HARMONIC MAASS FORMS AND MOCK MODULAR FORMS:OVERVIEW AND APPLICATIONSLARRY ROLEN

Summer Research Institute on q-Series, Nankai University, July 2018.
We have seen in Michael Mertens’ lectures that the “web of modularity” reverberates across many
areas of mathematics and physics. In these lectures, we will discuss an extension of this web and its
consequences and uses in different areas, and, in particular, in combinatorics and q-series. In fact, a
lot of the best applications in the theory of harmonic Maass forms and motivating examples which
historically motivated important results in our field have come from its interactions with these othersubjects, in particular combinatorics and physics. For example, combinatorialists have a different
set of tools and intuitions, and a knack for discovering interesting new examples of modular-type
objects. Thus, collaboration and communication between the two areas, such as that established at
this conference, is very fruitful in both directions.
Before we begin, I have two brief comments. Firstly, a survey of many of the things in these notes
and more generally the theory and applications of harmonic Maass forms can be found in the book on
harmonic Maass forms by Bringmann, Folsom, Ono, and myself. Secondly, if you have any comments(or questions) about any of these notes, such as typos or general questions about the material, proofs,
how to apply the techniques discussed here, etc., please feel free to email them to me.
1. History and motivation
There are numerous beautiful examples of combinatorial generating functions which are both qhypergeometric series and modular forms. As a fundamental example, consider the partition functionp(n), which counts the number of integer partitions of n, that is, the number of ways to write n as asum of a sequencePof non-increasing integers. As we have seen in the earlier lectures, the generating
function P (q) := n p(n)q n is essentially a modular form of weight −1/2, and this can be used toprove congruences and estimates/exact formulas for the numbers p(n). We saw this by using theclassical product formula for P . There is another formula for P , which arises from a counting ofpartitions while keeping track of sizes of Durfee squares in the Ferrers diagrams. Namely,
P (q) =

X q n2
(q)2n
n≥0

,

Q
j
where (a; q)n = (a)n := n−1
j=0 (1 − aq ). This form is beautiful and useful, but hard to deduce modularity from. Being able to do so would be incredibly useful, and is a problem frequently encounteredin this area, either explicitly or implicitly. This was summarized neatly in the following problem,
popularized by George Andrews.
Problem (Andrews). How can one directly prove the modularityof P (q) directly from the
PpropertiesQ
q-hypergeometric expression above, i.e., without first proving a=identity?
This problem is hard (in particular, unsolved). However, while modularity is difficult to pin down
exactly on the level of q-hypergeometric series, modular forms do leave their fingerprints behind. Thiscan be seen via the following elementary lemma, which follows directly from the transformations in
the definition of a modular form. A proof of this lemma in this form can be found in Vlasenko andZwegers’ paper on Nahm’s conjecture.
1

2

LARRY ROLEN

Lemma. If f ∈ Mk (Γ) for a subgroup Γ of finite index in SL2 (Z), then there are constants a, b suchthat as ε & 0,
 
iε
a/ε
e f
∼ be−k + o(εN )2πfor all N ≥ 0. That is, there is only one term in the asymptotic expansion to all orders.
The point is that we are letting the modular variable τ tend to 0 from above along the imaginary
axis, or letting q tend to 1 radially from within the unit disk, and this can be related to the behavior
at i∞ by modularity, and the behavior there is easy to determine using the Fourier expansion.
However, among q-hypergeometric series, this type of asymptotic expansion is very rare. Almost
all q-hypergeometric expansions have complicated expansions, usually of an infinite form, which arevery far from this elementary asymptotic behavior. In fact, in his original letter where Ramanujan
describes his mock theta functions, he begins by giving a “generic” example with an infinite type
expansion which doesn’t look related to modular forms.
Example 1. A famous conjecture of Nahm claims that certain q-hypergeometric series are modular
if and only if certain elements of the so-called Bloch group (an object from K-theory) are torsion.
The first case of this conjecture claims to predict exactly those A, B, C ∈ Q for which
2

FA,B,C (q) :=X q An2 +Bn+C
n≥0

(q)n

is modular. Zagier solved this by proving the following result.
Theorem 1 (Zagier). The series FA,B,C is modular for exactly 7 (explicitly given) triples (A, B, C) ∈Q3 .
For example, these examples include the famous Rogers-Ramanujan function:
X q n21=,5(q)n(q; q )∞ (q 4 ; q 5 )∞n
where the right-hand “product-side” can be directly shown to be modular. How did Zagier prove
this? Essentially, by combining the type of reasoning encoded in the elementary lemma above with abrilliant method for computing the asymptotic expansions near q = 1 of each of the functions FA,B,C ,he reduced the candidates P
for modularity
to finitely many cases. In each of these cases, modularity isQestablished by applying a=identity, such as that for the Rogers-Ramanujan function above.
2. A surprising discovery
Ramanujan was of course well aware of examples such as, well, the Rogers-Ramanujan identities.
Every example of such an identity, and correspondingly of a q-hypergeometric modular form, ishighly interesting. As mentioned above, he computed (at least one) examples of q-hypergeometricasymptotic expansions near roots of unity like q = 1. Shortly before his untimely death, while isolated
in India, he wrote a brief letter to Hardy describing a startling set of examples he discovered which heknew are not quite modular (in his language, are not “theta functions,” or quotients, products, andlinear combinations thereof), but which are so uncannily close that something must be very special
about them. For instance, one of his functions was the series
X q n2f (q) :=,(−q)2nnwhich looks similar to the expression for P above, but with a minus sign inserted. In particular, it
doesn’t satisfy the lemma above (which can be modified to approach other roots of unity, or, in τ ,for τ approaching rational numbers vertically from above), but it doesn’t have the kind of infinite,

HARMONIC MAASS FORMS AND MOCK MODULAR FORMS

3

wild expansions which are typical for non-modular series. In particular, he considered the (up to arational power of q) weight 1/2 weakly holomorphic modular formX2b(q) := (q; q 2 )∞(−1)n q n .
n∈ZRamanujan then described a shocking numerical near cancellation between these two functions nearq = −1, where they both explode. For example (using a modern computer we can compute that),f (−0.998) ≈ −6 · 1090 . However,f (q) + b(q) ≈ 3.992 ≈ 4.
N.B.: Where does the number 4 come from? We will address this question later.
This near miss example is what Ramanujan called a mock theta function, is it is “imitating” orpretending to be the “theta function” b(q) near q = −1, and also has predictable behavior near otherroots of unity. Ramanujan’s letter goes on to give a definition of mock theta functions as he conceivedof them, where he attempts to analytically describe the strangenesss of what he observed, and to givea number of further examples he claims satisfy similar properties. His examples gave tantalizing hintsof a new structure of modular-type objects, and this feeling served as a final challenge of Ramanujan
which tantalized number theorists for decades to come.
3. A framework for Ramanujan’s mock theta functions
Building on earlier work expanding Ramanujan’s examples and shedding light on their transformation properties, combinatorial implications, and various representations (such as those in a form
compatible with Zwegers’ three-fold path described below) and properties due to many people suchas Andrews, Berndt, Hickerson, Watson, and a number of others, a framework for what sorts of functions Ramanujan was studying finally emerged about 80 years after his original letter, in the early
2000’s. This can be explained using the theory of harmonic Maass forms, which was independently
developed in a seminal paper due to Bruinier and Funke. There, they defined a harmonic Maass
form, which can be roughly summarized as follows. Essentially, harmonic Maass forms are no longer
required to be holomorphic, but rather to satisfy a second order differential equation instead (and,being in the kernel of a “Laplacian,” they are termed “harmonic”).
“Definition”. A harmonic Maass form (HMF) is a function which satisfies the following.
(1) It transforms like a modular form.
(2) It is in the kernel of∆k := −ξ2−k ξk ,where∂
ξk := 2i Im(τ )k .
∂τ(3) It grows like at most as quickly as a weakly holomorphic modular form at the cusps (i.e., it
has a principal part, and once this is subtracted, it decays rapidly).
The space of HMFs of weight k is denoted by Hk (with the subgroup of SL2 (Z) suppressed freely forconvenience and clarity in these notes).
Note that classical modular forms satisfy the conditions of the definition of a HMF, as the operator
ξk kills holomorphic functions by definition, and so ∆k does too. The operator ξk plays a central
role in the theory of harmonic Maass forms. It shuttles between “dual weights” k and 2 − k, which
in many ways reflect dual worlds of existence. In particular, ξk applied to a HMF of weight k yields
a modular object of weight 2 − k. Since ∆k splits as a composition of two ξk operators (up to asign, which is there for classical reasons), the image under ξk must be in the kernel of ξ2−k . But this
precisely means that the image is a holomorphic function, and so one must obtain a holomorphic

4

LARRY ROLEN

modular form. By considering the growth condition at cusps, it turns out that one always obtains acusp form. Thus, one obtains a mapξk : Hk → S2−k .
Importantly, Bruinier and Funke used geometry to prove that this map is surjective (one can thinkof this as a sort of “Serre duality”), and, following Zagier, we commonly refer to the image of a HMF
under it as the shadow of the HMF. Moreover, just as holomorphicity and translation invarianceimply the existence of Fourier expansions of modular forms, harmonicity and translation invairance
prove that HMFs have a Fourier expansion which splits into two pieces (since its a second orderdifferential equation):f ∈ Hk =⇒ f = f + + f − .
Here, f + is called the holomorphic part and f − is called the non-holomorphic part. Fittingly, f +is holomorphic, and f − is not (assuming its non-zero, i.e., that we don’t have a classical modularform). In particular, f + is an ordinary q-series, whose coefficients often encode some important
object in whatever application we may be studying, and the coefficients of f − , while decorated bycertain non-holomorphic functions, are the same numbers which occur in the Fourier expansion of the
shadow. Thus, the non-holomorphic part “comes from” classical modular forms and understanding
it is equivalent to knowing about the shadow of f .
Definition 3.1. A mock modular form is the holomoprhic part of a harmonic Maass form. A mocktheta function is a mock modular form whose shadow is a unary theta series.
Around the same time as Bruinier and Funke developed this theory, Zwegers proved that thesesame properties can be applied to modifications of Ramanujan’s mock theta functions, and so, in
this language, he proved the following in his famous Ph.D. thesis.
Theorem 2 (Zwegers). Ramanujan’s mock theta functions are mock theta functions according to thedefinition above (up to multiplying by a rational power of q and a rescaling of q by a rational power).
Thus, Zwegers proved that Ramanujan’s functions are missing an extra piece, which, when added,“corrects” their modularity transformations. Moreover, this piece is “simpler” than the original mocktheta function, as it contains the same information as a unary theta function, and so this completionis particularly useful for understanding the original mock theta function. This realization led toan explosion of applications across mathematics and physics, including to representation theory,
combinatorics, black holes, and arithmetic geometry.
4. The three-fold path of Zwegers
Zwegers gave three related ways to realize Ramanujan’s mock theta functions as examples of broadclasses of functions. Firstly, he studied properties of Appell-Lerch series, which have a shape such as
X q Q(n)
,
n
1
−
aq
n∈Z
where the sum over Z can be replaced with other lattices, and Q(n) can be a quadratic form, forinstance. If you expand the denominator as a geometric series in a certain range, you can findexpressions for indefinite theta functions. As an example of the shape of such an indefinite theta
function, we have the following identity of Andrews for Ramanujan’s fifth order mock theta function
f0 (q):X
1 X
j n(5n+1)/2−j 2.
f0 (q) =− (−1) q(q)∞ n+j≥0 n+j<0
n−j≥0

n−j<0

HARMONIC MAASS FORMS AND MOCK MODULAR FORMS

5

This is called an indefinite theta function as the quadratic form in the exponents of q is indefinite(has both positive and negative eigenvalues). As mentioned in the earlier lectures, this means that asum over all choices of (n, j) would diverge, which is why we must do something more complicatedsuch as summing only in a certain range.
N.B.:Much more on these functions, with the explanations of the ideas of the proofs and a discussion of very recent developments on indefinite theta functions and generalizations of harmonic Maass
forms into a sequence of spaces of modular-type objects is discussed in detail (with exercises) in thenotes on indefinite theta functions on the front page of my personal website. In accordance with thecomment at the beginning of these notes, collaboration with combinatorics may play a key role inthe development of these new structures.
Finally, Zwegers showed mock modularity properties of coefficients of meromorphic Jacobi forms.
Important work of Dabholkar, Murthy, and Zagier follows up on this, and the reader (including
readers with interest in q-series) are strongly encouraged to read their work as well. It should also
be pointed out that coefficients of Jacobi forms, for example of infinite products in two variables, hasoften been important in combinatorics, for example, in using the constant term method. For instance,in his important memoir on generalized Frobenius partitions, Andrews proves formulas which canbe used to connect combinatorial generating functions to both coefficients of Jacobi forms and mock
modular forms, as they fit into the examples along the lines of what is briefly discussed here.
5. A few important applications
Here, we will survey just a few of the many topics where mock modular forms play a role.
5.1. Combinatorics. As a starting point, let us consider the Ramanujan congruences modulo 5 and
7 for p(n):p(5n + 4) ≡ 0 (mod 5),p(7n + 5) ≡ 0 (mod 7).
Dyson conjectured, and Atkin and Swinnerton-Dyer proved, that these congruences are combinatorially explained by the rank statistic of a partition π, r(π), which is the largest part minus the numberof parts. If we form the generating function of partitions refined to keep track of ranks:XX
q n ζ r(π) ,R(ζ; q) :=n

π is a partition of n2πiz

where ζ := eand z ∈ C, then we observe that R(1; q) = P (q) is modular, and R(−1; q) = f (q) is
mock modular. Bringmann and Ono proved that these two observations are instances of the following
fact.
Theorem 3 (Bringmann-Ono). For any root of unity ζ, R(ζ; q) is a mock modular form (up tominor considerations such as multiplying by a rational power of q).
This theorem immediately implies many congruences, and also allows one to study asymptotics ofranks very efficiently. We will see how these two properties, congruences and asymptotics, can bestudied for mock modular forms in generality below.
5.2. Arithmetic Geometry. As an example of the applications to geometry, we consider the picture
for elliptic curves. Suppose that E is an elliptic curve over Q. Then by the Modularity Theorem,
there is a weight 2 newform associated to the curve, which we will denote by fE . We denote byΛE the lattice such that E ∼=C C/ΛE . We will sketch an idea due to Guerzhoy, which allows us
to give a “canonical” lifting under the (∞-to-1) map ξ0 : H0 → S2 (recall that Bruinier and Funkeproved this is surjective). Crucially, this is compatible with “arithmetic,” such as p-adic properties.
Other methods for lifting, such as following Bruinier and Funke’s proof or using the method of
Poincaré series sketched below, either aren’t as explicit (the following method relies only on classicalmodular forms theory and classical elliptic functions and is direct), or destroy arithmetic (writing

6

LARRY ROLEN

a cusp form as a sum of cuspidal Poicnaré series is always possible in principle, but is not easy tounderstand explicitly and destroys the arithemetic properties by possibly expressing a q-series withintegral coefficients using coefficients with transcendental coefficients; this method is better-suited toexistence proofs and asymptotic analysis).
Consider the Weierstrass p function ℘, which is an elliptic function. The (negative of the) antiderivative is the Weierstrass zeta function:℘(ΛE ; z) = −Z 0 (ΛE ; z).
Now Z is not elliptic, but only nearly so. Now, considering the modular form fE =:we consider the formal antiderivative (known as the Eichler integral)X an
EE (z) :=
qn,
n
n≥1

P

n≥1 an qnagain,

so thatqd(EE ) = fE .
dq

Now this function is nearly modular of weight 0, but not quite. This is because slashing in weight 0
and 2 (and only in this case!) commutes with differentiation, in the sense the for any function g on
the upper half-plane and any γ ∈ SL2 (Z),D (g|0 γ) = (Dg)|2 γ
dwhere D := q dq
. Thus, for any γ in the group Γ under which fE transforms, the function EE |0 (1 − γ)
has derivative equal to zero, and so is a constant. Thus, we obtain a map

Ψ: Γ → C
by setting Ψ(γ) to be the constant obtained by evaluating EE |0 (1−γ) at any point in H. But classicaltheory of Eichler and Shimura says that the image of Ψ is exactly the lattice ΛE ! Thus, the “errors
to modularity” of EE all live in this lattice. Thus, if we consider the function Z(ΛE ; EE ), it would
actually be modular of weight 0 if Z were elliptic, as errors to modularity would be swallowed up bylattice invariance. However, since Z is so very nearly an elliptic function, it can be corrected with aminor non-holmorphic correction term. Working this out, one finds that Z(ΛE ; EE ) is really a mock
modular form. Moreover, the non-holomorphic part is especially simple, so computing the shadow of
the mock modular form is essentially trivial, and easily returns the original cusp form fE . Thus, wehave answered our question of how to find explicit lifts under ξ0 . In a paper of Alfes, Griffin, Ono,and myself, we explored the implications for elliptic curves. By applying work of Bruinier and Ono,
as well as Waldspurger and Kohnen-Zagier, it turns out that we can directly construct a single HMF
whose holomorphic part encodes the vanishing of L-derivatives of quadratic twists of E, and whose
non-holomorphic part has coefficients encoding the vanishing of the central L-values of the twists.
By Birch and Swinnerton-Dyer conjecture, these coefficients should thus determine the rank of thequadratic twists of E up to rank 2, that is, they “know” whether the rank is 0, 1, or greater than 1.
More generally, these types of lifting problems are central in many questions on harmonic Maass
forms, and as we now know, in classical modular forms. Returning to q-series, recall that Lehmer’s
conjecture states that all coefficients τ (n) of the Ramanujan Delta function ∆(τ ) are non-zero. Again
by Bruinier-Funke, we can show that “nice” lifts of ∆ to weight 2 − 12 = −10 exist. It turns out that
Lehmer’s conjecture is equivalent to the coefficients of the corresponding mock modular form beingtranscendental. Thus, an explicit understanding of the lifting problem in a similar manner which iscompatible with arithmetic for higher weights would be highly interesting.

HARMONIC MAASS FORMS AND MOCK MODULAR FORMS

7

6. Quantum Modular forms
Following Zagier, a quantum modular form of weight k, roughly speaking, is a function f : Q∪∞ →C such that the “errors to modularity” f |k(1 − γ) are “nice functions” for all γ in some subgroup of
SL2 (Z) (as this sounds similar to the previous section, it is worth pointing out that Eichler integrals
can be used to define quantum modular forms). What nice means depends on the context, andthe point is that this definition includes many different examples with a different flavor which Zagier
surveyed in his original paper on the subject and which have been developed since. A typical behavior
might be a function which is defined on Q, with no way or no obvious way to extend it to R, butwhose cocycles become not only defined on R but also become C 10 , C ∞ , analytic, etc.
As one example, recall that above we saw that Ramanujan’s mock theta function f (q) is almost
cancelled out by the (essentially) modular form b(q) near q = −1:
f (q) + b(q) ≈ 4.
Ramanujan conjectured this value 4 in his letter, as well as similar almost cancelling values off (q) ± b(q) at other roots of unity. It turns out that these constants which Ramanujan studied
encode a hidden structure, namely, a quantum modular form. This is a good example of the factthat going back to the original sources in a subject and really reading them closely can be very usefulin any field. The hidden structure behind Ramanujan’s calculations is revealed in the following
elegant result of Folsom, Ono, and Rhoades.
Theorem 4 (Folsom-Ono-Rhoades). For every even order 2k root of unity ζ, we have the following
radial limit formula:X
lim f (q) − (−1)k b(q) = −4
(1 + ζ)2 (1 + ζ 2 )2 · · · (1 + ζ n )2 ζ n+1 .
q→ζnNote the right hand side is actually a finite sum for any root of unity ζ. As an exercise, consider
what happens for odd order roots of unity. What is the meaning of the right hand side? We alreadyknow that f (q) is a specialization of the rank generating function R. It turns out that there is anelegant radial limit theorem for any other specialization of R, which are mock modular as discussed
above. To explain this, we need the it crank generating function C(ζ; q), which si defined in thesame way as R but with “rank” replaced by “crank,” another partition statistic which simultaneously
explains Ramanujan’s congruences modulo 5, 7, and 11. This was conjectured by Dyson and foundby Andrews and Garvan. The generating function enjoys the infinite product formula:
(q)∞
C=
.
(ζq)∞ (ζ −1 q)∞
This is essentially the inverse of the Jacobi theta function times η 2 , and so it is a meromorphic Jacobi
form of weight 1/2. Thus, as discussed in the previous lectures on modular forms, its specializations,
including b(q), are all weakly holomorphic modular forms of weight 1/2. Furthermore, if we consider
a generating function U(ζ; q) which encodes strongly unimodal sequences by size and a statistic onthem which is also called rank, then it turns out thatX
U(ζ; q) =(−ζq)n (−ζ −1 q)n · q n+1 .
nFolsom, Ono, and Rhoades then proved the following general limiting formula, which pleasinglyconnects three types of combinatorial generating functions in one and generalizes the above formulafor f (q) and b(q).
Theorem 5 (Folsom-Ono-Rhoades). Under certain conditions on the roots of unity ζ, ζ 0 , and forcertain elementary factors ∗, we have thatlim (R(ζ; q) − ∗C(ζ; q)) = U(−ζ; ζ 0 ).

q→ζ 0

8

LARRY ROLENThis can be proved using asymptotic calculations, which require significant work. The reader is also
referred to later, simpler proofs for such formulas by Zudilin, and later by Folsom, Ono, and Rhoades,
and I highly recommend these works especially for a combinatorial audience. Since specializationsof R are mock, and specializations of C are modular, it is natural to ask what kind of function U is.
We noted above that it becomes a finite
P sum for roots of unity ζ, so it is well defined as a function
on z ∈ Q. As an example, U(−1; q) = n (q)2n q n+1 , Byrson, Pitman, Ono, and Rhoades proved that
this function is actually (at roots of unity)U(−1; q) = F (q −1 ),where F (q) is Kontsevich’s function
F (q) =

X(q)n .
nThis function is known to be a quantum modular form, as Zagier proved by establishing his “strange
identity”
n2 −1
1X
nχ12 (n)q 24 .
F (q)“ = ” −2 n≥1Quotation marks are used here as the left hand side now really only makes sense at roots of unity,
while the partial theta function (it is summed over only half of a lattice) is only defined for |q| < 1.
Note that this expansion is very closely related to the theta function representation for η(τ ). The
world where F (q) lives, the Habiro ring which is known to encode invariants in knot theory, can bethought of as a set of “analytic functions at the roots of unity.” In particular, we can consider the
expansion of this function around roots of unity, such as the following one around q = 1:XXξn q n .
(1 − q; 1 − q)n =:nn

It is an instructive exercise to prove that the coefficients on the right hand side converge, as wellas that similar expansions make sense at other roots of unity. These coefficients ξn are importantnumbers in knot theory. They were used by Zagier in the study of Vassiliev invariants. Just as
modular and mock modular (as we will see) forms are very useful in studying asymptotics, Zagier
used quantum modularity (in current language) to study the growth properties of these numbers.
It also turns out that the same numbers count a number of interesting things in combinatorics,
and these were further studied by Andrews and Sellers. They observed and proved that they satisfy
infinitely many Ramanujan-type congruences. Further work by Guerzhoy, Kent, and myself explained
this from a general quantum modular perspective and for general types of partial theta functions.
Thus, in contrast to Ramanujan’s statement that “false” (essentially the same theory as partial) theta
functions “do not enter mathematics as beautifully” as his mock theta functions, they do enter into
mathematics, in fact Ramanujan’s mathematics, quite beautifully. Though, to be fair, it is difficult
for most things to match the beauty of the mock theta functions.
7. General philosophy of applications to combinatorics
We conclude with a general discussion of how the theory of harmonic Maass forms gives powerfultools for studying two types of properties of sequences of numbers when you are lucky enough tohave mock modular forms lurking: congruences and asymptotics.
7.1. Congruences. We have seen that modular forms are a powerful tool for proving congruencesof arithmetic sequences. Here, we briefly sketch an approach for obtaining congruences for mock
modular forms. The key idea is to reduce to the classical modular case. For instance, if the shadow isa linear combination of unary theta functions (for example, for a mock theta function; this property
is also almost required for most combinatorial sequences as mock modular forms don’t usually haveintegral Fourier coefficients otherwise). Then we are lucky, and we can twist away the shadow. As weHARMONIC MAASS FORMS AND MOCK MODULAR FORMS

9

saw in the earlier lectures, sieving operators, which restrict the Fourier coefficients of a modular formby throwing away all that don’t lie in a fixed arithmetic progression, preserve modularity. The proof ofthat fact only used the transformation properties, i.e., it is a statement on the level of slash operators.
Thus, sieving the coefficients of a HMF also yields a new HMF. But if the shadow is supported on
finitely many square classes, then we can choose a progression where the coefficients of the shadow
vanish. This then reduces us to the world of classical modular forms, and we can generically checkin many cases that this procedure doesn’t also kill the holomorphic part. By combining with theclassical theory of modular form congruences, we thus obtain many congruences for coefficients of themock modular form we may want to study. For instance, this already tells us, by applying Treneer’s
thesis, that mock theta functions satisfy infinitely many linear congruences. A number of papers
have applied this technique to obtain strong results.
7.2. Asymptotics. We have seen that modularity can be used to prove asymptotics, and indeed
exact formulas, for the partition function p(n) by means of a Tabuerian Theorem (for the asymptotic)or the circle method (for more precision, depending on which variant of the circle method is used).
Rademacher’s formula turns out to be a hint of a more general structure of modular forms. We wanta procedure to take any modular form (or mock modular form; as we shall see, answering the firstquestion naturally leads us to the second) and give a procedure for computing the asymptotics ofthe coefficients of the modular form quickly. In the case of the partition function, we saw that thecircle method consists of a careful study of the generating function near its singularities, and so its
principal parts at (all equivalent in this case) the cusps dictate the asymptotic growth properties.
The theory of harmonic Maass forms gives a way to diagonalize spaces based on principal parts. As
is well-known, for general congruence subgroups, weakly holomorphic modular forms with arbitrary
principal parts don’t exist. However, harmonic Maass forms are allowed to have arbitrary principal
part. Considering just the cusp ∞ for simplicity, for any k ≤ 0, any m ≥ 1, and any congruencesubgroup Γ, we have a Maass-Poincaré series Fk,m,Γ ∈ Hk (Γ) such that
Fk,m,Γ = q −m + O(1)at ∞, and Fk,m,Γ is holomorphic at all inequivalent cusps. This series is a lift of the classical Poincaré
cusp forms under ξk , and it is defined by a similar procedure as we saw for the Eisenstein series
and other Poincaré series before; namely, by averaging a certain function hit with slash operators toforce modularity while retaining good convergence and growth properties. Since there are no (nonconstant) holomorphic modular forms of weight k ≤ 0, this means that we can take the appropriate
linear combination of these Poincaré series which matches the principal part of any weakly holomrophic, and more generally any mock, modular form in these weights, and the difference is simply
a constant. The advantage here is that the series Fk,m,Γ have completely explicit formulas for their
Fourier coefficients, in terms of Kloosterman sums and Bessel functions, just as Rademacher foundfor p(n). Thus, we can quickly determine an exact formula for the coefficients of any of these forms
with a routine, finite check. We conclude by briefly considering other weights. For weights between 0
and 2, convergence of the Poincaré series is an issue. However, general work of Goldfeld and Sarnak
can be used to analytically continue the Poincaré series in weights 1/2 and 3/2 (which, as describedabove, are often the ones of combinatorial interest), and the same formulas for Fourier expansions
still hold. In these weights, as well as for methods of Poincaré series which could be applied in weights
larger that 3/2 (note that there are no cusp forms of non-positive weight, so modifications have to bemade for large weights), note that due to the existence of holomorphic modular forms, the methodof matching principal parts won’t yield exact formulas. However, the obstruction to obtaining exact
formulas will then only have polynomial growth, and if you started with a function with a pole atsome cusp, so that there will be rapid growth of the Fourier coefficients, the method of Poincaré
series still yields a strong estimate, and, in particular, an asymptotic.