# Modularity of $\mathrm{PGL}_{2}({\mathbb{F}}_{p})$-representations over
totally real fields

Patrick B. Allen,111Department of Mathematics and Statistics, McGill
University, Montreal, Canada. Email address<EMAIL_ADDRESS>Chandrashekhar B. Khare222Department of Mathematics, UCLA, Los Angeles, USA.
Email address<EMAIL_ADDRESS>and Jack A. Thorne333Department of Pure
Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge, United
Kingdom. Email address<EMAIL_ADDRESS>
###### Abstract

We study an analogue of Serre’s modularity conjecture for projective
representations
${\overline{\rho}}:\operatorname{Gal}(\overline{K}/K)\rightarrow\mathrm{PGL}_{2}(k)$,
where $K$ is a totally real number field. We prove new cases of this
conjecture when $k=\mathbb{F}_{5}$ by using the automorphy lifting theorems
over CM fields established in [AKT].4442010 Mathematics Subject
Classification: 11F41, 11F80.

###### Contents

  1. 1 Introduction
    1. 1.1 Notation
  2. 2 Lifting representations
    1. 2.1 Automorphy of linear and projective representations
    2. 2.2 Lifting to characteristic 0
    3. 2.3 Lifting projective representations
  3. 3 Modularity of mod $3$ representations
  4. 4 Modularity of mod $5$ representations

##  1 Introduction

Let $K$ be a number field, and consider a continuous representation

$\rho:G_{K}\to\mathrm{GL}_{2}(k),$

where $k$ is a finite field. (Here $G_{K}$ denotes the absolute Galois group
of $K$; for this and other notation, see §1.1 below.) We say that $\rho$ is of
Serre-type, or $S$-type, if it is absolutely irreducible and totally odd, in
the sense that for each real place $v$ of $K$ and each associated complex
conjugation $c_{v}\in G_{K}$, $\det\rho(c_{v})=-1$.

Serre’s conjecture and its generalisations assert that any $\rho$ of $S$-type
should be automorphic (see for example [Ser75, Ser87] in the case
$K=\mathbb{Q}$, [BDJ10] when $K$ is totally real, and [ŞS18] for a general
number field $K$). The meaning of the word ‘automorphic’ depends on the
context but when $K$ is totally real, for example, we can ask for $\rho$ to be
associated to a cuspidal automorphic representation $\pi$ of
$\mathrm{GL}_{2}(\mathbb{A}_{K})$ which is regular algebraic of weight 0 (see
§2.1 below). Serre’s conjecture is now a theorem when $K=\mathbb{Q}$ [KW09a,
KW09b]. For a totally real field $K$, some results are available when $k$ is
‘small’. These are summarised in the following theorem, which relies upon the
papers [Ser87, Tun81, SBT97, Man04, Ell05]:

######  Theorem 1.1.

Let $K$ be a totally real number field, and let
$\rho:G_{K}\to\mathrm{GL}_{2}(k)$ be a representation of $S$-type. Then $\rho$
is automorphic provided $|k|\in\\{2,3,4,5,7,9\\}$.

One can equally consider continuous representations

$\sigma:G_{K}\to\mathrm{PGL}_{2}(k),$

where again $k$ is a finite field. We say that $\sigma$ is of $S$-type if it
is absolutely irreducible and totally odd, in the sense that if $k$ has odd
characteristic then for each real place $v$ of $K$, $\sigma(c_{v})$ is non-
trivial. One could formulate a projective analogue of Serre’s conjecture,
asking that any representation $\sigma$ of $S$-type be automorphic. A theorem
of Tate implies that $\sigma$ lifts to a linear representation valued in
$\mathrm{GL}_{2}(k^{\prime})$ for some finite extension $k^{\prime}/k$, and by
$\sigma$ being automorphic we mean that a lift of it to a linear
representation is automorphic (see §2.1 below). Thus if $k$ is allowed to
vary, this conjecture is equivalent to Serre’s conjecture, since any
representation $\rho$ has an associated projective representation
$\mathrm{Proj}(\rho)$, and any projective representation $\sigma$ lifts to a
representation valued in $\mathrm{GL}_{2}(k^{\prime})$ for some finite
extension $k^{\prime}/k$; moreover, $\rho$ is of $S$-type if and only if
$\mathrm{Proj}(\rho)$ is, and $\rho$ is automorphic if and only if
$\mathrm{Proj}(\rho)$ is.

However, for fixed $k$ the two conjectures are not equivalent: certainly if
$\rho$ is valued in $\mathrm{GL}_{2}(k)$ then $\mathrm{Proj}(\rho)$ takes
values in $\mathrm{PGL}_{2}(k)$, but it is not true that any representation
$\sigma:G_{K}\to\mathrm{PGL}_{2}(k)$ admits a lift valued in
$\mathrm{GL}_{2}(k)$, and in fact in general the determination of the minimal
extension $k^{\prime}/k$ such that there is a lift to
$\mathrm{GL}_{2}(k^{\prime})$ is somewhat subtle. It is therefore of interest
to ask whether the consideration of projective representations allows one to
expand the list of ‘known’ cases of Serre’s conjecture.

Our main theorem affirms that this is indeed the case. Before giving the
statement we need to introduce one more piece of notation. We write
$\Delta:\mathrm{PGL}_{2}(k)\to k^{\times}/(k^{\times})^{2}$ for the
homomorphism induced by the determinant. We say that a homomorphism $G_{K}\to
k^{\times}/(k^{\times})^{2}$ is totally even (resp. totally odd) if each
complex conjugation in $G_{K}$ is trivial (resp. non-trivial) image.

######  Theorem 1.2.

Let $K$ be a totally real number field, and let
$\sigma:G_{K}\to\mathrm{PGL}_{2}(k)$ be a representation of $S$-type. Then
$\sigma$ is automorphic provided that one of the following conditions is
satisfied:

  1. 1.

$|k|\in\\{2,3,4\\}$.

  2. 2.

$|k|=5$, $[K(\zeta_{5}):K]=4$, and $\Delta\circ\sigma$ is totally even.

  3. 3.

$|k|=5$, $[K(\zeta_{5}):K]=4$, and $\Delta\circ\sigma$ is totally odd.

  4. 4.

$|k|=7$ and $\Delta\circ\sigma$ is totally odd.

  5. 5.

$|k|=9$ and $\Delta\circ\sigma$ is totally even.

We note the exceptional isomophisms
$\mathrm{PSL}_{2}({\mathbb{F}}_{9})=A_{6}$,
$\mathrm{PGL}_{2}({\mathbb{F}}_{5})=S_{5}$,
$\mathrm{PGL}_{2}({\mathbb{F}}_{3})=S_{4}$,
$\mathrm{PGL}_{2}({\mathbb{F}}_{2})=S_{3}$ which link our results to showing
that splitting fields of polynomials of small degree over $K$ arise
automorphically.

The proof of Theorem 1.2 falls into three cases. The first is when $|k|$ is
even or $k={\mathbb{F}}_{3}$. When $|k|$ is even, the homomorphism
$\mathrm{GL}_{2}(k)\to\mathrm{PGL}_{2}(k)$ splits, so we reduce easily to
Theorem 1.1. When $k={\mathbb{F}}_{3}$, the homomorphism
$\mathrm{PGL}_{2}({\mathbb{Z}}[\sqrt{-2}])\to\mathrm{PGL}_{2}({\mathbb{F}}_{3})$
splits and we can use the Langlands–Tunnell theorem [Tun81] to establish the
automorphy of $\sigma$.

The second case is when $|k|$ is odd and $-1$ is a square in $k$ (resp. a non-
square in $k$) and $\Delta\circ\sigma$ is totally even (resp. totally odd). In
this case we are able to construct the following data:

  * •

A solvable totally real extension $L/K$ and a representation
${\overline{\rho}}_{1}:G_{L}\to\mathrm{GL}_{2}(k)$ such that
$\mathrm{Proj}({\overline{\rho}}_{1})=\sigma|_{G_{L}}$ (by showing that $L/K$
can be chosen to kill the Galois cohomological obstruction to lifting).

  * •

A representation
$\rho_{2}:G_{K}\to\mathrm{GL}_{2}(\overline{{\mathbb{Q}}}_{p})$ such that
$\mathrm{Proj}(\overline{\rho}_{2})$ and $\sigma$ are conjugate in
$\mathrm{PGL}_{2}(\overline{{\mathbb{F}}}_{p})$ (by choosing an arbitrary lift
of $\sigma$ to $\mathrm{GL}_{2}(\overline{{\mathbb{F}}}_{p})$ and applying the
Khare–Wintenberger method).

We can then use Theorem 1.1 to verify the automorphy of
${\overline{\rho}}_{1}$, hence the residual automorphy of
${\overline{\rho}}_{2}|_{G_{L}}$. An automorphy lifting theorem then implies
the automorphy of $\rho_{2}|_{G_{L}}$, hence $\rho_{2}$ itself by solvable
descent, hence finally of $\sigma$.

The final case is when $k={\mathbb{F}}_{5}$ and $\Delta\circ\sigma$ is totally
odd. In this case there does not exist any totally real extension $L/K$ such
that $\sigma|_{G_{L}}$ lifts to a representation valued in
$\mathrm{GL}_{2}(k)$ (there is a local obstruction at the real places).
However, it is possible to find a CM extension $L/K$ such that
$\sigma|_{G_{L}}$ lifts to a representation valued in $\mathrm{GL}_{2}(k)$
with determinant the cyclotomic character. When $k={\mathbb{F}}_{5}$ such a
representation necessarily appears in the group of 5-torsion points of an
elliptic curve over $L$ (cf. [SBT97]) and so we can use the automorphy results
over CM fields established in [AKT] together with a solvable descent argument
to obtain the automorphy of $\sigma$. The main novelty in this paper is
contained in our treatment of this case.

######  Remark 1.3.

In the final case above of a representation
$\sigma:G_{K}\to\mathrm{PGL}_{2}({\mathbb{F}}_{5})$ with non-solvable image,
the residual automorphy of the lift
$\rho:G_{L}\to\mathrm{GL}_{2}({\mathbb{F}}_{5})$ ultimately depends on [AKT,
Theorem 7.1], which proves the automorphy of certain residually dihedral
2-adic Galois representations. The residual automorphy of these 2-adic
representations is verified using automorphic induction. In particular, our
proof in this case does not depend on the use of the Langlands–Tunnell
theorem. This is in contrast to the argument used in e.g. [SBT97, Theorem 4.1]
to establish the automorphy of representations
$\rho^{\prime}:G_{K}\to\mathrm{GL}_{2}({\mathbb{F}}_{5})$ with cyclotomic
determinant.

This ‘2-3 switch’ strategy can also be used to prove the automorphy of
representations $\sigma:G_{K}\to\mathrm{PGL}_{2}({\mathbb{F}}_{3})$ with
$\Delta\circ\sigma$ totally odd using the 2-adic automorphy theorems proved in
[All14], see Theorem 3.1 of the text. This class of representations includes
the projective representations associated to the Galois action on the
3-torsion points of an elliptic curve over $K$. This gives a way to verify the
modulo 3 residual automorphy of elliptic curves over $K$ which does not rely
on the Langlands–Tunnell theorem (and in particular the works [Lan80, JPSS81])
but only on the Saito–Shintani lifting for holomorphic Hilbert modular forms
[Sai75]. (We note that we do need to use the Langlands-Tunnell theorem to
prove the automorphy of representations
$\sigma:G_{K}\to\mathrm{PGL}_{2}({\mathbb{F}}_{3})$ with $\Delta\circ\sigma$
totally even, cf. Theorem 2.12.)

We now describe the structure of this note. We begin in §2 by studying the
lifts of projective representations and collecting various results about the
existence of characteristic 0 lifts of residual representations and their
automorphy. We are then able to give the proofs of Theorem 1.1 and the first
two cases in the proof of Theorem 1.2 described above. In §3, we expand on
Remark 1.3 by showing how the main theorems of [All14] can be used to give
another proof of the automorphy of $S$-type representations
$\sigma:G_{K}\to\mathrm{PGL}_{2}({\mathbb{F}}_{3})$ (still under the
hypothesis that $K$ is totally real and $\Delta\circ\sigma$ is totally odd).
Finally, in §4 we use similar arguments, now based on the main theorems of
[AKT], to complete the proof of Theorem 1.2.

### Acknowledgments

We would like to thank the anonymous referee for their comments and
corrections. P.A. was supported by Simons Foundation Collaboration Grant
527275 and NSF grant DMS-1902155. He would like to thank the third author and
Cambridge University for hospitality during a visit where some of this work
was completed. Parts of this work were completed while P.A. was a visitor at
the Institute for Advanced Study, where he was partially supported by the NSF.
He would like to thank the IAS for providing excellent working conditions
during his stay. J.T.’s work received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 714405). This research was begun during the
period that J.T. served as a Clay Research Fellow.

###  1.1 Notation

If $K$ is a perfect field then we write
$G_{K}=\operatorname{Gal}(\overline{K}/K)$ for the Galois group of $K$ with
respect to a fixed choice of algebraic closure. If $K$ is a number field and
$v$ is a place of $K$ then we write $K_{v}$ for the completion of $K$ at $v$,
and fix an embedding $\overline{K}\to\overline{K}_{v}$ extending the natural
embedding $K\to K_{v}$; this determines an injective homomorphism
$G_{K_{v}}\to G_{K}$. If $v$ is a finite place of $K$ then we write
$\operatorname{Frob}_{v}\in G_{K_{v}}$ for a lift of the geometric Frobenius,
$k(v)$ for the residue field of $K_{v}$, and $q_{v}$ for the cardinality of
$K_{v}$; if $v$ is a real place, then we write $c_{v}\in G_{K_{v}}$ for
complex conjugation. Any homomorphism from a Galois group $G_{K}$ to another
topological group will be assumed to be continuous.

If $p$ is a prime and $K$ is a field of characteristic 0, then we write
$\epsilon:G_{K}\to{\mathbb{Z}}_{p}^{\times}$ for the $p$-adic cyclotomic
character, $\overline{\epsilon}:G_{K}\to{\mathbb{F}}_{p}^{\times}$ for its
reduction modulo $p$, and
$\omega:G_{K}\to{\mathbb{F}}_{p}^{\times}/({\mathbb{F}}_{p}^{\times})^{2}$ for
the character $\overline{\epsilon}\text{ mod
}({\mathbb{F}}_{p}^{\times})^{2}$. More generally, if
$\rho:G_{K}\to\mathrm{GL}_{n}(\overline{{\mathbb{Q}}}_{p})$ is a
representation, then we write
${\overline{\rho}}:G_{K}\to\mathrm{GL}_{n}(\overline{{\mathbb{F}}}_{p})$ for
the associated semisimple residual representation (uniquely determined up to
conjugation).

If $k$ is a field then we write
$\mathrm{Proj}:\mathrm{GL}_{n}(k)\to\mathrm{PGL}_{n}(k)$ for the natural
projection and $\Delta:\mathrm{PGL}_{n}(k)\to k^{\times}/(k^{\times})^{n}$ for
the character induced by the determinant. We will only use these maps in the
case $n=2$.

If $K$ is a field of characteristic 0, $E$ is an elliptic curve curve over
$K$, and $p$ is a prime, then we write
${\overline{\rho}}_{E,p}:G_{K}\to\mathrm{GL}_{2}({\mathbb{F}}_{p})$ for the
representation associated to $H^{1}(E_{\overline{K}},{\mathbb{F}}_{p})$ after
a choice of basis. Thus
$\det{\overline{\rho}}_{E,p}=\overline{\epsilon}^{-1}$.

##  2 Lifting representations

In this section we study different kinds of liftings of representations:
liftings to characteristic 0 (and the automorphy of such liftings) and
liftings of projective representations to true (linear) representations. We
begin by discussing what it means for a (projective or linear) representation
to be automorphic.

###  2.1 Automorphy of linear and projective representations

Let $K$ be a CM or totally real number field. If $\pi$ is a cuspidal, regular
algebraic automorphic representation of $\mathrm{GL}_{2}(\mathbb{A}_{K})$ then
(see e.g. [Tay89, HLTT16]) for any isomorphism
$\iota:\overline{{\mathbb{Q}}}_{p}\to{\mathbb{C}}$, there exists a semisimple
representation
$r_{\iota}(\pi):G_{K}\to\mathrm{GL}_{2}(\overline{{\mathbb{Q}}}_{p})$
satisfying the following condition, which determines $r_{\iota}(\pi)$ uniquely
up to conjugation: for all but finitely many finite places $v$ of $K$ such
that $\pi_{v}$ is unramified, $r_{\iota}(\pi)|_{G_{K_{v}}}$ is unramified and
$r_{\iota}(\pi)|_{G_{K_{v}}}^{ss}$ is related to the representation
$\iota^{-1}\pi_{v}$ under the Tate-normalised unramified local Langlands
correspondence. (See [Tho16, §2] for an explanation of how the characterstic
polynomial of $r_{\iota}(\pi)|_{G_{K_{v}}}$ may be expressed in terms of the
eigenvalues explicit unramified Hecke operators.) In this paper we only need
to consider automorphic representations which are of regular algebraic
automorphic representations $\pi$ which are of weight 0, in the sense that for
each place $v|\infty$ of $K$, $\pi_{v}$ has the same infinitesimal character
as the trivial representation.

Let $k$ be a finite field of characteristic $p$, viewed inside its algebraic
closure $\overline{{\mathbb{F}}}_{p}$. In this paper, we say that a
representation $\rho:G_{K}\to\mathrm{GL}_{2}(k)$ is automorphic if it is
$\mathrm{GL}_{2}(\overline{{\mathbb{F}}}_{p})$-conjugate to a representation
of the form $\overline{r_{\iota}(\pi)}$, where $\pi$ is a cuspidal, regular
algebraic automorphic representation of $\mathrm{GL}_{2}({\mathbb{A}}_{K})$ of
weight 0. We say that a representation $\sigma:G_{K}\to\mathrm{PGL}_{2}(k)$ is
automorphic if it is $\mathrm{PGL}_{2}(\overline{{\mathbb{F}}}_{p})$-conjugate
to a representation of the form $\mathrm{Proj}(\overline{r_{\iota}(\pi)})$,
where $\pi$ is a cuspidal, regular algebraic automorphic representation of
$\mathrm{GL}_{2}({\mathbb{A}}_{K})$ of weight 0.

We say that a representation
$\rho:G_{K}\to\mathrm{GL}_{2}(\overline{{\mathbb{Q}}}_{p})$ is automorphic if
it is conjugate to a representation of the form $\overline{r_{\iota}(\pi)}$,
where $\pi$ is a cuspidal, regular algebraic automorphic representation of
$\mathrm{GL}_{2}({\mathbb{A}}_{K})$ of weight 0. We say that an elliptic curve
$E$ over $K$ is modular if the representation of $G_{K}$ afforded by
$H^{1}(E_{\overline{K}},{\mathbb{Q}}_{p})$ is automorphic in this sense.

######  Lemma 2.1.

Let $K$ be a CM or totally real number field, let
$\rho:G_{K}\to\mathrm{GL}_{2}(k)$ be a representation, and let
$\sigma=\mathrm{Proj}(\rho)$. Then:

  1. 1.

Let $\chi:G_{K}\to k^{\times}$ be a character. Then $\rho$ is automorphic if
and only if $\rho\otimes\chi$ is automorphic.

  2. 2.

$\sigma$ is automorphic if and only if $\rho$ is automorphic.

###### Proof.

If $\chi:G_{K}\to k^{\times}$ is a character then its Teichmüller lift
$X:G_{K}\to\overline{{\mathbb{Q}}}_{p}^{\times}$ is associated, by class field
theory, to a finite order Hecke character
$\Xi:{\mathbb{A}}_{K}^{\times}\to{\mathbb{C}}^{\times}$. If $\pi$ is a
cuspidal automorphic representation which is regular algebraic of weight 0 and
$\overline{r_{\iota}(\pi)}$ is conjugate to $\rho$, then
$\pi\otimes(\Xi\circ\det)$ is also cuspidal and regular algebraic of weight 0
and $\overline{r_{\iota}(\pi\otimes(\Xi\circ\det))}$ is conjugate to
$\rho\otimes\chi$.

It is clear from the definition that if $\rho$ is automorphic then so is
$\sigma$. Conversely, if $\sigma$ is automorphic then there is a cuspidal,
regular algebraic automorphic representation $\pi$ of
$\mathrm{GL}_{2}(\mathbb{A}_{K})$ and isomorphism
$\iota:\overline{{\mathbb{Q}}}_{p}\to{\mathbb{C}}$ such that
$\mathrm{Proj}(\overline{r_{\iota}(\pi}))=\mathrm{Proj}(\rho)$. It follows
that there exists a character
$\chi:G_{K}\to\overline{{\mathbb{F}}}_{p}^{\times}$ such that $\rho$ is
conjugate to $\overline{r_{\iota}(\pi)}\otimes\chi$. The automorphy of $\rho$
follows from the first part of the lemma. ∎

###  2.2 Lifting to characteristic 0

We recall a result on the existence of liftings with prescribed properties. We
first need to say what it means for a representation to be exceptional. If $K$
is a number field and $\sigma:G_{K}\to\mathrm{PGL}_{2}(k)$ is a projective
representation, we say that $\sigma$ is exceptional if it is
$\mathrm{PGL}_{2}(\overline{{\mathbb{F}}}_{p})$-conjugate to a representation
$\sigma^{\prime}:G_{K}\to\mathrm{PGL}_{2}({\mathbb{F}}_{5})$ such that
$\sigma^{\prime}(G_{K})$ contains $\mathrm{PSL}_{2}({\mathbb{F}}_{5})$ and the
character $(-1)^{\Delta\circ\sigma^{\prime}}\overline{\epsilon}$ is trivial.
(Here we write $(-1)^{\Delta\circ\sigma^{\prime}}$ for the composition of
$\Delta\circ\sigma^{\prime}$ with the unique isomorphism
${\mathbb{F}}_{5}^{\times}/({\mathbb{F}}_{5}^{\times})^{2}\cong\\{\pm 1\\}$.)
We say that a representation $\rho:G_{K}\to\mathrm{GL}_{2}(k)$ is exceptional
if $\mathrm{Proj}(\rho)$ is exceptional. If $K$ is totally real then this is
equivalent to the definition given in [KT17, §3]. The exceptional case is
often excluded in the statements of automorphy lifting theorems (the root
cause being the non-triviality of the group
$H^{1}(\sigma(G_{K}),\operatorname{Ad}^{0}\rho(1))$).

######  Theorem 2.2.

Let $K$ be a totally real field, let
${\overline{\rho}}:G_{K}\rightarrow\mathrm{GL}_{2}(k)$ be a representation of
$S$-type, and let $\psi:G_{K}\to\overline{{\mathbb{Z}}}_{p}^{\times}$ be a
continuous character lifting $\det{\overline{\rho}}$ such that $\psi\epsilon$
is of finite order. Suppose that the following conditions are satisfied:

  1. 1.

$p>2$ and ${\overline{\rho}}|_{G_{K(\zeta_{p})}}$ is absolutely irreducible.

  2. 2.

If $p=5$ then ${\overline{\rho}}$ is non-exceptional.

Then ${\overline{\rho}}$ lifts to a continuous representation
$\rho:G_{K}\to\mathrm{GL}_{2}(\overline{{\mathbb{Z}}}_{p})$ satisfying the
following conditions:

  1. 1.

For all but finitely many places $v$ of $F$, $\rho|_{G_{K_{v}}}$ is
unramified.

  2. 2.

$\det\rho=\psi$.

  3. 3.

For each place $v|p$ of $K$, $\rho|_{G_{K_{v}}}$ is potentially crystalline
and for each embedding $\tau:K_{v}\to\overline{{\mathbb{Q}}}_{p}$,
$\mathrm{HT}_{\tau}(\rho)=\\{0,1\\}$. Moreover, for any $v|p$ such that
${\overline{\rho}}|_{G_{K_{v}}}$ is reducible, we can assume that
$\rho|_{G_{K_{v}}}$ is ordinary, in the sense of [Tho16, §5.1].

###### Proof.

This follows from [Sno09, Theorem 7.6.1], on noting that the condition (A2)
there can be replaced by the more general condition that ${\overline{\rho}}$
is non-exceptional (indeed, the condition (A2) is used to invoke [Kis09,
Proposition 3.2.5], which is proved under this more general condition). To
verify the existence of a potentially crystalline lift of
${\overline{\rho}}|_{G_{K_{v}}}$ for each $v|p$ (or in the terminology of
_loc. cit._ , the compatibility of ${\overline{\rho}}|_{G_{K_{v}}}$ with type
$A$ or $B$) we apply [Sno09, Proposition 7.8.1] (when
${\overline{\rho}}|_{G_{K_{v}}}$ is irreducible) or [BLGG12, Lemma 6.1.6]
(when ${\overline{\rho}}|_{G_{K_{v}}}$ is reducible). ∎

We next recall an automorphy lifting theorem.

######  Theorem 2.3.

Let $K$ be a totally real number field, and let
$\rho:G_{K}\to\mathrm{GL}_{2}(\overline{{\mathbb{Z}}}_{p})$ be a continuous
representation satisfying the following conditions:

  1. 1.

$p>2$ and ${\overline{\rho}}|_{G_{K(\zeta_{p})}}$ is absolutely irreducible.

  2. 2.

For all but finitely many finite places $v$ of $K$, $\rho|_{G_{K_{v}}}$ is
unramified.

  3. 3.

For each place $v|p$ of $K$, $\rho|_{G_{K_{v}}}$ is de Rham and for each
embedding $\tau:K_{v}\to\overline{{\mathbb{Q}}}_{p}$,
$\mathrm{HT}_{\tau}(\rho)=\\{0,1\\}$.

  4. 4.

The representation ${\overline{\rho}}$ is automorphic.

Then $\rho$ is automorphic.

###### Proof.

This follows from [KT17, Theorem 9.3]. ∎

We now combine the previous two theorems to obtain a “solvable descent of
automorphy” theorem for residual representations, along similar lines to
[Kha00, Tay03].

######  Proposition 2.4.

Let $K$ be a totally real number field and let
${\overline{\rho}}:G_{K}\rightarrow\mathrm{GL}_{2}(k)$ be a representation of
$S$-type. Suppose that there exists a solvable totally real extension $L/K$
such that the following conditions are satisfied:

  1. 1.

$p>2$ and ${\overline{\rho}}|_{G_{L(\zeta_{p})}}$ is absolutely irreducible.
If $p=5$, then ${\overline{\rho}}$ is non-exceptional.

  2. 2.

${\overline{\rho}}|_{G_{L}}$ is automorphic.

Then ${\overline{\rho}}$ is automorphic.

###### Proof.

Let $\psi:G_{K}\to\overline{{\mathbb{Z}}}_{p}^{\times}$ be the character such
that $\psi\epsilon$ is the Teichmüller lift of
$(\det{\overline{\rho}})\overline{\epsilon}$, and let
$\rho:G_{K}\to\mathrm{GL}_{2}(\overline{{\mathbb{Z}}}_{p})$ be the lift of
$\rho$ whose existence is asserted by Theorem 2.2. Then Theorem 2.3 implies
the automorphy of $\rho|_{G_{L}}$, and the automorphy of $\rho$ itself and
hence of ${\overline{\rho}}$ follows by cyclic descent, using the results of
Langlands [Lan80]. ∎

We can now give the proof of Theorem 1.1, which we restate here for the
convenience of the reader:

######  Theorem 2.5.

Let $K$ be a totally real field and let
${\overline{\rho}}:G_{K}\rightarrow\mathrm{GL}_{2}(k)$ be a representation of
$S$-type. Suppose that $\lvert k\rvert\in\\{2,3,4,5,7,9\\}$. Then
${\overline{\rho}}$ is automorphic.

###### Proof.

Many of the results we quote here are stated in the case of $K={\mathbb{Q}}$
but hold more generally for totally real fields with minor modification. We
will apply them in the more general setting without further comment.

If ${\overline{\rho}}$ is dihedral then this is a consequence of results of
Hecke (see [Ser87, §5.1]). If $k={\mathbb{F}}_{3}$, it is a consequence of the
Langlands–Tunnell theorem [Tun81] (see the discussion following Theorem 5.1 in
[Wil95, Chapter 5]). We may thus assume for the remainder of the proof that
$\lvert k\rvert>3$. We may also assume that for any abelian extension $L/K$,
the restriction ${\overline{\rho}}|_{G_{L(\zeta_{p})}}$ is absolutely
irreducible (as otherwise ${\overline{\rho}}$ would be dihedral).

Next suppose that $k={\mathbb{F}}_{5}$. We note that ${\overline{\rho}}$ is
not exceptional, by [KT17, Lemma 3.1]. Let $L/K$ be the totally real cyclic
extension cut out by $(\det{\overline{\rho}})\overline{\epsilon}$. By [SBT97,
Theorem 1.2], there is an elliptic curve $E$ over $L$ such that
${\overline{\rho}}_{E,5}\cong{\overline{\rho}}|_{G_{L}}$ and
${\overline{\rho}}_{E,3}(G_{L})$ contains $\mathrm{SL}_{2}({\mathbb{F}}_{3})$.
By the $k={\mathbb{F}}_{3}$ case of the theorem and by Theorem 2.3, we see
that $E$ is automorphic, hence so is ${\overline{\rho}}|_{G_{L}}$. The
automorphy of ${\overline{\rho}}$ then follows from Proposition 2.4. The
$k={\mathbb{F}}_{7}$ case is similar, using [Man04, Proposition 3.1] instead
of [SBT97, Theorem 1.2].

Next suppose that $k={\mathbb{F}}_{4}$. We can twist $\rho$ to assume that it
is valued in $\mathrm{SL}_{2}({\mathbb{F}}_{4})$. Then [SBT97, Theorem 3.4]
shows that there is an abelian surface $A$ over $F$ with real multiplication
by ${\mathcal{O}}_{{\mathbb{Q}}(\sqrt{5})}$ such that the
$G_{K}$-representation on $A[2]\cong{\mathbb{F}}_{4}^{2}$ is isomorphic to
${\overline{\rho}}$ and such that the $G_{K}$-representation on
$A[\sqrt{5}]\cong{\mathbb{F}}_{5}^{2}$ has image containing
$\mathrm{SL}_{2}({\mathbb{F}}_{5})$. By the $k={\mathbb{F}}_{5}$ case of the
theorem, Theorem 2.2, and Theorem 2.3, we see that $A$ is automorphic, hence
so is ${\overline{\rho}}$.

Finally suppose that $k={\mathbb{F}}_{9}$. Let $L/K$ be the totally real
cyclic extension cut out by $(\det{\overline{\rho}})\overline{\epsilon}$. Then
the argument of [Ell05, §2.5] shows that there is a solvable totally real
extension $M/K$ containing $L/K$ and an abelian surface $A$ over $M$ with real
multiplication by ${\mathcal{O}}_{{\mathbb{Q}}(\sqrt{5})}$ such that the
$G_{M}$-representation on $A[3]\cong{\mathbb{F}}_{9}^{2}$ is isomorphic to
$\overline{\rho}|_{G_{M}}\otimes\overline{\epsilon}$ and such that the
$G_{M}$-representation on $A[\sqrt{5}]\cong{\mathbb{F}}_{5}^{2}$ has image
containing $\mathrm{SL}_{2}({\mathbb{F}}_{5})$. By the $k={\mathbb{F}}_{5}$
case of the theorem, Theorem 2.2, and Theorem 2.3, we see that $A$ is
automorphic, hence so is ${\overline{\rho}}|_{G_{M}}$. The automorphy of
${\overline{\rho}}$ follows from Proposition 2.4. ∎

######  Remark 2.6.

The ‘2-3 switch’ strategy employed in Theorem 3.1 below, can be used to prove
automorphy of totally odd representations
$\rho:G_{K}\to\mathrm{GL}_{2}({\mathbb{F}}_{3})$ without using the Langlands-
Tunnell theorem.

###  2.3 Lifting projective representations

We now consider the problem of lifting projective representations.

######  Lemma 2.7.

Let $K$ be a number field, and let $\sigma:G_{K}\to\mathrm{PGL}_{2}(k)$ be a
continuous homomorphism. Then there exists a finite extension $k^{\prime}/k$
such that $\sigma$ lifts to a homomorphism
$\rho:G_{K}\to\mathrm{GL}_{2}(k^{\prime})$.

###### Proof.

The obstruction to lifting a continuous homomorphism
$\sigma:G_{K}\to\mathrm{PGL}_{2}(k)$ to a continuous homomorphism
$\rho:G_{K}\to\mathrm{GL}_{2}(\overline{k})$ lies in
$H^{2}(G_{K},\overline{k}^{\times})$. Tate proved that
$H^{2}(G_{K},\overline{k}^{\times})=0$ (see [Ser77, §6.5]) so a lift always
exists. ∎

######  Lemma 2.8.

Suppose that $p>2$, let $K$ be a number field, and let
$\sigma:G_{K}\to\mathrm{PGL}_{2}(k)$ be a homomorphism. Let $S$ be a finite
set of places of $K$ such that for each $v\in S$, there exists a lift of
$\sigma|_{G_{K_{v}}}$ to a homomorphism $\rho_{v}:G_{K}\to\mathrm{GL}_{2}(k)$.
Then we can find the following data:

  1. 1.

A solvable $S$-split extension $L/K$.

  2. 2.

A homomorphism $\rho:G_{L}\to\mathrm{GL}_{2}(k)$ such that
$\mathrm{Proj}(\rho)=\sigma|_{G_{L}}$ and for each $v\in S$ and each place
$w|v$ of $L$, $\rho|_{G_{L_{w}}}=\rho_{v}$.

Moreover, if $K$ is a CM field we can choose $L$ also to be a CM field.

###### Proof.

Let $H$ denote the $2$-Sylow subgroup of $k^{\times}$, of order $2^{m}$, and
let $H^{\prime}\leq k^{\times}$ denote its prime-to-2 complement. If $0\leq
k\leq m$, we write $G_{k}=\mathrm{GL}_{2}(k)/(2^{m-k}H\times H^{\prime})$,
which is an extension

$1\to H/2^{m-k}H\to G_{k}\to\mathrm{PGL}_{2}(k)\to 1.$

We show by induction on $k\geq 0$ that we can find a solvable, $S$-split
extension $L_{k}/K$ and a homomorphism $\rho_{k}:G_{L_{k}}\to G_{k}$ lifting
$\sigma|_{G_{L_{k}}}$ and such that for each $v\in S$ and each place $w|v$ of
$L_{k}$, $\rho_{k}|_{G_{L_{k,w}}}=\rho_{v}\text{ mod }2^{m-k}H\times
H^{\prime}$. The case $k=0$ is the existence of $\sigma$. The case $k=m$
implies the statement of the lemma, since $\mathrm{GL}_{2}(k)=G_{m}\times
H^{\prime}$. (Note [AT09, Ch. X, Theorem 5] implies that any collection of
characters $\chi_{v}:G_{K_{v}}\to H^{\prime}$ can be globalised to a character
$\chi:G_{K}\to H^{\prime}$.)

For the induction step, suppose the induction hypothesis holds for a fixed
value of $k$. We consider the obstruction to lifting $\rho_{k}$ to a
homomorphism $\rho_{k+1}:G_{L_{k}}\to G_{k+1}$. This defines an element of
$H^{2}(G_{L_{k}},{\mathbb{Z}}/2{\mathbb{Z}})$ which is locally trivial at the
places of $L_{k}$ lying above $S$. We can therefore find an extension of the
form $L_{k+1}=L_{k}\cdot E_{k+1}$, where $E_{k+1}/K$ is a solvable $S$-split
extension, such that the image of this obstruction class in
$H^{2}(G_{L_{k+1}},{\mathbb{Z}}/2{\mathbb{Z}})$ vanishes and so there is a
homomorphism $\rho^{\prime}_{k+1}:G_{L_{k+1}}\to G_{k+1}$ lifting
$\rho_{k}|_{G_{L_{k+1}}}$.

If $v\in S$ and $w|v$ is a place of $L_{k+1}$ then there is a character
$\chi_{w}:G_{L_{k+1,w}}\to{\mathbb{Z}}/2{\mathbb{Z}}$ such that
$\rho^{\prime}_{k+1}|_{G_{L_{k+1,w}}}=(\rho_{v}\text{ mod }2^{m-(k+1)}H\times
H^{\prime})\cdot\chi_{w}$. We can certainly find a character
$\chi:G_{L_{k+1}}\to{\mathbb{Z}}/2{\mathbb{Z}}$ such that
$\chi|_{G_{L_{k+1,w}}}=\chi_{w}$ for each such place $w$. The induction step
is complete on taking $\rho_{k+1}=\rho^{\prime}_{k+1}\cdot\chi$.

It remains to explain why we can choose $K$ to be CM if $L$ is. Since the
extensions $E_{k}$ in the proof are required only to satisfy some local
conditions, which are vacuous if $K$ is CM, we can choose the fields $E_{k}$
to be of the form $KE_{k}’$ where $E_{k}’$ is a totally real extension, in
which case the field $L$ constructed in the proof is seen to be CM. ∎

######  Remark 2.9.

We remark that if $v$ is a real place of $K$ and $\sigma(c_{v})\neq 1$, then
there exists a lift of $\sigma|_{G_{K_{v}}}$ to $\mathrm{GL}_{2}(k)$ if and
only if either $-1$ is a square in $k^{\times}$ and
$\Delta\circ\sigma(c_{v})=1$, or $-1$ is not a square in $k^{\times}$ and
$\Delta\circ\sigma(c_{v})\neq 1$. We also note the utility of the ‘$S$-split’
condition: we can add any set of places at which $\sigma$ is unramified to
$S$, and in this way ensure that the $S$-split extension $L/K$ is linearly
disjoint from any other fixed finite extension of $K$.

Here is a variant.

######  Lemma 2.10.

Suppose that $p>2$. Let $K$ be a number field, let
$\sigma:G_{K}\to\mathrm{PGL}_{2}(k)$ be a homomorphism, and let $\chi:G_{K}\to
k^{\times}$ be a character. Suppose that the following conditions are
satisfied:

  1. 1.

$\Delta\circ\sigma=\chi\text{ mod }(k^{\times})^{2}$.

  2. 2.

For each finite place $v$ of $K$, $\sigma|_{G_{K_{v}}}$ and
$\chi|_{G_{K_{v}}}$ are unramified.

  3. 3.

For each real place $v$ of $K$, $\sigma(c_{v})\neq 1$ and $\chi(c_{v})=-1$.

Then there exists a homomorphism $\rho:G_{K}\to\mathrm{GL}_{2}(k)$ such that
$\mathrm{Proj}(\rho)=\sigma$ and $\det(\rho)=\chi$.

###### Proof.

We consider the short exact sequence of groups

$1\to\\{\pm
1\\}\to\mathrm{GL}_{2}(k)\to\mathrm{PGL}_{2}(k)\times_{\Delta}k^{\times}\to
1,$

where the last group is the subgroup of
$(g,\alpha)\in\mathrm{PGL}_{2}(k)\times k^{\times}$ such that
$\Delta(g)=\alpha\text{ mod }(k^{\times})^{2}$. By hypothesis the pair
$(\sigma,\chi)$ defines a homomorphism
$\Sigma:G_{K}\to\mathrm{PGL}_{2}(k)\times_{\Delta}k^{\times}$ such that for
every place $v$ of $K$, $\Sigma|_{G_{K_{v}}}$ lifts to $\mathrm{GL}_{2}(k)$
(see Remark 2.9). The subgroup of locally trivial elements of
$H^{2}(G_{K},\\{\pm 1\\})$ is trivial, by class field theory, so $\Sigma$
lifts to a homomorphism $\rho:G_{K}\to\mathrm{GL}_{2}(k)$, as required. ∎

We now prove an analogue of Proposition 2.4 for projective representations.

######  Proposition 2.11.

Let $K$ be a totally real number field and let
$\sigma:G_{K}\to\mathrm{PGL}_{2}(k)$ be a representation of $S$-type. Suppose
that there exists a solvable totally real extension $L/K$ satisfying the
following conditions:

  1. 1.

$p>2$ and $\sigma|_{G_{L(\zeta_{p})}}$ is absolutely irreducible. If $p=5$,
then $\sigma$ is non-exceptional.

  2. 2.

$\sigma|_{G_{L}}$ is automorphic.

Then $\sigma$ is automorphic.

###### Proof.

By Lemma 2.7, we can lift $\sigma$ to a representation
${\overline{\rho}}:G_{K}\to\mathrm{GL}_{2}(\overline{k})$. Then
${\overline{\rho}}|_{G_{L}}$ is automorphic and we can apply Proposition 2.4
to conclude that ${\overline{\rho}}$ is automorphic, hence that $\sigma$ is
automorphic. ∎

We are now in a position to establish a large part of Theorem 1.2.

######  Theorem 2.12.

Let $K$ be a totally real number field and let
$\sigma:G_{K}\to\mathrm{PGL}_{2}(k)$ be a representation of $S$-type. If one
of the following conditions holds, then $\sigma$ is automorphic:

  1. 1.

$|k|\in\\{2,3,4\\}$.

  2. 2.

$|k|=5$ or $9$ and $\Delta\circ\sigma$ is totally even. If $|k|=5$, then
$\sigma$ is non-exceptional.

  3. 3.

$|k|=7$ and $\Delta\circ\sigma$ is totally odd.

###### Proof.

When $k={\mathbb{F}}_{2}$ or ${\mathbb{F}}_{4}$, the map
$\mathrm{SL}_{2}(k)\to\mathrm{PGL}_{2}(k)$ is an isomorphism, so $\sigma$
trivially lifts to a $\mathrm{GL}_{2}(k)$ representation and we can apply
Theorem 2.5. The case when $|k|=3$ follows from [Tun81]. In the other cases,
we can assume that $\sigma|_{G_{K(\zeta_{p})}}$ is absolutely irreducible (as
otherwise $\sigma$ lifts to a dihedral representation). Let $S_{\infty}$ be
the set of infinite places of $K$ and choose a finite set $S^{\prime}$ of
finite places of $K$ at which $\sigma$ is unramified such that
$\operatorname{Gal}(\overline{K}^{\ker(\sigma|_{G_{K(\zeta_{p})}})}/K)$ is
generated by $\\{\operatorname{Frob}_{v}\\}_{v\in S^{\prime}}$. We can apply
Lemma 2.8, see also Remark 2.9, with $S=S_{\infty}\cup S^{\prime}$ to find a
solvable, totally real extension $L/K$ such that $\sigma$ lifts to a
representation ${\overline{\rho}}:G_{L}\to\mathrm{GL}_{2}(k)$ such that
${\overline{\rho}}|_{G_{L(\zeta_{p})}}$ is absolutely irreducible and
${\overline{\rho}}$ is not exceptional if $p=5$. Then Theorem 2.5 implies the
automorphy of ${\overline{\rho}}$ and Proposition 2.11 implies the automorphy
of $\sigma$, as desired. ∎

##  3 Modularity of mod $3$ representations

In this section, which is a warm-up for the next one, we give a proof of the
following theorem that does not depend on the Langlands–Tunnell theorem:

######  Theorem 3.1.

Let $K$ be a totally real number field, and let
$\sigma:G_{K}\to\mathrm{PGL}_{2}({\mathbb{F}}_{3})$ be a representation of
$S$-type such that $\Delta\circ\sigma$ is totally odd. Then $\sigma$ is
automorphic.

###### Proof.

We can assume that $\sigma$ is not dihedral; by the classification of finite
subgroups of $\mathrm{PGL}_{2}({\mathbb{F}}_{3})$, we can therefore assume
that $\sigma(G_{K})$ contains $\mathrm{PSL}_{2}({\mathbb{F}}_{3})$. By
Proposition 2.11, we can moreover assume, after replacing $K$ by a solvable
totally real extension, that $\sigma$ is everywhere unramified and that for
each place $v|2$ of $K$, $q_{v}\equiv 1\text{ mod }3$ and
$\sigma|_{G_{K_{v}}}$ is trivial.

######  Lemma 3.2.

There exists a solvable totally real extension $L/K$ and a modular elliptic
curve $E$ over $L$ satisfying the following conditions:

  1. 1.

$\sigma(G_{L})$ contains $\mathrm{PSL}_{2}({\mathbb{F}}_{3})$. In particular,
$\sigma|_{G_{L}}$ is of $S$-type.

  2. 2.

The homomorphism $\mathrm{Proj}({\overline{\rho}}_{E,3})$ is
$\mathrm{PGL}_{2}(\overline{{\mathbb{F}}}_{3})$-conjugate to
$\sigma|_{G_{L}}$.

###### Proof.

The character $(\Delta\circ\sigma)\omega:G_{K}\to{\mathbb{F}}_{3}^{\times}$ is
totally even, so cuts out a totally real (trivial or quadratic) extension
$L/K$, and $\sigma(G_{L})$ contains $\mathrm{PSL}_{2}({\mathbb{F}}_{3})$ and
satisfies $\Delta\circ\sigma|_{G_{L}}=\omega$. Using Lemma 2.10, we can find a
lift ${\overline{\rho}}:G_{L}\to\mathrm{GL}_{2}({\mathbb{F}}_{3})$ of
$\sigma|_{G_{L}}$ satisfying the following conditions:

  * •

$\det{\overline{\rho}}=\overline{\epsilon}^{-1}$.

  * •

For each place $v|2$ of $L$, ${\overline{\rho}}|_{G_{L}}$ is trivial.

  * •

${\overline{\rho}}(G_{L})$ contains $\mathrm{SL}_{2}({\mathbb{F}}_{3})$. In
particular, ${\overline{\rho}}|_{G_{L(\zeta_{3})}}$ is absolutely irreducible.

We can then apply [AKT, Lemma 9.7] to conclude that there exists an elliptic
curve $E/L$ satisfying the following conditions:

  * •

There is an isomorphism ${\overline{\rho}}_{E,3}\cong{\overline{\rho}}$.

  * •

For each place $v|2$ of $L$, $E$ has multiplicative reduction at $v$ and the
valuation at $v$ of the minimal discriminant of $E$ is 3.

  * •

${\overline{\rho}}_{E,2}(G_{L})=\mathrm{SL}_{2}({\mathbb{F}}_{2})$.

Then [All14, p. 1237, Corollary] implies that $E$ is modular, proving the
lemma. ∎

We see that $\sigma|_{G_{L}}$ is automorphic. We can then apply Proposition
2.11 to conclude that $\sigma$ itself is automorphic, as required. ∎

##  4 Modularity of mod $5$ representations

In this section we complete the proof of Theorem 1.2 by proving Theorem 4.1
below.

######  Theorem 4.1.

Let $K$ be a totally real field, and let
$\sigma:G_{K}\to\mathrm{PGL}_{2}({\mathbb{F}}_{5})$ be a representation of
$S$-type which is non-exceptional, and such that $\Delta\circ\sigma$ is
totally odd. Then $\sigma$ is automorphic.

###### Proof.

By the classification of subgroups of $\mathrm{PGL}_{2}({\mathbb{F}}_{5})$,
automorphic induction, and the Langlands–Tunnell theorem, we can assume that
$\sigma(G_{K})$ contains $\mathrm{PSL}_{2}({\mathbb{F}}_{5})$. By Theorem 2.2,
Lemma 2.7, and Proposition 2.11 we can assume, after possibly replacing $K$ be
a solvable totally real extension, that the following conditions are
satisfied:

  * •

There exists a representation
${\overline{\rho}}:G_{K}\to\mathrm{GL}_{2}(\overline{{\mathbb{F}}}_{5})$ such
that $\mathrm{Proj}({\overline{\rho}})$ is
$\mathrm{PGL}_{2}(\overline{{\mathbb{F}}}_{5})$-conjugate to $\sigma$.
Moreover, ${\overline{\rho}}$ is everywhere unramified.

  * •

There exists a representation
$\rho:G_{K}\to\mathrm{GL}_{2}(\overline{{\mathbb{Q}}}_{5})$ lifting
${\overline{\rho}}$, which is unramified almost everywhere.

  * •

For each place $v|5$ of $K$, $\zeta_{5}\in K_{v}$,
${\overline{\rho}}|_{G_{K_{v}}}$ is trivial, and $\rho|_{G_{K_{v}}}$ is
ordinary, in the sense of [Tho16, §5.1].

  * •

Let $\chi=\det\rho$. Then $\chi\epsilon$ has finite order prime to $5$ and for
each finite place $v$ of $K$, $\chi\epsilon|_{G_{K_{v}}}$ is unramified. In
particular, $\overline{\chi}$ is everywhere unramified.

Let $K^{\prime}/K$ denote the quadratic CM extension cut out by the character
$(\Delta\circ\sigma)\omega$.

######  Lemma 4.2.

The representation ${\overline{\rho}}|_{G_{K^{\prime}}}$ is decomposed generic
in the sense of [ACC+, Definition 4.3.1].

###### Proof.

It is enough to find a prime number $l$ such that $l$ splits in $K^{\prime}$
and for each place $v|l$ of $K^{\prime}$, $q_{v}\equiv 1\text{ mod }5$ and the
eigenvalues of ${\overline{\rho}}(\operatorname{Frob}_{v})$ are distinct. The
argument of [ACC+, Lemma 7.1.5, (3)] will imply the existence of such a prime
$l$ if we can show that if $M=K^{\prime}(\zeta_{5})$ and
$\widetilde{M}/{\mathbb{Q}}$ is the Galois closure of $M/{\mathbb{Q}}$, then
$\sigma(G_{\widetilde{M}})$ contains $\mathrm{PSL}_{2}({\mathbb{F}}_{5})$. To
see this, first let $\widetilde{K}/{\mathbb{Q}}$ be the Galois closure of
$K/{\mathbb{Q}}$. Then $\widetilde{K}$ is totally real, and so
$\sigma(G_{\widetilde{K}})=\sigma(G_{K})=\mathrm{PGL}_{2}({\mathbb{F}}_{5})$
because $\Delta\circ\sigma$ is totally odd. The extension
$M\widetilde{K}/\widetilde{K}$ is abelian, so $\widetilde{M}/\widetilde{K}$ is
abelian and $\sigma(G_{\widetilde{M}})$ must contain
$\mathrm{PSL}_{2}({\mathbb{F}}_{5})$. ∎

By construction,
$\Delta\circ\sigma|_{G_{K^{\prime}}}=\overline{\epsilon}^{-1}\text{ mod
}({\mathbb{F}}_{5}^{\times})^{2}$, so by Lemma 2.10,
$\sigma|_{G_{K^{\prime}}}$ lifts to a continuous homomorphism
$\tau:G_{K^{\prime}}\to\mathrm{GL}_{2}({\mathbb{F}}_{5})$ such that
$\det\tau=\overline{\epsilon}^{-1}$. In particular, there is a character
$\overline{\psi}:G_{K^{\prime}}\to\overline{{\mathbb{F}}}_{5}^{\times}$ such
that $\tau={\overline{\rho}}|_{G_{K^{\prime}}}\otimes\overline{\psi}$. Let
$\psi$ denote the Teichmüller lift of $\overline{\psi}$; then the determinant
of $\rho|_{G_{K^{\prime}}}\otimes\psi$ equals $\epsilon^{-1}$.

######  Lemma 4.3.

The representation $\tau$ satisfies the following conditions:

  1. 1.

$\tau|_{G_{K^{\prime}(\zeta_{5})}}$ is absolutely irreducible and $\tau$ is
non-exceptional.

  2. 2.

$\tau$ is decomposed generic.

###### Proof.

The representation $\tau|_{G_{K^{\prime}(\zeta_{5})}}$ is absolutely
irreducible because its projective image contains
$\mathrm{PSL}_{2}({\mathbb{F}}_{5})$. If $\zeta_{5}\in K^{\prime}$ then
$\sqrt{5}\in K$ and so $K^{\prime}=K(\Delta\circ\sigma)=K(\zeta_{5})$; this
possibility is ruled out because $\sigma$ is non-exceptional. It follows that
$\tau$ is non-exceptional. The representation $\tau$ is decomposed generic
because ${\overline{\rho}}|_{G_{K^{\prime}}}$ is (and this condition only
depends on the associated projective representation). ∎

Thanks to the lemma, we can apply [AKT, Lemma 9.7] and [AKT, Corollary 9.13]
to conclude the existence of a modular elliptic curve $E$ over $K^{\prime}$
such that ${\overline{\rho}}_{E,5}\cong\tau$ and for each place $v|5$ of
$K^{\prime}$, $E$ has multiplicative reduction at the place $v$. We can then
apply the automorphy lifting theorem [AKT, Theorem 8.1] to conclude that
$\rho|_{G_{K^{\prime}}}\otimes\psi$ is automorphic, hence that
$\rho|_{G_{K^{\prime}}}$ is automorphic. It follows by cyclic descent [Lan80]
that $\rho$ and hence $\sigma$ are also automorphic, and this completes the
proof. ∎

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