The Chevalley–Weil Formula for Orbifold Curves

The Chevalley–Weil Formula for Orbifold Curves††This paper is a contribution
to the Special Issue on Modular Forms and String Theory in honor of Noriko
Yui. The full collection is available at
http://www.emis.de/journals/SIGMA/modular-forms.html

Luca CANDELORI

L. Candelori

Department of Mathematics, University of Hawaii at Manoa, Honolulu, HI, USA
<EMAIL_ADDRESS>http://math.hawaii.edu/~candelori/

Received December 08, 2017, in final form July 02, 2018; Published online July
17, 2018

In the 1930s Chevalley and Weil gave a formula for decomposing the canonical
representation on the space of differential forms of the Galois group of a
ramified Galois cover of Riemann surfaces. In this article we prove an
analogous Chevalley–Weil formula for ramified Galois covers of orbifold
curves. We then specialize the formula to the case when the base orbifold
curve is the (reduced) modular orbifold. As an application of this latter
formula we decompose the canonical representations of modular curves of full,
prime level and of Fermat curves of arbitrary exponent.

orbifold curves; automorphisms; modular curves; Fermat curves

14H30; 14H37; 14H45

##  1 Introduction

Let $f\colon X\rightarrow Y$ be a ramified Galois cover of compact Riemann
surfaces of genus $g_{X}$ and $g_{Y}$, and let $G:=\mathrm{Aut}_{X}(Y)$ be the
Galois group of $f$. This is a finite group of automorphisms of $X$ that acts
by pull-back on the space of holomorphic differential 1-forms of $X$, giving a
$g_{X}$-dimensional complex representation

$\displaystyle\rho_{f}\colon\
G^{\rm{op}}\rightarrow\mathrm{GL}\big{(}H^{0}\big{(}X,\Omega^{1}_{X}\big{)}\big{)},$

the canonical representation of $G$. Given an irreducible representation
$\rho$ of $G$, Chevalley and Weil [6, 22] gave a formula for the multiplicity
of $\rho$ inside $\rho_{f}$ in terms of the genus of $Y$ and the ramification
data of $f$. This is called the Chevalley–Weil formula. A modern account of
this result in the more general setting of algebraic curves over an
algebraically closed field whose characteristic does not divide $|G|$ can be
found in [14]. The formula has subsequently been generalized by Nakajima [15]
to any coherent sheaf and any ramified cover of algebraic varieties over an
any algebraically closed field (the generalization of the Chevalley–Weil
formula for curves over any algebraically closed field was also treated in
[8]). The Chevalley–Weil formula can be used in conjunction with the character
table of $G$ to write down explicit matrices for $\rho_{f}$ without having to
compute a basis for $H^{0}\big{(}X,\Omega^{1}_{X}\big{)}$. As an application,
when $g_{X}\geq 3$ is small explicit equations for the canonical embedding
$X\rightarrow\mathbb{P}^{g_{X}-1}$ can be found using this model for
$\rho_{f}$ [19].

In this article we generalize the Chevalley–Weil formula to ramified Galois
covers of orbifold curves (also known as Deligne–Mumford curves or stacky
curves), under the mild assumption that the ramification locus is disjoint
from the locus of orbifold points in $Y$. We work over the complex numbers,
but the same exact arguments go through unchanged for an arbitrary
algebraically closed field whose characteristic does not divide $|G|$. The
proof is contained in Section 3. The structure of the proof follows the
excellent exposition in [14], with the necessary generalizations and extra
computations needed in the orbifold case. We have also simplified the
arguments when possible by referring to the literature.

Suppose now that the base orbifold curve $Y=X(1)$ is the compactification of
the orbifold quotient $\mathfrak{h}/\mathrm{PSL}_{2}(\mathbb{Z})$, the reduced
modular orbifold. The Galois covers of $X(1)$ are modular curves $X(\Gamma)$
corresponding to normal subgroups
$\Gamma\triangleleft\mathrm{PSL}_{2}(\mathbb{Z})$. The canonical
representations $\rho_{\Gamma}$ in this case can be viewed as representations
of $\mathrm{PSL}_{2}(\mathbb{Z})^{\rm{op}}$ factoring through the finite
quotient $G:=\mathrm{PSL}_{2}(\mathbb{Z})/\Gamma$. Our orbifold Chevalley–Weil
formula in this case applies and we are able to compute explicit matrices for
$\rho_{\Gamma}$ whenever all its factors are of dimension $\leq 6$, by
consulting the wealth of information available on the representation theory of
$\mathrm{PSL}_{2}(\mathbb{Z})$ (e.g., [11, 12, 20]). We do so in Section 4. In
Section 5 we specialize to the case of principal congruence subgroups
$\Gamma=\Gamma(p)$, for $p\geq 5$ a prime. In this case
$G\simeq\mathrm{PSL}_{2}(\mathbb{F}_{p})$ and the character tables of these
groups have been known since Frobenius and Schur. For each irreducible
representation $\rho$ appearing in the character table of
$\mathrm{PSL}_{2}(\mathbb{F}_{p})$, we find a formula for the multiplicity of
$\rho$ inside the canonical representation $\rho_{\Gamma(p)}$. We then apply
our computations to compute the canonical representation for the Klein
quartic, which can be uniformized by the principal congruence subgroup
$\Gamma(7)$. In Section 6, we present a uniformization of Fermat curves
$F_{N}$ of exponent $N$ by normal subgroups
$\Phi(N)\triangleleft\mathrm{PSL}_{2}(\mathbb{Z})$, which exhibits $F_{N}$ as
a Galois cover of $X(1)$ with Galois group isomorphic to
$(\mathbb{Z}/N\mathbb{Z})^{2}\rtimes S_{3}$. This is known to be the full
automorphism group of $F_{N}$ [21], and is thus interesting to compute the
decomposition of the canonical representation $\rho_{\Phi(N)}$. We do so using
our Chevalley–Weil formula in Theorems 6.2 and 6.4 below. As shown in [1], the
decomposition of $\rho_{\Phi(N)}$ into irreducible representations gives a
corresponding ‘group algebra’ decomposition of the Jacobian variety of
$F_{N}$. Decomposing the Jacobian variety of $F_{N}$ is an important problem
that has been previosuly considered by many in algebraic geometry and number
theory, including Noriko Yui [23].

##  2 Vector bundles over orbifold curves

In this section we recall a few basic facts about vector bundles on orbifold
curves. Good references for these facts are [7] and [16]. Alternatively, the
reader may consult the literature on locally free sheaves over Deligne–Mumford
curves or stacky curves [2]. In this article an orbifold curve $X$ is a
compact, connected, complex orbifold of dimension one with finitely many
orbifold points $(P_{1},\ldots,P_{n})$ with non-trivial cyclic stabilizers of
orders $(p_{1},\ldots,p_{n})$, respectively. We assume that $X$ is generically
a Riemann surface (thus a reduced orbifold). The genus of $X$, denoted by
$g_{X}$, is the genus of its underlying Riemann surface $\bar{X}$ (this
coincides with the coarse moduli space if $X$ is viewed as a Deligne–Mumford
curve). Let $\alpha_{1},\ldots,\alpha_{g_{X}},\beta_{1},\ldots,\beta_{g_{X}}$
be a set of standard generators for the fundamental group of $\bar{X}$. The
orbifold fundamental group of $X$ (with respect to some base-point $b$) can be
defined as

$\displaystyle\pi_{1}(X;b):=\big{\\{}\alpha_{1},\ldots,\alpha_{g_{X}},\beta_{1},\ldots,\beta_{g_{X}},\gamma_{P_{1}},\ldots,\gamma_{P_{n}}\colon$
$\displaystyle\hphantom{\pi_{1}(X;b):=\big{\\{}}{}\gamma_{P_{1}}^{p_{1}}=1,\ldots,\gamma_{P_{n}}^{p_{n}}=1,\gamma_{P_{1}}\cdots\gamma_{P_{n}}[\alpha_{1},\beta_{1}]\cdots[\alpha_{g_{X}},\beta_{g_{X}}]=1\big{\\}},$
(2.1)

where $\gamma_{P_{i}}$ is an oriented generator of the stabilizer of the
orbifold point $P_{i}$, $i=1,\ldots,n$. Let $\mathrm{Pic}(X)$ be the group
under $\otimes$ of all line bundles (i.e invertible sheaves) over $X$ up to
isomorphism, and let

$\displaystyle\deg\colon\
\mathrm{Pic}(X)\longrightarrow\frac{1}{m}\mathbb{Z},\qquad
m=\mathrm{lcm}(p_{1},\ldots,p_{n})$

be the usual ‘degree’ homomorphism, obtained by summing divisor
multiplicities, possibly rational numbers at the orbifold points [16, Section
1B]. For example, the sheaf $\Omega^{1}_{X}$ of holomorphic differential
1-forms on $X$ satisfies [7, Section 1], [16, Section 1A]

$\displaystyle\deg\Omega^{1}_{X}=2g_{X}-2+\sum_{i=1}^{n}\frac{p_{i}-1}{p_{i}}=2g_{X}-2+n-\sum_{i=1}^{n}\frac{1}{p_{i}}.$
(2.2)

Let $\mathcal{V}$ be a vector bundle over $X$ (i.e., a locally free sheaf of
finite rank). The degree of a vector bundle is the rational number
$\deg\det\mathcal{V}$. The fiber of $\mathcal{V}$ at each orbifold point
$P_{i}$ gives a $\mathrm{rk}(\mathcal{V})$-dimensional representation

$\displaystyle\mu(\mathcal{V},P_{i})\colon\
\langle\gamma_{P_{i}}\rangle\longrightarrow\mathrm{GL}_{\mathrm{rk}(\mathcal{V})}(\mathbb{C}),$

which is entirely determined by the linear transformation
$\mu(\mathcal{V},P_{i})(\gamma_{P_{i}})$. This is of finite order, hence
diagonalizable, with eigenvalues of the form $e^{\frac{2\pi
i}{p_{i}}\nu_{ij}}$, $\nu_{ij}\in\\{0,\ldots,p_{i}-1\\}$,
$j=1,\ldots,\mathrm{rk}(\mathcal{V})$.

######  Definition 2.1 ([7]).

The integers $\nu_{ij}\in\\{0,\ldots,p_{i}-1\\}$,
$j=1,\ldots,\mathrm{rk}(\mathcal{V})$ are called the isotropies of
$\mathcal{V}$ at $P_{i}$. The integer

$\displaystyle\iota(\mathcal{V},P_{i}):=\sum_{i=1}^{\mathrm{rk}(\mathcal{V})}\nu_{ij}\in\mathbb{Z}_{\geq
0}$

is called the isotropy trace of $\mathcal{V}$ at $P_{i}$.

The cohomology groups $H^{i}(X,\mathcal{V})$ of a vector bundle $\mathcal{V}$
over $X$ are defined as the sheaf cohomology over the site defined by $X$.
Denote by

$\displaystyle\chi(X,\mathcal{V}):=\dim H^{0}(X,\mathcal{V})-\dim
H^{1}(X,\mathcal{V})$

the Euler characteristic of the vector bundle $\mathcal{V}$ over $X$. We have:

######  Theorem 2.2 (Riemann–Roch theorem for orbifold curves).

Let $\mathcal{V}$ be a vector bundle over an orbifold curve $X$. Then

$\displaystyle\chi(X,\mathcal{V})=\mathrm{rk}(\mathcal{V})(1-g_{X})+\deg(\mathcal{V})-\left(\sum_{i=1}^{n}\frac{\iota(\mathcal{V},P_{i})}{p_{i}}\right).$

###### Proof.

See, e.g., [7, Theorem 1.5] for the case of line bundles. The general case
follows from the splitting principle. ∎

In addition, the Serre duality theorem for orbifold curves [16, Theorem 1.7]
gives a canonical isomorphism

$\displaystyle H^{1}(X,\mathcal{V})\simeq
H^{0}\big{(}X,\mathcal{V}^{*}\otimes\Omega^{1}_{X}\big{)}^{*},$ (2.3)

where
$\mathcal{V}^{*}:=\operatorname{Hom}_{\mathcal{O}_{X}}(\mathcal{V},\mathcal{O}_{X})$.

##  3 The Chevalley–Weil formula

Let $f\colon X\rightarrow Y$ be a degree $d$ ramified Galois cover of orbifold
curves of genus $g_{X}$, $g_{Y}$, respectively. By this we mean that $f$ is a
morphism that is generically a degree $d$ finite étale Galois cover, outside
finitely many ramification points $Q_{1},\ldots,Q_{r}\in Y$, with ramification
degrees $e_{1},\ldots,e_{r}$, respectively. We assume for simplicity that none
of the $Q_{i}$’s are orbifold points of $Y$.

Let

$\displaystyle G:=\mathrm{Aut}_{Y}(X)$

be the group of covering transformations of $f$. For each ramification point
$Q\in Y$, the stabilizer in $G$ of any point $R\stackrel{{\scriptstyle
f}}{{\longmapsto}}Q$ of $X$ is a cyclic subgroup $G_{R}\subseteq G$ of order
$e$, the ramification degree of $Q$. Choose generators $\gamma_{R}\in G_{R}$
for all such ‘local monodromy’ groups. Note that if
$R,R^{\prime}\stackrel{{\scriptstyle f}}{{\longmapsto}}Q$ are points of $X$
lying over $Q$, then $G_{R}$ and $G_{R^{\prime}}$ are conjugate to each other
and there exists $g\in G$ with

$\displaystyle g^{-1}\gamma_{R}g=\gamma_{R^{\prime}}.$ (3.1)

For each orbifold point $P_{j}\in Y$, $j=1,\ldots,n$, let
$\gamma_{P_{j}}\in\pi_{1}(Y\backslash\\{Q_{1},\ldots,Q_{r}\\};b)$ be an
oriented generator for the stabilizer of the point $P_{j}$, as in (2.1). We
will abuse notation and also denote by $\gamma_{P_{j}}$ the image via the
quotient map $\pi_{1}(Y\backslash\\{Q_{1},\ldots,Q_{r}\\};b)\rightarrow G$.

The sheaf $f_{*}\mathcal{O}_{X}$ is a vector bundle over $Y$ of rank $d$,
together with a right action of $G$ by pull-back of functions. This vector
bundle is generically a finite étale sheaf with right $G$-action, which on the
stalks is just the right regular representation of $G$ on itself. The group
$G$ also acts linearly on the line bundle $\Omega^{1}_{X}$ (and on
$f_{*}\Omega^{1}_{X}$) by pull-back

$\displaystyle g\omega:=g^{*}\omega,$

giving the canonical representation

$\displaystyle\rho_{f}\colon\
G^{\rm{op}}\longrightarrow\mathrm{GL}\big{(}H^{0}\big{(}X,\Omega^{1}_{X}\big{)}\big{)}\simeq\mathrm{GL}_{g_{X}}(\mathbb{C}),$
(3.2)

by taking global sections. For ease of notation we will drop the ‘op’ super-
script in what follows. Since $G$ is finite, the category of finite-
dimensional representations of $G$ is semi-simple. Let $\\{\rho_{i}\\}$ be the
set of irreducible representations of $G$, and let $d_{i}=d(\rho_{i})$ be the
multiplicity $d_{i}$ of each $\rho_{i}$ occurring in $\rho_{f}$. The
Chevalley–Weil formula for algebraic curves [6, 22] is a formula for each
$d_{i}$ in terms of global invariants of $f\colon X\rightarrow Y$ and
ramification data around each local monodromy $\gamma_{R}$. To generalize the
formula to orbifold curves, we first set some notation. For $g\in G$ an
element of exact order $N$, a representation $\rho$ of $G$, and
$k=0,\ldots,N-1$, let

$\displaystyle N_{k}(\rho;g):=\dim\ker\big{(}\rho(g)-e^{\frac{2\pi
i}{N}k}I\big{)}.$

Our goal is to prove the following:

######  Theorem 3.1 (Chevalley–Weil formula).

Let $\rho_{i}$ be an irreducible representation of $G$. For each ramification
point $Q_{j}\in Y$ choose some $R_{j}\stackrel{{\scriptstyle
f}}{{\mapsto}}Q_{j}$. Then the multiplicity $d_{i}=d(\rho_{i})$ of $\rho_{i}$
in $\rho_{f}$ is given by

$\displaystyle
d_{i}=\epsilon+\dim\rho_{i}\left(g_{Y}-1+n-\sum_{j=1}^{n}\frac{1}{p_{j}}\right)+\sum_{j=1}^{r}\sum_{k=1}^{e_{j}-1}N_{k}(\rho_{i};\gamma_{R_{j}})\left(1-\frac{k}{e_{j}}\right)$
$\displaystyle\hphantom{d_{i}=}{}-\sum_{j=1}^{n}\sum_{k=1}^{p_{j}-1}N_{k}\Big{(}e^{\frac{-2\pi
i}{p_{j}}}\rho_{i};\gamma_{P_{j}}\Big{)}\frac{k}{p_{j}},$

where $\epsilon=1$ if $\rho_{i}=1$ and $\epsilon=0$ otherwise.

######  Remark 3.2.

Note that by (3.1) the integers $N_{\rho_{i}}(\gamma_{R_{j}};k)$ do not depend
on the choice of point $R_{j}\stackrel{{\scriptstyle f}}{{\mapsto}}Q_{j}$.

To prove Theorem 3.1 we proceed by a series of reductions, as in the case of
algebraic curves [14]. First, since $f$ is generically étale we have an exact
sequence

$\displaystyle 0\rightarrow
f^{*}\Omega^{1}_{Y}\rightarrow\Omega^{1}_{X}\rightarrow\Omega^{1}_{X/Y}\rightarrow
0.$

Applying $f_{*}$ to this exact sequence and the projection formula on the
first term we get

$\displaystyle 0\rightarrow\Omega^{1}_{Y}\otimes
f_{*}\mathcal{O}_{X}\rightarrow f_{*}\Omega^{1}_{X}\rightarrow
f_{*}\Omega^{1}_{X/Y}\rightarrow 0.$ (3.3)

This is an exact sequence of $\mathcal{O}_{Y}$-modules with right
$\mathcal{O}_{Y}$-linear $G$-action. For any irreducible representation
$\rho_{i}$ of $G$ we can take the $\rho_{i}$-isotypical component of this
exact sequence and obtain (note that the action of $G$ on $Y$ is trivial):

$\displaystyle 0\rightarrow\Omega^{1}_{Y}\otimes
f_{*}\mathcal{O}_{X}^{\rho_{i}}\rightarrow\big{(}f_{*}\Omega^{1}_{X}\big{)}^{\rho_{i}}\rightarrow\big{(}f_{*}\Omega_{{}_{X/Y}}^{1}\big{)}^{\rho_{i}}\rightarrow
0.$

To prove Theorem 3.1 we have to compute

$\displaystyle\dim
H^{0}\big{(}Y,\big{(}f_{*}\Omega^{1}_{X}\big{)}^{\rho_{i}}\big{)}=\dim
H^{0}\big{(}X,\Omega^{1}_{X}\big{)}^{\rho_{i}}=d_{i}\cdot\dim\rho_{i}.$

To do so we compute the Euler characteristic:

$\displaystyle\chi\big{(}Y,\big{(}f_{*}\Omega^{1}_{X}\big{)}^{\rho_{i}}\big{)}=\dim
H^{0}\big{(}X,\Omega^{1}_{X}\big{)}^{\rho_{i}}-\dim
H^{1}\big{(}X,\Omega^{1}_{X}\big{)}^{\rho_{i}}.$

######  Proposition 3.3.

The action of $G$ on $H^{1}\big{(}X,\Omega^{1}_{X}\big{)}$ is trivial.
Therefore

$\displaystyle\dim
H^{1}\big{(}X,\Omega^{1}_{X}\big{)}^{\rho_{i}}=\epsilon=\begin{cases}1&\text{if
}\rho_{i}=1,\\\ 0&\text{if }\rho_{i}\neq 1,\end{cases}$

for any irreducible representation $\rho_{i}$ of $G$.

###### Proof.

By Serre duality (2.3) there is a canonical isomorphism
$H^{1}\big{(}X,\Omega^{1}_{X}\big{)}\simeq H^{0}(X,\mathcal{O}_{X})^{*}$,
commuting with the linear $G$-action on both sides. Since
$H^{0}(X,\mathcal{O}_{X})\simeq\mathbb{C}$ consists of constant functions, the
action of $G$ on this vector space is trivial. Therefore its contragradient
action on $H^{1}\big{(}X,\Omega^{1}_{X}\big{)}$ is trivial as well. ∎

By Proposition 3.3, to compute $d_{i}$ it suffices to compute
$\chi\big{(}Y,\big{(}f_{*}\Omega^{1}_{X}\big{)}^{\rho_{i}}\big{)}$. Since
Euler characteristics are additive with respect to exact sequences, we must
have

$\displaystyle\chi\big{(}Y,\big{(}f_{*}\Omega^{1}_{X}\big{)}^{\rho_{i}}\big{)}=\chi\big{(}Y,\big{(}f_{*}\Omega_{{}_{X/Y}}^{1}\big{)}^{\rho_{i}}\big{)}+\chi\big{(}Y,\Omega^{1}_{Y}\otimes
f_{*}\mathcal{O}_{X}^{\rho_{i}}\big{)},$

by applying $\chi(Y,-)$ to (3.3). We compute each summand in Propositions 3.4
through 3.7 below.

######  Proposition 3.4.

Let $\rho_{i}$ be an irreducible representation of $G$. For each ramification
point $Q_{j}\in Y$ choose some $R_{j}\stackrel{{\scriptstyle
f}}{{\rightarrow}}Q_{j}$. Then

$\displaystyle\chi\big{(}Y,\big{(}f_{*}\Omega_{X/Y}^{1}\big{)}^{\rho_{i}}\big{)}=\dim\rho_{i}\sum_{j=1}^{r}\mathrm{rk}(\rho_{i}(\gamma_{R_{j}})-I).$

###### Proof.

The proof given in [14] for algebraic curves goes through unchanged, since the
question is localized at the ramification locus (and disjoint from the
orbifold locus). ∎

It remains to compute the Euler characteristic
$\chi\big{(}Y,\Omega^{1}_{Y}\otimes f_{*}\mathcal{O}_{X}^{\rho_{i}}\big{)}$.
Let $\mathcal{V}=\Omega^{1}_{Y}\otimes f_{*}\mathcal{O}_{X}^{\rho_{i}}$. By
Theorem 2.2,

$\displaystyle\chi(Y,\mathcal{V})=\mathrm{rk}(\mathcal{V})(1-g_{Y})+\deg\mathcal{V}-\left(\sum_{i=1}^{n}\frac{\iota(\mathcal{V},P_{i})}{p_{i}}\right).$

We compute each term in turn.

######  Proposition 3.5.

$\displaystyle\mathrm{rk}\mathcal{V}=(\dim\rho_{i})^{2}.$

###### Proof.

The vector bundle $f_{*}\mathcal{O}_{X}$ is a finite étale sheaf of rank $d$
away from the ramification points, and the same holds for
$f_{*}\mathcal{O}_{X}^{\rho_{i}}$. At a point $P\in Y$,
$P\notin\\{Q_{1},\ldots,Q_{r}\\}$, the fiber of $f_{*}\mathcal{O}_{X}$ is the
right regular $G$-representation. As is well-known, the representation
$\rho_{i}$ occurs inside the regular representation $\dim\rho_{i}$ times.
Therefore the fiber at $P$ of $f_{*}\mathcal{O}_{X}^{\rho_{i}}$ has dimension
$(\dim\rho_{i})^{2}$ and that is also the rank of
$f_{*}\mathcal{O}_{X}^{\rho_{i}}$ as a vector bundle over $Y$. Since
$\Omega^{1}_{Y}$ has rank one, the rank of $\mathcal{V}$ is also of rank
$(\dim\rho_{i})^{2}$. ∎

To compute the degree of $\mathcal{V}$, note that

$\displaystyle\deg(\mathcal{V})=(\dim\rho_{i})^{2}\deg\big{(}\Omega^{1}_{Y}\big{)}+\deg\big{(}f_{*}\mathcal{O}_{X}^{\rho_{i}}\big{)}$
$\displaystyle\hphantom{\deg(\mathcal{V})}{}=(\dim\rho_{i})^{2}\left(2g_{Y}-2+n-\sum_{j=1}^{n}\frac{1}{p_{j}}\right)+\deg\big{(}f_{*}\mathcal{O}_{X}^{\rho_{i}}\big{)}$

by (2.2). It remains to compute $\deg(f_{*}\mathcal{O}_{X}^{\rho_{i}})$, as
follows.

######  Proposition 3.6.

$\displaystyle\deg\big{(}f_{*}\mathcal{O}_{X}^{\rho_{i}}\big{)}=-\dim\rho_{i}\sum_{j=1}^{r}\sum_{k=1}^{e_{j}-1}N_{k}(\rho_{i};\gamma_{R_{j}})\frac{k}{e_{j}}.$

###### Proof.

The vector bundle $f_{*}\mathcal{O}_{X}^{\rho_{i}}$ is a direct sum of
$\dim\rho_{i}$-copies of the parabolic bundle canonically associated to the
unitary representation $\rho_{i}$, viewed as a representation of
$\pi_{1}(Y\backslash\\{Q_{1},\ldots$, $Q_{r}\\};b)$ factoring through the
finite quotient $G$ [13]. The parabolic structure is supported at the
ramification points $Q_{1},\ldots,Q_{r}$ and for each $j$ it is determined by
the local monodromy $\gamma_{R_{j}}$, as follows. Since $\rho_{i}(R_{j})$ is a
matrix of order $e_{j}$, its eigenvalues are of the form $e^{2\pi
ik_{w}/e_{j}}$, $k_{w}\in\\{0,\ldots,e_{j}-1\\}$. The parabolic structure at
$Q_{j}$ is given by the decreasing filtration of eigenspaces of
$\gamma_{R_{j}}$, the weights are the rational numbers $k_{w}/e_{j}$, $0\leq
k_{w}/e_{j}<1$, and each weight $k_{w}$ has multiplicity
$\dim\rho_{i}N_{k_{w}}(\rho_{i};\gamma_{R_{j}})$. The formula for the degree
of $f_{*}\mathcal{O}_{X}^{\rho_{i}}$ then just follows from the well-known
formula for the degree of a parabolic bundle [13, Corollary 1.10]. ∎

Finally we compute the isotropy term in $\chi(Y,\mathcal{V})$:

######  Proposition 3.7.

$\displaystyle\left(\sum_{i=1}^{n}\frac{\iota(\mathcal{V},P_{i})}{p_{i}}\right)=\sum_{j=1}^{n}\sum_{k=1}^{p_{j}-1}N_{k}\big{(}e^{\frac{-2\pi
i}{p_{j}}}\rho_{i};\gamma_{P_{j}}\big{)}\frac{k}{p_{j}}.$

###### Proof.

Note that the action of $\gamma_{P_{j}}$ on the fiber of
$\mathcal{V}=\Omega^{1}_{Y}\otimes f_{*}\mathcal{O}_{X}^{\rho_{i}}$ over the
orbifold point $P_{j}$ is given by $e^{-2\pi i/p_{j}}\rho_{i}$. The identity
then follows from Definition 2.1. ∎

Theorem 3.1 now follows by putting together Propositions 3.3, 3.4, 3.5, 3.6
and 3.7.

##  4 Applications to modular curves

We now apply Theorem 3.1 to the special case when the base orbifold $Y$ is the
compactification $X(1)$ of the orbifold quotient
$\mathrm{PSL}_{2}(\mathbb{Z})\backslash\mathfrak{h}$ obtained by adding the
cusp $\infty$. The genus of this orbifold curve is zero,
$\pi_{1}(X(1)-\infty;b)\simeq\mathrm{PSL}_{2}(\mathbb{Z})$ and $X(1)$ has
$n=2$ orbifold points $P_{1}=[i]$, $P_{2}=[e^{2\pi i/3}]$ of orders $p_{1}=2$,
$p_{2}=3$, with oriented stabilizers generated by

$\displaystyle\gamma_{P_{1}}:=S=\left(\begin{matrix}0&-1\\\
1&0\end{matrix}\right),\qquad\gamma_{P_{2}}:=R^{-1}=\left(\begin{matrix}1&1\\\
-1&0\end{matrix}\right).$

Any finite-index normal subgroup
$\Gamma\triangleleft\mathrm{PSL}_{2}(\mathbb{Z})$ gives a Galois cover

$\displaystyle f(\Gamma)\colon\ X(\Gamma)\rightarrow X(1),$

where $X(\Gamma)$ is the compactification of the quotient
$\Gamma\backslash\mathfrak{h}$ obtained by adding finitely many cusps. The map
$f(\Gamma)$ is of degree $d=[\Gamma:\mathrm{PSL}_{2}(\mathbb{Z})]$, and it is
ramified only above the cusp $\infty$. The Galois group of the cover is
therefore $G=\mathrm{PSL}_{2}(\mathbb{Z})/\Gamma$. To compute the ramification
data, we may choose the cusp $\infty$ of $X(\Gamma)$, which lies above the
cusp $\infty$ of $X(1)$. The local monodromy at $\infty$ is the image of

$\displaystyle\gamma_{\infty}:=T=\left(\begin{matrix}1&1\\\
0&1\end{matrix}\right)$

under the quotient map $\mathrm{PSL}_{2}(\mathbb{Z})\rightarrow G$. The
ramification degree $e$ of $f(\Gamma)$ over $\infty$ is just the order of the
image of $T$ in $G$ (the ramification level of
$G\triangleleft\mathrm{PSL}_{2}(\mathbb{Z})$). There is a canonical
isomorphism

$\displaystyle S_{2}(\Gamma)$ $\displaystyle\longrightarrow
H^{0}\big{(}X(\Gamma),\Omega^{1}_{X(\Gamma)}\big{)},$ $\displaystyle f$
$\displaystyle\longmapsto fd\tau,$

so that the canonical representation $\rho_{\Gamma}:=\rho_{f(\Gamma)}$ (3.2)
goes over to the representation
$\mathrm{PSL}_{2}(\mathbb{Z})\rightarrow\mathrm{GL}(S_{2}(\Gamma))$ given by

$\displaystyle\rho_{\Gamma}(\gamma)f=f|_{2}\gamma=f\left(\frac{a\tau+b}{c\tau+d}\right)(c\tau+d)^{-2},\qquad\gamma=\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in\mathrm{PSL}_{2}(\mathbb{Z}),$

factoring through the finite group $G$. Note that if
$\rho_{\Gamma}(\gamma)f=f$ for all $\gamma\in\mathrm{PSL}_{2}(\mathbb{Z})$
then $f\in S_{2}(\mathrm{PSL}_{2}(\mathbb{Z}))=0$, therefore the trivial
representation can never occur inside $\rho_{\Gamma}$. For the remaining non-
trivial representations of $G$ Theorem 3.1 simplifies to

######  Theorem 4.1 (Chevalley–Weil formula for
$\Gamma\triangleleft\mathrm{PSL}_{2}(\mathbb{Z})$).

Let $\Gamma\triangleleft\mathrm{PSL}_{2}(\mathbb{Z})$ be a finite-index normal
subgroup, and let $G:=\mathrm{PSL}_{2}(\mathbb{Z})/\Gamma$. Let $\rho\neq 1$
be an irreducible representation of $G$. Then the multiplicity of $\rho$ in
$\rho_{\Gamma}$ is given by

$\displaystyle
d=-\frac{5}{12}\dim\rho+\sum_{k=1}^{e-1}N_{k}(\rho;T)\left(1-\frac{k}{e}\right)-\frac{\operatorname{Tr}(\rho(S))}{4}+\frac{\zeta^{2}_{3}\operatorname{Tr}\big{(}\rho\big{(}R^{-1}\big{)}\big{)}}{3(1-\zeta_{3}^{2})}+\frac{\zeta_{3}\operatorname{Tr}(\rho(R))}{3(1-\zeta_{3})},$

where $\zeta_{3}:=e^{2\pi i/3}$.

###### Proof.

Using the notation of Theorem 3.1, note first that $\epsilon=0$ always since
we are assuming $\rho\neq 1$. We also have that $g_{X(1)}=0$, $n=2$,
$p_{1}=2$, $p_{2}=3$, which gives a contribution of $\dim\rho/6$ in Theorem
4.1. As stated above, there is only one ramification point $\infty\in X(1)$ of
index $e$ and with local monodromy given by the image of $T$ in $G$, which
gives the second term. As for the remaining terms, since the order of
$\rho(S)$ and $\rho(R^{-1})$ is low it is more convenient computationally to
express the eigenvalue multiplicities in terms of traces. In particular, note
that

$\displaystyle\operatorname{Tr}(\rho(S))=N_{0}(\rho(S))-N_{1}(\rho(S))=N_{1}(-\rho(S))-N_{0}(-\rho(S))=2N_{1}(-\rho(S))-\dim\rho$

and similarly

$\displaystyle\frac{N_{1}\big{(}e^{-2\pi
i/3}\rho;R^{-1}\big{)}}{3}+2\cdot\frac{N_{2}\big{(}e^{-2\pi
i/3}\rho;R^{-1}\big{)}}{3}=\frac{\dim\rho}{3}-\frac{\zeta^{2}_{3}\operatorname{Tr}\big{(}\rho\big{(}R^{-1}\big{)}\big{)}}{3\big{(}1-\zeta_{3}^{2}\big{)}}-\frac{\zeta_{3}\operatorname{Tr}(\rho(R))}{3(1-\zeta_{3})},$

yielding the simplified formula. ∎

The formula of Theorem 4.1 severely restricts which irreducible
representations of $\mathrm{PSL}_{2}(\mathbb{Z})$ may occur inside a canonical
representation. For example of the six characters of
$\mathrm{PSL}_{2}(\mathbb{Z})$ only one may occur, namely the character $\chi$
defined by $\chi(T)=e^{2\pi i/6}$. This character must therefore be the
canonical representation $\rho_{\Gamma}$ corresponding to any genus one normal
subgroup $\Gamma\triangleleft\mathrm{PSL}_{2}(\mathbb{Z})$ (there are
infinitely many of them).

Of the 27 two-dimensional irreducible representations of
$\mathrm{PSL}_{2}(\mathbb{Z})$ which factor through a finite group (these have
been classified in [12]) only four may appear in a canonical representation,
namely the ones with $\rho(T)$ equivalent to

$\displaystyle\left(\begin{matrix}\zeta_{12}&0\\\
0&\zeta_{12}^{5}\end{matrix}\right),\qquad\left(\begin{matrix}\zeta_{8}&0\\\
0&\zeta_{8}^{3}\end{matrix}\right),\qquad\left(\begin{matrix}\zeta_{20}&0\\\
0&\zeta^{9}_{20}\end{matrix}\right),\qquad\left(\begin{matrix}\zeta^{3}_{20}&0\\\
0&\zeta^{7}_{20}\end{matrix}\right),$ (4.1)

where $\zeta_{n}:=e^{2\pi i/n}$. For any one of the four representations
$\rho$ above, we can use GAP and a list of normal subgroups of
$\mathrm{PSL}_{2}(\mathbb{Z})$ to identify the normal subgroup $\Gamma$ of
smallest index such that $\rho$ is a factor of $\rho_{\Gamma}$. The first
representation appears as a factor inside the canonical representation of the
unique genus three subgroup of index 48, the second one is the canonical
representation corresponding to the unique genus two subgroup (also of index
48), the third one appears as a factor inside the canonical representation of
the unique genus 15 subgroup of index 240 and the last representation appears
as a factor inside the canonical representation of the unique genus 55
subgroup of index 720.

A similar analysis could in principle be applied to higher-dimensional
irreducible representations of $\mathrm{PSL}_{2}(\mathbb{Z})$ factoring
through a finite group. If a given irreducible representation $\rho$ satisfies
$d(\rho)>0$ in Theorem 4.1, what is the minimal index of the normal subgroup
$\Gamma\triangleleft\mathrm{PSL}_{2}(\mathbb{Z})$ such that $\rho$ is a factor
of $\rho_{\Gamma}$?

######  Remark 4.2.

Let $\\{f_{1},\ldots,f_{g}\\}$ be a basis for $S_{2}(\Gamma)$,
$\Gamma\triangleleft\mathrm{PSL}_{2}(\mathbb{Z})$ of finite-index and genus
$g\geq 1$. Then $F:=(f_{1},\ldots,f_{g})$ is a (non-zero)
$\rho_{\Gamma}$-valued cusp form of weight two [5]. Since $\rho_{\Gamma}$ is
unitarizable, this means that its minimal weight should be equal to two [3].

######  Example 4.3.

Let $\Gamma(8)\triangleleft\mathrm{SL}_{2}(\mathbb{Z})$ be the principal
congruence subgroup of level 8, and let $\Gamma$ be its image inside
$\mathrm{PSL}_{2}(\mathbb{Z})$. The quotient
$G=\mathrm{PSL}_{2}(\mathbb{Z})/\Gamma$ is a finite group of order 192,
isomorphic to $\mathrm{SL}_{2}(\mathbb{Z}/8\mathbb{Z})$. Its character table
is easily computable using GAP, and for each irreducible character we may
compute its multiplicity inside $\rho_{\Gamma}$ using Theorem 4.1. We obtain
the decomposition

$\displaystyle\rho_{\Gamma}\simeq\rho_{1}\oplus\rho_{2},$

where $\rho_{1}$ is irreducible of dimension two and $\rho_{2}$ is irreducible
of dimension three. According to [20, Proposition 2.5], such irreducible
representations of $\mathrm{PSL}_{2}(\mathbb{Z})$ are entirely determined by
their $\rho_{i}(T)$-eigenvalues. These can easily be computed again from the
character table of $G$. Using the formulas of [20, Proposition 2.5] we obtain
the following model for $\rho_{\Gamma}$:

$\displaystyle\rho_{\Gamma}(T)=\left(\begin{matrix}\zeta_{8}&\zeta_{8}&0&0&0\\\
0&\zeta_{8}^{3}&0&0&0\\\ 0&0&\zeta_{8}&i-1&i\\\ 0&0&0&i&i\\\
0&0&0&0&\zeta^{5}_{8}\end{matrix}\right),\qquad\rho_{\Gamma}(S)=\left(\begin{matrix}0&\zeta^{5}_{8}&0&0&0\\\
\zeta_{8}^{3}&0&0&0&0\\\ 0&0&0&0&1\\\ 0&0&0&-1&0\\\
0&0&1&0&0\end{matrix}\right),$

where $\zeta_{8}:=e^{2\pi i/8}$. Note in particular that $\rho_{1}$ is
equivalent to the second representation in (4.1).

##  5 The canonical representation of modular curves
$\boldsymbol{X(\Gamma(p))}$

We now apply Theorem 4.1 to a well-known family of normal subgroups of
$\mathrm{PSL}_{2}(\mathbb{Z})$. Let $p$ be a prime and let
$\Gamma(p)\triangleleft\mathrm{SL}_{2}(\mathbb{Z})$ be the principal
congruence subgroup of level $p$. For $p\geq 3$ we have and isomorphism
$G\simeq\mathrm{PSL}_{2}(\mathbb{F}_{p})$. The map
$X(\Gamma(p))\longrightarrow X(1)$ is a Galois cover of degree
$|G|=\frac{(p-1)p(p+1)}{2}$, ramified above $\infty\in X(1)$. The local
monodromy at $\infty\in X(\Gamma(p))$ is generated by $T$, whose image in $G$
has order $p$. Therefore the ramification degree is $e=p$.

By Theorem 4.1 we can use the character table of
$\mathrm{PSL}_{2}(\mathbb{F}_{p})$ to decompose the representation

$\displaystyle\rho_{\Gamma(p)}\colon\
\mathrm{PSL}_{2}(\mathbb{Z})\longrightarrow\mathrm{GL}(S_{2}(\Gamma(p)))$

into irreducible representations. Note that $S_{2}(\Gamma(3))=0$, so assume
$p\geq 5$. In this case the images of $R$ and $R^{-1}$ inside $G$ are
conjugate, therefore
$\operatorname{Tr}(\rho(R))=\operatorname{Tr}(\rho(R^{-1}))$ for any
representation $\rho$ of $G$. The formula of Theorem 4.1 then simplifies to

$\displaystyle
d=-\frac{5}{12}\dim\rho+\sum_{k=1}^{p-1}N_{k}(\rho;T)\left(1-\frac{k}{p}\right)-\frac{\operatorname{Tr}(\rho(S))}{4}-\frac{\operatorname{Tr}(\rho(R))}{3}.$
(5.1)

The character table of $\mathrm{PSL}_{2}(\mathbb{F}_{p})$, $p\geq 5$ was
originally computed by Frobenius and Schur (a modern account can be found in
[9, Section 8]) and it depends on whether $p\equiv 1,3\,(4)$. We therefore
break down the computation into these two cases. For ease of notation, we let
$u_{p}$ be a generator for $\mathbb{F}_{p}^{\times}$ and we let $u_{p^{2}}$ be
a generator for $\mathbb{F}_{p^{2}}^{\times}$. We also let $\varphi\in G$ be
the linear transformation of $\mathbb{F}_{p^{2}}$ sending $x\mapsto
u_{p^{2}}^{p-1}x$.

###  5.1 Case $\boldsymbol{p\equiv 1\,(4)}$

In this case

$\displaystyle S\sim\left(\begin{matrix}u_{p}^{\frac{p-1}{4}}&0\\\
0&u_{p}^{-\frac{p-1}{4}}\end{matrix}\right),\qquad
R\sim\begin{cases}\left(\begin{matrix}u_{p}^{\frac{p-1}{3}}&0\\\
0&u_{p}^{-\frac{p-1}{3}}\end{matrix}\right)&\text{if }p\equiv 1\,(3),\\\ \\\
\varphi^{\frac{p+1}{3}}&\text{if }p\equiv 2\,(3),\end{cases}$

and from the character table of [9, Theorem 8.9] we obtain the following
table, displaying the traces of each irreducible character of $G$ evaluated at
the matrices $I_{2}$, $S$, $R$, $T$ and $\left(\begin{smallmatrix}1&0\\\
u_{p}&1\end{smallmatrix}\right)$:

$\displaystyle\begin{array}[]{@{}c|c|c|c| c|c|c}&\dim&S&R\;\text{ if }p\equiv
1\,(3)&R\;\text{ if }p\equiv 2\,(3)&T&\left(\begin{smallmatrix}1&0\\\
u_{p}&1\end{smallmatrix}\right)\\\ \hline\cr\lambda&p&1&1&-1&0&0\\\
\\{\mu_{s}\\}_{1\leq
s\leq\frac{p-5}{4}}&p+1&2\cdot(-1)^{s}&\epsilon_{3}(s)&0&1&1\\\
\\{\theta_{t}\\}_{1\leq t\leq\frac{p-1}{4}}&p-1&0&0&-\epsilon_{3}(t)&-1&-1\\\
\chi_{1}&\frac{p+1}{2}&(-1)^{\frac{p-1}{4}}&1&0&\frac{1+\sqrt{p}}{2}&\frac{1-\sqrt{p}}{2}\\\
\chi_{2}&\frac{p+1}{2}&(-1)^{\frac{p-1}{4}}&1&0&\frac{1-\sqrt{p}}{2}&\frac{1+\sqrt{p}}{2}\\\
\end{array}$ (5.8)

where

$\displaystyle\epsilon_{3}(n)=\begin{cases}\hphantom{-}2&\text{if }n\equiv
0\,(3),\\\ -1&\text{if }n\equiv 1,2\,(3),\end{cases}$

and the irreducible characters $\lambda$, $\mu_{s}$, $\theta_{t}$, $\chi_{1}$,
$\chi_{2}$ are labelled as in [9, Theorem 8.9]. The table contains all the
data required to compute formula (5.1), including the ramification data at
$\infty$, which is derived below in Proposition 5.1. In order to state this
computation, for any odd prime $p$ and any integer $a$ such that $\gcd(a,p)=1$
let

$\displaystyle\left(\frac{a}{p}\right)_{L}=\begin{cases}\hphantom{-}1&\text{if
$a$ is a square mod $p$},\\\ -1&\text{if $a$ is a not a square mod
$p$}\end{cases}$

be the Legendre symbol.

######  Proposition 5.1.

For each one of the irreducible representations appearing in Table 5.8, we
have

$\displaystyle N_{k}(\lambda;T)=1,\qquad N_{k}(\mu_{s};T)=1,\qquad
N_{k}(\theta_{t};T)=1,$ $\displaystyle
N_{k}(\chi_{1};T)=\begin{cases}1&\text{if
}\left(\dfrac{k}{p}\right)_{L}=1,\vspace{1mm}\\\ 0&\text{if
}\left(\dfrac{k}{p}\right)_{L}=-1,\end{cases}\qquad
N_{k}(\chi_{2};T)=\begin{cases}0&\text{if
}\left(\dfrac{k}{p}\right)_{L}=1,\vspace{1mm}\\\ 1&\text{if
}\left(\dfrac{k}{p}\right)_{L}=-1,\end{cases}$

where $k=1,\ldots,p-1$.

###### Proof.

For any representation $\rho$ of $G$, let $P(\rho;T)$ be the characteristic
polynomial of $\rho(T)$. Since $\rho(T)^{p}=I_{\dim\rho}$, we know that the
roots of $P(\rho;T)$ are necessarily $p$-th roots of unity. The polynomial
itself can be computed from Table 5.8 by first computing the traces of
$\rho(T^{k})$, $k=0,\ldots,p-1$ and then compute from these traces the
coefficients of $P(\rho;T)$ using Newton’s identities. Now the subgroup
$\big{\\{}T^{k}\big{\\}}$ breaks up into two conjugacy classes:

$\displaystyle\big{\\{}T^{k}\big{\\}}\cap[T]=\big{\\{}\left(\begin{smallmatrix}1&r\\\
0&1\end{smallmatrix}\right)\colon r\text{ is a square in
}\mathbb{F}_{p}\big{\\}},$
$\displaystyle\big{\\{}T^{k}\big{\\}}\cap\big{[}\left(\begin{smallmatrix}1&0\\\
u_{p}&1\end{smallmatrix}\right)\big{]}=\big{\\{}\left(\begin{smallmatrix}1&n\\\
0&1\end{smallmatrix}\right)\colon n\text{ is not a square in
}\mathbb{F}_{p}\big{\\}},$

therefore the coefficients of $P(\rho;T)$ are polynomials in the last two
columns of Table 5.8. For $\rho=\lambda,\mu_{s}$ and $\theta_{t}$, it follows
that $P(\rho;T)$ has coefficients in $\mathbb{Q}$, and therefore

$\displaystyle\gcd\big{(}P(\rho;T),x^{p}-1\big{)}=\begin{cases}x-1\\\
\Phi_{p}\\\ x^{p}-1\end{cases}\in\mathbb{Q}[x],$

where $\Phi_{p}$ is the $p$-th cyclotomic polynomial, which is irreducible
over $\mathbb{Q}$. Suppose first $\rho=\lambda$. If
$\gcd(P(\lambda;T),x^{p}-1)=(x-1)$ then necessarily $\lambda(T)=I_{p}$, which
is impossible since Table 5.8 gives $\operatorname{Tr}\lambda(T)=0\neq p$. If
$\gcd(P(\lambda;T),x^{p}-1)=\Phi_{p}$, then $P(\lambda;T)$ factors over
$\mathbb{Q}$ as $\Phi_{p}\cdot(x-\zeta_{p})$, for some $p$-th root of unity
$\zeta_{p}$. But since $p\geq 5$, the only such polynomial $(x-\zeta_{p})$
defined over $\mathbb{Q}$ is $(x-1)$, therefore $P(\lambda;T)=x^{p}-1$
necessarily, which gives the formula

$\displaystyle N_{k}(\lambda;T)=1,\qquad k=1,\ldots,p-1.$

For $\rho=\mu_{s}$, again we cannot have $\gcd(P(\mu_{s};T),x^{p}-1)=(x-1)$
since none of the $\mu_{s}(T)$ is the identity, the trace being $1\neq 0$. If
$\gcd(P(\mu_{s};T),x^{p}-1)=\Phi_{p}$ then $P(\mu_{s};T)$ factors as
$\Phi_{p}\cdot q(x)$, for some quadratic polynomial $q(x)$. The only
$\mathbb{Q}$-rational quadratic polynomials having only $p$-th roots as their
solutions are $(x-1)^{2}$ and $\Phi_{3}$. Since $p\geq 5$, the latter cannot
occur and $P(\mu_{s};T)=\Phi_{p}\cdot(x-1)^{2}$. A similar reasoning shows
that $P(\theta_{t};T)=\Phi_{p}$, so the first row of Proposition 5.1 is
proved. It remains to compute $P(\rho;T)$ for $\rho=\chi_{1},\chi_{2}$. Note
that for these two representations the last two columns of Table 5.8 have
coefficients in $\mathbb{Q}(\sqrt{p})$ and therefore $P(\rho;T)$ is
$\mathbb{Q}(\sqrt{p})$-rational. Now the $p$-th cyclotomic polynomial factors
over $\mathbb{Q}(\sqrt{p})$ as

$\displaystyle\Phi_{p}=\Phi^{(r)}_{p}\cdot\Phi^{(n)}_{p},\qquad\in\mathbb{Q}(\sqrt{p})[x],$

where $\Phi^{(r)}_{p}$ (resp. $\Phi^{(n)}_{p}$) is the irreducible monic
polynomial of degree $(p-1)/2$ over $\mathbb{Q}(\sqrt{p})$ whose roots are the
$p$-th roots of unity of the form $e^{2\pi ir/p}$ (resp. $e^{2\pi in/p}$),
with $r$ (resp. $n$) a quadratic residue mod $p$ (resp. quadratic non-residue
mod $p$). Let now $\rho=\chi_{1}$. Then $\gcd(P(\chi_{1};T),x^{p}-1)$ is a
non-constant $\mathbb{Q}(\sqrt{p})$-rational polynomial and it cannot be
$(x-1)$ for otherwise $\chi(T)$ would be the identity, which is impossible
since $\operatorname{Tr}(\chi_{1}(T))\neq p$. Since $\deg
P(\chi_{1};T)=(p+1)/2$, the only other possibilities are that
$\gcd(P(\chi_{1};T),x^{p}-1)=\Phi^{(r)}_{p}$ or $\Phi^{(n)}_{p}$, so that
$P(\chi_{1};T)=(x-1)\Phi^{(r)}_{p}$ or $(x-1)\Phi^{(n)}_{p}$. To determine
which one it is, note that the sum of the roots of $(x-1)\Phi^{(r)}_{p}$ can
be computed by a Gauss sum and it is equal to $1/2(-1-\sqrt{p})$. This must
coincide with the negative of $\operatorname{Tr}(\chi_{1}(T))$, so by Table
5.8 we see that this is indeed the correct factorization. Similarly we may
deduce that $P(\chi_{2};T)=(x-1)\Phi^{(n)}_{p}$, which concludes the proof. ∎

Putting everything into formula (5.1) we get:

######  Theorem 5.2.

Let $p\geq 5$ be a prime, $p\equiv 1\,(4)$. Then the multiplicities of the
irreducible representations of $\mathrm{PSL}_{2}(\mathbb{F}_{p})$ inside the
canonical representation $\rho_{\Gamma(p)}$ are given by

$\displaystyle
d(\lambda)=\frac{p-9}{12}-\frac{1}{3}\left(\frac{p}{3}\right)_{L},$
$\displaystyle
d(\mu_{s})=\frac{p-11}{12}-\frac{(-1)^{s}}{2}-\left(1+\left(\frac{p}{3}\right)_{L}\right)\frac{\epsilon_{3}(s)}{6},\qquad
1\leq s\leq\frac{p-5}{4},$ $\displaystyle
d(\theta_{t})=\frac{p-1}{12}+\left(1-\left(\frac{p}{3}\right)_{L}\right)\frac{\epsilon_{3}(t)}{6},\qquad
1\leq t\leq\frac{p-1}{4},$ $\displaystyle
d(\chi_{1})=\frac{p-11}{24}-\frac{(-1)^{\frac{p-1}{4}}}{4}-\frac{1}{6}\left(1+\left(\frac{p}{3}\right)_{L}\right),$
$\displaystyle
d(\chi_{2})=\frac{p-11}{24}-\frac{(-1)^{\frac{p-1}{4}}}{4}-\frac{1}{6}\left(1+\left(\frac{p}{3}\right)_{L}\right).$

###### Proof.

The formulas follow from a straightforward insertion of the data of Table 5.8
into (5.1). Note that for the multiplicities of $\chi_{1}$ and $\chi_{2}$ we
have used the formulas

$\displaystyle\sum_{\begin{subarray}{c}k=1\\\
\left(\frac{k}{p}\right)_{L}=1\end{subarray}}^{p-1}\frac{k}{p}=\frac{p-1}{4}=\sum_{\begin{subarray}{c}k=1\\\
\left(\frac{k}{p}\right)_{L}=-1\end{subarray}}^{p-1}\frac{k}{p},$

which are valid for $p\equiv 1\,(4)$. ∎

###  5.2 Case $\boldsymbol{p\equiv 3\,(4)}$

In this case

$\displaystyle S\sim\varphi^{\frac{p+1}{4}},\qquad
R\sim\begin{cases}\left(\begin{matrix}u_{p}^{\frac{p-1}{3}}&0\\\
0&u_{p}^{-\frac{p-1}{3}}\end{matrix}\right)&\text{if }p\equiv 1\,(3),\\\
\varphi^{\frac{p+1}{3}}&\text{ if }p\equiv 2\,(3),\end{cases}$

and from the character table of [9, Theorem 8.11] we obtain the following data
(note that $T\sim\left(\begin{smallmatrix}1&0\\\
u_{p}&1\end{smallmatrix}\right)$ in this case)

$\displaystyle\begin{array}[]{@{}c|c|c|c| c | c | c}&\dim&S&R\;\text{ if }p\equiv 1\,(3)&R\;\text{ if }p\equiv 2\,(3)&T&\left(\begin{smallmatrix}1&0\\\ 1&1\end{smallmatrix}\right)\\\ \hline\cr\lambda&p&-1&1&-1&0&0\\\ \\{\mu_{s}\\}_{1\leq s\leq\frac{p-3}{4}}&p+1&0&\epsilon_{3}(s)&0&1&1\\\ \\{\theta_{t}\\}_{1\leq t\leq\frac{p-3}{4}}&p-1&-2\cdot(-1)^{t}&0&-\epsilon_{3}(t)&-1&-1\\\ \gamma_{1}&\frac{p-1}{2}&-(-1)^{\frac{p+1}{4}}&0&-1&\frac{-1-\sqrt{-p}}{2}&\frac{-1+\sqrt{-p}}{2}\\\ \gamma_{2}&\frac{p-1}{2}&-(-1)^{\frac{p+1}{4}}&0&-1&\frac{-1+\sqrt{-p}}{2}&\frac{-1-\sqrt{-p}}{2}\end{array}$

where the irreducible characters $\lambda$, $\mu_{s}$, $\theta_{t}$,
$\chi_{1}$, $\chi_{2}$ are labelled as in [9, Theorem 8.11]. Exactly as in
Proposition 5.1 we can derive from this table the ramification data at
$\infty$:

$\displaystyle N_{k}(\lambda;T)=1,\qquad N_{k}(\mu_{s};T)=1,\qquad
N_{k}(\theta_{t};T)=1,$ $\displaystyle
N_{k}(\gamma_{1};T)=\begin{cases}1&\text{if
}\left(\dfrac{k}{p}\right)_{L}=-1,\vspace{1mm}\\\ 0&\text{if
}\left(\dfrac{k}{p}\right)_{L}=1,\end{cases}\qquad
N_{k}(\gamma_{2};T)=\begin{cases}0&\text{if
}\left(\dfrac{k}{p}\right)_{L}=-1,\vspace{1mm}\\\ 1&\text{if
}\left(\dfrac{k}{p}\right)_{L}=1,\end{cases}$

where $k=1,\ldots,p-1$. Putting everything into formula (5.1) we get:

######  Theorem 5.3.

Let $p\geq 5$ be a prime, $p\equiv 3\,(4)$. Then the multiplicities of the
irreducible representations of $\mathrm{PSL}_{2}(\mathbb{F}_{p})$ inside the
canonical representation $\rho_{\Gamma(p)}$ are given by

$\displaystyle
d(\lambda)=\frac{p-3}{12}-\frac{1}{3}\left(\frac{p}{3}\right)_{L},$
$\displaystyle
d(\mu_{s})=\frac{p-11}{12}-\left(1+\left(\frac{p}{3}\right)_{L}\right)\frac{\epsilon_{3}(s)}{6},\qquad
1\leq s\leq\frac{p-3}{4},$ $\displaystyle
d(\theta_{t})=\frac{p-1}{12}+\frac{(-1)^{t}}{2}+\left(1-\left(\frac{p}{3}\right)_{L}\right)\frac{\epsilon_{3}(t)}{6},\qquad
1\leq t\leq\frac{p-3}{4},$ $\displaystyle
d(\gamma_{1})=\frac{p-1}{24}-\frac{h(p)}{2}+\frac{(-1)^{\frac{p+1}{4}}}{4}+\frac{1}{6}\left(1-\left(\frac{p}{3}\right)_{L}\right),$
$\displaystyle
d(\gamma_{2})=\frac{p-1}{24}+\frac{h(p)}{2}+\frac{(-1)^{\frac{p+1}{4}}}{4}+\frac{1}{6}\left(1-\left(\frac{p}{3}\right)_{L}\right),$

where $h(p)$ is the class number of $\mathbb{Q}(\sqrt{-p})$.

###### Proof.

Only the formulas for $d(\gamma_{1})$, $d(\gamma_{2})$ require justification.
Note that since $p\equiv 3\,(4)$

$\displaystyle\sum_{\begin{subarray}{c}k=1\\\
\left(\frac{k}{p}\right)_{L}=-1\end{subarray}}^{p-1}\frac{k}{p}-\sum_{\begin{subarray}{c}k=1\\\
\left(\frac{k}{p}\right)_{L}=1\end{subarray}}^{p-1}\frac{k}{p}=L\left(0,\left(\frac{\cdot}{p}\right)_{L}\right)\neq
0$

and clearly $\sum\limits_{\begin{subarray}{c}k=1\\\
\left(\frac{k}{p}\right)_{L}=-1\end{subarray}}^{p-1}\frac{k}{p}+\sum\limits_{\begin{subarray}{c}k=1\\\
\left(\frac{k}{p}\right)_{L}=1\end{subarray}}^{p-1}\frac{k}{p}=(p-1)/2$. Using
the functional equation for the $L$-function and the analytic class number
formula, we get

$\displaystyle\sum_{\begin{subarray}{c}k=1\\\
\left(\frac{k}{p}\right)_{L}=1\end{subarray}}^{p-1}\frac{k}{p}=\frac{p-1}{4}-\frac{h(p)}{2},\qquad\sum_{\begin{subarray}{c}k=1\\\
\left(\frac{k}{p}\right)_{L}=-1\end{subarray}}^{p-1}\frac{k}{p}=\frac{p-1}{4}+\frac{h(p)}{2},$

which gives the formulas for $d(\gamma_{1})$, $d(\gamma_{2})$. ∎

######  Remark 5.4.

We thank Cameron Franc for showing us how to simplify $d(\gamma_{1})$ and
$d(\gamma_{2})$ using Dirichlet’s $L$-functions and the class number formula.
Similar computations, but in a different context, have also appeared in [4].

######  Example 5.5.

The formulas of Theorems 5.2 and 5.3 show that the only prime $p$ for which
the canonical representation $\rho_{\Gamma(p)}$ is irreducible is the prime
$p=7$. In this case the orbifold curve $X(\Gamma(7))$ is an actual algebraic
curve, and in fact it is a model for the Klein quartic, the unique Hurwitz
surface of genus 3. Applying Theorem 5.3 we get that $d(\gamma_{2})=1$, since
$h(7)=1$, while all the other multiplicities are zero. Therefore
$\rho_{\Gamma(p)}\simeq\gamma_{2}$. To write down explicit matrices for
$\rho_{\Gamma(p)}$, we know by [20] that a three-dimensional irreducible
representation of $\mathrm{PSL}_{2}(\mathbb{Z})$ is entirely determined by the
eigenvalues of the matrix $\rho_{\Gamma(p)}(T)$. As in Theorem 5.3, we know
that these are $\zeta_{7}$, $\zeta_{7}^{2}$, $\zeta_{7}^{4}$, where
$\zeta_{7}:=e^{2\pi i/7}$. According to [20, Proposition 2.5], there is a
basis where $\rho_{\Gamma(p)}$ is given by the matrices

$\displaystyle\rho_{\Gamma(7)}(T)=\left(\begin{matrix}\zeta_{7}&\zeta^{2}_{7}+\zeta^{3}_{7}&\zeta_{7}^{2}\\\
0&\zeta_{7}^{2}&\zeta_{7}^{2}\\\
0&0&\zeta_{7}^{4}\end{matrix}\right),\qquad\rho_{\Gamma(7)}(S)=\left(\begin{matrix}0&0&1\\\
0&-1&0\\\ 1&0&0\end{matrix}\right).$

##  6 The canonical representation of Fermat curves

Let $\Gamma(2)\subseteq\mathrm{SL}_{2}(\mathbb{Z})$ be the principal
congruence subgroup of level two. This is isomorphic to the free product on
two generators, which can be chosen to be

$\displaystyle A=\left(\begin{matrix}1&2\\\ 0&1\end{matrix}\right),\qquad
B=\left(\begin{matrix}1&0\\\ -2&1\end{matrix}\right).$

For any integer $N>1$, let $\Phi(N)$ be the kernel of the composition

$\displaystyle\Gamma(2)\stackrel{{\scriptstyle\varphi^{\rm{ab}}}}{{\longrightarrow}}\mathbb{Z}\times\mathbb{Z}\longrightarrow\mathbb{Z}/N\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z},$
$\displaystyle\left(\begin{smallmatrix}1&2\\\
0&1\end{smallmatrix}\right)\longmapsto(1,0),\qquad\left(\begin{smallmatrix}1&0\\\
-2&1\end{smallmatrix}\right)\longmapsto(0,1),$

where the first map $\varphi^{\rm{ab}}$ is projection onto the abelianization
and the second map is reduction modulo $N$. The composition is clearly
surjective, therefore $\Phi(N)$ is a normal subgroup of index $N^{2}$ in
$\Gamma(2)$. We also denote by $\Phi(N)$ the corresponding subgroup inside
$\mathrm{PSL}_{2}(\mathbb{Z})$.

######  Lemma 6.1.

For all $N>1$, $\Phi(N)$ is normal in $\mathrm{PSL}_{2}(\mathbb{Z})$.

###### Proof.

By definition, $\Phi(N)=\varphi^{\rm{ab},-1}(N\mathbb{Z}\times N\mathbb{Z})$.
For any $\gamma\in\mathrm{PSL}_{2}(\mathbb{Z})$, the conjugate subgroup
$g^{-1}\Phi(N)g$ is the kernel of a surjective homomorphism
$\Gamma(2)\rightarrow\mathbb{Z}/N\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}$. This
map must factor through a map
$\Gamma(2)^{\rm{ab}}=\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}/N\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}$,
by the universal property of abelianizations. Any such map must have kernel
equal to $N\mathbb{Z}\times N\mathbb{Z}$, therefore
$g^{-1}\Phi(N)g=\varphi^{\rm{ab},-1}(N\mathbb{Z}\times N\mathbb{Z})=\Phi(N)$.
∎

The normal subgroup $\Phi(N)$ can be described explicitly inside
$\mathrm{PSL}_{2}(\mathbb{Z})$ as the normal closure

$\displaystyle\Phi(N)=\big{\langle}U^{3},V^{2N}\big{\rangle}^{\mathrm{PSL}_{2}(\mathbb{Z})},\qquad
V:=SR^{-1},\qquad U:=TV\in\mathrm{PSL}_{2}(\mathbb{Z}).$

The quotient $G:=\mathrm{PSL}_{2}(\mathbb{Z})/\Phi(N)$ is the semi-direct
product $(\mathbb{Z}/N\mathbb{Z})^{2}\rtimes S_{3}$. The normal subgroup
$(\mathbb{Z}/N\mathbb{Z})^{2}$ is generated by the images of $A$ and $B$ while
the $S_{3}$-subgroup is generated by the images of $S$ and $U$. The action of
$S_{3}$ on $(\mathbb{Z}/N\mathbb{Z})^{2}$ is given explicitly by

$\displaystyle SAS=B,\qquad UAU^{2}=B,\qquad UBU^{2}=A^{-1}B^{-1}\mod\Phi(N).$

By Lemma 6.1, the curve $X(\Phi(N))$ is a Galois cover

$\displaystyle f(\Phi(N))\colon\ X(\Phi(N))\longrightarrow X(1)$

with Galois group isomorphic to $G$. Since $A=T^{2}$ and $A^{N}\in\Phi(N)$,
the image of $T$ in $G$ has order $2N$, so the ramification level is $2N$. For
all $N>1$, there is an embedding of $X(\Phi(N))$ into $\mathbb{P}^{2}$ cut out
by the equation $X^{N}+Y^{N}=Z^{N}$ in homogeneous coordinates [10]. In other
words, $X(\Phi(N))$ is a uniformization of the Fermat curve $F_{N}$ of
exponent $N$.

By construction, the group of automorphisms of the curve $X(\Phi(N))$ contains
$G=(\mathbb{Z}/N\mathbb{Z})^{2}\rtimes S_{3}$ and it is known that these are
indeed all the automorphisms of the curve [21]. We now apply Theorem 4.1 to
compute the decomposition of the canonical representation

$\displaystyle\rho_{\Phi(N)}\colon\
G\longrightarrow\mathrm{GL}(S_{2}(\Phi(N)))$

into irreducible representations. The genus of $X(\Phi(N))$ can easily be
worked out to be $\frac{1}{2}(N-2)(N-1)$, so we may assume that $N>2$. Note
that $\Phi(N)$ is a non-congruence subgroup for almost all $N$ [17], therefore
the decomposition of $\rho_{\Phi(N)}$ cannot be deduced from the calculations
of Section 5, even for $N=p$ a prime.

All the irreducible representations of $G$ can be computed using the ‘little
subgroups method’ for semi-direct products by an abelian group [18, Section
8.2], [1, Table 2], which can be applied as follows. Let $S_{3}$ act on
$\operatorname{Hom}((\mathbb{Z}/N\mathbb{Z})^{2},\mathbb{C}^{\times})$ by
$g\chi(x)=\chi(g^{-1}xg)$ and let $\\{\chi_{i}\\}$ be representatives for the
cosets under this action. Let $H_{i}\subseteq S_{3}$ be the stabilizer of
$\chi_{i}$, so that $\chi_{i}$ can be extended to a character of
$G_{i}=(\mathbb{Z}/N\mathbb{Z})^{2}\cdot H_{i}$. Let $\bar{\rho}$ be any
representation of $H_{i}$, viewed as a representation of $G_{i}$ under the
quotient map $G_{i}\rightarrow H_{i}$ and let
$\theta_{i,\bar{\rho}}=\mathrm{Ind}_{G_{i}}^{G}\bar{\rho}\otimes\chi_{i}$. It
can be shown that $\theta_{i,\bar{\rho}}$ is irreducible and that all the
irreducible representations of $G$ are obtained in this way [18, Proposition
25]. These calculations in our case break down into two cases, according to
whether $3\,|\,N$.

###  6.1 Case $\boldsymbol{3\nmid N}$

In this case the orbits of the action of $S_{3}$ on
$\operatorname{Hom}((\mathbb{Z}/N\mathbb{Z})^{2},\mathbb{C}^{\times})$ can
only have order 1, 3 or 6 with stabilizers $H_{i}$ isomorphic to $S_{3}$,
$C_{2}$ or trivial, respectively. Write $\chi_{\alpha,\beta}$ for the
character in
$\operatorname{Hom}((\mathbb{Z}/N\mathbb{Z})^{2},\mathbb{C}^{\times})$
uniquely determined by

$\displaystyle\chi_{\alpha,\beta}(A)=\zeta_{N}^{\alpha},\qquad\chi_{\alpha,\beta}(B)=\zeta_{N}^{\beta},\qquad\zeta_{N}=e^{2\pi
i/N}.$

Then the $S_{3}$-orbit of size one is $\\{\chi_{0,0}=1\\}$ and the
$S_{3}$-orbits of size three are of the form
$\\{\chi_{\alpha,\alpha},\chi_{\alpha,-2\alpha},\chi_{-2\alpha,\alpha}\\}$ for
$\alpha=1,\ldots,N-1$. There are a total of $3(N-1)+1$ characters in these
orbits. The remaining $N^{2}-3N+2$ elements are partitioned into orbits of
size 6, with trivial stabilizers. We let

$\displaystyle\\{\chi_{\alpha_{i},\beta_{i}}\\},\qquad
i=1,\ldots,\frac{N^{2}-3N+2}{6}=\frac{1}{6}(N-2)(N-1)$

be a set of representatives of these orbits. It is easy to see that
$\alpha_{i}$, $\beta_{i}$ can be chosen so that $\alpha_{i},\beta_{i}\neq 0$.
Assuming so in what follows slightly simplifies the formulas.

Now the orbit $\\{\chi_{0,0}=1\\}$ produces via the small subgroup method 3
representations, corresponding to the 3 irreducible representations of
$S_{3}$:

  * (i)

the trivial representation $\rho_{1}$,

  * (ii)

1 one-dimensional irreducible representation $\rho_{2}$,

  * (iii)

1 two-dimensional irreducible representation $\rho_{3}$.

The orbits of cyclic stabilizer $C_{2}$, with representatives
$\chi_{\alpha,\alpha}$ produce

  * (iv)

$(N-1)$irreducible 3-dimensional representations $\rho^{+}_{\alpha}$,
$\alpha\in\\{1,\ldots,N-1\\}$,

  * (v)

$(N-1)$ irreducible 3-dimensional representations $\rho^{-}_{\alpha}$,
$\alpha\in\\{1,\ldots,N-1\\}$.

Each class comes from inducing $\bar{\rho}^{\pm}\otimes\chi_{\alpha}$ to $G$,
where $\rho^{+}$ is the trivial representation of $C_{2}$ and $\rho^{-}$ is
the non-trivial. Finally, the orbits of representative
$\chi_{\alpha_{i},\beta_{i}}$ give

  * (vi)

$\frac{1}{6}(N-2)(N-1)$ irreducible 6-dimensional representations
$\rho_{\alpha_{i},\beta_{i}}$,

each representation being given by inducing $\chi_{\alpha_{i},\beta_{i}}$ to
all of $G$. Explicit matrices for the generators $S$, $U$, $A$ can easily be
written down as in [1], from which we also borrowed the notation (a slight
correction is needed in Table 2 of [1] for the matrices
$\rho_{\alpha}^{\pm}(A)$ and $\rho_{\alpha,\beta}(A)$). Using the relation

$\displaystyle R=A^{-1}U\mod\Phi(N)$

the traces of $\rho(S)$, $\rho(R)$, $\rho\big{(}R^{-1}\big{)}$ and the
eigenvalues of the $\rho(T)$-matrix can then be computed from the explicit
formulas and they are given in table

$\displaystyle\begin{array}[]{@{}c|c|c|c| c | c }&\dim&S&R&R^{-1}&T-\text{eigenvalues}\\\ \hline\cr\rho_{2}&1&-1&1&1&-1\\\ \rho_{3}&2&0&-1&-1&1,-1\\\ \rho^{+}_{\alpha}&3&1&0&0&\zeta_{N}^{-\alpha},\zeta^{\alpha}_{2N},-\zeta^{\alpha}_{2N}\\\ \rho^{-}_{\alpha}&3&-1&0&0&-\zeta_{N}^{-\alpha},-\zeta^{\alpha}_{2N},\zeta^{\alpha}_{2N}\\\ \rho_{\alpha_{i},\beta_{i}}&6&0&0&0&\pm\zeta_{2N}^{\alpha_{i}},\pm\zeta_{2N}^{\beta_{i}},\pm\zeta_{2N}^{-(\alpha_{i}+\beta_{i})}\end{array}$

where $\zeta_{N}=e^{2\pi i/N}$ and $\zeta_{2N}=e^{2\pi i/2N}$. Applying
Theorem 4.1 we obtain the following.

######  Theorem 6.2.

Let $N>2$ be an integer such that $3\nmid N$. Then the multiplicities of the
irreducible representations of $G=(\mathbb{Z}/N\mathbb{Z})^{2}\rtimes S_{3}$
inside the canonical representation $\rho_{\Phi(N)}$ are given by

$\displaystyle d(\rho_{1})=d(\rho_{2})=d(\rho_{3})=0,\qquad
d(\rho_{\alpha}^{+})=0,\qquad\alpha=1,\ldots,N-1,$ $\displaystyle
d(\rho_{\alpha}^{-})=\begin{cases}1&\text{if
}\alpha=1,\ldots,\left\lfloor\dfrac{N-1}{2}\right\rfloor,\vspace{1mm}\\\
0&\text{if
}\alpha=\left\lfloor\dfrac{N-1}{2}\right\rfloor+1,\ldots,N-1,\end{cases}$
$\displaystyle d(\rho_{\alpha_{i},\beta_{i}})=\begin{cases}1&\text{if
}\alpha_{i}+\beta_{i}<N,\\\ 0&\text{if }\alpha_{i}+\beta_{i}\geq
N,\end{cases}\qquad\alpha_{i},\beta_{i}\neq 0.$

######  Example 6.3.

Let $N=7$. There are 5 $S_{3}$-orbits of size 6 in
$\operatorname{Hom}((\mathbb{Z}/7\mathbb{Z})^{2},\mathbb{C}^{\times})$, and
the representatives for these orbits can be chosen to be
$\\{\chi_{6,1},\chi_{5,2},\chi_{4,3},\chi_{2,1},\chi_{6,3}\\}$. By Theorem 6.2
we deduce that

$\displaystyle\rho_{\Phi(7)}\sim\rho^{-}_{1}\oplus\rho^{-}_{2}\oplus\rho^{-}_{3}\oplus\rho_{2,1}.$

An explicit model for $\rho_{\Phi(7)}$ can then be written down using the
following formulas

$\displaystyle\rho_{\alpha}^{-}(T)=\left(\begin{matrix}0&0&\zeta_{7}^{-\alpha}\\\
0&\zeta_{7}^{-\alpha}&0\\\
-\zeta_{7}^{2\alpha}&0&0\end{matrix}\right),\qquad\rho_{\alpha}^{-}(S)=\left(\begin{matrix}-1&0&0\\\
0&0&-1\\\ 0&-1&0\end{matrix}\right),\qquad\alpha=1,2,3,$
$\displaystyle\rho_{2,1}(T)=\left(\begin{matrix}0&0&0&0&\zeta_{7}^{-1}&0\\\
0&0&0&0&0&\zeta_{7}^{-2}\\\ 0&0&0&\zeta_{7}^{3}&0&0\\\
0&0&\zeta_{7}^{-1}&0&0&0\\\ \zeta_{7}^{3}&0&0&0&0&0\\\
0&\zeta_{7}^{-2}&0&0&0&0\end{matrix}\right),\qquad\rho_{2,1}(S)=\left(\begin{matrix}0&0&0&1&0&0\\\
0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\ 1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&1&0&0&0\end{matrix}\right).$

###  6.2 Case $\boldsymbol{3\,|\,N}$

In this case there are 3 characters with full stabilizer $S_{3}$, given by
$\\{\chi_{0,0}\\}$, $\\{\chi_{N/3,N/3}\\}$ and $\\{\chi_{2N/3,2N/3}\\}$ (and
$N-3$ remaining orbits with $C_{2}$-stabilizers). The two additional
characters each produce

  * (vii)

2 one-dimensional (non-trivial) representations $\rho^{1}_{N/3}$ and
$\rho^{1}_{2N/3}$,

  * (viii)

2 one-dimensional (non-trivial) representations $\rho^{2}_{N/3}$ and
$\rho^{2}_{2N/3}$,

  * (ix)

2 two-dimensional representations $\rho^{3}_{N/3}$ and $\rho^{3}_{2N/3}$,

the first corresponding to the trivial representation of $S_{3}$, the second
to the sign representation, and the third to the 2-dimensional representation
of $S_{3}$. For each one of these new representations we can compute the
following data:

$\displaystyle\begin{array}[]{@{}c|c|c|c| c | c }&\dim&S&R&R^{-1}&T\text{-eigenvalues}\\\ \hline\cr\rho^{1}_{N/3}&1&1&\zeta_{3}^{-1}&\zeta_{3}&\zeta_{3}^{-1}\\\ \rho^{1}_{2N/3}&1&1&\zeta_{3}&\zeta_{3}^{-1}&\zeta_{3}\\\ \rho^{2}_{N/3}&1&-1&\zeta_{3}^{-1}&\zeta_{3}&-\zeta_{3}^{-1}\\\ \rho^{2}_{2N/3}&1&-1&\zeta_{3}&\zeta_{3}^{-1}&-\zeta_{3}\\\ \rho^{3}_{N/3}&2&0&-\zeta_{3}^{-1}&-\zeta_{3}&\zeta_{6},-\zeta_{6}\\\ \rho^{3}_{2N/3}&2&0&-\zeta_{3}&-\zeta_{3}^{-1}&-\zeta_{3},\zeta_{3}\\\ \end{array}$

Applying Theorem 4.1 we get

######  Theorem 6.4.

Let $N>2$ be an integer such that $3\,|\,N$. Then the multiplicities of the
irreducible representations of $G=(\mathbb{Z}/N\mathbb{Z})^{2}\rtimes S_{3}$
inside the canonical representation $\rho_{\Phi(N)}$ are given by

$\displaystyle d(\rho_{1})=d(\rho_{2})=d(\rho_{3})=0,$ $\displaystyle
d(\rho_{\alpha}^{+})=0,\qquad\alpha=1,\ldots,N-1,\qquad\alpha\neq N/3,2N/3,$
$\displaystyle d(\rho_{\alpha}^{-})=\begin{cases}1&\text{if
}\alpha=1,\ldots,\left\lfloor\dfrac{N-1}{2}\right\rfloor,\vspace{1mm}\\\
0&\text{if
}\alpha=\left\lfloor\dfrac{N-1}{2}\right\rfloor+1,\ldots,N-1,\end{cases}\qquad\alpha\neq
N/3,2N/3,$ $\displaystyle
d(\rho_{\alpha_{i},\beta_{i}})=\begin{cases}1&\text{if
}\alpha_{i}+\beta_{i}<N,\\\ 0&\text{if }\alpha_{i}+\beta_{i}\geq
N,\end{cases}\qquad\alpha_{i},\beta_{i}\neq 0,$ $\displaystyle
d\big{(}\rho^{1}_{N/3}\big{)}=d\big{(}\rho^{1}_{2N/3}\big{)}=d\big{(}\rho^{2}_{2N/3}\big{)}=0,\qquad
d\big{(}\rho^{2}_{N/3}\big{)}=1,\qquad
d\big{(}\rho^{3}_{N/3}\big{)}=d\big{(}\rho^{3}_{2N/3}\big{)}=0.$

######  Example 6.5.

Let $N=6$. There are 4 $S_{3}$-orbits of size 6 in
$\operatorname{Hom}\big{(}(\mathbb{Z}/6\mathbb{Z})^{2},\mathbb{C}^{\times}\big{)}$,
and representatives for these orbits can be chosen to be
$\\{\chi_{5,1},\chi_{4,2},\chi_{5,4},\chi_{3,2}\\}$. By Theorem 6.4 we deduce
that

$\displaystyle\rho_{\Phi(6)}\sim\rho_{2}^{2}\oplus\rho_{1}^{-}\oplus\rho_{3,2}$

and an explicit model for $\rho_{\Phi(6)}$ can be written down using

$\displaystyle\rho_{2}^{2}(T)=e^{2\pi i/6},\qquad\rho_{2}^{2}(S)=-1,$
$\displaystyle\rho_{\alpha}^{-}(T)=\left(\begin{matrix}0&0&\zeta_{6}^{-1}\\\
0&\zeta_{6}^{-1}&0\\\
-\zeta_{3}&0&0\end{matrix}\right),\qquad\rho_{\alpha}^{-}(S)=\left(\begin{matrix}-1&0&0\\\
0&0&-1\\\ 0&-1&0\end{matrix}\right),$
$\displaystyle\rho_{3,2}(T)=\left(\begin{matrix}0&0&0&0&\zeta_{6}^{-2}&0\\\
0&0&0&0&0&\zeta_{6}^{-3}\\\ 0&0&0&\zeta_{6}^{5}&0&0\\\
0&0&\zeta_{6}^{-2}&0&0&0\\\ \zeta_{6}^{5}&0&0&0&0&0\\\
0&\zeta_{6}^{-3}&0&0&0&0\end{matrix}\right),\qquad\rho_{3,2}(S)=\left(\begin{matrix}0&0&0&1&0&0\\\
0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\ 1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&1&0&0&0\end{matrix}\right).$

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