TALKS FOR THE SCHOOL ON MOCK MODULAR FORMS AND
RELATED TOPICS IN FUKUOKA, JAPANBEN KANE
Abstract. We investigate the modular completions of the mock theta functions.

1. Completions of the mock theta functions: from E2 to Zwegers
1.1. The Mock Theta functions. In his last letter to Hardy, Ramanujan introduced
a new family of functions which he dubbed “mock theta functions”. He stated thatthe mock theta functions were not themselves theta functions, but that they behave
like theta functions Historical note: in Ramanujan’s time, “theta functions” wasthe terminology for “modular forms”.
Recall the definition of modular forms: A function f is a modular form of weight k
on Γ ⊆ SL2 (Z) with multiplier system χ (having absolute value 1) if(1) For all M = ( ac db ) ∈ Γ, defining f |k M (τ ) = f |k,χ M (τ ) := χ(M )(cτ +d)−k f (M τ )with τ ∈ H (we omit χ when it is clear from the context)
f |k,χ M = f (τ ).
(2) The function f is holomorphic on H.
(3) The function is holomorphic at the cusps: for every cusp ρ = αγ , the functionfρ := f |k Mρ−1 , with Mρ = ( αγ ∗∗ ), satisfies (here q := e2πiτ and `ρ is the cusp
form)
X
n
fρ (τ ) =af,ρ (n)q `ρ .
n≥0Ramanujan included 17 examples of his new mock theta functions, and listed some
vague properties of the functions, but did not fully explain why they were like modular
forms, nor did he prove that they were indeed not modular forms.
One of the main properties that he listed is the following: At each root of unityζ = e2πiρ (with ρ ∈ Q), and q = e−t ζ with t → 0+ (i.e., approaching radially), there
exist M, N such that the function has the asymptotic
!
M
N
X
X
kµ
ν
t exp
cµν t + O(1).
(1.1)
µ=1

ν=1

Date: November 18, 2016.
2010 Mathematics Subject Classification. 11F11, 11F12, 11F30.
Key words and phrases. mock theta functions, harmonic Maass forms.
The research of the author was supported by grant project numbers 27300314, 17302515, and
17316416 of the Research Grants Council.
1

In addition, he required that at each cusp, there is a modular form which cancels thegrowth at that cusp, and that there is not a single modular form which cancels thegrowth at all cusps simultaneously.
An example of a mock theta function given by Ramanujan (he calls it a mock theta
function of order 3) is
∞
X

2

qn
f (q) :=.
(1 + q)2 (1 + q 2 )2 · · · (1 + q n )2
n=0
G.N. Watson (1935) later proved that f (q) indeed satisfies the asymptotic formula(1.1). In particular, he showed that
X (−1)n q 23 n2 + 12 n1
f (q) = Q∞.
`
1 + qn`=1 (1 − q ) n∈Z
The denominator

(1.2)

Q∞

`
`=1 (1 − q ) is essentially the η-function

η(τ ) := q

1
24

∞
Y


1 − q` .

`=1

Another mock theta function (of order 5) is given by
f0 (q) :=

∞
X

2

qn
.
2 ) · · · (1 + q n )
(1
+
q)(1
+
q
n=0

Watson did not obtain a formula similar to (1.2) for f0 , but Andrews (1986) proved
XX

5 2 1
1
2
(−1)j q 2 n + 2 n−j 1 − q 4n+2 .
f0 (q) = Q∞``=1 (1 − q ) n≥0|j|≤n

Question: How does this fit into the theory of modular forms?
“Rough” answer (Zwegers): It is part of a “modular object”.
What do we mean by that?
1.2. Completions: a simple example. Let’s start with a simple example:
Consider the Eisenstein seriesX
Ek (τ ) :=(cτ + d)−k .
(c,d)=1

Converges absolutely and locally uniformly for k > 2. Note that (here Γ∞ := {±T n :n ∈ Z} with T := ( 10 11 ))X
Ek =1|k M.
M ∈Γ∞ \SL2 (Z)For 2 < k ∈ 2N, when slashing with γ ∈ SL2 (Z), we may use |k M |k γ = |k (M γ) ot
obtain that Ek is a modular form of weight k. What about k = 2?
As is well-known, if one writes the Eisenstein series
X X∗ X
Ek (τ ) =(cτ + d + nc)−k ,c d(mod c) n∈Z

2

then
obtains the Fourier expansion (Bk is kth Bernoulli number and σ` (n) :=P one`
d|n d is the sum of divisors)∞

2k X
Ek (τ ) = 1 +σk−1 (n)q n .
Bk n=1
Similarly, by choosing the same ordering on the conditionally convergent series defining
E2 , we obtain (σ = σ1 )∞X
E2 (τ ) = 1 − 24
σ(n)q n .
n=1

Although E2 is holomorphic, it is not modular of weight 2 (this is a consequence ofthe fact that the sum is only conditionally convergent, and hence reordering is notpossible). However, it does satisfy a transformation property:
6i
E2 (M τ ) = (cτ + d)2 E2 (τ ) − c(cτ + d)(1.3)
π
Idea: Find something else with the same transformation property as (1.3). Note that(τ = u + iv throughout)|cτ + d|2(cτ + d)(cτ + d)
1==
Im(M τ )vv(cτ + d)(c(τ − 2iv) + d)(cτ + d)2==
− 2ic(cτ + d).
v
v
Hence
3
E2∗ (τ ) := E2 (τ ) −πv
is a non-holomorphic modular form and we conclude that E2 is “part” of a nonholomorphic modular form, i.e., the holomorphic function E2 has been completed to a
non-holomorphic modular form. This is the type of behavior meant above by “part ofa modular object”.
1.3. Appell–Lerch sums and their completions. To understand how Ramanujan’s mock theta functions are like E2 , we first need to understand sums like (1.2). We
begin by defining Jacobi’s theta functions
X
X
1
n2
2
ϑ(z; τ ) :=eπiν τ +2πiν (z+ 2 ) =(−1)n q 2 ζ n ,ν∈ 12 +Zn∈ 12 +Z

where ζ := e2πiz . A normalized Appell–Lerch sum is defined by
2eπiu X (−1)n eπi(n +n)τ +2πinv.
µ(u, v) = µ(u, v; τ ) :=ϑ(v; τ ) n∈Z
1 − e2πinτ +2πiu

Note: The Jacobi triple product formula states that
∞
Y


1
− 12
8
ϑ(z; τ ) = −iq ζ(1 − q n ) 1 − ζq n−1 1 − ζ −1 q n .
n=1

The µ-function satisfies the following properties.
3

Proposition 1.1 (Lerch/Zwegers).
(1)µ(u + 1, v) = −µ(u, v).
(2)µ(u, v + 1) = −µ(u, v)(3)µ(u, v) + e−2πi(u−v)−πiτ µ(u + τ, v) = −e−πi(u−v)−πiτ /4 .
(4)µ(u + τ, v + τ ) = µ(u, v)(5)µ(−u, −v) = µ(u, v)(6) The function u 7→ µ(u, v) is meromorphic with simple poles at u = nτ + m
1(n, m ∈ Z), and particularly has residue − 2πiϑ(v;τat u = 0.
)(7)∂ϑ(w; τ ) w=0 ϑ(u + v + z; τ )ϑ(z; τ )1 ∂w.
µ(u + z, v + z) − µ(u, v) =2πi ϑ(u; τ )ϑ(v; τ )ϑ(u + z; τ )ϑ(v + z; τ )(8)µ(v, u) = µ(u, v).
(9) The function µ has the transformation properties
πiµ(u, v; τ + 1) = e− 4 µ(u, v; τ ).
and(u−v)21√ eπi τ µ
iτ

where



u v 1
, ;−
τ τ
τ


+ µ(u, v; τ ) =1h(u − v; τ ),2i

2eπiτ x −2πizx
h(z; τ ) :=dx.
R cosh(πx)
Z

The last two properties are somewhat reminiscent of the transformation properties
3of E2 . If we can find another function (similar to − πyin the case of E2 ) which misses
modularity in the same way as µ, then we can “complete” it to obtain a “modular
object”.
We next show how to complete the Lerch sums. Define
Z z
∞
X

(−π)n z 2n+1−πu22E(z) := 2edu =1 = sgn(z) 1 − β(z ) ,n! n + 2
0
n=0
with

Z ∞
β(x) :=

1u− 2 e−πu du.

x

For u ∈ C and τ ∈ H we then define (set a = Im(u))
Im(τ )

X 
√ 
1
2
R(u) = R(u; τ ) :=sgn(v) − E (n + a) 2v (−1)n− 2 e−πin τ −2πinu .
n∈ 12 +Z4

Note that because of the appearance of a and v, the function R(u; τ ) is non-holomorphic,
both as a function of u and of τ , but it is real analytic. Indeed, its image under differentiation with respect to u essentially gives one of Jacobi’s theta functions. In
particular,r∂R
2 −2πa2 v(u; τ ) =eϑ(u; −τ )
∂u
v
and
X
1
∂R
i
2
2
(aτ − b; τ ) = − √ e−2πa v(−1)n− 2 (n + a)e−πin τ −2πin(aτ −b) .
∂τ2v
1n∈ 2 +Z

The right-hand side is essentially a weight 3/2 unary theta function. Define∂,∂τwhich maps functions satisfying weight κ modularity to functions satisfying weight2 − κ modularity (consider simultaneous modularity in (τ, τ ), then for f satisfying∂
weight (κ, 0) modularity, ∂τ(f ) satisfies weight (κ, 2) modularity, conjugating gives
weight (2, κ), and then multiplying by the weight (−κ, −κ) form v κ yields a weight.
(2 − κ, 0) form). Then (= means up to non-zero constant) for α ∈ (−1/2, 1/2) andβ ∈ R we have.
ξ1/2,τ (R(ατ − β; τ )) = gα+ 1 ,β+ 1 (τ ),(1.4)ξκ = ξκ,τ := 2iv κ

2

where
ga,b (τ ) :=

X

2

n2

nq 2 ζ bn

n∈a+Z

The function R satisfies certain transformation properties.
Proposition 1.2.
First, it satisfies certain elliptic transformation properties:(1)
R(u + 1) = −R(u).
(2)πiτ
R(u) + e−2πiu−πiτ R(u + τ ) = 2e−πiu− 4 .
(3)R(−u) = −R(u).
It also satisfies modular transformation properties:(1)
πi
R(u; τ + 1) = e− 4 R(u; τ ).
(2)


2
1
u 1
πi uτ
√
e
R
;−
+ R(u; τ ) = h(u; τ ).
ττ−iτComparing Proposition 1.2 with Proposition 1.1, we see that the functions essentially“miss modularity” in the same way. It is thus natural to defineiµe(u, v; τ ) := µ(u, v; τ ) + R(u − v; τ ).
2
5

Theorem 1.3. The function µe is real analytic (but not holomorphic) and satisfies thefollowing modularity properties:(1) The function µe satisfies the elliptic transformation property2µe(u + kτ + `, v + mτ + n) = (−1)k+`+m+n eπi(k−m) τ +2πi(k−m)(u−v) µe(u, v)(2) Let χ be the η-multiplier (i.e., χ(M ) :=η(M τ )
1(cτ +d) 2 η(τ )). Then for every M =( ac db ) ∈ SL2 (Z), the function µe satisfies the modular transformation propertyc(u−v)21v
aτ + bu,;= χ(M )−3 (cτ + d) 2 e−πi cτ +d µµee(u, v; τ ).
cτ + d cτ + d cτ + d(3)µ
e(−u, −v) = µe(v, u) = µe(u, v).
(4) We have∂ϑ(w; τ ) w=0 ϑ(u + v + z; τ )ϑ(z; τ )1 ∂w.
µ
e(u + z, v + z) − µe(u, v) =2πi ϑ(u; τ )ϑ(v; τ )ϑ(u + z; τ )ϑ(v + z; τ )Remark. There are important properties of the functions µ, R, and µe which closely
resemble the properties of E2 , y1 , and E2∗ . Namely, the first is meromorphic, the second
is “simpler”, and the third satisfies modularity properties.
We see that µ is like E2 and µe is like E2∗ . However, it also has elliptic transformation
properties like the Weierstrass elliptic functions (e.g., the Weierstrass ℘-function). Twovariable functions which satisfy both modularity and elliptic transformations are what
are known as Jacobi forms. More precisely, a Jacobi form of weight k and index m isa holomorphic function φ : H × C → C satisfying the transformation properties
2πimcz 2aτ + bzφ,= (cτ + d)k e cτ +d φ(τ, z)cτ + d cτ + dand2
φ(τ, z + λτ + µ) = e−2πim(λ τ +2λz) φ(τ, z).
Jacobi forms are so-named because the Jacobi theta functions ϑ(z; τ ) are Jacobi forms
of weight 1/2 and index 1/2.
Theorem 1.3 shows that µe has similar properties to those of a weight 1/2 Jacobi
1form, except that it is a vector-valued analogue with index −1
1 −1 (or index 1 withz = u − v).
One can then use formulas like (1.2) to realize f (q) as a certain µ-function. Hencef (q) is the “holomorphic part” µ of a weight 1/2 non-holomorphic function µe. Namely,1µ(2
The non-holomorphic piece R that needed to be added to f (q) is related to a weight
3/2 unary theta function. The non-holomorphic part may be realized as what is known
as a non-holomorphic Eichler integral (also known as a period integral)
Z i∞ g 1 1 (z)a+ ,b+−πia2 τ +2πi(b+ 21 )p 2 2
eR(aτ − b) =dz
−i(z + τ )−τ6

of the weight 3/2 unary theta function ga+1/2,b+1/2 .

7

2. Fourier coefficients of harmonic Maass forms and weakly
holomorphic modular forms with applications
2.1. Weakly holomorphic modular forms and harmonic Maass forms. In thelast lecture, we investigated how the mock theta functions are “part” of a “modular
object”. We now describe more precisely what we mean.
Recall the definition of modular forms: A function f is a modular form of weight k
on Γ ⊆ SL2 (Z) with multiplier system χ (having absolute value 1) if(1) For all M = ( ac db ) ∈ Γ, defining f |k M (τ ) := χ(M )(cτ + d)−k f (M τ ) with τ ∈ H
f |k M = f (τ ).
(2) The function f is holomorphic on H.
(3) The function is holomorphic at the cusps: for every cusp ρ = αγ , the functionfρ := f |k Mρ−1 , with Mρ = ( αγ ∗∗ ), satisfies (here q := e2πiτ and `ρ is the cusp
width)
Xnfρ (τ ) =af,ρ (n)q `ρ .
n≥0

The main aspect of the definition is the modularity property, and it is natural to relax
the other conditions. For example, we may relax the third condition by allowing afinite-order pole at the cusps (i.e., the Laurent expansion in q is supported in finitely
many negative powers of q). In other words, these are meromorphic modular forms
whose only poles occur at the cusps, and such functions are known as weakly holomorphic modular forms. Of course, if the condition that f is holomorphic on H is also
relaxed to allow finite-order poles in the upper half-plane, one obtains the space ofmeromorphic modular forms.
Recalling that Zwegers’s µ-function is part of a non-holomorphic modular form, one
can try to classify such functions by placing them into a more general framework. For
this, we need a natural family of non-holomorphic modular forms. Zwegers’s functions
are rather smooth, so it seems natural to require them to be real analytic. Recall
that µ is holomorphic in τ and ξ1/2 sends R to a weight 3/2 unary theta function.
Thus ξ1/2 sends µ
b to a weight 3/2 unary theta function. Since the weight 3/2 unary∂
theta function is holomorphic, it is annihilated by ∂τ, and hence by ξ3/2 , in particular.
Similarly,  
3 2i ∂ 1331∗
ξ2 (E2 (τ )) = − ξ2
= − 2iv= .
πyπ
2 ∂v vπ∗Hence ξ0 ◦ ξ2 (E2 ) = 0. These examples are part of a more general family of nonholomorphic modular forms satisfying a certain differential equation. For κ ∈ 12 Z,
define the weight κ hyperbolic Laplacian
 2



∂
∂2
∂
∂
2
+
+ ikv
+i
= −ξ2−κ ◦ ξκ .
∆κ := −v
∂x2 ∂v 2∂x∂v
Then a harmonic Maass form of weight κ on Γ is a real-analytic function F : H → C
satisfying the following.
(1) We have F|κ M = F for every M ∈ Γ.
(2) The function F is annihilated by ∆κ (i.e., ∆κ (F) = 0).
(3) The function F has at most linear exponential growth at all cusps of Γ.
8If the second condition is replaced with ∆κ (F) = λF, then we call F a weak Maass
form with eigenvalue λ.
Assuming that Γ is a congruence subgroup, a harmonic Maass form F has a Fourier
expansionXnF(τ ) =
aF ,v (n)q `∞ .
n∈Z

We assume without loss of generality that `∞ = 1. This Fourier expansion has anatural splitting into a holomorphic part F + and a non-holomorphic part F − , so thatF = F + + F −.
Since there are two independent solutions to the second-order differential equation∆κ (F) = 0, for κ < 1 one can writeXn
F + (τ ) =a+
F (n)qn−∞1−κF − (τ ) = a−+
F (0)vXna−
F (n)Γ(1 − κ, −4πny)q ,

n∞
n6=0

R∞
where Γ(s, y) := y ts−1 e−t dt is the (upper) incomplete gamma function. The holomorphic part is often called a mock modular form.
The kernel of ξκ is meromorphic forms, so in some sense harmonic Maass forms givea natural second-order extension of meromorphic modular forms. Indeed, if F is aharmonic Maass form, then ξκ (F) is annihilated by ξ2−κ , and is hence meromorphic.
The fact that F is real analytic on H implies that ξκ (F) is furthermore weakly holomorphic. We saw in the last lecture that ξκ (F) satisfies weight 2 − κ modularity, so it
is thus a weight 2 − κ weakly holomorphic modular form. If one is given only a mock
modular form (such as one of Ramanujan’s mock theta functions), then one calls theresulting weight 2 − κ form the shadow of the mock modular form (since it is “hidden”
if one is only given the mock modular form). The shadows of the mock theta functions
were all unary theta functions.
2.2. Growth of the coefficients. The coefficients of negative-weight weakly holomorphic modular forms and mock modular forms have been studied by a number ofauthors. Consider first a well-known example given by the partition function. Letp(n) denote the number of partitions of an integer n (i.e., the number of distinct ways
to write n as a sum of positive integers, where the order is irrelevant). Then thegenerating function is
P (z) := 1 +

∞
X

1
,
n
n=1 (1 − q )p(n)q n = Q∞

n=1

which is essentially the weight −1/2 weakly holomorphic modular form η1 . Hardy and
Ramanujan developed the Circle Method to obtain an asymptotic formula for p(n).
Note that by the Residue TheoremZ1
P (z)q −n−1 dq = p(n),2πi9where the integral is any contour around zero. In the Circle Method, one splits thisintegral into major and minor arcs in order to obtain the main asymptotic growth ofp(n) and an error term. Doing so, Hardy and Ramanujan obtained√ 2n1p(n) ∼ √ eπ 3 .
4n 3Rademacher refined Hardy and Ramanujan’s methods by choosing a better contour forwhich he could obtain better bounds. Rademacher then obtained a sequence of boundswith increasing accuracy (i.e., with decaying error term) and ultimately obtained what
is known as an exact formula for p(n). Namely, letting Is (x) denote the I-Bessel
function and defining the Kloosterman sum
r
X
πix
1 c
χ12 (x)e 6c ,Ac (n) :=2 12x (mod 24c)x2 ≡−24n+1 (mod 24c)

Rademacher showed thatp(n) =∞X
Ac (n)

2π3c(24n − 1) 4 c=1

 √
π 24n − 1
I3
.
2
6c

Rademacher and Zuckerman then later showed similar exact formulas for all negativeweight weakly holomorphic modular forms. Roughly speaking, they showed that thenth coefficient of a weight κ < 0 weakly holomorphic modular form is a sum ofKloosterman sums times I-Bessel functions.
Knopp then investigated a sort of converse question for this. The question that headdressed was whether any function whose Fourier coefficients had the same shapeas Rademacher’s formula were always the coefficients of a modular form. Knopp
answered this converse question in the negative. Namely, he found certain examples offorms which missed modularity in a “predictable way” (they are period polynomials)
which have formulas with the same shape as Rademacher and Zuckerman’s formulas
for negative weight weakly holomorphic modular forms. It turns out that Knopp had
uncovered families of (integral weight) mock modular forms, but their connections to
Ramanujan’s mock theta functions would not be discovered for many years. Namely,
the completions of the functions investigated by Knopp were not known at the time.
2.3. Formulas for the coefficients of the mock theta functions. Recall thethird-order mock theta function
∞
X

2
∞
X
qn
f (q) := 1 +=1+α(n)q n .
2
n2
(1 + q) · · · (1 + q )n=1
n=1

Ramanujan claimed that√(−1)n−1 6 π√ n − 1α(n) = √e 6 144 + O
24n − 1
10√n 1 !
πe 2 6 − 144√
24n − 1

Dragonette proved this claim in her Ph.D. thesis and Andrews extended her work in
his Ph.D. thesis to obtain the asymptotic formula√c(1+(−1)c )
nc
bX

 √
A2c n −
4π
π 24n − 1
I1
+ O(nε ) .
α(n) =
12c
12c(24n − 1) 4 c=1
Noting the similarity to the exact formula for the partition function, the Andrews–
Dragonette Conjecture claimed thatc(1+(−1)c ) √
∞
X A2c n −4
π 24n − 1πα(n) =
I1
.
(2.1)
1
2
c
12c
(24n − 1) 4 c=1
However, since f (q) is not modular, the techniques used by Rademacher and Zuckerman did not yield a sufficiently small error term to allow the sum to go to infinity.
Following Zwegers’s realization of the mock theta functions as part of a modular
object (namely, as the holomorphic part of a harmonic Maass form), Bringmann andOno realized that the modularity properties could be used to obtain the exact formula(2.1).
2.4. Poincaré series and Bringmann–Ono. Instead of arguing via the Circle Method,
Bringmann and Ono employed Poincaré series to obtain their exact formula. A
Poincaré series is formed by taking a test function φ : H → C and definingXPκ,Γ,φ :=φ|κ M.
M ∈Γ∞ \ΓWhenever the series converges absolutely and locally uniformly, The function Pκ,Γ,φsatisfies weight κ modularity on Γ. The choice φ(τ ) := 1 yields the Eisenstein series,
which is part of a family of weakly holomorphic modular forms Pκ,m given by taking
φ(τ ) = φm (τ ) := e2πimτ
Now setκMκ,s (w) := |w|− 2 Msgn(w) κ ,s− 1 (|w|),22

where Mν,µ is the M -Whittaker function and choose (recall that τ = u + iv)
ϕκ,m,s (τ ) := Mκ,s (4πmv)e2πimu .
One can check that ϕκ,m,s is an eigenfunction under ∆κ with eigenvalue (s − κ/2)(1 −s − κ/2). Hence the Poincaré seriesXPκ,m,Γ,s :=
ϕκ,m,s |κ M
M ∈Γ∞ \Γtaken from choosing φ = ϕκ,m,s is also an eigenfunction, as long as it converges absolutely and locally uniformly. In particular, choosing s = κ2 or s = 1 − κ2 yields a
harmonic Maass form Pκ,m . The series converges absolutely and locally uniformly forRe(s) > 1, and in particular for κ < 0 or κ > 2 we obtain a harmonic Maass form.
A key property about harmonic Maass forms is the fact that if they do not exhibit growth towards the cusps, then they must be holomorphic modular forms; thisis a result of Niebur, who studied the weight zero version of these Poincaré series.
Thus for weight < 0 they must exhibit growth towards the cusps. In other words,
11

a negative-weight harmonic Maass form is uniquely determined by the terms in its
Fourier expansion F + + F − which exhibit growth. The sum of such terms is known
as the principal part of F. In other words, the principal part is the terms in F + withn < 0 or in F − with n ≥ 0. For κ < 0, we see that the Poincaré series Pκ,m then forma basis of the space of harmonic Maass forms of weight κ (on Γ).
The principal part of each Pκ,m is given by the term coming from the identity matrix,
and is hence essentially ϕκ,m,1−κ/2 . We may further split ϕκ,m,1−κ/2 naturally into its
holomorphic and non-holomorphic parts to obtain the following.
Proposition 2.1. Suppose that κ < 0. For m < 0, the principal part of Pκ,m is aconstant multiple of q m . For m = 0, the principal part is a constant multiple of v 1−κ ,and for m > 0, the principal part is a constant multiple of Γ(1 − κ, −4πmv)q m .
The idea now to prove the Andrews–Dragonette Conjecture is to compute the coefficients of the Poincaré series and then write f (q) as an explicit linear combinationof the Poincaré series. One first needs to determine the group Γ under consideration.
Zwegers had investigated the completion and modularity properties of f (q) as partof a vector-valued modular form. As a consequence of his results, one obtains thefollowing:
Theorem 2.2 (Zwegers, Bringmann–Ono). The function q −1 f (q 24 ) is the holomorphic part of a weight 1/2 harmonic Maass form on Γ0 (144), where as usual we have
Γ0 (N ) := {( ac db ) ∈ SL2 (Z) : N | c}. Moreover, it has a simple pole at i∞ and no othersingularities.
Hence we are interested in taking Γ = Γ0 (N ). By Theorem 2.2 and the fact that the
principal part uniquely determines the form up to holomorphic modular forms, one
may use Proposition 2.1 to guess that q −1 f (q 24 ) is a constant multiple of P1/2,−1 . It
would then seem that the question is resolved after computing the Fourier expansionof P1/2,−1 . However, the Poincaré series does not converge absolutely in this case, sosome care is needed. In order to address this issue, one computes the Fourier expansionof Pκ,Γ0 (N ),m,s for Re(s) > 1 and then analytically continues to s = 3/4.
By rewriting the sum over Γ∞ \Γ0 (N ) as a sum over (c, d) = 1 (like the Eisensteinseries) and writing the inner sum in d (assuming that we have absolute convergence),
we may compute the Fourier expansion by Poisson summation,XX
g(n) =
gb(n),n∈Z

n∈Z

where gb is the Fourier transform. Assume that m < 0. Then one obtains that thecoefficient of q n have the same shape as (2.1) and the coefficient of Γ(1 − κ, −4πnv)q n
are essentially the same, except that the I-Bessel function has been replaced with the
J-Bessel function. More generally, for Re(s) > 1, the parameter of the Bessel functions
is replaced with 2s − 1. Using bounds for the Kloosterman sums, one can show that
the Fourier expansion may be analytically continued beyond s = 3/4, yielding theAndrews–Dragonette conjecture.
Applications of Fourier coefficients of mock modular forms do not end there. For
example, one may interpret f (q) as the generating function for a certain partitiontheoretic invariant, and further examples of this invariant yield more mock modular
12forms. After Ramanujan famously found the congruences for the partition functionp(5n + 4) ≡ 0(mod 5),p(7n + 5) ≡ 0(mod 7),p(11n + 6) ≡ 0(mod 11),in 1944, Dyson (an undergraduate at the time) tried to find a “combinatorial” explanation for these congruences. He defined a statistic known as the rank of a partition,
which is defined to be the size of the largest part minus the number of parts. He
conjectured that for the first 2 congruences, the rank modulo 5 (resp. modulo 7)
equally splits the partition into 5 (resp. 7) distinct groups each of the same size (thusgiving an explanation why the size is divisible by 5 (resp. 7). This was later provenby Atkin and Swinnerton-Dyer. Dyson also proposed that there was a similar statistic(that he called the “crank”) which explained all 3 congruences; this was later foundby Andrews and Garvan. Let N (m, n) denote the number of partitions of n with rank
m. Then the rank generating function is given by
2
∞
∞
∞
X
X
X
qn
m n
Qn
R(w; q) := 1 +
N (m, n)w q =`−1 ``=1 (1 − wq ) (1 − w q )n=1
n=1 m=−∞
Plugging in w = 1, we get
R(1; q) = P (z),while w = −1 yields
R(−1; q) = f (q).
In other words, f (q) is a generating function for the number of partitions of n witheven rank minus the number of partitions with odd rank. It turns out that this is nota coincidence.
Theorem 2.3 (Bringmann–Ono). If ζ is a root of unity, then R(ζ; q) is the holomorphic part of a weight 1/2 harmonic Maass form.
Bringmann and Ono gave the specific completion of R(ζ; q) and modularity properties of the completion.
As a consequence of their result, let N (r, t; n) be the number of partitions of n withrank congruent to r modulo t.
Theorem 2.4 (Bringmann–Ono). The generating function (`t := lcm(2c2 , 24))∞ 
X
p(n) `t (n− 241 )N (r, t; n) −qt
n=0
is the holomorphic part of a weight 1/2 harmonic Maass form.
Theorem 2.4 has a number of interesting consequences. Firstly, For 0 ≤ r < s < t
we may consider∞
X
1(N (r, t; n) − N (s, t; n)) q `t (n− 24 ) .
n=0This is also the holomorphic part of a weight 1/2 harmonic Maass form. There is anatural symmetry of partitions which implies that N (r, t; n) = N (t−r, t; n), so one can
13also restrict 0 ≤ r < s ≤ 2t . By investigating the asymptotics of the Fourier coefficients,
one can then determine inequalities like N (r, t; n) > N (s, t; n) for n sufficiently large.
This leads to the resolution of a conjecture of Andrews and Lewis (up to a finite
number of exceptions where equality holds).
Theorem 2.5 (Bringmann). For n > 0, we have
N (0, 3; n) < N (1, 3; n)

if n ≡ 0, 2

N (0, 3; n) > N (1, 3; n)

if n ≡ 1(mod 3),(mod 3).

This is part of a more general phenomenon.
Theorem 2.6 (Bringmann–K.). Assume that c > 9 is an odd integer. If 0 ≤ a < b ≤c−1, then for n sufficiently large (depending on a, b, c), we have2
N (a, c; n) > N (b, c; n).
Furthermore, if c = 3, 5, or 7, then for n sufficiently large, the inequality (> or <)is completely determined by a, b, and the congruence class of n modulo c.
As another application, the non-holomorphic part is closely related to a unary theta
function,
P and hence is only supported on certain square classes. For a Fourier expansion n∈Z av (n)q n , consider the “sieved” functionX
av (n)q n .
n≡An∈Z(mod B)

Then for A and B chosen appropriately so that the non-holomorphic part vanishes, theresulting function is weakly holomorphic. Since weakly holomorphic modular forms
with integral coefficients have bounded denominators, this has algebraic applications.
Theorem 2.7 (Bringmann–Ono). Let t be a positive odd integer, and let Q - 6t beprime. If j is a positive integer, then there are infinitely many non-nested arithmetic
progressions An + B such that for every 0 ≤ r < t we have
N (r, t; An + B) ≡ 0

14(mod Qj ).

3. Fourier coefficients of polar harmonic Maass forms andmeromorphic modular forms
3.1. Ramanujan–like expansions. Instead of negative-weight weakly holomorphic
modular forms like 1/η related to the partition function, we investigate here coefficientsof meromorphic modular forms. We relax the holomorphicity condition of modular
forms. A function f is a meromorphic modular form of weight k on Γ ⊆ SL2 (Z) withmultiplier system χ (having absolute value 1) if(1) For all M = ( ac db ) ∈ Γ, defining f |k M (τ ) = f |k,χ M (τ ) := χ(M )(cτ +d)−k f (M τ )with τ ∈ H (we omit χ when it is clear from the context)
f |k,χ M = f (τ ).
(2) The function f is meromorphic on H.
(3) The function is meromorphic at the cusps:
Xnfσ (τ ) =af,σ (n)q `σ .
n−∞If the growth at the cusps instead satisfies v κ/2 f (τ ) vanishes (i.e., it grows like that ofa cusp form towards cusps), then we call it a meromorphic cusp form.
Consider the example 1/E4 . The coefficients grow very fast:1= 1 − 240q + 55440q 2 − 12793920q 3 + 2952385680q 4
E4 (τ )
− 681306078240q 5 + 157221316739520q 6 + . . .
Specifically, since there is a pole of 1/E4 at ρ, the radius√of convergence around i∞ must
not include ρ. Specifically, it converges for Im(τ ) > 3/2. Ramanujan conjectured
and Bialek later proved that∞X1=βn q nE4 (τ ) n=0
πiwith (λ norms from OQ(ρ) , with ρ := e 6 )βn := (−1)n3 X X h(c,d) (n) πn√3e λ .
E6 (ρ)λ3(3.1)(λ) (c,d)Here (c, d) runs over “distinct” elements cρ + d with norm λ. Unlike the Fourier
expansions of weakly holomorphic modular forms, the expansion does not converge
everywhere. Finally, we let h(1,0) (n) := (−1)n /2, h(2,1) (n) := 1/2, and for λ ≥ 7 we let(a, b) ∈ Z2 be any solution to ad − bc = 1 and set
√ !!
πn
c 3
h(c,d) (n) := cos (ad + bc − 2ac − 2bd)
− 6 arctan.
λ2d − cAlthough it may not be clear from this definition, the h(c,d) are independent of the
choice of a and b.
15

Hardy and Ramanujan showed that 1/E6 has a similar shape. It is thus natural todefine (z = z1 + iz2 , b prim. ideals)∞
X
X∗ Ck (b, m)
2πmz2−jFk,j,r (τ, z) := z2(4πm)r e N (b) e2πimτ .
k−j2
m=0 b⊆OQ(z) N (b)Here Ck is similar to the h(c,d) .
Hardy and Ramanujan and then Bialek, Berndt–Bialek, and Berndt–Bialek–Yee
used the Circle Method to compute the coefficients. One has to be careful because theFourier expansion doesn’t converge everywhere. The calculations become increasingly
difficult as the order of the poles increase. Berndt–Bialek–Yee studied forms withsecond-order poles in particular and the relevant calculations are quite long.
3.2. Poincaré series. Petersson attacked the problem differently. He constructing
Poincaré series.
In particular, he computed the Fourier expansion of negative-weight meromorphic
cusp forms with only simple poles (throughout this section, k ≥ 2).
Theorem 3.1 (Petersson). If f is a weight 2 − 2k < 0 meromorphic cusp form with
only simple poles, then f has a Fourier expansion of the type∞
X
X.
P2k,−m (z)q m .
f (τ ) =Resτ =z (f (τ ))
m=0

z∈Γ\H
ordz (f )=−1

Here P2k,−m are the weakly holomorphic Poincaré seriesX
P2k,m (z) :=ϕm 2k M (z),M ∈Γ∞ \SL2 (Z)where k ∈ N≥2 and for m ∈ Z
ϕm (z) := e2πimz .
In the case of simple poles, it turns out that the formulas of Ramanujan are simply
evaluations of Poincaré series at the points where poles occur. The norm of cz + d isz2
λ = |cz + d|2 and Im(M z) = |cz+d|
2 , so the exponential becomes
e−2πimRe(M z) e2πmIm(M z) = e−2πimRe(M z) e

2πmz2
λ

;

we directly see the exponential which occurred in (3.1) (for z = ρ), while the otherfactor contributes to the cosine.
To obtain this result, letX
1
H2k (z, τ ) := 2πiM
1 − e2πi(τ −z) 2k,zM ∈Γ∞ \SL2 (Z)The function H2k is meromorphic in both z and τ (with simple poles, unless theresidue vanishes). It is a meromorphic modular form of weight 2k as a function of z,
and Petersson furthermore showed that∞. X
H2k (z, τ ) =P2k,−m (z)q m .
m=016

Hence, in order to obtain Theorem 3.1, the main step is the following.
Theorem 3.2 (Petersson). If f is a weight 2 − 2k meromorphic cusp form with simple
poles, then there exist z1 , . . . , zr and c` ∈ CrX
f (τ ) =c` H2k (z` , τ ) .
`=1

Indeed, Petersson obtained a converse theorem, classifying the choices of c` for which
resulting form is indeed modular. The classification depends on a certain consistency
condition from the Fourier coefficients and elliptic coefficients of cusp forms of weight)
2k. The elliptic expansion of a cusp form g around z is (Xz (τ ) := ττ−z−zXg(τ ) = (τ − z)−2kag,z (n)Xzn (τ ).
(3.2)
n−∞

Namely, if f is a weight 2 − 2k meromorphic modular form and g is a weight 2k cusp
form, then the product is a weight 2 meromorphic modular form. It turns out that theresidue of f g at every point must vanish. This gives a necessary consistency condition,
which also turns out to be sufficient for the sum of the H2k functions to be modular.
Petersson’s proof is useful for showing that the Fourier coefficients of forms withsimple poles have the desired shape, but other Poincaré series are needed to address
higher order poles. LetXz−`2
H2k,` (z, τ ) := 2πiM
1 − e2πi(τ −z) 2k,zM ∈Γ∞ \SL2 (Z)and(r)
H2k,` (z, τ ) :=∂r
H2k,` (z, τ ).
∂τ r
(r)

Lemma 3.3 (Bringmann–K.). The Fourier coefficients of H2k,` agree with F2k,`,r .
Hence in order to obtain a formula for coefficients of meromorphic cusp forms withhigher order poles, one needs a classification like Theorem 3.2 for higher order poles.
Petersson has provided such a classification. For ν ≤ −1, define1∂ −ν−1Y2−2k,ν (z, τ ) :=(α − z)2k H2k (α, τ ) α=z .
−ν−1(−ν − 1)! ∂Xz (α)
The differential operator is naturally connected to the elliptic expansion (3.2).
Petersson showed the more general classification below.
Theorem 3.4 (Petersson). If f (τ ) is a meromorphic modular form with poles of orderat most R, then f is a linear combination of Y2−2k,ν (z, τ ) with |ν| ≤ R.
(r)

By showing that Y2−2k,ν is itself a linear combination of the H2k,` whenever |ν| ≤ 3,
the following was shown.
Theorem 3.5 (Bringmann–K.). If f is a negative-weight meromorphic cusp form with
poles of order at most 3, then f is a linear combination of the Fm,j,r functions.
As the order of the poles increase, the proof becomes slightly more complicated, but
is still less involved than the calculations with the Circle Method.
17

3.3. Polar harmonic Maass forms. The results of the previous section turn out tobe much more general.
Theorem 3.6 (Bringmann–K.). If f is a negative-weight meromorphic cusp form or
quasi-modular form, then f is a linear combination of the Fm,j,r functions.
In order to obtain this more general result, one employs the theory of polar harmonic
Maass forms. These are harmonic Maass forms which can also have singularities in
the upper half-plane.
The key is to realize H2k (τ, z) as the meromorphic part of a polar harmonic Poincaré
Maass form v 2k−1 Ψ2k (z, τ ), where we have the Poincaré seriesXΨ2k (z, τ ) :=(z − τ )−2k Xτ (z)−1M.
2k,zM ∈SL2 (Z)The function v 2k−1 Ψ2k (z, τ ) is a weight 2k meromorphic modular form as a functionof z and a weight 2 − 2k polar harmonic Maass form as a function of τ .
By repeatedly applying the Maass raising operatorRκ,z := 2i

κ
∂
+
∂z z2

in the z variable, one obtains functions which are still harmonic as a function of τ , buthave higher order poles. Specifically, define repeated raising by
nR2k,z:= R2k+2n−2,z ◦ · · · ◦ R2k,z ;this sends a weight 2k object to a weight 2k+2n object. Using the image of v 2k−1 Ψ2k (z, τ )under these operators as a basis instead of Petersson’s Y -functions, we have the following.
Proposition 3.7. Every meromorphic cusp form of weight 2 − 2k is a linear combination of the functions

n
R2k,z
v 2k−1 Ψ2k (z, τ ) .
Furthermore, a linear combinationrz
XXncn,` R2k,zv 2k−1 Ψ2k (z, τ )z∈H n=0

is a meromorphic cusp form if and only if its non-meromorphic part vanishes.
This classification is like Theorem 3.4, except that while he was investigating whether
the a priori meromorphic linear combination was modular, the above proposition
checks whether an a prior modular object is meromorphic. A key step to using theabove classification to obtain formulas for all meromorphic modular forms is the following identity: nX(2k + n − 1)!
n
∂ n−j
n
R2k,z (Ψ2k (z, τ )) =(−2i)n−j n−j H2k+2n,j (z, τ ),(2k + n − 1 − j)! j
∂τ
j=0
18

where H2k,` is a polar harmonic completion of H2k,` . Namely, taking the meromorphic
parts of each side, we have
 
n
n−j
X
(2k + n − 1)!
n
n
n−j ∂
R2k,z (H2k (z, τ )) =(−2i)
H2k+2n,j (z, τ ).
(3.3)
(2k + n − 1 − j)! j∂τ n−jj=0
The right-hand sides have the correct shape to yield Theorem 3.6.
3.4. Divisors of modular forms. In the above, we assumed that k ≥ 2, but it isnatural to ask about the case k = 1. For this, we extend to Γ0 (N ), since the caseN = 1 is not as interesting. In this case, it is natural to attempt to use Hecke’s
trick to extend the Poincaré series to weight 2. Due to the poles occurring in theupper half-plane and the fact that the Fourier expansion does not always exist, there
is some care needed, however (specifically, one cannot analytically continue via the
Fourier expansion because this would only give a continuation in part of H). Define
for z, τ ∈ H and s ∈ C with Re(s) > 0
Xϕs (M z, τ )(3.4)
PN,s (z, τ ) :=j (M, z)2 |j (M, z) |2s
M ∈Γ (N )0with j(M, τ ) := (cτ + d) andϕs (z, τ ) := v 1+s (z − τ )−1 (z − τ )−1 |z − τ |−2s .
One splits PN,s into three pieces. The first piece is those elements of Γ∞ , the second isϕs (M z, τ ) − ϕs (M (i∞), τ )
,
2
2s
j
(M,
z)
|j
(M,
z)
|
M ∈Γ (N )\Γ∞X
0and the third isϕs (M (i∞), τ )
.
2
2s
j
(M,
z)
|j
(M,
z)
|
M ∈Γ (N )\Γ∞X
0

The first two pieces turn out to converge absolutely for Re(s) > −1, while the lastpiece can be analytically continued via its Fourier expansion (as a function of z).
The reason for this splitting is that the first two pieces have poles and hence their
Fourier expansions do not converge everywhere, while the third piece does not converge
absolutely, but its Fourier expansion exists everywhere and it can be analyticallycontinued.
The analytic continuation to s = 0 we write as vΨ2,N (z, τ ). We normalize to definev∗HN,τ(z) :=:= − Ψ2,N (z, τ ).
2πThis turns out to be harmonic both as a function of z (in weight 2) and in τ (in weight0). Its meromorphic part is a quasi-modular form (like E2 ), and its non-meromorphic
part is a constant multiple of 1/v. As z → τ , it satisfies
∗
HN,τ
(z) =

ωN,τ 1+ O(1),
2πi z − τ
19

(3.5)

where eN,τ is half of the size of the stabilizer of τ in Γ0 (N ). At cusps σ 6= i∞, one
also defines∗∗HN,σ(z) := lim HN,τ
(z).
τ →σ

This exhibits a constant term as z approaches the cusp σ. There is a similar definitionat the cusp i∞, although the limit does not directly exist.
Note that
D(f )
,
f
1 ∂
where D := 2πi(also written Θ), is also a quasi-modular form. To cancel off the∂τterms which grow, it is natural to define
X
1
∗
(z).
(3.6)
f div (z) :=ordτ (f )HN,τ
ωN,τ
τ ∈X0 (N )The following then holds.
Theorem 3.8 (Bringmann–K.–Löbrich–Ono–Rolen). If S2 (Γ0 (N )) denotes the spaceof weight 2 cusp forms on Γ0 (N ), thenkΘ(f (z))f div (z) ≡−(mod S2 (Γ0 (N ))).
4πz2
f (z)
∗
From Theorem 3.8 and the asymptotic growth of the coefficients of HN,τ(from aRamanujan-like formula in weight 2), one can determine the points τ for which there
are poles and zeros, and furthermore can determine the orders of these poles and zeros
because those the the multiplicity with which HN,τ occurs in f div . This allows one to(numerically) compute the divisor of f given only the Fourier expansion (since Θ(f )can also be directly computed via the Fourier expansion).

20

4. Regularized Petersson inner products for meromorphic modular
forms
For simplicity, we mostly assume integral weight and modularity for SL2 (Z) throughout.
4.1. Definition of the inner product. Considering the variables z and z as independent variables, note that for a weight 2k modular form f (z), the function f (z)satisfies weight 2k modularity as a function of z. Furthermore, writing z = x + iy ∈ H,
the function y 2k satisfies simultaneous weight −2k modularity in both z and z because
Im((az + b)(cz + d))
az + by=Im(M z) = Im=,2
cz + d|cz + d||cz + d|2
where we used the fact that ad − bc = 1.
Petersson [14] then realized that, for functions f and g satisfying modularity for all
M ∈ SL2 (Z), the function
f (z)g(z)y 2k
is SL2 (Z)-invariant. Moreover, the metricdxdyy2
is also SL2 (Z)-invariant. Hence the integralZdxdy
f (z)g(z)y 2k 2
hf, gi :=y
SL2 (Z)\H(4.1)

is well-defined whenever it converges absolutely. Using bounds for cusp forms (inparticular, they exponentially decay as y → ∞), one can show that the integral (4.1)
converges absolutely for f, g ∈ S2k . This exponential decay also suffices to show
convergence when taking the inner product between f ∈ S2K and the Eisenstein series
E2k .
4.2. Petersson coefficient formula. The Petersson coefficient formula uses an explicit evaluation of the inner product to compute the Fourier coefficients of modular
forms. To describe this result, we require the classical Poincaré series (see [16, 17])X
ϕm 2k M (z),P2k,m (z) :=(4.2)
M ∈Γ∞ \SL2 (Z)where k ∈ N≥2 and for m ∈ Z
ϕm (z) := e2πimz .
As discussed in the second talk, these converge locally and absolutely uniformly. For
m = 0, the Poincaré series is precisely the Eisenstein series, while for m > 0 we have!
P2k,m ∈ S2k and for m < 0 we have P2k,m ∈ M2k.
Theorem 4.1 (Petersson coefficient formula). If f ∈ S2k and m ∈ N, thenhf, P2k,m i =(2k − 2)!
af (m).
(4πm)2k−1
21Sketch of proof. Plugging in the definition (4.2) of the Poincaré series P2k,m and choosing a fundamental domain F for SL2 (Z)\H (a “nice” connected set of representativesz ∈ H of the orbits of SL2 (Z)\H under fractional linear transformations), we unfold
the integral on the left-hand side by rewriting (formally, but this is valid because of
the exponential decay of cusp forms towards the cusps)Zf (z)SL2 (Z)\H

ϕm (M z) 2k dxdyy(cz + d)2k
y2XM ∈Γ∞ \SL2 (Z)=XM ∈Γ∞ \SL2 (Z)=XM ∈Γ∞ \SL2 (Z)Z

f (M z)ϕm (M z)Im(M z)2k

F

Z
MF

dxdy
f (z)ϕm (z)y 2k 2 =
y

dxdy
y2

Z

f (z)ϕm (z)y 2k

Γ∞ \H

dxdy
. (4.3)y2

Since the fundamental domain for Γ∞ \H is very simple, this unfolding argument resultsin the double integral
Z ∞Z 1dxdy
f (z)ϕm (z)y 2k 2 .
y00
The integral over x essentially picks off the mth coefficient and then explicitly computing the integral over y yields the claim.

4.3. Orthogonal splitting. The inner product on S2k is positive-definite. Hence,by the Gram-Schmidt process, one can construct an orthonormal basis. A particular
choice of the basis elements turns out to be very natural. Namely, one can obtain an
orthonormal basis of Hecke eigenforms. Why are these orthogonal? Since the Hecke
operators are Hermitian, we have
λf (n) hf, gi = hλf (n)f, gi = hf |2k Tn , gi = hf, g|2k Tn i = hf, λg (n)gi = λg (n) hf, gi .
Since λf (n) 6= λg (n), this leads to a contradiction if hf, gi 6= 0. We thus conclude
that f and g are orthogonal to each other. Hence the splitting of S2k into eigenspaces
precisely yields the orthogonal splitting, with the orthonormal basis given by the Hecke
eigenforms normalized such that kf k2 = 1.
We note that the other normalization af (1) = 1 is also natural. Under this normalization (and appropriately normalizing the Hecke operators), the coefficients af (n) andthe eigenvalues λf (n) coincide. This realization “de-mystifies” the coefficients of the
Hecke eigenforms and plays an important role in understanding Fourier expansions.
4.4. Inner products for weakly holomorphic modular forms.
4.4.1. The regularization of Petersson, Harvey–Moore, and Borcherds and its exten!
sion. For f, g ∈ M2k, the integral (4.1) generally diverges. Petersson established a
Cauchy principal value for the integral as a partial solution to this problem. Insteadof integrating over the full fundamental domain, we integrate over a cut-off fundamental domain whose closure does not include the cusp on the boundary of the chosen22fundamental domain. In our case, the cusp is i∞ and the cut-off fundamental domain
is given by11
FT := z ∈ H : |z| ≥ 1, y ≤ T, − ≤ x ≤.
2
2!
, Petersson then defined the regularized inner product (see [15])For f, g ∈ M2k
Zdxdy
hf, gi := lim
f (z)g(z)y 2k 2 .
(4.4)
T →∞ FyT
The key to the above regularization is that it essentially gives an ordering to theintegrals over x and y.
This construction was further independently rediscovered and extended by Harvey–
Moore [11] and Borcherds [2] by multiplying the integrand by y s for some s ∈ Cwith Re(s)  0 and then taking the constant term of the Laurent expansion of themeromorphic continuation (in s) at s = 0.
One can use the regularized inner product to show that for m < 0 the Poincaré series
P2k,m , defined in (4.2), is orthogonal to cusp forms. This was shown by Petersson in amuch more general setting in [15, Satz 4].
The regularization of Petersson/Harvey–Moore/Borcherds does not always converge,
however. In particular, Petersson found a necessary and sufficient condition for his
regularization (4.4)to converge (see [15, Satz 1]) and Petersson norms once again pose a
problem, as they did for the Eisenstein series. This problem has been recently resolved
by Bringmann, Diamantis, and Ehlen [3], who were able to extend the regularization in!
a way so that the inner product hf, gi is well-defined and finite for all f, g ∈ M2k. The
basic idea is to multiply the integrand by an invariant function hs (z) which satisfies
h0 (z) = 1 and such that the integral converges for Re(s) sufficiently large. One thentakes the analytic continuation to s = 0 as the definition of the inner product. We do
not give any of the technical details here, but the reader is encouraged to look at [3,Section 3, and in particular Theorem 3.2].
4.4.2. Theta lifts. The inner product has been used by many authors (for example, in[2] and [6]) to obtain theta lifts from modular forms of one type to modular forms ofanother type. To give a rough idea, one defines a two-variable theta function Θ(z, τ )which is modular in both variables (one calls this function the theta kernel), but which
satisfies a different kind of modularity in each variable (for example, suppose that it
satisfies weight 2k modularity as a function of z and weight k + 1/2 modularity as afunction of τ ). Taking the inner product in one variable against another function fsatisfying the same type of modularity then yields a new function in the other variablesatisfying the other type of modularity. In other words, in the example above, if fsatisfies weight 2k modularity, thenΦ(f )(τ ) := hΘ(·, τ ), f isatisfies weight k + 1/2 modularity. This yields a theta lift Φ from weight 2k modular
forms to weight k + 1/2 modular forms. The example illustrated above is Shintani’sconstruction [19] of his lift from integral weight to half-integral weight modular forms
and the lift in the opposite direction can be shown to be one of Shimura’s lifts [18] fromhalf-integral weight to integral weight (see [13] and [12] for two alternative options forthe theta kernel).
23

Lifts from “simpler” spaces with special properties often yield strange or exceptional
modular forms which can be used to understand or narrow down conjectures that areoften precisely false on the image or pre-image of such lifts. For example, the Shimura
lift generally sends cusp forms to cusp forms, but there is an exceptional class offorms known as unary theta functions in weight 3/2 which are cusp forms but whose
image under the Shimura lift is an Eisenstein series. These unary theta functions arealso counter-examples to the Ramanujan–Petersson conjecture, which states that thecoefficients of weight κ ∈ 21 Z cusp forms f satisfy
κ−1|af (n)| f,ε n 2 +ε .
The coefficients of the unary theta functions grow like n1/2 , contradicting the conjecture in this wide of generality. However, for integral weight cusp forms f ∈ S2k ,the conjecture is a celebrated result of Deligne [7] and it is conjectured that the
Ramanujan–Petersson conjecture holds in half-integral weight as long as f is orthogonal to unary theta functions.
!
4.4.3. Computation of the inner product by the Brunier–Funke pairing. For f, g ∈ M2k,
we next describe a way to compute the inner product between these two forms. Let
G be a harmonic Maass form for whichξ2−2k (G) = g.
The inner product between f and g is then given by the Bruinier–Funke pairing
between the function G and f , given byX{f, G} :=af (−n)a+(4.5)
G (n),
n∈Z

where a+G (n) is the nth coefficient of the holomorphic part of the Fourier expansion.
In particular, we havehf, gi = {f, G}.
(4.6)
The pairing is useful for computing inner products because only finitely many termsin (4.5) are non-zero.
Roughly speaking, the pairing is shown by using Stokes Theorem to evaluate theintegral instead of the unfolding method described in Section 4.2. The pre-image G ofg under ξ2−2k naturally appears in this context.
4.5. Inner products for meromorphic modular forms. We would now like todefine an inner product on arbitrary meromorphic modular forms f, g ∈ S2k . However,an arbitrary meromorphic modular form f ∈ S2k may be decomposed into two pieces,!
one of which only has poles at the cusps (i.e., it is in M2k) and one of which only haspoles in the upper half-plane (vanishing towards all cusps); we call forms of the second
type weight 2k meromorphic cusp forms and denote the subspace of such forms by
S2k . It thus essentially suffices to consider inner products between forms f, g ∈ S2k!
(technically, we also have to take inner products between forms f ∈ M2k
and g ∈ S2k ,but hybrid approaches for the regularizations will work in full generality and we ignore
the details here).
244.5.1. Regularization of Petersson. The idea that Petersson used to generalize (4.1)
is very similar to the idea used in the regularization (4.4). Instead of cutting off thefundamental domain away from i∞, one cuts out small neighborhoods around eachpole z of f or g and then shrinks the hyperbolic volume of the neighborhoods to zeroin a limit. In particular, for z ∈ H define the ballBε (z) := {z ∈ H : rz (z) < ε} ,where rz (z) := |Xz (z)| with
z−z
.
z−z
The functions rz (z) are naturally connected to the hyperbolic distance d(z, z) betweenz and z = z1 + iz2 in H via the formulad(z, z)rz (z) = tanh.
2
Xz (z) :=Let [z1 ], . . . , [zr ] ∈ PSL2 (Z)\H be the distinct SL2 (Z)-equivalence classes of all of
the poles of f, g ∈ S2k and choose a fundamental domain F ∗ such that all z` lie in theinterior of Γz` F ∗ , where Γz is the stabilizer of z in PSL2 (Z). Petersson’s regularized
inner product is then defined byZ
dxdy
hf, gi :=lim +f (z)g(z)y 2k 2 .
(4.7)
S
ε1 ,...,εr →0yF ∗ \( r`=1 Bε` (z` ))A necessary and sufficient condition for the convergence of the regularization (4.7) is
given by Petersson in [15, Satz 1]. Furthermore, certain Poincaré series related to theelliptic expansions (Petersson proved an elliptic coefficient formula as well; cf. [15, Satz9]) with poles in the upper half-plane were also shown to be orthogonal to cusp forms
in [15, Satz 7]. Once again, Petersson’s necessary and sufficient condition implies that
his regularization diverges in particular when evaluating Petersson norms for elementsof S2k which are not cusp forms.
4.5.2. A new regularization. Since Petersson’s regularization still sometimes diverges,
one requires a further regularization; we recall the construction from [5]. The idea issimilar to the regularization of Bringmann–Diamantis–Ehlen; roughly speaking, theintegrand in (4.1) is multiplied by an SL2 (Z)-invariant function Hs (τ ) which removes
the poles of the integrand whenever Re(s) is sufficiently large. We then take theconstant term of the Laurent expansion around s = 0 to be our regularization. To be
more precise, let [z1 ], . . . , [zr ] ∈ PSL2 (Z)\H be the distinct SL2 (Z)-equivalence classesof all of the poles of f and g and defineZ
2k dxdyhf, gi := CTs=0f (z)Hs (z)g(z)y
,
(4.8)
y2
SL2 (Z)\H
where
Hs (z) = Hs1 ,...,sr ,z1 ,...,zr (z) :=

r
Y
`=1

Here
hs` ,z` (z) := rz2s` ` (M z),
25hs` ,z` (z).

with M ∈ SL2 (Z) chosen such that M z ∈ F ∗ . Moreover CTs=0 denotes the constantterm in the Laurent expansion around s1 = s2 = · · · = sr = 0 of the meromorphic
continuation (if existent).
In the same sense that the results in [3] may be viewed as an analytic definition
for a regularized inner product satisfying the Bruinier–Funke pairing for arbitrary!
, the above regularization may be viewed as an analytic definition for af, g ∈ M2k
regularized integral giving a similar pairing for all f, g ∈ S2k . However, instead ofdefining the pairing via the Fourier expansions, the pairing is defined via the elliptic
expansions of f and a weight 2 − 2k polar harmonic Maass form (i.e., a harmonic
Maass form with singularities in the upper half-plane) G which is a pre-image of gunder the ξ-operator. To describe the pairing, the elliptic expansion of f ∈ S2k aroundz ∈ H is given byX
f (z) = (z − z)−2k(4.9)
af,z (n)Xzn (z).
n−∞For the polar harmonic Maass form G, we again denote the coefficients of its meromorphic part (i.e., of the form in (4.9)) by a+G,z (n).
Denoting z2 := Im(z) and writing ωz for the size of the stabilizer Γz of z in PSL2 (Z),the pairing is given by (see [4, Proposition 6.1])
X
π X{f, G} :=af,z (n)a+(4.10)G,z (−n − 1).
z2 ωz n∈Zz∈SL2 (Z)\H

It is again important to emphasize that the pairing gives a formula for the inner product with only finitely many coefficients in (4.10) non-zero. In comparison, Petersson
evaluated his inner product (4.4) (resp. (4.7)) on [15, pages 42–43] via the Fourier(res. elliptic) coefficients of the forms f and g themselves, but his evaluation is given
as an infinite sum, so one can only obtain an approximation for the inner product bycomputing the Fourier (resp. elliptic) coefficients. In other words, Petersson’s constructions are better in the sense that they are given in terms of the coefficients of theoriginal functions, while one is required to introduce new functions to determine (4.5)
and (4.10), but the sums in these pairings are instead finite.
4.5.3. Higher Greens functions. The regularization (4.8) was used in [5] to computethe inner product betweenXkfQ (z) = fk,−D,[Q] (z) := D 2
Q(z, 1)−kQ∈[Q]

for positive-definite integral binary quadratic forms Q of discriminant −D. These areweight 2k meromorphic modular forms which have poles of order k at the unique zeroτQ of Q in H. The evaluation of the inner product between two such functions is doneby again using Stokes Theorem to rewrite the inner product as the pairing (4.10) in
terms of the elliptic coefficients of fQ and the elliptic coefficients of the meromorphic
part of a polar harmonic Maass form GQ associated with fQ via the ξ-operator. It
then remains to explicitly compute the elliptic coefficients occurring in (4.10).
In particular, choosing two such binary quadratic forms Q and Q, the inner productbetween fQ and fQ is related to the higher Green’s function Gk : H × H → C, whichis uniquely characterized by the following properties:
26

(1) Gk is a smooth real-valued function on H × H \ {(z, γz)|γ ∈ Γ, z ∈ H}.
(2) For γ1 , γ2 ∈ Γ, we have Gk (γ1 z, γ2 z) = Gk (z, z).
∂ ∂(3) Denoting ∆0,z := −4y 2 ∂z
, we have∂z∆0,z (Gk (z, z)) = ∆0,z (Gk (z, z)) = k(1 − k)Gk (z, z) .
(4) As z → z
Gk (z, z) = 2ωz log (rz (z)) + O(1).
(5) As z approaches a cusp, Gk (z, z) → 0.
These higher Green’s functions have a long history, appearing as special cases of theresolvent kernel studied by Fay [8] and investigated thoroughly by Hejhal in [10], forexample. Gross and Zagier [9] conjectured that their evaluations at CM-points are
essentially logarithms of algebraic numbers, which has been since provenR 1 a−1in a numberof cases. To state the connection with inner products, let β(a, b) := 0 t (1 − t)b−1 dt
be the beta function, and let Q−D denote the set of positive-definite integral binary
quadratic forms of discriminant −D < 0. Evaluating the elliptic coefficients in (4.10)
for fQ and GQ then yields the following theorem.
Theorem 4.2 (Theorem 1.5 of [5]). For Q ∈ Q−D1 and Q ∈ Q−D2 (−D1 , −D2 < 0discriminants) with [τQ ] 6= [τQ ], we have
hfQ , fQ i = −π(−4)1−k Gk (τQ , τQ ).
(2k − 1)β(k, k) ωτQ ωτQ
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1995.
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[15] H. Petersson, Über automorphe Orthogonalfunktionen und die Konstruktion der automorphen
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Department of Mathematics, University of Hong Kong, Pokfulam, Hong Kong
E-mail address: bkane@maths.hku.hk

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