# Pre-modular fusion categories of global dimensions $p^{2}$

Zhiqiang Yu

###### Abstract

Let $p\geq 5$ be a prime, we show that a non-pointed modular fusion category
$\mathcal{C}$ is Grothendieck equivalent to
$\mathcal{C}(\mathfrak{sl}_{2},2(p-1))_{A}^{0}$ if and only if
$\dim(\mathcal{C})=p\cdot u$, where $u$ is a certain totally positive
algebraic unit and $A$ is the regular algebra of the Tannakian subcategory
$\text{Rep}(\mathbb{Z}_{2})\subseteq\mathcal{C}(\mathfrak{sl}_{2},2(p-1))$. As
a direct corollary, we classify non-simple modular fusion categories of global
dimensions $p^{2}$.

Keywords: Global dimension; pre-modular fusion category; modular fusion
category

Mathematics Subject Classification 2020: 18M20

##  1 Introduction

A fusion category $\mathcal{C}$ over the complex field $\mathbb{C}$ is a
semisimple finite tensor category, fusion categories are widely studied with
restrictions on their Frobenius-Perron dimensions [10, 11], ranks (i.e., the
number of the isomorphism classes of simple objects) [4, 21] etc. The global
dimension of a fusion category is an important concept that deserved more
attentions. Unlike the classification of fusion categories by Frobenius-Perron
dimensions, however, we even don’t know the structures of fusion categories
with a prime global dimension.

Recently, some progresses are made in topics related to the classifications of
spherical fusion categories by their global dimensions. In [22], V. Ostrik
gave lower bounds of the dimensions (more generally, the formal codegrees) of
fusion categories, as a direct result, he classified spherical fusion
categories of integer global dimensions less than $6$. Braided spherical (or,
pre-modular) fusion categories of global dimension less than or equal to 10
were completely classified by the author in [27], and spherical fusion
categories of dimension $6$ were also shown to be weakly integral. It was
conjectured in [27] that pre-modular fusion categories of prime dimension $p$
are always pointed if $p\neq 5$, this is answered affirmatively in [25]
lately. Moreover, by using a classical Siegel’s trace theorem about totally
positive algebraic integers [26], spherical fusion categories of prime
dimension $p$ are proved to be pointed if $(p-1)/2$ is also an odd prime.

As a special class of fusion categories, modular fusion categories connect
deeply with conformal field theory [2], vertex operator algebras [6], quantum
groups at root of unity [2, 10]. The $S$-matrix and $T$-matrix of modular
fusion categories (see section 2), which reflect many important properties of
modular fusion categories, also enjoy interesting algebraic and arithmetic
properties [3, 4, 6]. Therefore, modular fusion categories are inseparable
with algebraic number theory and representations of the modular group
$\text{SL}(2,\mathbb{Z})$ [3, 4, 10], in particular, one can peer into their
properties by considering the number of Galois orbits of the simple objects
[16, 23] and the representation type of $\text{SL}(2,\mathbb{Z})$ associated
to a modular category [17], for example.

Let $p$ be a prime, $\mathcal{C}$ a modular fusion category of global
dimension $p^{2}$. Then $\mathcal{C}$ is tensor equivalent to an Ising
category $\mathcal{C}(\mathfrak{sl}_{2},2)$ or $\mathcal{C}$ is pointed [22]
if $p=2$. Modular fusion categories of global dimension $9$ are either pointed
or braided tensor equivalent to a Galois conjugate of the non-pointed simple
modular fusion category
$\mathcal{C}(\mathfrak{so}_{5},\frac{3}{2})_{\text{ad}}$ [27, Theorem 4.8].
Let $\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}$ denote the Yang-Lee (or,
Fibonacci) fusion category, which is a rank $2$ transitive modular fusion
category [16], and its Galois conjugate
$\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}$ has global dimension
$\frac{5-\sqrt{5}}{2}$, where
$\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{5})/\mathbb{Q})$ such that
$\sigma(\zeta_{5})=\zeta_{5}^{2}$. So modular fusion categories

$\displaystyle\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma},~{}\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}\boxtimes\mathcal{C}(\mathbb{Z}_{5},\eta)$

both have global dimension $25$, where $\mathcal{C}(\mathbb{Z}_{5},\eta)$ is a
pointed modular fusion category of dimension $5$. When $p=7$, modular fusion
category
$\mathcal{C}(\mathfrak{sl}_{2},5)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},5)_{\text{ad}}^{\tau}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},5)_{\text{ad}}^{\tau^{2}}$
and its Galois conjugates have global dimension $49$, where
$\tau\in\text{Gal}(\mathbb{Q}(\zeta_{7}))$ such that
$\tau(\zeta_{7})=\zeta_{7}^{2}$,
$\mathcal{C}(\mathfrak{sl}_{2},5)_{\text{ad}}$ is a transitive modular fusion
categories of rank $3$ [21, 16]. It was asked in [27, Question 4.9] whether
the modular fusion categories (and their Galois conjugates) mentioned above
are all non-pointed modular fusion categories of global dimension $p^{2}$.

In this paper, from both the views of algebraic and arithmetic properties of
modular fusion categories, we continue to investigate the structure of
(pre-)modular fusion categories of global dimensions $p^{2}$ with $p\geq 5$.
Let $\mathcal{C}$ be such a modular fusion category, if $\mathcal{C}$ does not
contains a non-trivial fusion subcategory of integer dimension, we show that
$\mathcal{C}$ always contains a Galois conjugate of the transitive modular
fusion category $\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}}$ (Theorem
4.3), then we give a complete classification of non-simple modular fusion
category of global dimension $p^{2}$ (Theorem 4.15). Moreover, we find that
there exists another non-pointed modular fusion category of global dimension
$11^{2}$, which then gives a negative answer to [27, Question 4.9].

Table 1: Some Notations Notation | Meaning
---|---
$\zeta_{n}$ | the $n$-th primitive root of unity $e^{\frac{2\pi i}{n}}$
$N(f)$ | the norm of $f$, i.e., $N(f)=\Pi_{\sigma\in\text{Gal}(\mathbb{Q}(f)/\mathbb{Q})}\sigma(f)$
$\mathcal{O}_{X}(\mathcal{C})$ | the Galois orbit of simple object $X$ of a modular fusion category $\mathcal{C}$
$\mathcal{C}(\mathfrak{g},k)$ | the modular fusion category obtained from representation category
|  $\text{Rep}(U_{q}(\mathfrak{g}))$ of quantum group $U_{q}(\mathfrak{g})$ at
root of unity

The paper is organized as follows. In section 2, we recall some basic notions
and notations of (modular) fusion categories, such as global dimensions,
formal codegrees and $d$-numbers, modular data, and the congruence
representations of the modular group $\text{SL}(2,\mathbb{Z})$. In section 3,
we consider modular fusion categories whose norm of global dimensions are
powers of a prime $p$, in particular, if $p\geq 5$, we show that a non-pointed
modular fusion category $\mathcal{C}$ is Grothendieck equivalent to
$\mathcal{C}(\mathfrak{sl}_{2},2(p-1))_{A}^{0}$ if and only if
$\dim(\mathcal{C})=p\cdot u$ in Theorem 3.12 and Corollary 3.13, where $u$ is
a certain algebraic unit. In section 4, we first show that a non-simple
modular fusion category $\mathcal{C}$ of global dimension $p^{2}$ contains a
Galois conjugate of a transitive modular fusion subcategory if $\mathcal{C}$
does not contain a non-trivial fusion subcategory with an integer global
dimension in Theorem 4.3, then we give a complete classification of non-simple
modular fusion categories of dimension $p^{2}$ in Theorem 4.15.

##  2 Preliminaries

In this section, we recall we will recall some important definitions and
properties about fusion categories and modular fusion categories, we refer the
readers to [2, 4, 7, 10, 11, 15, 20].

###  2.1 Fusion categories and dimensions

Let $\mathcal{C}$ be a fusion category, $\mathcal{O}(\mathcal{C})$ the set of
isomorphism classes of simple objects of a fusion category $\mathcal{C}$. Then
the Frobenius-Perron homomorphism
$\text{FPdim}(-):\text{Gr}(\mathcal{C})\to\mathbb{C}$ is the unique ring
homomorphism such that $\text{FPdim}(X)\geq 1$ is an algebraic integer,
$\forall X\in\mathcal{O}(\mathcal{C})$, $\text{FPdim}(X)$ is called the
Frobenius-Perron dimension of the object $X$, and the sum

$\displaystyle\text{FPdim}(\mathcal{C}):=\sum_{X\in\mathcal{O}(\mathcal{C})}\text{FPdim}(X)^{2}$

is defined as the Frobenius-Perron dimension of $\mathcal{C}$.

A fusion category $\mathcal{C}$ is weakly integral if
$\text{FPdim}(\mathcal{C})\in\mathbb{Z}$, $\mathcal{C}$ is integral if
$\text{FPdim}(X)\in\mathbb{Z}$ for all $X\in\mathcal{O}(\mathcal{C})$. We use
$\mathcal{C}_{\text{int}}$ to denote the maximal integral fusion subcategory
of $\mathcal{C}$. The adjoint subcategory $\mathcal{C}_{\text{ad}}$ of a
weakly integral fusion category is always integral [10]. A simple object of
$\mathcal{C}$ is called invertible if $X\otimes X^{*}=I$, the unit object,
equivalently, $\text{FPdim}(X)=1$. A fusion category $\mathcal{C}$ is pointed
if $\mathcal{O}(\mathcal{C})$ is a finite group, where the group
multiplication is induced by the tensor product $\otimes$. In the following,
we use $\mathcal{C}_{\text{pt}}$ to denote the maximal pointed subcategory of
$\mathcal{C}$, that is, the fusion subcategory generated by invertible simple
objects of $\mathcal{C}$. And we say a fusion category $\mathcal{C}$ is non-
pointed if $\mathcal{C}_{\text{pt}}\neq\mathcal{C}$. In addition, two fusion
categories $\mathcal{C}$ and $\mathcal{D}$ are Grothendieck equivalent if
$\text{Gr}(\mathcal{C})\cong\text{Gr}(\mathcal{D})$ as fusion rings.

Let $\mathcal{C}$ be a spherical fusion category $\mathcal{C}$ with spherical
structure $j$, which is a natural isomorphism
$j=\\{j_{X}|j_{X}:X\overset{\sim}{\to}X^{**},X\in\mathcal{C}\\}$ such that
$\dim_{j}(X)=\dim_{j}(X^{*})$, $\dim_{j}(X)$ is called the quantum (or,
categorical) dimension of $X$ determined by $j$, where $\dim_{j}(X)$ is
defined as the (categorical) trace of $\text{id}_{X}$, that is,

$\displaystyle\dim_{j}(X)=\text{Tr}(\text{id}_{X}):=\text{ev}_{X}\circ(j_{X}\circ\text{id}_{X})\otimes\text{id}_{X^{*}}\circ\text{coev}_{X},$

where $(X^{*},\text{ev}_{X},\text{coev}_{X})$ is the dual object of $X$ and we
suppress the associativity and unit constraints of $\mathcal{C}$. We define
the global (or, categorical) dimension of the fusion category $\mathcal{C}$ as

$\displaystyle\dim(\mathcal{C}):=\sum_{X\in\mathcal{O}(\mathcal{C})}\dim_{j}(X)^{2},$

the global dimension $\dim(\mathcal{C})$ is independent of the choice of the
spherical structure of $\mathcal{C}$ and $\dim_{j}(-)$ induces a homomorphism
from $\text{Gr}(\mathcal{C})$ to $\mathbb{C}$ [10, Proposition 4.7.12].

Given an arbitrary spherical fusion category $\mathcal{C}$, we can consider
the twist (or Galois conjugate) $\mathcal{C}^{\sigma}$ of $\mathcal{C}$, where
$\sigma\in\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ and
$\overline{\mathbb{Q}}$ is the algebraic closure of $\mathbb{Q}$. More
precisely, $\mathcal{C}^{\sigma}$ is a fusion category with the same monoidal
functor $\otimes$ as $\mathcal{C}$ and the associator of
$\mathcal{C}^{\sigma}$ is obtained by composing the one of $\mathcal{C}$ with
automorphism $\sigma$. Moreover,
$\text{dim}(\mathcal{C}^{\sigma})=\sigma(\text{dim}(\mathcal{C}))$. A fusion
category $\mathcal{C}$ is said to be pseudo-unitary if
$\text{dim}(\mathcal{C})=\text{FPdim}(\mathcal{C})$. For more properties of
global dimension, we refer the readers to references [10, 11, 22]. In this
paper, we will fix a spherical structure $j$ and denote $\dim_{j}(-)$ by
$\dim(-)$.

###  2.2 Formal codegrees of fusion categories

Let $\mathcal{C}$ be a fusion category, then the complexified ring
$\text{Gr}(\mathcal{C})\otimes_{\mathbb{Z}}\mathbb{C}$ is semisimple. Given an
irreducible representation $\chi$ of
$\text{Gr}(\mathcal{C})\otimes_{\mathbb{Z}}\mathbb{C}$, the element

$\displaystyle\alpha_{\chi}=\sum_{X\in\mathcal{O}(\mathcal{C})}\text{tr}_{\chi}(X)X^{*}$

is central, where $\text{tr}_{\chi}$ is the ordinary trace function on the
representation $\chi$, moreover $\chi^{\prime}(\alpha_{\chi})=0$ if
$\chi\ncong\chi^{\prime}$ and $f_{\chi}:=\chi(\alpha_{\chi})$ is a positive
algebraic integer [14], $f_{\chi}$ is called a formal codegree of fusion
category $\mathcal{C}$ [20]. For example, $\text{FPdim}(\mathcal{C})$ and
$\dim(\mathcal{C})$ are formal codegrees of $\mathcal{C}$.

It was showed that $\frac{\dim(\mathcal{C})}{f_{\chi}}$ is always a totally
positive algebraic integer [20, Corollary 2.14] and the set of formal
codegrees of $\mathcal{C}$ satisfy the following equation [20, Proposition
2.10]

(1) $\displaystyle\sum_{\chi}\frac{\chi(1)}{f_{\chi}}=1.$

If $\mathcal{C}\ncong\text{Vec}$, then
$f_{\chi}>\sqrt{\frac{2\text{rank}(\mathcal{C})}{\text{rank}(\mathcal{C})+1}}\geq\sqrt{\frac{4}{3}}$
for all irreducible representations $\chi$ [22, Theorem 4.2.1]. Moreover,
$\sigma(\dim(\mathcal{C}))>\frac{4\sqrt{3}}{5}$ if the non-trivial fusion
category $\mathcal{C}$ is not a Galois conjugate of
$\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}$ [22, Proposition A.1.1], where
$\sigma\in\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.

Let $\alpha$ be an algebraic integer with the minimal polynomial
$g(x)=x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}$, then $\alpha$ is called a
$d$-number if $(a_{n})^{j}$ divides $(a_{j})^{n}$ for all $1\leq j\leq n$ [20,
Definition 1.1], see [20, Lemma 2.7] for more equivalent conditions about
$d$-numbers. The formal codegrees of a fusion category are $d$-numbers [20,
Theorem 1.2], for example. In addition, the Frobenius-Perron dimensions and
quantum dimensions of objects, and formal codegrees of a fusion category are
cyclotomic algebraic integers [11, Corollary 8.53]. Hence, in order to
determine whether a totally positive algebraic integer $\alpha$ is a formal
codegree of a fusion category $\mathcal{C}$, we can use the Program GAP to
test whether $\alpha$ is a $d$-number and the Galois group of the minimal
polynomial of $\alpha$ is abelian, this is called the $d$-number tests and
cyclotomic test [20].

###  2.3 Modular fusion categories and representations of
$\text{SL}(2,\mathbb{Z})$

Let $\mathcal{C}$ be a braided fusion category with braiding $c$ and
$\mathcal{D}$ a fusion subcategory of $\mathcal{C}$. Then the centralizer of
$\mathcal{D}$ in $\mathcal{C}$ is the following fusion subcategory

$\displaystyle\mathcal{D}_{\mathcal{C}}^{\prime}={\\{X\in\mathcal{C}|c_{Y,X}c_{X,Y}=\text{id}_{X\otimes
Y},\forall Y\in\mathcal{D}}\\}.$

We call $\mathcal{C}^{\prime}:=\mathcal{C}_{\mathcal{C}}^{\prime}$ the Müger
center of $\mathcal{C}$ [15].

Let $\mathcal{C}$ be a braided spherical (i.e., pre-modular) fusion category
with ribbon structure $\theta$, the matrices $S=(S_{X,Y})$ and
$T=(\delta_{X,Y}\theta_{X})$ are called the $S$-matrix and $T$-matrix of
$\mathcal{C}$, respectively, where $S_{X,Y}=\text{Tr}(c_{Y,X}c_{X,Y})$,
$X,Y\in\mathcal{O}(\mathcal{C})$. A modular fusion category is a pre-modular
fusion category such that the $S$-matrix $S$ is non-degenerate, equivalently
$\mathcal{C}^{\prime}=\text{Vec}$ [7, 10, 15]. Moreover, the $S$-matrix a
modular fusion category $\mathcal{C}$ determines the multiplication of the
Grothendieck ring $\text{Gr}(\mathcal{C})$ by the famous Verlinde formula
[10], i.e., for $X,Y,Z\in\mathcal{O}(\mathcal{C})$,

(2) $\displaystyle N_{X,Y}^{Z}:=\dim_{\mathbb{C}}(\text{Hom}(X\otimes
Y,Z))=\sum_{W\in\mathcal{O}(\mathcal{C})}\frac{S_{X,W}S_{Y,W}S_{Z^{*},W}}{\dim(W)}.$

Moreover, given a modular fusion category $\mathcal{C}$, the Verlinde formula
(2) also implies that the set of ring homomorphism from
$\text{Gr}(\mathcal{C})$ to $\mathbb{C}$ is in bijective correspondence with
$\mathcal{O}(\mathcal{C})$ [10]. Explicitly, let
$X\in\mathcal{O}(\mathcal{C})$, then $h_{X}(Y):=\frac{S_{X,Y}}{\dim(X)}$
defines a ring homomorphism from $\text{Gr}(\mathcal{C})$ to $\mathbb{C}$,
$\forall Y\in\mathcal{O}(\mathcal{C})$; notice that the set of formal
codegrees of $\mathcal{C}$ is
$\\{\frac{\dim(\mathcal{C})}{\dim(X)^{2}}|X\in\mathcal{O}(\mathcal{C})\\}$ due
to the Verlinde formula (2). Since $\sigma\circ h_{X}(-)$ is also a ring
homomorphism of $\text{Gr}(\mathcal{C})$, where
$\sigma\in\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, there is a unique
simple object $Y$ such that $\sigma\circ h_{X}(-)=h_{Y}(-)$. Hence, there is a
unique permutation $\hat{\sigma}$ of $\mathcal{O}(\mathcal{C})$ such that
$\hat{\sigma}(X)=Y$ and $\sigma\circ h_{X}(-)=h_{\hat{\sigma}(X)}(-)$. We call
the subset

$\displaystyle\mathcal{O}_{X}(\mathcal{C}):=\\{Y\in\mathcal{O}(\mathcal{C})|Y=\hat{\sigma}(X)~{}\text{for
some $\sigma\in\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$}\\}$

the Galois orbit of the simple object $X$. When
$\mathcal{O}_{I}(\mathcal{C})=\mathcal{O}(\mathcal{C})$, then $\mathcal{C}$ is
said to be transitive; transitive modular fusion categories are classified
explicitly [16, Theorem II].

The modular data $(S,T)$ of a modular fusion category $\mathcal{C}$ is also
connected closely with the congruence representations of the modular group
$\text{SL}(2,\mathbb{Z})$ [3, 4, 6], which is generated by
$\mathfrak{s}=\left(\begin{array}[]{cc}0&1\\\ -1&0\\\ \end{array}\right)$ and
$\mathfrak{t}=\left(\begin{array}[]{cc}1&1\\\ 0&1\\\ \end{array}\right)$ with
relations $\mathfrak{s}^{4}=1$ and
$(\mathfrak{s}\mathfrak{t})^{3}=\mathfrak{s}^{2}$. Explicitly, the morphism
$\mathfrak{s}\mapsto\frac{1}{\sqrt{\dim(\mathcal{C})}}S$, $\mathfrak{t}\mapsto
T$ defines a projective representation of $\text{SL}(2,\mathbb{Z})$ [10,
Theorem 8.16.1], where $\sqrt{\dim(\mathcal{C})}$ is the positive square root
of $\dim(\mathcal{C})$.

Let
$\tau^{\pm}(\mathcal{C}):=\sum_{X\in\mathcal{O}(\mathcal{C})}\dim(X)^{2}\theta_{X}^{\pm}$
be the Gauss sums of $\mathcal{C}$ [7, 10], then
$\xi(\mathcal{C}):=\frac{\tau^{+}(\mathcal{C})}{\sqrt{\dim(\mathcal{C})}}$ is
called the multiplicative central charge of $\mathcal{C}$. It follows from [4,
6] that there always exists a third root $\xi$ of $\xi(\mathcal{C})$ such that
$s:=\rho_{\mathcal{C}}(\mathfrak{s})=\frac{1}{\sqrt{\dim(\mathcal{C})}}S$ and
$t:=\rho_{\mathcal{C}}(\mathfrak{t})=\frac{1}{\xi}T$ defines a finite-
dimensional congruence representation $\rho_{\mathcal{C}}$ of
$\text{SL}(2,\mathbb{Z})$ of level $N$, that is, $\ker(\rho_{\mathcal{C}})$ is
a congruence subgroup of level $N=\text{ord}(t)$. Moreover,
$\mathbb{Q}(S)\subseteq\mathbb{Q}(T)\subseteq\mathbb{Q}(t)$ [6, Theorem II].
Let $t_{X}:=\theta_{X}/\xi$, the normalized ribbon structure of the simple
object $X$, $\forall X\in\mathcal{O}(\mathcal{C})$, then we have the Galois
symmetry [6, Theorem II], that is, $\sigma^{2}(t_{X})=t_{\hat{\sigma}(X)}$,
$\forall\sigma\in\text{Gal}(\mathbb{Q}(t)/\mathbb{Q})$.

Let $\rho$ be a finite-dimensional congruence representation of
$\text{SL}(2,\mathbb{Z})$ of level $n$, where $n$ is a positive integer. Then
$\rho$ factors through the finite groups

$\displaystyle\text{SL}(2,\mathbb{Z}_{n})\cong\text{SL}(2,\mathbb{Z}_{p_{1}^{n_{1}}})\times\cdots\times\text{SL}(2,\mathbb{Z}_{p_{r}^{n_{r}}})$

and $\rho\cong\otimes_{j=1}^{r}\rho_{p_{j}}$ by the Chinese Reminder Theorem,
where $n=\prod_{j=1}^{r}p_{j}^{n_{j}}$ and $p_{j}$ are distinct primes,
$\rho_{p_{j}}$ are finite-dimensional representations of subgroups
$\text{SL}(2,\mathbb{Z}_{p_{j}^{n_{j}}})$. Moreover, given an arbitrary prime
$p$, all finite-dimensional irreducible representations of the group
$\text{SL}(2,\mathbb{Z}_{p^{m}})$ are completely classified and constructed
explicitly in [18, 19]. A finite-dimensional congruence representation $\rho$
of the modular group $\text{SL}(2,\mathbb{Z})$ is said to be non-degenerate if
the eigenvalues of $\rho(t)$ are distinct; non-degenerate finite-dimensional
congruence representations are irreducible [9, Lemma 1]. In addition, the set
of eigenvalues of $\rho(t)$ is called the $\mathfrak{t}$-spectrum of $\rho$
following [4, 17, 23]; we note that the $\mathfrak{t}$-spectrum of any finite-
dimensional irreducible representation of $\text{SL}(2,\mathbb{Z}_{p^{m}})$ is
produced in [23, Appendix].

The following remark is known to experts, we list it here for the reader’s
convenience, and it will play a key role in the arguments of this paper.

######  Remark 2.1.

Let $\mathcal{C}$ be a non-trivial modular fusion category with
$N(\dim(\mathcal{C}))=p^{N}$, where $p$ is an odd prime. Then there exists a
simple object $X$ such that $p$ divides $\text{ord}(t_{X})$.

Indeed, if $t_{Y}=1$ for all $Y\in\mathcal{O}(\mathcal{C})$, then $\xi=1$ as
$t_{I}=1/\xi=1$, so $\theta_{Y}=1$ $(\forall Y\in\mathcal{O}(\mathcal{C}))$.
Notice that the balancing equation [10] then implies that
$s_{X,Y}=\dim(X)\dim(Y)$, particularly the $S$-matrix of $\mathcal{C}$ can’t
be non-degenerate, it is a contradiction.

Moreover, let $X$ be the simple object of $\mathcal{C}$ such that
$\text{ord}(t_{X})=p^{n}$ is maximal. Then

$\displaystyle\text{rank}(\mathcal{C})\geq\frac{p^{n-1}(p-1)}{2}.$

By the Galois symmetry of modular fusion categories [6], we have
$\sigma^{2}(t_{X})=t_{\hat{\sigma}(X)}$, then

$\displaystyle\text{Gal}(\mathbb{Q}(t_{X})/\mathbb{Q})\cdot
t_{X}={\\{t_{\hat{\sigma}(X)}=\sigma^{2}(t_{X})|\sigma\in\text{Gal}(\mathbb{Q}(t_{X})/\mathbb{Q})}\\},$

hence the number of Galois conjugates of $X$ is greater or equal to the order
of the following subgroup

$\displaystyle\text{Gal}(\mathbb{Q}(t_{X})/\mathbb{Q})^{2}:={\\{\sigma^{2}|\sigma\in\text{Gal}(\mathbb{Q}(t_{X})/\mathbb{Q})}\\}$

of $\text{Gal}(\mathbb{Q}(t_{X})/\mathbb{Q})$. It follows immediately that the
order of $\text{Gal}(\mathbb{Q}(t_{X})/\mathbb{Q})^{2}$ is exactly
$\frac{p^{n-1}(p-1)}{2}$, since $\text{Gal}(\mathbb{Q}(t_{X})/\mathbb{Q})$ is
a cyclic group of order $p^{n-1}(p-1)$.

Throughout this paper, we always use $\rho_{0}$ to denote the trivial
representation of $\text{SL}(2,\mathbb{Z})$.

######  Example 2.2.

Let $p\geq 5$ be a prime, and let
$\mathcal{C}:=\mathcal{C}(\mathfrak{sl}_{2},2(p-1))$. Then $\mathcal{C}$
contains a unique non-trivial connected étale algebra $A$ [12, Theorem 6.5],
i.e., the regular algebra of Tannakian subcategory
$\text{Rep}(\mathbb{Z}_{2})\subseteq\mathcal{C}$, such that
$\mathcal{C}_{A}^{0}$ is a modular fusion category and

$\displaystyle\dim(\mathcal{C}_{A}^{0})=\frac{\dim(\mathcal{C})}{\dim(A)^{2}}=\frac{p}{4\cos^{2}(\frac{d\pi}{p})}$

by [12, Theorem 4.5], where $d=\frac{p+1}{2}$. Moreover, the simple objects of
$\mathcal{C}_{A}^{0}$ are also characterized explicitly in [12, Theorem 7.1],
i.e.,
$\mathcal{O}(\mathcal{C}_{A}^{0})=\\{V_{0},V_{2},\cdots,V_{p-3},V_{p-1}^{+},V_{p-1}^{-}\\}$,
and

$\displaystyle\dim(V_{j})=\frac{\zeta_{4p}^{j+1}-\zeta_{4p}^{-(j+1)}}{\zeta_{4p}-\zeta_{4p}^{-1}}=\frac{\cos(\frac{(d-(j+1))\pi}{p})}{\cos(\frac{(d-1)\pi}{p})},0\leq
j\leq p-3,\dim(V_{p-1}^{\pm})=\frac{1}{2\cos(\frac{(d-1)\pi}{p})}.$

Hence, the formal codegrees of $\mathcal{C}(\mathfrak{sl}_{2},2(p-1))_{A}^{0}$
are

$\displaystyle
p(\text{twice}),\sigma\left(\frac{p}{4\cos^{2}(\frac{d\pi}{p})}\right)~{}\text{where}~{}\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q}).$

In particular, if $p=5$, then
$\mathcal{C}(\mathfrak{sl}_{2},8)_{A}^{0}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}$
as modular fusion category.

By [12, Theorem 1.17], $\theta_{V_{j}}=e^{2\pi i\cdot\frac{j(j+2)}{8p}}$ and
$\theta_{V_{p-1}^{\pm}}=\zeta_{p}^{\frac{p^{2}-1}{8}}$. Notice that the
multiplicative central charge
$\xi(\mathcal{C}_{A}^{0})=\xi(\mathcal{C})=e^{2\pi i\cdot\frac{6(p-1)}{16p}}$,
let $\xi$ be a third root of $\xi(\mathcal{C}_{A}^{0})$, so $\xi$ is an
$24$-th root of $\zeta_{p}^{3(p-1)}$, while
$\zeta_{p}^{p-1}=\zeta_{p}^{(p-1)+p(p-1)}=\zeta_{p}^{p^{2}-1}$, hence we can
choose $\xi=\zeta_{p}^{\frac{p^{2}-1}{8}}$ to be the third root of
$\xi(\mathcal{C}_{A}^{0})$, then the $\mathfrak{t}$-spectrum of the normalized
$t$-matrix is

$\displaystyle\left\\{\sigma^{2}(\zeta_{p}^{-\frac{p^{2}-1}{8}})\bigg{|}\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})/\mathbb{Q})\right\\}\cup\\{1(\text{twice})\\}.$

Therefore, the associated modular representation
$\rho_{\mathcal{C}_{A}^{0}}\cong\rho_{1}\oplus\rho_{0}$, where $\rho_{1}$ is a
$d$-dimensional irreducible representation of $\text{SL}(2,\mathbb{Z})$ of
level $p$ and $\rho_{0}$ is the trivial representation of
$\text{SL}(2,\mathbb{Z})$.

##  3 Modular fusion categories with $N(\dim(\mathcal{C}))=p^{m}$

In this section, we always assume $p$ is a prime, and we study the structures
of modular fusion categories $\mathcal{C}$ such that $N(\dim(\mathcal{C}))$ is
a power of $p$. Notice that if $N(\dim(\mathcal{C}))=p$ and
$\dim(\mathcal{C})\notin\mathbb{Z}$, then $\mathcal{C}$ is braided tensor
equivalent to a Galois conjugate of
$\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}$ [25].

###  3.1 Modular fusion category $\mathcal{C}$ with
$N(\dim(\mathcal{C}))=p^{2}$

Let $\mathcal{C}$ be a modular fusion category with
$N(\dim(\mathcal{C}))=p^{2}$. Then [25, Proposition 4.11] says that
$\frac{1}{2}d_{\dim(\mathcal{C})}\leq 2$, where
$d_{\dim(\mathcal{C})}=[\mathbb{Q}(\dim(\mathcal{C})):\mathbb{Q}]$, that is,
$\dim(\mathcal{C})=1,2,3,4$. Note that
$\mathbb{Q}(\dim(\mathcal{C}))\subseteq\mathbb{Q}(T_{\mathcal{C}})$ and the
Cauchy’s Theorem [3, Theorem 3.9] shows that
$\text{ord}(T_{\mathcal{C}})=p^{m}$ for some positive integer $m$. Since
$\mathbb{Q}(\dim(\mathcal{C}))$ is a real subfield of
$\mathbb{Q}(T_{\mathcal{C}})$,
$\mathbb{Q}(\dim(\mathcal{C}))\subseteq\mathbb{Q}(\zeta_{p^{n}})^{+}$, the
maximal real subfield of $\mathbb{Q}(\zeta_{p^{n}})$, and the Galois group
$\text{Gal}(\mathbb{Q}(\zeta_{p^{n}})/\mathbb{Q})$ is

$\displaystyle\text{Gal}(\mathbb{Q}(\zeta_{p^{n}})/\mathbb{Q})\cong(\mathbb{Z}_{p^{n}})^{*}=\left\\{\begin{array}[]{ll}\mathbb{Z}_{p^{n-1}}\times\mathbb{Z}_{p-1},&\hbox{$p>2$;}\\\
\mathbb{Z}_{2},&\hbox{$p=2$ and $n=2$;}\\\
\mathbb{Z}_{2}\times\mathbb{Z}_{2^{n-2}},&\hbox{$p=2$ and $n\geq
3$.}\end{array}\right.$

and $[\mathbb{Q}(\zeta_{p^{n}}):\mathbb{Q}(\zeta_{p^{n}})^{+}]=2$. Moreover,
for any odd prime $p$, it is well-known that the cyclotomic field
$\mathbb{Q}(\zeta_{p^{n}})$ contains a unique quadratic subfield
$\mathbb{Q}(\sqrt{p^{*}})$, where $p^{*}:=(-1)^{\frac{p-1}{2}}p$.

We first prove the following lemma, which is a direct corollary of [25,
Theorem 4.4].

######  Lemma 3.1.

Let $\mathcal{C}$ be a modular fusion category such that
$N(\dim(\mathcal{C}))=p^{m}$, where $p>3$ is a prime. If
$d_{\dim(\mathcal{C})}>m$, then $m=\frac{p-3}{2}$ and $\mathcal{C}$ is braided
equivalent to a Galois conjugate of the transitive modular fusion category
$\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}}$.

###### Proof.

Since $N(\dim(\mathcal{C}))=p^{m}$ and $m<d_{\dim(\mathcal{C})}$, $p$ does not
divide $\dim(\mathcal{C})$ by [25, Proposition 4.2]. Thus, for any integer
$q\in\mathbb{Z}_{\geq 2}$, $q$ does not divides $\dim(\mathcal{C})$. Hence,
all formal codegrees of $\mathcal{C}$ are Galois conjugates of
$\dim(\mathcal{C})$ [25, Theorem 4.4]. Note that $\mathcal{C}$ is a modular
fusion category, formal codegrees of $\mathcal{C}$ all have the form
$\frac{\dim(\mathcal{C})}{\dim(X)^{2}}$ for simple objects $X$ of
$\mathcal{C}$, so for any object $X\in\mathcal{O}(\mathcal{C})$, there exists
a $\sigma\in\text{Gal}(\mathbb{Q}(\dim(\mathcal{C}))/\mathbb{Q})$ such that

$\displaystyle\frac{\dim(\mathcal{C})}{\dim(X)^{2}}=\sigma(\dim(\mathcal{C}))=\sigma\left(\frac{\dim(\mathcal{C})}{\dim(I)^{2}}\right)=\frac{\dim(\mathcal{C})}{\dim(\hat{\sigma}(I))^{2}},$

and orthogonality of $S$-matrix of $\mathcal{C}$ means $\hat{\sigma}(I)=X$.
Therefore, $\mathcal{C}$ is a transitive modular fusion category in sense of
[16], then $\mathcal{C}$ must be a prime transitive modular fusion category
for $N(\dim(\mathcal{C}))=p^{m}$. Hence it follows from [16, Theorem 1.1] that
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},p-2)^{\sigma}_{\text{ad}}$ as
modular fusion category, where
$\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})/\mathbb{Q})$. Since

$\displaystyle\dim(\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}})=\frac{p}{4\text{sin}^{2}(\pi/p)},~{}N\left(\frac{p}{4\text{sin}^{2}(\pi/p)}\right)=p^{(p-3)/2},$

which implies $m=\frac{p-3}{2}$ and $d_{\dim(\mathcal{C})}=\frac{p-1}{2}$. ∎

######  Corollary 3.2.

Let $\mathcal{C}$ be a modular fusion category such that
$N_{\mathbb{Q}_{\mathcal{C}}}(\dim(\mathcal{C}))=p^{2}$. If
$d_{\dim(\mathcal{C})}>2$, then $p=7$ and
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},5)^{\tau}_{\text{ad}}$, where
$\tau\in\text{Gal}(\mathbb{Q}(\zeta_{7})/\mathbb{Q})$.

Let $\mathcal{C}$ be a modular fusion category,
$\mathcal{D}:=\boxtimes_{\sigma\in\text{Gal}(\mathbb{Q}(\dim(\mathcal{C}))/\mathbb{Q})}\mathcal{C}^{\sigma}$,
then

$\displaystyle\dim(\mathcal{D})=N(\dim(\mathcal{C}))=\Pi_{\sigma\in\text{Gal}(\mathbb{Q}(\dim(\mathcal{C}))/\mathbb{Q})}\dim(\mathcal{C}^{\sigma})=\Pi_{\sigma\in\text{Gal}(\mathbb{Q}(\dim(\mathcal{C}))/\mathbb{Q})}\sigma(\dim(\mathcal{C})).$

Hence, [22, Lemma 4.2.2, Remark 4.2.3] say that

(3)
$\displaystyle\text{rank}(\mathcal{D})=\text{rank}(\mathcal{C})^{d_{\dim(\mathcal{C})}}\leq\dim(\mathcal{D}),$

and $\text{rank}(\mathcal{D})=\dim(\mathcal{D})$ if and only if $\mathcal{D}$
is pointed. In particular, if $N(\dim(\mathcal{C}))=p^{2}$, then
$\text{rank}(\mathcal{C})\leq p^{2}$. When $p=2$, it is easy to see
$\dim(\mathcal{C})=4$, then $\mathcal{C}\cong\mathcal{I}$ is an Ising category
or $\mathcal{C}$ is pointed [22, Example 5.1.2]; when $p=3$,
$d_{\dim(\mathcal{C})}=1$, that is, $\dim(\mathcal{C})=9$, then $\mathcal{C}$
is either pointed or $\mathcal{C}$ is braided equivalent to a Galois conjugate
of $\mathcal{C}(\mathfrak{so}_{5},9,\zeta_{18})_{\text{ad}}$ by [27, Theorem],
where $\zeta_{18}$ is a primitive $18$-root of unity.

######  Lemma 3.3.

Let $\mathcal{C}$ be a modular fusion category such that
$N(\dim(\mathcal{C}))=p^{2}$, where $p$ is a prime. If
$d_{\dim(\mathcal{C})}=2$, then $p=5$, and as a modular fusion category

$\displaystyle\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}~{}\text{or}~{}\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma},$

where $\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{5})/\mathbb{Q})$ such that
$\sigma(\sqrt{5})=-\sqrt{5}$.

###### Proof.

By using the Cauchy theorem [3, Theorem 3.9] and [25, Proposition 4.2], we
know that $\dim(\mathcal{C})\in\mathbb{Q}(\zeta_{p}^{s})$ for some positive
integer $s$. If $d_{\dim(\mathcal{C})}=2$, then
$\frac{\dim{(\mathcal{C})}}{p}$ is a totally positive algebraic unit in the
unique quadratic subfield $\mathbb{Q}(\sqrt{p})$ of
$\mathbb{Q}(\zeta_{p}^{s})$, in particular, $p\equiv 1(\text{mod}~{}4)$ and
$N(\frac{\dim{(\mathcal{C})}}{p})=1$. Let
$\sigma\in\text{Gal}(\mathbb{Q}(\sqrt{p})/\mathbb{Q})$ be the unique non-
trivial element, then $\sigma(\epsilon_{p})\epsilon_{p}=-1$ since
$N_{\mathbb{Q}(\sqrt{p})}(\epsilon_{p})=-1$, where $\epsilon_{p}$ is the
fundamental unit in the quadratic field $\mathbb{Q}(\sqrt{p})$. As
$\dim(\mathcal{C})$ is a totally positive algebraic integer, we have
$\sigma(\dim(\mathcal{C}))>0$, so
$\frac{\dim{(\mathcal{C})}}{p}=\epsilon_{p}^{n}$ for a nonzero even integer
$n$ by the Fundamental unit theorem [1, Theorem 11.5.1]. Besides, for the
Galois conjugate $\sigma(\dim(\mathcal{C}))$ of $\dim(\mathcal{C})$, we have
$\sigma(\dim(\mathcal{C}))>\frac{4\sqrt{3}}{5}$ and
$\dim(\mathcal{C})>\frac{4\sqrt{3}}{5}$ by [22, Proposition A.1.1]. Hence
without loss of generality, we can assume $n\geq 2$ below, then

$\displaystyle\sigma(\dim(\mathcal{C}))=p\epsilon_{p}^{-n}>\frac{4\sqrt{3}}{5},$

also $\epsilon_{p}^{-n}\leq\epsilon_{p}^{-2}$ as $n\geq 2$, thus

$\displaystyle p\epsilon_{p}^{-2}\geq
p\epsilon_{p}^{-n}=\sigma(\dim(\mathcal{C}))>\frac{4\sqrt{3}}{5}.$

which then implies $\epsilon_{p}<\sqrt{\frac{5p}{4\sqrt{3}}}$.

Let $\epsilon_{p}=\frac{a_{p}+b_{p}\sqrt{p}}{2}$ for positive integers
$a_{p},b_{p}$. If $b_{p}\geq 2$, then
$\epsilon_{p}>\sqrt{\frac{5p}{4\sqrt{3}}}$; if $b_{p}=1$, then
$a_{p}^{2}-pb_{p}^{2}=-4$, i.e., $a_{p}=\sqrt{p-4}$, we also have
$\epsilon_{p}>\sqrt{\frac{5p}{4\sqrt{3}}}$ if $p>5$. In fact, let $p>5$ and
$p\equiv 1(\text{mod}~{}4)$, $\epsilon_{p}>\sqrt{\frac{5p}{4\sqrt{3}}}$ if and
only if
$\frac{\sqrt{p-4}}{2}>\left(\sqrt{\frac{5}{4\sqrt{3}}}-\frac{1}{2}\right)\sqrt{p}$,
which is equivalent to
$\sqrt{\frac{5}{4\sqrt{3}}}-\frac{5}{4\sqrt{3}}>\frac{1}{p}$, while we have
inequalities

$\displaystyle\sqrt{\frac{5}{4\sqrt{3}}}-\frac{5}{4\sqrt{3}}>0.12>\frac{1}{13}\geq\frac{1}{p},~{}\text{when
$p\geq 13$},$

which is a contradiction for $p\geq 13$. Then $p=5$.

If $p=5$ and $d_{\dim(\mathcal{C})}=2$, then
$\mathbb{Q}(\dim(\mathcal{C}))=\mathbb{Q}(\sqrt{5})$. Since $\mathcal{C}$ is
not a transitive modular fusion category, $5\mid\text{FPdim}(\mathcal{C})$ by
[25, Theorem 4.4]. Meanwhile, [25, Proposition 4.2] implies
$N(\text{FPdim}(\mathcal{C}))=25$ as $N(\dim(\mathcal{C}))=25$. Therefore,
$\text{FPdim}(\mathcal{C})=5\epsilon_{5}^{n}$, where $n$ is a positive even
integer. Let $\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{5}/\mathbb{Q})$ such that
$\sigma(\sqrt{5})=-\sqrt{5}$, notice that $\text{FPdim}(\mathcal{C})$ and
$\sigma(\text{FPdim}(\mathcal{C}))$ are formal codegrees of $\mathcal{C}$, so
$\sigma(\text{FPdim}(\mathcal{C}))>\sqrt{\frac{4}{3}}$ by [22, Theorem 4.2.1],
which implies $n=2$ and $\text{FPdim}(\mathcal{C})=\frac{15+5\sqrt{5}}{2}$.
Since $\text{rank}(\mathcal{C})\leq 4$ by equation (3), by using the argument
of [22, Example 5.1.2(v)], we obtain that
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}$
or
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}$
as modular fusion category. ∎

######  Remark 3.4.

It follows from the proof of Lemma 3.3 and [22, Example 5.1.2] that a non-
pointed modular fusion category $\mathcal{C}$ is Grothendieck equivalent to
$\mathcal{C}(\mathfrak{sl}_{2},8)_{A}^{0}$ if and only if
$\dim(\mathcal{C})=5\epsilon_{5}^{m}$, where $m=0,\pm 2$.

In summary, Corollary 3.2 and Lemma 3.3 imply the following theorem:

######  Theorem 3.5.

Let $\mathcal{C}$ be a modular fusion category with
$N(\dim(\mathcal{C}))=p^{2}$, where $p$ is a prime. Then either
$\dim(\mathcal{C})=p^{2}$, or $p=5,7$. Moreover, if $\dim(\mathcal{C})\neq
p^{2}$, then

$\displaystyle\mathcal{C}\cong\left\\{\begin{array}[]{ll}\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}},&\hbox{if
$p=5$;}\\\ \mathcal{C}(\mathfrak{sl}_{2},5)^{\tau}_{\text{ad}},&\hbox{if
$p=7$.}\end{array}\right.$

as modular fusion category, where
$\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{5})/\mathbb{Q})$ and
$\tau\in\text{Gal}(\mathbb{Q}(\zeta_{7})/\mathbb{Q})$.

Let $\mathcal{C}$ be a spherical fusion category of global dimension $p$,
where $p$ is a prime. Then for any formal codegree $f$ of $\mathcal{C}$, it
follows from [25, Lemma 5.1] that $f=p\cdot u_{f}$, where $u_{f}$ is an
algebraic unit in the field $\mathbb{Q}(\zeta_{p})^{+}$ [25, Proposition 5.2],
thus $d_{f}:=[\mathbb{Q}(f):\mathbb{Q}]$ divides $\frac{p-1}{2}$.

######  Proposition 3.6.

Assume that $\mathcal{C}$ is not pointed. Then
$2<d_{\text{FPdim}(\mathcal{C})}<\frac{p-1}{2}$ if $p>5$.

###### Proof.

Spherical fusion categories of integer dimensions less or equal to $5$ are
classified in [22, Example 5.2.2], so we assume $p>5$ below. If
$d_{\text{FPdim}(\mathcal{C})}=2$, then
$u_{\text{FPdim}(\mathcal{C})}=\epsilon_{p}^{n}$ [1, Theorem 11.5.1], where
$n\neq 0$ is an even integer, as $\mathcal{C}$ is not pointed and
$u_{\text{FPdim}(\mathcal{C})}$ is a totally positive algebraic unit. Thus
$\text{FPdim}(\mathcal{C})=p\epsilon_{p}^{n}$, while
$\text{FPdim}(\mathcal{C})>\sqrt{\frac{8}{5}}$ and
$\sigma(\text{FPdim}(\mathcal{C}))>\sqrt{\frac{8}{5}}$ [22, Theorem 4.2.1],
where $\langle\sigma\rangle=\text{Gal}(\mathbb{Q}(\epsilon_{p})/\mathbb{Q})$.
By using the same argument of Lemma 3.3, we see $p\leq 11$. However, spherical
fusion categories of global dimensions $7,11$ are pointed [25, Theorem 5.5],
it is a contradiction. Hence $d_{\text{FPdim}(\mathcal{C})}>2$ if $p>5$.

If $d_{\text{FPdim}(\mathcal{C})}=\frac{p-1}{2}$, then [21, Corollary 2.15]
says

$\displaystyle\mathbb{Q}(\text{FPdim}(\mathcal{C}))=\mathbb{Q}(\zeta_{p})^{+}=\mathbb{Q}(\dim(X),X\in\mathcal{O}(\mathcal{C})).$

That is, each of the homomorphisms $\dim(-)$ and $\text{FPdim}(-)$ has
$\frac{p-1}{2}$ Galois conjugates. Since $\mathcal{C}$ is not a pointed fusion
category, $\text{rank}(\mathcal{C})<p$ by [22, Remark 4.2.3], so
$\text{rank}(\mathcal{C})=p-1$ and the Grothendieck ring
$\text{Gr}(\mathcal{C})$ is commutative. Note that

$\displaystyle\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q})}\frac{1}{\sigma(\dim(\mathcal{C}))}+\frac{1}{\sigma(\text{FPdim}(\mathcal{C}))}=1,$

by [20, Proposition 2.10], then we have

$\displaystyle\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q})}\frac{1}{\sigma(u_{\text{FPdim}(\mathcal{C})})}=\frac{p+1}{2}.$

While $\frac{1}{u_{\text{FPdim}(\mathcal{C})}}\neq 1$ is a totally positive
algebraic unit and $\mathbb{Q}(\zeta_{p})^{+}\neq\mathbb{Q}(\sqrt{5})$, the
Siegel’s trace theorem [26, Theorem III] shows

$\displaystyle\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q})}\frac{1}{\sigma(u_{\text{FPdim}(\mathcal{C})})}=\text{tr}(\frac{1}{u_{\text{FPdim}(\mathcal{C})}})>\frac{p-1}{2}\cdot\frac{3}{2}.$

Thus we obtain $\frac{p+1}{2}>\frac{p-1}{2}\cdot\frac{3}{2}$, which then
implies $p<5$, it is impossible. ∎

######  Remark 3.7.

In fact, let $f_{\chi}$ be an arbitrary formal codegree of $\mathcal{C}$,
where $\mathcal{C}$ is a spherical fusion category of global dimension $p$
with $p\neq 5$. If $f_{\chi}\notin\mathbb{Z}$, then the method of the above
proposition says that $2<[\mathbb{Q}(f_{\chi}):\mathbb{Q}]<\frac{p-1}{2}$.

###  3.2 Modular fusion category $\mathcal{C}$ with
$N(\dim(\mathcal{C}))=p^{3}$

Let $\mathcal{C}$ be a modular fusion category such that
$N(\dim(\mathcal{C}))=p^{3}$ and $\dim(\mathcal{C})\notin\mathbb{Z}$. Then
$d_{\dim(\mathcal{C})}\leq 6$ by [25, Proposition 4.11], moreover Lemma 3.1
says $d_{\dim(\mathcal{C})}=2,3$. Hence, we know either $4$ or $6$ divides
$p-1$. In particular, there does not exist modular fusion categories such that
$N(\dim(\mathcal{C}))=p^{3}$ and $d_{\dim(\mathcal{C})}>1$ when $(p-1,6)=2$.
The inequality (3) implies $\text{rank}(\mathcal{C})\leq p-1$, and we know
there does not exist such modular fusion category when $p=2,3$ [21], so we
assume $p\geq 5$ below. Moreover,

######  Lemma 3.8.

Let $\mathcal{C}$ be a modular fusion category such that
$N(\dim(\mathcal{C}))=p^{3}$ and $d_{\dim(\mathcal{C})}>1$. Then $\mathcal{C}$
is simple when $p>5$.

###### Proof.

Assume
$\mathcal{C}\cong\mathcal{D}\boxtimes\mathcal{D}_{\mathcal{C}}^{\prime}$ where
$\mathcal{D}$ is a modular fusion subcategory of $\mathcal{C}$, then the proof
of Lemma 3.1 shows that either $\mathcal{D}$ or
$\mathcal{D}_{\mathcal{C}}^{\prime}$ is transitive, as $p$ can not divide both
$\dim(\mathcal{D})$ and $\dim(\mathcal{D}_{\mathcal{C}}^{\prime})$. Without
loss of generality, assume that $\mathcal{D}$ is braided equivalent to a
Galois conjugate of $\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}}$, then
$N(\dim(\mathcal{D}))=p^{\frac{p-3}{2}}$ and

$\displaystyle
N_{\mathbb{Q}(\zeta_{p})^{+}}(\dim(\mathcal{C}))=\left\\{\begin{array}[]{ll}p^{\frac{3(p-1)}{4}},&\hbox{if
$d_{\dim(\mathcal{C})}=2$;}\\\ p^{\frac{(p-1)}{2}},&\hbox{if
$d_{\dim(\mathcal{C})}=3$.}\end{array}\right.$

thus we obtain
$N_{\mathbb{Q}(\zeta_{p})^{+}}(\dim(\mathcal{D}_{\mathcal{C}}^{\prime}))=p^{\frac{(p+3)}{4}}$
if $d_{\dim(\mathcal{C})}=2$, or $p$ if $d_{\dim(\mathcal{C})}=3$. Obviously,
in each case $p$ can’t divide $\dim(\mathcal{D}_{\mathcal{C}}^{\prime})$ by
[25, Proposition 4.2], so $\mathcal{D}_{\mathcal{C}}^{\prime}$ is also
transitive by [25, Theorem 4.4]. However, there is a contradiction when $p>5$.
∎

######  Lemma 3.9.

Let $\mathcal{C}$ be a modular fusion category such that
$N(\dim(\mathcal{C}))=5^{3}$ and $d_{\dim(\mathcal{C})}\neq 1$. Then
$\dim(\mathcal{C})=\frac{25\pm 5\sqrt{5}}{2}$ or $25\pm 10\sqrt{5}$, and
$\text{FPdim}(\mathcal{C})=25+10\sqrt{5}$ or $\frac{25+5\sqrt{5}}{2}$.

###### Proof.

It is easy to see $\mathbb{Q}(\sqrt{5})=\mathbb{Q}(\dim(\mathcal{C}))$. Since
$d_{\dim(\mathcal{C})}=2$ and
$N\left(\frac{\dim(\mathcal{C})}{5\sqrt{5}}\right)=-1$,
$\dim(\mathcal{C})=5\sqrt{5}\epsilon^{m}_{5}$ with an odd integer $m$, where
$\epsilon_{5}=\frac{\sqrt{5}+1}{2}$ is the fundamental algebraic unit of
$\mathbb{Q}(\sqrt{5})$. By [22, Proposition A.1.1],
$\sigma(\dim(\mathcal{C}))>\frac{4\sqrt{3}}{5}$ and
$\dim(\mathcal{C})>\frac{4\sqrt{3}}{5}$, where $\sigma(\sqrt{5})=-\sqrt{5}$.
Hence, if $m\leq-1$, then
$\dim(\mathcal{C})=5\sqrt{5}(\frac{\sqrt{5}-1}{2})^{-m}>\frac{4\sqrt{3}}{5}$,
i.e., $(\frac{\sqrt{5}-1}{2})^{-m}>\frac{4\sqrt{3}}{25\sqrt{5}}$, thus
$m=-1,-3$; if $m>0$, then
$\sigma(\dim(\mathcal{C}))=-5\sqrt{5}\sigma(\epsilon^{m}_{5})=5\sqrt{5}\epsilon^{-m}_{5}>\frac{4\sqrt{3}}{5}$,
so $m=1,3$. In summary, $m=\pm 1,\pm 3$.

Assume that $\dim(-)$ takes value in a subfield $\mathbb{F}$ of the totally
real $\mathbb{Q}(\zeta_{5^{n}})^{+}$ for some positive integer $n$ with $n$
being minimal. If $n>1$, then $[\mathbb{F}:\mathbb{Q}]\geq 10$ as
$\mathbb{Q}(\sqrt{5})\subseteq\mathbb{F}$, for any
$\sigma\in\text{Gal}(\mathbb{F}/\mathbb{Q})$, there exists a unique simple
object $\hat{\sigma}(I)$ such that

$\displaystyle\sigma(\dim(\mathcal{C}))=\frac{\dim(\mathcal{C})}{\dim(\hat{\sigma}(I))^{2}},$

while $\sigma(\dim(\mathcal{C}))\in{\\{\frac{25\pm 5\sqrt{5}}{2}}\\}$ or
${\\{25\pm 10\sqrt{5}}\\}$, then $\mathcal{C}$ has at least $5$ formal
codegrees equal to $\frac{25+5\sqrt{5}}{2}$ and $\frac{25-5\sqrt{5}}{2}$ if
$\dim(\mathcal{C})=\frac{25\pm 5\sqrt{5}}{2}$, however
$5(1/\frac{25+5\sqrt{5}}{2}+1/\frac{25-5\sqrt{5}}{2})=1$, so these are all
formal codegrees of $\mathcal{C}$, it is a contradiction as $n=1$ in this
case; if $\dim(\mathcal{C})=\frac{25\pm 5\sqrt{5}}{2}$, then
$5(1/(25+10\sqrt{5})+1/(25-10\sqrt{5}))=2$, impossible. Thus, $n=1$. Since
$\frac{\dim(\mathcal{C})}{\text{FPdim}(\mathcal{C})}$ is a totally positive
algebraic integer that is less than or equal to $1$ [10, Proposition 9.4.2],
we see $N(\text{FPdim}(\mathcal{C}))=25,125$ by Lemma 3.1 and [25, Proposition
4.2].

If $N(\text{FPdim}(\mathcal{C}))=25$, then
$\text{FPdim}(\mathcal{C})=5\epsilon_{5}^{2}=\frac{15+5\sqrt{5}}{2}$ and
$\mathcal{C}$ is braided equivalent to a Deligne product of two Fibonacci
fusion categories by [22, Example 5.1.2], however, in this case
$N(\dim(\mathcal{C}))\neq 125$. Therefore, $N(\text{FPdim}(\mathcal{C}))=125$,
then $\text{FPdim}(\mathcal{C})=\frac{25+5\sqrt{5}}{2}$ or $25+10\sqrt{5}$, as
$\text{FPdim}(\mathcal{C})$ is the largest among the set of formal codegrees
of $\mathcal{C}$. ∎

######  Proposition 3.10.

Let $\mathcal{C}$ be a modular fusion category such that
$N(\dim(\mathcal{C}))=5^{3}$ and $d_{\dim(\mathcal{C})}\neq 1$. Then as
modular fusion category,

$\displaystyle\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma_{1}}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma_{2}}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma_{3}}_{\text{ad}}~{}\text{or}~{}\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{5},\eta)\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma_{4}}_{\text{ad}},$

where $\sigma_{i}\in\text{Gal}(\mathbb{Q}(\zeta_{5})/\mathbb{Q})$ for all
$1\leq i\leq 4$.

###### Proof.

By Lemma 3.9, we know $\text{FPdim}(\mathcal{C})=25+10\sqrt{5}$ or
$\frac{25+5\sqrt{5}}{2}$.

If $\text{FPdim}(\mathcal{C})=\frac{25+5\sqrt{5}}{2}$, then
$\dim(\mathcal{C})=\frac{25\pm 5\sqrt{5}}{2}$, so $\mathcal{C}$ is conjugated
to a pseudo-unitary fusion category. Notice that $\mathcal{C}$ contains a
self-dual simple object $X=\hat{\sigma}(I)$ of FP-dimension
$\frac{1+\sqrt{5}}{2}$, $X\otimes X=I\oplus Z$ where $Z$ is also a self-dual
simple object of FP-dimension $\frac{1+\sqrt{5}}{2}$. If $Z=X$, then
$\mathcal{C}$ contains $\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}$ as a
fusion subcategory, hence
$\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{5},\eta)\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}$;
if not, $X\otimes Z=X\oplus g$, where $g$ is a non-trivial invertible object,
so $\mathcal{C}_{\text{pt}}$ is non-trivial. While
$\text{FPdim}(\mathcal{C}_{\text{pt}})$ divides $\text{FPdim}(\mathcal{C})$ by
[11, Proposition 8.15], which implies
$\text{FPdim}(\mathcal{C}_{\text{pt}})=5$ and
$\mathcal{C}_{\text{pt}}\cong\mathcal{C}(\mathbb{Z}_{5},\eta)$. By [7, Theorem
3.13], we have

$\displaystyle\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{5},\eta)\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}~{}\text{if}\dim(\mathcal{C})=\frac{25+5\sqrt{5}}{2},$
$\displaystyle\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{5},\eta)\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}}~{}\text{if}\dim(\mathcal{C})=\frac{25-5\sqrt{5}}{2}.$

If $\text{FPdim}(\mathcal{C})=25+10\sqrt{5}$, then
$\mathcal{C}_{\text{pt}}=\text{Vec}$. In fact, if
$\mathcal{C}_{\text{pt}}\neq\text{Vec}$, then
$\mathcal{C}_{\text{pt}}\cong\mathcal{C}(\mathbb{Z}_{5},\eta)$ and
$\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{5},\eta)\boxtimes\mathcal{D}$ with
$\dim(\mathcal{D})=5+2\sqrt{5}$. However, note that $5-2\sqrt{5}$ is a formal
codegree of $\mathcal{D}$, which contradicts to [22, Theorem 1.1.2]. If
$\dim(\mathcal{C})=25\pm 10\sqrt{5}$, by using the same argument as above, we
know $\mathcal{C}$ contains a Galois conjugate of
$\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}$ as a modular fusion
subcategory. Consequently,
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}\boxtimes\mathcal{D}$
with $\mathcal{D}$ being a modular fusion subcategory and
$N(\dim(\mathcal{D}))=5^{2}$, where
$\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{5})/\mathbb{Q})$, and Lemma 3.3 implies

$\displaystyle\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}~{}\text{if}\dim(\mathcal{C})=25+10\sqrt{5},$
$\displaystyle\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}}~{}\text{if}\dim(\mathcal{C})=25-10\sqrt{5}.$

Notice that if $\dim(\mathcal{C})=\frac{25-5\sqrt{5}}{2}$, then
$\dim(\mathcal{C}^{\sigma})=\frac{25+5\sqrt{5}}{2}$, hence it suffices to
consider when $\dim(\mathcal{C})=\frac{25+5\sqrt{5}}{2}$ below. We know that
$\mathcal{C}$ contains a self-dual simple object $X=\hat{\sigma}(I)$ with
$\dim(X)^{2}=\frac{3+\sqrt{5}}{2}$, simple objects $Y$ and $Z=\hat{\sigma}(Y)$
such that
$\dim(Y)^{2}=\frac{\dim(\mathcal{C})}{\text{FPdim}(\mathcal{C})}=\frac{3-\sqrt{5}}{2}$
and
$\dim(Z)^{2}=\frac{\dim(\mathcal{C})}{\sigma(\text{FPdim}(\mathcal{C}))}=\frac{7+3\sqrt{5}}{2}$.
Let $f$ be another formal codegree of $\mathcal{C}$, then $N(f)=25$ or $125$.
If $N(f)=25$, then $f$ is a root of $x^{2}-ax+25=0$ with $a^{2}\geq 100$, we
obtain $f=5$ by equation (1), thus $\mathcal{C}$ contains a simple object $V$
with $\dim(V)^{2}=\frac{5+\sqrt{5}}{2}$, which is impossible as
$\dim(V)\notin\mathbb{Q}(\zeta_{5^{n}})^{+}$. Thus, $N(f)=125$ for all other
formal codegrees of $\mathcal{C}$, a direct computation shows
$\mathcal{O}(\mathcal{C})=\\{I,X,Y,Z,V_{1},V_{2},W_{1},W_{2}\\}$ with
$\dim(V_{1})^{2}=\dim(V_{2})^{2}=1$ and
$\dim(W_{1})^{2}=\dim(W_{2})^{2}=\frac{3+\sqrt{5}}{2}$.

By decomposing $\text{FPdim}(\mathcal{C})$ into the sum of squares of
Frobenius-Perron dimensions of eight simple objects over field
$\mathbb{Q}(\sqrt{5})$, we find that $\mathcal{C}$ always contains a simple
object of Frobenius-Perron dimension $\frac{1+\sqrt{5}}{2}$, therefore,
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{D}$
as modular fusion subcategory by [7, Theorem 3.13]. Thus, again by Lemma 3.3

$\displaystyle\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}}~{}\text{if}\dim(\mathcal{C})=\frac{25+5\sqrt{5}}{2},$
$\displaystyle\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}~{}\text{if}\dim(\mathcal{C})=\frac{25-5\sqrt{5}}{2},$

$\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{5})/\mathbb{Q})$ such that
$\sigma(\sqrt{5})=-\sqrt{5}$, this completes the proof of the proposition. ∎

Let $\mathcal{C}$ be a modular fusion category. Let $p\geq 5$, and let
$\rho_{1}$ be a $\frac{p+1}{2}$-dimensional irreducible representation of
$\text{SL}(2,\mathbb{Z})$ with level equal to $p$, it is proved in [17,
Proposition 3.22] that if the associated modular representation
$\rho_{\mathcal{C}}$ is equivalent to $m\rho_{0}\oplus\rho_{1}$, then $m=1$.

We strengthen the above conclusion in the following proposition and theorem.

######  Proposition 3.11.

Let $p\geq 7$ be a prime, and let $\mathcal{C}$ be a modular fusion category
such that $\rho_{\mathcal{C}}\cong\rho_{0}\oplus\rho_{1}$, where $\rho_{1}$ is
the $\frac{p+1}{2}$-dimensional irreducible representation of
$\text{SL}(2,\mathbb{Z})$ of level $p$. Then $\mathcal{C}$ is a Galois
conjugate of a pseudo-unitary fusion category.

###### Proof.

Let $\mathcal{O}(\mathcal{C})=\\{X_{0},X_{1},\cdots,X_{d}\\}$ and
$S=(S_{X_{i},X_{j}})$ be the un-normalized $S$-matrix of $\mathcal{C}$. Let
$d:=\frac{p+1}{2}$, $p^{*}=\left(\frac{-1}{p}\right)p$ and
$\beta:=\left(\frac{a}{p}\right)/\sqrt{p^{*}}$, where $a$ is an integer
comprime to $p$. By [17, Proposition 3.22] there exists an real orthogonal
matrix $U=\left(\begin{array}[]{cc}A_{1}&0\\\ 0&A_{2}\\\ \end{array}\right)$
such that

$\displaystyle\rho_{\mathcal{C}}(\mathfrak{s})=\frac{1}{\sqrt{\dim(\mathcal{C})}}S=U\left(\begin{array}[]{cc}1&0\\\
0&\rho_{1}(\mathfrak{s})\\\
\end{array}\right)U^{T},\rho_{\mathcal{C}}(\mathfrak{t})=\text{diag}(1,\zeta_{p}^{a\cdot
0},\cdots,\zeta_{p}^{a\cdot(d-1)^{2}}),$

$\rho_{1}(\mathfrak{s})=\beta\left(\begin{array}[]{cccc}1&\sqrt{2}&\cdots&\sqrt{2}\\\
\sqrt{2}&\\\ \vdots&&2\cos(\frac{4\pi ajk}{p})\\\ \sqrt{2}&\\\
\end{array}\right)$, $1\leq j,k,l\leq d-1$,
$A_{1}=\left(\begin{array}[]{cc}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{array}\right)$,
$A_{2}=\text{diag}(\lambda_{1},\cdots,\lambda_{d-1})$, where
$\lambda_{l}\in\\{\pm 1\\}$. Moreover,
$\sqrt{2}a_{12},\sqrt{2}a_{22}\in\mathbb{Q}$, $0<a_{12}^{2}<1$ and
$0<a_{22}^{2}<1$.

Up to isomorphism, there are exactly two $\frac{p+1}{2}$-dimensional
irreducible representations of level $p$ [8, 19] depending on the value
$\left(\frac{a}{p}\right)$. We assume $a=1$ below, the other case is the same.
A direct computation shows

$\displaystyle\rho_{\mathcal{C}}(\mathfrak{s})=\left(\begin{array}[]{cccccc}1+a_{12}^{2}(\beta-1)&a_{12}a_{22}(\beta-1)&\sqrt{2}\beta
a_{12}\lambda_{1}&\cdots\quad\sqrt{2}\beta a_{12}\lambda_{d-1}\\\
a_{12}a_{22}(\beta-1)&1+a_{22}^{2}(\beta-1)&\sqrt{2}\beta
a_{22}\lambda_{1}&\cdots\quad\sqrt{2}\beta a_{22}\lambda_{d-1}\\\
\sqrt{2}\beta a_{12}\lambda_{1}&\sqrt{2}\beta a_{22}\lambda_{1}&\\\
\vdots&\vdots&&2\beta\lambda_{j}\lambda_{k}\cos(\frac{4\pi ajk}{p})\\\
\sqrt{2}\beta a_{12}\lambda_{d-1}&\sqrt{2}\beta a_{22}\lambda_{d-1}\\\
\end{array}\right),$

$1\leq j,k\leq d-1$. We show that either $X_{0}$ or $X_{1}$ don’t represent
the unit object $I$.

On the contrary, without loss of generality, let $X_{0}=I$, then
$\frac{1}{\sqrt{\dim(\mathcal{C})}}=|1+(\beta-1)a_{12}^{2}|$. Notice that

$\displaystyle\beta=\left(\frac{1}{p}\right)/\sqrt{p^{*}}=\left\\{\begin{array}[]{ll}\frac{1}{p},&\hbox{$p\equiv
1\mod 4$;}\\\ \frac{i}{p},&\hbox{$p\equiv 3\mod 4$.}\end{array}\right.$

then
$|1+(\beta-1)a_{12}^{2}|^{2}=\left\\{\begin{array}[]{ll}1-2a_{12}^{2}+\frac{p+1}{p}a_{12}^{4}+2(a_{12}^{2}-a_{12}^{4})\frac{1}{\sqrt{p}},&\hbox{$p\equiv
1\mod 4$;}\\\ 1-2a_{12}^{2}+\frac{p+1}{p}a_{12}^{4},&\hbox{$p\equiv 3\mod
4$.}\end{array}\right.$, and

$\displaystyle|\beta-1|^{2}=\left\\{\begin{array}[]{ll}\frac{p+1-2\sqrt{p}}{p},&\hbox{$p\equiv
1\mod 4$;}\\\ \frac{p+1}{p},&\hbox{$p\equiv 3\mod 4$.}\end{array}\right.$

When $p\equiv 3\mod 4$, the set of formal codegrees of $\mathcal{C}$ is

$\displaystyle\frac{\dim(\mathcal{C})}{(\sqrt{\dim(\mathcal{C})}|S_{1,l}|)^{2}}\in\\{\dim(\mathcal{C}),\frac{1}{a_{12}^{2}a_{22}^{2}|\beta-1|^{2}},\frac{p}{(\sqrt{2}a_{12}\lambda_{j})^{2}},1\leq
j\leq d\\},1\leq l\leq d+1,$

so the formal codegrees of $\mathcal{C}$ are all rational, which implies they
are integers. In particular, $\text{FPdim}(\mathcal{C})=p^{m}$ by Cauchy’s
theorem [3, Theorem 3.9], hence $\mathcal{C}$ must be integral as $p$ is odd
[10, Corollary 3.5.8]. However, in this case $\mathcal{C}_{\text{pt}}$ must be
non-trivial, so $\text{rank}(\mathcal{C})\geq p$, it is impossible.

When $p\equiv 1\mod 4$,
$\frac{1}{\dim(\mathcal{C})}=1-2a_{12}^{2}+\frac{p+1}{p}a_{12}^{4}+2(a_{12}^{2}-a_{12}^{4})\frac{1}{\sqrt{p}}\notin\mathbb{Z}$,
again the Cauchy’s theorem [3, Theorem 3.9] implies that
$N(\dim(\mathcal{C}))=p^{m}$ for $m\geq 2$. Let
$\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})/\mathbb{Q})$ be a generator, then
$\sigma$ has no invariant simple objects [17, Proposition 3.22]. Hence

$\displaystyle\sigma(\dim(\mathcal{C}))=\frac{\dim(\mathcal{C})}{\dim(X_{2})^{2}}=\frac{1}{a_{12}^{2}a_{22}^{2}(\beta-1)^{2}}=\frac{p}{a_{12}^{2}a_{22}^{2}}\cdot\frac{(p+1+2\sqrt{p})}{(p-1)^{2}}.$

Therefore,
$N(\dim(\mathcal{C}))=\dim(\mathcal{C})\sigma(\dim(\mathcal{C}))=\frac{p^{2}}{(p-1)^{2}a_{12}^{4}a_{22}^{4}}$,
that is, $a_{12}^{4}a_{22}^{4}(p-1)^{2}=p^{-m}$ with $2\mid m$ and $m\geq 0$,
since $a_{12}^{2}a_{22}^{2}\in\mathbb{Q}$. Assume $m=2l$, so
$a_{12}^{2}a_{22}^{2}=\frac{1}{p^{l}(p-1)}$; meanwhile
$a_{12}^{2}+a_{22}^{2}=1$, so $a_{12}^{2}$ and $a_{22}^{2}$ are roots of
equation $x^{2}-x+\frac{1}{p^{l}(p-1)}=0$, consequently

$\displaystyle
a_{12}^{2},a_{22}^{2}=\frac{\sqrt{(p-1)p^{l}}\pm\sqrt{(p-1)p^{l}-4}}{2\sqrt{(p-1)p^{l}}}.$

Since $p\geq 7$ and $p\equiv 1\mod 4$, $((p-1)p^{l}-4,(p-1)p^{l})=4$; as
$a_{12}^{2},a_{22}^{2}$ are rational, we have $(p-1)p^{l}-4=q_{1}^{2}$ and
$(p-1)p^{l}=q_{2}^{2}$ for nonnegative integers $q_{1},q_{2}$. Note that
$(q_{2}-q_{1})(q_{2}+q_{1})=4$, then $l=0$ and $p=5$, it is a contradiction.

Therefore, the unit object $I\in\\{X_{2}\cdots,X_{d}\\}$. Meanwhile the Galois
symmetry [6, Theorem II] implies that the unit object $I$ has exactly
$\frac{p-1}{2}$ Galois conjugates, so
$\mathcal{O}_{I}(\mathcal{C})=\\{X_{2},\cdots,X_{d}\\}$. If the simple object
$X_{1}$ or $X_{2}$ determines the homomorphism $\text{FPdim}(-)$, then
$\text{FPdim}(\mathcal{C})=\frac{\dim(\mathcal{C})}{\dim(X_{1})^{2}}$ or
$\text{FPdim}(\mathcal{C})=\frac{\dim(\mathcal{C})}{\dim(X_{2})^{2}}$. Note
that $\text{FPdim}(\mathcal{C})\in\mathbb{Q}$ in both cases, again
$\mathcal{C}$ must be an integral fusion category, so
$\text{rank}(\mathcal{C})\geq p$, which is absurd. Therefore, the simple
object which determines the homomorphism $\text{FPdim}(-)$ belongs to
$\mathcal{O}_{I}(\mathcal{C})$, so $\mathcal{C}$ is Galois conjugate to a
pseudo-unitary fusion category. ∎

######  Theorem 3.12.

Let $p\geq 5$ be a prime and $\mathcal{C}$ a modular fusion category such that
$\rho_{\mathcal{C}}\cong\rho_{0}\oplus\rho_{1}$, where $\rho_{1}$ is the
$d$-dimensional irreducible representation of $\text{SL}(2,\mathbb{Z})$ of
level $p$. Then $\mathcal{C}$ is Grothendieck equivalent to
$\mathcal{C}(\mathfrak{sl}_{2},2(p-1))_{A}^{0}$. Moreover,
$\dim(\mathcal{C})=p\cdot u$ where $u$ is a Galois conjugate of algebraic unit
$4\cos^{2}(\frac{d\pi}{p})$.

###### Proof.

By Proposition 3.11 we know that $\mathcal{C}$ is a Galois conjugate of a
pseudo-unitary fusion category if $p\geq 7$. Without loss of generality, we
assume that $\mathcal{C}$ is pseudo-unitary. As we have proved in Proposition
3.11 that $X_{0},X_{1}\notin\mathcal{O}_{I}(\mathcal{C})$, and
$\rho(\mathfrak{s})=\frac{1}{\sqrt{\dim(\mathcal{C})}}S$,

$\displaystyle\dim(\mathcal{C})=\text{FPdim}(\mathcal{C})=\frac{1}{|S_{X_{k},X_{k}}|^{2}}~{}\text{for
some $2\leq k\leq d$}.$

Meanwhile, $\text{FPdim}(\mathcal{C})$ is the largest among its Galois
conjugates, we see $\dim(\mathcal{C})=\frac{p}{4\cos^{2}(\frac{d\pi}{p})}$. In
addition, if $p=5$, then the argument of Proposition 3.11 also shows that
$\dim(\mathcal{C})=5\epsilon_{5}^{\pm 2}$ if the unit object
$I\in\\{X_{0},X_{1}\\}$ and that $\mathcal{C}$ is a Galois conjugate of a
pseudo-unitary fusion category if $I\in\\{X_{2},X_{3}\\}$. Hence, in both
cases, we see that $\mathcal{C}$ is braided tensor equivalent to a Galois
conjugate of $\mathcal{C}(\mathfrak{sl}_{2},8)_{A}^{0}$ by Lemma 3.3.

We assume $p\geq 7$ below, then $\mathcal{C}$ is a simple modular fusion
category. Indeed, if $\mathcal{C}$ is not simple, then fusion subcategories of
$\mathcal{C}$ are modular by [27, Theorem 3.1]. Let $\mathcal{D}$ be an
arbitrary non-trivial modular fusion subcategory of $\mathcal{C}$, then
$\mathcal{C}\cong\mathcal{D}\boxtimes\mathcal{D}_{\mathcal{C}}^{\prime}$ by
[7, Theorem 3.13]. As $p$ can’t divide both $\dim(\mathcal{D})$ and
$\dim(\mathcal{D}_{\mathcal{C}}^{\prime})$, therefore $\mathcal{D}$ and
$\mathcal{D}_{\mathcal{C}}^{\prime}$ are transitive modular fusion categories.
By [16], we know
$\mathcal{D}\cong\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}}^{\sigma_{1}}$
and
$\mathcal{D}_{\mathcal{C}}^{\prime}\cong\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}}^{\sigma_{2}}$
for $\sigma_{1},\sigma_{2}\in\text{Gal}(\mathbb{Q}(\zeta_{p})/\mathbb{Q})$.
However,
$\dim((\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}})=\frac{p}{4\sin^{2}(\frac{\pi}{p})}$
and

$\displaystyle\dim(\mathcal{C})=\dim(\mathcal{D})\dim(\mathcal{D}_{\mathcal{C}}^{\prime})=\sigma_{1}(\dim(\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}}))\sigma_{1}(\dim(\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}})),$

so
$p^{\frac{p-1}{2}}=N(\dim(\mathcal{C}))=N(\dim(\mathcal{D}))N(\dim(\mathcal{D}_{\mathcal{C}}^{\prime}))=p^{p-3}$,
i.e., $p=5$, it is impossible.

A direct computation shows that the set of formal codegrees of $\mathcal{C}$
is

$\displaystyle\left\\{\frac{\dim(\mathcal{C})}{|\sqrt{\dim(\mathcal{C})}S_{X_{k},X_{0}}|^{2}},\frac{\dim(\mathcal{C})}{|\sqrt{\dim(\mathcal{C})}S_{X_{k},X_{1}}|^{2}},\sigma(\dim(\mathcal{C})),\forall\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})/\mathbb{Q})\right\\},$

while norms of
$\frac{\dim(\mathcal{C})}{|\sqrt{\dim(\mathcal{C})}S_{X_{k},X_{0}}|^{2}}=\frac{p}{(\sqrt{2}a_{12})^{2}}$
and
$\frac{\dim(\mathcal{C})}{|\sqrt{\dim(\mathcal{C})}S_{X_{k},X_{0}}|^{2}}=\frac{p}{(\sqrt{2}a_{22})^{2}}$
divide that of $\dim(\mathcal{C})$, which is power of $p$, consequently
$(\sqrt{2}a_{12})^{2}=\frac{1}{p^{m_{1}}}$ and
$(\sqrt{2}a_{22})^{2}=\frac{1}{p^{m_{2}}}$ for nonnegative integers
$m_{1},m_{2}$, while $(\sqrt{2}a_{12})^{2}+(\sqrt{2}a_{22})^{2}=2$, so
$a_{12}=\frac{\mu_{1}}{\sqrt{2}}$ and $a_{22}=\frac{\mu_{2}}{\sqrt{2}}$,
$\mu_{1},\mu_{2}\in\\{\pm 1\\}$. Hence, we obtain that

$\displaystyle\rho_{\mathcal{C}}(\mathfrak{s})=\left(\begin{array}[]{cccccc}\frac{\beta+1}{2}&\frac{\mu_{1}\mu_{2}(\beta-1)}{2}&\beta\mu_{1}\lambda_{1}&\cdots\quad\beta\mu_{1}\lambda_{d-1}\\\
\frac{\mu_{1}\mu_{2}(\beta-1)}{2}&\frac{\beta+1}{2}&\beta\mu_{2}\lambda_{1}&\cdots\quad\beta\mu_{2}\lambda_{d-1}\\\
\beta\mu_{1}\lambda_{1}&\beta\mu_{2}\lambda_{1}&\\\
\vdots&\vdots&&2\beta\lambda_{j}\lambda_{k}\cos(\frac{4\pi ajk}{p})\\\
\beta\mu_{1}\lambda_{d-1}&\beta\mu_{2}\lambda_{d-1}\\\
\end{array}\right),1\leq j,k\leq d-1.$

We choose $-a=\frac{p-1}{2}$ below, the other case is same. Since
$4\left(\frac{p-1}{2}\right)^{3}=(p-1)^{2}\frac{p-1}{2}\equiv\frac{p-1}{2}\mod
p$, when $-a=j=k=\frac{p-1}{2}$,

$\displaystyle|2\beta\lambda_{j}\lambda_{k}\cos(\frac{4\pi
ajk}{p})|^{2}=\frac{4\cos^{2}(\frac{(d-1)\pi}{p})}{p}=\frac{1}{\dim(\mathcal{C})},$

therefore $X_{d}$ represents the isomorphism class of the unit object.

Consequently, the set of the Frobenius-Perron dimensions of simple objects of
$\mathcal{C}$ is:

$\displaystyle\left\\{\frac{\mu_{1}\lambda_{d-1}}{2\cos(\frac{(d-1)\pi}{p})},\frac{\mu_{2}\lambda_{d-1}}{2\cos(\frac{(d-1)\pi}{p})},\frac{\lambda_{j}\lambda_{d-1}\cos(\frac{j\pi}{p})}{\cos(\frac{(d-1)\pi}{p})},1\leq
j\leq d-1\right\\},$

since the Frobenius-Perron dimensions of simple objects are positive,
$\mu_{1}=\mu_{2}=\lambda_{j}$ for all $1\leq j\leq d-1$. By comparing the
Frobenius-Perron dimensions of simple objects of $\mathcal{C}$ and
$\mathcal{C}(\mathfrak{sl}_{2},2(p-1))_{A}^{0}$ (see Example 2.2), let
$\phi:\text{Gr}(\mathcal{C}(\mathfrak{sl}_{2},2(p-1))_{A}^{0})\to\text{Gr}(\mathcal{C})$
be a morphism such that $\phi(V_{p-1}^{\pm})=X_{0},X_{1}$ and
$\phi(V_{2(d-j)})=X_{j}$, $1\leq j\leq d$, respectively. Notice that
$\mathcal{C}$ and $\text{Gr}(\mathcal{C})$ share the same modular data and
$\phi$ preserves the Frobenius-Perron dimensions of simple objects, hence
$\phi$ is an isomorphism of fusion ring by the Verlinde formula (2). ∎

######  Corollary 3.13.

Let $\mathcal{C}$ be a modular fusion category such that
$\dim(\mathcal{C})=p\cdot u$, where $u\neq 1$ and
$u\in\mathbb{Q}(\zeta_{p})^{+}$ is a totally positive algebraic unit. Then
$\mathcal{C}$ is Grothendieck equivalent to
$\mathcal{C}(\mathfrak{sl}_{2},2(p-1))_{A}^{0}$.

###### Proof.

Let
$\mathcal{D}:=\boxtimes_{\sigma\in\text{Gal}(\mathbb{Q}(u)/\mathbb{Q})}\mathcal{C}^{\sigma}$.
Since $\mathcal{C}$ is not pointed, the inequality (3) and [22, Remark 4.2.3]
imply $\text{rank}(\mathcal{C})\leq p-1$; so we can assume $p\geq 5$. If
$[\mathbb{Q}(u):\mathbb{Q}]=2$, then $N(\dim(\mathcal{C}))=p^{2}$, $p=5$ and
$\dim(\mathcal{C})=\frac{5(3\pm\sqrt{5})}{2}$ by Lemma 3.3. Assume
$[\mathbb{Q}(u):\mathbb{Q}]>2$ and $p>5$ below. By Remark 2.1, we know
$\text{ord}(t_{\mathcal{C}})=p$ and at least one of the Galois orbits of
simple objects of $\mathcal{C}$ contains $\frac{p-1}{2}$ simple objects.
Meanwhile $\mathcal{C}$ can not be a transitive modular fusion category [16,
Theorem I], therefore, $\text{rank}(\mathcal{C})>\frac{p-1}{2}$.

Assume $\rho_{\mathcal{C}}$ is the modular representation associated to
$\mathcal{C}$, then $\rho_{\mathcal{C}}$ is a direct sum of level $p$
irreducible representations of $\text{SL}(2,\mathbb{Z})$. However,
$\rho_{\mathcal{C}}$ can’t be decomposed as direct sum of irreducible
representation of disjoint $\mathfrak{t}$-spectrum [4, Lemma 3.18] and
$\rho_{\mathcal{C}}$ also can’t decomposed into direct sum of one-dimensional
representations [9, Lemma 4]. Therefore, by comparing the dimensions of level
$p$ irreducible representations of $\text{SL}(2,\mathbb{Z})$, we obtain that
either $\rho_{\mathcal{C}}$ is an irreducible representation of dimension
$p-1$ or $\frac{p+1}{2}$, or $\rho$ is a direct sum of two irreducible
representations of dimension $\frac{p-1}{2}$, or $\rho$ is a direct sum of
one-dimensional representations and an irreducible representation of dimension
$\frac{p+1}{2}$.

If $\rho_{\mathcal{C}}\cong\rho_{1}\oplus\rho_{2}$ with
$\dim(\rho_{1})=\dim(\rho_{2})=\frac{p-1}{2}$, since the
$\mathfrak{t}$-spectrums of $\rho_{1}$ and $\rho_{2}$ intersect non-trivially
[4, Lemma 3.18], $\rho_{1}=\rho_{2}$, which is impossible by [23, Lemma
5.2.2]. And the irreducible representations of dimension $\frac{p+1}{2}$ can’t
be realized as representations of modular fusion categories [8]. Therefore,
$\rho_{\mathcal{C}}$ is an irreducible representation of dimension $p-1$ or
$\rho_{\mathcal{C}}=\rho_{0}\oplus\rho_{1}$ by [17, Lemma 3.20, Proposition
3.22], where $\rho_{0}$ is the trivial representation and $\rho_{1}$ is the
irreducible representation of dimension $\frac{p+1}{2}$.

If $\rho_{\mathcal{C}}$ is irreducible, then $\text{rank}(\mathcal{C})=p-1$.
In particular, the Galois symmetry [6, Theorem II] implies that
$\mathcal{O}(\mathcal{C})$ splits into two orbits and each Galois orbit have
exactly $\frac{p-1}{2}$ simple objects. Notice that all formal codegrees of
$\mathcal{C}$ are divided by $p$. Indeed, if not, then $\mathcal{C}$ is a
transitive modular fusion category [25, Theorem 4.4] and

$\displaystyle\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},p-2)^{\sigma}_{\text{ad}}~{}\text{
for some $\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})/\mathbb{Q})$},$

however, $\text{rank}(\mathcal{C}(\mathfrak{sl}_{2},p-2))=\frac{p-1}{2}$, it
is a contradiction. Meanwhile, $p^{2}\nmid f$ for any formal codegree $f$ of
$\mathcal{C}$, otherwise, $p^{2}\mid\dim(\mathcal{C})$, which implies $p\mid
u$, it is impossible by [25, Proposition 4.2]. Thus, for any formal codegree
$f$ of $\mathcal{C}$, we have $f=p\cdot u_{f}$, where
$u_{f}\in\mathbb{Q}(\dim(X),X\in\mathcal{O}(\mathcal{C}))$ is a totally
positive algebraic integer as $f\mid\dim(\mathcal{C})$. Since
$f=\frac{\dim(\mathcal{C})}{\dim(X)^{2}}$ for some simple object
$X\in\mathcal{O}(\mathcal{C})$, we see $u_{X}:=u_{f}=\frac{u}{\dim(X)^{2}}$.

By definition,

$\displaystyle p\cdot
u=\dim(\mathcal{C})=\sum_{X\in\mathcal{O}(\mathcal{C})}\dim(X)^{2}=\sum_{X\in\mathcal{O}(\mathcal{C})}\frac{u}{u_{X}},$

and for any $\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q})$,
$\sigma\left(\frac{\dim(\mathcal{C})}{\dim(X)^{2}}\right)=\frac{\dim(\mathcal{C})}{\dim(\hat{\sigma}(X))^{2}}$,
therefore,

$\displaystyle\dim(\hat{\sigma}(X))^{2}=\frac{u\sigma(\dim(X)^{2})}{\sigma(u)}=\frac{u}{\sigma(u_{X})},~{}\text{and}~{}\sigma(u_{X})=\frac{u}{\dim(\hat{\sigma}(X))^{2}}=u_{\hat{\sigma}(X)}.$

Thus, we have the following equation

$\displaystyle\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q})}\dim(\hat{\sigma}(X))^{2}=\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q})}\frac{u}{\sigma(u_{X})}.$

Hence, let $\mathcal{O}(\mathcal{C})=\Gamma_{1}\cup\Gamma_{2}$, where
$\Gamma_{i}:={\\{\hat{\sigma}(X_{i})|\sigma\in\text{Gal}(\mathbb{Q}(u_{X_{i}})/\mathbb{Q})}\\}$,
then

$\displaystyle
p=\sum^{2}_{i=1}\sum_{\sigma\in\text{Gal}(\mathbb{Q}(u_{X_{i}})/\mathbb{Q})}\frac{1}{\sigma(u_{X_{i}})}.$

Since $u_{I}=u\neq 1$, the Siegel’ trace theorem [26] says that

$\displaystyle\sum_{\sigma\in\text{Gal}(\mathbb{Q}(u_{X_{i}})/\mathbb{Q})}\frac{1}{\sigma(u_{X_{i}})}>1.79\cdot\frac{p-1}{2},$

except for roots of the equation $x^{3}-5x^{2}+6x-1=0$ when $p=7$ [13, Theorem
1.1]. If $u_{X_{i}}=1$, then
$\sum_{\sigma\in\text{Gal}(\mathbb{Q}(u_{X_{i}})/\mathbb{Q})}\frac{1}{\sigma(u_{X_{i}})}=\frac{p-1}{2}$.
Hence,

$\displaystyle p\geq\frac{5}{3}\cdot\frac{p-1}{2}+\frac{p-1}{2},$

that is, $p\leq 3$, it is impossible.

When $\rho_{\mathcal{C}}=\rho_{0}\oplus\rho_{1}$, we deduce from Theorem 3.12
that $\mathcal{C}$ is Grothendieck equivalent to the modular fusion category
$\mathcal{C}(\mathfrak{sl}_{2},2(p-1))_{A}^{0}$. This finishes the proof of
the corollary. ∎

######  Corollary 3.14.

Let $\mathcal{C}$ be a modular fusion category such that
$N(\dim(\mathcal{C}))=p^{3}$ and $d_{\dim(\mathcal{C})}=3$. Then $p=7$ and
$\mathcal{C}$ is Grothendieck equivalent to
$\mathcal{C}(\mathfrak{sl}_{2},12)_{A}^{0}$.

##  4 Modular fusion category of global dimension $p^{2}$

In this section, we always assume $p$ is a prime and we study the structure of
modular fusion categories $\mathcal{C}$ of global dimension $p^{2}$.

######  Proposition 4.1.

Let $\mathcal{C}$ be a pre-modular fusion category of global dimension
$p^{2}$. If $\mathcal{C}^{\prime}\ncong\text{Vec}$, then either $\mathcal{C}$
is pointed or
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}\boxtimes\text{Rep}(\mathbb{Z}_{5})$,
where $\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{5})/\mathbb{Q})$ such that
$\sigma(\zeta_{5})=\zeta_{5}^{2}$.

###### Proof.

Since $\mathcal{C}^{\prime}\ncong\text{Vec}$,
$\dim(\mathcal{C}^{\prime})=p,p^{2}$ by [27, Theorem 3.1]. If
$\dim(\mathcal{C}^{\prime})=p^{2}$, then $\mathcal{C}$ is symmetric, so it is
pointed [11, Proposition 8.32]. If $\dim(\mathcal{C}^{\prime})=p$, then
$\mathcal{C}^{\prime}=\text{Rep}(\mathbb{Z}_{p})$ is Tannakian, or
$\mathcal{C}^{\prime}=\text{sVec}$ and $p=2$. When $p=2$, it is obviously
$\mathcal{C}$ is pointed [22, Example 5.1.2]. If
$\mathcal{C}^{\prime}=\text{Rep}(\mathbb{Z}_{p})$, then
$\mathcal{C}_{\mathbb{Z}_{p}}$ is a modular fusion category of dimension $p$,
hence $\mathcal{C}_{\mathbb{Z}_{p}}$ is pointed or
$\mathcal{C}_{\mathbb{Z}_{p}}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}$
by [25, Theorem 5.12]. Therefore, $\mathcal{C}$ is a pointed fusion category
or
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}\boxtimes\text{Rep}(\mathbb{Z}_{5})$.
∎

######  Proposition 4.2.

Let $\mathcal{D}$ be a non-trivial fusion subcategory of $\mathcal{C}$. If
$\dim(\mathcal{D})\in\mathbb{Z}$, then either $\mathcal{C}$ is pointed, or
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},2)$, or
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}$,
or
$\mathcal{C}\cong\mathcal{C}(\mathbb{Z}_{5},\eta)\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}$,
where $\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{5})/\mathbb{Q})$ such that
$\sigma(\zeta_{5})=\zeta_{5}^{2}$.

###### Proof.

Since $\mathcal{D}$ is a non-trivial fusion subcategory of $\mathcal{C}$ with
integral global dimension, by [27, Theorem 3.1] $\dim(\mathcal{D})=p$. Then we
consider the global dimension of the Müger center $\mathcal{D}^{\prime}$ of
$\mathcal{D}$, [27, Theorem 3.1] again implies that either
$\mathcal{D}=\mathcal{D}^{\prime}$ is symmetric or $\mathcal{D}$ is a modular
fusion category. If $\mathcal{D}$ is modular, then
$\mathcal{D}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}$
or $\mathcal{D}$ is pointed by [25, Theorem 5.12]. Notice that
$\mathcal{C}\cong\mathcal{D}\boxtimes\mathcal{D}_{\mathcal{C}}^{\prime}$ as
modular fusion category [7, Theorem 3.13], so
$\mathcal{D}_{\mathcal{C}}^{\prime}$ is also a modular fusion category of
dimension $p$, thus the structure of $\mathcal{C}$ is known. If $\mathcal{D}$
is symmetric, then either it is a Tannakian fusion category or
$\mathcal{D}\cong\text{sVec}$. In the first case, $\mathcal{D}$ must be a
Lagrange fusion category as
$\mathcal{D}\subseteq\mathcal{D}_{\mathcal{C}}^{\prime}$ and
$\dim(\mathcal{D})\dim(\mathcal{D}_{\mathcal{C}}^{\prime})=p^{2}$ [7, Theorem
3.10], which implies
$\mathcal{C}\cong\mathcal{Z}(\text{Vec}^{\omega}_{\mathbb{Z}_{p}})$ by [7,
Theorem 4.64], where $\omega\in Z^{3}(\mathbb{Z}_{p},\mathbb{C}^{*})$ is a
$3$-cocycle; in the second case $\dim(\mathcal{C})=4$, then
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},2)$ is an Ising category or
$\mathcal{C}$ is pointed [22, Example 5.1.2]. ∎

Let $\mathcal{C}$ be a modular fusion category of global dimension $p^{2}$,
then $\text{ord}(T_{\mathcal{C}})$ divides $p^{5}$ [10, Corollary 8.18.2].
Since the structures of modular fusion categories of global dimension $4$ and
$9$ are known [22, 27], we always assume $p\geq 5$ below. Let
$\text{ord}(t_{\mathcal{C}})=p^{n}$ where $t$ is the normalized $T$-matrix of
$\mathcal{C}$, since $\text{rank}(\mathcal{C})\leq p^{2}-1$ [22, Lemma 4.2.2],
$n\leq 2$ by Remark 2.1. In particular, the number of simple objects in each
Galois orbit is less than or equal to
$|\text{Gal}(\mathbb{Q}(\zeta_{p^{2}})/\mathbb{Q})^{2}|=\frac{p^{2}-p}{2}$.

Assume that $\mathcal{C}$ does not contain a non-trivial fusion subcategory of
integer global dimension below, and let $\mathcal{D}$ be a non-trivial modular
fusion subcategory of $\mathcal{C}$. Note that we can assume that
$\mathcal{D}$ to be a simple modular fusion category and that $\mathcal{C}$
can be decomposed as Deligne product of simple modular fusion subcategories.
Indeed, since $\mathcal{D}^{\prime}$ is an integral fusion subcategory,
$\mathcal{D}^{\prime}=\text{Vec}$, hence fusion subcategories of $\mathcal{C}$
are modular. Assume $\dim(\mathcal{D})$ be the largest among its Galois
conjugates, otherwise we can replace $\mathcal{D}$ with one of its Galois
conjugates $\mathcal{D}^{\sigma}$.

######  Theorem 4.3.

Let $\mathcal{C}$ be a non-simple modular fusion category of dimension
$p^{2}$, where $p>3$. If $\mathcal{C}$ does not contain a non-trivial fusion
subcategory with integer global dimension, then $\mathcal{C}$ contains a
modular fusion subcategory that is braided equivalent to a Galois conjugate of
$\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}}$.

###### Proof.

Let $\mathcal{D}\subseteq\mathcal{C}$ be a simple modular fusion subcategory
with $\dim(\mathcal{D})\notin\mathbb{Z}$, then
$\mathcal{C}\cong\mathcal{D}\boxtimes\mathcal{D}_{\mathcal{C}}^{\prime}$ and
$\mathcal{D}_{\mathcal{C}}^{\prime}$ is also a modular fusion category by [7,
Theorem 3.13]. By Lemma 3.1, it suffices to show that $\mathcal{C}$ contains a
non-trivial fusion subcategory whose global dimension is not divided by $p$ in
sense of algebraic integers. If $p\nmid\dim(\mathcal{D})$, then we are done;
assume $\dim(\mathcal{D})=pa_{\mathcal{D}}$ for some totally positive
algebraic integer $a_{\mathcal{D}}$ below. If
$\mathcal{D}_{\mathcal{C}}^{\prime}$ is not simple, let
$\mathcal{D}_{\mathcal{C}}^{\prime}\cong\mathcal{A}\boxtimes\mathcal{B}$,
where $\mathcal{A},\mathcal{B}$ are non-trivial modular fusion subcategories,
obviously $p$ can’t divide both $\dim(\mathcal{A})$ and $\dim(\mathcal{B})$,
thus the argument of Lemma 3.1 says that either $\mathcal{A}$ or $\mathcal{B}$
is a transitive modular fusion category, so $\mathcal{C}$ contains a Galois
conjugate of $\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}}$ [16, Theorem
1.1].

Assume that $\mathcal{D}_{\mathcal{C}}^{\prime}$ is a simple modular fusion
category such that
$\dim(\mathcal{D}_{\mathcal{C}}^{\prime})=pb_{\mathcal{D}}$, $a_{\mathcal{D}}$
and $b_{\mathcal{D}}$ are totally positive algebraic units with
$1=a_{\mathcal{D}}b_{\mathcal{D}}$. Notice that
$a_{\mathcal{D}}\in\mathbb{Q}(\zeta_{p})^{+}$, if not, each Galois orbit of
the unit objects $I_{\mathcal{D}}$ and
$I_{\mathcal{D}_{\mathcal{C}}^{\prime}}$ has at least $p$ simple objects, then
$\text{rank}(\mathcal{C})\geq p^{2}$, it is impossible. By Corollary 3.13,
$\mathcal{D}$ and $\mathcal{D}_{\mathcal{C}}^{\prime}$ are Grothendieck
equivalent to modular fusion category
$\mathcal{C}(\mathfrak{sl}_{2},2(p-1))_{A}^{0}$. Up to Galois conjugates, we
can assume $\dim(\mathcal{D})=\frac{p}{4\cos^{2}(\frac{d\pi}{p})}$ and
$\dim(\mathcal{D}_{\mathcal{C}}^{\prime})=4p\cos^{2}(\frac{d\pi}{p})$ with
$d=\frac{p+1}{2}$.

If there exists a $\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})/\mathbb{Q})$ such
that

$\displaystyle\sigma\left(\frac{p}{4\cos^{2}(\frac{d\pi}{p})}\right)=\dim(\mathcal{D}_{\mathcal{C}}^{\prime})=4p\cos^{2}(\frac{d\pi}{p}),$

then $p=5$; in addition, when $p=5$ we have a braided tensor equivalence

$\displaystyle\mathcal{D}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\quad\text{or}\quad\mathcal{D}\cong\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}^{\sigma}$

by Theorem 3.5, where $\sigma(\zeta_{5})=\zeta_{5}^{2}$. However,
$\mathcal{C}$ must contains a fusion subcategory of dimension $5$ by
Proposition 3.10, which contradicts to the assumption. Therefore,
$\mathcal{C}$ contains a modular fusion subcategory that is braided equivalent
to a Galois conjugate of $\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}}$. ∎

######  Lemma 4.4.

Let $f(x)=\frac{4\pi\sin^{2}(x)}{x}$, where $x\in(0,\frac{\pi}{5}]$. Then
$f(x)$ is an increasing function.

###### Proof.

We have $f^{\prime}(x)=\frac{4\pi\sin(x)[2x\cos(x)-\sin(x)]}{x^{2}}$; since
$x\in(0,\frac{\pi}{5}]$, $\frac{4\pi\sin(x)}{x^{2}}>0$. Let

$\displaystyle g(x):=2x\cos(x)-\sin(x),\quad x\in[0,\frac{\pi}{5}],$

then $g^{\prime}(x)=\cos(x)-2x\sin(x)$,
$g^{\prime\prime}(x)=-\sin(x)-2x\cos(x)$, hence $g^{\prime\prime}(x)<0$ for
all $x\in[0,\frac{\pi}{5}]$, so $g^{\prime}(x)\geq
g^{\prime}(\frac{\pi}{5})>0$. Thus $g(x)\geq g(0)=0$ for
$x\in[0,\frac{\pi}{5}]$, which shows $f^{\prime}(x)>0$ when
$x\in(0,\frac{\pi}{5}]$. Consequently, $f(x)$ is a strictly increasing
function. ∎

######  Theorem 4.5.

Let $\mathcal{C}$ be a modular fusion category of global dimension $p^{2}$,
where $p>3$ is a prime. If $\mathcal{C}$ contains a modular fusion category
that is a Galois conjugate of
$\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}}$, then $p\leq 23$ and
$\mathbb{Q}(S_{\mathcal{C}})\subseteq\mathbb{Q}(T_{\mathcal{C}})=\mathbb{Q}(\zeta_{p})$.

###### Proof.

Let $\mathcal{D}:=\mathcal{C}(\mathfrak{sl}_{2},p-2)_{\text{ad}}$, then
$\mathcal{C}\cong\mathcal{D}\boxtimes\mathcal{D}_{\mathcal{C}}^{\prime}$ as
modular fusion category [7, Theorem 3.13], thus
$\dim(\mathcal{D})=\frac{p}{4\text{sin}^{2}(\pi/p)}$ and
$\dim(\mathcal{D}_{\mathcal{C}}^{\prime})=4p\text{sin}^{2}(\pi/p)$. By [22,
Proposition A.1.1], we obtain that
$\dim(\mathcal{D}_{\mathcal{C}}^{\prime})>\frac{4\sqrt{3}}{5}$, since
$\dim(\mathcal{D}_{\mathcal{C}}^{\prime})=f(\pi/p)$, where
$f(x)=\frac{4\pi\sin^{2}(x)}{x}$, and $f(\pi/29)<1.38<\frac{4\sqrt{3}}{5}$,
Lemma 4.4 implies $p\leq 23$.

Since $\mathcal{C}$ is not pointed, $\text{rank}(\mathcal{C})\leq p^{2}-1$ by
[22, Remark 4.2.3]; while $\text{rank}(\mathcal{D})=\frac{p-1}{2}$, so
$\text{rank}(\mathcal{D}_{\mathcal{C}}^{\prime})\leq 2(p+1)$. Assume that
$\mathbb{Q}(t_{\mathcal{C}})=\mathbb{Q}(\zeta_{p^{n}})$ with $n$ being
minimal. When $p>5$, if $n>1$, then $\text{rank}(\mathcal{C})>p^{2}$ by Remark
2.1, which is a contradiction. When $p=5$, the structure of $\mathcal{C}$ is
known by Proposition 3.10, so $n=1$. ∎

######  Lemma 4.6.

Let $\mathcal{C}$ be a modular fusion category of global dimension
$p\beta_{p}$, where $p$ is a prime and
$\beta_{p}:=2-(\zeta_{p}+\zeta_{p}^{-1})$. Then $\dim(X)$ is an algebraic unit
for all objects $X\in\mathcal{O}(\mathcal{C})$.

###### Proof.

Let $f=\frac{\dim(\mathcal{C})}{\dim(X)^{2}}$ be an arbitrary formal codegree
of $\mathcal{C}$ corresponding to a simple object $X$. Since $\mathcal{C}$ is
not transitive, $p\mid f$ by [25, Theorem 4.4], hence
$\frac{\beta_{p}}{\dim(X)^{2}}$ is an algebraic integer, i.e.,
$N_{\mathbb{Q}(\zeta_{p})^{+}}(\dim(X)^{2})\mid
N_{\mathbb{Q}(\zeta_{p})^{+}}(\beta_{p})$ by [25, Proposition 4.2]. While
$N_{\mathbb{Q}(\zeta_{p})^{+}}(\dim(X)^{2})=N_{\mathbb{Q}(\zeta_{p})^{+}}(\dim(X))^{2}$
and $N_{\mathbb{Q}(\zeta_{p})^{+}}(\beta_{p})=p$, so
$N_{\mathbb{Q}(\zeta_{p})^{+}}(\dim(X)^{2})=1$, which means that $\dim(X)$
must be an algebraic unit. ∎

######  Corollary 4.7.

Let $\mathcal{D}$ be a modular fusion category of global dimension
$p\beta_{p}$ where $p>3$ is a prime. Then $\mathcal{C}$ does not contain
simple object that is fixed by
$\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q})$.

###### Proof.

Let $X$ be a simple object that is fixed by the Galois group
$\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q})$, then for any
$\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q})$, we have

$\displaystyle\frac{\dim(\mathcal{D})}{\dim(X)^{2}}=\sigma\left(\frac{\dim(\mathcal{D})}{\dim(X)^{2}}\right)=\frac{\dim(\mathcal{D})}{\dim(\hat{\sigma}(X))^{2}}\in\mathbb{Z},$

which implies $\dim(X)^{2}=\beta_{p}$, it contradicts to Lemma 4.6. ∎

######  Proposition 4.8.

Let $\mathcal{C}$ be a modular fusion category of global dimension
$p\beta_{p}$, where $p\geq 7$. Then $\text{rank}(\mathcal{C})=p-1$ or
$\frac{3(p-1)}{2}$.

###### Proof.

Since $\mathbb{Q}(\dim(\mathcal{C}))=\mathbb{Q}(\zeta_{p})^{+}$ and
$\text{ord}(t)=p$ by Theorem 4.5, the orbit of $I$ has exactly $\frac{p-1}{2}$
simple objects, it is easy to see that $\mathcal{C}$ can’t be a transitive
modular fusion category, so $\text{rank}(\mathcal{C})>\frac{p-1}{2}$. By
Corollary 4.5, $p\leq 23$; Corollary 4.7 says that each Galois orbit of simple
objects of $\mathcal{C}$ has exactly $\frac{p-1}{2}$ simple objects when
$p=7,11,23$.

When $p=13,17,19$, for any formal codegree $f$ of $\mathcal{C}$, and let
$g(x)=x^{n}-a_{1}x^{n-1}+\cdots+(-1)^{n}a_{n}$ be the minimal polynomial of
$f$, where $a_{j}$ are positive integers and $n$ is a divisor of
$\frac{p-1}{2}$. Notice that $a_{0}=N(f)$ is a power of $p$ and that

$\displaystyle\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q})}\frac{1}{\sigma(\dim(\mathcal{C}))}=\frac{a_{n-1}}{a_{n}}=\left\\{\begin{array}[]{ll}\frac{7}{13},&\hbox{if
$p=13$;}\\\ \frac{12}{17},&\hbox{if $p=17$;}\\\ \frac{15}{19},&\hbox{if
$p=19$.},\end{array}\right.$

equation (1) implies
$\sum_{\sigma\in\text{Gal}(\mathbb{Q}(f))/\mathbb{Q}}\frac{1}{\sigma(f)}\leq\frac{a_{n}-a_{n-1}}{a_{n}}$.
By using the program GAP, we can show that the $d$-number test or cyclotomic
test fail when $[\mathbb{Q}(f):\mathbb{Q}]<\frac{p-1}{2}$ for any formal
codegree $f$ of $\mathcal{C}$, hence each Galois orbit of simple object of
$\mathcal{C}$ has $\frac{p-1}{2}$ simple objects. Since
$N(\dim(\mathcal{C}))=p^{\frac{p+1}{2}}$, the inequality (3) shows that
$\text{rank}(\mathcal{C})\leq\left[p^{\frac{p+1}{p-1}}\right]=\left[p\cdot
p^{\frac{2}{p-1}}\right]$, where $[\alpha]$ is the integer part of a positive
algebraic integer $\alpha$. Therefore, $\text{rank}(\mathcal{C})\leq 12$ if
$p=7$, $\text{rank}(\mathcal{C})=p-1,\frac{3(p-1)}{2}$ when $p=11,13,17$ and
$\text{rank}(\mathcal{C})=p-1$ when $p=19,23$.

When $p=7$, for any formal codegree $f$ of $\mathcal{C}$, let
$g(x)=x^{3}-ax^{2}+bx-7^{4}$ be the minimal polynomial of $f$, where $a,b$ are
positive integers. Then the $d$-number test and cyclotomic test show that
$g(x)=x^{3}-49x^{2}+686x-7^{4}$, which is the minimal polynomial of
$7\beta_{7}$, or $g(x)=x^{3}-98x^{2}+1029x-7^{4}$, the minimal polynomial of
the totally positive algebraic integer $\frac{49}{\beta_{7}^{2}}$. Then the
set of formal codegrees of $\mathcal{C}$ are exactly the Galois conjugates of
$\dim(\mathcal{C})$ (with multiplicity equals two) and
$\text{FPdim}(\mathcal{C})=\frac{49}{\beta_{7}^{2}}$ by equation (1), so
$\text{rank}(\mathcal{C})=9$. ∎

######  Lemma 4.9.

Let $\mathcal{C}$ be a modular fusion category of global dimension
$p\beta_{p}$ where $p=11,13$. Then
$\text{rank}(\mathcal{C})\neq\frac{3(p-1)}{2}$.

###### Proof.

Assume $\text{rank}(\mathcal{C})=\frac{3(p-1)}{2}$. By Proposition 4.8 each
Galois orbits of the simple objects has exactly $\frac{p-2}{2}$ simple
objects. Let $X_{1}=I,X_{2},X_{3}$ be the representatives of each Galois
orbits. Let $d:=\frac{p-1}{2}$, let
$g_{2}(x)=x^{d}+\sum_{j=1}^{d}(-1)^{j}a_{j}x^{d-j}$ and
$g_{3}(x)=x^{d}+\sum_{j=1}^{d}(-1)^{j}b_{j}x^{d-j}$ be the minimal polynomials
of the formal codegrees $\frac{\dim(\mathcal{C})}{\dim(X_{i})^{2}}$ of
$\mathcal{C}$, respectively, where $a_{j},b_{j}$ are positive integers for
$1\leq j\leq d$. Since $\dim(X_{i})^{2}$ are algebraic units by Lemma 4.6, we
obtain $a_{d}=b_{d}=N(\dim(\mathcal{C}))=p^{d+1}$. The $d$-number condition
[20] says $a_{d}^{d-1}\mid a_{d-1}^{d}$ and $b_{d}^{d-1}\mid b_{d-1}^{d}$,
that is, $p^{d}$ divides both $a_{d-1}$ and $b_{d-1}$.

Assume $b_{d-1}\geq a_{d-1}$ and $a_{d-1}=mp^{d}$ with $m$ being a positive
integer. As
$\sigma\left(\frac{\dim(\mathcal{C})}{\dim(X_{i})^{2}}\right)=\frac{\dim(\mathcal{C})}{\dim(\hat{\sigma}(X_{i}))^{2}}$
for all $\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q})$, then

$\displaystyle\dim(\mathcal{C})$
$\displaystyle=\sum_{i=1}^{3}\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q})}\dim(\hat{\sigma}(X_{i}))^{2}$
$\displaystyle=\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{17})^{+}/\mathbb{Q})}\frac{\dim(\mathcal{C})}{\sigma(\dim(\mathcal{C}))}+\sum_{i=2}^{3}\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{17})^{+}/\mathbb{Q})}\frac{\dim(\mathcal{C})}{\sigma(\dim(\mathcal{C}))}\sigma(\dim(X_{i})^{2}),$

meanwhile
$\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{p})^{+}/\mathbb{Q})}\frac{\dim(\mathcal{C})}{\sigma(\dim(\mathcal{C}))}=\left\\{\begin{array}[]{ll}5\beta_{11},&\hbox{if
$p=11$;}\\\ 7\beta_{13},&\hbox{if $p=13$.}\end{array}\right.$, hence
$\frac{6}{p}\geq\frac{2m}{p}$.

We assume $p=11$ below, the other case is same. Let $f_{1}\leq\cdots\leq
f_{5}$ be the Galois conjugates of
$\frac{\dim(\mathcal{C})}{\dim(X_{2})^{2}}$. If $m=1$, then $f_{j}\geq j\cdot
11$ for all $1\leq j\leq 5$, consequently $a_{5}\geq 11^{5}\cdot 5!>11^{6}$,
it is impossible. If $m=2$, the $d$-number test shows $11^{2}|a_{1}$,
$11^{3}|a_{2}$ and $11^{4}\mid a_{4}$; also note $f_{j}\geq\frac{j\cdot
11}{2}$ ($1\leq j\leq 5$), by using a similar restriction of [27, Theorem
4.2], we see $27a_{1}\leq a_{2}$, $11a_{2}\leq a_{3}$ and $4a_{3}\leq
a_{4}=2\cdot 11^{5}$; the cyclotomic test fails for all possible cases,
however. If $m=3$, then $f_{j}\geq\frac{j\cdot 11}{3}$ ($1\leq j\leq 5$),
$18a_{1}\leq a_{2}$, $7a_{2}\leq a_{3}$ and $7a_{3}\leq 2a_{4}=6\cdot 11^{5}$,
again the cyclotomic test fails for all possible cases. Therefore,
$\text{rank}(\mathcal{C})\neq\frac{3(p-1)}{2}$. ∎

######  Lemma 4.10.

Let $\mathcal{C}$ be a modular fusion category of global dimension
$17\beta_{17}$, then $\text{rank}(\mathcal{C})\neq 24$.

###### Proof.

On the contrary, assume $\text{rank}(\mathcal{C})=24$. Let
$X_{1}=I,X_{2},X_{3}$ be the representatives of each Galois orbits. Therefore,
same as Lemma 4.9

$\displaystyle\dim(\mathcal{C})$
$\displaystyle=\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{17})^{+}/\mathbb{Q})}\frac{\dim(\mathcal{C})}{\sigma(\dim(\mathcal{C}))}+\sum_{i=2}^{3}\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{17})^{+}/\mathbb{Q})}\frac{\dim(\mathcal{C})}{\sigma(\dim(\mathcal{C}))}\sigma(\dim(X_{i})^{2})$
$\displaystyle>\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{17})^{+}/\mathbb{Q})}\frac{\dim(\mathcal{C})}{\sigma(\dim(\mathcal{C}))}+\frac{\dim(\mathcal{C})}{M}\sum_{i=2}^{3}\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{17})^{+}/\mathbb{Q})}\sigma(\dim(X_{i})^{2}),$

where $M$ is the maximal Galois conjugate of $\dim(\mathcal{C})$. Note that

$\displaystyle\sum_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{17})^{+}/\mathbb{Q})}\frac{\dim(\mathcal{C})}{\sigma(\dim(\mathcal{C}))}=12\beta_{17},$

we have $5\beta_{17}>\frac{\dim(\mathcal{C})}{M}\cdot 2\cdot 8\cdot 1.79$ by
the Siegel’s trace theorem [13], which then implies
$\sin^{2}\frac{8\pi}{17}>1$, it is a contradiction. ∎

######  Lemma 4.11.

Let $\mathcal{C}$ be a modular fusion category of global dimension
$p\beta_{p}$, where $p\geq 7$. If $\mathcal{C}$ is not simple, then $p=7$ and
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},5)^{\sigma}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},5)^{\sigma^{2}}_{\text{ad}}$,
where $\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{7})/\mathbb{Q})$ such that
$\sigma(\zeta_{7})=\zeta_{7}^{2}$.

###### Proof.

Indeed, if $\mathcal{C}\cong\mathcal{B}_{1}\boxtimes\mathcal{B}_{2}$ with
$\mathcal{B}_{1},\mathcal{B}_{2}$ being non-trivial modular fusion
subcategories, then it is easy to see that $p$ cannot divide both
$\dim(\mathcal{B}_{1})$ and $\dim(\mathcal{B}_{2})$, so $\mathcal{B}_{1}$ and
$\mathcal{B}_{2}$ are prime transitive modular fusion categories [25, Theorem
4.4]. Note that

$\displaystyle
N(\dim(\mathcal{C}))=p^{\frac{p+1}{2}}=N(\dim(\mathcal{B}_{1}))N(\dim(\mathcal{B}_{2}))=p^{p-3},$

that is, $p=7$, and
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},5)^{\sigma}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},5)^{\sigma^{2}}_{\text{ad}}$
as modular fusion category. ∎

For the modular fusion category
$\mathcal{C}(\mathfrak{sl}_{2},5)_{\text{ad}}$, let
$\mathcal{O}(\mathcal{C}(\mathfrak{sl}_{2},5)_{\text{ad}})=\\{I,X,Y\\}$, then
the fusion rules $\mathcal{C}(\mathfrak{sl}_{2},5)_{\text{ad}}$ are: $Y\otimes
Y=I\oplus X,X\otimes X=I\oplus X\oplus Y,X\otimes Y=X\oplus Y$. We denote
$d_{Y}:=\text{FPdim}(Y)=2\cos\frac{\pi}{7}$ and
$d_{X}:=\text{FPdim}(X)=d_{Y}^{2}-1$ below.

######  Proposition 4.12.

Let $\mathcal{C}$ be a modular fusion category of global dimension
$7\beta_{7}$, then as a modular fusion category
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},5)^{\sigma}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},5)^{\sigma^{2}}_{\text{ad}}$.

###### Proof.

By Proposition 4.8, we know $\text{rank}(\mathcal{C})=9$ and
$\text{FPdim}(\mathcal{C})=\frac{49}{\beta_{7}^{2}}$. Let $X_{1}=I$, and
$\mathcal{O}_{X_{1}}(\mathcal{C})=\\{X_{1},X_{2},X_{3}\\}$, that is,
$\sigma(\dim(\mathcal{C}))=\frac{\dim(\mathcal{C})}{\dim(X_{2})^{2}}$,
$\sigma^{2}(\dim(\mathcal{C}))=\frac{\dim(\mathcal{C})}{\dim(X_{3})^{2}}$,
$\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{7})/\mathbb{Q})$ such that
$\sigma(\zeta_{7})=\zeta_{7}^{2}$;
$\mathcal{O}_{X_{4}}(\mathcal{C})=\\{X_{4},X_{5},X_{6}\\}$, where
$\dim(X_{4})^{2}=1$,

$\displaystyle\sigma(\dim(\mathcal{C}))=\frac{\dim(\mathcal{C})}{\dim(X_{5})^{2}},\sigma^{2}(\dim(\mathcal{C}))=\frac{\dim(\mathcal{C})}{\dim(X_{6})^{2}};$

and $\mathcal{O}_{X_{7}}(\mathcal{C})=\\{X_{7},X_{8},X_{9}\\}$, where
$\dim(X_{7})^{2}=\frac{\dim(\mathcal{C})}{\text{FPdim}(\mathcal{C})}=\frac{\beta_{7}^{3}}{7}$,

$\displaystyle\frac{\dim(\mathcal{C})}{\dim(X_{8})^{2}}=\sigma\left(\frac{\dim(\mathcal{C})}{\dim(X_{7})^{2}}\right),\frac{\dim(\mathcal{C})}{\dim(X_{9})^{2}}=\sigma^{2}\left(\frac{\dim(\mathcal{C})}{\dim(X_{7})^{2}}\right).$

Then we obtain that

$\displaystyle\epsilon_{2}\dim(X_{2})=\epsilon_{5}\dim(X_{5})=\frac{1}{d_{Y}},\epsilon_{3}\dim(X_{3})=\epsilon_{6}\dim(X_{6})=\frac{1}{d_{X}},\dim(X_{4})=\epsilon_{4},$
$\displaystyle\dim(X_{7})=\frac{\epsilon_{7}}{d_{X}d_{Y}},\dim(X_{8})=\frac{\epsilon_{8}d_{Y}}{d_{X}},\dim(X_{9})=\frac{\epsilon_{9}d_{X}}{d_{Y}},~{}\epsilon_{j}\in\\{\pm
1\\},2\leq j\leq 9.$

Since $\hat{\sigma}(X_{j})=X_{j+1}$ and $\hat{\sigma}^{2}(X_{j})=X_{j+2}$
where $j\in\\{1,4,7\\}$, for $1\leq k\leq 9$,

$\displaystyle\sigma\left(\frac{S_{X_{j},X_{k}}}{\dim(X_{j})}\right)=\frac{S_{X_{j+1},X_{k}}}{\dim(X_{j+1})},\quad\sigma^{2}\left(\frac{S_{X_{j},X_{k}}}{\dim(X_{j})}\right)=\frac{S_{X_{j+2},X_{k}}}{\dim(X_{j+2})}.$

Meanwhile
$\text{FPdim}(\mathcal{C})=\frac{\dim(\mathcal{C})}{\dim(X_{7})^{2}}$, so
$\text{FPdim}(X_{j})=\frac{S_{X_{j},X_{7}}}{\dim(X_{7})}$ for all $1\leq j\leq
9$. Note that $\sigma(d_{X})=\frac{1}{d_{Y}}$ and
$\sigma(d_{Y})=-\frac{d_{X}}{d_{Y}}$, hence
$\text{FPdim}(X_{2})=\frac{\dim(X_{2})\sigma(\dim(X_{7}))}{\dim(X_{7})}=-\epsilon_{2}d_{Y}^{2}$
and
$\text{FPdim}(X_{3})=\frac{\dim(X_{3})\sigma^{2}(\dim(X_{7}))}{\dim(X_{7})}=-\epsilon_{3}d_{X}^{2}$,
we see $\epsilon_{2}=\epsilon_{3}=-1$.

Notice that the $S$-matrix of $\mathcal{C}$ are presented by $\epsilon_{j}$
($4\leq j\leq 9$), $d_{X},d_{Y}$, $\text{FPdim}(X_{4})$,
$\text{FPdim}(X_{7})$, $S_{X_{4},X_{4}}$ and their Galois conjugates. In
particular,

$\displaystyle\text{FPdim}(X_{5})=-\epsilon_{4}\epsilon_{5}d_{Y}^{2}\sigma(\text{FPdim}(X_{4})),\text{FPdim}(X_{6})=-\epsilon_{4}\epsilon_{6}d_{X}^{2}\sigma^{2}(\text{FPdim}(X_{4})),$
$\displaystyle\text{FPdim}(X_{8})=\epsilon_{7}\epsilon_{8}d_{Y}^{2}\sigma(\text{FPdim}(X_{7})),\text{FPdim}(X_{8})=\epsilon_{7}\epsilon_{9}d_{X}^{2}\sigma^{2}(\text{FPdim}(X_{7})).$

The Verlinde formula (2) implies $X_{2}\otimes X_{2}=I\oplus X_{3}\oplus A$,
where $A$ is an object with $\text{FPdim}(A)=2d_{X}$, so either $A$ is a
simple object or $A$ is a direct sum of two simple objects by [10, Corollary
3.3.16] and [5, Theorem 1.0.1], note that $A$ contains a simple object of
Frobenius-Perron dimension $d_{X}$ or $d_{Y}$ as a direct summand in the
latter case [5, Theorem 1.0.1]. We claim that $A$ is a direct sum of two non-
isomorphic simple objects of Frobenius-Perron dimension $d_{X}$. Indeed, if
$A$ is simple or $A=V\oplus W$ with $\text{FPdim}(V)=d_{Y}$ and
$\text{FPdim}(W)=2d_{X}-d_{Y}$, then a direct computation shows that
$\text{FPdim}(\mathcal{C})>\frac{49}{\beta_{7}^{2}}$; if $A=2M$ with
$\text{FPdim}(M)=d_{X}$, then $M\otimes X_{2}=2X_{2}\oplus N$, however
$\text{FPdim}(N)=d_{Y}(d_{X}-d_{Y})<1$, it is a contradiction.

Since the Frobenius-Perron dimensions of simple objects in the Galois orbits
of $X_{4}$ and $X_{7}$ are distinct, it is easy to show they are
$d_{X},d_{Y},d_{X}d_{Y}$, respectively. Assume $A=W_{1}\oplus W_{2}$ and let
$V_{1},V_{2}$ be simple objects of Frobenius-Perron dimension $d_{Y}$. Then
$V_{1}\otimes V_{2}$ and $W_{1}\otimes W_{2}$ must be simple, as $\mathcal{C}$
does not contain non-trivial invertible simple objects. Assume $V_{1}\otimes
V_{1}=I\oplus W_{1}$, if $V_{2}\otimes V_{2}=I\oplus W_{1}$, then
$2I\subseteq(V_{1}\otimes V_{2})\otimes(V_{1}\otimes V_{2})$, it is
impossible, so $V_{2}\otimes V_{2}=I\oplus W_{2}$; note that if $V_{1}\otimes
W_{1}=V_{1}\oplus W_{2}$, then $I\subseteq W_{2}\otimes(V_{1}\otimes
W_{1})\cong V_{1}\otimes(W_{1}\otimes W_{2})$, it is impossible. Therefore,
$V_{1}\otimes W_{1}=V_{1}\oplus W_{1}$ and $W_{1}\otimes W_{1}=I\oplus
V_{1}\oplus W_{1}$, so $\mathcal{C}$ contains a fusion subcategory that is
Grothendieck equivalent to $\mathcal{C}(\mathfrak{sl}_{2},5)_{\text{ad}}$.
Consequently,
$\mathcal{C}\cong\mathcal{C}(\mathfrak{sl}_{2},5)^{\sigma}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},5)^{\sigma^{2}}_{\text{ad}}$
as modular fusion category by Lemma 4.11. ∎

######  Proposition 4.13.

Let $\mathcal{C}$ be a modular fusion category of global dimension
$p\beta_{p}$, where $11\leq p\leq 23$. If $\text{rank}(\mathcal{C})=p-1$, then
$p=11$ and $\mathcal{C}$ is braided tensor equivalent to a Galois conjugate of
modular fusion category
$\mathcal{C}(\mathfrak{so}_{5},\frac{5}{2})_{\text{ad}}$.

###### Proof.

Let $\rho_{\mathcal{C}}$ be the associated modular representation of
$\mathcal{C}$. We know that $\rho_{\mathcal{C}}$ can’t be decomposed as direct
sum of sub-representations with disjoint $\mathfrak{t}$-spectrums [4, Lemma
3,18] and also that $\rho_{\mathcal{C}}$ can’t be decomposed as direct sum of
non-degenerate sub-representations of same type [23, Lemma 5.2.2]. Since
$\text{rank}(\mathcal{C})=p-1$ and $\text{ord}(t)=p$, by comparing the
dimensions of level $p$ irreducible representations of
$\text{SL}(2,\mathbb{Z})$, we obtain either $\rho_{\mathcal{C}}$ is
irreducible or $\rho_{\mathcal{C}}=\rho_{1}\oplus(\oplus_{m}\rho_{0})$, where
$\rho_{1}$ is an irreducible representation of rank $\frac{p+1}{2}$ and
$m=\frac{p-3}{2}$. However, [17, Proposition 3.22] states $m=1$ and then
$p=5$, which is impossible. Hence, $\rho_{\mathcal{C}}$ is irreducible.

As $\rho_{\mathcal{C}}$ is non-degenerate and $\text{rank}(\mathcal{C})\leq
23$, [9, Main Theorem 4] implies that $p=11,17,23$. Moreover, when $p=17,23$,
the $(p-1)$-dimensional non-degenerate representations are realized as modular
representations of modular fusion categories
$\mathcal{C}(\mathfrak{g}_{2},\frac{5}{3})$ and
$\mathcal{C}(\mathfrak{e}_{7},5)_{\text{ad}}$, respectively. However, neither
of the Galois conjugates of $\dim(\mathcal{C}(\mathfrak{g}_{2},\frac{5}{3}))$
equal to $17\beta_{17}$, and the norm of
$\dim(\mathcal{C}(\mathfrak{e}_{7},5)_{\text{ad}})$ is $23^{14}$, it is also
impossible. When $p=11$, the $10$-dimensional non-degenerate representations
can only be realized as modular representation of
$\mathcal{C}(\mathfrak{so}_{5},\frac{5}{2})_{\text{ad}}$. Therefore,
$\mathcal{C}$ is braided tensor equivalent to a Galois conjugate of
$\mathcal{C}(\mathfrak{so}_{5},\frac{5}{2})_{\text{ad}}$. ∎

######  Theorem 4.14.

Let $\mathcal{C}$ be a modular fusion category of global dimension
$p\beta_{p}$ where $p>3$ is a prime. If $\mathcal{C}_{\text{pt}}=\text{Vec}$,
then $\mathcal{C}$ is braided equivalent to one of the following modular
fusion categories

$\displaystyle\left\\{\begin{array}[]{ll}\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}},&\hbox{
if $p=5;$}\\\
\mathcal{C}(\mathfrak{sl}_{2},5)^{\tau}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},5)^{\tau^{2}}_{\text{ad}},&\hbox{if
$p=7;$}\\\
\mathcal{C}(\mathfrak{so}_{5},\frac{5}{2})_{\text{ad}}^{\nu},&\hbox{if
$p=11$.}\end{array}\right.$

where $\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{5})/\mathbb{Q})$ such that
$\sigma(\zeta_{5})=\zeta_{5}^{2}$,
$\tau\in\text{Gal}(\mathbb{Q}(\zeta_{7})/\mathbb{Q})$ such that
$\tau(\zeta_{7})=\zeta_{7}^{2}$ and
$\nu\in\text{Gal}(\mathbb{Q}(\zeta_{11})/\mathbb{Q})$ such that
$\nu(\zeta_{11})=\zeta_{11}^{9}$.

###### Proof.

We know $p\leq 23$ by Theorem 4.5. When $p=5$, this is the conclusion of
Proposition 3.10, and Proposition 4.8 shows $\text{rank}(\mathcal{C})=p-1$ or
$\frac{3(p-1)}{2}$ if $p>5$. If $\text{rank}(\mathcal{C})=\frac{3(p-1)}{2}$,
then there exists such a modular fusion category $\mathcal{C}$ only for $p=7$
by Lemma 4.9, Lemma 4.10 and Proposition 4.12; if
$\text{rank}(\mathcal{C})=p-1$, we have $p=11$ by Proposition 4.13. ∎

Recall that modular fusion categories of global dimension $4$ and $9$ are
either pointed, or braided equivalent to a Galois conjugate of Ising category
$\mathcal{C}(\mathfrak{sl}_{2},2)$ and
$\mathcal{C}(\mathfrak{so}_{5},\frac{3}{2})_{\text{ad}}$ [27], respectively.
Hence, combining with the conclusions of [27], Proposition 4.2, Theorem 4.3
and Theorem 4.14 together imply the following theorem:

######  Theorem 4.15.

Let $\mathcal{C}$ be a modular fusion category of global dimension $p^{2}$,
where $p\geq 5$ is a prime. If $\mathcal{C}$ contains a non-trivial fusion
subcategory, then either $\mathcal{C}$ is pointed, or $\mathcal{C}$ is braided
tensor equivalent to a Galois conjugate of one of the following modular fusion
categories

$\displaystyle\left\\{\begin{array}[]{ll}\left\\{\begin{array}[]{ll}\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}}\boxtimes\mathcal{C}(\mathbb{Z}_{5},\eta),&\\\
\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},3)^{\sigma}_{\text{ad}},&\end{array}\right.&\hbox{if
$p=5;$}\\\
\mathcal{C}(\mathfrak{sl}_{2},5)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},5)^{\tau}_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},5)^{\tau^{2}}_{\text{ad}},&\hbox{if
$p=7;$}\\\
\mathcal{C}(\mathfrak{sl}_{2},9)_{\text{ad}}\boxtimes\mathcal{C}(\mathfrak{so}_{5},\frac{5}{2})_{\text{ad}}^{\nu}&\hbox{if
$p=11$.}\end{array}\right.$

where $\sigma\in\text{Gal}(\mathbb{Q}(\zeta_{5})/\mathbb{Q})$ such that
$\sigma(\zeta_{5})=\zeta_{5}^{2}$,
$\tau\in\text{Gal}(\mathbb{Q}(\zeta_{7})/\mathbb{Q})$ such that
$\tau(\zeta_{7})=\zeta_{7}^{2}$ and
$\nu\in\text{Gal}(\mathbb{Q}(\zeta_{11})/\mathbb{Q})$ such that
$\nu(\zeta_{11})=\zeta_{11}^{9}$.

This completes the classification of non-simple modular fusion categories of
global dimension $p^{2}$. In addition, it is worth to note that
$\mathcal{C}(\mathfrak{so}_{5},\frac{3}{2})_{\text{ad}}$ and its Galois
conjugates are simple modular fusion categories of global dimension $9$.

######  Question 4.16.

Let $p>3$ be a prime. Is there a simple modular fusion category $\mathcal{C}$
of global dimension $p^{2}$?

Moreover, let $\mathcal{D}$ be a spherical fusion category of global dimension
$p$, then its Drinfeld center $\mathcal{Z}(\mathcal{D})$ is a modular fusion
category of global dimension $p^{2}$. Therefore, a negative answer to Question
4.16 will also result in a complete classification of spherical fusion
categories of prime global dimensions.

## Acknowledgements

The author is supported by the National Natural Science Foundation of China
(no.12101541), the Natural Science Foundation of Jiangsu Province
(no.BK20210785), and the Natural Science Foundation of Jiangsu Higher
Institutions of China (no.21KJB110006). The author is grateful to Y. Wang for
conversations on the representations of the modular group
$\text{SL}(2,\mathbb{Z})$.

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Zhiqiang Yu
Email<EMAIL_ADDRESS>
School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China