) = cΓ N/p −1, e(`pjz) 0( p ) ` 0( (`,N/p) ) ` j>0 j>0 `|mpr−1 (2.2.31)

Then, we can subtract these sums.

40 X r X r−1 cΓ0(N) (−mp , j) e(jz) − c N −mp , j e(pjz) Γ0( p ) j>0 j>0 X X mpr mprj = cΓ N −1, e(`jz) ` 0( (`,N) ) ` (2.2.32) j>0 `|mpr X X mpr−1 mpr−1j − cΓ N/p −1, e(`pjz) ` 0( (`,N/p) ) ` j>0 `|mpr−1

Since we assume p6 |m, we can split the sums over the divisors of mpr and the divisors of mpr−1 into two sums each.

X r X r−1 cΓ0(N) (−mp , j) e(jz) − c N −mp , j e(pjz) Γ0( p ) j>0 j>0 X X X mpr mprj = cΓ N −1, e(`djz) `d 0( (`d,N) ) `d (2.2.33) j>0 `|m d|pr X X X mpr−1 mpr−1j − cΓ N/p −1, e(`dpjz) `d 0( (`d,N/p ) `d j>0 `|m d|pr−1

We have cancellation of terms between the sums over the divisors of pr and the divisors of pr−1 except in the case that d = 1 in the ﬁrst sum.

X r X r−1 cΓ0(N) (−mp , j) e(jz) − c N −mp , j e(pjz) Γ0( p ) j>0 j>0 (2.2.34) X X mpr mprj = cΓ N −1, e(`jz) ` 0( (`,N) ) ` j>0 `|m

41 Applying Proposition 2.2.3 to the left-hand side of the equation we have,

X X X pr prj c r (−m, j)e(jz) = c −m, e(jz). M(p ,N) ` Γ0(N) `2 (2.2.35) j>0 j>0 `|(pr,j) (`,N)=1

Since we assume p|N, the sum over the divisors of (pr, j) has a single term corresponding to the case ` = 1.

X X r r r cM(p ,N)(−m, j)e(jz) = p cΓ0(N) (−m, p j) e(jz) (2.2.36) j>0 j>0

Now we apply Proposition 2.2.4 to the resulting sum.

X X X mpr mpr cM(pr,N)(−m, j)e(jz) = cΓ N −1, j e(`jz) (2.2.37) ` 0( (`,N) ) ` j>0 j>0 `|m

Therefore,

X X r X r−1 r cM(p ,N)(−m, j)e(jz) = cΓ0(N) (−mp , j) e(jz) − c N −mp , j e(pjz). Γ0( p ) j>0 j>0 j>0 (2.2.38)

42 Chapter 3

Rademacher Sums and Hecke Operators

We now determine the connection between Hecke Operators and Rademacher sums associated to Γ0(N).

Lemma 3.0.1. Let m and k be positive integers. Then,

1 T (k, 1)R(−m) (z) = R(−m) (z). (3.0.1) Γ0(1) k M(k,1)

43 Proof. We have

T (k, 1)R(−m) (z) Γ0(1) 1 X (−m) az + b = R k Γ0(1) d ad=k 0≤b

1 X (−m) = R (σz) k Γ0(1) (3.0.2) σ∈Γ0(1)\M(k,1) 1 X = e (−mσ · z) k σ∈Γ∞\M(k,1)∞ 1 X X + lim e (−mγσ · z) − e (−mγ · ∞) L→∞ k ∗ σ∈Γ0(1)\M(k,1) γ∈Γ0(1)

∗ ∗ space Γ0(1)\M(k, 1), and γ ∈ Γ0(1)

T (k, 1)R(−m) (z) Γ0(1) 1 X = e (−mσ · z) k σ∈Γ∞\M(k,1)∞ 1 X X + lim e (−mγσ · z) − e (−mγσ · ∞) L→∞ (3.0.3) k ∗ σ∈Γ0(1)\M(k,1) γ∈Γ0(1)

44 We generalize the Hecke operator as follows.

Deﬁnition 3.0.2. Let `, k and N be positive integers such that `|(k, N). For a

th modular function f(z) for Γ0(N), the `-modiﬁed k Hecke operator of level N acts on f(z) by

1 X az + b (T (k, N)f)(z) = f . ` k d (a,N)=` (3.0.4) ad=k 0≤b

We deﬁne the total kth Hecke operator of level N by

X Tb(k, N)f (z) = (T`(k, N)f)(z). (3.0.5) `|(k,N)

Corollary 3.0.3. Let `, m, k and N be positive integers. Then,

(−m) 1 (−m) T`(k, N)R (z) = R (z) Γ0(N) M`(k,N) k (3.0.6) 1 = R(−m) (`z). k M(k/`,N/`)

Proof. The ﬁrst equality follows directly from Lemma 3.0.1. Proceed as in the proof above, carrying the congruency conditions with respect to N and ` throughout the summations. The second equality follows from Proposition 2.1.1.

We note that Corollary 3.0.3 holds after change of variable with respect to z. In particular,

(−m) 1 (−m) T`(k, N)R (αz + β) = R (`αz + `β) (3.0.7) Γ0(N) k M(k/`,N/`)

45 Our primary goal is to apply a Hecke operator to a Rademacher sum associated to Γ0(N) to obtain one or more Rademacher sums associated to Γ0(N) for various values of N. We ﬁrst formulate a result to represent a Rademacher sum associated to M(k, N) in terms of the coeﬃcient function. This theorem will require the use of the following formulae known as the Lipschitz summation formulae. These hold for <(s) > 1 and =(z) > 0.

X −1 X −1 −1 e(nz) = + lim (−2πi) (z + n) (3.0.8) 2 L→∞ n>0 −L

X ns−1 X e(nz) = (−2πi)−s(z + n)−s (3.0.9) (s − 1)! n>0 n∈Z

Proposition 3.0.4. Let m, k and N be positive integers. Then,

(−m) X az 1 X R (z) = de −m + c (−m, 0) + c (−m, j)e(jz). M(k,N) d 2 M(k,N) M(k,N) (a,N)=1 j>0 d|(m,k) ad=k (3.0.10)

In particular, if m = 1, then,

  1 P e (−kz) + 2 cM(k,N)(−1, 0) + cM(k,N)(−1, j)e(jz)(k, N) = 1 (−1)  j>0 RM(k,N)(z) =  1 P  2 cM(k,N)(−1, 0) + cM(k,N)(−1, j)e(jz)(k, N) 6= 1.  j>0 (3.0.11)

46 Proof. Recall Lemma 2.1.3.

(−m) X az X X az + b a R (z) = de −m + lim e −m − e −m M(k,N) d L→∞ cz + d c (a,N)=1 0

Thus, it suﬃces to examine the sum

X X az + b a lim e −m − e −m . L→∞ cz + d c (3.0.13) 0

X X az + b a lim e −m − e −m L→∞ cz + d c 0

We can consider rewriting this to be a sum over a complete and irredundant set of representatives for the double coset space Γ∞\M(k, N)/Γ∞. In this case, we can restrict 0 ≤ d ≤ c − 1. To compensate for the restriction on values for d, we introduce the sum over s below.

47 We can also rewrite the sum over j by separating the ﬁrst term. When j = 0, we have

−1 X X X a −2πik d + sc lim e −m (−m) z + L→∞ c c2 c 0

  −1 X X a −2πik X d 2 lim e −m (−m)  z + + s + O(c/L ) L→∞ c c2 c 00 c≡0(N) (c,d)|k (3.0.16)

48 Hence, the error term tends toward zero in the limit. Thus, we have

! X X a −4π2k 1 X d lim e −m (−m) + e (sz) e s L→∞ c c2 2 c 00 c≡0(N) (c,d)|k X 1 a −4π2k = lim e −m (−m) L→∞ 2 ∗ 2 c c γ∈[Γ∞\M(k,N)/Γ∞]≤L X d a −4π2k (3.0.17) + e(sz)e s e −m (−m) c c c2 s>0 1 X X d a = cM(k,N)(−m, 0) + lim e(sz)e s e −m 2 L→∞ c c ∗ s>0 γ∈[Γ∞\M(k,N)/Γ∞]≤L −4π2k · (−m). c2

We now deal with the remaining terms of the sum. As before, we will rewrite the sum in terms of a complete and irredundant set of representatives for the double coset space Γ∞\M(k, N)/Γ∞ before applying the second Lipschitz summation formula. We notice the sum is absolutely convergent, so we let s be any integer.

j+1 −j−1 X a X −2πik (−m)j+1 X d lim e −m z + + s L→∞ c c2 (j + 1)! c ∗ j≥1 γ∈[Γ∞\M(k,N)/Γ∞]≤L s∈Z j+1 X X X a d −4π2k (−m)j+1 sj = lim e −m e s e(sz) L→∞ c c c2 (j + 1)! j! ∗ j≥1 s>0 γ∈[Γ∞\M(k,N)/Γ∞]≤L (3.0.18)

49 Combining terms, we have

(−m) X az 1 X R (z) = de −m + c (−m, 0) + c (−m, j)e(jz). M(k,N) d 2 M(k,N) M(k,N) (a,N)=1 j>0 d|(m,k) ad=k (3.0.19)

The following lemmas will be used to prove Theorem 3.0.7.

Lemma 3.0.5. Let m, k and N be positive integers such that (k, N) = 1 and (m, k) = 1. Then,

1 T (k, N)R(−m) (z) = R(−mk)(z). (3.0.20) Γ0(N) k Γ0(N)

Proof. Applying Corollary 3.0.3 to the left-hand side of the equation,

1 T (k, N)R(−m) (z) = R(−m) (z). (3.0.21) Γ0(N) k M(k,N)

Now applying Proposition 3.0.4,

T (k, N)R(−m) (z) = Γ0(N)     (3.0.22) 1  X az 1 X   de −m + cM(k,N)(−m, 0) + cM(k,N)(−m, j)e(jz) . k  d 2  (a,N)=1 j>0  d|(m,k) ad=k

Since we assume (m, k) = 1, we must have d = 1 and a = k in the ﬁrst summation.

50 Then,

! (−m) 1 1 X T (k, N)R (z) = e (−mkz) + cM(k,N)(−m, 0) + cM(k,N)(−m, j)e(jz) . Γ0(N) k 2 j>0 (3.0.23)

Now we can apply Proposition 2.2.5 directly. As in the proof of the Proposition

2.2.5, we have the relationship cM(k,N)(−m, 0) = cΓ0(N)(−mk, 0). Hence,

! (−m) 1 1 X T (k, N)R (z) = e (−mkz) + cΓ (N)(−mk, 0) + cΓ (N)(−mk, j)e(jz) Γ0(N) k 2 0 0 j>0 1 = R(−mk)(z). k Γ0(N) (3.0.24)

Lemma 3.0.6. Let m, N and r be positive integers and let p be a prime such that p|N and p6 |m. Then,

r−1 1 r (−mp ) T (pr,N) R(−m) (z) = R(−mp )(z) − R (pz) . (3.0.25) Γ0(N) r Γ0(N) N p Γ0( p )

Proof. Applying Corollary 3.0.3 to the left-hand side of the equation,

r (−m) 1 (−m) T (p ,N) R (z) = R r (z). (3.0.26) Γ0(N) pr M(p ,N)

51 Now applying Proposition 3.0.4,

T (pr,N) R(−m) (z) = Γ0(N)     (3.0.27) 1  X az 1 X   de −m + cM(pr,N)(−m, 0) + cM(pr,N)(−m, j)e(jz) . pr  d 2  (a,N)=1 j>0  d|(pr,m) ad=pr

Since we assume p6 |m, we must have d = 1 in the ﬁrst summation. This forces a = pr, however, the condition (a, N) = 1 eliminates that possibility. So the sum has no terms and hence evaluates to zero. Then,

! r (−m) 1 1 X T (p ,N) R (z) = cM(pr,N)(−m, 0) + cM(pr,N)(−m, j)e(jz) . Γ0(N) pr 2 j>0 (3.0.28)

Now we can apply Proposition 2.2.6 directly. As in the proof of the Proposition

r r−1 r 2.2.6, we have the relationship cM(p ,N)(−m, 0) = cΓ0(N) (−mp , 0)−c N (−mp , 0) . Γ0( p ) Hence,

! r (−m) 1 1 r X r T (p ,N) R (z) = cΓ (N) (−mp , 0) + cΓ (N) (−mp , j) e(jz) Γ0(N) pr 2 0 0 j>0 ! 1 1 r−1 X r−1 − cΓ N −mp , 0 + cΓ N −mp , j e(pjz) pr 2 0( p ) 0( p ) j>0

r−1 1 r (−mp ) = R(−mp )(z) − R (pz) . r Γ0(N) N p Γ0( p ) (3.0.29)

52 For the next theorem, we need to recall the Moebius function. For a positive integer n, the Moebius function is given by

  1 if n is square-free and has an even number of prime factors   µ(n) = −1 if n is square-free and has an odd number of prime factors (3.0.30)    0 if n has a squared prime factor.

Theorem 3.0.7. Let m, k and N be positive integers such that (m, k) = 1. Then,

k (−m) 1 X (−m ) T (k, N)R (z) = µ(`)R ` (`z). Γ0(N) N (3.0.31) k Γ0( ` ) `|(k,N)

In particular, if (k, N) = 1, then,

1 T (k, N)R(−m) (z) = R(−mk)(z). (3.0.32) Γ0(N) k Γ0(N)

Proof. Suppose for positive integers, r1, r2, . . . rl, and primes p1, p2, . . . pl, that k has

r1 r2 rl the prime factorization k = p1 p2 ··· pl . Then,

T (k, N)R(−m) (z) = T (pr1 pr2 ··· prl ,N) R(−m) (z) Γ0(N) 1 2 l Γ0(N) (3.0.33) = T (pr1 ,N) T (pr2 ,N) ··· T (prl ,N) R(−m) (z). 1 2 l Γ0(N)

For each of the primes pi for i = 1, 2, . . . l, either (pi,N) = 1 or (pi,N) = pi. If

(pi,N) = 1, then we apply Lemma 3.0.5, and if (pi,N) = pi, then we apply Lemma 3.0.6. Thus, we obtain the desired result.

53 k (−m) 1 X (−m ) T (k, N)R (z) = µ(`)R ` (`z) Γ0(N) N (3.0.34) k Γ0( ` ) `|(k,N)

Now set (k, N) = 1.

1 T (k, N)R(−m) (z) = R(−mk)(z) (3.0.35) Γ0(N) k Γ0(N)

The following result provides a generalization of Proposition 2.1.5 for Rademacher sums associated to M(k, N).

Corollary 3.0.8. Let m, k and N be positive integers such that (m, k) = 1. For

σ ∈ Γ0(N),

∞ Z (−m) (−m) X −mk (mk/`) R (σz)−R (z) = −2πi µ(`) P (`z)dz. (3.0.36) M(k,N) M(k,N) ` Γ0(N/`) σ−1·∞ `|(k,N)

(−m) Thus, RM(k,N)(z) is a weight zero mock modular form for Γ0(N) with shadow equal to P µ(`) −mk P (mk/`) (`z). ` Γ0(N/`) `|(k,N) Proof. By Theorem 3.0.7, we have

k (−m) X (−m ` ) RM(k,N)(z) = µ(`)R N (`z). (3.0.37) Γ0( ` ) `|(k,N)

Then, apply Proposition 2.1.5 to each of the summands.

54 Corollary 3.0.9. Let m, k and N be positive integers such that (m, k) = 1. Then,

(−m) 1 (−mk) Tb(k, N)R (z) = R (z). (3.0.38) Γ0(N) k Γ0(N)

Proof. We apply Corollary 3.0.3 followed by Theorem 3.0.7.

(−m) X (−m) T (k, N)R (z) = T (k, N)R (z) b Γ0(N) ` Γ0(N) `|(k,N)

X 1 (−m) = R (`z) k M(k/`,N/`) `|(k,N) (3.0.39) k 1 X X (−m `r ) = µ(r)R N (`rz) k Γ0( `r ) `|(k,N) r|(k/`,N/`) 1 = R(−mk)(z) k Γ0(N)

55 Chapter 4

Rademacher Sums, Hecke Operators and Moonshine

In Proposition 2.1.5 and Corollary 3.0.8, we consider the action of Γ0(N) on a particular Rademacher sum. We concluded that these Rademacher sums are mock modular forms and obtained formulae for their shadows. Thus, we naturally investigate the

(−m) (−m) conditions under which RM(k,N)(σz) = RM(k,N)(z) for σ ∈ Γ0(N).

Proposition 4.0.1. Let m and k be positive integers such that (m, k) = 1 and let

N ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25}. For σ ∈ Γ0(N),

(−m) (−m) (4.0.1) RM(k,N)(σz) =RM(k,N)(z)

(−m) Thus, RM(k,N)(z) is a modular function for Γ0(N).

Proof. Recall that a modular function is a mock modular form with shadow equal to

(−m) zero. In Corollary 3.0.8, we compute the shadow for RM(k,N)(z). In particular, it is

56 a ﬁnite linear combinations of weight two cusp forms for Γ0(N). However, for N as above, all weight two cusp forms are trivial. Hence the shadows of these Rademacher sums must be equal to zero.

Monstrous moonshine associates a group Γg to each conjugacy class of elements g of the monster group M. The following theorem considers elements of M for which

Γg = Γ0(N) when |g| = N, as in Theorem 6.5.1 of [DF11]. For the purposes of monstrous moonshine, we modify our Rademacher sums slightly. In particular, we take our usual Rademacher sum and eliminate the constant term. We denote the normalized Rademacher sum of weight zero and order m

(−m) associated to M(k, N) as TM(k,N)(z). Explicitly, we set

1 T (−m) (z) = R(−m) (z) − c (−m, 0). (4.0.2) M(k,N) M(k,N) 2 M(k,N)

Normalized Rademacher sums inherit a number of attributes from Rademacher sums, the primary of which are summarized in the following proposition.

Proposition 4.0.2. Proposition 2.1.2, Lemma 3.0.1, Theorem 3.0.7 and Proposition 4.0.1 remain true when R(−m) (z) is replaced by T (−m) (z) and R(−m) (z) is replaced M(k,N) M(k,N) Γ0(N) by T (−m) (z). In particular, if m, k and N are positive integers such that (m, k) = 1 Γ0(N) and z lies in the upper half-plane, then,

k (−m) 1 X (−m ) T (k, N)T (z) = µ(`)T ` (`z). Γ0(N) N (4.0.3) k Γ0( ` ) `|(k,N)

Proof. The analog of Proposition 2.1.2 follows directly from Proposition 2.2.1, as normalized Rademacher sums diﬀer from Rademacher sums by a constant.

57 For the analog of Lemma 3.0.1, we compute T (k, N)T (−m) (z). Γ0(N)

(−m) (−m) 1 T (k, N)T (z) = T (k, N) R (z) − cΓ (N)(−m, 0) (z) Γ0(N) Γ0(N) 2 0 1 1 = R(−m) (z) − T (k, N) c (−m, 0) (z) k M(k,N) 2 Γ0(N)

1 (−m) 1 X 1 = R (z) − c (−m, 0) k M(k,N) k 2 Γ0(N) (a,N)=1 ad=k 0≤b

The analog of Theorem 3.0.7 follows directly from the proofs of Lemmas 3.0.5 and 3.0.6. The analog of Proposition 4.0.1 follows from the deﬁnition of normalized Rademacher sums.

For the purposes of monstrous moonshine, we are primarily interested in T (−1) (z) Γ0(N) for particular values of N.

Proposition 4.0.3. Let k be a positive integer and let g ∈ M so that g belongs to one of the following conjugacy classes of M:

{1A, 2B, 3B, 4C, 5B, 6E, 7B, 8E, 9B, 10E, 12I, 13B, 16B, 18D}. Denote Ng = |g|.

58 Then,

−k (−1) 1 X ( ) T (k, N ) T (z) = µ(`)T ` (`z). g Γg Γ ` (4.0.5) k g `|(k,Ng)

In particular, if (k, Ng) = 1, then,

(−1) 1 (−k) T (k, Ng) T (z) = T (z). (4.0.6) Γg k Γg

Proof. Take g ∈ M as above and apply Proposition 4.0.2. Note that for g in the conjugacy classes listed, we have Γg = Γ0 (Ng) .

T (k, N ) T (−1) (z) = T (k, N ) T (−1) (z) g Γg g Γ0(Ng) 1 X ( −k ) (4.0.7) = µ(`)T ` (`z) k Γ0(Ng/`) `|(k,Ng)

Now we consult [CN79] to conﬁrm that Γg` = Γ0 (Ng/`) for each ` in equation (4.0.7). Thus,

−k (−1) 1 X ( ` ) T (k, Ng) T (z) = µ(`)T (`z) Γg k Γ0(Ng/`) `|(k,N ) g (4.0.8) −k 1 X ( ` ) = µ(`)TΓ (`z). k g` `|(k,Ng)

Now set (k, Ng) = 1.

(−1) 1 (−k) T (k, Ng) T (z) = T (z). (4.0.9) Γg k Γg

59 For particular values of k and N, we notice that T (k, N ) T (−1) (z) vanishes. g Γg Therefore we revisit Proposition 4.0.3 with total Hecke operators.

Proposition 4.0.4. Let k be a positive integer and let g ∈ M so that g belongs to one of the following conjugacy classes of M:

{1A, 2B, 3B, 4C, 5B, 6E, 7B, 8E, 9B, 10E, 12I, 13B, 16B, 18D}. Denote Ng = |g|. Then,

(−1) 1 (−k) Tb (k, Ng) T (z) = T (z). (4.0.10) Γg k Γg

Proof. Applying Corollary 3.0.9,

(−1) (−1) 1 Tb (k, Ng) T (z) = Tb (k, Ng) R (z) − cΓ0(Ng)(−1, 0) (z) Γg Γ0(Ng) 2 (4.0.11) 1 (−k) 1 = R (z) − Tb (k, Ng) cΓ0(Ng)(−1, 0) (z). k Γ0(Ng) 2

Now apply the remaining total Hecke operator.

(−1) 1 (−k) 1 X 1 Tb (k, Ng) T (z) = R (z) − cM(k/`,Ng/`)(−1, 0) Γg k Γ0(Ng) k 2 `|(k,Ng) 1 (−k) 1 X X 1 k = R (z) − µ(r) c Ng − , 0 Γ0(Ng) Γ k k 2 0 `r `r `|(k,Ng) r|(k/`,Ng/`)

1 (−k) 1 1 = R (z) − cΓ (N )(−k, 0) k Γ0(Ng) k 2 0 g 1 = T (−k)(z) k Γg (4.0.12)

60 The groups associated to the elements of conjugacy classes of M not listed in

Proposition 4.0.4 are not of the form Γ0(N). To investigate those groups, we consider the following.

2 N Let N and h be positive integers such that h|24 and h |N. Set n = h . Then,

Γ0 (n|h) denotes the subgroup of SL2 (R) deﬁned by

    a b   h  Γ0 (n|h) =   ∈ SL2 (R) | a, b, c, d ∈ Z . (4.0.13)  nc d 

  h 0   Note that Γ0 (n|h) is the conjugate of Γ0 (n/h) by the matrix   . Further, when 0 1

h = 1, we recover the groups Γ0(N).

For γ ∈ Γ0 (n|h) , let

  1 if γ ∈ Γ0(N)   λ (γ) = −1 (4.0.14) e h if a ≡ b ≡ d ≡ 1 and c ≡ 0 (mod h)    1 e h if a ≡ c ≡ d ≡ 1 and b ≡ 0 (mod h).

Then, the subgroup of SL2 (R) of interest is

Γ0 (n||h) = ker λ. (4.0.15)

For the remaining conjugacy classes, we need matrices known as Atkin-Lehner involutions. For positive integers e and N, we say that e is an exact divisor of N if e

N divides N, and e, e = 1. We denote this by e||N. Then, Atkin-Lehner involutions

61 for Γ0(N) are given by

    ae b    2 We =   | a, b, c, d ∈ Z, ade − bcN = e, e||N . (4.0.16)  Nc de 

Similarly, the Atkin-Lehner involutions for Γ0(n||h) are given by

    ae b n n W =  h  | a, b, c, d ∈ , ade2 − bc = e, e|| . (4.0.17) e   Z h h  nc de 

Note that for an Atkin-Lehner involution for Γ0 (n||h) , when e = 1, we have We =

Γ0 (n||h) . The case that e = N is referred to as the Fricke involution for Γ0(N). There are some well-known properties of Atkin-Lehner involutions. In particular,

WeWf = Wf We = Wg, (4.0.18)

ef where g = (e,f)2 , and,

WeΓ0(N) = Γ0(N)We. (4.0.19)

These involutions can be rescaled to have determinant 1. Thus, there are groups formed by adjoining rescaled Atkin-Lehner involutions to groups of the form Γ0(n||h).

n For an exact divisor e of h , denote the group formed by adjoining We to Γ0(n||h) by

Γ0(n||h) + e. That is,

1 Γ (n||h) + e = Γ (n||h) ∪ √ · W . (4.0.20) 0 0 e e

62 We can also adjoin multiple Atkin-Lehner involutions to Γ0(n||h). For exact divisors

n e, f, . . . of h , denote the group,

1 1 Γ (n||h) + e, f = Γ (n||h) ∪ √ · W ∪ √ · W ∪ · · · . (4.0.21) 0 0 e e f f

In the case that all of the Atkin-Lehner involutions have been adjoined to a group, we denote the resulting group by Γ0(n||h) + . To investigate the Atkin-Lehner involutions, we begin with the following lemma.

Lemma 4.0.5. Let k, e and N be positive integers such that e||N. Then,

Mc(k, N)We = WeMc(k, N), (4.0.22)

S where Mc(k, N) = M`(k, N). `|(k,N)     a b Ae B     Proof. Let γ =   ∈ Mc(k, N) and let we =   ∈ We Nc d NC De

From equation (4.0.18), we see that WeWe = Γ0(N). Thus, we compute

1 we · γ · we ∈ Mc(k, N). (4.0.23) e

Hence,

1 weMc(k, N)we = Mc(k, N). (4.0.24) e

Equivalently, we have

Mc(k, N)we = weMc(k, N). (4.0.25)

63 Thus, we obtain the desired result.

We use this lemma to prove the following proposition.

Proposition 4.0.6. Let k, e and N be positive integers such that e||N. Then,

(−1) 1 (−k) Tb (k, N) T (z) = T (z). (4.0.26) We k We

Proof. Apply the total Hecke operator to T (−1). Let w ∈ W . We e e

(−1) 1 X (−1) Tb (k, N) T (z) = T (z) We k WeM`(k,N) `|(k,N) 1 = T (−1) (z) k WeMc(k,N) 1 (−1) = T (z) (4.0.27) k Mc(k,N)We 1 X (−1) = T (z) k M`(k,N)We `|(k,N)

1 X (−1) = T (we · z) k M`(k,N) `|(k,N)

Now we can apply Corollary 3.0.9 to obtain the desired result.

1 T (k, N) T (−1) (z) = T (−k) (w · z) b We Γ0(N) e k (4.0.28) 1 = T (−k)(z) k We

We apply our established results to groups of the form Γ0 (n||h) by recognizing

64 that Γ0 (n||h) is expressible as a disjoint union of a ﬁnite number of cosets for Γ0(N). For example, when h = 2, we can write

    1 0 1 1    2  Γ0 (n||2) = Γ0(N) ∪   Γ0(N)   . (4.0.29) n 1 0 1

    1 0 1 1    2  Thus, if we set x =   and y =   we can write the normalized Rademacher n 1 0 1 sum associated to Γ0 (n||2) as

T (−m) (z) = T (−m) (z) + T (−m) (z). (4.0.30) Γ0(n||2) Γ0(N) xΓ0(N)y

We will use the following proposition to rewrite T (−m) (z) in terms of normalized xΓ0(N)y

Rademacher sums associated to Γ0(N).

Proposition 4.0.7. Let N and h be positive integers. Let M be a coset of Γ0(N)   1 1  h  and let y =  . Then, 0 1

1 R(−m)(z) = R(−m) z + . (4.0.31) My M h

65 Proof. By equation (2.1.1),

(−m) X RMy (z) = e (−mγ · z)

γ∈Γ∞\(My)∞ X + lim e (−mγ · z) − e (−mγ · ∞) L→∞ ∗ γ∈Γ∞\(My)

γ∈Γ∞\(M)∞ X + lim e (−mγy · z) − e (−mγy · ∞) . L→∞ ∗ γ∈Γ∞\(M)

1 Now, we notice that y · ∞ = ∞ and that y · z = z + h . Hence,

(−m) X RMy (z) = e (−mγy · z)

γ∈Γ∞\(M)∞ X + lim e (−mγy · z) − e (−mγy · ∞) L→∞ ∗ γ∈Γ∞\(M)

With Proposition 4.0.7 in mind, we can rewrite equation (4.0.30) in the following way.

1 T (−m) (z) = T (−m) (z) + T (−m) z + (4.0.34) Γ0(n||2) Γ0(N) xΓ0(N) 2

66 Thus, we examine the coset xΓ0(N), and notice the following relationship.

  1 0     Γ0(N) = {γ ∈ Γ0(n)|γ 6∈ Γ0(N)} (4.0.35) n 1

  1 0   That is, Γ0(n) is the disjoint union of Γ0(N) and   Γ0(N). Then, we can n 1 improve equation (4.0.34) further.

1 1 T (−m) (z) = T (−m) (z) + T (−m) z + − T (−m) z + (4.0.36) Γ0(n||2) Γ0(N) Γ0(n) 2 Γ0(N) 2

This relationship allows us to prove the following proposition.

Proposition 4.0.8. Let k and n be positive integers and set N = 2n. Then,

(−1) 1 (−k) Tb (k, N) T (z) = T (z). (4.0.37) Γ0(n||2) k Γ0(n||2)

Proof. First, we apply equation (4.0.36).

(−1) (−1) (−1) 1 Tb (k, N) T (z) = Tb (k, N) T (z) + Tb (k, N) T z + Γ0(n||2) Γ0(N) Γ0(n) 2 (−1) 1 − Tb (k, N) T z + Γ0(N) 2 (4.0.38)

It follows directly from the proof of Proposition 4.0.4 that

T (k, N) T (−1) (z) = T (−k) (z), (4.0.39) b Γ0(N) Γ0(N)

67 and

(−1) 1 (−k) 1 Tb (k, N) T z + = T z + . (4.0.40) Γ0(N) 2 Γ0(N) 2

We also notice that

(−1) 1 (−1) 1 Tb (k, N) T z + = Tb (k, n) T z + . (4.0.41) Γ0(n) 2 Γ0(n) 2

Thus, by the proof of Proposition 4.0.4 we see also that

(−1) 1 (−k) 1 Tb (k, N) T z + = T z + . (4.0.42) Γ0(n) 2 Γ0(n) 2

Combining equations (4.0.38), (4.0.39), (4.0.40) and (4.0.42), we have

1 1 T (k, N) T (−1) (z) =T (−k) (z) + T (−k) z + − T (−k) z + b Γ0(n||2) Γ0(N) Γ0(n) Γ0(N) 2 2 (4.0.43) =T (−k) (z), Γ0(n||2)

achieving the desired result.

With Proposition 4.0.6 and Proposition 4.0.8 in mind, we formulate a result to

improve Proposition 4.0.4 to include more conjugacy classes of M.

Proposition 4.0.9. Let h and k be positive integers and let g ∈ M so that g belongs to one of the following conjugacy classes of M : {1A, 2A, 2B, 3A, 3B, 4A, 4B, 4C, 4D, 5A, 5B, 6A, 6B, 6C, 6D, 6E, 7A, 7B, 8A, 8B, 8D, 8E, 9A, 9B, 10A, 10B, 10C, 10D, 10E, 11A, 12A, 12B, 12C, 12E, 12F, 12G, 12H, 12I, 13A, 13B, 14A, 14B, 14C, 15A, 15B, 15C, 16A, 16B, 16C, 17A, 18A, 18B, 18C, 18D,

68 18E, 19A, 20A, 20B, 20C, 20D, 20E, 20F, 21A, 21B, 21D, 22A, 22B, 23AB, 24A, 24B, 24C, 24D, 24H, 24I, 25A, 26A, 26B, 27A, 27B, 28A, 28B, 28C, 28D, 29A, 30A, 30B, 30C, 30D, 30F, 30G, 31AB, 32A, 32B, 33A, 33B, 34A, 35A, 35B, 36A, 36B, 36C, 36D, 38A, 39A, 39CD, 40B, 40CD, 41A, 42A, 42B, 42D, 44AB, 45A, 46AB, 46CD, 47AB, 48A, 50A, 51A, 52A, 52B, 54A, 55A, 56A, 59AB, 60A, 60B, 60C, 60D, 60E, 62AB, 66A, 66B, 68A, 69AB, 70A, 70B, 71AB, 78A, 78BC, 84A, 84B, 87AB, 88AB, 92AB,

94AB, 95AB, 105A, 110A, 119AB}. Denote Ng = h|g|. Then,

(−1) 1 (−k) Tb (k, Ng) T (z) = T (z). (4.0.44) Γg k Γg

Proof. For g ∈ M as above, Γg resembles one of the groups appearing in Proposition 4.0.4, Proposition 4.0.6 or Proposition 4.0.8.

Thus, we have shown that the total Hecke operators give the desired result for 150 of the 194 conjugacy classes of the monster.

69 Chapter 5

Conclusions

The theory of modular forms is rich and intriguing. In Chapter 1, we describe the modular group and state its properties. We consider modular functions and give explicit formulae for Eisenstein series, which are examples of modular forms. A discussion of the space of modular forms leads us to the modular invariant, which is a modular function. The modular invariant has inspired much curiosity among mathematicians. In the 1930s Rademacher proved an explicit formula for its Fourier series coefficients. This proof motivated a generalization of the modular invariant, creating the family of functions known as Rademacher sums. Thanks to the work of Duncan and Frenkel, much is known about these sums. In particular, we know that Rademacher sums are generally mock modular forms. We then investigate the actions of Hecke operators on Rademacher sums. To do so, we first generalize Rademacher sums to consider certain sets of matrices. Then, we derive explicit formulae for their Fourier series coefficients. We notice

70 some fascinating relationships between diﬀerent coeﬃcient functions, and use them to derive relationships between certain Rademacher sums. We prove that certain Hecke operators acting on certain Rademacher sums yield other Rademacher sums. Combining our analysis, we prove our main result, Theorem 3.0.7, which holds for positive integers m, k and N where (m, k) = 1.

k (−m) 1 X (−m ) T (k, N)R (z) = µ(`)R ` (`z) Γ0(N) N (5.0.1) k Γ0( ` ) `|(k,N)

This motivates the definition of the modified Hecke operator, and subsequently, the total modified Hecke operator. We use the new Hecke operators to rewrite Theorem 3.0.7 as Corollary 3.0.9.

(−m) 1 (−mk) Tb(k, N)R (z) = R (z) (5.0.2) Γ0(N) k Γ0(N)

The monster group, the largest of the sporadic groups, exhibits a number of curious characteristics. Since the 1970s, mathematicians have been hypothesizing and justifying these curiosities. Monstrous moonshine associates a group and modular function to each element of a particular conjugacy class of the monster. In [DF11], we see that these functions are equal to certain Rademacher sums. Then, we can write equation (5.0.2) to apply to certain groups associated to elements of the monster group. Explicitly, if g ∈ M is in one of the 150 conjugacy classes listed in Proposition 4.0.9, then,

(−1) 1 (−k) Tb (k, Ng) T (z) = T (z). (5.0.3) Γg k Γg

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