be a non trivial solution of (3.12) for $m_{0}$. For each $\gamma\in\Gamma$,
we define $a_{N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)}$ in such a way that
it satisfies (3.32). Then, $a_{N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)}$ is
a non trivial solution of (3.12) for
$m=N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)$.

###### Proof.

Suppose that there exists $m_{0}\in\mathbb{Z}^{n}$ with $\frac{m_{0}}{N}\in F$
such that (3.13) holds. By Lemma 3.10, (3.12) for $m_{0}$ has a non trivial
solution $a_{m_{0}}$. Then, for each $\gamma\in\Gamma$, define
$a_{N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)}$ by (3.32). By Lemma 3.10
again, in order to show this lemma, it is sufficient to prove
$a_{N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)}$ is a solution of (3.12) for
$m=N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)$. Let us compute the Jacobi
matrix of the both sides of (3.32). The left hand side is

(3.36) $\displaystyle
D\left(a_{N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)}\circ\rho_{\gamma}\right)_{x}$
$\displaystyle=\left(Da_{N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)}\right)_{\rho_{\gamma}(x)}\left(D\rho_{\gamma}\right)_{x}$
$\displaystyle=\left(\partial_{x_{1}}a_{N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)},\ldots,\partial_{x_{n}}a_{N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)}\right)_{\rho_{\gamma}(x)}A_{\gamma}.$

The right hand side is

(3.37) $\displaystyle
D\left(g_{\gamma}a_{m}(x)e^{2\pi\sqrt{-1}N\left\\{{\widetilde{g}}_{\gamma}(x)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}(x)\right\\}}\right)_{x}$
$\displaystyle=g_{\gamma}e^{2\pi\sqrt{-1}N\left\\{{\widetilde{g}}_{\gamma}(x)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}(x)\right\\}}(Da_{m})_{x}+g_{\gamma}a_{m}(x)D\left(e^{2\pi\sqrt{-1}N\left\\{{\widetilde{g}}_{\gamma}(x)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}(x)\right\\}}\right)$
$\displaystyle\stackrel{{\scriptstyle\eqref{Dirac3}}}{{=}}-2\pi\sqrt{-1}g_{\gamma}e^{2\pi\sqrt{-1}N\left\\{{\widetilde{g}}_{\gamma}(x)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}(x)\right\\}}a_{m}(x){}^{t}\left(\Omega_{x}(m-Nx)\right)$
$\displaystyle\quad\quad+2\pi\sqrt{-1}Ng_{\gamma}a_{m}(x)e^{2\pi\sqrt{-1}N\left\\{{\widetilde{g}}_{\gamma}(x)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}(x)\right\\}}D\left({\widetilde{g}}_{\gamma}(x)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}(x)\right)$ $\displaystyle\stackrel{{\scriptstyle\eqref{Gamma-
invariance}}}{{=}}-2\pi\sqrt{-1}a_{N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)}\left(\rho_{\gamma}(x)\right){}^{t}\left(\Omega_{x}A_{\gamma}^{-1}\left(N\rho_{\gamma}\left(\frac{m}{N}\right)-N\rho_{\gamma}(x)\right)\right)$
$\displaystyle\quad\quad+2\pi\sqrt{-1}Na_{N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)}\left(\rho_{\gamma}(x)\right)D\left({\widetilde{g}}_{\gamma}(x)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}(x)\right).$

For each $i=1,\ldots,n$, the direct computation shows

$\displaystyle\partial_{x_{i}}\left({\widetilde{g}}_{\gamma}(x)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}(x)\right)$
$\displaystyle=(\partial_{x_{i}}u_{\gamma})_{x}\cdot\left(\rho_{\gamma}(x)-\rho_{\gamma}\left(\frac{m}{N}\right)\right)+\left({}^{t}A_{\gamma}u_{\gamma}(x)\right)_{i}-\left({}^{t}A_{\gamma}u_{\gamma}(0,\ldots,0,x_{i},\ldots,x_{n})\right)_{i}$
$\displaystyle\quad-\sum_{j<i}\int_{0}^{x_{j}}\left({}^{t}A_{\gamma}Du_{\gamma}\right)_{ji}(0,\ldots,0,x_{j},\ldots,x_{n})dx_{j}$
$\displaystyle=(\partial_{x_{i}}u_{\gamma})_{x}\cdot\left(\rho_{\gamma}(x)-\rho_{\gamma}\left(\frac{m}{N}\right)\right)$
$\displaystyle\quad+\sum_{j<i}\int_{0}^{x_{j}}\partial_{x_{j}}\left({}^{t}A_{\gamma}u_{\gamma}(0,\ldots,0,x_{j},\ldots,x_{n})\right)_{i}dx_{j}-\sum_{j<i}\int_{0}^{x_{j}}\left({}^{t}A_{\gamma}Du_{\gamma}\right)_{ji}(0,\ldots,0,x_{j},\ldots,x_{n})dx_{j}$
$\displaystyle=(\partial_{x_{i}}u_{\gamma})_{x}\cdot\left(\rho_{\gamma}(x)-\rho_{\gamma}\left(\frac{m}{N}\right)\right)$
$\displaystyle\quad+\sum_{j<i}\int_{0}^{x_{j}}\left({}^{t}A_{\gamma}Du_{\gamma}\right)_{ij}(0,\ldots,0,x_{j},\ldots,x_{n})dx_{j}-\sum_{j<i}\int_{0}^{x_{j}}\left({}^{t}A_{\gamma}Du_{\gamma}\right)_{ji}(0,\ldots,0,x_{j},\ldots,x_{n})dx_{j}$
$\displaystyle=-(\partial_{x_{i}}u_{\gamma})_{x}\cdot\left(\rho_{\gamma}\left(\frac{m}{N}\right)-\rho_{\gamma}(x)\right).$

In the last equality, we used
${}^{t}\left({}^{t}A_{\gamma}Du_{\gamma}\right)={}^{t}A_{\gamma}Du_{\gamma}$.
Hence, we have

(3.38)
$D\left({\widetilde{g}}_{\gamma}(x)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}(x)\right)=-{}^{t}\left(\rho_{\gamma}\left(\frac{m}{N}\right)-\rho_{\gamma}(x)\right)\left(Du_{\gamma}\right)_{x}.$

By (3.36), (3.37), and (3.38), we obtain

$\displaystyle{}^{t}A_{\gamma}{}^{t}\left(Da_{N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)}\right)_{\rho_{\gamma}(x)}=-2\pi\sqrt{-1}a_{N\rho_{\gamma}\left(\frac{m_{0}}{N}\right)}\left(\rho_{\gamma}(x)\right)\left(\Omega_{x}A_{\gamma}^{-1}+{}^{t}(Du_{\gamma})_{x}\right)\left(N\rho_{\gamma}\left(\frac{m}{N}\right)-N\rho_{\gamma}(x)\right).$

On the other hand, by (3.4) and (3.5), we have

(3.39)
${}^{t}A_{\gamma}\Omega_{\rho_{\gamma}(x)}=\Omega_{x}A_{\gamma}^{-1}+{}^{t}\left(Du_{\gamma}\right)_{x}.$

This proves the lemma. ∎

######  Remark 3.16.

By Remark 3.3 and Lemma 3.15, the condition (3.13) holds for all
$\frac{m}{N}\in F\cap\frac{1}{N}\mathbb{Z}^{n}$ if and only if the condition
(1), hence all conditions in Proposition 3.12 holds.

##  4\. The integrable case

###  4.1. Definition and properties of $\vartheta_{\frac{m}{N}}$

We use the settting and the notations introduced in the previous section. Let
$\frac{m}{N}\in F\cap\frac{1}{N}\mathbb{Z}^{n}$ be the point for which the
condition (3.13) holds, and $a_{m}$ the non trivial solution of (3.12) of the
form $\eqref{a(x)}$ with $a_{m}\left(\frac{m}{N}\right)=1$. For each
$\gamma\in\Gamma$, define $a_{N\rho_{\gamma}\left(\frac{m}{N}\right)}$ in such
a way that it satisfies (3.32). As we showed in Lemma 3.15,
$a_{N\rho_{\gamma}\left(\frac{m}{N}\right)}$ is a non trivial solution of
(3.12) for $N\rho_{\gamma}\left(\frac{m}{N}\right)$. Then, we can define the
formal Fourier series $\vartheta_{\frac{m}{N}}$ by

(4.1)
$\vartheta_{\frac{m}{N}}(x,y):=\sum_{\gamma\in\Gamma}a_{N\rho_{\gamma}\left(\frac{m}{N}\right)}(x)e^{2\pi\sqrt{-1}N\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
y}.$

######  Proposition 4.1.

(1) $\vartheta_{\frac{m}{N}}$ has the following expression

$\displaystyle\vartheta_{\frac{m}{N}}(x,y)=\sum_{\gamma\in\Gamma}g_{\gamma}e^{2\pi\sqrt{-1}\left[-\sum_{i=1}^{n}G_{m}^{i}\left(\frac{m_{1}}{N},\ldots,\frac{m_{i-1}}{N},\left(\rho_{\gamma^{-1}}(x)\right)_{i},\ldots,\left(\rho_{\gamma^{-1}}(x)\right)_{n}\right)+N\left\\{{\widetilde{g}}_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)+\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
y\right\\}\right]}.$

(2) $\vartheta_{\frac{m}{N}}$ can be described as
$\vartheta_{\frac{m}{N}}=\sum_{\gamma\in\Gamma}{\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}_{\gamma}\circ
s_{m}\circ{\widetilde{\rho}}_{\gamma^{-1}}$, where $s_{m}$ is the section
defined by (3.16).
(3) If $Y+XY^{-1}X$ is constant, then, $\vartheta_{\frac{m}{N}}$ converges
absolutely and uniformly on any compact set.

###### Proof.

(1) and (2) are obtained by (3.32), (3.14), (2.5), and (3.16). Let us prove
(3). By (2.4) and (3.5), we obtain

${}^{t}A_{\gamma^{-1}}\left(Y+XY^{-1}X\right)^{-1}A_{\gamma^{-1}}=\left(Y+XY^{-1}X\right)^{-1}.$

By using this formula together with the assumption, the expression in (1) can
be rewritten as

$\vartheta_{\frac{m}{N}}(x,y)=\sum_{\gamma\in\Gamma}g_{\gamma}e^{2\pi\sqrt{-1}\left[\frac{\sqrt{-1}N}{2}\left(x-\rho_{\gamma}\left(\frac{m}{N}\right)\right)\cdot\left(Y+XY^{-1}X\right)^{-1}\left(x-\rho_{\gamma}\left(\frac{m}{N}\right)\right)+\text{real
part}\right]}.$

Since $\left(Y+XY^{-1}X\right)^{-1}$ is positive definite, there exists a
positive constant $c>0$ such that $\left(Y+XY^{-1}X\right)^{-1}\geq cI$. Then,

$\displaystyle\left|g_{\gamma}e^{2\pi\sqrt{-1}\left[\frac{\sqrt{-1}N}{2}\left(x-\rho_{\gamma}\left(\frac{m}{N}\right)\right)\cdot\left(Y+XY^{-1}X\right)^{-1}\left(x-\rho_{\gamma}\left(\frac{m}{N}\right)\right)+\text{real
part}\right]}\right|$
$\displaystyle=e^{-N\pi\left(x-\rho_{\gamma}\left(\frac{m}{N}\right)\right)\cdot\left(Y+XY^{-1}X\right)^{-1}\left(x-\rho_{\gamma}\left(\frac{m}{N}\right)\right)}$
$\displaystyle\leq e^{-cN\pi\lVert
x-\rho_{\gamma}\left(\frac{m}{N}\right)\rVert^{2}}$
$\displaystyle=e^{-cN\pi\lVert x-\frac{l}{N}\rVert^{2}}\quad(\text{put
}l:=N\rho_{\gamma}\left(\frac{m}{N}\right))$
$\displaystyle=\prod_{i=1}^{n}e^{-cN\pi\left(x_{i}-\frac{l_{i}}{N}\right)^{2}}.$

Hence, the series is dominated by
$\prod_{i=1}^{n}\sum_{l_{i}\in\mathbb{Z}}e^{-cN\pi(\frac{l_{i}}{N}-x_{i})^{2}}$.
Any compact set is contained in a product of closed intervals
$I_{1}\times\cdots\times I_{n}$, so it is sufficient to show that
$\sum_{l\in\mathbb{Z}}e^{-cN\pi(\frac{l}{N}-x)^{2}}$ converges uniformly on
any closed interval $I$. Suppose that $I$ is of the form $I:=[x_{m},x_{M}]$.
Set $l_{M}:=\max\\{l\in\mathbb{Z}\mid\frac{l}{N}\in I\\}$ and
$l_{m}:=\min\\{l\in\mathbb{Z}\mid\frac{l}{N}\in I\\}$. On $I$, $\sum_{-k\leq
l\leq k}e^{-cN\pi(\frac{l}{N}-x)^{2}}$ can be estimated as

$\displaystyle\sum_{-k\leq l\leq k}e^{-cN\pi(\frac{l}{N}-x)^{2}}$
$\displaystyle=\left(\sum_{-k\leq l<l_{m}}+\sum_{l_{m}\leq l\leq
l_{M}}+\sum_{l_{M}\leq l\leq k}\right)e^{-cN\pi(\frac{l}{N}-x)^{2}}$
$\displaystyle\leq\sum_{-k\leq
l<l_{m}}e^{\frac{-c\pi}{N}(l-Nx_{m})^{2}}+(l_{M}-l_{n}+1)+\sum_{l_{M}<l\leq
k}e^{\frac{-c\pi}{N}(l-Nx_{M})^{2}}$
$\displaystyle\leq\int_{-k}^{l_{m}}e^{\frac{-c\pi}{N}(\tau-
Nx_{m})^{2}}d\tau+(l_{M}-l_{n}+1)+\int_{l_{M}}^{k}e^{\frac{-c\pi}{N}(\tau-
Nx_{M})^{2}}d\tau,$

and it is well-known that both $\int_{-k}^{l_{m}}e^{\frac{-c\pi}{N}(\tau-
Nx_{m})^{2}}d\tau$ and $\int_{l_{M}}^{k}e^{\frac{-c\pi}{N}(\tau-
Nx_{M})^{2}}d\tau$ converges as $k\to+\infty$. ∎

######  Theorem 4.2.

If $s\in\Gamma\left({\widetilde{L}}^{\otimes N}\right)$ is a non trivial
$\Gamma$-equivariant solution of $0=Ds$, then, there exists
$m\in\mathbb{Z}^{n}$ with $\frac{m}{N}\in F$ such that the condition (3.13)
holds. Conversely, suppose that the condition (3.13) holds for some
$\frac{m}{N}\in F\cap\frac{1}{N}\mathbb{Z}^{n}$ and $\vartheta_{\frac{m}{N}}$
converges absolutely and uniformly. Then, $\vartheta_{\frac{m}{N}}$ is a non
trivial $\Gamma$-equivariant solution of $0=Ds$. In particular, if $J$ is
integrable and $\vartheta_{\frac{m}{N}}$ converges absolutely and uniformly
for any $\frac{m}{N}\in F\cap\frac{1}{N}\mathbb{Z}^{n}$, then,
$\left\\{\vartheta_{\frac{m}{N}}\right\\}_{\frac{m}{N}\in
F\cap\frac{1}{N}\mathbb{Z}^{n}}$ is a basis of the space of
$\Gamma$-equivariant holomorphic sections of
$\left({\widetilde{L}},\nabla^{\widetilde{L}}\right)^{\otimes
N}\to({\widetilde{M}},N\omega_{0},J)$.

###### Proof.

Since $s=\sum_{l\in\mathbb{Z}^{n}}a_{l}(x)e^{2\pi\sqrt{-1}l\cdot y}$ is non
trivial solution of $0=Ds$, by Proposition 3.11, there exists
$l\in\mathbb{Z}^{n}$ such that $a_{l}\not=0$. On the other hand, as is noticed
in Remark 3.3, there exists
$\left(\gamma,\frac{m}{N}\right)\in\Gamma\times\left(F\cap\frac{1}{N}\mathbb{Z}^{n}\right)$
such that $l=N\rho_{\gamma}\left(\frac{m}{N}\right)$. Since $s$ is
$\Gamma$-equivariant, by (3.32),
$0\not=a_{l}=a_{N\rho_{\gamma}\left(\frac{m}{N}\right)}$ implies
$a_{m}\not=0$.

Let us prove the latter. It is trivial that
$\left\\{\vartheta_{\frac{m}{N}}\right\\}_{\frac{m}{N}\in
F\cap\frac{1}{N}\mathbb{Z}^{n}}$ is linear independent. Let $s$ be a
$\Gamma$-equivariant holomorphic section of ${\widetilde{L}}^{\otimes N}$. By
Lemma 3.14, $s$ can be written as in (3.33). Then,

$\displaystyle s(x,y)$ $\displaystyle\stackrel{{\scriptstyle\eqref{Gamma-equiv
section}}}{{=}}\sum_{\left(\gamma,\frac{m}{N}\right)\in\Gamma\times\left(F\cap\frac{1}{N}\mathbb{Z}^{n}\right)}g_{\gamma}a_{m}\left(\rho_{\gamma^{-1}}(x)\right)e^{2\pi\sqrt{-1}N\left\\{{\widetilde{g}}_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)+\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
y\right\\}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{a(x)}}}{{=}}\sum_{\left(\gamma,\frac{m}{N}\right)\in\Gamma\times\left(F\cap\frac{1}{N}\mathbb{Z}^{n}\right)}g_{\gamma}a_{m}\left(\frac{m}{N}\right)e^{2\pi\sqrt{-1}\left[-\sum_{i=1}^{n}G_{m}^{i}\left(\frac{m_{1}}{N},\ldots,\frac{m_{i-1}}{N},\left(\rho_{\gamma^{-1}}(x)\right)_{i},\ldots,\left(\rho_{\gamma^{-1}}(x)\right)_{n}\right)\right.}$
$\displaystyle\hskip
142.26378pt{}^{\left.+N\left\\{{\widetilde{g}}_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)+\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
y\right\\}\right]}$ $\displaystyle=\sum_{\frac{m}{N}\in
F\cap\frac{1}{N}\mathbb{Z}^{n}}a_{m}\left(\frac{m}{N}\right)\sum_{\gamma\in\Gamma}g_{\gamma}e^{2\pi\sqrt{-1}\left[-\sum_{i=1}^{n}G_{m}^{i}\left(\frac{m_{1}}{N},\ldots,\frac{m_{i-1}}{N},\left(\rho_{\gamma^{-1}}(x)\right)_{i},\ldots,\left(\rho_{\gamma^{-1}}(x)\right)_{n}\right)\right.}$
$\displaystyle\hskip
142.26378pt{}^{\left.+N\left\\{{\widetilde{g}}_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)+\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
y\right\\}\right]}$ $\displaystyle=\sum_{\frac{m}{N}\in
F\cap\frac{1}{N}\mathbb{Z}^{n}}a_{m}\left(\frac{m}{N}\right)\vartheta_{\frac{m}{N}}(x,y).$

This proves the theorem. ∎

By Corollary 2.25, any Lagrangian fibration $\pi\colon(M,\omega)\to B$ on a
connected complete base $B$ with prequantum line bundle
$(L,\nabla^{L})\to(M,\omega)$ are obtained as the quotient of the action of
$\Gamma:=\pi_{1}(B)$. Let $J$ be a compatible almost complex structure on
$(M,\omega)$ which is invariant along the fiber in the sense of Lemma 3.6 and
$D^{M}$ the associated Spinc Dirac operator on $(M,N\omega)$ with coefficients
in $L^{\otimes N}$. Since the $\Gamma$-action preserves all the data,
$\Gamma(L^{\otimes N})\cap\ker D^{M}$ is identified with
$\left(\Gamma\left({\widetilde{L}}^{\otimes N}\right)\cap\ker
D\right)^{\Gamma}$. Moreover, $F\cap\frac{1}{N}\mathbb{Z}^{n}$ is identified
with $B_{BS}$ as is noticed in Remark 3.3. Thus, we obtain the following
corollary.

######  Corollary 4.3.

Let $\pi\colon(M,\omega)\to B$ be a Lagrangian fibration on a connected
complete base $B$ and $(L,\nabla^{L})\to(M,\omega)$ a prequantum line bundle.
Let $J$ be a compatible almost complex structure on $(M,\omega)$ which is
invariant along the fiber in the sense of Lemma 3.6. Assume that $J$ is
integrable and $\vartheta_{\frac{m}{N}}$ converges absolutely and uniformly
for each $\frac{m}{N}\in F\cap\frac{1}{N}\mathbb{Z}^{n}$. Then,
$\left\\{\vartheta_{\frac{m}{N}}\right\\}_{\frac{m}{N}\in
F\cap\frac{1}{N}\mathbb{Z}^{n}}$ gives a basis of the space of holomorphic
sections of $(L,\nabla^{L})^{\otimes N}\to(M,N\omega,J)$ indexed by the Bohr-
Sommerfeld points.

######  Remark 4.4.

When $M$ is compact as well as the assumption of Corollary 4.3, we choose the
orientation on $M$ so that
$(-1)^{\frac{n(n-1)}{2}}\frac{\left(N\omega\right)^{n}}{n!}$ is a positive
volume form, and define the Hermitian inner product of the space of sections
of $L^{\otimes N}$ by

$\displaystyle\left(s,s^{\prime}\right)_{L^{\otimes L}}:=\int_{M}\langle
s,s^{\prime}\rangle_{L^{\otimes
N}}(-1)^{\frac{n(n-1)}{2}}\dfrac{\left(N\omega\right)^{n}}{n!},$

where $\langle\cdot,\ \cdot\rangle_{L^{\otimes N}}$ is the Hermitian metric of
$L^{\otimes N}$. Then it is clear that
$\left\\{\vartheta_{\frac{m}{N}}\right\\}_{\frac{m}{N}\in
F\cap\frac{1}{N}\mathbb{Z}^{n}}$ are orthogonal basis.

######  Example 4.5.

For Example 2.30, $Z=X+\sqrt{-1}Y$ can be chosen so that $Y+XY^{-1}X$ is a
constant map and $XY^{-1}$ and $Y^{-1}$ satisfy

$\displaystyle(XY^{-1})_{x}=(Y+XY^{-1}X)\begin{pmatrix}u_{11}\cdot
C^{-1}x&\cdots&u_{1n}\cdot C^{-1}x\\\ \vdots&&\vdots\\\ u_{n1}\cdot
C^{-1}x&\cdots&u_{nn}\cdot C^{-1}x\end{pmatrix},$
$\displaystyle(Y^{-1})_{x}=\begin{pmatrix}u_{11}\cdot
C^{-1}x&\cdots&u_{1n}\cdot C^{-1}x\\\ \vdots&&\vdots\\\ u_{n1}\cdot
C^{-1}x&\cdots&u_{nn}\cdot
C^{-1}x\end{pmatrix}(Y+XY^{-1}X)\begin{pmatrix}u_{11}\cdot
C^{-1}x&\cdots&u_{1n}\cdot C^{-1}x\\\ \vdots&&\vdots\\\ u_{n1}\cdot
C^{-1}x&\cdots&u_{nn}\cdot C^{-1}x\end{pmatrix}$ $\displaystyle\hskip
48.36958pt+Y+XY^{-1}X.$

In this case, $Y+XY^{-1}X$ is necessarily $I$ and $\Omega$ can be written as

$\Omega_{x}=\begin{pmatrix}u_{11}\cdot C^{-1}x&\cdots&u_{1n}\cdot C^{-1}x\\\
\vdots&&\vdots\\\ u_{n1}\cdot C^{-1}x&\cdots&u_{nn}\cdot
C^{-1}x\end{pmatrix}+\sqrt{-1}(Y+XY^{-1}X)^{-1},$

and the condition $(2)$ in Proposition 3.12 is equivalent to the following
condition

$\left({}^{t}C^{-1}u_{jk}\right)_{i}=\left({}^{t}C^{-1}u_{ik}\right)_{j}\
\text{for all }i,j,k=1,\ldots,n.$

Assume this condition as well as the condition $\frac{N}{2}v_{i}\cdot
U_{j}v_{i}\in\mathbb{Z}$ for all $i,j=1,\ldots,n$. Then, for each
$\frac{m}{N}\in F\cap\frac{1}{N}\mathbb{Z}^{n}$, $\vartheta_{\frac{m}{N}}$ is
described by

$\displaystyle\vartheta_{\frac{m}{N}}(x,y)=$
$\displaystyle\sum_{\gamma\in\Gamma}g_{\gamma}\exp{2\pi\sqrt{-1}N\left[\sum_{i=1}^{n}\sum_{j>i}\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)_{i}\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)_{j}\left({}^{t}C^{-1}u_{ij}\right)\cdot\begin{pmatrix}\frac{m_{1}}{N}\\\
\vdots\\\ \frac{m_{i-1}}{N}\\\
\frac{1}{2}\left(\rho_{\gamma^{-1}}(x)+\frac{m}{N}\right)_{i}\\\
\left(\rho_{\gamma^{-1}}(x)\right)_{i+1}\\\ \vdots\\\
\left(\rho_{\gamma^{-1}}(x)\right)_{n}\end{pmatrix}\right.}$
$\displaystyle{+\frac{1}{2}\sum_{i=1}^{n}\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)_{i}^{2}\left({}^{t}C^{-1}u_{ii}\right)\cdot\begin{pmatrix}\frac{m_{1}}{N}\\\
\vdots\\\ \frac{m_{i-1}}{N}\\\
\frac{1}{3}\left(2\rho_{\gamma^{-1}}(x)+\frac{m}{N}\right)_{i}\\\
\left(\rho_{\gamma^{-1}}(x)\right)_{i+1}\\\ \vdots\\\
\left(\rho_{\gamma^{-1}}(x)\right)_{n}\end{pmatrix}}$
$\displaystyle+\frac{1}{2}\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)\cdot\left(\begin{pmatrix}u_{11}\cdot\gamma&\cdots&u_{1n}\cdot\gamma\\\
\vdots&&\vdots\\\
u_{n1}\cdot\gamma&\cdots&u_{nn}\cdot\gamma\end{pmatrix}+\sqrt{-1}\left(Y+XY^{-1}X\right)^{-1}\right)\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)$
$\displaystyle{\left.-\frac{1}{2}\frac{m}{N}\cdot\begin{pmatrix}u_{11}\cdot\gamma&\cdots&u_{1n}\cdot\gamma\\\
\vdots&&\vdots\\\
u_{n1}\cdot\gamma&\cdots&u_{nn}\cdot\gamma\end{pmatrix}\frac{m}{N}+\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
y\right]}.$

By Proposition 4.1 (3), $\vartheta_{\frac{m}{N}}$ converges absolutely and
uniformly on any compact set.

###  4.2. The case when $Z$ is constant

Let $\pi\colon(M,\omega)\to B$ be a Lagrangian fibration on a complete
$n$-dimensional $B$ with prequantum line bundle $(L,\nabla^{L})\to(M,\omega)$.
Then, it is obtained as the quotient of the $\Gamma:=\pi_{1}(B)$-action on
$\pi_{0}\colon({\widetilde{M}},\omega_{0})\to\mathbb{R}^{n}$ with prequantum
line bundle
$\left({\widetilde{L}},\nabla^{\widetilde{L}}\right)\to({\widetilde{M}},\omega_{0})$.
Suppose that the $\Gamma$-actions are described by (2.3) and (2.5) as before.
Let $J$ be a compatible almost complex structure on $(M,\omega)$ and $Z\in
C^{\infty}({\widetilde{M}},{\mathcal{S}}_{n})$ be the map corresponding to the
pull-back of $J$ to ${\widetilde{M}}$. A situation in which $(2)$ in
Proposition 3.12 holds occurs when $Z$ is a constant map. In this subsection,
we discuss this case in detail. Note that in this case, $Du_{\gamma}$ is a
constant map for each $\gamma\in\Gamma$. It is obtained by (3.4). Moreover, as
a special case of the setting in the previous subsection, we can obtain the
following theorem.

######  Theorem 4.6.

(1) For each $\frac{m}{N}\in F\cap\frac{1}{N}\mathbb{Z}^{n}$,
$\vartheta_{\frac{m}{N}}$ can be described as follows

$\vartheta_{\frac{m}{N}}(x,y)=\sum_{\gamma\in\Gamma}g_{\gamma}e^{2\pi\sqrt{-1}N\left[\frac{1}{2}\left\\{\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)\cdot(\Omega+{}^{t}A_{\gamma}Du_{\gamma})\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)-\frac{m}{N}\cdot({}^{t}A_{\gamma}Du_{\gamma})\frac{m}{N}\right\\}-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}(0)+\rho_{\gamma}\left(\frac{m}{N}\right)\cdot y\right]}.$

(2) For each $\frac{m}{N}\in F\cap\frac{1}{N}\mathbb{Z}^{n}$,
$\vartheta_{\frac{m}{N}}$ converges absolutely and uniformly on any compact
set.
(3) $J$ is integrable and
$\left\\{\vartheta_{\frac{m}{N}}\right\\}_{\frac{m}{N}\in
F\cap\frac{1}{N}\mathbb{Z}^{n}}$ gives a basis of the space of holomorphic
sections of $(L,\nabla^{L})^{\otimes N}\to(M,N\omega,J)$.

###### Proof.

(1) is obtained from Proposition 4.1 (1). (2) is obtained by the assumption
and Proposition 4.1 (3). The first half of (3) is true since $J$ is covariant
constant with respect to the associated Levi-Civita connection. The other half
is obtained by Corollary 4.3. ∎

When $Z$ is constant, the associated Riemannian metric of $M$ is flat. So, by
Bieberbach’s theorem, if $M$ is compact, then, $M$ is finitely covered by the
$2n$-dimensional torus $T^{2n}$, hence, $\vartheta_{\frac{m}{N}}$’s should be
obtained from classical theta functions. So, let us see how
$\vartheta_{\frac{m}{N}}$’s relate with classical theta functions for Example
2.28 with $C=I$, in which $M$ itself is $T^{2n}$. First, let us briefly recall
classical theta functions. For each $T\in{\mathcal{S}}_{n}$ and
$a,b\in\mathbb{Q}^{n}$, the theta function with characteristics is a
holomorphic section on the trivial holomorphic line bundle
$\mathbb{C}^{n}\times\mathbb{C}\to\mathbb{C}^{n}$ which is defined by

$\vartheta\begin{bmatrix}a\\\
b\end{bmatrix}\left(z,T\right):=\sum_{\gamma\in\mathbb{Z}^{n}}e^{\pi\sqrt{-1}\left(\gamma+a\right)\cdot
T\left(\gamma+a\right)+2\pi\sqrt{-1}\left(\gamma+a\right)\cdot\left(z+b\right)}.$

It is well-known that $\vartheta\begin{bmatrix}a\\\
b\end{bmatrix}\left(z,T\right)$ has the following quasi-periodicity

$\begin{split}&\vartheta\begin{bmatrix}a\\\
b\end{bmatrix}\left(z+m,T\right)=e^{2\pi\sqrt{-1}a\cdot
m}\vartheta\begin{bmatrix}a\\\ b\end{bmatrix}\left(z,T\right),\\\
&\vartheta\begin{bmatrix}a\\\
b\end{bmatrix}\left(z+Tm,T\right)=e^{-2\pi\sqrt{-1}b\cdot
m}e^{-\pi\sqrt{-1}m\cdot Tm-2\pi\sqrt{-1}m\cdot z}\vartheta\begin{bmatrix}a\\\
b\end{bmatrix}\left(z,T\right)\end{split}$

for $m\in\mathbb{Z}^{n}$. For more detail, see [28, Chapter II, §1] and [29,
ß2]. Here we need the case when $T=N\Omega$, $a=\frac{m}{N}$, and $b=0$. In
this case, define the
$\mathbb{Z}^{2n}=\mathbb{Z}^{n}\times\mathbb{Z}^{n}$-action on
$\mathbb{C}^{n}\times\mathbb{C}\to\mathbb{C}^{n}$ by

$(\gamma,\gamma^{\prime})\cdot(z,w):=\left(z+N(-\Omega\gamma+\gamma^{\prime}),e^{-\pi\sqrt{-1}N\gamma\cdot\Omega\gamma+2\pi\sqrt{-1}\gamma\cdot
z}w\right)$

for $(\gamma,\gamma^{\prime})\in\mathbb{Z}^{2n}$ and
$(z,w)\in\mathbb{C}^{n}\times\mathbb{C}$. Also define the
$\mathbb{Z}^{2n}$-action on the trivial complex line bundle
$\mathbb{R}^{2n}\times\mathbb{C}\to\mathbb{R}^{2n}$ by

(4.2)
$(\gamma,\gamma^{\prime})\cdot(x,y,w):=\left(x+\gamma,y+\gamma^{\prime},e^{2\pi\sqrt{-1}N\gamma\cdot
y}w\right)$

for $(\gamma,\gamma^{\prime})\in\mathbb{Z}^{2n}$ and
$(x,y,w)\in\mathbb{R}^{2n}\times\mathbb{C}$. Note that by taking the quotient
of the latter $\mathbb{Z}^{n}$-action of (4.2), we can recover Example 2.28
with $C=I$ and $g_{\gamma}=1$. Let $F\colon\mathbb{R}^{2n}\to\mathbb{C}^{n}$
and
${\widetilde{F}}\colon\mathbb{R}^{2n}\times\mathbb{C}\to\mathbb{C}^{n}\times\mathbb{C}$
be the $\mathbb{R}$-linear isomorphism and the bundle isomorphism covering $F$
which are defined by

$\displaystyle F(x,y)$ $\displaystyle:=N\left(-\Omega x+y\right),$
$\displaystyle{\widetilde{F}}(x,y,w)$ $\displaystyle:=\left(N\left(-\Omega
x+y\right),e^{-\pi\sqrt{-1}Nx\cdot\Omega x}w\right).$

Then, the direct computation shows the following theorem.

######  Theorem 4.7.

(1) $J_{\sqrt{-1}I}\circ F=F\circ(J_{Z})$, i.e., $F$ is a $\mathbb{C}$-linear
isomorphism from $(\mathbb{R}^{2n},J_{Z})$ to the standard complex vector
space $(\mathbb{C}^{n},J_{\sqrt{-1}I})$.
(2) ${\widetilde{F}}$ is equivariant with respect to the
$\mathbb{Z}^{2n}$-actions defined above.
(3) $\vartheta_{\frac{m}{N}}$ satisfies
${\widetilde{F}}\circ\vartheta_{\frac{m}{N}}(x,y)=\vartheta\left[\begin{matrix}\frac{m}{N}\\\
0\end{matrix}\right]\left(F(x,y),N\Omega\right)$, i.e.,

$\vartheta_{\frac{m}{N}}(x,y)=e^{\pi\sqrt{-1}Nx\cdot\Omega
x}\vartheta\left[\begin{matrix}\frac{m}{N}\\\
0\end{matrix}\right]\left(N(-\Omega x+y),N\Omega\right).$

###  4.3. Adiabatic-type limit

In this subsection let us consider a one parameter family
$\\{(g^{t},J^{t})\\}_{t>0}$ of the Riemannian metrics and the almost complex
structures on a Lagrangian fibration so that the fiber shrinks as $t$ goes to
$\infty$, and investigate the behavior of $\vartheta_{\frac{m}{N}}$ defined by
(4.1) when $t$ goes to $\infty$. We use the same notations introduced in the
previous sections.

Let $Z=X+\sqrt{-1}Y\in
C^{\infty}\left({\widetilde{M}},{\mathcal{S}}_{n}\right)$ be the map
independent of $y_{1},\ldots,y_{n}$. Let $J=J_{Z}$ be the corresponding
compatible almost complex structure on
$\left({\widetilde{M}},\omega_{0}\right)$. For each $t>0$, we define the
almost complex structure $J^{t}$ by

$\displaystyle
J^{t}u:=\left(-J\partial_{y},\,\partial_{y}\right)\begin{pmatrix}0&\frac{-1}{t}\\\
t&0\end{pmatrix}\begin{pmatrix}u_{H}\\\ u_{V}\end{pmatrix}$

for $u=\left(-J\partial_{y},\,\partial_{y}\right)\begin{pmatrix}u_{H}\\\
u_{V}\end{pmatrix}\in T_{(x,y)}{\widetilde{M}}$. It is easy to see the
following lemma.

######  Lemma 4.8.

(1) For any $t>0$, $J^{t}$ is compatible with $\omega_{0}$. The map $Z^{t}\in
C^{\infty}\left({\widetilde{M}},{\mathcal{S}}_{n}\right)$ corresponding to
$J^{t}$ is described as

$Z^{t}=\left(\frac{1}{t}X+\sqrt{-1}Y\right)Y^{-1}\left(Y+XY^{-1}X\right)\left(tY+\frac{1}{t}XY^{-1}X\right)^{-1}Y.$

$J^{t}$ can be also written as

$\displaystyle
J^{t}\left(\left(\partial_{x},\partial_{y}\right)\begin{pmatrix}u_{x}\\\
u_{y}\end{pmatrix}\right)=\left(\partial_{x},\partial_{y}\right)\frac{1}{t}\begin{pmatrix}XY^{-1}&-Y-XY^{-1}X\\\
Y^{-1}\left(t^{2}Y+XY^{-1}X\right)\left(Y+XY^{-1}X\right)^{-1}&-Y^{-1}X\end{pmatrix}\begin{pmatrix}u_{x}\\\
u_{y}\end{pmatrix}.$

(2) For any $t>0$, let $g^{t}$ be the Riemannian metric corresponding to
$\omega_{0}$ and $J^{t}$. Then, for
$u=\left(-J\partial_{y},\,\partial_{y}\right)\begin{pmatrix}u_{H}\\\
u_{V}\end{pmatrix}$,
$v=\left(-J\partial_{y},\,\partial_{y}\right)\begin{pmatrix}v_{H}\\\
v_{V}\end{pmatrix}\in T_{(x,y)}{\widetilde{M}}$, $g^{t}$ can be written by

$\displaystyle g^{t}(u,v)$ $\displaystyle=\omega_{0}\left(u,J^{t}v\right)$
$\displaystyle=t(0,{}^{t}u_{H})\begin{pmatrix}Y^{-1}&-Y^{-1}X\\\
-XY^{-1}&Y+XY^{-1}X\end{pmatrix}\begin{pmatrix}0\\\
v_{H}\end{pmatrix}+\frac{1}{t}(0,{}^{t}u_{V})\begin{pmatrix}Y^{-1}&-Y^{-1}X\\\
-XY^{-1}&Y+XY^{-1}X\end{pmatrix}\begin{pmatrix}0\\\ v_{V}\end{pmatrix}.$

Suppose that a group $\Gamma$ acts on
$\pi_{0}\colon({\widetilde{M}},\omega_{0})\to\mathbb{R}^{n}$ and the
$\Gamma$-actions $\rho$ on $\mathbb{R}^{n}$ and $\widetilde{\rho}$ on
$({\widetilde{M}},\omega_{0})$ are written as in (2.3).

######  Lemma 4.9.

The $\Gamma$-action $\widetilde{\rho}$ preserves $J^{t}$ (hence, $g^{t}$) for
all $t>0$ if and only if $\widetilde{\rho}$ preserves $J$.

For $J^{t}$ and $g^{t}$ defined as above, the same arguments in Section 3.3
goes well, just by replacing $J$, $g$ by $J^{t}$, $g^{t}$. For each $t>0$, let
$\vartheta^{t}_{\frac{m}{N}}$ be the one defined by (4.1) for $J^{t}$ and
$g^{t}$. Let us investigate the behavior of $\vartheta^{t}_{\frac{m}{N}}$ as
$t$ goes to infinity. For $t>0$, $\Omega^{t}$ defined by (3.7) for $Z^{t}$ can
be described as

(4.3)
$\Omega^{t}=\left(Y+XY^{-1}X\right)^{-1}\left(X+t\sqrt{-1}Y\right)Y^{-1}.$

Let $D^{t}$ be the corresponding Spinc Dirac operator. Then, for a section $s$
of ${\widetilde{L}}^{\otimes N}$, $D^{t}s$ can be described as

(4.4) $\displaystyle
D^{t}s=-\frac{\sqrt{-1}}{N}\sum_{i=1}^{n}\partial_{y_{i}}\otimes_{\mathbb{C}}\left\\{\partial_{x_{i}}s+\sum_{j=1}^{n}\left(\Omega^{t}\right)_{ij}\left(\partial_{y_{j}}s-2\pi\sqrt{-1}Nx_{j}s\right)\right\\}.$

It is clear that

######  Lemma 4.10.

For any $t>0$, the condition $(2)$ in Proposition 3.12 holds for $\Omega^{t}$
if and only if it holds for $\Omega=\Omega^{1}$. In particular, $J^{t}$ is
integrable if and only if $J$ is integrable.

Suppose that $\pi_{0}\colon({\widetilde{M}},N\omega_{0},J)\to\mathbb{R}^{n}$
with prequantum line bundle
$\left({\widetilde{L}},\nabla^{\widetilde{L}}\right)^{\otimes
N}\to({\widetilde{M}},N\omega_{0},J)$ is equipped with an action of a group
$\Gamma$ which preserves all the data, and the $\Gamma$-actions are described
by (2.3) and (2.5) as before. We assume that the $\Gamma$-action $\rho$ on
$\mathbb{R}^{n}$ is properly discontinuous, free, and cocompact. Let
$\pi\colon(M,N\omega)\to B$ and $(L,\nabla^{L})^{\otimes N}\to(M,N\omega)$ be
the Lagrangian fibration and the prequantum line bundle on it obtained by the
quotient of the $\Gamma$-action. On $M$, we consider the orientation so that
$(-1)^{\frac{n(n-1)}{2}}\frac{\left(N\omega\right)^{n}}{n!}$ is a positive
volume form, and define the $L^{p}$-norm of a section $s$ of $L^{\otimes N}$
by

(4.5) $\displaystyle\lVert s\rVert_{L^{p}}:=\left(\int_{M}\langle
s,s\rangle_{L^{\otimes
N}}^{\frac{p}{2}}(-1)^{\frac{n(n-1)}{2}}\dfrac{\left(N\omega\right)^{n}}{n!}\right)^{\frac{1}{p}},$

where $\langle\cdot,\ \cdot\rangle_{L^{\otimes N}}$ is the Hermitian metric of
$L^{\otimes N}$ which is induced from the Hermitian metric $\langle\cdot,\
\cdot\rangle_{{\widetilde{L}}^{\otimes N}}$ of ${\widetilde{L}}^{\otimes N}$.
As noticed in Remark 2.26, there exists a positive constant $C$ such that
$\langle\cdot,\ \cdot\rangle_{{\widetilde{L}}^{\otimes N}}$ can be written as
$\langle\cdot,\ \cdot\rangle_{{\widetilde{L}}^{\otimes N}}=C\langle\cdot,\
\cdot\rangle_{\mathbb{C}}$, where $\langle\cdot,\ \cdot\rangle_{\mathbb{C}}$
is the standard Hermitian inner product on $\mathbb{C}$.

For each $t>0$ and each point $\frac{m}{N}\in F\cap\frac{1}{N}\mathbb{Z}^{n}$
for which the condition (3.13) holds, the corresponding
$\vartheta^{t}_{\frac{m}{N}}$ is defined by (4.1) for $\Omega^{t}$. We
identify $F\cap\frac{1}{N}\mathbb{Z}^{n}$ with $B_{BS}$ the set of Bohr-
Sommerfeld points of $\pi\colon(M,N\omega)\to B$ with prequantum line bundle
$(L,\nabla^{L})^{\otimes N}\to(M,N\omega)$ and identify
$\vartheta^{t}_{\frac{m}{N}}$ with the section of $(L,\nabla^{L})^{\otimes
N}\to(M,N\omega)$ which is induced from $\vartheta^{t}_{\frac{m}{N}}$. Then,
concerning the $L^{p}$-norm, we have the following lemma.

######  Lemma 4.11.

Suppose that $Y+XY^{-1}X$ is constant. Then, the $L^{p}$-norm of
$\vartheta^{t}_{\frac{m}{N}}$ can be calculated as follows

$\lVert\vartheta^{t}_{\frac{m}{N}}\rVert_{L^{p}}^{p}=C\sqrt{\det\left(Y+XY^{-1}X\right)}\left(\frac{N}{pt}\right)^{\frac{n}{2}}.$

###### Proof.

Let $o(B)$ be the orientation bundle of $B$ which is defined as the quotient
bundle of the trivial real line bundle
$\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}^{n}$ on the universal cover of
$B$ by the $\Gamma$-action
$\rho^{\prime}_{\gamma}(x,r):=\left(\rho_{\gamma}(x),(\det
A_{\gamma})r\right)$ for $\gamma\in\Gamma$ and
$(x,r)\in\mathbb{R}^{n}\times\mathbb{R}$. Then, we have a push-forward map
$\pi_{*}\colon\Omega^{k}(M)\to\Omega^{k-n}(B,o(B))$, where
$\Omega^{\bullet}(B,o(B))$ is the de Rham complex twisted by $o(B)$. $B$ has a
natural density which we denote by $\lvert dx\rvert$. For densities, see [8,
Chapter I, §7]. Then,

$\displaystyle\lVert\vartheta^{t}_{\frac{m}{N}}\rVert_{L^{p}}^{p}$
$\displaystyle=\int_{M}\langle\vartheta^{t}_{\frac{m}{N}},\vartheta^{t}_{\frac{m}{N}}\rangle_{L^{\otimes
N}}^{\frac{p}{2}}(-1)^{\frac{n(n-1)}{2}}\dfrac{\left(N\omega\right)^{n}}{n!}$
$\displaystyle=\int_{B}\pi_{*}\left(\langle\vartheta^{t}_{\frac{m}{N}},\vartheta^{t}_{\frac{m}{N}}\rangle_{L^{\otimes
N}}^{\frac{p}{2}}(-1)^{\frac{n(n-1)}{2}}\dfrac{\left(N\omega\right)^{n}}{n!}\right)$
(4.6) $\displaystyle=CN^{n}\sum_{\gamma\in\Gamma}\int_{F}e^{-pN\pi
t\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)\cdot\left(Y+XY^{-1}X\right)^{-1}\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)}\lvert
dx\rvert.$

By changing the coordinates as $x^{\prime}=\rho_{\gamma^{-1}}(x)$,

$\displaystyle\eqref{L^p theta-1}$
$\displaystyle=CN^{n}\sum_{\gamma\in\Gamma}\int_{\rho_{\gamma^{-1}}(F)}e^{-pN\pi
t\left(x^{\prime}-\frac{m}{N}\right)\cdot\left(Y+XY^{-1}X\right)^{-1}\left(x^{\prime}-\frac{m}{N}\right)}\lvert
dx^{\prime}\rvert$ (4.7) $\displaystyle=CN^{n}\int_{\mathbb{R}^{n}}e^{-pN\pi
t\left(x^{\prime}-\frac{m}{N}\right)\cdot\left(Y+XY^{-1}X\right)^{-1}\left(x^{\prime}-\frac{m}{N}\right)}\lvert
dx^{\prime}\rvert.$

Since $Y+XY^{-1}X$ is positive definite, symmetric, there exists $P\in O(n)$
such that

$Y+XY^{-1}X={}^{t}P\begin{pmatrix}\lambda_{1}&&\\\ &\ddots&\\\
&&\lambda_{n}\end{pmatrix}P.$

Then, we define a positive definite symmetric matrix $\sqrt{Y+XY^{-1}X}$ by

(4.8) $\sqrt{Y+XY^{-1}X}:={}^{t}P\begin{pmatrix}\sqrt{\lambda_{1}}&&\\\
&\ddots&\\\ &&\sqrt{\lambda_{n}}\end{pmatrix}P,$

and put $\tau:=\sqrt{(Y+XY^{-1}X)^{-1}}\left(x^{\prime}-\frac{m}{N}\right)$.
Then,

$\displaystyle\eqref{L^p theta-2}$
$\displaystyle=C\sqrt{\det\left(Y+XY^{-1}X\right)}N^{n}\int_{\mathbb{R}^{n}}e^{-pN\pi
t\lVert\tau\rVert^{2}}\lvert d\tau\rvert$
$\displaystyle=C\sqrt{\det\left(Y+XY^{-1}X\right)}N^{n}\prod_{i=1}^{n}\int_{-\infty}^{\infty}e^{-pN\pi
t\tau_{i}^{2}}d\tau_{i}$
$\displaystyle=C\sqrt{\det\left(Y+XY^{-1}X\right)}N^{n}\left(\sqrt{\dfrac{1}{pNt}}\right)^{n}.$

∎

We define the section $\delta_{\frac{m}{N}}$ of
$\left(L,\nabla^{L}\right)^{\otimes N}|_{\pi^{-1}\left(\frac{m}{N}\right)}$ by

(4.9) $\delta_{\frac{m}{N}}(y):=\frac{1}{C}e^{2\pi\sqrt{-1}m\cdot y}.$

By Lemma 3.2, $\delta_{\frac{m}{N}}$ is a covariant constant section of
$\left(L,\nabla^{L}\right)^{\otimes N}|_{\pi^{-1}\left(\frac{m}{N}\right)}$.
Let $T_{\pi}^{*}M$ be the cotangent bundle along the fiber of $\pi$. On
$\left(\wedge^{n}T_{\pi}^{*}M\right)\otimes\pi^{*}o(B)^{*}$, there exists a
natural section, i.e., a density along the fiber of $\pi$, say $\lvert
dy\rvert$, which satisfies $\int_{\pi^{-1}\left(x\right)}\lvert dy\rvert=1$ on
each fiber of $\pi$. Then, we obtain the following theorem.

######  Theorem 4.12.

Suppose that $Y+XY^{-1}X$ is constant. Then, the section
$\displaystyle\frac{\vartheta^{t}_{\frac{m}{N}}}{\lVert\vartheta^{t}_{\frac{m}{N}}\rVert_{L^{1}}}$
converges to a delta-function section supported on the fiber
$\pi^{-1}\left(\frac{m}{N}\right)$ as $t$ goes to $\infty$ in the following
sense: for any section $s$ of $L^{\otimes N}$,

$\lim_{t\to\infty}\int_{M}\left\langle
s,\frac{\vartheta^{t}_{\frac{m}{N}}}{\lVert\vartheta^{t}_{\frac{m}{N}}\rVert_{L^{1}}}\right\rangle_{L^{\otimes
N}}(-1)^{\frac{n(n-1)}{2}}\dfrac{\left(N\omega\right)^{n}}{n!}=\int_{\pi^{-1}\left(\frac{m}{N}\right)}\left\langle
s,\delta_{\frac{m}{N}}\right\rangle_{L^{\otimes N}}\lvert dy\rvert.$

###### Proof.

We denote by ${\widetilde{s}}$ the pull-back of $s$ to
${\widetilde{L}}^{\otimes N}\to{\widetilde{M}}$. Since ${\widetilde{s}}$ is
$\Gamma$-equivariant, the Fourier expansion of ${\widetilde{s}}$ can be
written as in (3.33). Then, by using Proposition 4.1 (1),

$\displaystyle\int_{M}\left\langle
s,\frac{\vartheta^{t}_{\frac{m}{N}}}{\lVert\vartheta^{t}_{\frac{m}{N}}\rVert_{L^{1}}}\right\rangle_{L^{\otimes
N}}(-1)^{\frac{n(n-1)}{2}}\dfrac{\left(N\omega\right)^{n}}{n!}$
$\displaystyle=\int_{B}\pi_{*}\left(\left\langle
s,\frac{\vartheta^{t}_{\frac{m}{N}}}{\lVert\vartheta^{t}_{\frac{m}{N}}\rVert_{L^{1}}}\right\rangle_{L^{\otimes
N}}(-1)^{\frac{n(n-1)}{2}}\dfrac{\left(N\omega\right)^{n}}{n!}\right)$ (4.10)
$\displaystyle=\frac{CN^{n}}{\lVert\vartheta^{t}_{\frac{m}{N}}\rVert_{L^{1}}}\sum_{\gamma\in\Gamma}\int_{F}a_{m}\left(\rho_{\gamma^{-1}}(x)\right)\overline{e^{-2\pi\sqrt{-1}\sum_{i=1}^{n}G_{m}^{i}\left(\frac{m_{1}}{N},\ldots,\frac{m_{i-1}}{N},\left(\rho_{\gamma^{-1}}(x)\right)_{i},\ldots,\left(\rho_{\gamma^{-1}}(x)\right)_{n}\right)}}\lvert
dx\rvert.$

By putting $x^{\prime}=\rho_{\gamma^{-1}}(x)$, we have

$\displaystyle\eqref{delta-1}$
$\displaystyle=\frac{CN^{n}}{\lVert\vartheta^{t}_{\frac{m}{N}}\rVert_{L^{1}}}\sum_{\gamma\in\Gamma}\int_{\rho_{\gamma^{-1}}(F)}a_{m}\left(x^{\prime}\right)\overline{e^{-2\pi\sqrt{-1}\sum_{i=1}^{n}G_{m}^{i}\left(\frac{m_{1}}{N},\ldots,\frac{m_{i-1}}{N},x^{\prime}_{i},\ldots,x^{\prime}_{n}\right)}}\lvert
dx^{\prime}\rvert$
$\displaystyle=\frac{CN^{n}}{\lVert\vartheta^{t}_{\frac{m}{N}}\rVert_{L^{1}}}\int_{\mathbb{R}^{n}}a_{m}\left(x^{\prime}\right)\overline{e^{-2\pi\sqrt{-1}\sum_{i=1}^{n}G_{m}^{i}\left(\frac{m_{1}}{N},\ldots,\frac{m_{i-1}}{N},x^{\prime}_{i},\ldots,x^{\prime}_{n}\right)}}\lvert
dx^{\prime}\rvert$ (4.11)
$\displaystyle=\frac{CN^{n}}{\lVert\vartheta^{t}_{\frac{m}{N}}\rVert_{L^{1}}}\int_{\mathbb{R}^{n}}a_{m}\left(x^{\prime}\right)e^{2\pi\sqrt{-1}\sum_{i=1}^{n}\operatorname{Re}G_{m}^{i}\left(\frac{m_{1}}{N},\ldots,\frac{m_{i-1}}{N},x^{\prime}_{i},\ldots,x^{\prime}_{n}\right)}e^{-\pi
Nt\left(x^{\prime}-\frac{m}{N}\right)\cdot\left(Y+XY^{-1}X\right)^{-1}\left(x^{\prime}-\frac{m}{N}\right)}\lvert
dx^{\prime}\rvert.$

We put
$f(x^{\prime}):=a_{m}\left(x^{\prime}\right)e^{2\pi\sqrt{-1}\sum_{i=1}^{n}\operatorname{Re}G_{m}^{i}\left(\frac{m_{1}}{N},\ldots,\frac{m_{i-1}}{N},x^{\prime}_{i},\ldots,x^{\prime}_{n}\right)}$
and $\tau:=\sqrt{(Y+XY^{-1}X)^{-1}}\left(x^{\prime}-\frac{m}{N}\right)$. By
using Lemma 4.11 for $p=1$, (4.11) can be written as follows

(4.12) $\displaystyle\eqref{delta-2}$
$\displaystyle=\frac{CN^{n}}{\lVert\vartheta^{t}_{\frac{m}{N}}\rVert_{L^{1}}}\int_{\mathbb{R}^{n}}f(x^{\prime})e^{-\pi
Nt\left(x^{\prime}-\frac{m}{N}\right)\cdot\left(Y+XY^{-1}X\right)^{-1}\left(x^{\prime}-\frac{m}{N}\right)}\lvert
dx^{\prime}\rvert$
$\displaystyle=\frac{CN^{n}}{\lVert\vartheta^{t}_{\frac{m}{N}}\rVert_{L^{1}}}\sqrt{\det\left(Y+XY^{-1}X\right)}\int_{\mathbb{R}^{n}}f\left(\sqrt{Y+XY^{-1}X}\tau+\frac{m}{N}\right)e^{-\pi
Nt\lVert\tau\rVert^{2}}\lvert d\tau\rvert$
$\displaystyle=\left(Nt\right)^{\frac{n}{2}}\int_{\mathbb{R}^{n}}f\left(\sqrt{Y+XY^{-1}X}\tau+\frac{m}{N}\right)e^{-\pi
Nt\lVert\tau\rVert^{2}}\lvert d\tau\rvert.$

It is well-known that

$\lim_{t\to\infty}\eqref{delta-3}=f\left(\frac{m}{N}\right)=a_{m}\left(\frac{m}{N}\right).$

On the other hand, by using the expression

${\widetilde{s}}=\sum_{\left(\gamma,\frac{m^{\prime}}{N}\right)\in\Gamma\times\left(F\cap\frac{1}{N}\mathbb{Z}^{n}\right)}a_{N\rho_{\gamma}\left(\frac{m^{\prime}}{N}\right)}(x)e^{2\pi\sqrt{-1}N\rho_{\gamma}\left(\frac{m^{\prime}}{N}\right)\cdot
y},$

the right hand side can be computed as

$\displaystyle\int_{\pi^{-1}\left(\frac{m}{N}\right)}\left\langle
s,\delta_{\frac{m}{N}}\right\rangle_{L^{\otimes N}}\lvert dy\rvert$
$\displaystyle=\int_{T^{n}}\left\langle{\widetilde{s}},\delta_{\frac{m}{N}}\right\rangle_{{\widetilde{L}}^{\otimes
N}}\lvert dy\rvert$
$\displaystyle=\sum_{\left(\gamma,\frac{m^{\prime}}{N}\right)\in\Gamma\times\left(F\cap\frac{1}{N}\mathbb{Z}^{n}\right)}a_{N\rho_{\gamma}\left(\frac{m^{\prime}}{N}\right)}\left(\frac{m}{N}\right)\int_{T^{n}}e^{2\pi\sqrt{-1}\left(N\rho_{\gamma}\left(\frac{m^{\prime}}{N}\right)-m\right)\cdot
y}\lvert dy\rvert.$

$\int_{T^{n}}e^{2\pi\sqrt{-1}\left(N\rho_{\gamma}\left(\frac{m^{\prime}}{N}\right)-m\right)\cdot
y}\lvert dy\rvert$ vanishes unless
$\rho_{\gamma}\left(\frac{m^{\prime}}{N}\right)=\frac{m}{N}$. Since both
$\frac{m^{\prime}}{N}$ and $\frac{m}{N}$ lie in the fundamental domain $F$,
this implies $\gamma=e$ and $m^{\prime}=m$, and in this case,
$\int_{T^{n}}e^{2\pi\sqrt{-1}\left(N\rho_{\gamma}\left(\frac{m^{\prime}}{N}\right)-m\right)\cdot
y}\lvert dy\rvert=1$. Thus,

$\int_{\pi^{-1}\left(\frac{m}{N}\right)}\left\langle
s,\delta_{\frac{m}{N}}\right\rangle_{L^{\otimes N}}\lvert
dy\rvert=a_{m}\left(\frac{m}{N}\right).$

This proves the theorem. ∎

##  5\. The non-integrable case

We still use the same notations introduced in the previous sections. By Lemma
3.10, the equation (3.12) has no smooth solution for $\frac{m}{N}\in
F\cap\frac{1}{N}\mathbb{Z}^{n}$ such that (3.13) does not holds. For such
$\frac{m}{N}$, instead of (3.12), let us consider the following equation which
is obtained from (3.12) by replacing $\Omega$ by its value
$\Omega_{\frac{m}{N}}$ at $\frac{m}{N}$

(5.1) $0=\begin{pmatrix}\partial_{x_{1}}{\widetilde{a}}_{m}\\\ \vdots\\\
\partial_{x_{n}}{\widetilde{a}}_{m}\end{pmatrix}+2\pi\sqrt{-1}{\widetilde{a}}_{m}\Omega_{\frac{m}{N}}(m-Nx).$

The equation (5.1) has a solution of the form

${\widetilde{a}}_{m}(x)={\widetilde{a}}_{m}\left(\frac{m}{N}\right)e^{\pi\sqrt{-1}N\left(x-\frac{m}{N}\right)\cdot\Omega_{\frac{m}{N}}\left(x-\frac{m}{N}\right)}.$

We put the initial condition ${\widetilde{a}}_{m}\left(\frac{m}{N}\right)=1$
on the above ${\widetilde{a}}_{m}$, and for each $\gamma\in\Gamma$, define
${\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}$ in such a way that
it satisfies (3.32).

######  Lemma 5.1.

${\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}$ satisfies the
following equality

(5.2) $\displaystyle 0=$
$\displaystyle\begin{pmatrix}\partial_{x_{1}}{\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}(x)\\\
\vdots\\\
\partial_{x_{n}}{\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}(x)\end{pmatrix}+2\pi\sqrt{-1}{\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}(x)\Omega_{x}\left({N\rho_{\gamma}\left(\frac{m}{N}\right)}-Nx\right)$
$\displaystyle+2\pi\sqrt{-1}{\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}(x){}^{t}A_{\gamma}^{-1}\left(\Omega_{\frac{m}{N}}-\Omega_{\rho_{\gamma^{-1}}(x)}\right)A_{\gamma}^{-1}\left({N\rho_{\gamma}\left(\frac{m}{N}\right)}-Nx\right).$

###### Proof.

By the same calculation as in the proof of Lemma 3.15, we have

${}^{t}A_{\gamma}\begin{pmatrix}\partial_{x_{1}}{\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}\left(\rho_{\gamma}(x)\right)\\\
\vdots\\\
\partial_{x_{n}}{\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}\left(\rho_{\gamma}(x)\right)\end{pmatrix}=-2\pi\sqrt{-1}{\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}\left(\rho_{\gamma}(x)\right)\left(\Omega_{\frac{m}{N}}A^{-1}_{\gamma}+{}^{t}\left(Du_{\gamma}\right)_{x}\right)\left({N\rho_{\gamma}\left(\frac{m}{N}\right)}-N\rho_{\gamma}(x)\right).$

(5.2) can be obtained from this equation and (3.39). ∎

By using ${\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}$’s, we
define ${\widetilde{\vartheta}}_{\frac{m}{N}}$ in the same manner as
$\vartheta_{\frac{m}{N}}$, i.e.,

${\widetilde{\vartheta}}_{\frac{m}{N}}(x,y)=\sum_{\gamma\in\Gamma}{\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}(x)e^{2\pi\sqrt{-1}N\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
y}.$

${\widetilde{\vartheta}}_{\frac{m}{N}}$ converges absolutely and uniformly on
any compact set and can be written as

${\widetilde{\vartheta}}_{\frac{m}{N}}=\sum_{\gamma\in\Gamma}{\widetilde{\raisebox{0.0pt}[0.85pt]{$\widetilde{\rho}$}}}_{\gamma}\circ
s^{\prime}_{m}\circ{\widetilde{\rho}}_{\gamma^{-1}},$

where $s^{\prime}_{m}$ is the section defined by

$s^{\prime}_{m}(x,y):=e^{\pi\sqrt{-1}N\left(x-\frac{m}{N}\right)\cdot\Omega_{\frac{m}{N}}\left(x-\frac{m}{N}\right)+2\pi\sqrt{-1}m\cdot
y}.$

In particular, when $M$ is compact, it defines a section of $L^{\otimes N}\to
M$. Moreover, these two sections with different $\frac{m}{N}$ and
$\frac{m^{\prime}}{N}$ are orthogonal to each other. These can be proved by
the same way as Proposition 4.1. In the rest of this section, we assume that
$M$ is compact.

Next let us consider the one parameter family of $J^{t}$ and $g^{t}$ defined
in Section 4.3. Then, corresponding to $J^{t}$ and $g^{t}$, we can obtain
${\widetilde{\vartheta}}_{\frac{m}{N}}^{t}$, which can be explicitly describe
as

${\widetilde{\vartheta}}_{\frac{m}{N}}^{t}(x,y)=\sum_{\gamma\in\Gamma}g_{\gamma}e^{2\pi\sqrt{-1}N\left[\frac{1}{2}\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)\cdot\Omega^{t}_{\frac{m}{N}}\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)+{\widetilde{g}}_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)-\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
u_{\gamma}\left(\rho_{\gamma^{-1}}(x)\right)\right]}e^{2\pi\sqrt{-1}N\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
y},$

where $\Omega^{t}_{\frac{m}{N}}$ is the value of $\Omega^{t}$ given in (4.3)
at $\frac{m}{N}$. Then, ${\widetilde{\vartheta}}_{\frac{m}{N}}^{t}$ has the
following property. The proof is same as Theorem 4.12.

######  Theorem 5.2.

For each $\frac{m}{N}\in F\cap\frac{1}{N}\mathbb{Z}^{n}$, the section
$\displaystyle\frac{{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}}{\lVert{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}\rVert_{L^{1}}}$
converges to a delta-function section supported on the fiber
$\pi^{-1}\left(\frac{m}{N}\right)$ as $t$ goes to $\infty$ in the following
sense: for any section $s$ of $L^{\otimes N}$,

$\lim_{t\to\infty}\int_{M}\left\langle
s,\frac{{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}}{\lVert{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}\rVert_{L^{1}}}\right\rangle_{L^{\otimes
N}}(-1)^{\frac{n(n-1)}{2}}\dfrac{\left(N\omega\right)^{n}}{n!}=\int_{\pi^{-1}\left(\frac{m}{N}\right)}\left\langle
s,\delta_{\frac{m}{N}}\right\rangle_{L^{\otimes N}}\lvert dy\rvert.$

${\widetilde{\vartheta}}^{t}_{\frac{m}{N}}$ is not a solution of $0=D^{t}s$,
but we can show that ${\widetilde{\vartheta}}^{t}_{\frac{m}{N}}$ approximates
the solution of this equation in the following sense:

######  Theorem 5.3.

$\lim_{t\to\infty}\lVert
D^{t}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}\rVert_{L^{2}\left(\left(TM,J^{t}\right)\otimes_{\mathbb{C}}L\right)}=0.$

Before proving the theorem, we have to make sure the meaning of $L^{2}$-norm
in the left hand side. $D^{t}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}$ is a
section of $\left(TM,J^{t}\right)\otimes_{\mathbb{C}}L$, and
$\left(TM,J^{t}\right)\otimes_{\mathbb{C}}L$ admits a Hermitian metric
$\langle\cdot,\ \cdot\rangle_{\left(TM,J^{t}\right)\otimes_{\mathbb{C}}L}$
induced by the one parameter version of (3.10) of $\left(TM,J^{t}\right)$ and
the Hermitian metric of $L$. In terms of this Hermitian metric, the
$L^{2}$-norm is defined as

$\lVert
D^{t}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}\rVert_{L^{2}\left(\left(TM,J^{t}\right)\otimes_{\mathbb{C}}L\right)}^{2}:=\int_{M}\langle
D^{t}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}},\
D^{t}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}\rangle_{\left(TM,J^{t}\right)\otimes_{\mathbb{C}}L}(-1)^{\frac{n(n-1)}{2}}\frac{(N\omega)^{n}}{n!}.$

###### Proof.

For $n=1$, it is clear that the condition 3.13 automatically holds for all
$m\in\mathbb{Z}$. Thus, it is sufficient to prove the theorem for $n\geq 2$.
By the definition of ${\widetilde{\vartheta}}^{t}_{\frac{m}{N}}$ and (5.2),
$D^{t}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}$ can be written as

$\displaystyle D^{t}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}$
$\displaystyle=-\frac{\sqrt{-1}}{N}\sum_{i=1}^{n}\partial_{y_{i}}\otimes_{\mathbb{C}}\left\\{\partial_{x_{i}}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}+\sum_{j=1}^{n}\left(\Omega^{t}_{x}\right)_{ij}\left(\partial_{y_{j}}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}-2\pi\sqrt{-1}Nx_{j}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}\right)\right\\}$
$\displaystyle=-\frac{\sqrt{-1}}{N}\sum_{i=1}^{n}\partial_{y_{i}}\otimes_{\mathbb{C}}\sum_{\gamma\in\Gamma}\left\\{\partial_{x_{i}}{\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}(x)+2\pi\sqrt{-1}{\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}(x)\left(\Omega^{t}_{x}\left(N\rho_{\gamma}\left(\frac{m}{N}\right)-Nx\right)\right)_{i}\right\\}e^{2\pi\sqrt{-1}N\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
y}$
$\displaystyle=-2\pi\sum_{i=1}^{n}\partial_{y_{i}}\otimes_{\mathbb{C}}\sum_{\gamma\in\Gamma}{\widetilde{a}}_{N\rho_{\gamma}\left(\frac{m}{N}\right)}(x)\left({}^{t}A_{\gamma}^{-1}\left(\Omega^{t}_{\frac{m}{N}}-\Omega^{t}_{\rho_{\gamma^{-1}}(x)}\right)\left(\frac{m}{N}-\rho_{\gamma^{-1}}(x)\right)\right)_{i}e^{2\pi\sqrt{-1}N\rho_{\gamma}\left(\frac{m}{N}\right)\cdot
y}.$

Then,

$\displaystyle\langle D^{t}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}},\
D^{t}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}\rangle_{\left(TM,J^{t}\right)\otimes_{\mathbb{C}}L}$
$\displaystyle=(2\pi)^{2}\sum_{\gamma_{1},\gamma_{2}\in\Gamma}\sum_{i_{1},i_{2}}\langle{\widetilde{a}}_{N\rho_{\gamma_{1}}\left(\frac{m}{N}\right)}(x)e^{2\pi\sqrt{-1}N\rho_{\gamma_{1}}\left(\frac{m}{N}\right)\cdot
y},{\widetilde{a}}_{N\rho_{\gamma_{2}}\left(\frac{m}{N}\right)}(x)e^{2\pi\sqrt{-1}N\rho_{\gamma_{2}}\left(\frac{m}{N}\right)\cdot
y}\rangle_{L^{\otimes
N}}Ng^{t}\left(\partial_{y_{i_{1}}},\partial_{y_{i_{2}}}\right)$
$\displaystyle\times\left({}^{t}A_{\gamma_{1}}^{-1}\left(\Omega^{t}_{\frac{m}{N}}-\Omega^{t}_{\rho_{\gamma_{1}^{-1}}(x)}\right)\left(\frac{m}{N}-\rho_{\gamma_{1}^{-1}}(x)\right)\right)_{i_{1}}\overline{\left({}^{t}A_{\gamma_{2}}^{-1}\left(\Omega^{t}_{\frac{m}{N}}-\Omega^{t}_{\rho_{\gamma_{2}^{-1}}(x)}\right)\left(\frac{m}{N}-\rho_{\gamma_{2}^{-1}}(x)\right)\right)_{i_{2}}}$
$\displaystyle=(2\pi)^{2}\frac{N}{t}\sum_{\gamma_{1},\gamma_{2}\in\Gamma}\langle{\widetilde{a}}_{N\rho_{\gamma_{1}}\left(\frac{m}{N}\right)}(x)e^{2\pi\sqrt{-1}N\rho_{\gamma_{1}}\left(\frac{m}{N}\right)\cdot
y},{\widetilde{a}}_{N\rho_{\gamma_{2}}\left(\frac{m}{N}\right)}(x)e^{2\pi\sqrt{-1}N\rho_{\gamma_{2}}\left(\frac{m}{N}\right)\cdot
y}\rangle_{L^{\otimes N}}$
$\displaystyle\times\left({}^{t}A_{\gamma_{1}}^{-1}\left(\Omega^{t}_{\frac{m}{N}}-\Omega^{t}_{\rho_{\gamma_{1}^{-1}}(x)}\right)\left(\frac{m}{N}-\rho_{\gamma_{1}^{-1}}(x)\right)\right)\cdot\left(Y+XY^{-1}X\right)_{x}\overline{\left({}^{t}A_{\gamma_{2}}^{-1}\left(\Omega^{t}_{\frac{m}{N}}-\Omega^{t}_{\rho_{\gamma_{2}^{-1}}(x)}\right)\left(\frac{m}{N}-\rho_{\gamma_{2}^{-1}}(x)\right)\right)}.$

For each $x\in F$ and $u\in\mathbb{C}^{n}$, define the norm of $u$ with
respect to $(Y+XY^{-1}X)_{x}$ by

$\lVert u\rVert^{2}_{(Y+XY^{-1}X)_{x}}:=u\cdot(Y+XY^{-1}X)_{x}\overline{u}.$

By (3.5), for each $\gamma\in\Gamma$, $\lVert u\rVert^{2}_{(Y+XY^{-1}X)_{x}}$
satisfies

$\lVert{}^{t}A_{\gamma}u\rVert^{2}_{(Y+XY^{-1}X)_{x}}=\lVert
u\rVert^{2}_{(Y+XY^{-1}X)_{\rho_{\gamma}(x)}}.$

In terms of this norm, we obtain

$\displaystyle\lVert
D^{t}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}\rVert_{L^{2}\left(\left(TM,J^{t}\right)\otimes_{\mathbb{C}}L\right)}^{2}$
$\displaystyle=(2\pi)^{2}\frac{CN^{n+1}}{t}\sum_{\gamma\in\Gamma}\int_{F}e^{-2\pi
Nt\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)\cdot\left(Y+XY^{-1}X\right)^{-1}_{\frac{m}{N}}\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)}$
$\displaystyle\times\left({}^{t}A_{\gamma}^{-1}\left(\Omega^{t}_{\frac{m}{N}}-\Omega^{t}_{\rho_{\gamma^{-1}}(x)}\right)\left(\frac{m}{N}-\rho_{\gamma^{-1}}(x)\right)\right)\cdot\left(Y+XY^{-1}X\right)_{x}\overline{\left({}^{t}A_{\gamma}^{-1}\left(\Omega^{t}_{\frac{m}{N}}-\Omega^{t}_{\rho_{\gamma^{-1}}(x)}\right)\left(\frac{m}{N}-\rho_{\gamma^{-1}}(x)\right)\right)}\lvert
dx\rvert$
$\displaystyle=(2\pi)^{2}\frac{CN^{n+1}}{t}\sum_{\gamma\in\Gamma}\int_{F}e^{-2\pi
Nt\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)\cdot\left(Y+XY^{-1}X\right)^{-1}_{\frac{m}{N}}\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)}$
$\displaystyle\times\lVert{}^{t}A_{\gamma^{-1}}\left(\Omega^{t}_{\frac{m}{N}}-\Omega^{t}_{\rho_{\gamma^{-1}}(x)}\right)\left(\frac{m}{N}-\rho_{\gamma^{-1}}(x)\right)\rVert^{2}_{\left(Y+XY^{-1}X\right)_{x}}\lvert
dx\rvert$
$\displaystyle=(2\pi)^{2}\frac{CN^{n+1}}{t}\sum_{\gamma\in\Gamma}\int_{F}e^{-2\pi
Nt\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)\cdot\left(Y+XY^{-1}X\right)^{-1}_{\frac{m}{N}}\left(\rho_{\gamma^{-1}}(x)-\frac{m}{N}\right)}$
$\displaystyle\times\lVert\left(\Omega^{t}_{\frac{m}{N}}-\Omega^{t}_{\rho_{\gamma^{-1}}(x)}\right)\left(\frac{m}{N}-\rho_{\gamma^{-1}}(x)\right)\rVert^{2}_{\left(Y+XY^{-1}X\right)_{\rho_{\gamma^{-1}}(x)}}\lvert
dx\rvert$
$\displaystyle=(2\pi)^{2}\frac{CN^{n+1}}{t}\sum_{\gamma\in\Gamma}\int_{\rho_{\gamma^{-1}}(F)}e^{-2\pi
Nt\left(x^{\prime}-\frac{m}{N}\right)\cdot\left(Y+XY^{-1}X\right)^{-1}_{\frac{m}{N}}\left(x^{\prime}-\frac{m}{N}\right)}$
$\displaystyle\times\lVert\left(\Omega^{t}_{\frac{m}{N}}-\Omega^{t}_{x^{\prime}}\right)\left(\frac{m}{N}-x^{\prime}\right)\rVert^{2}_{(Y+XY^{-1}X)_{x^{\prime}}}\lvert
dx^{\prime}\rvert\quad(\because x^{\prime}:=\rho_{\gamma^{-1}}(x))$
$\displaystyle=(2\pi)^{2}\frac{CN^{n+1}}{t}\int_{\mathbb{R}^{n}}\lVert\left(\Omega^{t}_{\frac{m}{N}}-\Omega^{t}_{x^{\prime}}\right)\left(\frac{m}{N}-x^{\prime}\right)\rVert^{2}_{(Y+XY^{-1}X)_{x^{\prime}}}e^{-2\pi
Nt\left(x^{\prime}-\frac{m}{N}\right)\cdot\left(Y+XY^{-1}X\right)^{-1}_{\frac{m}{N}}\left(x^{\prime}-\frac{m}{N}\right)}\lvert
dx^{\prime}\rvert.$

Since $\Omega^{t}$ can be described as (4.3),

$\displaystyle\lVert\left(\Omega^{t}_{\frac{m}{N}}-\Omega^{t}_{x^{\prime}}\right)\left(\frac{m}{N}-x^{\prime}\right)\rVert^{2}_{(Y+XY^{-1}X)_{x^{\prime}}}$
$\displaystyle=\lVert\left(\operatorname{Re}\left(\Omega_{\frac{m}{N}}-\Omega_{x^{\prime}}\right)\right)\left(\frac{m}{N}-x^{\prime}\right)\rVert^{2}_{(Y+XY^{-1}X)_{x^{\prime}}}+t^{2}\lVert\left(\operatorname{Im}\left(\Omega_{\frac{m}{N}}-\Omega_{x^{\prime}}\right)\right)\left(\frac{m}{N}-x^{\prime}\right)\rVert^{2}_{(Y+XY^{-1}X)_{x^{\prime}}}.$

We put

$\displaystyle R(x^{\prime})$
$\displaystyle:=\lVert\left(\operatorname{Re}\left(\Omega_{\frac{m}{N}}-\Omega_{x^{\prime}}\right)\right)\left(\frac{m}{N}-x^{\prime}\right)\rVert^{2}_{(Y+XY^{-1}X)_{x^{\prime}}},$
$\displaystyle I(x^{\prime})$
$\displaystyle:=\lVert\left(\operatorname{Im}\left(\Omega_{\frac{m}{N}}-\Omega_{x^{\prime}}\right)\right)\left(\frac{m}{N}-x^{\prime}\right)\rVert^{2}_{(Y+XY^{-1}X)_{x^{\prime}}}.$

By changing coordinates as
$\tau:=\sqrt{(Y+XY^{-1}X)_{\frac{m}{N}}^{-1}}\left(x^{\prime}-\frac{m}{N}\right)$,
$\lVert
D^{t}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}\rVert_{L^{2}\left(\left(TM,J^{t}\right)\otimes_{\mathbb{C}}L\right)}^{2}$
can be written by

$\displaystyle\lVert
D^{t}{\widetilde{\vartheta}}^{t}_{\frac{m}{N}}\rVert_{L^{2}\left(\left(TM,J^{t}\right)\otimes_{\mathbb{C}}L\right)}^{2}$
$\displaystyle=2^{2-\frac{n}{2}}\pi^{2}CN^{\frac{n}{2}+1}\sqrt{\det(Y+XY^{-1}X)_{\frac{m}{N}}}$
$\displaystyle\times\left\\{t^{-1-\frac{n}{2}}\int_{\mathbb{R}^{n}}R\left(\sqrt{(Y+XY^{-1}X)_{\frac{m}{N}}}\tau+\frac{m}{N}\right)\left(2Nt\right)^{\frac{n}{2}}e^{-2\pi
Nt\lVert\tau\rVert^{2}}\lvert d\tau\rvert\right.$
$\displaystyle\left.+t^{1-\frac{n}{2}}\int_{\mathbb{R}^{n}}I\left(\sqrt{(Y+XY^{-1}X)_{\frac{m}{N}}}\tau+\frac{m}{N}\right)\left(2Nt\right)^{\frac{n}{2}}e^{-2\pi
Nt\lVert\tau\rVert^{2}}\lvert d\tau\rvert\right\\}.$

It is well-known that

$\displaystyle\lim_{t\to\infty}\int_{\mathbb{R}^{n}}R\left(\sqrt{(Y+XY^{-1}X)_{\frac{m}{N}}}\tau+\frac{m}{N}\right)\left(2Nt\right)^{\frac{n}{2}}e^{-2\pi
Nt\lVert\tau\rVert^{2}}\lvert d\tau\rvert=R\left(\frac{m}{N}\right)=0,$
$\displaystyle\lim_{t\to\infty}\int_{\mathbb{R}^{n}}I\left(\sqrt{(Y+XY^{-1}X)_{\frac{m}{N}}}\tau+\frac{m}{N}\right)\left(2Nt\right)^{\frac{n}{2}}e^{-2\pi
Nt\lVert\tau\rVert^{2}}\lvert d\tau\rvert=I\left(\frac{m}{N}\right)=0.$

Since $n\geq 2$, this proves Theorem 5.2. ∎

######  Example 5.4.

For Example 2.29, let us consider the compatible almost complex structure
associated with

$Z:=\begin{pmatrix}0&0\\\
0&x_{1}\end{pmatrix}+\sqrt{-1}\begin{pmatrix}\frac{1}{x_{1}^{2}+1}&0\\\
0&1\end{pmatrix}.$

The corresponding $\Omega$ is

$\Omega_{x}=\begin{pmatrix}\sqrt{-1}&0\\\ 0&x_{1}+\sqrt{-1}\end{pmatrix}.$

This $Z$ does not satisfies $(2)$ in Proposition 3.12, nor the condition 3.13
for any $m\in\mathbb{Z}^{2}$. In fact, for any $m\in\mathbb{Z}^{2}$,
$\left(\left(\partial_{x_{1}}\Omega\right)(m-Nx)\right)_{2}=m_{2}-Nx_{2}$
while $\left(\left(\partial_{x_{2}}\Omega\right)(m-Nx)\right)_{1}=0$. In this
case, ${\widetilde{\vartheta}}_{\frac{m}{N}}^{t}$ can be written as

${\widetilde{\vartheta}}_{\frac{m}{N}}^{t}(x,y)=\sum_{\gamma\in\mathbb{Z}^{2}}g_{\gamma}e^{2\pi\sqrt{-1}N\left[\frac{1}{2}\left\\{t\sqrt{-1}(x_{1}-\gamma_{1}-\frac{m_{1}}{N})^{2}+\left(\frac{m_{1}}{N}+t\sqrt{-1}\right)(x_{2}-\gamma_{2}-\frac{m_{2}}{N})^{2}\right\\}\right.}\\\
{}^{\left.+\left(x_{2}-\gamma_{2}\right)\left\\{\frac{1}{2}\gamma_{1}\left(x_{2}+\gamma_{2}\right)-\left(\frac{m_{2}}{N}+\gamma_{2}\right)\gamma_{2}\right\\}\right]}e^{2\pi\sqrt{-1}(m+N\gamma)\cdot
y}.$

######  Example 5.5.

In the case where $n=2$ of Example 2.31, we can take the compatible almost
complex structure associated with

$Z:=\frac{1}{x_{2}^{2}+1}\begin{pmatrix}\lambda^{2}x_{2}^{3}&\lambda
x_{2}^{2}\\\ \lambda
x_{2}^{2}&x_{2}\end{pmatrix}+\dfrac{\sqrt{-1}}{x_{2}^{2}+1}\begin{pmatrix}(1+\lambda^{2})x_{2}^{2}+1&\lambda
x_{2}\\\ \lambda x_{2}&1\end{pmatrix}.$

The corresponding $\Omega$ is

$\Omega_{x}=\begin{pmatrix}\sqrt{-1}&-\sqrt{-1}\lambda x_{2}\\\
-\sqrt{-1}\lambda x_{2}&x_{2}+\sqrt{-1}(\lambda^{2}x_{2}^{2}+1)\end{pmatrix}.$

In this case, $\partial_{x_{2}}\Omega_{12}=-\sqrt{-1}\lambda$ and
$\partial_{x_{1}}\Omega_{22}=0$. So, $Z$ satisfies $(2)$ in Proposition 3.12
if and only if $\lambda=0$, which is the special case of Example 4.5.
Equivalently, $Z$ does not satisfy the condition 3.13 for any
$m\in\mathbb{Z}^{2}$ unless $\lambda=0$. In fact, for any
$m\in\mathbb{Z}^{2}$,
$\left(\left(\partial_{x_{1}}\Omega\right)(m-Nx)\right)_{2}=0$ while
$\left(\left(\partial_{x_{2}}\Omega\right)(m-Nx)\right)_{1}=-\sqrt{-1}\lambda(m_{2}-Nx_{2})$.
In this case, ${\widetilde{\vartheta}}_{\frac{m}{N}}^{t}$ can be written as

${\widetilde{\vartheta}}_{\frac{m}{N}}^{t}(x,y)=\sum_{\gamma\in\Gamma}g_{\gamma}e^{2\pi\sqrt{-1}N\left[\frac{t\sqrt{-1}}{2}\left\\{x_{1}-\gamma_{1}-\gamma_{2}\lambda(x_{2}-\gamma_{2})-\frac{m_{1}}{N}\right\\}^{2}-t\sqrt{-1}\lambda\frac{m_{2}}{N}\left\\{x_{1}-\gamma_{1}-\gamma_{2}\lambda(x_{2}-\gamma_{2})-\frac{m_{1}}{N}\right\\}\left(x_{2}-\gamma_{2}-\frac{m_{2}}{N}\right)\right.}\\\
{}^{\left.+\frac{1}{2}\left\\{\frac{m_{2}}{N}+t\sqrt{-1}\left(\lambda^{2}\frac{m_{2}^{2}}{N^{2}}+1\right)\right\\}\left(x_{2}-\gamma_{2}-\frac{m_{2}}{N}\right)^{2}+\frac{1}{2}\gamma_{2}(x_{2}-\gamma_{2})(x_{2}+\gamma_{2})-\left(\frac{m_{2}}{N}+\gamma_{2}\right)\gamma_{2}(x_{2}-\gamma_{2})\right]}\\\
\times e^{2\pi\sqrt{-1}\left\\{(m_{1}+\gamma_{2}\lambda
m_{2}+N\gamma_{1})y_{1}+(m_{2}+N\gamma_{2})y_{2}\right\\}}.$

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