INDEFINITE THETA FUNCTIONS OF TYPE (n, 1) II: FURTHERDISCUSSION OF ZWEGERS’ FUNCTIONS AND THE PROOF OFTHEIR MODULARITYLARRY ROLEN

In the last lecture, we motivated and described Zwegers’ type (n, 1) indefinite theta
functions. Several key objectives remain. Firstly, we have to describe their exact relation
to the theory of harmonic Maass forms, and in particular how to compute their shadows.
After all, something should really only count as a completion if it is “simpler” than thething you started with (without some restriction like this, you could always “complete”
a function by adding minus itself to it!). Finally, we have to discuss the proof. Although
we will not give all details of the proof, and although we won’t follow Zwegers’ original
proof, we shall try to emphasize the intuition behind it and explain “why” functions
like E(z) should arise in our theta functions. This will be crucial when we discuss moregeneral types of indefinite theta functions in the next lecture as well.
1. Relation to harmonic Maass forms
In order to explicitly describe Zwegers’ indefinite theta functions in terms of BruinierFunke’s theory of harmonic Maass forms, we first decompose the function ρ from the lastlecture. For this, note that E can be related to the beta function, which is essentially anincomplete Gamma function (as required in the Fourier expansions of harmonic Maassforms). Specifically, for all x ∈ R, we have (cf. Exercise 1)
(1)E(x) = sgn(x) 1 − β x2 ,where for real, non-negative yZ ∞β(y) :=1t− 2 e−πt dt,yand we have11β(y) = √ Γ, πy .
2πThe humble identity (1) will turn out to play several key roles for us.
For any c1 , c2 ∈ CQ we may now write ρ(n; τ ) as a sum of the expressionsB(cj , n)2cjj
ρj (n; τ ) = ρj (n; τ ) := (−1) sgn (B(cj , n)) β −v ,Q(cj )ρ3 (n; τ ) = ρc31 ,c2 (n; τ ) := sgn (B(c1 , n)) − sgn (B(c2 , n)) .
12LARRY ROLEN

Specifically, we may decompose ρ as
ρ1 (n; τ ) + ρ2 (n; τ ) + ρ3 (n; τ )ρ (n; τ ) + ρ (n; τ )13
ρ(n; τ ) =ρ2 (n; τ ) + ρ3 (n; τ )
ρ3 (n; τ )if c1 , c2 ∈ CQ ,if c1 ∈ CQ c2 ∈ SQ ,if c1 ∈ SQ c2 ∈ CQ ,if c1 , c2 ∈ SQ .

Then we define for any c1 , c2 ∈ C Q the series
X+Θ+ρ3 (n; τ )e(B(n, b))q Q(n) ,a,b,c1 ,c2 (τ ) = ΘA,a,b,c1 ,c2 (τ ) :=n∈a+Zr−Θ−
a,b,c1 ,c2 (τ ) = ΘA,a,b,c1 ,c2 (τ ) :=X(ρ(n; τ ) − ρ3 (n; τ )) e(B(n, b))q Q(n) .

n∈a+Zr

The point of this definition, as the notation suggests, is that Θ+a,b,c1 ,c2 is the holomor−
phic part of Θa,b,c1 ,c2 , and, similarly, Θa,b,c1 ,c2 is the non-holomorphic part.
We want to discuss exactly how we can see Θ+ as a mixed mock modular form and how
to compute its shadow. For this, we require special non-holomophic functions, which
are like those used in the construction of Zwegers’ µb. For any a, b ∈ R and τ ∈ H, letX
n2sgn(n)β 2n2 v e(−nb)q − 2 .
Ra,b (τ ) :=n∈a+Z

Recall that ga,b denotes a special unary theta function
Xn2ne(nb)q 2 .
ga,b (τ ) :=n∈a+Z

Similarly to the case of the R function from Chapter 1 of Zwegers’ thesis, one can simply
differentiate to show the following, where∂∂τ
is the shadow operator of weight k (Exercise 2 will ask you to show this):r−1 √(2)ξ r2 (Ra,b ) = −v 2 2ga,b .
ξk := 2iv kThe non-holomoprhic parts (or equivalently, shadows) have shapes dictated by the following decompositions. If cj ∈ C Q ∩ Zr has relatively prime coordinates, then we write
B(c ,`)n = ` + N cj with ` ∈ a + Zr , N ∈ Z, such that 2Q(cj j ) ∈ [0, 1). As cj ∈ Zr , we may write
the set of such ` as a disjoint union [˙r B(cj , `)`∈a+Z :∈ [0, 1) =`0 + hcj i⊥Z ,`0 ∈P0,j2Q(cj )INDEFINITE THETA FUNCTIONS OF TYPE (n, 1) II3

rwhere P0,j is a suitable finite set and where hcj i⊥
Z := {λ ∈ Z : B(cj , λ) = 0}. Then theB(cj ,`0 )following can be shown by a direct computation, where `⊥0 := `0 − 2Q(cj ) cj andJ := {j ∈ {1, 2} : cj ∈ CQ }.
Proposition 1. Suppose that c1 , c2 ∈ C Q and that cj ∈ Zr have relatively prime coefficients. Then we have√ξ r2 (Θa,b,c1 ,c2 (τ )) = 2v r−1XXX×(−1)jg cB(cj , `0 )e (B (ν, b⊥ )) q Q(ν) .
(−2Q (cj ) τ )j∈I`0 ∈P0,j2Q(cj ),B(cj ,b)⊥ν∈`⊥
0 +hcj iZ

Here, the c denotes taking complex conjugation of the Fourier coefficients. The rightmost sum in the last equation is a theta function on a shifted (r − 1)-dimensional lattice,and so in particular is a modular form of weight (r − 1)/2. Thus, the shadow has asimple representation in terms of ordinary modular forms.
Example 1. Andrews provided a “Hecke-type double sum” representation of Ramanujan’s fifth order mock theta function f0 , which, after rewriting, yields
X
21 X
j 5n + n −j 2f0 (q) =− (−1) q 2 2 .
(q)∞ n+j≥0 n+j<0n−j<0n−j≥0

A short calculation then shows that we may write this in terms of the type (1, 1) indefinite
theta function
1q 60 X(sgn(B(n, c1 )) − sgn(B(n, c2 ))) q Q(n) e (B(n, b)) ,2η(τ )2n∈a+Zwhere1   
102−22210Q(j, k) :=5j − 2k , a :=, b := 1 , c1 :=, c2 :=.
5502
4As c1 , c2 ∈ CQ , we can apply Zwegers’ theory, and we find that
1q 60 +
Θ(τ ).
2η(τ ) a,b,c1 ,c2
Exercise (3) will ask you to compute the shadow of this mock theta function. Similar
examples can be used to show mock modularity and compute the shadows of all ofRamanujan’s original mock theta functions.
f0 (q) =Example 2. Sometimes, indefinite theta functions have enough internal symmetries
to become ordinary modular forms. For example, indefinite theta functions on higher
dimensional lattices can be built out of products of simpler (in)-definite theta series.
We will see an important example of such a decomposition in the next lecture, with4LARRY ROLEN

an application to Gromov-Witten theory. For now, we record the following well-known
Hecke-type expansion for the weight 1 cusp form!
XX
n2
m2
1
1
1
η(τ )2 =−(−1)n+m q 2 +2nm+ 2 + 2 n+ 2 m+ 12 .
n,m≥0n,m<0This can be seen by writing down the associated Artin representation, along the lines of(and in fact related to) the discussion of Cohen’s results the indefinite theta series σ, σ ?
studied in the next lecture.
2. Vignéras’ theorem and the proof of modularity of Zwegers’
functions
The main theoretical tool for showing modularity of theta series such as Zwegers’
is the following result of Vignéras, which determines the appropriate conditions for
multiplicative factors to add into the theta functions while retaining modularity.
Lemma. Let Q be an integral quadratic form of type (r − s, s), and suppose p : Rr → C
satisfies the following differential equation for some λ ∈ Z∆QE−(3)p = λp,2π
where E is the Euler operatorrX∂
E :=xj.
∂xj
j=1Assuming convergence, the theta functionX √ λy pv n+e(B(n, z))q Q(n)ΘA,a (z; τ ) = v − 2vn∈a+Znis a vector-valued Jacobi form of weight r/2 + λ.
How might one find nice solutions to this differential equation? It is difficult to findall solutions of large-order differential equations, but there is a trick to relate finding
solutions of Vigéras’ equation to a one-dimensional problem. Specifically, one can showthat if c is a fixed vector with Q(c) < 0, then a function!
B(x, c)g=f p−Q(c)satisfies the differential equation for a one-variable function f (z) if(4)zf 0 − f 00 /(2π)

returns a multiple of f (cf. Exercise 4). This is almost the famous differential equation
y 00 + 2xy 0 = 2ny,

INDEFINITE THETA FUNCTIONS OF TYPE (n, 1) II5

whose solutions are known. In particular, they can be written in terms of iterated
integrals of the complementary error function erfc, which when translated correctlyreturns our error function E(z) as a solution. Another family of solutions is given by
the Hermite polynomials, which are discussed in relation to indefinite theta series in
Stopples’ paper “Theta series for indefinite quadratic forms over real number fields.”
These Hermite polynomials are neatly defined by can be defined in terms of iterated
derivatives byim h2πz 2 ∂−2πz 2Hm (z) := ee.
∂z m
We thus have an idea where the function E comes from, and what makes it special.
By (1), it also has the right shape to produce harmonic functions (with respect to theweight k Laplacian) when plugged into Vignéras’ framework. Doing so directly will fail,as one gets a divergent series (indeed, the sign term in (1) doesn’t decay at alltowards
infinity). However, Zwegers made the brilliant observation that the difference of two
such functions does. Although this proof takes significant effort (and is one of the moredifficult parts of his proof as a whole), we can get an inkling of what is going on using
our decompositions above. The function β in (1) decays as y → ∞, so in some sense thenon-holomorphic part should have an easier time converging. The holomorphic part,on the other hand, inherits a difference of sign functions (note that by definition thesign function occurs in every case regardless of whether cj ∈ SQ or not). Thus, in thesummation over the lattice, only certain terms do not vanish in this difference of signfunctions, and the theta function is only really summing over a cone. If this cone is aregion where Q is positive-definite, our theta series will then be convergent, and it will
be modular as we have still summed the original E function over the whole lattice. Avery simple example of such convergence, which you can play with by hand, is given in
Exercise 7.
Finally, we have to make a remark about the cuspidal case. If cj ∈ SQ , then thediscontinuities of the sign function present a problem. Here, we must actually express
this as a limit of vectors cj,t and prove modularity of the limit of Vignéras theta functions.
This is facilitated, once again, by the fact that the sign function is the limiting function
remaining as y → ∞ in (1).

3. Exercises
R z −πt2
R |z|√2(1) Show (1) by writing 0 e
dt as sign(z) 0 e−πt dt and subsituting t = u.
(2) Prove (2).
(3) Compute the shadow of the fifth order mock theta function f0 in Example 1 (ifyou need a hint, an equivalent expression is given on page 77 of Zwegers’ thesis).
(4) Fill in the details for the claimed relation of the differential equation (4) toVignéras’ differential equation.

6LARRY ROLEN(5) Let



1 00
A := 0 −1 0 0 0 −13and c ∈ R such that Q(c) = −1. Write x ∈ R3 as x = (x1 , x2 , x3 ). The
associated bilinear form is (n = (n1 , n2 , n3 ))B(r, n) = n1 x1 − n2 x2 − n3 x3 .
In this case, the associate Laplacian is∂2∂2∂2∆Q =−−,∂x21 ∂x22 ∂x23
and the Euler operator is∂∂∂+ x2+ x3.
E = x1∂x1∂x2∂x3
Then show that[E, ∆] = −2∆.
Show that if p satisfies Vignéras’ equation with eigenvalue d, we thus have that
∆Q p satisfies it with eigenvalue d − 2.
(6) A function f : Rn \ {0} → R is called positive homogenous of degree k iff (tx) = tk f (x)(5)

for all t > 0. Show Euler’s homogenous function theorem, which states that a C 1
function f is positive homogenous of degree k if and only if
E(f ) = kf.
Hint: Differentiate both sides of (5) with respect to t and substitute t = 1.
Conversely, integrate.
In particular, this shows that spherical polynomials (recall that these satisfy∆Q p = 0 and are assumed to be homogenous) satisfy Vignéras’ equation, confirming our earlier claim that (positive definite) spherical theta functions are
modular (as long as they converge, which is clear due to the exponential decayof terms q Q(n) ).
(7) Consider the very simple type (1, 1) form Q(x1 , x2 ) := x21 −x22 , and let c1 = (1, 2),c2 = (1, 3), a = (1/3, 0), b = (0, 0). Then with x = (n + 1/3, j) ∈ 1/3 + Z2 , wehave (you should write this down explicitly to check)
X22(τ)=(sign(−4j + 2n + 2/3) − sign(−6j + 2n + 2/3)) q (n+1/3) −j .
Θ+A,a,b,c1 ,c2n,j∈Z

The only terms that are non-vanishing in this sum are those for whichsign(−4j + 2n + 2/3) = −sign(−6j + 2n + 2/3)

INDEFINITE THETA FUNCTIONS OF TYPE (n, 1) II7(note that neither is ever 0 due to the presence of the 2/3). Show that if thislast condition holds, then Q(n + 1/3, j) > 0, and conclude that Θ+A,a,b,c1 ,c2 (τ ) isconvergent.