Witten–Reshetikhin–Turaev Invariants, Homological Blocks, and Quantum Modular
Forms

Witten–Reshetikhin–Turaev Invariants,
Homological Blocks, and Quantum Modular Forms
for Unimodular Plumbing H-Graphs

Akihito MORI and Yuya MURAKAMI A. Mori and Y. Murakami Mathematical Institute,
Tohoku University, 6-3, Aoba, Aramaki, Aoba-Ku,
Sendai 980-8578, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>https://sites.google.com/view/yuya-
murakami/home

Received November 23, 2021, in final form April 28, 2022; Published online May
07, 2022

Gukov–Pei–Putrov–Vafa constructed $q$-series invariants called homological
blocks in a physical way in order to categorify Witten–Reshetikhin–Turaev
(WRT) invariants and conjectured that radial limits of homological blocks are
WRT invariants. In this paper, we prove their conjecture for unimodular
H-graphs. As a consequence, it turns out that the WRT invariants of H-graphs
yield quantum modular forms of depth two and of weight one with the quantum
set $\mathbb{Q}$. In the course of the proof of our main theorem, we first
write the invariants as finite sums of rational functions. We second carry out
a systematic study of weighted Gauss sums in order to give new vanishing
results for them. Combining these results, we finally prove that the above
conjecture holds for H-graphs.

quantum invariants; Witten–Reshetikhin–Turaev invariants; homological blocks;
quantum modular forms; plumbed manifolds; false theta funcitons; Gauss sums

57K31; 57K10; 57K16; 11F27; 11L05; 11T24

##  1 Introduction

Many authors have studied Witten–Reshetikhin–Turaev (WRT) invariants for
3-manifolds. Lawrence–Zagier [16, Theorem 1] proved that the WRT invariant for
the Poincaré homology sphere $\Sigma(2,3,5)$ coincides with a radial limit of
a false theta function. Hikami [11, Theorem 9] generalized it for Brieskorn
homology spheres $\Sigma(p_{1},p_{2},p_{3})$. Subsequently, Hikami [12]
expressed the WRT invariants for Seifert fibered homology spheres with four
singular fibers as a sum of limiting values of Eichler integrals. Furthermore,
Hikami [13] generalised it for Seifert fibered homology spheres with
$M$-singular fiberes.

These works allow one to expect that WRT invariants are radial limits for some
$q$-series with integer coefficients.

For plumbed 3-manifolds, Gukov–Pei–Putrov–Vafa gave a definition of a
candidate, called homological blocks from a physical viewpoint, and
conjectured that they categorify WRT invariants, that is, WRT invariants of
any closed orientable $3$-manifolds are linear combinations of radial limits
of homological blocks [10, Conjecture 2.1].

For Seifert fibered homology spheres, Fuji–Iwaki–Murakami–Terashima [8]
introduced $q$-series called the WRT functions and proved that their radial
limits are WRT invariants. They are identified with homological blocks by
Andersen–Mistegård [1]. Independently in a different way, Andersen–Mistegård
[1, Theorem 4] proved that radial limits of homological blocks coincide with
WRT invariants for Seifert fibered homology spheres. To our knowledge, the
precise relation between homological blocks and Eichler integrals for the
cases treated in Hikami [13] is not known except for [5, 6, 9, 19].

Seifert manifolds are special cases of plumbed manifolds. Plumbed graphs yield
Seifert manifolds if they have the only one vertex with the degree at least
three. H-graphs shown in Figure 1 are the simplest graphs which are not such
graphs. It is expected that H-graphs yield non-Siefert manifolds.

$w_{1}$$w_{2}$$w_{3}$$w_{4}$$w_{5}$$w_{6}$ Figure 1: The H-graph $\Gamma$.

In the case of H-graphs, Bringmann–Mahlburg–Milas [3] showed that radial
limits of homological blocks coincide with sums of radial limits of multiple
Eichler integrals. Moreover, they proved that the radial limits of a
homological block make up a quantum modular form of depth two and of weight
one with the quantum set $\mathbb{Q}$.

Meanwhile, for non-Seifert manifolds, it remains to be seen whether WRT
invariants relate to homological blocks [10, Conjecture 2.1] and have
modularity.

In this paper, we study WRT invariants of H-graphs. We will write them
explicitly as weighted Gauss sums. Moreover, we will prove that they relate to
homological blocks and they yield quantum modular forms of depth two and of
weight one with the quantum set $\mathbb{Q}$.

Let us explain our setting. Let $\Gamma$ be the H-graph with vertices
$1,\dots,6$ and weights $w_{v}\in\mathbb{Z}_{<0}$ for each vertex $v$ shown in
Figure 1 such that its linking matrix

$\displaystyle W=\begin{pmatrix}w_{1}&1&1&1&0&0\\\ 1&w_{2}&0&0&1&1\\\
1&0&w_{3}&0&0&0\\\ 1&0&0&w_{4}&0&0\\\ 0&1&0&0&w_{5}&0\\\
0&1&0&0&0&w_{6}\end{pmatrix}$

is negative definite and its determinant is $1$.

Then we obtain the link $\mathcal{L}(\Gamma)$ defined by $\Gamma$ shown in
Figure 2. Let $M_{3}(\Gamma)$ be the plumed 3-manifold obtained from $S^{3}$
through the surgery along the diagram $\mathcal{L}(\Gamma)$. We obtain
$H_{1}(M_{3}(\Gamma),\mathbb{Z})\allowbreak\cong\mathbb{Z}^{6}/W\big{(}\mathbb{Z}^{6}\big{)}$
by use of the Mayer–Vietoris sequence. Since $\det W=1$, $M_{3}(\Gamma)$ is an
integral homology sphere.

$w_{1}$$w_{2}$$w_{3}$$w_{4}$$w_{5}$$w_{6}$ Figure 2: The surgery diagram
$\mathcal{L}(\Gamma)$ corresponding to $\Gamma$.

Such H-graphs yield Seifert manifolds if $w_{v}=-1$ for some $3\leq v\leq 6$.
Bringmann–Mahlburg–Milas [3, Theorem 1.2] showed that there exist precisely
$39$ equivalence classes of such H-graphs with $w_{v}\leq-2$ for each $3\leq
v\leq 6$ up to graph isomorphism.

For a positive integer $k$, let $\tau_{k}$ be the WRT invariant normalised as
$\tau_{k}\big{(}S^{3}\big{)}=1$. We study a relation between the WRT invariant
$\tau_{k}(M_{3}(\Gamma))$ and the homological block $\widehat{Z}_{\Gamma}(q)$.

For each complex number $z$, we denote $\bm{e}(z):={\rm e}^{2\pi{\rm i}z}$.
For a point $\tau\in\mathbb{H}$, let $q:=\bm{e}(\tau)$. Gukov–Pei–Putrov–Vafa
[10, formula (3.145)] defined the homogical block $\widehat{Z}(q)$ of the
plumed graph. In our case, it can be written as

$\displaystyle\widehat{Z}_{\Gamma}(q)=\widehat{Z}_{\Gamma,\delta}(q)=\widehat{Z}_{\delta}(q):=q^{(18-w_{1}-\cdots-
w_{6})/4}$
$\displaystyle\hphantom{\widehat{Z}_{\Gamma}(q)=\widehat{Z}_{\Gamma,\delta}(q)=\widehat{Z}_{\delta}(q):=}{}\times\mathrm{v.p.}\int_{\left\lvert
z_{v}\right\rvert=1,1\leq v\leq
6}\frac{\big{(}z_{3}-z_{3}^{-1}\big{)}\big{(}z_{4}-z_{4}^{-1}\big{)}\big{(}z_{5}-z_{5}^{-1}\big{)}\big{(}z_{6}-z_{6}^{-1}\big{)}}{\big{(}z_{1}-z_{1}^{-1}\big{)}\big{(}z_{2}-z_{2}^{-1}\big{)}}$
$\displaystyle\hphantom{\widehat{Z}_{\Gamma}(q)=\widehat{Z}_{\Gamma,\delta}(q)=\widehat{Z}_{\delta}(q):=}{}\times\Theta_{-W,\delta}(q;z_{1},\dots,z_{6})\prod_{1\leq
v\leq 6}\frac{{\rm d}z_{v}}{2\pi{\rm i}z_{v}},$ (1.1)

where $\delta:=(1,1,1,1,1,1)^{\top}$ is the signature of indexes of vertices
of $\Gamma$, $\mathrm{v.p.}$ denotes the Cauchy principal value, and

$\displaystyle\Theta_{-W,\delta}(q;z_{1},\dots,z_{6}):=\sum_{\bm{l}=(l_{1},\dots,l_{6})^{\top}\in
2\mathbb{Z}^{6}+\delta}q^{-\bm{l}^{\top}W^{-1}\bm{l}/4}\prod_{1\leq v\leq
6}z_{v}^{l_{v}}$

is the theta function of the linking matrix $W$.

[10, Conjecture 2.1 and relation (A.28)] states a conjectural relation between
the WRT invariants and radial limits of the homological blocks. In our case,
their relation is

$\tau_{k}(M_{3}(\Gamma))=\frac{1}{2\big{(}\zeta_{2k}-\zeta_{2k}^{-1}\big{)}}\lim_{\begin{subarray}{c}q\to\zeta_{k}\\\
\left\lvert q\right\rvert<1\end{subarray}}\widehat{Z}_{\Gamma}(q),$ (1.2)

where $\zeta_{k}:={\rm e}^{2\pi{\rm i}/k}$. In this paper, we prove this
relation.

######  Theorem 1.1.

The notation being as above, $(\ref{eq:GPPV})$ holds.

The proof of Theorem 1.1 takes several steps as below:

  1. $Step~{}1.$

We give a formula (Proposition 6.1) for the WRT invariant
$\tau_{k}(M_{3}(\Gamma))$ analogous to the formula given by Lawrence–Rozansky
[15, formula (4.2)] for any Seifert fibered manifold. This formula involves a
certain type of quadratic forms of rank two which appear in false theta
functions in [3, Proposition 5.3] related to the homological blocks.

  2. $Step~{}2.$

The sum involved in the formula (Proposition 6.1) is incomplete, that is, it
is like a form

$\displaystyle\sum_{n\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/kN\mathbb{Z}}f(n).$

That means, we can not apply usual techniques in number theory. To avoid this
situation, we interpolate the sum in the above formula for
$\tau_{k}(M_{3}(\Gamma))$ by adding terms which diverge a priori. These terms
turn out to be zero by vanishing results of “weighted” Gauss sums (Proposition
4.2).

  3. $Step~{}3.$

In Proposition 5.3, we prove that the radial limit of the homological block
coincides with the interpolated formula of the WRT invariant by using the
Euler–Maclaurin summation formula (Lemma 5.1) and vanishing results of
weighted Gauss sums (Proposition 4.2).

As for our formula in _Step_ 1, we need to handle rank two quadratic forms
which do not appear in Seifert case. In _Step_ 2, we need some vanishing
results for Gauss sums which we develop in this article.

We need more notation to explain the next main result. Let $S=(l_{jk})_{1\leq
j,k\leq 2}$ be the submatrix of $-W^{-1}=(l_{jk})_{1\leq j,k\leq 6}$. Let

$\displaystyle Q(m,n):=(m,n)S(m,n)^{\top},$ $\displaystyle M:=w_{3}w_{4},$
$\displaystyle N:=w_{5}w_{6},$ $\displaystyle
a:=-w_{2}w_{5}w_{6}+w_{5}+w_{6},$ $\displaystyle
c:=-w_{1}w_{3}w_{4}+w_{3}+w_{4}.$

Then we have $Q(m,n)=Mam^{2}-2MNmn+Ncn^{2}$ by a direct calculation and
$S^{-1}\big{(}\mathbb{Z}^{2}\big{)}=\frac{1}{M}\mathbb{Z}\oplus\frac{1}{N}\mathbb{Z}$
by Remark 3.1.

For non-zero integers $w$ and $w^{\prime}$, we define a map
$\chi^{(w,w^{\prime})}\colon\mathbb{Z}\to\mathbb{Z}/2ww^{\prime}\mathbb{Z}\to\\{\pm
1,0\\}$ by

$\displaystyle\chi^{(w,w^{\prime})}(n):=\begin{cases}ee^{\prime}&\text{if}\
n\equiv w^{\prime}e+we^{\prime}+ww^{\prime}\pmod{2ww^{\prime}}\ \text{for
some}\ e,e^{\prime}\in\\{\pm 1\\},\\\ 0&\text{otherwise}.\end{cases}$

We note that the generating function of this map appears in a calculation of
the WRT invariant of the H-graph $\Gamma$ (Proposition 6.1). We also define a
map

$\varepsilon\colon\
(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}\to(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}/\mathbb{Z}^{2}\to\\{\pm
1,0\\}$

by

$\varepsilon(\alpha,\beta):=\chi^{(w_{3},w_{4})}(2M\alpha)\chi^{(w_{5},w_{6})}(2N\beta).$

Bringmann–Mahlburg–Milas [3, Theorems 1.1, 1.3 and 4.1] shows that the
homological block $\widehat{Z}_{\Gamma}(q)$ forms a depth two quantum modular
form. Combined with their results, Theorem 1.1 implies the following theorem.

######  Theorem 1.2.

Let $f_{\Gamma}\colon\mathbb{Q}\to\mathbb{C}$ be a map defined as

$\displaystyle
f_{\Gamma}\bigg{(}\frac{h}{k}\bigg{)}=\frac{1}{k^{2}}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\,\bm{e}\bigg{(}\frac{h}{k}Q(\gamma)\bigg{)}\alpha\beta$

for an irreducible fraction $h/k\in\mathbb{Q}$. Then $f_{\Gamma}$ is a quantum
modular form of depth two and of weight one with the quantum set $\mathbb{Q}$.
Moreover, for any $k\in\mathbb{Z}_{>0}$ we have

$\displaystyle
f_{\Gamma}\left(\frac{1}{k}\right)=\frac{2\left(\zeta_{2k}-\zeta_{2k}^{-1}\right)}{\zeta_{k}^{-(18+\sum_{v=1}^{6}w_{v}+\sum_{v=3}^{6}1/w_{v})/4}}\tau_{k}(M_{3}(\Gamma)).$

This theorem yields new examples of quantum modular forms. These examples have
two interesting properties. They have explicit expressions and WRT invariants
appear in their special values.

This paper will be organised as follows. In Section 2, we discuss WRT
invariants, homological blocks, and quantum modular forms. In Section 3, we
set notation which we use throughout this paper. We also define the rank two
false theta function $F_{Q,\varepsilon}(\tau)$ which coincides with the
homological block $\widehat{Z}_{\Gamma}(q)$ without a simple factor. In
Section 4, we prove vanishing results for several types of Gauss sums, which
we need to calculate a radial limit of $F_{Q,\varepsilon}(\tau)$ and
$\widehat{Z}_{\Gamma}(q)$ in Section 5. Finally, in Section 6, we calculate
the WRT invariant of $M_{3}(\Gamma)$ and prove Theorem 1.1.

##  2 Preliminaries

In this section, we prepare basic materials.

###  2.1 Witten–Reshetikhin–Turaev invariants

Reshetikhin–Turaev [17] constructed the invariant $\tau_{k}$ for a pair
$(M,L)$ of a 3-manifold $M$ and a framed link $L$ in $M$. This invariant is
parametrised by a positive integer $k$. In our situation $L=\varnothing$, so
$\tau_{k}$ is an invariant for 3-manifolds. This invariant is called the
$\operatorname{SU}(2)$ Witten–Reshetikhin–Turaev (WRT) invariant. Turaev gives
a detailed description [18].

For plumbed manifolds, Gukov–Pei–Putrov–Vafa caluculated the WRT invariant
explicitly [10, formula (A.10)]:

$\displaystyle\tau_{k}(M_{3}(\Gamma))={}$
$\displaystyle\frac{\bm{e}(-\sigma/8)\zeta_{k}^{3\sigma/4}}{2(2k)^{V/2}\big{(}\zeta_{2k}-\zeta_{2k}^{-1}\big{)}}\sum_{n\in(\mathbb{Z}^{V}/2k\mathbb{Z}^{V})\smallsetminus(k\mathbb{Z}^{V}/2k\mathbb{Z}^{V})}\prod_{v=1}^{V}\zeta_{k}^{w_{v}(n_{v}^{2}-1)/4}$
$\displaystyle\times\bigg{(}\frac{1}{\zeta_{k}^{n_{v}/2}-\zeta_{k}^{-n_{v}/2}}\bigg{)}^{\deg(v)-2}\prod_{(v,v^{\prime})\in\mathrm{Edges}}\frac{\zeta_{k}^{n_{v}n_{v^{\prime}}/2}-\zeta_{k}^{-n_{v}n_{v^{\prime}}/2}}{2},$

where $b_{+}$ and $b_{-}$ are the number of positive and negative eigenvalues
counted with multiplicities of the linking matrix of a plumbed graph $\Gamma$,
$\sigma:=b_{+}-b_{-}$ is the signature, $V$ is the number of vertices of
$\Gamma$, $\deg(v)$ is the degree for an $v$th vertex, and $w_{v}$ is a weight
for an $v$th vertex. We use this formula in Section 6.

###  2.2 Homological blocks

In this section, we show a definition of homological blocks and express them
as false theta functions. Though any mathematical definitions of these
invariants are not given for general cases, they provided a rigorous
definition for a particular case in which 3-manifolds are plumbed.

A plumbed graph is a tree whose vertices are weighted by integers. A
3-manifold is said to be plumbed if it is obtained by the surgery along a
plumbed graph.

######  Definition 2.1 ([10, Section 3.4]).

Let $G$ be a plumbed graph with vertices $1,\dots,N$ and $\delta$ be as in
Section 1. For the plumbed $3$-manifold $M_{3}(G)$ and
$b\in(H_{1}(M_{3}(G),\mathbb{Z})+\delta)/\\{\pm
1\\}\cong\big{(}2\mathbb{Z}^{N}/2W\big{(}\mathbb{Z}^{N}\big{)}+\delta\big{)}/\\{\pm
1\\}$, the homological block $\widehat{Z}_{G,b}(q)$ is defined as follows:

$\displaystyle\widehat{Z}_{G,b}(q)=q^{-\sum_{v=1}^{N}(w_{v}+3)/4}\mathrm{v.p.}\int_{\left\lvert
z_{v}\right\rvert=1,v=1,\dots,N}\prod_{v=1}^{N}\frac{{\rm d}z_{v}}{2\pi{\rm
i}z_{v}}(z_{v}-1/z_{v})^{2-\deg(v)}\Theta_{-W,b}(z),$

where $w_{v}$ is the weight of the vertex $v$, $\mathrm{v.p.}$ denotes the
Cauchy principal value, $\deg(v)$ is the degree of $v$, $W$ is the linking
matrix of $\mathcal{L}(G)$, and

$\displaystyle\Theta_{-W,b}(q;z_{1},\dots,z_{N}):=\sum_{\bm{l}=(l_{1},\dots,l_{N})^{\top}\in
2W(\mathbb{Z}^{N})+b}q^{-\bm{l}^{\top}W^{-1}\bm{l}/4}\prod_{1\leq v\leq
N}z_{v}^{l_{v}}.$

If $G$ is an H-graph $\Gamma$, the above definition of $\widehat{Z}_{G,b}(q)$
coincides with (1.1).

######  Definition 2.2.

For $\tau\in\mathbb{H}$, we define false theta functions
$F_{Q_{+},\varepsilon}(\tau)$ and $F_{Q_{-},\varepsilon}(\tau)$ by

$\displaystyle
F_{Q_{+},\varepsilon}(\tau):=\sum_{0\leq\gamma\in(2S)^{-1}(\mathbb{Z}^{2})}\varepsilon(\gamma)q^{Q_{+}(\gamma)},\qquad
F_{Q_{-},\varepsilon}(\tau):=\sum_{0\leq\gamma\in(2S)^{-1}(\mathbb{Z}^{2})}\varepsilon(\gamma)q^{Q_{-}(\gamma)},$

where $Q_{+}(m,n):=Q(m,n)$, $Q_{-}(m,n):=Q(m,-n)$.

In order to calculate radial limits of the homological block
$\widehat{Z}_{\Gamma}(q)$, we need an expression of $\widehat{Z}_{\Gamma}(q)$
as a sum of false theta functions given by Bringmann–Mahlburg–Millas.

######  Proposition 2.3 ([3, Proposition 5.3]).

$\displaystyle\widehat{Z}_{\Gamma}(q)=\frac{1}{2}q^{(-18-w_{1}-\cdots-
w_{6}-1/w_{3}-1/w_{4}-1/w_{5}-1/w_{6})/4}\left(F_{Q_{+},\varepsilon}(\tau)-F_{Q_{-},\varepsilon}(\tau)\right).$

###  2.3 Quantum modular forms

Zagier [20] introduced quantum modular forms. We describe its formal
definition as follows.

######  Definition 2.4 ([20], [4, Section 3.2]).

A map $f\colon\mathcal{Q}\to\mathbb{C}$ is a quantum modular form of weight
$k\in\frac{1}{2}\mathbb{Z}$ for a subgroup
$\Gamma\subset\operatorname{SL}_{2}(\mathbb{Z})$ (we assume
$\Gamma\subset\Gamma_{0}(4)$ if $k\in 1/2+\mathbb{Z}$ ) and the quantum set
$\mathcal{Q}\subset\mathbb{Q}$ if for any $\gamma\in\Gamma$, the function
$f-\left.f\right|_{k}\gamma$ can be extended to an open set of $\mathbb{R}$
real analytically.

We denote the $\mathbb{R}$-vector space of such forms by
$\mathcal{Q}_{k}(\Gamma)$.

Zagier [20, Examples 0–5] gave 6 examples of quantum modular forms. Several
authors studied quantum modular property of the WRT invariants for Seifert
manifolds. For example, Lawrence–Zagier [16] considered the Poincaré homology
sphere $\Sigma(2,3,5)$, Hikami [11, Proposition 8 and Theorem 9] considered
the Brieskorn homology sphere $\Sigma(p_{1},p_{2},p_{3})$ with
$p_{1}^{-1}+p_{2}^{-1}+p_{3}^{-1}<1$, and Hikami [14, Propositions 2, 4 and 8]
considered spherical Seifert manifolds. Bringmann–Rolen [4, Theorem 1.1]
constructed quantum modular forms by special values of the $L$-functions of
cusp forms.

In our case, homological blocks are essentially rank two false theta functions
and are not quantum modular forms. They are depth two quantum modular forms
defined by Bringmann–Kaszian–Milas [2, Definition 3].

######  Definition 2.5 ([2, Definition 3]).

A map $f\colon\mathcal{Q}\to\mathbb{C}$ is a quantum modular form of depth
two, weight $k\in\frac{1}{2}\mathbb{Z}$ for a subgroup
$\Gamma\subset\operatorname{SL}_{2}(\mathbb{Z})$ $(\text{we assume
$\Gamma\subset\Gamma_{0}(4)$ if $k\in 1/2+\mathbb{Z}$})$ and the quantum set
$\mathcal{Q}\subset\mathbb{Q}$ if for any $\gamma\in\Gamma$, there exist
$k_{1},\dots,k_{r}\in\frac{1}{2}\mathbb{Z}$ and an open subset
$R\subset\mathbb{R}$ such that

$\displaystyle
f-\left.f\right|_{k}\gamma\in\mathcal{O}(R)\oplus\mathcal{Q}_{k_{1}}(\Gamma)\mathcal{O}(R)\oplus\cdots\oplus\mathcal{Q}_{k_{r}}(\Gamma)\mathcal{O}(R),$

where $\mathcal{O}(R)$ is an $\mathbb{R}$-vector space of real analytic
functions on $R$.

Bringmann–Mahlburg–Milas [3, Theorems 1.1, 1.3, 3.1 and 4.1] proved the
following theorem.

######  Theorem 2.6.

Let $S\in\operatorname{Sym}_{2}^{+}(\mathbb{Z})$ be a positive definite
symmetric matrix and $Q(m,n):=(m,n)S(m,n)^{\top}=Am^{2}+2Bmn+Cn^{2}$ be the
corresponding quadratic form such that $A,B,C\in\mathbb{Z}$. Let
$\varepsilon\colon(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}\to(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}/\mathbb{Z}^{2}\to\mathbb{C}$
be a map such that
$\varepsilon(\alpha,\beta)=\varepsilon(1-\alpha,\beta)=\varepsilon(\alpha,1-\beta)=\varepsilon(1-\alpha,1-\beta)$.
Let $K\in\mathbb{Z}_{>0}$ and

$\displaystyle
F_{K,Q,\varepsilon}(\tau):=\sum_{0\leq\gamma\in(2S)^{-1}(\mathbb{Z}^{2})}\varepsilon(\gamma)q^{KQ(\gamma)}.$

Then the followings hold:

  1. $(i)$

([3, Theorems 1.1 and 3.1]) The function $F_{K,Q,\varepsilon}$ is a quantum
modular form of depth two and of weight one for the group
$\Gamma(8\operatorname{lcm}(A,C)K\det S)$ with the quantum set

$\displaystyle\mathcal{Q}_{K,Q,\varepsilon}:=\left\\{\\!\frac{h}{k}\in\mathbb{Q}\bigg{|}\gcd(h,k)=1,\,k\in\mathbb{Z}_{>0},\\!\sum_{0\leq\gamma\in(2S)^{-1}(\mathbb{Z}^{2})/k\mathbb{Z}^{2}}\\!\\!\\!\varepsilon(\gamma)\bm{e}\bigg{(}\frac{h}{k}KQ(\gamma)\bigg{)}=0\\!\right\\}\\!.$

  2. $(ii)$

([3, Theorems 1.3 and 4.1]) Suppose that there exist positive integers
$M,N,a,c$ and an integer $b$ such that $A=Ma,B=2MNb,C=Nc$ and $ac-MNb^{2}=1$.
Let $\alpha_{1}\neq\alpha_{2},\beta_{1}\neq\beta_{2}$ be rational numbers in
the open interval $(0,1)$ such that the denominators of irreducble fraction
expansions of $\alpha_{j}$ and $\beta_{j}$ are $2M$ and $2N$ respectively,
$\alpha_{1}^{2}\equiv\alpha_{2}^{2}\pmod{\frac{1}{2M}\mathbb{Z}}$, and
$\beta_{1}^{2}\equiv\beta_{2}^{2}\pmod{\frac{1}{2N}\mathbb{Z}}$. For a
rational number $\alpha$, let $\mathcal{S}(\alpha):=\\{\alpha,1-\alpha\\}$.
Let
$\mathcal{S}_{1}:=\mathcal{S}(\alpha_{1})\times\mathcal{S}(\beta_{1})\cup\mathcal{S}(\alpha_{2})\times\mathcal{S}(\beta_{2})$,
$\mathcal{S}_{2}:=\mathcal{S}(\alpha_{1})\times\mathcal{S}(\beta_{2})\cup\mathcal{S}(\alpha_{2})\times\mathcal{S}(\beta_{1})$,
$\mathcal{S}:=\mathcal{S}_{1}\cup\mathcal{S}_{2}$, and
$\varepsilon\colon(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}\to(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}/\mathbb{Z}^{2}\to\mathbb{C}$
be a map such that

$\displaystyle\varepsilon(\gamma)=\begin{cases}(-1)^{j+1}&\text{if}\
\gamma\in\mathcal{S}_{i}+\mathbb{Z}^{2},\\\ 0&\text{if}\
\gamma\notin\mathcal{S}+\mathbb{Z}^{2}.\end{cases}$

Then we have $\mathcal{Q}_{2,Q,\varepsilon}=\mathbb{Q}$. In particular, the
function $F_{2,Q,\varepsilon}$ is a quantum modular form of depth two and of
weight one for the group $\Gamma(16MN\operatorname{lcm}(a,c)\det S)$ and the
quantum set $\mathbb{Q}$.

Here we remark that in the setting in Theorem 2.6$(ii)$, we have
$(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}=\frac{1}{2M}\mathbb{Z}\oplus\frac{1}{2N}\mathbb{Z}$
by the same argument as in Remark 3.1. In Theorem 5.4, we will treat the
quantum modularity of $F_{Q,\varepsilon}$ instead of $F_{2Q,\varepsilon}$ in
Theorem 2.6$(i)$.

###  2.4 A reciprocity formula for Gauss sums

To calculate the WRT invariant of $M_{3}(\Gamma)$, we need the reciprocity
formula in the following proposition.

######  Proposition 2.7 ([7, Theorem 1]).

Let $L$ be a lattice of finite rank $n$ equipped with a non-degenerated
symmetric $\mathbb{Z}$-valued bilinear form
$\left\langle\cdot,\cdot\right\rangle$ and let $\sigma$ be the signature of
the quadratic form $\left\langle x,h(y)\right\rangle$. We write

$\displaystyle L^{\prime}:=\\{y\in L\otimes\mathbb{R}\mid\left\langle
x,y\right\rangle\in\mathbb{Z}\text{ for all }x\in L\\}$

for the dual lattice. Let $0<k\in\left\lvert
L^{\prime}/L\right\rvert\mathbb{Z}$, $z\in\frac{1}{k}L$, and $h\colon
L\otimes\mathbb{R}\to L\otimes\mathbb{R}$ be a self-adjoint automorphism such
that $h(L^{\prime})\subset L^{\prime}$ and $\frac{k}{2}\left\langle
y,h(y)\right\rangle\in\mathbb{Z}$ for all $y\in L^{\prime}$. Then it holds

$\displaystyle\sqrt{\left\lvert L^{\prime}/L\right\rvert}\\!\sum_{x\in
L/kL}\\!\\!\bm{e}\bigg{(}\frac{1}{2k}\left\langle
x,h(x)\right\rangle+\left\langle
x,z\right\rangle\\!\\!\bigg{)}=\frac{\bm{e}(\sigma/8)k^{n/2}}{\sqrt{\left\lvert\det
h\right\rvert}}\\!\\!\sum_{y\in
L^{\prime}/h(L^{\prime})}\\!\\!\bm{e}\bigg{(}{-}\frac{k}{2}\left\langle
y+z,h^{-1}(y+z)\right\rangle\\!\\!\bigg{)}.$

##  3 Notation

In this section, we prepare notation which we use throughout this paper. The
setting in Section 1 is a particular case of the setting in this section.

We fix an integer $h$ and a positive integer $k$ such that $\gcd(h,k)=1$. Let
$M$, $N$, $a$, $c$ be positive integers and an integer $b$ such that $ac-
MNb^{2}=1$. Let

$\displaystyle S=\begin{pmatrix}Ma&MNb\\\ MNb&Nc\end{pmatrix}$

be a positive definite symmetric matrix. The symbol
$\gamma=(\alpha,\beta)^{\top}$ denotes any element in $\mathbb{Q}^{2}$. Here,
$t$ is the transpose of vectors. Let
$Q(\alpha,\beta):=(\alpha,\beta)S(\alpha,\beta)^{\top}=Ma\alpha^{2}+2MNb\alpha\beta+Nc\beta^{2}$
be the corresponding quadratic form. Let
$\varepsilon\colon(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}\to(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}/\mathbb{Z}^{2}\to\mathbb{C}$
be a map. For a complex number $z$, we denote $\bm{e}(z):={\rm e}^{2\pi{\rm
i}z}$. Let $\mathbb{H}$ be the upper half plane, that is, the set of complex
numbers whose imaginary part are positive. For a point $\tau\in\mathbb{H}$,
let

$\displaystyle
F_{Q,\varepsilon}(\tau):=\sum_{0\leq\gamma\in(2S)^{-1}(\mathbb{Z}^{2})}\varepsilon(\gamma)q^{Q(\gamma)}$

be a false theta function.

We can rewrite a lattice $S^{-1}\big{(}\mathbb{Z}^{2}\big{)}$ explicitly as
below.

######  Remark 3.1.

Let

$\displaystyle A:=\begin{pmatrix}c&-Nb\\\
-Mb&a\end{pmatrix}\in\operatorname{SL}_{2}(\mathbb{Z}).$

Then we have

$\displaystyle SA=\begin{pmatrix}M&0\\\ 0&N\end{pmatrix}$

and $S^{-1}(\mathbb{Z}^{2})=\frac{1}{M}\mathbb{Z}\oplus\frac{1}{N}\mathbb{Z}$.
Thus, we obtain
$\varepsilon\colon\frac{1}{M}\mathbb{Z}/\mathbb{Z}\oplus\frac{1}{N}\mathbb{Z}/\mathbb{Z}\to\mathbb{C}$.

##  4 Vanishing results of Gauss sums

In this section, we prove several claims which state vanishing results of
weighted Gauss sums. We will use the results in this section to compute the
WRT invariants and limit values of the homological blocks. Let us keep the
notation in Section 3, for example,
$h,k,Q(\gamma),\varepsilon\colon(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}/\allowbreak\mathbb{Z}^{2}\to\mathbb{C}$.

###  4.1 Main results in this section

Throughout this section, we argue that Gauss sums vanish in the setting in
Section 3 and assume the following assumption for a map
$\varepsilon\colon(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}/\mathbb{Z}^{2}\to\mathbb{C}$.

######  Assumption 4.1.

  1. $(i)$

It holds

$\displaystyle\sum_{\gamma\in(2S)^{-1}(\mathbb{Z}^{2})/\mathbb{Z}^{2}}\varepsilon(\gamma)=0.$

  2. $(ii)$

For $\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}$
such that $\varepsilon(\gamma)\neq 0$, the denominator of the irreducible
fractions of $\alpha$ and $\beta$ are $2M$ and $2N$ respectively.

  3. $(iii)$

For $\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}$
such that $\varepsilon(\gamma)\neq 0$,

$\displaystyle M\alpha\pmod{\mathbb{Z}},\qquad N\beta\pmod{\mathbb{Z}},\qquad
M\alpha^{2}\pmod{\mathbb{Z}},$ $\displaystyle
N\beta^{2}\pmod{\mathbb{Z}},\qquad 2MN\alpha\beta\pmod{\mathbb{Z}}$

are independent of $\gamma$.

Here we remark that the map
$\varepsilon\colon(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}/\mathbb{Z}^{2}\to\\{\pm
1,0\\}$ defined in Section 1 satisfies the above assumption.

Our main results in this section are the following.

######  Proposition 4.2.

Under Assumption 4.1, the followings are true:

  1. $(i)$

It holds

$\displaystyle\sum_{\gamma\in(2S)^{-1}(\mathbb{Z}^{2})/k\mathbb{Z}^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{h}{k}Q(\gamma)\bigg{)}=0.$

  2. $(ii)$

We assume that there exist two maps
$\chi\colon\frac{1}{2M}\mathbb{Z}\to\frac{1}{2M}\mathbb{Z}/\mathbb{Z}\to\mathbb{C}$
and
$\psi\colon\frac{1}{2N}\mathbb{Z}\to\frac{1}{2N}\mathbb{Z}/\mathbb{Z}\to\mathbb{C}$
such that

$\displaystyle\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}/\mathbb{Z}}\chi(\alpha)=\sum_{\beta\in\frac{1}{2N}\mathbb{Z}/\mathbb{Z}}\psi(\beta)=0$

and $\varepsilon\big{(}(\alpha,\beta)^{\top}\big{)}=\chi(\alpha)\psi(\beta)$
for any $\alpha\in\frac{1}{2M}\mathbb{Z}/\mathbb{Z}$ and
$\beta\in\frac{1}{2N}\mathbb{Z}/\mathbb{Z}$. Then for any map
$C\colon\mathbb{Q}/k\mathbb{Z}\to\mathbb{C}$, we have

$\displaystyle\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})/k\mathbb{Z}^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{h}{k}Q(\gamma)\bigg{)}C(\alpha)$
$\displaystyle\qquad{}=\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})/k\mathbb{Z}^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{h}{k}Q(\gamma)\bigg{)}C(\beta)=0.$

  3. $(iii)$

Under the assumption in $(ii)$, let
$B\colon\mathbb{Q}/k\mathbb{Z}\to\mathbb{C}$ be a map such that

$\displaystyle\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}/\mathbb{Z}}\chi(\alpha)\widetilde{B}(\alpha)=0,$

where $\widetilde{B}(\alpha):=\sum_{0\leq m<k}B(\alpha+m)$. Then it holds

$\displaystyle\sum_{\mu\in
k\mathbb{Z}\oplus\mathbb{Z}/2kS(\mathbb{Z}^{2})}\bm{e}\bigg{(}\frac{1}{4k}{\mu^{\top}}S^{-1}\mu\bigg{)}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})/k\mathbb{Z}^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}{\gamma^{\top}}\mu\bigg{)}B(\alpha)C(\beta)$
$\displaystyle\qquad{}=\sum_{\mu\in
k\mathbb{Z}^{2}/2kS(\mathbb{Z}^{2})}\bm{e}\bigg{(}\frac{1}{4k}{\mu^{\top}}S^{-1}\mu\bigg{)}\\!\\!\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})/k\mathbb{Z}^{2}}\\!\\!\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}{\gamma^{\top}}\mu\bigg{)}B(\alpha)C(\beta)$
$\displaystyle\qquad{}=0.$

  4. $(iv)$

Under the assumption in $(ii)$, we have

$\displaystyle\sum_{\mu\in
k\mathbb{Z}\oplus\mathbb{Z}/2kS(\mathbb{Z}^{2})}\bm{e}\bigg{(}\frac{1}{4k}{\mu^{\top}}S^{-1}\mu\bigg{)}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}{\gamma^{\top}}\mu\bigg{)}B_{1}(\alpha)B_{1}(\beta)$
$\displaystyle\qquad{}=\sum_{\mu\in\mathbb{Z}\oplus
k\mathbb{Z}/2kS(\mathbb{Z}^{2})}\\!\\!\\!\\!\\!\\!\bm{e}\bigg{(}\frac{1}{4k}{\mu^{\top}}S^{-1}\mu\bigg{)}\\!\\!\\!\\!\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\\!\\!\\!\\!\\!\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}{\gamma^{\top}}\mu\bigg{)}B_{1}(\alpha)B_{1}(\beta)$
$\displaystyle\qquad{}=\sum_{\mu\in
k\mathbb{Z}^{2}/2kS(\mathbb{Z}^{2})}\\!\\!\\!\\!\bm{e}\bigg{(}\frac{1}{4k}{\mu^{\top}}S^{-1}\mu\bigg{)}\\!\\!\\!\\!\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\\!\\!\\!\\!\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}{\gamma^{\top}}\mu\bigg{)}B_{1}(\alpha)B_{1}(\beta)$
$\displaystyle\qquad{}=0,$

where $B_{1}(x):=x-1/2$ is the first Bernoulli polynomial.

  5. $(v)$

Under the assumption in $(ii)$, we have

$\displaystyle\sum_{\mu\in
k\mathbb{Z}\oplus\mathbb{Z}/2kS(\mathbb{Z}^{2})}\bm{e}\bigg{(}\frac{1}{4k}{\mu^{\top}}S^{-1}\mu\bigg{)}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}{\gamma^{\top}}\mu\bigg{)}\alpha\beta$
$\displaystyle\qquad{}=\sum_{\mu\in\mathbb{Z}\oplus
k\mathbb{Z}/2kS(\mathbb{Z}^{2})}\bm{e}\bigg{(}\frac{1}{4k}{\mu^{\top}}S^{-1}\mu\bigg{)}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}{\gamma^{\top}}\mu\bigg{)}\alpha\beta$
$\displaystyle\qquad{}=\sum_{\mu\in
k\mathbb{Z}^{2}/2kS(\mathbb{Z}^{2})}\bm{e}\bigg{(}\frac{1}{4k}{\mu^{\top}}S^{-1}\mu\bigg{)}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}{\gamma^{\top}}\mu\bigg{)}\alpha\beta$
$\displaystyle\qquad{}=0.$

Our proof of Proposition 4.2 is based on the proof of [3, Theorem 4.1].
However, our assumptions are slightly different from it in [3, Theorem 4.1].

###  4.2 A proof of Proposition 4.2$\bm{(i)}$

Firstly, we prove Proposition 4.2$(i)$. To begin with, we need the result of
the quadratic Gauss sum. For integers $a,b,c\in\mathbb{Z}$ with $c>0$, we
define the quadratic Gauss sum

$\displaystyle
G(a,b,c):=\sum_{n\in\mathbb{Z}/c\mathbb{Z}}\bm{e}\bigg{(}\frac{an^{2}+bn}{c}\bigg{)}.$

The following result is elementary.

######  Lemma 4.3.

If $\gcd(a,c)\nmid b$, then $G(a,b,c)=0$.

###### Proof.

Let $g:=\gcd(a,c)$, $a^{\prime}:=a/g$, and $c^{\prime}:=c/g$. We have

$\displaystyle G(a,b,c)$ $\displaystyle=\sum_{0\leq l<c^{\prime}}\sum_{0\leq
m<g}\bm{e}\bigg{(}\frac{ga^{\prime}(l+c^{\prime}m)^{2}+b(l+c^{\prime}m)}{gc^{\prime}}\bigg{)}$
$\displaystyle=\sum_{0\leq
l<c^{\prime}}\bm{e}\bigg{(}\frac{ga^{\prime}l^{2}+bl}{gc^{\prime}}\bigg{)}\sum_{0\leq
m<g}\bm{e}\bigg{(}\frac{bm}{g}\bigg{)}.$

By assumption, the last sum vanishes. ∎

Proposition 4.2$(i)$ follows from the following two lemmas.

######  Lemma 4.4.

If $\gcd(M,k)>1$ or $\gcd(N,k)>1$, then for
$\gamma=(\alpha,\beta)^{\top}\in\mathbb{Q}^{2}$ we have

$\displaystyle\sum_{\mu\in\mathbb{Z}^{2}/k\mathbb{Z}^{2}}\bm{e}\bigg{(}\frac{h}{k}Q(\mu+\gamma)\bigg{)}=0.$

######  Lemma 4.5.

If $\gcd(M,k)=\gcd(N,k)=1$, then

$\displaystyle\sum_{\mu\in\mathbb{Z}^{2}/k\mathbb{Z}^{2}}\bm{e}\bigg{(}\frac{h}{k}Q(\mu+\gamma)\bigg{)}$

is independent of $\gamma\in(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}$ such that
$\varepsilon(\gamma)\neq 0$.

###### Proof of Lemma 4.4.

We give a proof for the case when $\gcd(M,k)>1$. Another case is similarly
handled. We have

$\displaystyle\sum_{\mu\in\mathbb{Z}^{2}/k\mathbb{Z}^{2}}\bm{e}\bigg{(}\frac{h}{k}Q(\mu+\gamma)\bigg{)}$
$\displaystyle=\sum_{\mu=(m,n)^{\top}\in\mathbb{Z}^{2}/k\mathbb{Z}^{2}}\bm{e}\bigg{(}\frac{h}{k}\big{(}Q(\mu)+2\mu^{\top}S\gamma+Q(\gamma)\big{)}\bigg{)}$
$\displaystyle=\bm{e}\bigg{(}\frac{h}{k}Q(\gamma)\bigg{)}\sum_{n\in\mathbb{Z}/k\mathbb{Z}}\bm{e}\bigg{(}\frac{h}{k}\big{(}Ncn^{2}+2Nn(Mb\alpha+c\beta)\big{)}\bigg{)}G_{n},$

where $r:=2M\alpha,s:=2N\beta\in\mathbb{Z}$, and
$G_{n}:=G(hMa,h(ar+Mb(s+2Nn)),k)$. Let $g:=\gcd(hMa,k)$. We have $G_{n}=0$ if
$g\nmid h(ar+Mb(s+2Nn))$ by Lemma 4.3. Since $g^{\prime}:=\gcd(M,k)$ divides
$g$, it suffices to show $g\nmid h(ar+Mb(s+2Nn))$.

We have $\gcd(g^{\prime},a)=\gcd(g^{\prime},r)=\gcd(g^{\prime},h)=1$ since
$ac-MNb^{2}=1$ and $\gcd(r,2M)=\gcd(h,k)=1$ respectively. Thus, it holds
$\gcd(g,har)=1$. Therefore, we obtain

$\displaystyle h(ar+Mb(s+2Nn))\equiv ar\not\equiv 0\pmod{g^{\prime}}.$ ∎

###### Proof of Lemma 4.5.

Since $\gcd(M,k)=\gcd(N,k)=1$, there exist integers $M^{*},N^{*}\in\mathbb{Z}$
such that $MM^{*}\equiv NN^{*}\equiv 1\pmod{k}$. For
$\gamma\in(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}$ such that
$\varepsilon(\gamma)\neq 0$, let
$\gamma^{*}:=(MM^{*}\alpha,NN^{*}\beta)^{\top}\in\frac{1}{2}\mathbb{Z}^{2}$.
Since

$\displaystyle 2S(\gamma-\gamma^{*})=2\begin{pmatrix}Ma&MNb\\\
MNb&Nc\end{pmatrix}\begin{pmatrix}(1-MM^{*})\alpha\\\
(1-NN^{*})\beta\end{pmatrix}\in k\mathbb{Z}^{2},$

we have

$\displaystyle
Q(\mu+\gamma)-Q(\mu+\gamma^{*})=Q(\gamma)-Q(\gamma^{*})+2\mu^{\top}S(\gamma-\gamma^{*})\equiv
Q(\gamma)-Q(\gamma^{*})\pmod{k\mathbb{Z}}.$

Thus, we obtain

$\displaystyle\sum_{\mu\in\mathbb{Z}^{2}/k\mathbb{Z}^{2}}\bm{e}\bigg{(}\frac{h}{k}Q(\mu+\gamma)\bigg{)}$
$\displaystyle=\sum_{\mu\in\mathbb{Z}^{2}/k\mathbb{Z}^{2}}\bm{e}\bigg{(}\frac{h}{k}(Q(\mu+\gamma^{*})+Q(\gamma)-Q(\gamma^{*}))\bigg{)}$
$\displaystyle=\bm{e}\bigg{(}\frac{h}{k}(Q(\gamma)-Q(\gamma^{*}))\bigg{)}\sum_{\mu\in\mathbb{Z}^{2}/k\mathbb{Z}^{2}}\bm{e}\bigg{(}\frac{h}{k}Q(\mu)\bigg{)}.$

By Assumption 4.1$(iii)$, $Q(\gamma)-Q(\gamma^{*})\\!\pmod{k\mathbb{Z}}$ is
independent of $\gamma$. Since
$(M\alpha,N\beta)^{\top}\\!\\!\pmod{\mathbb{Z}^{2}}$ is also independent of
$\gamma$ by Assumption 4.1$(iii)$, there exists
$\delta_{0}\in\frac{1}{2}\mathbb{Z}^{2}/\mathbb{Z}^{2}$ which is independent
of $\gamma$ and $\gamma^{*}\equiv\delta_{0}\pmod{\mathbb{Z}^{2}}$. Thus,

$\displaystyle\sum_{\mu\in\mathbb{Z}^{2}/k\mathbb{Z}^{2}}\bm{e}\bigg{(}\frac{h}{k}Q(\mu+\gamma^{*})\bigg{)}=\sum_{\mu\in\mathbb{Z}^{2}/k\mathbb{Z}^{2}}\bm{e}\bigg{(}\frac{h}{k}Q(\mu+\delta_{0})\bigg{)}$

is also independent of $\gamma$. ∎

###  4.3 A proof of Proposition 4.2$\bm{(ii)}$

Secondly, we prove Proposition 4.2$(ii)$.

###### Proof of Proposition 4.2$\bm{(ii)}$.

We only prove the second equality. By assumption, we have

$\displaystyle\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})/k\mathbb{Z}^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{h}{k}Q(\gamma)\bigg{)}C(\beta)$
$\displaystyle\qquad{}=\sum_{\beta\in\frac{1}{2N}\mathbb{Z}/k\mathbb{Z}}\psi(\beta)\bm{e}\bigg{(}\frac{h}{k}c\beta^{2}\bigg{)}C(\beta)\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}/k\mathbb{Z}}\chi(\alpha)\bm{e}\bigg{(}\frac{h}{k}MQ_{2N\beta}(\alpha)\bigg{)},$

where $Q_{s}(\alpha):=a\alpha^{2}+bs\alpha$. Then the second sum in the right
hand side vanishes by the following lemma. ∎

######  Lemma 4.6.

Under the same assumption as in Proposition $\ref{prop:Gauss_sum}(ii)$, for a
quadratic form $Q_{0}(\alpha)\allowbreak=a_{0}\alpha^{2}+b_{0}\alpha$ such
that $a_{0},b_{0}\in\mathbb{Z},\gcd(M,a_{0})=1$ and a map
$\widetilde{B}\colon\frac{1}{2M}\mathbb{Z}/\mathbb{Z}\to\mathbb{C}$ such that

$\displaystyle\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}/\mathbb{Z}}\chi(\alpha)\widetilde{B}(\alpha)=0,$

it holds

$\displaystyle\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}/k\mathbb{Z}}\chi(\alpha)\bm{e}\bigg{(}\frac{h}{k}MQ_{0}(\alpha)\bigg{)}\widetilde{B}(\alpha)=0.$

Lemma 4.6 follows from the following two lemmas.

######  Lemma 4.7.

If $\gcd(M,k)>1$, then for $\alpha\in\frac{1}{2M}\mathbb{Z}$ such that
$\chi(\alpha)\neq 0$, we have

$\displaystyle\sum_{m\in\mathbb{Z}/k\mathbb{Z}}\bm{e}\bigg{(}\frac{h}{k}MQ_{0}(m+\alpha)\bigg{)}=0.$

######  Lemma 4.8.

If $\gcd(M,k)=1$, then

$\displaystyle\sum_{m\in\mathbb{Z}/k\mathbb{Z}}\bm{e}\bigg{(}\frac{h}{k}MQ_{0}(m+\alpha)\bigg{)}$

is independent of $\alpha\in\frac{1}{2M}\mathbb{Z}$ such that
$\chi(\alpha)\neq 0$.

###### Proof of Lemma 4.7.

Let $r:=2M\alpha$. We can write

$\displaystyle\sum_{m\in\mathbb{Z}/k\mathbb{Z}}\\!\\!\\!\\!\bm{e}\bigg{(}\frac{h}{k}MQ_{0}\bigg{(}\\!m\\!+\\!\frac{r}{2M}\bigg{)}\\!\bigg{)}\\!=\bm{e}\bigg{(}\frac{h(a_{0}r^{2}\\!+\\!2Mb_{0}r)}{4Mk}\bigg{)}\\!\\!\\!\sum_{m\in\mathbb{Z}/k\mathbb{Z}}\\!\\!\\!\\!\bm{e}\bigg{(}\frac{h(Ma_{0}m^{2}\\!+\\!(a_{0}r\\!+\\!Mb_{0})m)}{k}\bigg{)}.$

Since $\gcd(hMa,k)\mid\gcd(M,k)$ and $\gcd(M,k)\nmid a_{0}r+Mb_{0}$, we obtain
the claim. ∎

###### Proof of Lemma 4.8.

Since $\gcd(M,k)=1$, there exists an integer $M^{*}\in\mathbb{Z}$ such that
$MM^{*}\equiv 1\pmod{k}$. For any integer $m\in\mathbb{Z}$, we have

$\displaystyle Q_{0}(m+\alpha)-Q_{0}(m+MM^{*}\alpha)$
$\displaystyle\qquad{}\equiv(2a_{0}m+b)(1-MM^{*})\alpha+\big{(}1-(MM^{*})^{2}\big{)}a_{0}\alpha^{2}\pmod{k\mathbb{Z}}.$

Thus, we obtain

$\displaystyle\sum_{m\in\mathbb{Z}/k\mathbb{Z}}\\!\\!\bm{e}\bigg{(}\frac{h}{k}MQ_{0}(m+\alpha)\bigg{)}={}$
$\displaystyle\bm{e}\bigg{(}\frac{h(1-(MM^{*})^{2})}{k}a_{0}M\alpha^{2}+\frac{h(1-MM^{*})}{k}(2a_{0}m+b)M\alpha\bigg{)}$
$\displaystyle\times\sum_{m\in\mathbb{Z}/k\mathbb{Z}}\bm{e}\bigg{(}\frac{h}{k}MQ_{0}(m+MM^{*}\alpha)\bigg{)}.$

This sum is independent of $\alpha$ by Assumption 4.1$(iii)$. ∎

###  4.4 A proof of Proposition 4.2$\bm{(iii)}$, $\bm{(iv)}$ and $\bm{(v)}$

Finally, we prove Proposition 4.2$(iii)$, $(iv)$ and $(v)$.

###### Proof of Proposition 4.2${\bm{(iii)}}$, $\bm{(iv)}$ and $\bm{(v)}$.

Firstly, we prove Proposition 4.2$(iii)$. We only prove that the most left
hand side vanishes. It can be written as

$\displaystyle\sum_{\beta\in\frac{1}{2N}\mathbb{Z}/k\mathbb{Z}}\\!\\!\\!\\!\psi(\beta)C(\beta)\\!\\!\\!\sum_{n\in\mathbb{Z}/2kN\mathbb{Z}}\\!\\!\\!\\!\bm{e}\bigg{(}\frac{an^{2}}{4kN}+\frac{\beta
n}{k}\bigg{)}\\!\\!\\!\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}/\mathbb{Z}}\\!\\!\\!\\!\chi(\alpha)\widetilde{B}(\alpha)\\!\\!\\!\sum_{m\in\mathbb{Z}/2M\mathbb{Z}}\\!\\!\\!\\!\bm{e}\bigg{(}\frac{kcm^{2}}{4M}-\frac{1}{2}bmn+\alpha
m\bigg{)}.$

It suffices to show that the second double sum for $\alpha$ and $m$ vanishes.
By Proposition 2.7, it equals to

$\displaystyle\frac{2M{\rm
i}}{\sqrt{kc}}\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}/\mathbb{Z}}\chi(\alpha)\widetilde{B}(\alpha)\sum_{m\in\mathbb{Z}/kc\mathbb{Z}}\bm{e}\bigg{(}{-}\frac{M}{kc}\bigg{(}m+\alpha-\frac{bn}{2}\bigg{)}^{2}\bigg{)}$
$\displaystyle\qquad{}=\frac{2M{\rm
i}}{\sqrt{kc}}\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}/kc\mathbb{Z}}\chi\bigg{(}\alpha+\frac{Nbn}{2M}\bigg{)}\widetilde{B}\bigg{(}\alpha+\frac{Nbn}{2M}\bigg{)}\bm{e}\bigg{(}{-}\frac{M}{kc}\alpha^{2}\bigg{)}.$

This vanishes by Lemma 4.6.

Secondly, we prove Proposition 4.2$(iv)$. By the multiplication theorem of
Bernoulli polynomials, we have

$\displaystyle B_{1}(\alpha)=\sum_{0\leq
m<k}B\bigg{(}\frac{\alpha}{k}+m\bigg{)}.$

Since

$\displaystyle\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}\cap[0,1)}\chi(\alpha)B_{1}(\alpha)=\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}\cap[0,1)}\chi(\alpha)B_{1}(1-\alpha)=-\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}\cap[0,1)}\chi(\alpha)B_{1}(\alpha)$

is equal to $0$, we obtain the claim by Proposition 4.2$(iii)$.

Finally, we prove Proposition 4.2$(v)$. By 4.1$(i)$, we have

$\displaystyle\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}\cap[0,k)}\chi(\alpha)\alpha$
$\displaystyle=\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}\cap[0,1)}\chi(\alpha)\bigg{(}k\alpha+\frac{1}{2}k(k-1)\bigg{)}=k\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}\cap[0,1)}\chi(\alpha)\alpha$
$\displaystyle=k\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}\cap[0,1)}\chi(\alpha)(1-\alpha)=-k\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}\cap[0,1)}\chi(\alpha)\alpha.$

Since it vanishes, we obtain the claim by Proposition 4.2$(iii)$. ∎

##  5 A limit value of a false theta function

In this section, we will calculate a radial limit of the false theta function
$F_{Q,\varepsilon}(\tau)$. We use notation in Section 3 throughout this
section. A key fact is the following lemma which follows from the
Euler–Maclaurin summation formula.

######  Lemma 5.1 ([3, Lemma 2.2]).

For two real numbers $\alpha$ and $\beta$ and $C^{\infty}$-function
$f\colon\mathbb{R}^{2}\to\mathbb{C}$ which has rapid decay, we have an
asymptotic evaluation

$\displaystyle\sum_{m,n=0}^{\infty}f(t(m+\alpha,n+\beta))\sim\sum_{j,l=-1}^{\infty}\frac{B_{j+1}(\alpha)}{(j+1)!}\frac{B_{l+1}(\beta)}{(l+1)!}f^{(j,l)}(0,0)t^{j+l}$

as $t\to+0$, where $F(t)\sim G(t)$ means that $F(t)=G(t)+O\big{(}t^{N}\big{)}$
for any positive integer $N$, $B_{n}(x)$ is the $n$-th Bernoulli polynomial,
and for integers $j,l\geq 0$ let

$\displaystyle
f^{(-1,-1)}(0,0):=\int_{0}^{\infty}\int_{0}^{\infty}f(x,y)\,{\rm d}x{\rm d}y,$
$\displaystyle
f^{(j,-1)}(0,0):=-\int_{0}^{\infty}\frac{\partial^{j}f}{\partial
x^{j}}(0,y)\,{\rm d}y,$ $\displaystyle
f^{(-1,l)}(0,0):=-\int_{0}^{\infty}\frac{\partial^{l}f}{\partial
y^{l}}(x,0)\,{\rm d}x,$ $\displaystyle
f^{(j,l)}(0,0):=\frac{\partial^{j+l}f}{\partial x^{j}\partial y^{l}}(0,0).$

We can calculate a radial limit of the false theta function
$F_{Q,\varepsilon}(\tau)$ as follows.

######  Lemma 5.2.

If

$\displaystyle\varepsilon(\gamma)=\varepsilon\big{(}(1,1)^{\top}-\gamma\big{)}=\varepsilon\big{(}(1-\alpha,\beta)^{\top}\big{)}=\varepsilon\big{(}(\alpha,1-\beta)^{\top}\big{)}$

for any
$\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}\big{(}\mathbb{Z}^{2}\big{)}$, then
we have an asymptotic evaluation

$F_{Q,\varepsilon}\bigg{(}\frac{h}{k}+\frac{t{\rm
i}}{2\pi}\bigg{)}\sim\sum_{r=-1}^{\infty}a_{h,k}(r)t^{r}$ (5.1)

as $t\to+0$, where $f(x,y):={\rm e}^{-Q(x,y)}$ and

$\displaystyle a_{h,k}(r):={}$ $\displaystyle
k^{2r}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{h}{k}Q(\gamma)\bigg{)}$
$\displaystyle\times\sum_{j+l=2r,\,j,l\geq-1}\frac{1}{(j+1)!(l+1)!}B_{j+1}\bigg{(}\frac{\alpha}{k}\bigg{)}B_{l+1}\bigg{(}\frac{\beta}{k}\bigg{)}f^{(j,l)}(0,0).$

Moreover, under Assumption 4.1 we have

$\displaystyle a_{h,k}(r)={}$ $\displaystyle
k^{2r}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{h}{k}Q(\gamma)\bigg{)}$
$\displaystyle{}\times\sum_{j+l=2r,\,j,l\geq
0}\frac{1}{(j+1)!(l+1)!}B_{j+1}\bigg{(}\frac{\alpha}{k}\bigg{)}B_{l+1}\bigg{(}\frac{\beta}{k}\bigg{)}f^{(j,l)}(0,0).$

###### Proof.

Our proof is based on the proof of [3, Lemma 3.2]. Since we can write

$\displaystyle F_{Q,\varepsilon}\bigg{(}\frac{h}{k}+\frac{t{\rm
i}}{2\pi}\bigg{)}=\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{h}{k}Q(\gamma)\bigg{)}\sum_{\mu\in\mathbb{Z}_{\geq
0}^{2}}f(\sqrt{t}(\mu+\gamma)),$

we have an asymptotic evaluation

$\displaystyle F_{Q,\varepsilon}\bigg{(}\frac{h}{k}+\frac{t{\rm
i}}{2\pi}\bigg{)}\sim{}$
$\displaystyle\sum_{j,l=-1}^{\infty}k^{j+l}t^{(j+l)/2}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{h}{k}Q(\gamma)\bigg{)}$
$\displaystyle\times\frac{1}{(j+1)!(l+1)!}B_{j+1}\bigg{(}\frac{\alpha}{k}\bigg{)}B_{l+1}\bigg{(}\frac{\beta}{k}\bigg{)}f^{(j,l)}(0,0)$

by Lemma 5.1. By assumption, the right hand side yields

$\displaystyle\sum_{j,l=-1}^{\infty}k^{j+l}t^{(j+l)/2}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{h}{k}Q(k-\alpha,k-\beta)\bigg{)}$
$\displaystyle\qquad{}\times\frac{1}{(j+1)!(l+1)!}B_{j+1}\bigg{(}\frac{k-\alpha}{k}\bigg{)}B_{l+1}\bigg{(}\frac{k-\beta}{k}\bigg{)}f^{(j,l)}(0,0).$

Since $Q(k-\alpha,k-\beta)\equiv Q(\gamma)\pmod{k\mathbb{Z}}$ and
$B_{n}(1-x)=(-1)^{n}B_{n}(x)$, this becomes

$\displaystyle\sum_{j,l=-1}^{\infty}k^{j+l}t^{(j+l)/2}(-1)^{j+l+2}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{h}{k}Q(\gamma)\bigg{)}$
$\displaystyle\qquad{}\times\frac{1}{(j+1)!(l+1)!}B_{j+1}\bigg{(}\frac{\alpha}{k}\bigg{)}B_{l+1}\bigg{(}\frac{\beta}{k}\bigg{)}f^{(j,l)}(0,0).$

Since it vanishes for the odd values of $j+l$, we obtain the asymptotic
evaluation (5.1).

The latter claim follows from Proposition 4.2$(i)$. ∎

Thus, we obtain the following proposition by Proposition 4.2$(i)$ and $(ii)$.

######  Proposition 5.3.

If Assumption 4.1 and the assumption in Lemma 5.2 and Proposition
$\ref{prop:Gauss_sum}(ii)$ hold, then we have

$\displaystyle\lim_{t\to
0}F_{Q,\varepsilon}\bigg{(}\frac{h}{k}+ti\bigg{)}=\frac{1}{k^{2}}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{h}{k}Q(\gamma)\bigg{)}\alpha\beta.$

We obtain the following quantum modular form by this proposition and Theorem
2.6$(i)$ which is proved in [3, Theorems 1.1 and 3.1].

######  Theorem 5.4.

Let $f_{\Gamma}\colon\mathbb{Q}\to\mathbb{C}$ be a map defined by

$\displaystyle
f_{\Gamma}\bigg{(}\frac{h}{k}\bigg{)}=\frac{1}{k^{2}}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{h}{k}Q(\gamma)\bigg{)}\alpha\beta$

for an irreducible fraction $h/k\in\mathbb{Q}$. If Assumption 4.1 and the
assumption in Lemma 5.2 and Proposition $\ref{prop:Gauss_sum}(ii)$ hold, then
$f_{\Gamma}$ is a quantum modular form of depth two and of weight one with the
quantum set $\mathbb{Q}$.

##  6 Witten–Reshetikhin–Turaev invariants

In this section, we prove that the WRT invariant is the radial limit of the
homological block. Firstly, we write the WRT invariant as a sum of rational
functions. Secondly, the Euler–Maclaurin summation formula is used to express
the WRT invariant as a sum of Bernoulli polynomials. Finally, we write the WRT
invariant as the radial limit of the homological block calculated in Section
5.

We start with an expression for the WRT invariant given by
Gukov–Pei–Putrov–Vafa [10, formula (A.10)]. In our situation it is written
down as follows:

$\displaystyle\tau_{k}(M_{3}(\Gamma))={}$ $\displaystyle\frac{-{\rm
i}\zeta_{k}^{-9/2}}{16k^{3}\big{(}\zeta_{2k}-\zeta_{2k}^{-1}\big{)}}\sum_{l\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})^{6}/2k\mathbb{Z}^{6}}\prod_{v=1}^{6}\zeta_{k}^{w_{v}(l_{v}^{2}-1)/4}\bigg{(}\frac{1}{\zeta_{k}^{l_{v}/2}-\zeta_{k}^{-l_{v}/2}}\bigg{)}^{\deg(v)-2}$
$\displaystyle\times\prod_{(v,v^{\prime})\in\mathrm{Edges}}\frac{\zeta_{k}^{l_{v}l_{v^{\prime}}/2}-\zeta_{k}^{-l_{v}l_{v^{\prime}}/2}}{2}.$
(6.1)

We can write the WRT invariant as a sum of rational functions.

######  Proposition 6.1.

It holds

$\displaystyle\tau_{k}(M_{3}(\Gamma))={}$ $\displaystyle\frac{{\rm
i}\zeta_{k}^{-(18+\sum_{v=1}^{6}w_{v}+\sum_{v=3}^{6}1/w_{v})/4}}{4k\big{(}\zeta_{2k}-\zeta_{2k}^{-1}\big{)}\sqrt{MN}}\sum_{\begin{subarray}{c}m\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2kw_{3}w_{4}\mathbb{Z}\\\ n\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2kw_{5}w_{6}\mathbb{Z}\end{subarray}}\zeta_{k}^{-(m,n)S^{-1}(m,n)^{\top}/4}$
$\displaystyle\times\frac{\big{(}\zeta_{k}^{m/(2w_{3})}-\zeta_{k}^{-m/(2w_{3})}\big{)}\big{(}\zeta_{k}^{m/(2w_{4})}-\zeta_{k}^{-m/(2w_{4})}\big{)}}{\zeta_{k}^{m/2}-\zeta_{k}^{-m/2}}$
$\displaystyle{}\times\frac{\big{(}\zeta_{k}^{n/(2w_{5})}-\zeta_{k}^{-n/(2w_{5})}\big{)}\big{(}\zeta_{k}^{n/(2w_{6})}-\zeta_{k}^{-n/(2w_{6})}\big{)}}{\zeta_{k}^{n/2}-\zeta_{k}^{-n/2}},$

and thus,

$\displaystyle\begin{split}&\tau_{k}(M_{3}(\Gamma))=\frac{{\rm
i}\zeta_{k}^{-(18+\sum_{v=1}^{6}w_{v}+\sum_{v=3}^{6}1/w_{v})/4}}{4k\big{(}\zeta_{2k}-\zeta_{2k}^{-1}\big{)}\sqrt{MN}}\\\
&\hphantom{\tau_{k}(M_{3}(\Gamma))=}{}\times\sum_{\mu=(m,n)^{\top}\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})^{2}/2kS(\mathbb{Z}^{2})}\zeta_{k}^{-\mu^{\top}S^{-1}\mu/4}G^{\omega}\big{(}\zeta_{k}^{m/(2M)}\big{)}G^{\varpi}\big{(}\zeta_{k}^{n/(2N)}\big{)},\end{split}$

where $\omega:=(w_{3},w_{4})$, $\varpi:=(w_{5},w_{6})$, and

$\displaystyle
G^{(w,w^{\prime})}(z):=\frac{(z^{w}-z^{-w})(z^{w^{\prime}}-z^{-w^{\prime}})}{(z^{ww^{\prime}}-z^{-ww^{\prime}})}=-\sum_{n=1}^{\infty}\chi^{(w,w^{\prime})}(n)z^{n}.$

###### Proof.

By (6.1), we obtain

$\displaystyle\tau_{k}(M_{3}(\Gamma))={}$ $\displaystyle\frac{-{\rm
i}\zeta_{k}^{-(18+\sum_{v=1}^{6}w_{v})/4}}{16k^{3}\big{(}\zeta_{2k}-\zeta_{2k}^{-1}\big{)}}\sum_{l_{1},l_{2}\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2k\mathbb{Z}}\bigg{(}\prod_{v=1}^{2}\frac{\zeta_{k}^{w_{v}l_{v}^{2}/4}}{\zeta_{k}^{l_{v}/2}-\zeta_{k}^{-l_{v}/2}}\bigg{)}\frac{\zeta_{k}^{l_{1}l_{2}/2}-\zeta_{k}^{-l_{1}l_{2}/2}}{2}$
$\displaystyle\times\prod_{v=3}^{4}\sum_{l_{v}\in\mathbb{Z}/2k\mathbb{Z}}\frac{1}{2}\zeta_{k}^{w_{v}l_{v}^{2}/4}\big{(}\zeta_{k}^{(l_{1}+1)l_{v}/2}-\zeta_{k}^{(l_{1}-1)l_{v}/2}-\zeta_{k}^{(-l_{1}+1)l_{v}/2}+\zeta_{k}^{(-l_{1}-1)l_{v}/2}\big{)}$
$\displaystyle\times\prod_{v=5}^{6}\sum_{l_{v}\in\mathbb{Z}/2k\mathbb{Z}}\frac{1}{2}\zeta_{k}^{w_{v}l_{v}^{2}/4}\big{(}\zeta_{k}^{(l_{2}+1)l_{v}/2}-\zeta_{k}^{(l_{2}-1)l_{v}/2}-\zeta_{k}^{(-l_{2}+1)l_{v}/2}+\zeta_{k}^{(-l_{2}-1)l_{v}/2}\big{)}.$

Applying symmetries $l_{v}\mapsto-l_{v}$, we can show that

$\displaystyle\tau_{k}(M_{3}(\Gamma))={}$ $\displaystyle\frac{-{\rm
i}\zeta_{k}^{-(18+\sum_{v=1}^{6}w_{v})/4}}{16k^{3}\big{(}\zeta_{2k}-\zeta_{2k}^{-1}\big{)}}\sum_{l_{1},l_{2}\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2k\mathbb{Z}}\bigg{(}\prod_{v=1}^{2}\frac{\zeta_{k}^{w_{v}l_{v}^{2}/4}}{\zeta_{k}^{l_{v}/2}-\zeta_{k}^{-l_{v}/2}}\bigg{)}\frac{\zeta_{k}^{l_{1}l_{2}/2}-\zeta_{k}^{-l_{1}l_{2}/2}}{2}$
$\displaystyle\times\prod_{v=3}^{4}\sum_{l_{v}\in\mathbb{Z}/2k\mathbb{Z}}\zeta_{k}^{w_{v}l_{v}^{2}/4}\big{(}\zeta_{k}^{(l_{1}+1)l_{v}/2}-\zeta_{k}^{(l_{1}-1)l_{v}/2}\big{)}$
$\displaystyle\times\prod_{v=5}^{6}\sum_{l_{v}\in\mathbb{Z}/2k\mathbb{Z}}\zeta_{k}^{w_{v}l_{v}^{2}/4}\big{(}\zeta_{k}^{(l_{2}+1)l_{v}/2}-\zeta_{k}^{(l_{2}-1)l_{v}/2}\big{)}.$

holds. Applying Proposition 2.7 for the case when $L=\mathbb{Z}$ equipped with
the standard inner product $\left\langle\cdot,\cdot\right\rangle$ and

$\displaystyle(h,z):=\begin{cases}(w_{v},(l_{1}\pm 1)/2k)&\text{if}\ v=3,4,\\\
(w_{v},(l_{2}\pm 1)/2k)&\text{if}\ v=5,6.\end{cases}$

we obtain the following equation:

$\displaystyle\tau_{k}(M_{3}(\Gamma))=\,$ $\displaystyle\frac{{\rm
i}\zeta_{k}^{-(18+\sum_{v=1}^{6}w_{v})/4}}{4k\big{(}\zeta_{2k}-\zeta_{2k}^{-1}\big{)}\prod_{v=3}^{6}\sqrt{\left\lvert
w_{v}\right\rvert}}$
$\displaystyle\times\sum_{l_{1},l_{2}\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2k\mathbb{Z}}\bigg{(}\prod_{v=1}^{2}\frac{\zeta_{k}^{w_{v}l_{v}^{2}/4}}{\zeta_{k}^{l_{v}/2}-\zeta_{k}^{-l_{v}/2}}\bigg{)}\frac{\zeta_{k}^{l_{1}l_{2}/2}-\zeta_{k}^{-l_{1}l_{2}/2}}{2}$
$\displaystyle\times\prod_{v=3}^{4}\sum_{l_{v}\in\mathbb{Z}/w_{v}\mathbb{Z}}\big{(}\zeta_{k}^{-(2kl_{v}+l_{1}+1)^{2}/(4w_{v})}-\zeta_{k}^{-(2kl_{v}+l_{1}-1)^{2}/(4w_{v})}\big{)}$
$\displaystyle\times\prod_{v=5}^{6}\sum_{l_{v}\in\mathbb{Z}/w_{v}\mathbb{Z}}\big{(}\zeta_{k}^{-(2kl_{v}+l_{2}+1)^{2}/(4w_{v})}-\zeta_{k}^{-(2kl_{v}+l_{2}-1)^{2}/(4w_{v})}\big{)}.$

For any map
$f\colon\mathbb{Z}/w_{3}\mathbb{Z}\times\mathbb{Z}/w_{4}\mathbb{Z}\to\mathbb{C}$,
we have

$\displaystyle\sum_{\begin{subarray}{c}l_{3}\in\mathbb{Z}/w_{3}\mathbb{Z}\\\
l_{4}\in\mathbb{Z}/w_{4}\mathbb{Z}\end{subarray}}f(l_{3},l_{4})=\sum_{n_{1}\in\mathbb{Z}/w_{3}w_{4}\mathbb{Z}}f(n_{1},n_{1})$

by the Chinese remainder theorem. Hence, for any map
$\phi\colon\mathbb{Z}/2k\mathbb{Z}\times\mathbb{Z}/2kw_{3}\mathbb{Z}\times\mathbb{Z}/2kw_{4}\mathbb{Z}\to\mathbb{C}$,
we have

$\displaystyle\sum_{l_{1}\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2k\mathbb{Z}}\sum_{\begin{subarray}{c}l_{3}\in\mathbb{Z}/w_{3}\mathbb{Z}\\\
l_{4}\in\mathbb{Z}/w_{4}\mathbb{Z}\end{subarray}}\phi(l_{1},l_{1}+2kl_{3},l_{1}+2kl_{4})$
$\displaystyle\qquad{}=\sum_{l_{1}\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2k\mathbb{Z}}\sum_{n_{1}\in\mathbb{Z}/w_{3}w_{4}\mathbb{Z}}\phi(l_{1},l_{1}+2kn_{1},l_{1}+2kn_{1})$

by letting $f(l_{3},l_{4}):=\phi(l_{1},l_{1}+2kl_{3},l_{1}+2kl_{4})$. This sum
equals

$\displaystyle\sum_{m\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2kw_{3}w_{4}\mathbb{Z}}\phi(m,m,m)$

by letting $m:=l_{1}+2kn_{1}$. Similarly, for any map
$\psi\colon\mathbb{Z}/2k\mathbb{Z}\times\mathbb{Z}/2kw_{5}\mathbb{Z}\times\mathbb{Z}/2kw_{6}\mathbb{Z}\to\mathbb{C}$,
we have

$\displaystyle\sum_{l_{2}\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2k\mathbb{Z}}\sum_{\begin{subarray}{c}l_{5}\in\mathbb{Z}/w_{5}\mathbb{Z}\\\
l_{6}\in\mathbb{Z}/w_{6}\mathbb{Z}\end{subarray}}\psi(l_{2},l_{2}+2kl_{5},l_{2}+2kl_{6})=\sum_{n\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2kw_{5}w_{6}\mathbb{Z}}\psi(n,n,n).$

Thus, we obtain

$\displaystyle\tau_{k}(M_{3}(\Gamma))={}$ $\displaystyle\frac{{\rm
i}\zeta_{k}^{-(18+\sum_{v=1}^{6}w_{v}+\sum_{v=3}^{6}1/w_{v})/4}}{8k\big{(}\zeta_{2k}-\zeta_{2k}^{-1}\big{)}\sqrt{MN}}$
$\displaystyle\times\sum_{\begin{subarray}{c}m\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2kw_{3}w_{4}\mathbb{Z}\\\ n\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2kw_{5}w_{6}\mathbb{Z}\end{subarray}}\big{(}\zeta_{k}^{-(m,-n)S^{-1}(m,-n)^{\top}/4}-\zeta_{k}^{-(m,n)S^{-1}(m,n)^{\top}/4}\big{)}$
$\displaystyle\times\frac{\big{(}\zeta_{k}^{m/(2w_{3})}-\zeta_{k}^{-m/(2w_{3})}\big{)}\big{(}\zeta_{k}^{m/(2w_{4})}-\zeta_{k}^{-m/(2w_{4})}\big{)}}{\zeta_{k}^{m/2}-\zeta_{k}^{-m/2}}$
$\displaystyle\times\frac{\big{(}\zeta_{k}^{n/(2w_{5})}-\zeta_{k}^{-n/(2w_{5})}\big{)}\big{(}\zeta_{k}^{n/(2w_{6})}-\zeta_{k}^{-n/(2w_{6})}\big{)}}{\zeta_{k}^{n/2}-\zeta_{k}^{-n/2}}$

since

$\displaystyle-(m,n)S^{-1}(m,n)^{\top}$ $\displaystyle=-c/Mm^{2}+2mn-a/Nn^{2}$
$\displaystyle=\bigg{(}w_{1}-\frac{1}{w_{3}}-\frac{1}{w_{4}}\bigg{)}m^{2}+2mn+\bigg{(}w_{2}-\frac{1}{w_{5}}-\frac{1}{w_{6}}\bigg{)}n^{2}.$

By replacing $n\mapsto-n$, we have

$\displaystyle\sum_{n\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2kw_{5}w_{6}\mathbb{Z}}\zeta_{k}^{-(m,-n)S^{-1}(m,-n)^{\top}/4}\frac{\big{(}\zeta_{k}^{n/(2w_{5})}-\zeta_{k}^{-n/(2w_{5})}\big{)}\big{(}\zeta_{k}^{n/(2w_{6})}-\zeta_{k}^{-n/(2w_{6})}\big{)}}{\zeta_{k}^{n/2}-\zeta_{k}^{-n/2}}$
$\displaystyle\qquad{}=-\sum_{n\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})/2kw_{5}w_{6}\mathbb{Z}}\zeta_{k}^{-(m,n)S^{-1}(m,n)^{\top}/4}\frac{\big{(}\zeta_{k}^{n/(2w_{5})}-\zeta_{k}^{-n/(2w_{5})}\big{)}\big{(}\zeta_{k}^{n/(2w_{6})}-\zeta_{k}^{-n/(2w_{6})}\big{)}}{\zeta_{k}^{n/2}-\zeta_{k}^{-n/2}}.$

Therefore we prove the claim. ∎

Then we calculate the asymptotic expansion of
$G^{\omega}\big{(}\zeta_{k}^{m/(2M)}{\rm
e}^{-tK/(2M)}\big{)}G^{\varpi}\big{(}\zeta_{k}^{n/(2N)}{\rm
e}^{-tL/(2N)}\big{)}$.

######  Lemma 6.2.

Let $K$ and $L$ be positive real numbers and $\mu=(m,n)^{\top}$ be a pair of
integers. For sufficiently small $t>0$,

$\displaystyle G^{\omega}\big{(}\zeta_{k}^{m/(2M)}{\rm
e}^{-tK/(2M)}\big{)}G^{\varpi}\big{(}\zeta_{k}^{n/(2N)}{\rm
e}^{-tL/(2N)}\big{)}$
$\displaystyle\qquad{}=\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})/k\mathbb{Z}^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}{\gamma^{\top}}\mu\bigg{)}\sum_{j,l=-1}^{\infty}\frac{B_{j+1}(\alpha/k)B_{l+1}(\beta/k)}{(j+1)!(l+1)!}K^{j}L^{l}(-kt)^{j+l}$

holds.

###### Proof.

By the definitions of $G^{\omega}$ and $G^{\varpi}$, the left hand side of the
above equation equals to

$\displaystyle\sum_{m^{\prime},n^{\prime}=0}^{\infty}\chi^{\omega}(m^{\prime})\chi^{\varpi}(n^{\prime})\big{(}\zeta_{k}^{m}{\rm
e}^{-tK}\big{)}^{m^{\prime}/2M}\big{(}\zeta_{k}^{n}{\rm
e}^{-tL}\big{)}^{n^{\prime}/2N}.$

Letting $\gamma=(m^{\prime}/(2M),n^{\prime}/(2N))^{\top}$ leads to

$\displaystyle\sum_{0\leq\gamma\in(2S)^{-1}(\mathbb{Z}^{2})}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}\gamma^{\top}\mu\bigg{)}{\rm
e}^{-t(K,L)\gamma}$

and the definition of Bernoulli polynomial shows the lemma. ∎

The constant term in the above expansion is simplified in the following lemma.

######  Proposition 6.3.

For $\mu=(m,n)^{\top}\in(\mathbb{Z}\smallsetminus k\mathbb{Z})^{2}$

$\displaystyle
G^{\omega}\big{(}\zeta_{k}^{m/(2M)}\big{)}G^{\varpi}\big{(}\zeta_{k}^{n/(2N)}\big{)}=\frac{1}{k^{2}}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}\gamma^{\top}\mu\bigg{)}\alpha\beta$

holds.

###### Proof.

Lemma 6.2 shows that for any $K,L>0$

$\displaystyle
G^{\omega}\big{(}\zeta_{k}^{m/(2M)}\big{)}G^{\varpi}\big{(}\zeta_{k}^{n/(2N)}\big{)}={}$
$\displaystyle\lim_{t\to 0}G^{\omega}\big{(}\zeta_{k}^{m/(2M)}{\rm
e}^{-tK/(2M)}\big{)}G^{\varpi}\big{(}\zeta_{k}^{n/(2N)}{\rm
e}^{-tL/(2N)}\big{)}$ $\displaystyle={}$
$\displaystyle\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}\gamma^{\top}\mu\bigg{)}$
$\displaystyle\times\bigg{(}\frac{K}{2L}B_{2}\bigg{(}\frac{\alpha}{k}\bigg{)}+\frac{L}{2K}B_{2}\bigg{(}\frac{\beta}{k}\bigg{)}+B_{1}\bigg{(}\frac{\alpha}{k}\bigg{)}B_{1}\bigg{(}\frac{\beta}{k}\bigg{)}\bigg{)}$

holds. The claim follows from the following lemma. ∎

######  Lemma 6.4.

For any $\mu=(m,n)^{\top}\in(\mathbb{Z}\smallsetminus
k\mathbb{Z})\times\mathbb{Z}$ and any map $B\colon\mathbb{Q}\to\mathbb{C}$, we
have

$\displaystyle\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}\gamma^{\top}\mu\bigg{)}B(\beta)=0.$

###### Proof.

The left hand side can be written as

$\displaystyle\sum_{\beta\in\frac{1}{2N}\mathbb{Z}/k\mathbb{Z}}\chi^{\varpi}(\beta)\bm{e}\bigg{(}\frac{\beta
n}{k}\bigg{)}B(\beta)\sum_{\alpha\in\frac{1}{2M}\mathbb{Z}/\mathbb{Z}}\chi^{\omega}(\beta)\bm{e}\bigg{(}\frac{\alpha
m}{k}\bigg{)}\sum_{l\in\mathbb{Z}/k\mathbb{Z}}\bm{e}\bigg{(}\frac{ml}{k}\bigg{)}.$

Since $k\nmid m$, the last sum vanishes. Thus, we obtain the claim. ∎

Propositions 6.1 and 6.3 assert that the following proposition holds.

######  Proposition 6.5.

$\displaystyle\tau_{k}(M_{3}(\Gamma))=\,\frac{\zeta_{k}^{-(18+\sum_{v=1}^{6}w_{v}+\sum_{v=3}^{6}1/w_{v})/4}}{2\big{(}\zeta_{2k}-\zeta_{2k}^{-1}\big{)}}\frac{1}{k^{2}}\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}Q(\gamma)\bigg{)}\alpha\beta.$

###### Proof.

Propositions 4.2$(v)$, 6.1 and 6.3 show

$\displaystyle\tau_{k}(M_{3}(\Gamma))={}$ $\displaystyle\frac{{\rm
i}\zeta_{k}^{-(18+\sum_{v=1}^{6}w_{v}+\sum_{v=3}^{6}1/w_{v})/4}}{4k\big{(}\zeta_{2k}-\zeta_{2k}^{-1}\big{)}\sqrt{MN}}\frac{1}{k^{2}}\sum_{\mu=(m,n)^{\top}\in\mathbb{Z}^{2}/2kS(\mathbb{Z}^{2})}\zeta_{k}^{-\mu^{\top}S^{-1}\mu/4}$
$\displaystyle\times\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}\gamma^{\top}\mu\bigg{)}\alpha\beta.$

Replacing $\mu$ with $\mu+2S\gamma$, we obtain

$\displaystyle\tau_{k}(M_{3}(\Gamma))={}$ $\displaystyle\frac{{\rm
i}\zeta_{k}^{-(18+\sum_{v=1}^{6}w_{v}+\sum_{v=3}^{6}1/w_{v})/4}}{4k\big{(}\zeta_{2k}-\zeta_{2k}^{-1}\big{)}\sqrt{MN}}G(2kS)\frac{1}{k^{2}}$
$\displaystyle\times\sum_{\gamma=(\alpha,\beta)^{\top}\in(2S)^{-1}(\mathbb{Z}^{2})\cap[0,k)^{2}}\varepsilon(\gamma)\bm{e}\bigg{(}\frac{1}{k}Q(\gamma)\bigg{)}\alpha\beta,$

where

$\displaystyle
G(2kS):=\sum_{\mu=(m,n)^{\top}\in\mathbb{Z}^{2}/2kS(\mathbb{Z}^{2})}\bm{e}\bigg{(}{-}\frac{1}{2}\mu^{\top}(2kS)^{-1}\mu\bigg{)}.$

By Proposition 2.7, $G(2S)=-2k{\rm i}\sqrt{MN}$ and this completes the proof.
∎

### Acknowledgements

The first and second author are supported by JSPS KAKENHI Grant Number JP
21J10271 and 20J20308. The first author was supported by a Scholarship of
Tohoku University, Division for Interdisciplinary Advanced Research and
Education. We would like to show their greatest appreciation to Professor Yuji
Terashima and Takuya Yamauchi for giving many pieces of advice. We are deeply
grateful to Professor Kazuhiro Hikami for giving many comments. The first
author thanks his family for all the support. We thank the referees for
helpful suggestions and comments which substantially improved the presentation
of our paper.

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