# Deformations of Certain Reducible Galois Representations, III

Anwesh Ray  Cornell University
Department of Mathematics
Malott Hall, Ithaca, NY 14853-4201 USA<EMAIL_ADDRESS>
###### Abstract.

Let $p$ be an odd prime and $q$ a power of $p$. We examine the deformation
theory of reducible and indecomposable Galois representations
$\bar{\rho}:\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GSp}_{2n}(\mathbb{F}_{q})$
that are unramified outside a finite set of primes $S$ and whose image lies in
a Borel subgroup. We show that under some additional hypotheses, such
representations have geometric lifts to the Witt vectors
$\text{W}(\mathbb{F}_{q})$. The main theorem of this manuscript is a higher
dimensional generalization of the result of Hamblen-Ramakrishna [6].

##  1\. Introduction

Let $p$ be an odd prime number and let $\mathbb{F}_{q}$ be the finite field
with $q=p^{M}$ elements. Denote by $\text{W}(\mathbb{F}_{q})$ the ring Witt
vectors of $\mathbb{F}_{q}$. For each prime number $l$, choose an embedding
$\iota_{l}:\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{l}$. Denote by
$\operatorname{G}_{l}$ the absolute Galois group
$\operatorname{Gal}(\bar{\mathbb{Q}}_{l}/\mathbb{Q}_{l})$. Let $S$ be a finite
set of prime numbers containing $p$. Denote by $\mathbb{Q}_{S}$ the maximal
algebraic extension of $\mathbb{Q}$ which is unramified at each prime $l\notin
S$ and set
$\operatorname{G}_{\mathbb{Q},S}=\operatorname{Gal}(\mathbb{Q}_{S}/\mathbb{Q})$.
The weak form of Serre’s conjecture states that an odd and irreducible two-
dimensional Galois representation

$\bar{\varrho}:\operatorname{G}_{\mathbb{Q},S}\rightarrow\text{GL}_{2}(\mathbb{F}_{q})$

is modular, i.e., lifts to a characteristic zero representation $\varrho$
attached to an eigencuspform. The strong form asserts that the eigencuspform
may be chosen to have optimal level equal to the prime to $p$ part of the
Artin conductor of $\bar{\varrho}$. Ribet [17] proved via a level lowering
argument that the weak form implies the strong form. Khare-Wintenberger [9]
went on to prove the full statement, building on Ribet’s work.

In [6], Hamblen and Ramakrishna prove a generalization of the weak form of
Serre’s conjecture for reducible two-dimensional Galois representations. They
impose some conditions on a two-dimensional representation
$\bar{\varrho}:\operatorname{G}_{\mathbb{Q},S}\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{q})$,
namely,

  1. (1)

$\bar{\varrho}$ is reducible of the form

$\bar{\varrho}=\left({\begin{array}[]{cc}\varphi&\ast\\\ &1\\\
\end{array}}\right),$

where
$\varphi:\operatorname{G}_{\mathbb{Q},S}\rightarrow\mathbb{F}_{q}^{\times}$ is
a Galois character.

  2. (2)

The representation $\bar{\varrho}$ is odd, i.e. if
$c\in\operatorname{G}_{\mathbb{Q}}$ denotes complex conjugation,

$\det\bar{\varrho}(c)=-1.$

  3. (3)

The representation $\bar{\varrho}$ is indecomposable, i.e. the cohomology
class

$\ast\in H^{1}(\operatorname{G}_{\mathbb{Q},S},\mathbb{F}_{q}(\varphi))$

is non-zero.

  4. (4)

The Galois character $\varphi$ is stipulated to satisfy some conditions, for
instance, $\varphi\neq\bar{\chi},\bar{\chi}^{-1}$ where $\bar{\chi}$ is the
mod $p$ cyclotomic character and $\varphi^{2}\neq 1$. Further, the image of
$\varphi$ is stipulated to span $\mathbb{F}_{q}$ over $\mathbb{F}_{p}$.

  5. (5)

There are further conditions on the restriction
$\bar{\varrho}_{\restriction\operatorname{G}_{p}}$. The reader may refer to
condition 5 of Theorem 2 in [6] for further details.

Hamblen and Ramakrishna show that if $\bar{\varrho}$ satisfies the above
mentioned conditions, then on enlarging the set of ramification $S$ by a
finite set of primes $X$, $\bar{\varrho}$ has an odd, irreducible,
$p$-ordinary lift $\varrho$ which is unramified outside $S\cup X$

$\operatorname{G}_{\mathbb{Q},S\cup
X}$$\operatorname{G}_{\mathbb{Q},S}$$\text{GL}_{2}(\mathbb{F}_{q}).$$\text{GL}_{2}(\text{W}(\mathbb{F}_{q}))$$\bar{\varrho}$$\varrho$

This lift is geometric in the sense of Fontaine and Mazur [4]. By the result
of Skinner and Wiles in [18], the representation $\varrho$ arises from a
$p$-ordinary eigencuspform. This settles the weak form of Serre’s conjecture
for such $\bar{\varrho}$.

The prospect of generalizing this lifting result leads us to examine higher
dimensional Galois representations with image in a smooth group-scheme
$\operatorname{G}$ over $\text{W}(\mathbb{F}_{q})$. Assume that
$\operatorname{G}_{\restriction\mathbb{F}_{q}}$ is split and reductive and
choose a split Borel
$\operatorname{B}_{/\mathbb{F}_{q}}\subset\operatorname{G}_{\restriction\mathbb{F}_{q}}$.
Let $\bar{\rho}$ be a homomorphism

$\bar{\rho}:\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{G}(\mathbb{F}_{q}).$

Let $\mathfrak{g}$ denote the Lie-algebra of the adjoint group
$\operatorname{G}_{\restriction\mathbb{F}_{q}}^{ad}$ and
$\Phi(\operatorname{G}_{\restriction\mathbb{F}_{q}}^{ad})$ be a root system
compatible with the choice of Borel. Denote by
$\mathfrak{n}\subset\mathfrak{g}$ the span of the positive roots. The
$\mathbb{F}_{q}$-vector space $\mathfrak{g}$ acquires an adjoint Galois action

$\operatorname{Ad}^{0}\bar{\rho}:\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{Aut}_{\mathbb{F}_{q}}(\mathfrak{g}).$

Denote by $\operatorname{Ad}^{0}\bar{\rho}$ the Galois module with underlying
vector space $\mathfrak{g}$. It is imperative that $\bar{\rho}$ is odd. For an
involution
$\tau\in\operatorname{Aut}(\operatorname{G}_{\restriction\mathbb{F}_{q}})$,
let $(\operatorname{Ad}^{0}\bar{\rho})^{\tau}$ denote the subspace of
$\operatorname{Ad}^{0}\bar{\rho}$ fixed by $\tau$. It was shown by E. Cartan
that

$\dim(\operatorname{Ad}^{0}\bar{\rho})^{\tau}\geq\dim\mathfrak{n}$

(see [21, Proposition 2.2] for further details). The representation
$\bar{\rho}$ is odd if equality is achieved for the involution
$\operatorname{ad}\bar{\rho}(c)$, i.e.

(1.1)
$\dim(\operatorname{Ad}^{0}\bar{\rho})^{\operatorname{ad}\bar{\rho}(c)}=\dim\mathfrak{n}.$

In particular, the group $\operatorname{G}$ must contain an element
$h=\bar{\rho}(c)$ for which equality 1.1 is achieved. Such an element is said
to induce a Chevalley involution. When $n>2$, the general linear group
$\operatorname{GL}_{n}(\mathbb{F}_{q})$ contains no such element. Hence there
are no odd representations for the group
$\operatorname{GL}_{n}(\mathbb{F}_{q})$ when $n>2$. On the other hand, the
general symplectic group $\operatorname{GSp}_{2n}(\mathbb{F}_{q})$ does
contain elements which induce Chevalley involutions.

Ramakrishna in [13] and [14] showed that odd, irreducible representations
$\bar{\rho}:\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(\bar{\mathbb{F}}_{p})$
satisfying some additional hypotheses exhibit characteristic zero lifts which
are geometric in the sense of Fontaine and Mazur. These results provided
evidence for Serre’s conjecture, before it was proved by Khare and
Wintenberger. Taylor in [19] introduced a reformulation of Ramakrishna’s
method, by showing that the vanishing of a certain dual Selmer group is
sufficient in asserting the existence of global Galois deformations with fixed
local conditions. This new formulation paved the way for higher dimensional
generalizations. In [11], Patrikis generalized Ramakrishna’s lifting theorem
to odd representations with big image in
$\operatorname{GSp}_{2n}(\mathbb{F}_{q})$. Fakhruddin, Khare and Patrikis
studied more general odd, irreducible representations in [2] and [3].

We assume that our representation has image in
$\operatorname{GSp}_{2n}(\mathbb{F}_{q})$ for $n\geq 2$. Associate to a
commutative $\text{W}(\mathbb{F}_{q})$-algebra $R$, a non-degenerate
alternating form on $R^{2n}$ prescribed by the matrix

$J:=\left({\begin{array}[]{cc}&\operatorname{Id}_{n}\\\
-\operatorname{Id}_{n}&\\\ \end{array}}\right).$

The group of general symplectic matrices $\operatorname{GSp}_{2n}(R)$ consists
of matrices $X$ which preserve this form up to a scalar i.e. satisfy
$X^{t}JX\in R^{\times}\cdot J$. The similitude character
$\nu:\operatorname{GSp}_{2n}(R)\rightarrow R^{\times}$ is defined by the
relation $X^{t}JX=\nu(X)\cdot J$. The space $\operatorname{Ad}^{0}\bar{\rho}$
is an $\mathbb{F}_{q}[\operatorname{G}_{\mathbb{Q},S}]$-module with underlying
space $\operatorname{sp}_{2n}(\mathbb{F}_{q})$. The Galois action is
prescribed by

$g\cdot X=\bar{\rho}(g)X\bar{\rho}(g)^{-1}$

where $g\in\operatorname{G}_{\mathbb{Q},S}$ and
$X\in\operatorname{sp}_{2n}(\mathbb{F}_{q})$. Let $\operatorname{B}(R)$ be the
Borel subgroup consisting of matrices

$M=\left({\begin{array}[]{cc}C&CD\\\ &\xi(C^{t})^{-1}\\\ \end{array}}\right)$

where $C\in\operatorname{GL}_{n}(R)$ is upper triangular,
$D\in\operatorname{GL}_{n}(R)$ is symmetric and $\xi\in R^{\times}$. Note that
in this setting, $\operatorname{B}$ is defined over
$\text{W}(\mathbb{F}_{q})$. Denote by $U_{1}\subset\operatorname{B}$ the
unipotent subgroup.

Let
$\bar{\rho}:\operatorname{G}_{\mathbb{Q},S}\rightarrow\operatorname{GSp}_{2n}(\mathbb{F}_{q})$
be a continuous Galois representation with image in
$\operatorname{B}(\mathbb{F}_{q})$. Composing $\bar{\rho}$ with the
similitude-character $\bar{\nu}$ defines a Galois character denoted by
$\bar{\kappa}$. Denote by $\mathcal{T}\subseteq\operatorname{GSp}_{2n}$ the
diagonal torus and $e_{i,j}\in\operatorname{GL}_{2n}(\mathbb{F}_{q})$ the
matrix with $1$ in the $(i,j)$-position and $0$ in all other positions. Set
$\mathfrak{t}$ for the $\mathbb{F}_{q}$-span of $H_{1},\dots,H_{n}$, where
$H_{i}:=e_{i,i}-e_{n+i,n+i}$. Let $L_{1},\dots,L_{n}\in\mathfrak{t}^{*}$ be
the dual basis of $H_{1},\dots,H_{n}$. An integer linear combination $\lambda$
of $L_{1},\dots,L_{n}$ is viewed as character on the torus
$\mathcal{T}(\mathbb{F}_{q})$, which is trivial on the center of
$\operatorname{GSp}_{2n}(\mathbb{F}_{q})$. Via the natural quotient map
$\operatorname{B}\rightarrow\mathcal{T}$, a character on $\mathcal{T}$ induces
a character on $\operatorname{B}$. The character on $\operatorname{B}$ induced
by $\lambda$ is denoted by

$\omega_{\lambda}:\operatorname{B}(\mathbb{F}_{q})\rightarrow\mathbb{F}_{q}^{\times}.$

Associated to $\omega_{\lambda}$ is the Galois character

$\sigma_{\lambda}=\omega_{\lambda}\circ\bar{\rho}:\operatorname{G}_{\mathbb{Q},S}\rightarrow\mathbb{F}_{q}^{\times}.$

Let ”$1$” be a formal symbol for the trivial linear combination of
$L_{1},\dots,L_{n}$ and set $\sigma_{1}$ equal to the trivial character. The
roots $\Phi=\Phi(\operatorname{Ad}^{0}\bar{\rho},\mathfrak{t})$ are specified
by

$\begin{split}\Phi=&\\{\pm 2L_{1},\dots,\pm 2L_{n}\\}\\\
\cup&\\{\pm(L_{i}+L_{j})\mid 1\leq i<j\leq n\\}\\\
\cup&\\{\pm(L_{i}-L_{j})\mid 1\leq i<j\leq n\\}.\end{split}$

The choice of the Borel $\operatorname{B}\subset\operatorname{GSp}_{2n}$
prescribes the following choice of simple roots $\Delta=\\{\lambda_{i}\mid
i=1,\dots,n\\}$, with $\lambda_{i}:=L_{i}-L_{i+1}$ for $i<n$ and
$\lambda_{n}:=2L_{n}$. The root
$2L_{1}=2\left(\sum_{i=1}^{n-1}\lambda_{i}\right)+\lambda_{n}$ is the highest
root and the unique root of height $2n-1$. Denote by $\mathfrak{b}$ and
$\mathfrak{n}$ the Lie subalgebras of $\operatorname{Ad}^{0}\bar{\rho}$
corresponding to the Borel and unipotent subgroups respectively. Set $\chi$
for the $p$-adic cyclotomic character, $\bar{\chi}$ its mod-$p$ reduction and
let $c$ denote complex conjugation.

######  Theorem 1.1.

Let
$\bar{\rho}:\operatorname{G}_{\mathbb{Q},S}\rightarrow\operatorname{B}(\mathbb{F}_{q})$
be a Galois representation of the form:

(1.2) $\bar{\rho}=\begin{pmatrix}\varphi_{1}&*&*&\cdots&*&*&\cdots&*\\\
&\varphi_{2}&*&\cdots&*&*&\cdots&*\\\
&&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\\ &&&\varphi_{n}&*&*&\cdots&*\\\
&&&&\varphi_{1}^{-1}\bar{\kappa}&&&\\\ &&&&*&\varphi_{2}^{-1}\bar{\kappa}&&\\\
&&&&\vdots&\vdots&\ddots&\\\ &&&&*&*&\cdots&\varphi_{n}^{-1}\bar{\kappa}\\\
\end{pmatrix}$

and let $S$ be a finite set of primes which contains $p$. Assume that the
following conditions are satisfied:

  1. (1)

$p>2n$,

  2. (2)

$\bar{\rho}$ is odd, i.e.
$\dim(\operatorname{Ad}^{0}\bar{\rho})^{\operatorname{ad}\bar{\rho}(c)}=\dim\mathfrak{n}$.

  3. (3)

The image of $\bar{\rho}$ contains the unipotent subgroup
$\operatorname{U}_{1}(\mathbb{F}_{q})$.

  4. (4)

Both the following conditions on the distinctness of the characters
$\\{\sigma_{\lambda}\\}$ are satisfied:

    1. (a)

For $\lambda,\lambda^{\prime}\in\Phi\cup\\{1\\}$ such that
$\lambda\neq\lambda^{\prime}$, $\sigma_{\lambda}$ is not a
$\operatorname{Gal}(\mathbb{F}_{q}/\mathbb{F}_{p})$-twist of
$\sigma_{\lambda^{\prime}}$.

    2. (b)

Moreover for $\lambda,\lambda^{\prime}\in\Phi\cup\\{1\\}$ not necessarily
distinct, $\sigma_{\lambda}$ is not a
$\operatorname{Gal}(\mathbb{F}_{q}/\mathbb{F}_{p})$-twist of
$\bar{\chi}\sigma_{\lambda^{\prime}}$.

  5. (5)

For each of the roots $\lambda\in\Phi$, the $\mathbb{F}_{p}$-linear span of
the image of $\sigma_{\lambda}$ in $\mathbb{F}_{q}$ is $\mathbb{F}_{q}$.

  6. (6)

At each prime $v\in S$ such that $v\neq p$, there is a liftable local
deformation condition $\mathcal{C}_{v}$ with tangent space $\mathcal{N}_{v}$
of dimension

$\dim\mathcal{N}_{v}=h^{0}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho}).$

  7. (7)

Tilouine’s regularity conditions $(\operatorname{REG})$ and
$(\operatorname{REG})^{*}$ are satisfied, i.e.

$H^{0}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho}/\mathfrak{b})=0\text{
and
}H^{0}(\operatorname{G}_{p},(\operatorname{Ad}^{0}\bar{\rho}/\mathfrak{b})(\bar{\chi}))=0.$

Let $\kappa$ be a fixed choice of a lift of the character $\bar{\kappa}$ such
that $\kappa=\kappa_{0}\chi^{k}$, where $k$ is a positive integer divisible by
$p(p-1)$ and $\kappa_{0}$ is the Teichmüller lift of $\bar{\kappa}$. Then
$\exists$ a finite set of auxiliary primes $X$ disjoint from $S$ and a lift
$\rho$

$\operatorname{G}_{\mathbb{Q},S\cup
X}$$\operatorname{G}_{\mathbb{Q},S}$$\operatorname{GSp}_{2n}(\mathbb{F}_{q}).$$\operatorname{GSp}_{2n}(\text{W}(\mathbb{F}_{q}))$$\bar{\rho}$$\rho$

for which

  1. (1)

$\rho$ is irreducible,

  2. (2)

$\rho$ is $p$-ordinary (in the sense of [12, section 4.1]),

  3. (3)

$\nu\circ\rho=\kappa$,

  4. (4)

for $v\in S\backslash\\{p\\}$, the restriction to the decomposition group
$\rho_{\restriction\operatorname{G}_{v}}\in\mathcal{C}_{v}$.

The lift $\rho$ is geometric in the sense of Fontaine and Mazur. For
$\lambda\in\Phi$, setting $\lambda=-\lambda^{\prime}$ in condition
$\eqref{thc4}$, we have that $\sigma_{\lambda}^{2}\neq 1$. Note that the
conditions also imply that $\sigma_{\lambda}\neq\bar{\chi},\bar{\chi}^{-1}$.
This is reminiscent of the condition $\varphi^{2}\neq 1$ of Hamblen-
Ramakrishna. In particular, condition $\eqref{thc4}$ implies that $p>2$. The
requirement $p>2n$ is primarily made so that we may suitably work with the
exponential map in various places.

It is a consequence of Tilouine’s regularity conditions that the ordinary
deformations of $\bar{\rho}_{\restriction\operatorname{G}_{p}}$ constitute a
liftable deformation condition $\mathcal{C}_{p}$ for which the tangent space
$\mathcal{N}_{p}$ has dimension:

$\dim\mathcal{N}_{p}=h^{0}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho})+\dim\mathfrak{n}.$

For a discussion on the ordinary deformation condition and Tilioune’s
regularity conditions, the reader may refer to [12, section 4]. With reference
to condition $\eqref{thc7}$, the reader may consult [12, sections 4.3 and 4.4]
for examples of such deformation conditions.

When $n=1$, the unipotent group is abelian and examples of such Galois
representations $\bar{\rho}$ are constructed via class field theory. The
reader may, for instance, refer to [16], [6, section 7] and [15] for more
details. In section 8, examples of Galois representations
$\bar{\rho}:\operatorname{G}_{\mathbb{Q},\\{p\\}}\rightarrow\operatorname{GSp}_{4}(\mathbb{F}_{p})$
satisfying the conditions of Theorem 1.1 are constructed. It is shown that if
$p\geq 23$ is a regular prime, there exists a Galois representation

$\bar{\rho}=\left({\begin{array}[]{cccc}\bar{\chi}^{3}&\ast&\ast&\ast\\\
&1&\ast&\ast\\\ &&\bar{\chi}^{6}&\\\
&&\ast&\bar{\chi}^{9}\end{array}}\right):\operatorname{G}_{\mathbb{Q},\\{p\\}}\rightarrow\operatorname{GSp}_{4}(\mathbb{F}_{p})$

which satisfies the conditions of Theorem 1.1.

### Acknowledgements

The author is very grateful to his advisor Ravi Ramakrishna for introducing
him to the fascinating subject of Galois deformations. He thanks Brian Hwang
and Nicolas Templier for fruitful conversations. The author also appreciates
the suggestions made by the anonymous referee which have led to significant
improvement of the article.

##  2\. Notation

  * •

For an $\mathbb{F}_{q}$-vector space $M$, set $\dim
M:=\dim_{\mathbb{F}_{q}}M$.

  * •

At every prime $v$, choose an embedding
$\iota_{v}:\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{v}$. The absolute
Galois group
$\operatorname{G}_{v}=\operatorname{Gal}(\bar{\mathbb{Q}}_{v}/\mathbb{Q}_{v})$
is identified with the decomposition group of the prime dividing $v$
determined by $\iota_{v}$.

  * •

Let $e_{i,j}$ denote the $2n\times 2n$ square matrix with coefficients in
$\mathbb{F}_{q}$ with $1$ in the $(i,j)$-th position and $0$ at all other
positions.

  * •

The space $\operatorname{Ad}^{0}\bar{\rho}$ is an
$\mathbb{F}_{q}[\operatorname{G}_{\mathbb{Q},S}]$-module with underlying space
$\operatorname{sp}_{2n}(\mathbb{F}_{q})$. The Galois action is prescribed by

$g\cdot X=\bar{\rho}(g)X\bar{\rho}(g)^{-1}$

where $g\in\operatorname{G}_{\mathbb{Q},S}$ and
$X\in\operatorname{sp}_{2n}(\mathbb{F}_{q})$.

  * •

The space of diagonal matrices in $\operatorname{Ad}^{0}\bar{\rho}$ is denoted
by $\mathfrak{t}$. Let $H_{1},\dots,H_{n}$ be the basis of $\mathfrak{t}$
defined by $H_{i}:=e_{i,i}-e_{n+i,n+i}$. Let
$L_{1},\dots,L_{n}\in\mathfrak{t}^{*}$ be the dual basis.

  * •

Let $\Phi$ be the set of roots of $\operatorname{sp}_{2n}(\mathbb{F}_{q})$ and
$\lambda_{1},\dots,\lambda_{n}\in\Phi$ be the simple roots defined as follows

$\lambda_{i}:=\begin{cases}L_{i}-L_{i+1}\text{ for }i<n\\\ 2L_{n}\text{ for
}i=n.\end{cases}$

Let $\Delta$ be the simple roots $\\{\lambda_{1},\dots,\lambda_{n}\\}$. Let
$\Phi^{+}$ and $\Phi^{-}$ denote the positive and negative roots respectively.

  * •

For $\lambda\in\Phi$, let $(\operatorname{Ad}^{0}\bar{\rho})_{\lambda}$ be the
$\lambda$ root-subspace of $\operatorname{Ad}^{0}\bar{\rho}$. For
$\lambda\in\Phi$, let $X_{\lambda}$ be a choice a root vector generating the
one-dimensional space $(\operatorname{Ad}^{0}\bar{\rho})_{\lambda}$. For
instance when $n=2$, we may choose root vectors as follows,

${\small X_{2L_{1}}}:={\tiny\left({\begin{array}[]{cccc}0&&1&\\\ &0&&\\\
&&0&\\\ &&&0\end{array}}\right)},{\small
X_{2L_{2}}}:={\tiny\left({\begin{array}[]{cccc}0&&&\\\ &0&&1\\\ &&0&\\\
&&&0\end{array}}\right)},$ ${\small
X_{L_{1}+L_{2}}}:={\tiny\left({\begin{array}[]{cccc}0&&&1\\\ &0&1&\\\ &&0&\\\
&&&0\end{array}}\right)}\text{ and }{\small
X_{L_{1}-L_{2}}}:={\tiny\left({\begin{array}[]{cccc}0&1&&\\\ &0&&\\\ &&0&\\\
&&-1&0\end{array}}\right).}$

For $\lambda\in\Phi^{-}$, we may choose $X_{\lambda}$ to be the transpose of
$X_{-\lambda}$.

  * •

Let $\lambda$ be a root. There is a unique presentation of
$\lambda=\sum\alpha_{i}\lambda_{i}$, where $\alpha_{i}$ are all non-negative
or all non-positive. The height of $\lambda$ is defined by
$\operatorname{ht}(\lambda):=\sum_{i}\alpha_{i}$. For instance, the root
$2L_{1}=2\left(\sum_{i=1}^{n-1}\lambda_{i}\right)+\lambda_{n}$ has height
equal to $2n-1$. Every other root has height less than $2n-1$.

  * •

For any integer $k$, let $(\operatorname{Ad}^{0}\bar{\rho})_{k}$ be the
$\mathbb{F}_{q}[\operatorname{G}_{\mathbb{Q},S}]$-submodule defined by

$(\operatorname{Ad}^{0}\bar{\rho})_{k}:=\bigoplus_{\begin{subarray}{c}\alpha\in\Phi\\\
\text{ht}\alpha\geq
k\end{subarray}}(\operatorname{Ad}^{0}\bar{\rho})_{\alpha}.$

Set $\mathfrak{b}:=(\operatorname{Ad}^{0}\bar{\rho})_{0}$ and
$\mathfrak{n}:=(\operatorname{Ad}^{0}\bar{\rho})_{1}$.

  * •

Associated with any root $\lambda$ is a Galois character denoted by
$\sigma_{\lambda}:\operatorname{G}_{\mathbb{Q},S}\rightarrow\mathbb{F}_{q}^{\times}$
obtained by composing $\bar{\rho}$ with the character induced on
$\operatorname{B}(\mathbb{F}_{q})$ by the root $\lambda$. Denote by
$\sigma_{1}$ the trivial character and set $\operatorname{ht}(1)=0$. For
$\lambda\in\Phi\cup\\{1\\}$, we have that

$g\cdot
X_{\lambda}-\sigma_{\lambda}(g)X_{\lambda}\in(\operatorname{Ad}^{0}\bar{\rho})_{\operatorname{ht}(\lambda)+1}.$

  * •

Let $Q\subseteq\operatorname{Ad}^{0}\bar{\rho}$ be an
$\mathbb{F}_{q}[\operatorname{G}_{\mathbb{Q},S}]$-submodule, the
$\sigma_{2L_{1}}$-eigenspace of $Q$ is the
$\mathbb{F}_{q}[\operatorname{G}_{\mathbb{Q},S}]$-submodule defined by
$Q_{\sigma_{2L_{1}}}:=Q\cap(\operatorname{Ad}^{0}\bar{\rho})_{2L_{1}}$.
Likewise, if $P\subseteq\operatorname{Ad}^{0}\bar{\rho}^{*}$ is an
$\mathbb{F}_{q}[\operatorname{G}_{\mathbb{Q},S}]$-submodule, the
$\bar{\chi}\sigma_{2L_{1}}$-eigenspace $P_{\bar{\chi}\sigma_{2L_{1}}}$ is
defined by

$P_{\bar{\chi}\sigma_{2L_{1}}}:=\\{v\in P\mid v(X)=0\text{ for
}X\in(\operatorname{Ad}^{0}\bar{\rho})_{-2n+2}\\}.$

  * •

For $k\geq 1$, let $U_{k}\subset\operatorname{B}(\mathbb{F}_{q})$ be the
exponential subgroup generated by
$\operatorname{exp}((\operatorname{Ad}^{0}\bar{\rho})_{k})$. Note that the
exponential map here is well-defined since $p>2n$. The group $U_{1}$ is the
unipotent subgroup of $\operatorname{B}(\mathbb{F}_{q})$.

  * •

Throughout, $h^{i}$ will be an abbreviation for $\dim H^{i}$. For instance,
$h^{i}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$ is an
abbreviation for $\dim
H^{i}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$.

  * •

Let $M$ be an $\mathbb{F}_{q}[\operatorname{G}_{S}]$-module, let
${}^{i}_{S}(M)$ consist of cohomology classes $f\in
H^{i}(\operatorname{G}_{\mathbb{Q},S},M)$ such that
$f_{\restriction\operatorname{G}_{v}}=0$ for all $v\in S$.

##  3\. The General Lifting Strategy

Let $\bar{\varrho}$ be a Galois representation
$\bar{\varrho}:\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(\bar{\mathbb{F}}_{p})$
which is irreducible, odd and unramified outside finitely many primes.
Ramakrishna in [14] and [13] showed that if $\bar{\varrho}$ satisfies
additional conditions, it lifts to a continuous Galois representation
$\varrho$ which is geometric in the sense of Fontaine and Mazur. In other
words, $\varrho$ is odd, unramified outside finitely many primes and
$\varrho_{\restriction\operatorname{G}_{p}}$ is de Rham. This geometric
lifting theorem provided evidence for the weak version of Serre’s conjecture
before it was proved by Khare and Wintenberger. The geometric lifting
construction was adapted to the reducible setting in [6]. The main result of
this manuscript is a higher dimensional generalization of the lifting theorem
of Hamblen-Ramakrishna. The basic strategy involves successively lifting
$\bar{\rho}$ to a characteristic zero irreducible geometric representation
$\rho$ by successively lifting $\rho_{m}$ to $\rho_{m+1}$

$\operatorname{G}_{\mathbb{Q},S\cup
X}$$\operatorname{GSp}_{2n}(\mathbb{F}_{q}).$$\operatorname{GSp}_{2n}(\text{W}(\mathbb{F}_{q})/p^{m})$$\operatorname{GSp}_{2n}(\text{W}(\mathbb{F}_{q})/p^{m+1})$$\bar{\rho}$$\rho_{m}$$\rho_{m+1}$

######  Definition 3.1.

Let $\mathcal{C}$ be the category of coefficient rings over
$\text{W}(\mathbb{F}_{q})$ with residue field $\mathbb{F}_{q}$. The objects of
this category consist of local $\text{W}(\mathbb{F}_{q})$-algebras
$(R,\mathfrak{m})$ for which

  * •

$R$ is complete and Noetherian,

  * •

$R/\mathfrak{m}$ is isomorphic to $\mathbb{F}_{q}$ as a
$\text{W}(\mathbb{F}_{q})$-algebra. The residual map

$\phi:R\rightarrow\mathbb{F}_{q}$

is the composite of the quotient map $R\rightarrow R/\mathfrak{m}$ with the
unique isomorphism of $W(\mathbb{F}_{q})$-algebras
$R/\mathfrak{m}\xrightarrow{\sim}\mathbb{F}_{q}$.

A morphism $F:(R_{1},\mathfrak{m}_{1})\rightarrow(R_{2},\mathfrak{m}_{2})$ is
a homorphism of local rings which is also a $\text{W}(\mathbb{F}_{q})$-algebra
homorphism. Recall that $\kappa$ is a fixed choice of lift of $\bar{\kappa}$.
Let $\kappa_{v}$ denote the restriction of $\kappa$ to $\operatorname{G}_{v}$.

Let $v$ be a prime and $R\in\mathcal{C}$. Denote by
$\phi^{*}:\operatorname{GSp}_{2n}(R)\rightarrow\operatorname{GSp}_{2n}(\mathbb{F}_{q})$
the group homomorphism induced by the residual homomorphism
$\phi:R\rightarrow\mathbb{F}_{q}$. We say that
$\rho_{R}:\operatorname{G}_{v}\rightarrow\operatorname{GSp}_{2n}(R)$ is an
$R$-lift of $\bar{\rho}_{\restriction\operatorname{G}_{v}}$ if
$\phi^{*}\circ\rho_{R}=\bar{\rho}_{\restriction\operatorname{G}_{v}}$, i.e.
the following diagram commutes

$\operatorname{G}_{v}$$\operatorname{GSp}_{2n}(\mathbb{F}_{q})$.$\operatorname{GSp}_{2n}(R)$$\bar{\rho}_{\restriction\operatorname{G}_{v}}$$\phi^{*}$$\rho_{R}$

Further, we shall require that the similitude character of $\rho_{R}$
coincides with the composite of $\kappa_{v}$ with the homomorphism
$W(\mathbb{F}_{q})^{\times}\rightarrow R^{\times}$ induced by the structure
map.

Two lifts $\rho_{R}$ and $\rho_{R}^{\prime}$ are said to be strictly-
equivalent if there is

$A\in\text{ker}\\{\operatorname{GSp}_{2n}(R)\xrightarrow{\phi^{*}}\operatorname{GSp}_{2n}(\mathbb{F}_{q})\\}$

such that $\rho_{R}=A\rho_{R}^{\prime}A^{-1}$. A deformation is a strict
equivalence class of lifts. Let $\operatorname{Def}_{v}(R)$ be the set of
$R$-deformations of $\bar{\rho}_{\restriction\operatorname{G}_{v}}$. The
association $R\mapsto\operatorname{Def}_{v}(R)$ defines a covariant functor

$\operatorname{Def}_{v}:\mathcal{C}\rightarrow\operatorname{Sets}.$

The tangent space
$\operatorname{Def}_{v}(\mathbb{F}_{q}[\epsilon]/(\epsilon^{2}))$ naturally
acquires the structure of an $\mathbb{F}_{q}$-vector space and is isomorphic
to $H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$. Under this
association, a cohomology class $f$ is identified with the deformation
$(\operatorname{Id}+\epsilon f)\bar{\rho}_{\restriction\operatorname{G}_{v}}$.

For $m\in\mathbb{Z}_{\geq 2}$, the deformations
$\operatorname{Def}_{v}(\text{W}(\mathbb{F}_{q})/p^{m})$ are equipped with
action of the cohomology group
$H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$. For
$\varrho_{m}\in\operatorname{Def}_{v}(\text{W}(\mathbb{F}_{q})/p^{m})$ and
$f\in H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$, the twist
of $\varrho_{m}$ by $f$ is defined by the formula
$(\operatorname{Id}+p^{m-1}f)\varrho_{m}$. The set of deformations
$\varrho_{m}$ of a fixed
$\varrho_{m-1}\in\operatorname{Def}_{v}(\text{W}(\mathbb{F}_{q})/p^{m-1})$ is
either empty or in bijection with
$H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$.

######  Definition 3.2.

(see [19]) We say that a sub-functor $\mathcal{C}_{v}$ of
$\operatorname{Def}_{v}$ is a deformation condition if (1) to (3) below are
satisfied. If condition (4) is satisfied, $\mathcal{C}_{v}$ is said to be
liftable.

  1. (1)

First, we require that
$\mathcal{C}_{v}(\mathbb{F}_{q})=\\{\bar{\rho}_{\restriction\operatorname{G}_{v}}\\}.$

  2. (2)

For $R_{1}$ and $R_{2}$ be $\mathcal{C}$, let
$\rho_{1}\in\mathcal{C}_{v}(R_{1})$ and $\rho_{2}\in\mathcal{C}_{v}(R_{2})$.
Let $I_{1}$ be an ideal in $R_{1}$ and $I_{2}$ an ideal in $R_{2}$ such that
there is an isomorphism $\alpha:R_{1}/I_{1}\xrightarrow{\sim}R_{2}/I_{2}$
satisfying

$\alpha(\rho_{1}\;\text{mod}{I_{1}})=\rho_{2}\;\text{mod}{I_{2}}.$

Let $R_{3}$ be the fibred product

$R_{3}=\\{(r_{1},r_{2})\mid\alpha(r_{1}\text{mod}I_{1})=r_{2}\text{mod}I_{2}\\}$

and $\rho_{3}$ the $R_{3}$-deformation induced from $\rho_{1}$ and $\rho_{2}$.
Then $\rho_{3}$ satisfies $\mathcal{C}_{v}(R_{3})$.

  3. (3)

Let $R\in\mathcal{C}$ with maximal ideal $\mathfrak{m}_{R}$. If
$\rho\in\operatorname{Def}_{v}(R)$ is such that
$\rho\mod{\mathfrak{m}_{R}^{n}}$ satisfies $\mathcal{C}_{v}$ for all
$n\in\mathbb{Z}_{\geq 1}$, then $\rho$ also satisfies $\mathcal{C}_{v}$.

  4. (4)

Let $R\in\mathcal{C}$ and $I$ an ideal such that $I.\mathfrak{m}_{R}=0$. For
$\rho\in\mathcal{C}_{v}(R/I)$, there exists
$\tilde{\rho}\in\mathcal{C}_{v}(R)$ such that $\rho=\tilde{\rho}\mod{I}$.

Let $\mathcal{C}_{v}$ be a local deformation condition at the prime $v$. The
tangent space $\mathcal{N}_{v}$ consists of $f\in
H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$, such that
$(\operatorname{Id}+\epsilon
f)\bar{\rho}_{\restriction\operatorname{G}_{v}}\in\mathcal{C}_{v}(\mathbb{F}_{q}[\epsilon]/(\epsilon^{2}))$.
The action of $\mathcal{N}_{v}$ on
$\operatorname{Def}_{v}(\text{W}(\mathbb{F}_{q})/p^{m})$ stabilizes
$\mathcal{C}_{v}(\text{W}(\mathbb{F}_{q})/p^{m})$. In other words, if
$\varrho_{m}\in\mathcal{C}_{v}(\text{W}(\mathbb{F}_{q})/p^{m})$ and
$f\in\mathcal{N}_{v}$, then

$(\operatorname{Id}+p^{m-1}f)\varrho_{m}\in\mathcal{C}_{v}(\text{W}(\mathbb{F}_{q})/p^{m}).$

It is assumed that each prime $v\in S\backslash\\{p\\}$ is equipped with a
liftable local deformation condition $\mathcal{C}_{v}$ such that

$\dim\mathcal{N}_{v}=h^{0}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho}).$

The reader may consult [12, sections 4.3 and 4.4] for examples of such
deformation conditions. The deformation condition $\mathcal{C}_{p}$ is the
ordinary deformation condition. Since we have assumed that Tilouine’s
regularity conditions are satisfied (cf. [12, section 4.1]), $\mathcal{C}_{p}$
is liftable and the tangent space $\mathcal{N}_{p}$ has dimension equal to

$\dim\mathcal{N}_{p}=h^{0}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho})+\dim\mathfrak{n},$

see [12, Proposition 4.4]. We allow the successive lifts $\rho_{m}$ to be
ramified at a set of primes $S\cup X$. Each auxiliary prime $v\in X$ is
equipped with a liftable subfunctor $\mathcal{C}_{v}$ of
$\operatorname{Def}_{v}$. These auxiliary primes are referred to as trivial
primes and were introduced by Hamblen and Ramakrishna in the two-dimensional
setting [6, section 4]. These are primes $v\equiv 1\mod{p}$, not contained in
$S$, at which $\bar{\rho}_{\restriction\operatorname{G}_{v}}$ the trivial
representation and $v\not\equiv 1\mod{p^{2}}$. We use a higher dimensional
generalization of the deformation functor $\mathcal{C}_{v}$ at a trivial prime
$v$, due to Fakhruddin, Khare and Patrikis [2, Definition 3.1]. At each
trivial prime $v$ there is a subspace $\mathcal{N}_{v}$ of
$H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$ of dimension
$h^{0}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$ which behaves
like a versal tangent space. For $m\geq 3$, the action of $\mathcal{N}_{v}$ on
$\operatorname{Def}(\text{W}(\mathbb{F}_{q})/p^{m})$ stabilizes
$\mathcal{C}_{v}(\text{W}(\mathbb{F}_{q})/p^{m})$. This is proved in the
$\operatorname{GL}_{2}$-case by Hamblen-Ramakrishna, see [6, Corollory 25,
29]. For more general groups, we refer to Fakhruddin-Khare-Patrikis [2, Lemma
3.6,3.10] for the precise statement. However, this is not the case for $m=2$.

Let $X$ be a finite set of trivial primes disjoint from $S$. For $v\in S\cup
X$, set $\mathcal{N}_{v}^{\perp}\subseteq
H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho}^{*})$ to be the
orthogonal complement of $\mathcal{N}_{v}$ with respect to the non-degenerate
Tate pairing

$H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})\times
H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho}^{*})\rightarrow
H^{2}(\operatorname{G}_{v},\mathbb{F}_{q}(\bar{\chi}))\xrightarrow{\sim}\mathbb{F}_{q}.$

Set $\mathcal{N}_{\infty}=0$ and $\mathcal{N}_{\infty}^{\perp}=0$. The Selmer-
condition $\mathcal{N}$ is the tuple $\\{\mathcal{N}_{v}\\}_{v\in S\cup
X\cup\\{\infty\\}}$ and the dual Selmer condition $\mathcal{N}^{\perp}$ is
$\\{\mathcal{N}_{v}^{\perp}\\}_{v\in S\cup X\cup\\{\infty\\}}$. Attached to
$\mathcal{N}$ and $\mathcal{N}^{\perp}$ are the Selmer and dual-Selmer groups
defined as follows:

$H^{1}_{\mathcal{N}}(\operatorname{G}_{\mathbb{Q},S\cup
X},\operatorname{Ad}^{0}\bar{\rho}):=\text{ker}\left\\{H^{1}(\operatorname{G}_{\mathbb{Q},S\cup
X},\operatorname{Ad}^{0}\bar{\rho})\xrightarrow{\operatorname{res}_{S\cup
X}}\bigoplus_{v\in S\cup
X}\frac{H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})}{\mathcal{N}_{v}}\right\\}$

and

$H^{1}_{\mathcal{N}^{\perp}}(\operatorname{G}_{\mathbb{Q},S\cup
X},\operatorname{Ad}^{0}\bar{\rho}^{*}):=\text{ker}\left\\{H^{1}(\operatorname{G}_{\mathbb{Q},S\cup
X},\operatorname{Ad}^{0}\bar{\rho}^{*})\xrightarrow{\operatorname{res}_{S\cup
X}^{\prime}}\bigoplus_{v\in S\cup
X}\frac{H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho}^{*})}{\mathcal{N}_{v}^{\perp}}\right\\}$

respectively. The following formula is due to Wiles (see [10, Theorem 8.7.9]):

(3.1) $\begin{split}h^{1}_{\mathcal{N}}(\operatorname{G}_{\mathbb{Q},S\cup
X},\operatorname{Ad}^{0}\bar{\rho})-h^{1}_{\mathcal{N}^{\perp}}(\operatorname{G}_{\mathbb{Q},S\cup
X},\operatorname{Ad}^{0}\bar{\rho}^{*})&=h^{0}(\operatorname{G}_{\mathbb{Q}},\operatorname{Ad}^{0}\bar{\rho})-h^{0}(\operatorname{G}_{\mathbb{Q}},\operatorname{Ad}^{0}\bar{\rho}^{*})\\\
&+\sum_{v\in S\cup
X\cup\\{\infty\\}}\left(\dim\mathcal{N}_{v}-h^{0}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})\right).\\\
\end{split}$

Since $\bar{\rho}$ is odd, one has that
$h^{0}(\operatorname{G}_{\infty},\operatorname{Ad}^{0}\bar{\rho})=\dim\mathfrak{n}$.
It follows from the above formula that the dimensions of the Selmer group and
dual Selmer group coincide, i.e.

$h^{1}_{\mathcal{N}}(\operatorname{G}_{\mathbb{Q},S\cup
X},\operatorname{Ad}^{0}\bar{\rho})=h^{1}_{\mathcal{N}^{\perp}}(\operatorname{G}_{\mathbb{Q},S\cup
X},\operatorname{Ad}^{0}\bar{\rho}^{*}).$

The Selmer and dual Selmer groups fit into a long exact sequence called the
Poitou-Tate sequence. We only point out that the cokernel of
$\operatorname{res}_{S\cup X}$ injects into
$H^{1}_{\mathcal{N}^{\perp}}(\operatorname{S}_{S\cup
X},\operatorname{Ad}^{0}\bar{\rho}^{*})^{\vee}$. In particular, if the Selmer
group is zero, then so is the dual Selmer group, in which case the restriction
map $\operatorname{res}_{S\cup X}$ is an isomorphism. Since the spaces
$\mathcal{N}_{v}$ at a trivial prime $v$ stabilize lifts only past mod
$p^{3}$, it becomes necessary to produce a mod $p^{3}$ lift $\rho_{3}$ of
$\bar{\rho}$ before applying the general lifting-strategy. All deformations
$\rho_{m}$ discussed in this paper will have similitude character equal to
$\kappa\mod{p^{m}}$.

The three main steps are as follows:

  1. (1)

first it is shown that there is a finite set of trivial primes $X_{1}$
disjoint from $S$ such that the representation $\bar{\rho}$ lifts to a mod
$p^{2}$ representation $\rho_{2}$ which is unramified outside $S\cup X_{1}$.

  2. (2)

It is shown in [6, section 5] that there is a finite set of trivial primes
$X_{2}\supset X_{1}$ disjoint from $S$ and a mod $p^{3}$ lift $\rho_{3}$ of
$\rho_{2}$ which satisfies the following conditions

     * •

$\rho_{3}$ is irreducible, i.e. does not contain a free rank one Galois stable
$\text{W}(\mathbb{F}_{q})/p^{3}$-submodule.

     * •

It is unramified outside $S\cup X_{2}$.

     * •

The lift $\rho_{3}$ is also arranged to satisfy conditions $\mathcal{C}_{v}$
at each prime $v\in S\cup X_{2}$.

This strategy for getting past mod-$p^{2}$ is based on the methods developed
by Khare, Larsen and Ramakrishna in [8].

  3. (3)

At this stage, all that remains to be shown is that the set of primes $X_{2}$
may be further enlarged to a set of trivial primes $X$ which is disjoint from
$S$ such that the Selmer group
$H^{1}_{\mathcal{N}}(\operatorname{G}_{\mathbb{Q},S\cup
X},\operatorname{Ad}^{0}\bar{\rho})$ is equal to zero.

The rest of the argument warrants some explanation. Since the Selmer group is
zero, the map $\operatorname{res}_{S\cup X}$ is an isomorphism. Suppose for
$m\geq 3$ and $\rho_{m}$ is a mod $p^{m}$ lift of $\rho_{3}$ which is
unramified outside $S\cup X$ and satisfies the conditions $\mathcal{C}_{v}$ at
each prime $v\in S\cup X$. We show that $\rho_{m}$ may be lifted to
$\rho_{m+1}$ which satisfies the same conditions. Since the dual Selmer group
is zero, so is ${}^{1}_{S\cup X}(\operatorname{Ad}^{0}\bar{\rho}^{*})$, and it
follows from global-duality that ${}^{2}_{S\cup
X}(\operatorname{Ad}^{0}\bar{\rho})$ is zero. Since local condition
$\mathcal{C}_{v}$ is liftable, there are no local obstructions to lifting
${\rho_{m}}$. The cohomological obstruction to lifting $\rho_{m}$ to
$\rho_{m+1}$ is a class in ${}^{2}_{S\cup X}(\operatorname{Ad}^{0}\bar{\rho})$
and hence is zero. As a result, $\rho_{m}$ does lift one more step to
$\rho_{m+1}$. In order to complete the inductive argument, it is shown that
$\rho_{m+1}$ can be chosen to satisfy the conditions $\mathcal{C}_{v}$. After
picking a suitable global cohomology class $z\in
H^{1}(\operatorname{G}_{\mathbb{Q},S\cup X},\operatorname{Ad}^{0}\bar{\rho})$
and replacing $\rho_{m+1}$ by its twist
$(\operatorname{Id}+p^{m}z)\rho_{m+1}$, this may be arranged. At each prime
$v\in S\cup X$, there is a cohomology class $z_{v}\in
H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$ such that the
twist
$(\operatorname{Id}+p^{m}z_{v}){\rho_{m+1}}_{\restriction\operatorname{G}_{v}}$
satisfies $\mathcal{C}_{v}$. Since we assume that $m\geq 3$, we have that
$\mathcal{N}_{v}$ stabilizes $\mathcal{C}_{v}$. For $v\in S\cup X$, the
elements $z_{v}$ are defined modulo $\mathcal{N}_{v}$. Since
$\operatorname{res}_{S\cup X}$ is an isomorphism, the tuple
$(z_{v})\in\bigoplus_{v\in S\cup
X}H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})/\mathcal{N}_{v}$
arises from a unique global cohomology class $z$ which is unramified outside
$S\cup X$. After replacing $\rho_{m+1}$ by
$(\operatorname{Id}+p^{m}z)\rho_{m+1}$, it satisfies the conditions
$\mathcal{C}_{v}$ at each prime $v\in S\cup X$. This completes the inductive
lifting argument.

##  4\. Preliminaries

In this section, we prove a number of Galois theoretic results which will be
applied in later sections. Let $M$ be a finite abelian group with
$\operatorname{G}_{\mathbb{Q}}$-action and $E$ be a number field. Denote by
$E(M)$ the extension of $E$ cut out by $M$. In other words, it is the Galois
extension of $E$ which is fixed by the kernel of the action of
$\operatorname{G}_{E}$ on $M$. Let $M_{1},\dots,M_{k}$ be finite abelian
groups on which $\operatorname{G}_{\mathbb{Q}}$ acts. Denote by
$E(M_{1},\dots,M_{k})$ the composite of the fields $E(M_{1})\cdots E(M_{k})$.
Let $K:=\mathbb{Q}(\bar{\rho},\mu_{p})$ and $L:=\mathbb{Q}(\bar{\rho})$ and
set $\operatorname{G}^{\prime}:=\operatorname{Gal}(K/\mathbb{Q})$ and
$\operatorname{G}:=\operatorname{Gal}(L/\mathbb{Q})$. Let $F$ be the subfield
$\mathbb{Q}(\varphi_{1},\dots,\varphi_{n},\bar{\kappa})$ of $L$, where we
recall that the characters $\varphi_{1},\dots,\varphi_{n}$ are as in
$\eqref{introducingbarrho}$. Denote by
$N^{\prime}:=\operatorname{Gal}(K/F(\mu_{p}))$ and
$N:=\operatorname{Gal}(L/F)$. The groups
$\operatorname{G},\operatorname{G}^{\prime},N$ and $N^{\prime}$ are depicted
in the following field diagram

$\mathbb{Q}.$$F$$F(\mu_{p})$$\mathbb{Q}(\bar{\rho})$$\mathbb{Q}(\bar{\rho},\mu_{p})$$N$$N^{\prime}$

Condition $\eqref{thc3}$ of Theorem 1.1 asserts that the image of $\bar{\rho}$
contains $U_{1}(\mathbb{F}_{q})$. Therefore $N$ may be identified with
$\bar{\rho}(N)=U_{1}(\mathbb{F}_{q})$. In particular the abelianization
$N^{ab}$ may be identified with $U_{1}(\mathbb{F}_{q})/U_{2}(\mathbb{F}_{q})$.
Since $N$ is a $p$-group and $[F(\mu_{p}):F]$ is coprime to $p$, it follows
that $\mathbb{Q}(\bar{\rho})$ and $F(\mu_{p})$ are linearly disjoint over $F$.
It follows that $N$ is canonically isomorphic to $N^{\prime}$. The inclusion
of $\mathcal{T}$ into $\operatorname{B}$ is a section of the quotient map
$\operatorname{B}\rightarrow\mathcal{T}$. This induces a semi-direct product
decomposition $\operatorname{B}=U_{1}\rtimes\mathcal{T}$. Let
$\mathcal{T}^{\prime}$ be the intersection of the image of $\bar{\rho}$ with
$\mathcal{T}$. The group $\operatorname{G}$ may be identified with the image
of $\bar{\rho}$. It is easy to see that $\operatorname{G}$ has a semi-direct
product decomposition
$\operatorname{G}\simeq\bar{\rho}(\operatorname{G})=U_{1}(\mathbb{F}_{q})\rtimes\mathcal{T}^{\prime}$.

######  Lemma 4.1.

Suppose $0<|k|\leq 2n-1$, there is an isomorphism of
$\mathbb{F}_{q}[\operatorname{G}_{\mathbb{Q},S}]$-modules

$(\operatorname{Ad}^{0}\bar{\rho})_{k}/(\operatorname{Ad}^{0}\bar{\rho})_{k+1}\simeq\bigoplus_{ht\lambda=k}\mathbb{F}_{q}(\sigma_{\lambda}).$

On the other hand,

$(\operatorname{Ad}^{0}\bar{\rho})_{0}/(\operatorname{Ad}^{0}\bar{\rho})_{1}=\mathfrak{b}/\mathfrak{n}\simeq\mathfrak{t}.$

###### Proof.

Let $\lambda$ be of height $k$. Let
$X\in(\operatorname{Ad}^{0}\bar{\rho})_{\lambda}$, we observe that

$\bar{\rho}(g)\cdot
X\cdot\bar{\rho}(g)^{-1}\equiv\sigma_{\lambda}(g)X\mod{(\operatorname{Ad}^{0}\bar{\rho})_{k+1}}.$

Likewise, for $X\in\mathfrak{b}$, the conjugation action on $X$ modulo
$\mathfrak{n}$ is trivial. ∎

Let $\zeta$ be a non-zero element of $\mathbb{F}_{q}(\bar{\chi})$. For
$i=1,\dots,n$, set $\delta_{i,j}=\zeta$ if $i=j$ and $0$ otherwise. Likewise,
for roots $\lambda$ and $\gamma$, set $\delta_{\lambda,\gamma}$ to equal
$\zeta$ if $\lambda=\gamma$ and $0$ otherwise. Denote by $X_{\lambda}^{*}$ and
$H_{i}^{*}$ the elements of $\operatorname{Ad}^{0}\bar{\rho}^{*}$ which are
defined by the following relations:

(4.1) $\begin{split}&X_{\lambda}^{*}(X_{\gamma})=\delta_{\lambda,\gamma}\text{
and }X_{\lambda}^{*}(H_{i})=0,\\\ &H_{i}^{*}(X_{\lambda})=0\text{ and
}H_{i}^{*}(H_{j})=\delta_{i,j}.\end{split}$

The element $H_{i}^{*}\in\operatorname{Ad}^{0}\bar{\rho}^{*}$ should not be
confused with $L_{i}\in\mathfrak{t}^{*}$. Let
$(\operatorname{Ad}^{0}\bar{\rho}^{*})_{\bar{\chi}}$ be the span of
$H_{1}^{*},\dots,H_{n}^{*}$ and
$(\operatorname{Ad}^{0}\bar{\rho}^{*})_{\bar{\chi}\sigma_{\lambda}}$ the span
of $X_{-\lambda}^{*}$. Let $P$ be a Galois-stable subgroup of
$\operatorname{Ad}^{0}\bar{\rho}^{*}$. Associated to $P$ are its eigenspaces
for the action of $\bar{\rho}^{-1}(\mathcal{T})$. For
$\lambda\in\Phi\cup\\{1\\}$, set $P_{\bar{\chi}\sigma_{\lambda}}$ to be the
intersection of $P$ with
$(\operatorname{Ad}^{0}\bar{\rho}^{*})_{\bar{\chi}\sigma_{\lambda}}$.
Likewise, associate to a Galois stable subgroup
$Q\subseteq\operatorname{Ad}^{0}\bar{\rho}$, an eigenspace
$Q_{\sigma_{\lambda}}$. Define $Q_{1}$ to be the intersection
$Q\cap\mathfrak{t}$. For $\lambda\in\Phi$, denote by $Q_{\sigma_{\lambda}}$
the intersection $Q\cap(\operatorname{Ad}^{0}\bar{\rho})_{\lambda}$.

The representation $\bar{\rho}$ factors through $\operatorname{G}$. Let
$\mathbb{T}$ be the subgroup of $\operatorname{G}^{\prime}$ consisting of $g$
such that $\bar{\rho}(g)\in\mathcal{T}$. For $\lambda\in\Phi\cup\\{1\\}$,
$\mathbb{T}$ acts on $P_{\bar{\chi}\sigma_{\lambda}}$ by the character
$\bar{\chi}\sigma_{\lambda}$ and on $Q_{\sigma_{\lambda}}$ by the character
$\sigma_{\lambda}$. Since the characters $\sigma_{\lambda}$ are assumed to be
distinct, it is easy to see that

$P_{\bar{\chi}\sigma_{\lambda}}=\\{p\in P|t\cdot
p=\bar{\chi}\sigma_{\lambda}(t)p\text{ for }t\in\mathbb{T}\\}$
$Q_{\sigma_{\lambda}}=\\{q\in Q|t\cdot q=\sigma_{\lambda}(t)q\text{ for
}t\in\mathbb{T}\\}.$

The order of $\mathbb{T}$ is coprime to $p$, hence Maschke’s theorem asserts
that any finite dimensional $\mathbb{F}_{p}[\operatorname{G}^{\prime}]$-module
$M$ decomposes into a direct sum $M=\bigoplus_{\tau}M_{\tau}$, where $\tau$ is
a character of $\mathbb{T}$ and $M_{\tau}$ is the $\tau$-eigenspace
$M_{\tau}:=\\{m\in M|g\cdot m=\tau(g)m\\}$. Thus, we have the next Lemma,
which follows from the discussion above.

######  Lemma 4.2.

Let $P\subseteq\operatorname{Ad}^{0}\bar{\rho}^{*}$ and
$Q\subseteq\operatorname{Ad}^{0}\bar{\rho}$ be Galois-stable subgroups.

  1. (1)

As a $\mathbb{T}$-module, $P$ decomposes into a direct sum of eigenspaces:

$P=\bigoplus_{\lambda\in\Phi\cup\\{1\\}}P_{\bar{\chi}\sigma_{\lambda}}.$

  2. (2)

As a $\mathbb{T}$-module, $Q$ decomposes into a direct sum of eigenspaces:

$Q=\bigoplus_{\lambda\in\Phi\cup\\{1\\}}Q_{\sigma_{\lambda}}.$

######  Corollary 4.3.

The following statements hold:

  1. (1)

let $P_{1}$ and $P_{2}$ be Galois-stable subgroups of
$\operatorname{Ad}^{0}\bar{\rho}^{*}$ such that there is an isomorphism
$\phi:P_{1}\xrightarrow{\sim}P_{2}$ of Galois modules. Then $P_{1}=P_{2}$ and
$\phi$ is multiplication by a scalar.

  2. (2)

Let $Q_{1}$ and $Q_{2}$ be Galois-stable subgroups of
$\operatorname{Ad}^{0}\bar{\rho}$ such that there is an isomorphism
$\phi:Q_{1}\xrightarrow{\sim}Q_{2}$ of Galois modules. Then $Q_{1}=Q_{2}$ and
$\phi$ is multiplication by a scalar.

###### Proof.

We prove part $\eqref{coradd1}$, part $\eqref{coradd2}$ is identical. Let
$\iota_{P_{i}}:P_{i}\hookrightarrow\operatorname{Ad}^{0}\bar{\rho}^{*}$ be the
inclusion. By Proposition LABEL:y1, the two inclusions $\iota_{P_{1}}$ and
$\iota_{P_{2}}\circ\phi$ are the same upto a scalar. The assertion follows. ∎

Let $Q$ be a $\operatorname{G}$-submodule of
$\operatorname{Ad}^{0}\bar{\rho}$, by Lemma 4.2, the projection of $Q$ to
$(\operatorname{Ad}^{0}\bar{\rho})_{-2L_{1}}$ equals $Q_{\sigma_{-2L_{1}}}$.
For convenience of notation, let $Q_{-2L_{1}}$ denote $Q_{\sigma_{-2L_{1}}}$.

######  Lemma 4.4.

Let $Q$ be a Galois-stable submodule of $\operatorname{Ad}^{0}\bar{\rho}$ for
which $Q_{-2L_{1}}\neq 0$, then $Q=\operatorname{Ad}^{0}\bar{\rho}$.

###### Proof.

Let $P:=\\{\gamma\in\operatorname{Ad}^{0}\bar{\rho}^{*}\mid\gamma(x)=0\text{
for }x\in Q\\}$. The assumption on $Q$ implies that
$P_{\bar{\chi}\sigma_{2L_{1}}}\neq(\operatorname{Ad}^{0}\bar{\rho}^{*})_{\bar{\chi}\sigma_{2L_{1}}}$.
Since the image of $\bar{\chi}\sigma_{2L_{1}}$ spans $\mathbb{F}_{q}$,
$P_{\bar{\chi}\sigma_{2L_{1}}}\neq(\operatorname{Ad}^{0}\bar{\rho}^{*})_{\bar{\chi}\sigma_{2L_{1}}}$
implies that $P_{\bar{\chi}\sigma_{2L_{1}}}=0$ By Lemma LABEL:mainin, $P=0$,
and therefore, $Q=\operatorname{Ad}^{0}\bar{\rho}$. ∎

######  Lemma 4.5.

The following statements hold:

  1. (1)

the fields $K=\mathbb{Q}(\bar{\rho},\mu_{p})$ and $\mathbb{Q}(\mu_{p^{2}})$
are linearly disjoint over $\mathbb{Q}(\mu_{p})$.

  2. (2)

Let $\mathcal{J}\supseteq S$ be a finite set of prime numbers,
$\psi_{1},\dots,\psi_{t}\in
H^{1}(\operatorname{G}_{\mathbb{Q},\mathcal{J}},\operatorname{Ad}^{0}\bar{\rho}^{*})$
and set $K_{j}:=K_{\psi_{j}}$ for $j=1,\dots,t$. Then the composite
$K_{1}\cdots K_{t}$ and $\mathbb{Q}(\mu_{p^{2}})$ are linearly disjoint over
$\mathbb{Q}(\mu_{p})$.

###### Proof.

Suppose by way of contradiction that $\mathbb{Q}(\mu_{p^{2}})\subseteq K$. Set
$V:=\operatorname{Gal}(K/F(\mu_{p^{2}}))$ and
$\mathcal{A}:=\operatorname{G}^{\prime}/N^{\prime}=\operatorname{Gal}(F(\mu_{p})/\mathbb{Q})$.
For $n\in N^{\prime ab}$ and $g\in\mathcal{A}$, let $\tilde{n}$ and
$\tilde{g}$ be lifts of $n$ and $g$ to $N^{\prime}$ and
$\operatorname{G}^{\prime}$ respectively. The action of $\mathcal{A}$ on
$N^{\prime ab}$ is induced by conjugation, defined by $g\cdot
n:=\tilde{g}\tilde{n}\tilde{g}^{-1}\mod{[N^{\prime},N^{\prime}]}$. The groups
$N$ and $N^{\prime}$ are isomorphic and the image of $\bar{\rho}$ is assumed
to contain $U_{1}(\mathbb{F}_{q})$ (condition 3 of Theorem 1.1). The quotient
$N^{\prime}/V=\operatorname{Gal}(F(\mu_{p^{2}})/F(\mu_{p}))\simeq\mathbb{F}_{p}$.
Let $\pi:\operatorname{N}^{\prime ab}\rightarrow\mathbb{F}_{p}$ denote the map
induced by the mod-$V$ quotient. Being the composite of Galois extensions,
$F(\mu_{p^{2}})$ is Galois over $\mathbb{Q}$. As a result, $\pi$ is
$\mathcal{A}$-equivariant. Furthermore, since $F(\mu_{p^{2}})$ is an abelian
extension of $\mathbb{Q}$, the $\mathcal{A}$-action on $N^{\prime}/V$ is
trivial. On the other hand, as an $\mathcal{A}$-module, $N^{\prime
ab}\simeq\bigoplus_{\lambda\in\Delta}\mathbb{F}_{q}(\sigma_{\lambda})$. It
follows from condition 4 of Theorem 1.1 that $\sigma_{\lambda}\neq 1$ for
$\lambda\in\Phi$. As a result,

$\operatorname{Hom}(N^{\prime
ab},\mathbb{F}_{p})^{\operatorname{G}^{\prime}}\simeq\bigoplus_{\lambda\in\Delta}\operatorname{Hom}(\mathbb{F}_{q}(\sigma_{\lambda}),\mathbb{F}_{p})^{\operatorname{G}^{\prime}}=0.$

This is a contradiction which concludes the proof of the first part.

Set $\mathcal{K}_{j}$ to be $K_{1}\dots K_{j}$ and $\mathcal{K}_{0}:=K$.
Setting $E:=\mathbb{Q}(\mu_{p^{2}})$, it suffices to show that
$\mathcal{K}_{j}\cap E=\mathcal{K}_{j-1}\cap E$. We begin with the case $j=1$.
For $\psi\in
H^{1}(\operatorname{G}_{\mathbb{Q},\mathcal{J}},\operatorname{Ad}^{0}\bar{\rho}^{*})$,
regard $\operatorname{Gal}(K_{\psi}/K)$ as an
$\mathbb{F}_{q}[\operatorname{G}^{\prime}]$-module, where the Galois action is
induced via conjugation. The $\operatorname{G}^{\prime}$-module
$P_{1}:=\operatorname{Gal}(K_{1}/K)$ is identified with
$\psi_{1}(\operatorname{G}_{K})$. Let $Q_{1}\subseteq P_{1}$ be the
$\operatorname{G}^{\prime}$-stable subgroup defined by
$Q_{1}:=\operatorname{Gal}(K_{1}/(K_{1}\cap E)\cdot K)$. The action of
$\operatorname{G}^{\prime}$ on $P_{1}/Q_{1}=\operatorname{Gal}((K_{1}\cap
E)\cdot K/K)$ is trivial. By Lemma 4.2, the quotient $P_{1}/Q_{1}$ decomposes
into subgroups

$P_{1}/Q_{1}=\bigoplus_{\lambda\in\Phi\cup\\{1\\}}(P_{1})_{\bar{\chi}\sigma_{\lambda}}/(Q_{1})_{\bar{\chi}\sigma_{\lambda}}.$

The characters $\sigma_{\lambda}\neq\bar{\chi}^{-1}$ and hence $P_{1}=Q_{1}$.
We have thus shown that $K_{1}\cap E=K\cap E$.

Let $P_{j}$ be defined by
$P_{j}:=\operatorname{Gal}(\mathcal{K}_{j}/\mathcal{K}_{j-1})$. The
$\operatorname{G}^{\prime}$-module $P_{j}$ is isomorphic to

$\operatorname{Gal}(K_{j}/K_{j}\cap\mathcal{K}_{j-1})\subseteq\psi_{j}(\operatorname{G}_{K})\subseteq\operatorname{Ad}^{0}\bar{\rho}^{*}.$

Let $Q_{j}$ be the $\operatorname{G}^{\prime}$-stable subgroup
$\operatorname{Gal}(\mathcal{K}_{j}/(\mathcal{K}_{j}\cap
E)\cdot\mathcal{K}_{j-1})$ and note that the $\operatorname{G}^{\prime}$
action on $P_{j}/Q_{j}$ is trivial. Invoking the same argument as in the case
when $j=1$, we have that $P_{j}=Q_{j}$ and hence $\mathcal{K}_{j}\cap
E=\mathcal{K}_{j-1}\cap E$. This completes the proof. ∎

######  Definition 4.6.

  1. (1)

Let $M_{1}$ and $M_{2}$ be
$\mathbb{F}_{p}[\operatorname{G}^{\prime}]$-modules. We say that $M_{1}$ is
unrelated to $M_{2}$ if for every
$\mathbb{F}_{p}[\operatorname{G}^{\prime}]$-submodule $N$ of $M_{1}$,

$\operatorname{Hom}(N,M_{2})^{\operatorname{G}^{\prime}}=0.$

  2. (2)

Let $E$ be a finite extension of $K$ such that $E$ is Galois over $\mathbb{Q}$
and $\operatorname{Gal}(E/K)$ is an $\mathbb{F}_{p}$-vector space. Let $M$ be
an $\mathbb{F}_{p}[\operatorname{G}^{\prime}]$-module. We say that $E$ is
unrelated to $M$ if $\operatorname{Gal}(E/K)$ is
$\operatorname{G}^{\prime}$-unrelated to $M$. Here, the
$\operatorname{G}^{\prime}$-action on $\operatorname{Gal}(E/K)$ is induced via
conjugation (let $x\in\operatorname{Gal}(E/K)$ and
$g\in\operatorname{G}^{\prime}$, pick a lift $\tilde{g}$ of $g$, set $g\cdot
x:=\tilde{g}x\tilde{g}^{-1}$).

######  Proposition 4.7.

Let $\mathcal{J}\supseteq S$ be a finite set of prime numbers and

$\theta_{0},\dots,\theta_{t}\in
H^{1}(\operatorname{G}_{\mathbb{Q},\mathcal{J}},\operatorname{Ad}^{0}\bar{\rho}^{*})$

be linearly independent over $\mathbb{F}_{q}$. Set $K_{i}:=K_{\theta_{i}}$ and
let $\mathbb{L}_{1},\dots,\mathbb{L}_{k}$ be a (possibly empty) set of Galois
extensions of $\mathbb{Q}$. Assume that $\mathbb{L}_{i}$ contains $K$ and
$\operatorname{Gal}(\mathbb{L}_{i}/K)$ is an $\mathbb{F}_{p}$-vector space for
$i=1,\dots,k$. Suppose that $\mathbb{L}_{i}$ is unrelated to
$\operatorname{Ad}^{0}\bar{\rho}^{*}$ for $i=1,\dots,k$. Denote by
$\mathcal{L}$ the composite $\mathbb{L}_{1}\cdots\mathbb{L}_{k}$. If the set
$\\{\mathbb{L}_{1},\dots,\mathbb{L}_{k}\\}$ is empty, set $\mathcal{L}=K$. The
field $K_{0}$ is not contained in the composite of the fields $K_{1}\cdots
K_{t}\cdot\mathcal{L}$.

###### Proof.

Let $\mathcal{K}$ denote the composite of the fields $K_{1},\dots,K_{t}$. If
$K_{0}$ is contained in $\mathcal{K}\cdot\mathcal{L}$, then
$\theta_{0},\dots,\theta_{t}\in
H^{1}(\operatorname{Gal}(\mathcal{K}\cdot\mathcal{L}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})$
and hence

$h^{1}(\operatorname{Gal}(\mathcal{K}\cdot\mathcal{L}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})\geq
t+1.$

Hence it suffices to show that

$h^{1}(\operatorname{Gal}(\mathcal{K}\cdot\mathcal{L}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})\leq
t.$

First we show that

$h^{1}(\operatorname{Gal}(\mathcal{L}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})=0.$

Denote by $\mathcal{L}_{i}$ the composite of the fields
$\mathbb{L}_{1}\cdots\mathbb{L}_{i}$ and set $\mathcal{L}_{0}:=K$. Note that
$\operatorname{Gal}(\mathcal{L}_{i}/\mathcal{L}_{i-1})$ is isomorphic to
$\operatorname{Gal}(\mathbb{L}_{i}/\mathbb{L}_{i}\cap\mathcal{L}_{i-1})$,
which is an $\mathbb{F}_{p}[\operatorname{G}^{\prime}]$-submodule of
$\operatorname{Gal}(\mathbb{L}_{i}/K)$. Since $\mathbb{L}_{i}$ is unrelated to
$\operatorname{Ad}^{0}\bar{\rho}^{*}$,

$\rm{Hom}(\operatorname{Gal}(\mathcal{L}_{i}/\mathcal{L}_{i-1}),\operatorname{Ad}^{0}\bar{\rho}^{*})^{\operatorname{G}^{\prime}}=0.$

Hence the inflation map

$H^{1}(\operatorname{Gal}(\mathcal{L}_{i-1}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})\xrightarrow{\operatorname{inf}}H^{1}(\operatorname{Gal}(\mathcal{L}_{i}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})$

is an isomorphism. We deduce that
$H^{1}(\operatorname{Gal}(\mathcal{L}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})$
is isomorphic to
$H^{1}(\operatorname{G}^{\prime},\operatorname{Ad}^{0}\bar{\rho}^{*})$ and
hence, is zero.

Let $\mathcal{K}_{i}$ denote the composite $K_{1}\cdots K_{i}$ and
$\mathcal{K}_{0}$ denote $K$. Note that
$\operatorname{Gal}(\mathcal{K}_{i}\cdot\mathcal{L}/\mathcal{K}_{i-1}\cdot\mathcal{L})$
is an $\mathbb{F}_{p}[\operatorname{G}^{\prime}]$-submodule of
$\operatorname{Gal}(K_{i}/K)$, and hence, of
$\operatorname{Ad}^{0}\bar{\rho}^{*}$. Lemma LABEL:y1 asserts that

$\dim\rm{Hom}(\operatorname{Gal}(\mathcal{K}_{i}\cdot\mathcal{L}/\mathcal{K}_{i-1}\cdot\mathcal{L}),\operatorname{Ad}^{0}\bar{\rho}^{*})^{\operatorname{G}^{\prime}}\leq
1.$

Therefore, by inflation-restriction,

$h^{1}(\operatorname{Gal}(\mathcal{K}_{i}\cdot\mathcal{L}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})\leq
h^{1}(\operatorname{Gal}(\mathcal{K}_{i-1}\cdot\mathcal{L}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})+1.$

Consequently, we deduce that
$h^{1}(\operatorname{Gal}(\mathcal{K}\cdot\mathcal{L}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})\leq
t$ and the proof is complete. ∎

######  Lemma 4.8.

Let $\mathcal{J}\supseteq S$ be a finite set of primes.

  1. (1)

Let $M$ be a nontrivial quotient of $\operatorname{Ad}^{0}\bar{\rho}^{*}$ and
$\eta\in H^{1}(\operatorname{G}_{\mathbb{Q},\mathcal{J}},M)$ be non-zero. Let
$K_{\eta}$ be the field extension of $K$ cut out by $\eta$. The field
$K_{\eta}$ is unrelated to $\operatorname{Ad}^{0}\bar{\rho}^{*}$.

  2. (2)

The field $K(\mu_{p^{2}})$ is unrelated to
$\operatorname{Ad}^{0}\bar{\rho}^{*}$.

  3. (3)

Let $f\in
H^{1}(\operatorname{G}_{\mathbb{Q},\mathcal{J}},\operatorname{Ad}^{0}\bar{\rho})$,
then the extension $K_{f}$ is unrelated to
$\operatorname{Ad}^{0}\bar{\rho}^{*}$.

  4. (4)

Suppose that we are given a lift
$\zeta_{2}:\operatorname{G}_{\mathbb{Q},\mathcal{J}}\rightarrow\operatorname{GSp}_{2n}(\text{W}(\mathbb{F}_{q})/p^{2})$
of $\bar{\rho}$ with similitude character $\kappa\mod{p^{2}}$. The field
extension $K(\zeta_{2})$ cut out by the kernel of $\zeta_{2}$, is unrelated to
$\operatorname{Ad}^{0}\bar{\rho}^{*}$.

###### Proof.

For part $\eqref{415c1}$, it suffices to show that $M$ is unrelated to
$\operatorname{Ad}^{0}\bar{\rho}^{*}$. Since $M$ is a non-trivial quotient of
$\operatorname{Ad}^{0}\bar{\rho}^{*}$, it follows from Lemma LABEL:mainin that
the $\bar{\chi}\sigma_{2L_{1}}$-eigenspace of $M$ is zero. Let $N\subseteq M$
be a $\operatorname{G}^{\prime}$-submodule and
$f:N\rightarrow\operatorname{Ad}^{0}\bar{\rho}^{*}$ be a homomorphism. The
$\bar{\chi}\sigma_{2L_{1}}$-eigenspace of $f(N)$ is zero, hence by Lemma
LABEL:mainin, the map $f=0$.

Since $\mathbb{Q}(\mu_{p^{2}})$ is an abelian extension of $\mathbb{Q}$, the
$G^{\prime}$-action on $\operatorname{Gal}(K(\mu_{p^{2}})/K)$ is trivial. On
the other hand, $\operatorname{Ad}^{0}\bar{\rho}^{*}$ has no trivial
$\mathbb{T}$-eigenspace. It follows that $K(\mu_{p^{2}})$ is unrelated to
$\operatorname{Ad}^{0}\bar{\rho}^{*}$ and part $\eqref{415c2}$ follows.

Let $Q$ be a $\operatorname{G}^{\prime}$-submodule of
$\operatorname{Ad}^{0}\bar{\rho}$, Lemma 4.2 asserts that
$Q=\bigoplus_{\lambda\in\Phi\cup\\{1\\}}Q_{\sigma_{\lambda}}$. On the other
hand,
$\operatorname{Ad}^{0}\bar{\rho}^{*}=\bigoplus_{\gamma\in\Phi\cup\\{1\\}}(\operatorname{Ad}^{0}\bar{\rho}^{*})_{\bar{\chi}\sigma_{\lambda}}$.
It follows from condition $\eqref{thc4}$ of Theorem 1.1 that

$\operatorname{Hom}(Q_{\sigma_{\lambda}},(\operatorname{Ad}^{0}\bar{\rho}^{*})_{\bar{\chi}\sigma_{\gamma}})^{\mathbb{T}}=0.$

As a result,
$\operatorname{Hom}(Q,\operatorname{Ad}^{0}\bar{\rho}^{*})^{\operatorname{G}^{\prime}}=0$
and hence part $\eqref{415c3}$ follows.

For part (4), identify $\operatorname{Ad}^{0}\bar{\rho}$ with the kernel of
the mod-$p$ reduction map

$\operatorname{Sp}_{2n}(\text{W}(\mathbb{F}_{q})/p^{2})\rightarrow\operatorname{Sp}_{2n}(\mathbb{F}_{q}),$

by identifying $X\in\operatorname{Ad}^{0}\bar{\rho}$ with
$\operatorname{Id}+pX$. Recall that $\kappa=\kappa_{0}\chi^{k}$, where $k$ is
a positive integer which is divisible by $p(p-1)$. Setting
$\kappa_{2}:=\kappa\mod{p^{2}}$, we see that the restriction
$\kappa_{2\restriction\operatorname{G}_{K}}$ is trivial. Therefore,
$\operatorname{Gal}(K(\zeta_{2})/K)$ may be identified with a Galois submodule
of $\operatorname{Ad}^{0}\bar{\rho}$. Here,
$g\in\operatorname{Gal}(K(\zeta_{2})/K)$ is identified with

$\zeta_{2}(g)\in\operatorname{ker}\left(\operatorname{Sp}_{2n}(\text{W}(\mathbb{F}_{q})/p^{2})\rightarrow\operatorname{Sp}_{2n}(\mathbb{F}_{q})\right)\simeq\operatorname{Ad}^{0}\bar{\rho}.$

The same reasoning as in the previous case shows that $K(\zeta_{2})$ is indeed
unrelated to $\operatorname{Ad}^{0}\bar{\rho}^{*}$. ∎

######  Lemma 4.9.

Let $L_{1},\dots,L_{k}$ and $K_{1},\dots,K_{l}$ be Galois extensions of
$\mathbb{Q}$ which contain $K$. Assume that:

  * •

$\operatorname{Gal}(L_{i}/K)$ and $\operatorname{Gal}(K_{i}/K)$ are finite
dimensional $\mathbb{F}_{p}$-vector spaces.

  * •

As a $\operatorname{G}^{\prime}$-module, $\operatorname{Gal}(L_{i}/K)$ is
isomorphic to a subquotient of $\operatorname{Ad}^{0}\bar{\rho}$ for
$i=1,\dots,k$.

  * •

As a $\operatorname{G}^{\prime}$-module, $\operatorname{Gal}(K_{i}/K)$ is
isomorphic to a subquotient of $\operatorname{Ad}^{0}\bar{\rho}^{*}$ for
$i=1,\dots,l$.

Then the composite $L_{1}\cdots L_{k}$ is linearly disjoint from
$K_{1},\dots,K_{l}$.

###### Proof.

The order of $\mathbb{T}$ is coprime to $p$, hence Maschke’s theorem asserts
that any finite dimensional $\mathbb{F}_{p}[\operatorname{G}^{\prime}]$-module
$M$ decomposes into a direct sum

$M=\oplus_{\tau}M_{\tau}.$

Here, $\tau$ is a character of $\mathbb{T}$ and $M_{\tau}$ is the
$\tau$-eigenspace

$M_{\tau}:=\\{m\in M|g\cdot m=\tau(g)m\\}.$

The action of $\operatorname{G}^{\prime}$ on $\operatorname{Gal}(L_{i}/K)$ and
$\operatorname{Gal}(K_{i}/K)$ is induced by conjugation. By assumption,
$\operatorname{Gal}(L_{i}/K)$ is isomorphic to a subquotient of
$\operatorname{Ad}^{0}\bar{\rho}$, i.e. there exist
$\operatorname{G}^{\prime}$-submodules $Q_{1}\subseteq Q_{2}$ of
$\operatorname{Ad}^{0}\bar{\rho}$ such that $\operatorname{Gal}(L_{i}/K)\simeq
Q_{2}/Q_{1}$. By Lemma 4.2, the module $Q_{i}$ decomposes into
$\mathbb{T}$-eigenspaces

$Q_{i}=\bigoplus_{\lambda\in\Phi\cup\\{1\\}}(Q_{i})_{\sigma_{\lambda}}$

for $i=1,2$. Therefore, the quotient $\operatorname{Gal}(L_{i}/K)$ decomposes
into

$\operatorname{Gal}(L_{i}/K)=\bigoplus_{\lambda\in\Phi\cup\\{1\\}}(\operatorname{Gal}(L_{i}/K))_{\sigma_{\lambda}}$

where
$(\operatorname{Gal}(L_{i}/K))_{\sigma_{\lambda}}:=(Q_{2})_{\sigma_{\lambda}}/(Q_{1})_{\sigma_{\lambda}}$
is the $\sigma_{\lambda}$-eigenspace for the action of $\mathbb{T}$ on
$\operatorname{Gal}(L_{i}/K)$. Likewise, $\operatorname{Gal}(K_{i}/K)$
decomposes into

$\operatorname{Gal}(K_{i}/K)=\bigoplus_{\lambda\in\Phi\cup\\{1\\}}(\operatorname{Gal}(K_{i}/K))_{\bar{\chi}\sigma_{\lambda}}.$

Let $\mathcal{L}$ be the composite $L_{1}\cdots L_{k}$ and $\mathcal{K}$ be
the composite $K_{1}\cdots K_{l}$. Letting $\mathcal{L}_{i}$ be the composite
$L_{1}\cdots L_{i}$, filter $\mathcal{L}$ by

$\mathcal{L}\supseteq\mathcal{L}_{k-1}\cdots\supseteq\mathcal{L}_{1}\supseteq
K.$

The Galois group

$\operatorname{Gal}(\mathcal{L}_{i}/\mathcal{L}_{i-1})\simeq\operatorname{Gal}(L_{i}/L_{i}\cap\mathcal{L}_{i-1})$

is a $\operatorname{G}^{\prime}$-submodule of $\operatorname{Gal}(L_{i}/K)$.
Hence the characters for the action of $\mathbb{T}$ on
$\operatorname{Gal}(\mathcal{L}_{i}/\mathcal{L}_{i-1})$ are each of the form
$\sigma_{\lambda}$. Similar reasoning shows that the characters for the action
of $\mathbb{T}$ on $\operatorname{Gal}(\mathcal{K}/K)$ are each of the form
$\bar{\chi}\sigma_{\lambda}$. Set $E=\mathcal{K}\cap\mathcal{L}$ and
$M=\operatorname{Gal}(E/K)$. Being a quotient of
$\operatorname{Gal}(\mathcal{L}/K)$, $M$ decomposes into eigenspaces for the
action of the torus

$M=\bigoplus_{\lambda\in\Phi\cup\\{1\\}}M_{\sigma_{\lambda}}.$

Since $M$ is a quotient of $\operatorname{Gal}(\mathcal{K}/K)$,

$M=\bigoplus_{\gamma\in\Phi\cup\\{1\\}}M_{\bar{\chi}\sigma_{\gamma}}.$

It is assumed that the image of $\sigma_{\lambda}$ spans $\mathbb{F}_{q}$ and
that $\sigma_{\lambda}$ is not a
$\operatorname{Gal}(\mathbb{F}_{q}/\mathbb{F}_{p})$ twist of
$\bar{\chi}\sigma_{\gamma}$. Hence, it follows that

$\operatorname{Hom}(\mathbb{F}_{q}(\sigma_{\lambda}),\mathbb{F}_{q}(\bar{\chi}\sigma_{\gamma}))^{\mathbb{T}}=0.$

Therefore, $\rm{Hom}(M,M)^{\operatorname{G}^{\prime}}=0$ and in particular,
the identity map is zero. This implies that $\mathcal{K}\cap\mathcal{L}=K$. ∎

##  5\. Deformation conditions at Auxiliary Primes

We introduce the auxiliary primes $v$ and the liftable deformation problem
$\mathcal{C}_{v}$ at $v$.

######  Definition 5.1.

A prime number $v$ is a trivial prime if the following splitting conditions
are satisfied:

  * •

$\operatorname{G}_{v}\subseteq\ker\bar{\rho}$,

  * •

$v\equiv 1\mod{p}$ and $v\not\equiv 1\mod{p^{2}}$.

In other words, a prime number $v$ is trivial if it splits in
$\mathbb{Q}(\bar{\rho},\mu_{p})$ and does not split in
$\mathbb{Q}(\mu_{p^{2}})$. By Lemma 4.5, $\mathbb{Q}(\bar{\rho},\mu_{p})$ does
not contain $\mathbb{Q}(\mu_{p^{2}})$. This is a Chebotarev condition, i.e.
defined by a finite union of sets that are defined by applying the Chebotarev
density theorem. Therefore, the set of trivial primes has positive Dirichlet
density, in particular, it is infinite.

Let $v$ be a trivial prime. The deformations of the trivial representation
$\bar{\rho}_{\restriction\operatorname{G}_{v}}$ are tamely ramified. The
Galois group of the maximal pro-p extension of $\mathbb{Q}_{v}$ is generated
by a Frobenius $\sigma_{v}$ and a generator of tame pro-$p$ inertia
$\tau_{v}$. These satisfy the relation
$\sigma_{v}\tau_{v}\sigma_{v}^{-1}=\tau_{l}^{v}$. We define the deformation
functor $\mathcal{C}_{v}$. The functor $\mathcal{C}_{v}$ will be liftable,
however, it will not be a deformation condition. Let $\alpha$ be a root which
shall be specified later. The root-subgroup
$\text{U}_{\alpha}\subset\operatorname{GSp}_{2n}$ is the subgroup generated by
the image of the root-subspace $(\operatorname{sp}_{2n})_{\alpha}$ under the
exponential map. We let $\text{Z}(\text{U}_{\alpha})$ be the subgroup of
$\operatorname{GSp}_{2n}$ consisting of elements which commute with
$\text{U}_{\alpha}$.

######  Definition 5.2.

[2, Definition 3.1] Let $\mathcal{D}_{v}^{\alpha}$ consist of the deformation
classes of lifts such that some representative $\varrho$ satisfies:

  1. (1)

$\varrho(\sigma_{v})\in\mathcal{T}\cdot\text{Z}(\text{U}_{\alpha})$ and
$\varrho(\tau_{v})\in\text{U}_{\alpha}$,

  2. (2)

under the composite

$\mathcal{T}\cdot\text{Z}(\text{U}_{\alpha})\rightarrow\mathcal{T}/(\mathcal{T}\cap\text{Z}(\text{U}_{\alpha}))\xrightarrow{\alpha}\text{GL}_{1}$

$\varrho(\sigma_{v})$ maps to $v$.

######  Remark 5.3.

When $n=1$ and $\alpha$ is the positive root of $\operatorname{sl}_{2}$, the
deformation functor $\mathcal{D}_{v}^{\alpha}$ consists of $\varrho$ such that
there exists $x$ and $y$ such that

$\varrho(\sigma_{v})=c\left({\begin{array}[]{cc}v&x\\\ 0&1\\\
\end{array}}\right)\text{ and
}\varrho(\tau_{v})=\left({\begin{array}[]{cc}1&y\\\ 0&1\\\
\end{array}}\right).$

Here $c$ is equal to $(\kappa(\sigma_{v})/v)^{\frac{1}{2}}$.

We shall denote by the kernel of $\alpha$ restricted to $\mathfrak{t}$ by
$\mathfrak{t}_{\alpha}$. Since the action of $\operatorname{G}_{v}$ on
$\operatorname{Ad}^{0}\bar{\rho}$ is trivial,

$H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})=\text{Hom}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho}).$

Let $\mathcal{P}_{v}^{\alpha}$ be the subspace of
$H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$ consisting of
$\phi$ such that

$\phi(\sigma_{v})\in\mathfrak{t}_{\alpha}+\text{Cent}((\operatorname{Ad}^{0}\bar{\rho})_{\alpha})$
$\phi(\tau_{v})\in(\operatorname{Ad}^{0}\bar{\rho})_{\alpha}.$

Let $\Phi^{\alpha}$ denote the subset of roots $\beta\in\Phi$ such that
$[(\operatorname{Ad}^{0}\bar{\rho})_{\alpha},(\operatorname{Ad}^{0}\bar{\rho})_{\beta}]\neq
0$. Recall that $X_{\alpha}$ is a choice of root vector for $\alpha$.

######  Definition 5.4.

  1. (1)

Let $v$ be a trivial prime which is unramified mod $p^{2}$ in our lifting
argument. Set $\alpha=2L_{1}$ and $\mathcal{C}_{v}=\mathcal{C}_{v}^{nr}$
consist of deformations with a representative

$\varrho^{\prime}=(\operatorname{Id}+X_{-\alpha})\varrho(\operatorname{Id}+X_{-\alpha})^{-1}$

where $\varrho$ is a representative for a deformation in
$\mathcal{D}_{v}^{\alpha}$ which satisfies further conditions. In accordance
with [2, Definition 3.5], we assume that the mod-$p^{2}$ reduction
$\varrho_{2}:=\varrho\mod{p^{2}}$ satisfies the following conditions:

    1. (a)

$\varrho_{2}$ is unramified, with
$\varrho_{2}(\sigma_{v})\in\mathcal{T}(\text{W}(\mathbb{F}_{q})/p^{2})$,

    2. (b)

for all $\beta\in\Phi^{\alpha}$,

$\beta(\varrho_{2}(\sigma_{v}))\neq 1\mod{p^{2}}.$

Let $\mathcal{S}_{v}^{\alpha}$ consist of $\phi\in
H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$ such that
$\phi(\sigma_{v})\in\bigoplus_{\beta\in\Phi^{\alpha}}(\operatorname{Ad}^{0}\bar{\rho})_{\beta}$
and $\phi(\tau_{v})=0$. Let $\mathcal{N}_{v}$ be specified by

$\mathcal{N}_{v}=\mathcal{N}_{v}^{nr}:=(\operatorname{Id}+X_{-\alpha})(\mathcal{P}_{v}^{\alpha}+\mathcal{S}_{v}^{\alpha})(\operatorname{Id}+X_{-\alpha})^{-1}.$

  2. (2)

Let $v$ be a trivial prime which will be ramified mod $p^{2}$ in our lifting
argument. Let $\alpha=-2L_{1}$ and $\mathcal{C}_{v}=\mathcal{C}_{v}^{ram}$
consist of deformations in $\mathcal{D}_{v}^{\alpha}$ with representative
$\varrho$ satisfying some additional conditions, which we specify. In
accordance with [2, Definition 3.9], assume that the mod-$p^{2}$ reduction
$\varrho_{2}$ satisfies the following conditions:

    1. (a)

$\varrho_{2}(\tau_{v})\in u_{\alpha}(py)$ where
$y\in\text{W}(\mathbb{F}_{q})^{\times}$, and
$u_{\alpha}:(\operatorname{Ad}^{0}\bar{\rho})_{\alpha}\rightarrow\operatorname{GSp}_{2n}$
is the root group homomorphism over $\operatorname{W}(\mathbb{F}_{q})$.

    2. (b)

For all $\beta\in\Phi^{\alpha}$,

$\beta(\varrho_{2}(\sigma_{v}))\neq 1\mod{p^{2}}.$

Let $\mathcal{S}_{v}^{\alpha}$ denote the space of cohomology classes
specified in the proof of [2, Lemma 3.10]. Let
$\mathcal{N}_{v}=\mathcal{N}_{v}^{ram}$ be defined by

$\mathcal{N}_{v}:=\mathcal{P}_{v}^{\alpha}+\mathcal{S}_{v}^{\alpha}.$

The following gives us a criterion for an element $f\in
H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$ to not be
contained in $\mathcal{N}_{v}^{nr}$. This criterion is used in the proof of
Proposition 7.1.

######  Lemma 5.5.

Let $v$ be a trivial prime and $\mathcal{C}_{v}=\mathcal{C}_{v}^{nr}$. Let
$f\in\mathcal{N}_{v}$, express
$f(\sigma_{v})=\sum_{\lambda\in\Phi}{a_{\lambda}}X_{\lambda}+\sum_{i=1}^{n}a_{i}H_{i}$.
Write $X_{-2L_{1}}=ce_{n+1,1}$ and $X_{2L_{1}}=de_{1,n+1}$. We have that
$a_{2L_{1}}=-(cd)^{-1}a_{1}$.

###### Proof.

Set $g:=(\operatorname{Id}+X_{-2L_{1}})^{-1}f(\operatorname{Id}+X_{-2L_{1}})$
and express
$g(\sigma_{v})=\sum_{\lambda\in\Phi}{b_{\lambda}}X_{\lambda}+\sum_{i=1}^{n}b_{i}H_{i}$.
Note that for $\phi\in\mathcal{P}_{v}^{2L_{1}}$,

$\phi(\sigma_{v})\in\mathfrak{t}_{2L_{1}}+\text{Cent}((\operatorname{Ad}^{0}\bar{\rho})_{2L_{1}})$

and hence has zero $H_{1}$-component. For $\phi\in\mathcal{S}_{v}^{2L_{1}}$,
we have that

$\phi(\sigma_{v})\in\bigoplus_{\beta\in\Phi^{2L_{1}}}(\operatorname{Ad}^{0}\bar{\rho})_{\beta}.$

We deduce that the $H_{1}$-component $b_{1}$ is equal to zero. We show that
the $H_{1}$-component of $g(\sigma_{v})$ is equal to $a_{1}+cda_{2L_{1}}$ from
the relation
$g(\sigma_{v})=(\operatorname{Id}+X_{-2L_{1}})^{-1}f(\sigma_{v})(\operatorname{Id}+X_{-2L_{1}})$.
Note that $X_{-2L_{1}}^{2}=0$ and thus,
$(\operatorname{Id}+X_{-2L_{1}})^{-1}=(\operatorname{Id}-X_{-2L_{1}})$. One
has that

$\begin{split}g(\sigma_{v})=&(\operatorname{Id}-X_{-2L_{1}})f(\sigma_{v})(\operatorname{Id}+X_{-2L_{1}})\\\
=&(\operatorname{Id}-ce_{n+1,1})f(\sigma_{v})(\operatorname{Id}+ce_{n+1,1})\\\
=&f(\sigma_{v})+c[f(\sigma_{v}),e_{n+1,1}]-c^{2}e_{n+1,1}f(\sigma_{v})e_{n+1,1}.\end{split}$

Note that

$c^{2}e_{n+1,1}f(\sigma_{v})e_{n+1,1}=a_{2L_{1}}c^{2}e_{n+1,1}X_{2L_{1}}e_{n+1,1}=a_{2L_{1}}c^{2}de_{n+1,1}=a_{2L_{1}}cdX_{-2L_{1}},$

and thus does not contribute to the $H_{1}$-component of $g(\sigma_{v})$. The
contribution of

$c[f(\sigma_{v}),e_{n+1,1}]=\sum_{\lambda\in\Phi}{ca_{\lambda}}[X_{\lambda},e_{n+1,1}]+\sum_{i=1}^{n}ca_{i}[H_{i},e_{n+1,1}]$

to the $H_{1}$-component of $g(\sigma_{v})$ is from the term

$ca_{2L_{1}}[X_{2L_{1}},e_{n+1,1}]=ca_{2L_{1}}[de_{1,n+1},e_{n+1,1}]=cda_{2L_{1}}H_{1}.$

Thus, we have shown that $b_{1}=a_{1}+cda_{2L_{1}}$. Since, $b_{1}=0$, we
deduce that $a_{1}=-cda_{2L_{1}}$. ∎

######  Lemma 5.6.

[2, Lemma 3.2, 3.6,3.10] Let $v$ be a trivial prime (for which either
$\mathcal{C}_{v}=\mathcal{C}_{v}^{ram}$ or $\mathcal{C}_{v}^{nr}$ is the
chosen deformation condition) and $X\in\mathcal{N}_{v}$,

  1. (1)

$\dim\mathcal{N}_{v}=\dim\operatorname{Ad}^{0}\bar{\rho}=h^{0}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$.

  2. (2)

Let $m\geq 3$ and
$\rho_{m}\in\mathcal{C}_{v}(\text{W}(\mathbb{F}_{q})/p^{m})$, then

$(\operatorname{Id}+p^{n-1}X)\rho_{m}\in\mathcal{C}_{v}(\text{W}(\mathbb{F}_{q})/p^{m}).$

  3. (3)

The deformation functor $\mathcal{C}_{v}$ is liftable.

Prior to lifting $\bar{\rho}$ to characteristic zero, we show that
$\bar{\rho}$ lifts to $\rho_{2}$ after increasing the set of ramification from
$S$ to $S\cup X_{1}$. One may choose a continuous lift $\tau$ of $\bar{\rho}$
as depicted

$\operatorname{G}_{\mathbb{Q},S\cup
X_{1}}$$\operatorname{GSp}_{2n}(\mathbb{F}_{q})$$\operatorname{GSp}_{2n}(\text{W}(\mathbb{F}_{q})/p^{2})$$\bar{\rho}$$\tau$

such that the composite $\nu\circ\tau=\psi\mod{p^{2}}$. The obstruction class

$\mathcal{O}(\bar{\rho})_{\restriction S\cup X_{1}}\in
H^{2}(\operatorname{G}_{S\cup X_{1}},\operatorname{Ad}^{0}\bar{\rho})$

is represented by the $2$-cocycle

$(g_{1},g_{2})\mapsto\tau(g_{1}g_{2})\tau(g_{2})^{-1}\tau(g_{1})^{-1}.$

The residual representation $\bar{\rho}$ lifts to a representation $\rho_{2}$
ramified only at primes in $S\cup X_{1}$ if and only if this obstruction is
zero. For $v\in S$, the local representation
$\bar{\rho}_{\restriction\operatorname{G}_{v}}$ satisfies $\mathcal{C}_{v}$
which is a liftable deformation condition (by assumption) and thus lifts to
mod $p^{2}$. The residual representation $\bar{\rho}$ is unramified at each
prime $v\in X_{1}$ and thus it is easy to see that
$\bar{\rho}_{\restriction\operatorname{G}_{v}}$ lifts to mod $p^{2}$ for $v\in
X_{1}$. As a consequence, $\mathcal{O}(\bar{\rho})_{\restriction S\cup X_{1}}$
is contained in ${}^{2}_{S\cup X_{1}}(\operatorname{Ad}^{0}\bar{\rho})$. We
will show that a set of finitely many trivial primes $X_{1}$ can be chosen so
that

${}^{2}_{S\cup X_{1}}(\operatorname{Ad}^{0}\bar{\rho})=0.$

For such a choice of $X_{1}$, there is a deformation $\rho_{2}$

$\operatorname{G}_{\mathbb{Q},S\cup
X_{1}}$$\operatorname{GSp}_{2n}(\mathbb{F}_{q})$.$\operatorname{GSp}_{2n}(\text{W}(\mathbb{F}_{q})/p^{2})$$\bar{\rho}$$\rho_{2}$

######  Proposition 5.7.

Let $\mathscr{M}$ denote the finite set of
$\operatorname{G}_{\mathbb{Q}}$-modules defined by

$\begin{split}\mathscr{M}:=&\\{(\operatorname{Ad}^{0}\bar{\rho})/(\operatorname{Ad}^{0}\bar{\rho})_{k},\mid-2n+1\leq
k\leq 2n\\}\\\
&\cup\\{(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp},\mid-2n+1\leq k\leq
2n\\}.\\\ \end{split}$

There is a finite set $T\supset S$ such that $T\backslash S$ consists of only
trivial primes such that for all $M\in\mathscr{M}$,

(5.1)
$\ker\\{H^{1}(\operatorname{G}_{\mathbb{Q},T},M)\rightarrow\bigoplus_{w\in
T\backslash S}H^{1}(\operatorname{G}_{w},M)\\}=0$

and so in particular,

${}_{T}^{1}(M)=0.$

###### Proof.

We show that $T$ can be chosen for which

${}_{T}^{1}(\operatorname{Ad}^{0}\bar{\rho}^{*})=0,$

the argument for any $M\in\mathscr{M}$ is identical. For $0\neq\psi\in
H^{1}(\operatorname{G}_{\mathbb{Q},S},\operatorname{Ad}^{0}\bar{\rho}^{*})$,
let $K_{\psi}\supset\mathbb{Q}(\operatorname{Ad}^{0}\bar{\rho}^{*})$ be the
field extension cut out by $\psi$. By Lemma LABEL:l4, the extension $K_{\psi}$
is not equal to $\mathbb{Q}(\operatorname{Ad}^{0}\bar{\rho}^{*})$. The
extension $K(\mu_{p^{2}})$ is linearly disjoint with $K_{\psi}$ over $K$. By
Lemma 4.5, $K(\mu_{p^{2}})$ is not contained in $K$ and $K(\mu_{p^{2}})\cap
K_{\psi}=K$. As a result, there is a nonempty Chebotarev class of primes which
split in $K$ and are non-split in $K_{\psi}$ and $K(\mu_{p^{2}})$. If $v$ is
such a prime, it must be a trivial prime since it splits in $K$ and is non-
split in $\mathbb{Q}(\mu_{p^{2}})$. On the other hand, since $v$ is non-split
in $K_{\psi}$, deduce that $\psi_{\restriction\operatorname{G}_{v}}\neq 0$. We
may therefore choose a finite set of primes $T$ such that

  * •

$T$ is finite,

  * •

$T\backslash S$ consists of only trivial primes,

  * •

$\ker\\{H^{1}(\operatorname{G}_{\mathbb{Q},T},\operatorname{Ad}^{0}\bar{\rho}^{*})\rightarrow\bigoplus_{w\in
T\backslash
S}H^{1}(\operatorname{G}_{w},\operatorname{Ad}^{0}\bar{\rho}^{*})\\}=0$.

∎

The set of trivial primes $X_{1}$ is taken to be $T\backslash S$.

##  6\. Lifting to mod $p^{3}$

By Proposition 5.7, there is a finite set of primes $T$ containing $S$ such
that $T\backslash S$ consists of trivial primes and
${}_{T}^{1}(\operatorname{Ad}^{0}\bar{\rho}^{*})=0$. Let $X_{1}$ be the set of
trivial primes $T\backslash S$. At each prime $v\in X_{1}$, let
$\mathcal{C}_{v}$ be the liftable deformation problem $\mathcal{C}_{v}^{nr}$.
By global duality, ${}_{T}^{2}(\operatorname{Ad}^{0}\bar{\rho})=0$ and thus
the cohomological obstruction to lifting $\bar{\rho}$ to a representation
$\zeta_{2}$

(6.1)
$\operatorname{G}_{\mathbb{Q},T}$$\operatorname{GSp}_{2n}(\mathbb{F}_{q})$$\operatorname{GSp}_{2n}(\text{W}(\mathbb{F}_{q})/p^{2})$$\bar{\rho}$$\zeta_{2}$

vanishes. Here, $\zeta_{2}$ is stipulated to have similitude character
$\kappa\mod{p^{2}}$. Let $v\in T$, recall that the set of
$W(\mathbb{F}_{q})/p^{2}$ lifts of
$\bar{\rho}_{\restriction\operatorname{G}_{v}}$ is an
$H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$-torsor.
Therefore there exists $z_{v}\in
H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho})$ such that the
twist
$(\operatorname{Id}+z_{v}p){\zeta_{2}}_{\restriction\operatorname{G}_{v}}$
satisfies $\mathcal{C}_{v}$. Further, for $v\in X_{1}$, the class $z_{v}$ may
is chosen so that this twist is unramified. We show that there is a set $W$ of
at most two trivial primes such that on increasing the set $T$ to $Z=T\cup W$
there exists a global cohomology class $h\in
H^{1}(\operatorname{G}_{\mathbb{Q},Z},\operatorname{Ad}^{0}\bar{\rho})$ such
that

  * •

$h_{\restriction\operatorname{G}_{v}}=z_{v}$ for $v\in T$,

  * •

$(1+ph)\zeta_{2}|_{\operatorname{G}_{v}}\in\mathcal{C}_{v}^{ram}$ for $v\in
W$.

Further, letting $\rho_{2}$ be the twist
$\rho_{2}=(\operatorname{Id}+ph)\zeta_{2}$, each local representation
${\rho_{2}}_{\restriction\operatorname{G}_{v}}$ satisfies $\mathcal{C}_{v}$
for $v\in Z$. As a consequence, the obstruction class $\mathcal{O}(\rho_{2})$
is in ${}_{Z}^{2}(\operatorname{Ad}^{0}\bar{\rho})$. Since $Z$ contains $T$,
the group ${}_{Z}^{2}(\operatorname{Ad}^{0}\bar{\rho})$ is zero. As a result,
$\rho_{2}$ must lift to $W(\mathbb{F})/p^{3}$. Assume that there is no such
class $h$ for a set $W$ such that $\\#W\leq 1$. It is shown that there is a
pair of trivial primes $v_{1},v_{2}\notin T$ such that $W$ can be chosen to be
equal to $\\{v_{1},v_{2}\\}$. The set of trivial primes $X_{2}$ is then chosen
to be $Z\backslash S$. For $v\in W$, choose $\mathcal{C}_{v}$ to be equal to
$\mathcal{C}_{v}^{ram}$. In what follows, a Chebotarev class refers to a
nonempty collection of primes defined by the application of the Chebotarev
density theorem. Note that a Chebotarev class has positive Dirichlet density,
and is in particular, infinite.

######  Proposition 6.1.

Let $T$ be as in Proposition 5.7 and $\psi$ be a nonzero element in
$H^{1}(\operatorname{G}_{\mathbb{Q},T},\operatorname{Ad}^{0}\bar{\rho}^{*})$
and let $W\subset
H^{1}(\operatorname{G}_{\mathbb{Q},T},\operatorname{Ad}^{0}\bar{\rho}^{*})$ be
a subspace not containing $\psi$. Then, there exists a Chebotarev class of
trivial primes $v$ such that

$\begin{split}&{\psi}_{\restriction\operatorname{G}_{v}}\neq 0\\\
&{\beta}_{\restriction\operatorname{G}_{v}}=0\text{ for all }\beta\in
W.\end{split}$

Moreover we may choose $v$ so that $v$ does not split completely in the
$\bar{\chi}\sigma_{2L_{1}}$-eigenspace of $\operatorname{Gal}(K_{\psi}/K)$
when viewed as a Galois submodule of $\operatorname{Ad}^{0}\bar{\rho}^{*}$.

###### Proof.

Let $\\{\psi_{1},\dots,\psi_{m}\\}$ be a basis of $W$. Since $\psi$ is not
contained in the span of $W$, the classes $\psi,\psi_{1},\dots,\psi_{m}$ are
linearly independent. Extend $\psi_{1},\dots,\psi_{m}$ to
$\psi_{1},\dots,\psi_{r}$, so that $\psi,\psi_{1},\dots,\psi_{r}$ is a basis
of
$H^{1}(\operatorname{G}_{\mathbb{Q},T},\operatorname{Ad}^{0}\bar{\rho}^{*})$.
Let $\widetilde{W}$ be the span of $\\{\psi_{1},\dots,\psi_{r}\\}$. It
suffices to prove the statement for $\widetilde{W}$ in place of $W$, since $W$
is contained in $\widetilde{W}$. Let $\mathfrak{F}$ denote the composite
$K_{\psi_{1}}\cdots K_{\psi_{r}}$. Set $P:=\operatorname{Gal}(K_{\psi}/K)$ and
recall that $J_{\psi}\subset K_{\psi}$ is the field fixed by
$P_{\bar{\chi}\sigma_{2L_{1}}}$. Lemma LABEL:l4 asserts that $J_{\psi}\neq
K_{\psi}$. We will show that $\mathfrak{F}\cap K_{\psi}\subseteq J_{\psi}$.
First, we show how the result follows from this.

Set $\mathfrak{L}:=\mathfrak{F}\cdot K_{\psi}=K_{\psi_{1}}\cdots
K_{\psi_{r}}\cdot K_{\psi}$. We consider the following field diagram,

$\mathbb{Q}(\mu_{p}).$$\mathfrak{F}\cap
K_{\psi}$$J_{\psi}$$K_{\psi}$$\mathfrak{F}$$\mathbb{Q}(\mu_{p^{2}})$$\mathfrak{L}$

By Lemma 4.5, the intersection
$K\cap\mathbb{Q}(\mu_{p^{2}})=\mathbb{Q}(\mu_{p})$. In fact, Lemma 4.5 asserts
that $\mathfrak{F}\cap\mathbb{Q}(\mu_{p^{2}})=\mathbb{Q}(\mu_{p})$. Therefore
there is a prime $v$ which is

  1. (1)

split in $\operatorname{Gal}(\mathfrak{F}/\mathbb{Q})$,

  2. (2)

nonsplit in $\operatorname{Gal}(\mathbb{Q}(\mu_{p^{2}})/\mathbb{Q}(\mu_{p}))$,

  3. (3)

nonsplit in $\operatorname{Gal}(K_{\psi}/J_{\psi})$.

Since $K=\mathbb{Q}(\bar{\rho},\mu_{p})$ is contained in $\mathfrak{F}$, the
prime $v$ is a trivial prime. Since $v$ splits in
$\operatorname{Gal}(\mathfrak{F}/\mathbb{Q})$, we have that
$\psi_{i\restriction\operatorname{G}_{v}}=0$ for $i=1,\dots,r$. Since $v$ does
not split in $\operatorname{Gal}(K_{\psi}/K)$, we have that
$\psi_{\restriction\operatorname{G}_{v}}\neq 0$.

We begin by showing that $K_{\psi}$ is not contained in $\mathfrak{F}$. This
is equivalent to the assertion that $\mathfrak{L}$ is not equal to
$\mathfrak{F}$. Each of the classes $\psi,\psi_{1},\dots,\psi_{r}$ is in the
image of the inflation map

$H^{1}(\operatorname{Gal}(\mathfrak{L}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})\xrightarrow{\operatorname{inf}}H^{1}(\operatorname{G}_{\mathbb{Q},T},\operatorname{Ad}^{0}\bar{\rho}^{*}),$

and hence the above map is an isomorphism. It follows that

$h^{1}(\operatorname{Gal}(\mathfrak{L}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})=h^{1}(\operatorname{G}_{\mathbb{Q},T},\operatorname{Ad}^{0}\bar{\rho}^{*})\geq
r+1.$

It suffices to show that
$h^{1}(\operatorname{Gal}(\mathfrak{F}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})\leq
r$. We show by induction on $i$ that

$h^{1}(\operatorname{Gal}(K_{\psi_{1}}\cdots
K_{\psi_{i}}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})\leq i.$

Lemma LABEL:l3 asserts that
$H^{1}(\operatorname{G}^{\prime},\operatorname{Ad}^{0}\bar{\rho}^{*})=0$ and
hence by inflation-restriction,

$H^{1}(\operatorname{Gal}(K_{\psi_{1}}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})\simeq\operatorname{Hom}(P_{1},\operatorname{Ad}^{0}\bar{\rho}^{*})^{\operatorname{G}^{\prime}}.$

Lemma LABEL:y1 asserts that

$\dim\operatorname{Hom}(P_{1},\operatorname{Ad}^{0}\bar{\rho}^{*})^{\operatorname{G}^{\prime}}\leq
1$

and hence the case $i=1$ follows.

For the induction step, set $\mathfrak{F}_{i}:=K_{\psi_{1}}\cdots
K_{\psi_{i}}$ and

$P_{i}:=\operatorname{Gal}(\mathfrak{F}_{i}/\mathfrak{F}_{i-1})\simeq\operatorname{Gal}(K_{\psi_{i}}/K_{\psi_{i}}\cap\mathfrak{F}_{i-1}).$

Lemma LABEL:y1 asserts that

$\dim\operatorname{Hom}(P_{i},\operatorname{Ad}^{0}\bar{\rho}^{*})^{\operatorname{G}^{\prime}}\leq
1$

from which we see from inflation-restriction

$h^{1}(\operatorname{Gal}(\mathfrak{F}_{i}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})\leq
h^{1}(\operatorname{Gal}(\mathfrak{F}_{i-1}/\mathbb{Q}),\operatorname{Ad}^{0}\bar{\rho}^{*})+1.$

We conclude that $\mathfrak{L}\neq\mathfrak{F}$ and thus we have deduced that
$K_{\psi}\cap\mathfrak{F}\neq K_{\psi}$. Set
$Q:=\operatorname{Gal}(K_{\psi}/K_{\psi}\cap\mathfrak{F})$, by Lemma
LABEL:mainin,

$Q_{\bar{\chi}\sigma_{2L_{1}}}\simeq(\operatorname{Ad}^{0}\bar{\rho}^{*})_{\bar{\chi}\sigma_{2L_{1}}}\simeq
P_{\bar{\chi}\sigma_{2L_{1}}}.$

We deduce that $K_{\psi}\cap\mathfrak{F}$ is contained in $J_{\psi}$. This
completes the proof. ∎

######  Definition 6.2.

Let $\mathcal{J}$ be a set of trivial primes that contains the set $S$ and
$v\notin\mathcal{J}$ be a trivial prime. Denote by $\Psi_{\mathcal{J}}^{k}$
and $\Psi_{\mathcal{J},v}^{k}$ the maps defined by

$\Psi_{\mathcal{J}}^{k}:H^{1}(\operatorname{G}_{\mathbb{Q},\mathcal{J}},(\operatorname{Ad}^{0}\bar{\rho})_{k})\xrightarrow{res_{\mathcal{J}}}\bigoplus_{w\in\mathcal{J}}H^{1}(\operatorname{G}_{w},(\operatorname{Ad}^{0}\bar{\rho})_{k})$

and

$\Psi_{\mathcal{J},v}^{k}:H^{1}(\operatorname{G}_{\mathbb{Q},\mathcal{J}\cup\\{v\\}},(\operatorname{Ad}^{0}\bar{\rho})_{k})\xrightarrow{res_{\mathcal{J}}}\bigoplus_{w\in\mathcal{J}}H^{1}(\operatorname{G}_{w},(\operatorname{Ad}^{0}\bar{\rho})_{k}).$

Let $\tau_{v}$ be a generator of the maximal pro-$p$ quotient of the tame
inertia at $v$, denote by

$\pi_{\mathcal{J},v}^{k}:H^{1}(\operatorname{G}_{\mathbb{Q},\mathcal{J}\cup\\{v\\}},(\operatorname{Ad}^{0}\bar{\rho})_{k})\rightarrow(\operatorname{Ad}^{0}\bar{\rho})_{k}$

the evaluation map defined by

$\pi_{\mathcal{J},v}^{k}(f):=f(\tau_{v}).$

######  Lemma 6.3.

Let $T$ be a set of primes as in Proposition 5.7 that contains the set $S$ and
$k$ an integer. Suppose $v\notin T$ is a trivial prime with the property that
for all $\beta\in
H^{1}(\operatorname{G}_{\mathbb{Q},T},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{*})$,
the restriction $\beta_{\restriction\operatorname{G}_{v}}=0$. The following
are exact:

(6.2)
$0\rightarrow\operatorname{ker}\Psi_{T}^{k}\xrightarrow{inf}\operatorname{ker}\Psi_{T,v}^{k}\xrightarrow{\pi_{v}^{k}}(\operatorname{Ad}^{0}\bar{\rho})_{k}\rightarrow
0,$ (6.3) $0\rightarrow
H^{1}(\operatorname{G}_{\mathbb{Q},T},(\operatorname{Ad}^{0}\bar{\rho})_{k})\xrightarrow{inf}H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},(\operatorname{Ad}^{0}\bar{\rho})_{k})\xrightarrow{\pi_{v}^{k}}(\operatorname{Ad}^{0}\bar{\rho})_{k}\rightarrow
0.$

Further, the image of $\Psi_{T}$ is equal to the image of $\Psi_{T,v}$.

###### Proof.

Clearly the composite of the maps is zero and $\eqref{shortexact1}$ is exact
in the middle. Denote by $\operatorname{res}_{v}$ the restriction map:

$\operatorname{res}_{v}:H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{*})\rightarrow
H^{1}(\operatorname{G}_{v},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{*}).$

By assumption,
$H^{1}(\operatorname{G}_{\mathbb{Q},T},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{*})$
and $\operatorname{ker}\operatorname{res}_{v}$ are equal. By the local Euler
characteristic formula and local duality,

$h^{1}(\operatorname{G}_{v},(\operatorname{Ad}^{0}\bar{\rho})_{k})-h^{0}(\operatorname{G}_{v},(\operatorname{Ad}^{0}\bar{\rho})_{k})$
$=h^{2}(\operatorname{G}_{v},(\operatorname{Ad}^{0}\bar{\rho})_{k})=h^{0}(\operatorname{G}_{v},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{*})=\dim(\operatorname{Ad}^{0}\bar{\rho})_{k}.$

By Wiles’ Formula $\eqref{wilesformula}$,

$\begin{split}\dim\operatorname{ker}\Psi_{T,v}^{k}=&\dim\operatorname{ker}\Psi_{T}^{k}+\dim\operatorname{ker}\operatorname{res}_{v}-h^{1}(\operatorname{G}_{\mathbb{Q},T},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{*})\\\
+&h^{1}(\operatorname{G}_{v},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{*})-h^{0}(\operatorname{G}_{v},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{*})\\\
=&\dim\operatorname{ker}\Psi_{T}^{k}+\dim(\operatorname{Ad}^{0}\bar{\rho})_{k}\\\
\end{split}$

and the exactness of $\eqref{shortexact1}$ follows. The exactness of
$\eqref{shortexact2}$ follows by the same arguments. Therefore,

$\begin{split}\dim\operatorname{im}\Psi_{T,v}=&h^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},(\operatorname{Ad}^{0}\bar{\rho})_{k})-\dim\operatorname{ker}\Psi_{T,v}\\\
=&h^{1}(\operatorname{G}_{\mathbb{Q},T},(\operatorname{Ad}^{0}\bar{\rho})_{k})-\dim\operatorname{ker}\Psi_{T}=\dim\operatorname{im}\Psi_{T}.\\\
\end{split}$

∎

Let $M$ be an $\mathbb{F}_{q}[\operatorname{G}_{w}]$-module which is a finite
dimensional $\mathbb{F}_{q}$-vector space. The cup product induces the map

$H^{1}(\operatorname{G}_{w},M)\times
H^{1}(\operatorname{G}_{w},M^{*})\rightarrow
H^{2}(\operatorname{G}_{w},\mathbb{F}_{q}(\bar{\chi}))\xrightarrow{\sim}\mathbb{F}_{q}$

taking $f_{1}\in H^{1}(\operatorname{G}_{w},M)$ and $f_{2}\in
H^{1}(\operatorname{G}_{w},M^{*})$ to $\operatorname{inv}_{w}(f_{1}\cup
f_{2})\in\mathbb{F}_{q}$. Define the non-degenerate pairing

$\left(\bigoplus_{w\in
T}H^{1}(\operatorname{G}_{w},\operatorname{Ad}^{0}\bar{\rho})\right)\times\left(\bigoplus_{w\in
T}H^{1}(\operatorname{G}_{w},\operatorname{Ad}^{0}\bar{\rho}^{*})\right)\rightarrow\mathbb{F}_{q}$

defined by $a\cup b=\sum_{w\in T}\operatorname{inv}_{w}(a_{w}\cup b_{w})$.
Denote by $\operatorname{Ann}((z_{w})_{w\in T})$ the annihilator of the tuple
$(z_{w})_{w\in T}$. Recall that we assume that $(z_{w})_{w\in T}$ does not
arise from a global class unramified outside $T$. In particular, the tuple
$(z_{w})_{w\in T}$ is not zero, and as a result,
$\operatorname{Ann}((z_{w})_{w\in T})$ is a codimension one subspace of
$\bigoplus_{w\in
T}H^{1}(\operatorname{G}_{w},\operatorname{Ad}^{0}\bar{\rho}^{*})$. Let
$\Psi_{T}$ and $\Psi_{T}^{*}$ denote the restriction maps

$\begin{split}&\Psi_{T}:H^{1}(\operatorname{G}_{\mathbb{Q},T},\operatorname{Ad}^{0}\bar{\rho})\rightarrow\bigoplus_{w\in
T}H^{1}(\operatorname{G}_{w},\operatorname{Ad}^{0}\bar{\rho}),\\\
&\Psi_{T}^{*}:H^{1}(\operatorname{G}_{\mathbb{Q},T},\operatorname{Ad}^{0}\bar{\rho}^{*})\rightarrow\bigoplus_{w\in
T}H^{1}(\operatorname{G}_{w},\operatorname{Ad}^{0}\bar{\rho}^{*}).\end{split}$

From the exactness of the Poitou-Tate sequence [10, Theorem 8.6.14], it
follows that the images of $\Psi_{T}$ and $\Psi_{T}^{*}$ are exact
annihilators of one another. Since it is assumed that $(z_{w})_{w\in T}$ is
not in the image $\Psi_{T}^{*}$, it follows that the image of $\Psi_{T}$ is
not contained in $\operatorname{Ann}((z_{w})_{w\in T})$. As a result,
${\Psi_{T}^{*}}^{-1}(\operatorname{Ann}(z_{w})_{w\in T})$ has codimension one
in
$H^{1}(\operatorname{G}_{\mathbb{Q},T},\operatorname{Ad}^{0}\bar{\rho}^{*})$.
Set $(\operatorname{Ad}^{0}\bar{\rho})_{-2L_{1}}$ for the $\mathbb{F}_{q}$
span of the root vector $X_{-2L_{1}}$.

######  Proposition 6.4.

Let $T$ be as in Proposition 5.7. There exists a Chebotarev class
$\mathfrak{l}$ of trivial primes $v$ such that

  1. (1)

$\beta_{\restriction\operatorname{G}_{v}}=0$ for all $\beta\in
H^{1}(\operatorname{G}_{\mathbb{Q},T},(\operatorname{Ad}^{0}\bar{\rho})_{d}^{*})$
for $d\geq-2n+2$,

  2. (2)

there exists an $\mathbb{F}_{q}$ basis $\\{\psi,\psi_{1},\dots,\psi_{r}\\}$ of
$H^{1}(\operatorname{G}_{\mathbb{Q},T},\operatorname{Ad}^{0}\bar{\rho}^{*})$
such that

     * •

$\\{\psi_{1},\dots,\psi_{r}\\}$ is a basis of
${\Psi_{T}^{*}}^{-1}(Ann(z_{w})_{w\in T})$

     * •

$\psi_{\restriction\operatorname{G}_{v}}\neq 0$ and
${\psi_{j}}_{\restriction\operatorname{G}_{v}}=0$ for all $j\geq 1$.

Furthermore, there is, for each $v\in\mathfrak{l}$, an element $h^{(v)}\in
H^{1}(\operatorname{G}_{T\cup\\{v\\}},\operatorname{Ad}^{0}\bar{\rho})$ such
that

$h^{(v)}|_{\operatorname{G}_{w}}=z_{w}$

for all $w\in T$ and

(6.4)
$h^{(v)}(\tau_{v})\in(\operatorname{Ad}^{0}\bar{\rho})_{-2L_{1}}\backslash\\{0\\}.$

###### Proof.

First, we analyze condition $\eqref{22Decc1}$. Recall that
$(\operatorname{Ad}^{0}\bar{\rho})_{d}^{*}$ is the quotient of
$\operatorname{Ad}^{0}\bar{\rho}^{*}$ by the Galois stable subspace
$(\operatorname{Ad}^{0}\bar{\rho})_{d}^{\perp}$, see Definition LABEL:perpdef.
Its $\mathbb{T}$-eigenspaces consist of
$(\operatorname{Ad}^{0}\bar{\rho})_{d,\bar{\chi}\sigma_{\lambda}^{-1}}^{*}$,
where $\lambda$ ranges through $\Phi\cup\\{1\\}$ with
$\operatorname{ht}(\lambda)\geq d$. Condition $\eqref{thc4}$ of Theorem 1.1
asserts that $\bar{\chi}\sigma_{-\lambda}\neq\sigma_{1}$, i.e.,
$\sigma_{\lambda}\neq\bar{\chi}$. Therefore,
$(\operatorname{Ad}^{0}\bar{\rho})_{d}^{*}$ contains no trivial eigenspace.
Hence, the splitting conditions imposed by $\eqref{22Decc1}$ are independent
of the non-splitting condition in $\mathbb{Q}(\mu_{p^{2}})$ imposed by the
fact that trivial primes are not $1\mod{p^{2}}$. On the other hand, by
Proposition 6.1, condition $\eqref{22Decc2}$ can be satisfied by a Chebotarev
class of trivial primes.

Next, we show that conditions $\eqref{22Decc1}$ and $\eqref{22Decc2}$ can be
satisfied simultaneously. To show this, note that the condition requiring
$\psi_{\restriction{\operatorname{G}_{v}}}\neq 0$ is a non-splitting condition
of $v$ in $\operatorname{Gal}(K_{\psi}/K)$. By Lemma LABEL:l4, the
$\bar{\chi}\sigma_{2L_{1}}$-eigenspace for the $\mathbb{T}$-action on
$\operatorname{Gal}(K_{\psi}/K)$ is nontrivial. We shall require that $v$ does
not split in the $\bar{\chi}\sigma_{2L_{1}}$-eigenspace of
$\operatorname{Gal}(K_{\psi}/K)$. On the other hand,

$(\operatorname{Ad}^{0}\bar{\rho}^{*})_{d}=\bigoplus_{\operatorname{ht}(\lambda)\geq
d}(\operatorname{Ad}^{0}\bar{\rho}^{*})_{d,\bar{\chi}\sigma_{\lambda}^{-1}},$

does not contain the $\bar{\chi}\sigma_{2L_{1}}$-eigenspace of
$\operatorname{Ad}^{0}\bar{\rho}^{*}$. Note that the character
$\bar{\chi}\sigma_{2L_{1}}$ is not twist equivalent to any of the characters
occurring in the $\mathbb{T}$-eigenspace decomposition of
$(\operatorname{Ad}^{0}\bar{\rho}^{*})_{d}$. As a result, it follows via an
argument identical to that in proof of Lemma 4.9, that the non-splitting
condition of $v$ in
$\operatorname{Gal}(K_{\psi}/K)_{\bar{\chi}\sigma_{2L_{1}}}$ may be
simultaneously satisfied along with the rest of the splitting conditions.

Let $v$ be a trivial prime which satisfies conditions $\eqref{22Decc1}$ and
$\eqref{22Decc2}$ and moreover is non-split in
$\operatorname{Gal}(K_{\psi}/K)_{\bar{\chi}\sigma_{2L_{1}}}$. Let
$d\geq-2n+2$, Lemma 6.3 asserts that the image of

$\Psi_{T}^{d}:H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},(\operatorname{Ad}^{0}\bar{\rho})_{d})\rightarrow\bigoplus_{w\in
T}H^{1}(\operatorname{G}_{w},(\operatorname{Ad}^{0}\bar{\rho})_{d})$

is the same as the image of

$\Psi_{T,v}^{d}:H^{1}(\operatorname{G}_{\mathbb{Q},T},(\operatorname{Ad}^{0}\bar{\rho})_{d})\rightarrow\bigoplus_{w\in
T}H^{1}(\operatorname{G}_{w},(\operatorname{Ad}^{0}\bar{\rho})_{d}).$

For a trivial prime $v$ for which condition $\eqref{22Decc2}$ is satisfied, it
follows from an application of Wiles’ formula $\eqref{wilesformula}$ that the
image of the map

$\Psi_{T,v}:H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},\operatorname{Ad}^{0}\bar{\rho})\rightarrow\bigoplus_{w\in
T}H^{1}(\operatorname{G}_{w},\operatorname{Ad}^{0}\bar{\rho})$

is greater than that of the map

$\Psi_{T}:H^{1}(\operatorname{G}_{\mathbb{Q},T},\operatorname{Ad}^{0}\bar{\rho})\rightarrow\bigoplus_{w\in
T}H^{1}(\operatorname{G}_{w},\operatorname{Ad}^{0}\bar{\rho}).$

We next deduce the existence of $h^{(v)}\in
H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},\operatorname{Ad}^{0}\bar{\rho})$
satisfying the specified properties. Since the image of $\Psi_{T,v}$ is
greater than the image of $\Psi_{T}$, there is a class $g$ in
$H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},\operatorname{Ad}^{0}\bar{\rho})$
such that $\Psi_{T,v}(g)\notin\text{Image}(\Psi_{T})$. Let

$W_{1}:=\text{Image}(\Psi_{T})+\mathbb{F}_{q}\cdot\Psi_{T,v}(g)$

and

$W_{2}:=\text{Image}(\Psi_{T})+\mathbb{F}_{q}\cdot(z_{w})_{w\in T}.$

The argument in [6, Proposition 34] applies verbatim to imply that
$W_{1}=W_{2}$ and so we deduce the existence of $h^{(v)}\in
H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},\operatorname{Ad}^{0}\bar{\rho})$
for which

$h^{(v)}_{\restriction\operatorname{G}_{w}}={z_{w}}_{\restriction\operatorname{G}_{w}}$

for all $w\in T$. As we have observed,

$\text{Image}(\Psi_{T}^{-2n+2})=\text{Image}(\Psi_{T,v}^{-2n+2})$

since $h^{(v)}\notin\text{Image}(\Psi_{T})$ it follows that
$h^{(v)}(\tau_{v})$ is not contained in
$(\operatorname{Ad}^{0}\bar{\rho})_{-2n+2}$. Invoking Lemma 6.3, we deduce
that on adding a suitable linear combination of elements to $h^{(v)}$ from
$\ker\Psi_{T,v}^{d}$ for $d>-2n+1$, we modify the class $h^{(v)}$ so that

$h^{(v)}(\tau_{v})\in(\operatorname{Ad}^{0}\bar{\rho})_{-2L_{1}}\backslash\\{0\\}$

as required. ∎

For $d\in\mathbb{Z}$, the natural inclusion
$(\operatorname{Ad}^{0}\bar{\rho})_{d}^{\perp}\hookrightarrow\operatorname{Ad}^{0}\bar{\rho}^{*}$
induces a natural map of cohomology groups
$H^{1}(\operatorname{G}_{\mathbb{Q}},(\operatorname{Ad}^{0}\bar{\rho})_{d}^{\perp})\rightarrow
H^{1}(\operatorname{G}_{\mathbb{Q}},\operatorname{Ad}^{0}\bar{\rho}^{*})$.

######  Lemma 6.5.

Let $\mathfrak{l}$ be the Chebotarev class of trivial primes in the
Proposition 6.4. Let $\\{X_{\lambda}^{*}\\}_{\lambda\in\Phi}$ and
$\\{H_{1}^{*},\dots,H_{n}^{*}\\}$ be as in $\eqref{XHdual}$. There exists an
$\mathbb{F}_{q}$-independent set

$\\{\eta_{\lambda}^{(v)}\mid\lambda\in\Phi\\}\cup\\{\
\eta_{1}^{(v)},\dots,\eta_{n}^{(v)}\\}$

contained in
$H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})$,
satisfying the following properties:

  1. (1)

$\eta_{\lambda}^{(v)}$ is in the image of the natural map

$H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},(\operatorname{Ad}^{0}\bar{\rho})_{h+1}^{\perp})\rightarrow
H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},\operatorname{Ad}^{0}\bar{\rho}^{*}),$

where $h=\operatorname{ht}(\lambda)$.

  2. (2)

For $i=1,\dots,n$, the cohomology class $\eta_{i}^{(v)}$ is in the image of
the natural map

$H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},(\operatorname{Ad}^{0}\bar{\rho})_{1}^{\perp})\rightarrow
H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},\operatorname{Ad}^{0}\bar{\rho}^{*}).$

  3. (3)

For $\lambda\in\Phi$, we have that
$\eta_{\lambda}^{(v)}(\tau_{v})=X_{\lambda}^{*}$.

  4. (4)

For $i=1,\dots,n$, we have that $\eta_{i}^{(v)}(\tau_{v})=H_{i}^{*}$.

  5. (5)

The images of the elements $\eta_{\lambda}^{(v)}$ are a basis for the cokernel
of the inflation map

$H^{1}(\operatorname{G}_{\mathbb{Q},T},\operatorname{Ad}^{0}\bar{\rho}^{*})\rightarrow
H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},\operatorname{Ad}^{0}\bar{\rho}^{*}).$

###### Proof.

The dual to $(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp}$ is
$\operatorname{Ad}^{0}\bar{\rho}/(\operatorname{Ad}^{0}\bar{\rho})_{k}$.
Proposition 5.7 asserts that

${}_{T}^{1}((\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp*})=0$

for all $k\in\mathbb{Z}$. Wiles’ formula (3.1) asserts that

$\begin{split}&h^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp})-\dim{}_{T\cup\\{v\\}}^{1}((\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp*})\\\
=&h^{1}(\operatorname{G}_{\mathbb{Q},T},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp})-\dim{}_{T}^{1}((\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp*})\\\
+&h^{1}(\operatorname{G}_{v},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp})-h^{0}(\operatorname{G}_{v},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp}).\end{split}.$

Also, by Proposition 5.7, we have that

${}_{T}^{1}(\operatorname{Ad}^{0}\bar{\rho}/(\operatorname{Ad}^{0}\bar{\rho})_{k})=0.$

On applying the local Euler characteristic formula and Tate duality we have
that

$h^{1}(\operatorname{G}_{v},(\operatorname{Ad}^{0}\bar{\rho})_{k})^{\perp}-h^{0}(\operatorname{G}_{v},(\operatorname{Ad}^{0}\bar{\rho})_{k})^{\perp})=h^{0}(\operatorname{G}_{v},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp*})=\dim(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp}.$

For the last equality, note that
$\bar{\chi}_{\restriction\operatorname{G}_{v}}=1$ since $v\equiv 1\mod{p}$ and
that the action on $(\operatorname{Ad}^{0}\bar{\rho})_{k})^{\perp}$ is
trivial. It follows that

$h^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp})=h^{1}(\operatorname{G}_{\mathbb{Q},T},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp})+\dim(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp}$

and the evaluation map at $\tau_{v}$

$H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp})\rightarrow(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp}$

induces a short exact sequence

$0\rightarrow
H^{1}(\operatorname{G}_{\mathbb{Q},T},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp})\rightarrow
H^{1}(\operatorname{G}_{\mathbb{Q},T\cup\\{v\\}},(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp})\rightarrow(\operatorname{Ad}^{0}\bar{\rho})_{k}^{\perp}\rightarrow
0.$

The assertion of the Lemma follows. ∎

Let $v$ a trivial prime in the Chebotarev class $\mathfrak{l}$ of Proposition
6.4. For $\lambda\in\Phi$, denote by
$K_{\lambda}^{(v)}:=K_{\eta_{\lambda}^{(v)}}$ and for $i=1,\dots,n$, set
$K_{i}^{(v)}:=K_{\eta_{i}^{(v)}}$. Let $J_{i}^{(v)}\subsetneq K_{i}^{(v)}$ and
$J_{\lambda}^{(v)}\subsetneq K_{\lambda}^{(v)}$ denote $J_{\eta_{i}^{(v)}}$
and $J_{\eta_{\lambda}^{(v)}}$ respectively. If $E=K_{i}^{(v)}$ (resp.
$K_{\lambda}^{(v)}$), denote by $J_{E}$ the sub-extension $J_{i}^{(v)}$ (resp.
$J_{\lambda}^{(v)}$). Let $\mathcal{F}^{(v)}$ denote the collection of fields
consisting of $K_{i}^{(v)}$ for $i=1,\dots,n$ and $K_{\lambda}^{(v)}$ for
$\lambda\in\Phi$. The Chebotarev class $\mathfrak{l}$ from Proposition 6.4 is
defined by Chebotarev classes in a collection of fields
$\mathcal{F}_{\mathfrak{l}}$. More specifically, $\mathcal{F}_{\mathfrak{l}}$
is the collection of fields:

  * •

$K_{\psi},K_{\psi_{1}},\dots,K_{\psi_{r}}$ from Proposition 6.4,

  * •

$K_{\beta}$ as $\beta$ runs through all cohomology classes
$H^{1}(\operatorname{G}_{\mathbb{Q},T},(\operatorname{Ad}^{0}\bar{\rho}^{*})_{d})$,
where $d\geq-2n+2$,

  * •

$K(\zeta_{2})$ (with $\zeta_{2}$ defined at the start of the section),

  * •

$K(\mu_{p^{2}})$.

For $v\in\mathfrak{l}$ from Proposition 6.4, recall that $L_{h^{(v)}}$ is the
field extension of $L$ cut out by

$h^{(v)}_{\restriction\operatorname{G}_{L}}:\operatorname{G}_{L}\rightarrow\operatorname{Ad}^{0}\bar{\rho}.$

Associate to a set of trivial primes $A=\\{v_{1},\dots,v_{k}\\}$ in
$\mathfrak{l}$,

$\mathcal{F}_{A}:=\cup_{i=1}^{k}\mathcal{F}^{(v_{i})}\text{, and
}\mathcal{L}_{A}:=\\{L_{h^{(v_{1})}},\dots,L_{h^{(v_{k})}}\\}.$

######  Lemma 6.6.

Let $A=\\{v_{1},\dots,v_{k}\\}\subset\mathfrak{l}$.

  1. (1)

Let $F_{1}$ be a field in the collection $\mathcal{F}_{A}$ and $F_{2}$ be the
composite of all the other fields in
$\mathcal{F}_{A}\cup\mathcal{L}_{A}\cup\mathcal{F}_{\mathfrak{l}}$. Then
$F_{1}$ is not contained in $F_{2}$. Moreover, the intersection $F_{1}\cap
F_{2}$ is contained in $J_{F_{1}}$.

  2. (2)

Let $M_{1}$ be a field in the collection $\mathcal{L}_{A}$ and $M_{2}$ denote
the composite of all the other fields in
$\mathcal{F}_{A}\cup\mathcal{L}_{A}\cup\mathcal{F}_{\mathfrak{l}}$. The
intersection $M_{1}\cap M_{2}=L$.

###### Proof.

Part $\eqref{66c1}$ is obtained from an application of Proposition 4.7, as we
now explain. In accordance with the statement of Proposition 4.7, we define a
sequence of linearly independent classes

$\theta_{0},\dots,\theta_{t}\in H^{1}(\operatorname{G}_{\mathbb{Q},T\cup
A},\operatorname{Ad}^{0}\bar{\rho}^{*}),$

and a sequence of fields $\mathbb{L}_{1},\dots,\mathbb{L}_{b}$, each of which
is unrelated to $\operatorname{Ad}^{0}\bar{\rho}^{*}$.

Consider the classes $\eta_{i}^{(v_{j})}$ and $\eta_{\lambda}^{(v_{j})}$ as
$i=1,\dots,n$, $\lambda\in\Phi$ and $j=1,\dots,k$. Enumerate these classes by
$\theta_{0},\dots,\theta_{a}$, so that $\theta_{0}$ is the cohomology class
specified in the description of $F_{1}$, i.e. $F_{1}=K_{\theta_{0}}$. The
number $a$ is equal to $(nk\\#\Phi)-1$ and
$\mathcal{F}_{A}=\\{K_{\theta_{0}},K_{\theta_{1}},\dots,K_{\theta_{a}}\\}$.
Let $\theta_{a+1},\dots,\theta_{t}$ be a basis of
$H^{1}(\operatorname{G}_{\mathbb{Q},T},\operatorname{Ad}^{0}\bar{\rho}^{*})$.
Recall that for $\lambda\in\Phi$, we have that
$\eta_{\lambda}^{(v_{j})}(\tau_{v_{j}})=X_{\lambda}^{*}$, and for
$i=1,\dots,n$, we have that $\eta_{i}^{(v_{j})}(\tau_{v_{j}})=H_{i}^{*}$. The
classes $\theta_{a+1},\dots,\theta_{t}$ are unramified at each of the primes
$v_{j}\in A$. It is thus, easy to see that $\theta_{0},\dots,\theta_{t}$ are
linearly independent. Let $\mathbb{L}_{1},\dots,\mathbb{L}_{l}$ be an
enumeration for the fields $K_{\beta}$, as $\beta$ runs through all cohomology
classes
$H^{1}(\operatorname{G}_{\mathbb{Q},T},(\operatorname{Ad}^{0}\bar{\rho}^{*})_{d})$
for $d\geq-2n+2$. Let $\mathbb{L}_{l+1}$ be the field $K(\zeta_{2})$ and
$\mathbb{L}_{l+2}$ the field $K(\mu_{p^{2}})$. The collection of fields
$\mathcal{F}_{\mathfrak{l}}$ consists of $K_{\theta_{i}}$ for $i=a+1,\dots,t$
and $\mathbb{L}_{i}$ for $i=1,\dots,l+2$. Next, we have to account for the
fields in $\mathcal{L}_{A}$. Let $\mathbb{L}_{l+3},\dots,\mathbb{L}_{b}$ be an
enumeration of the fields $L_{h^{(v_{1})}},\dots,L_{h^{(v_{k})}}$. Thus, the
collection of fields $\mathcal{L}_{A}$ is
$\\{\mathbb{L}_{l+3},\dots,\mathbb{L}_{b}\\}$. In order to apply Proposition
4.7, it suffices to show that each of the fields
$\mathbb{L}_{1},\dots,\mathbb{L}_{b}$ is unrelated to
$\operatorname{Ad}^{0}\bar{\rho}^{*}$. Note that:

  * •

by Lemma 4.8, part $\eqref{415c1}$, each of the fields
$\mathbb{L}_{1},\dots,\mathbb{L}_{l}$ is unrelated to
$\operatorname{Ad}^{0}\bar{\rho}^{*}$,

  * •

by part $\eqref{415c2}$, $\mathbb{L}_{l+2}$ is unrelated to
$\operatorname{Ad}^{0}\bar{\rho}^{*}$,

  * •

by part $\eqref{415c3}$, $\mathbb{L}_{l+3},\dots,\mathbb{L}_{b}$ are unrelated
to $\operatorname{Ad}^{0}\bar{\rho}^{*}$,

  * •

and by part $\eqref{415c4}$, $\mathbb{L}_{l+1}$ is unrelated to
$\operatorname{Ad}^{0}\bar{\rho}^{*}$.

By Proposition 4.7, $F_{1}$ is not contained in $F_{2}$ and it follows from
Lemma LABEL:mainin that $F_{1}\cap F_{2}\subseteq J_{F_{1}}$.

Assume without loss of generality that $M_{1}=L_{h^{(v_{1})}}$. Recall that by
(6.4), we have that

$h^{(v_{1})}(\tau_{v_{1}})\in(\operatorname{Ad}^{0}\bar{\rho})_{-2L_{1}}\backslash\\{0\\}.$

As a result, $v_{1}$ is ramified in the $\sigma_{-2L_{1}}$-eigenspace of
$\operatorname{Gal}(M_{1}/L)$. On the other hand, $v_{1}$ is unramified in
each of the field extensions in $\mathcal{F}_{\mathfrak{l}}$ and
$\mathcal{L}_{A}\backslash\\{M_{1}\\}$. Since the classes $\eta_{i}^{(v_{j})}$
and $\eta_{\lambda}^{(v_{j})}$ are valued in
$\operatorname{Ad}^{0}\bar{\rho}^{*}$, there is no
$\sigma_{-2L_{1}}$-eigenspace for the action of $\mathbb{T}$ on
$\operatorname{Gal}(K_{\theta_{i}}/K)$ for $K_{\theta_{i}}\in\mathcal{F}_{A}$.
As a result, $v_{1}$ is unramified in the $\sigma_{-2L_{1}}$-eigenspace of
$\operatorname{Gal}(M_{2}/L)$. Therefore, $M_{1}\not\subseteq M_{2}$. Identify
$Q:=\operatorname{Gal}(M_{1}/M_{1}\cap M_{2})$ with a subgroup of
$h^{(v_{1})}(\operatorname{G}_{L})\subseteq\operatorname{Ad}^{0}\bar{\rho}$.
By Lemma 4.4 it suffices to show that $Q_{-2L_{1}}\neq 0$. Since $v_{1}$ is
unramified in $M_{2}$, the image of $\tau_{v_{1}}$ in
$\operatorname{Gal}(M_{1}/L)$ lies in $\operatorname{Gal}(M_{1}/M_{1}\cap
M_{2})$. Since $h^{(v_{1})}(\tau_{v_{1}})$ is in
$(\operatorname{Ad}^{0}\bar{\rho})_{-2L_{1}}\backslash\\{0\\}$, we deduce that
$Q_{-2L_{1}}\neq 0$. The assertion $\eqref{66c2}$ follows. ∎

######  Lemma 6.7.

Let $v\in\mathfrak{l}$ and $h^{(v)}$ be as in Proposition 6.4. Then the
$\operatorname{Gal}(L_{h^{(v)}}/L)\simeq\operatorname{Ad}^{0}\bar{\rho}$.

###### Proof.

Let
$Q:=\operatorname{Gal}(L_{h^{(v)}}/L)\subseteq\operatorname{Ad}^{0}\bar{\rho}$.
Since $Q_{-2L_{1}}\neq 0$, the assertion follows from Lemma 4.4. ∎

######  Proposition 6.8.

For a pair $(v_{1},v_{2})$ of trivial primes in $\mathfrak{l}$ in Proposition
6.4 set $h=-h^{(v_{1})}+2h^{(v_{2})}$ and $\rho_{2}:=(I+ph)\zeta_{2}$. There
is a pair $(v_{1},v_{2})$ such that
$\rho_{2\restriction\operatorname{G}_{w}}\in\mathcal{C}_{w}$ for all $w\in T$
and $\rho_{2\restriction\operatorname{G}_{v_{i}}}\in\mathcal{C}_{v_{i}}^{ram}$
for $i=1,2$.

###### Proof.

For $i=1,2,$ we set $\mathcal{C}_{v_{i}}:=\mathcal{C}_{v_{i}}^{ram}$. Note
that $h_{\restriction\operatorname{G}_{w}}=z_{w}$ for all $w\in T$ and hence
$\rho_{2\restriction\operatorname{G}_{w}}\in\mathcal{C}_{w}$ for all $w\in T$.
This is not the case at the primes $v_{1}$ and $v_{2}$. We show that one may
indeed find a pair $(v_{1},v_{2})\in\mathfrak{l}\times\mathfrak{l}$ so that
$(I+pz_{v_{i}})\zeta_{2}\in\mathcal{C}_{v_{i}}^{ram}$ for $i=1,2$. Consider
for $v\in\mathfrak{l}$, the pair of elements
$(\zeta_{2}(\sigma_{v}),h^{(v)}(\sigma_{v}))$, and let $A=(A_{1},A_{2})$ be
the pair of matrices which occurs most frequently, that is, with maximal upper
density. The choice of $A$ is not necessarily unique. Let
$\mathfrak{l}_{1}=\\{v\in\mathfrak{l}\mid\zeta_{2}(\sigma_{v})=A_{1},h^{(v)}(\sigma_{v})=A_{2}\\}$.
Since there are finitely many choices for $A$, the set of primes
$\mathfrak{l}_{1}$ has positive upper-density. Since
$h(\tau_{v_{i}})\in(\operatorname{Ad}^{0}\bar{\rho})_{-2L_{1}}$ and
$\zeta_{2}$ is unramified at $v_{i}$, we have that

$(\operatorname{Id}+ph(\tau_{v_{i}}))\zeta_{2}(\tau_{v_{i}})=(\operatorname{Id}+ph(\tau_{v_{i}}))\in\operatorname{U}_{-2L_{1}}.$

Furthermore, since $h(\tau_{v_{i}})\neq 0$, the additional condition on
$(\operatorname{Id}+ph(\tau_{v_{i}}))\zeta_{2}(\tau_{v_{i}})$ (see Definition
5.4 part $\eqref{defconditions2}$) is satisfied. Since $\zeta_{2}(\sigma_{v})$
is fixed throughout $\mathfrak{l}_{1}$, there are (not necessarily unique)
matrices $C_{i}$ such that if $h(\sigma_{v_{i}})=C_{i}$, we will have
$(\operatorname{Id}+ph){\zeta_{2}}_{\restriction\operatorname{G}_{v_{i}}}\in\mathcal{C}_{v_{i}}$
for $i=1,2$. The values $h^{(v_{i})}(\sigma_{v_{j}})$ are represented in the
table below:

| $\sigma_{v_{1}}$ | $\sigma_{v_{2}}$
---|---|---
$h^{(v_{1})}$ | $A_{2}$ | $R$
$h^{(v_{2})}$ | $E$ |  $A_{2}$.

We need $E=(A_{2}+C_{1})/2$ and $R=2A_{2}-C_{2}$. Note that for an arbitrary
pair $(v_{1},v_{2})\in\mathfrak{l}_{1}\times\mathfrak{l}_{1}$, this need not
be the case. What follows is a recipe for producing a pair $(v_{1},v_{2})$
such that $E=(A_{2}+C_{1})/2$ and $R=2A_{2}-C_{2}$.

For $v\in\mathfrak{l}_{1}$, let $\delta^{(v)}\in
H^{1}(\operatorname{G}_{v},\operatorname{Ad}^{0}\bar{\rho}^{*})$ be the
cohomology class given by $\delta^{(v)}(\sigma_{v})=X_{-2L_{1}}^{*}$ and
$\delta^{(v)}(\tau_{v})=0$. Let $y$ be the element that occurs most frequently
among the elements $\operatorname{inv}_{v}(\delta^{(v)}\cup h^{(v)})$ among
primes $v$ of $\mathfrak{l}_{1}$. Set

$\mathfrak{l}_{2}=\\{v\in\mathfrak{l}_{1}\mid\operatorname{inv}_{v}(\delta^{(v)}\cup
h^{(v)})=y\\},$

$\mathfrak{l}_{2}$ has positive upper density. Suppose we first choose
$v_{1}\in\mathfrak{l}_{2}$. Recall that
$h^{(v_{1})}(\tau_{v_{1}})\in(\operatorname{Ad}^{0}\bar{\rho})_{-2L_{1}}$. By
Lemma 4.4, the class $h^{(v_{1})}$ has full rank, i.e.
$h^{(v_{1})}(\operatorname{G}_{K})=\operatorname{Ad}^{0}\bar{\rho}$. In
particular, $2A_{2}-C_{2}$ is contained in
$h^{(v_{1})}(\operatorname{G}_{K})$. Choosing $v_{2}$ such that
$h^{(v_{1})}(\sigma_{v_{2}})=2A_{2}-C_{2}$ is a Chebotarev condition on the
splitting of $v_{2}$ in $L_{h^{(v_{1})}}$. We show that
$h^{(v_{2})}(\sigma_{v_{1}})$ is determined by how $v_{2}$ splits in the
$\bar{\chi}\sigma_{2L_{1}}$-eigenspace each of the fields in
$\mathcal{F}^{(v_{1})}$. Since $h^{(v_{2})}$ is unramified at $v_{1}$, the
values $\eta_{\lambda}^{(v_{1})}(\tau_{v_{1}})$ and
$h^{(v_{2})}(\sigma_{v_{1}})$ determine $(\eta_{\lambda}^{(v_{1})}\cup
h^{(v_{2})})_{\restriction\operatorname{G}_{v_{1}}}$. Express
$h^{(v_{2})}(\sigma_{v_{1}})=\sum_{\lambda}a_{\lambda}X_{\lambda}+\sum_{i=1}^{n}a_{i}H_{i}$.
As $\eta_{\lambda}^{(v_{1})}(\tau_{v_{1}})=X_{\lambda}^{*}$, we see that
$\operatorname{inv}_{v_{1}}(\eta_{\lambda}^{(v_{1})}\cup h^{(v_{2})})$
determines $a_{\lambda}$. Likewise,
$\operatorname{inv}_{v_{1}}(\eta_{i}^{(v_{1})}\cup h^{(v_{2})})$ determines
$a_{i}$. For $v\in\mathfrak{l}$ and $\lambda\in\Phi$, set $z_{\lambda}^{(v)}$
to be equal to $\operatorname{inv}_{v}(\eta_{\lambda}^{(v)}\cup h^{(v)})$. The
global reciprocity law asserts that

$\sum_{w\in
T\cup\\{v_{1},v_{2}\\}}\operatorname{inv}_{w}(\eta_{\lambda}^{(v_{1})}\cup
h^{(v_{2})})=0,\text{ and }\sum_{w\in
T\cup\\{v_{1}\\}}\operatorname{inv}_{w}(\eta_{\lambda}^{(v_{1})}\cup
h^{(v_{1})})=0.$

Since
$h^{(v_{2})}_{\restriction\operatorname{G}_{w}}=z_{w}=h^{(v_{1})}_{\restriction\operatorname{G}_{w}}$
for $w\in T$, we deduce that

$\begin{split}\operatorname{inv}_{v_{1}}(\eta_{\lambda}^{(v_{1})}\cup
h^{(v_{2})})&=-\sum_{w\in
T}\operatorname{inv}_{w}(\eta_{\lambda}^{(v_{1})}\cup
h^{(v_{2})})-\operatorname{inv}_{v_{2}}(\eta_{\lambda}^{(v_{1})}\cup
h^{(v_{2})})\\\ &=-\sum_{w\in
T}\operatorname{inv}_{w}(\eta_{\lambda}^{(v_{1})}\cup
h^{(v_{1})})-\operatorname{inv}_{v_{2}}(\eta_{\lambda}^{(v_{1})}\cup
h^{(v_{2})})\\\ &=\operatorname{inv}_{v_{1}}(\eta_{\lambda}^{(v_{1})}\cup
h^{(v_{1})})-\operatorname{inv}_{v_{2}}(\eta_{\lambda}^{(v_{1})}\cup
h^{(v_{2})})\\\
&=z_{\lambda}^{(v_{1})}-\operatorname{inv}_{v_{2}}(\eta_{\lambda}^{(v_{1})}\cup
h^{(v_{2})}).\end{split}$

Since $z_{\lambda}^{(v_{1})}$ depends on $v_{1}$ which is fixed, the variance
of the right hand side of the equation comes from the term
$\operatorname{inv}_{v_{2}}(\eta_{\lambda}^{(v_{1})}\cup h^{(v_{2})})$. The
specification of $h^{(v_{2})}(\sigma_{v_{1}})$ amounts to the specification of
$\operatorname{inv}_{v_{1}}(\eta_{\lambda}^{(v_{1})}\cup h^{(v_{2})})$ for
$\lambda\in\Phi$ and $\operatorname{inv}_{v_{1}}(\eta_{i}^{(v_{1})}\cup
h^{(v_{2})})$ for $i=1,\dots,n$. Set $u_{\lambda}$ to be
$\eta_{\lambda}^{(v_{1})}(\sigma_{v_{2}})(X_{-2L_{1}})$ for $\lambda\in\Phi$
and set $u_{i}$ to be $\eta_{i}^{(v_{1})}(\sigma_{v_{2}})(X_{-2L_{1}})$ for
$i=1,\dots,n$. Since $h^{(v_{2})}(\tau_{v_{2}})$ is a multiple of
$X_{-2L_{1}}$, we see that

$\begin{split}&\operatorname{inv}_{v_{2}}(\eta_{\lambda}^{(v_{1})}\cup
h^{(v_{2})})=\operatorname{inv}_{v_{2}}(u_{\lambda}\delta^{(v_{2})}\cup
h^{(v_{2})})\\\ &\operatorname{inv}_{v_{2}}(\eta_{i}^{(v_{1})}\cup
h^{(v_{2})})=\operatorname{inv}_{v_{2}}(u_{i}\delta^{(v_{2})}\cup
h^{(v_{2})}).\\\ \end{split}$

Moreover since $h^{(v_{2})}(\tau_{v_{2}})$ is non-zero, we can choose
$b\in\mathbb{F}_{q}$ such that
$\operatorname{inv}_{v_{2}}(b\delta^{(v_{2})}\cup h^{(v_{2})})$ takes on any
desired value. Note that $\operatorname{inv}_{v_{2}}(\delta^{(v_{2})}\cup
h^{(v_{2})})$ is set to equal $y$ for all $v_{2}\in\mathfrak{l}_{2}$, i.e.,
does not depend on the choice of $v_{2}\in\mathfrak{l}_{2}$. As a result, for
$v_{1}\in\mathfrak{l}_{2}$, there exist values
$\\{b_{\lambda}\\}_{\lambda\in\Phi}$ and $\\{b_{i}\\}_{i=1,\dots,n}$ depending
only on $v_{1}$ such that if $u_{\lambda}=b_{\lambda}$ for $\lambda\in\Phi$
and $u_{i}=b_{i}$ for $i=1,\dots,n$, then,

$h^{(v_{2})}(\sigma_{v_{1}})=(A_{2}+C_{1})/2.$

The condition requiring $h^{(v_{2})}(\sigma_{v_{1}})=(A_{2}+C_{1})/2$, is
determined by $\eta_{\lambda}^{(v_{1})}(\sigma_{v_{2}})(X_{-2L_{1}})$ for
$\lambda\in\Phi$ and by $\eta_{i}^{(v_{1})}(\sigma_{v_{2}})(X_{-2L_{1}})$ for
$i=1,\dots,n$. Note that $v_{2}$ is unramified in
$K_{\eta_{\lambda}^{(v_{1})}}$ and the value of
$\eta_{\lambda}^{(v_{1})}(\sigma_{v_{2}})(X_{-2L_{1}})$ is determined by the
projection of $\sigma_{v_{2}}$ to the $\bar{\chi}\sigma_{2L_{1}}$-eigenspace
of $\operatorname{Gal}(K_{\eta_{\lambda}^{(v_{1})}}/K)$, when viewed as a
$\mathbb{T}$-module. Recall that $J_{\eta_{\lambda}^{(v_{1})}}$ is the
subextension $K\subseteq J_{\eta_{\lambda}^{(v_{1})}}\subsetneq
K_{\eta_{\lambda}^{(v_{1})}}$ such that

$\operatorname{Gal}(K_{\eta_{\lambda}^{(v_{1})}}/K)_{\bar{\chi}\sigma_{2L_{1}}}\simeq\operatorname{Gal}(K_{\eta_{\lambda}^{(v_{1})}}/J_{\eta_{\lambda}^{(v_{1})}}).$

Since $v_{2}$ is a trivial prime, it is split in $K$. One may indeed insist
that $v_{2}$ is split in $J_{\eta_{\lambda}^{(v_{1})}}$ and $\sigma_{v_{2}}$
takes on the appropriate value in
$\operatorname{Gal}(K_{\eta_{\lambda}^{(v_{1})}}/J_{\eta_{\lambda}^{(v_{1})}})$
so that $u_{\lambda}=b_{\lambda}$. Hence, the condition
$u_{\lambda}=b_{\lambda}$ is simply a condition on the
$\bar{\chi}\sigma_{2L_{1}}$-eigenspace of $K_{\eta_{\lambda}^{(v_{1})}}/K$.
Likewise, the condition $u_{i}=b_{i}$ is a condition on the
$\bar{\chi}\sigma_{2L_{1}}$-eigenspace of $K_{\eta_{i}^{(v_{1})}}/K$. To
summarize, the condition requiring
$h^{(v_{2})}(\sigma_{v_{1}})=(A_{2}+C_{1})/2$, is equivalent to Chebotarev
conditions on the splitting of $v_{2}$ in the
$\bar{\chi}\sigma_{2L_{1}}$-eigenspaces of the fields in
$\mathcal{F}^{(v_{1})}$, in the sense made precise in the preceding
discussion.

Suppose that for the choice of $v_{1}\in\mathfrak{l}_{2}$, there is a
$v_{2}\in\mathfrak{l}_{2}$ for which the required conditions are satisfied:

  1. (1)

the condition on the splitting of $v_{2}$ in $L_{h^{(v_{1})}}$ which amounts
to specifying $h^{(v_{1})}(\sigma_{v_{2}})$,

  2. (2)

the condition on the splitting of $v_{2}$ in the fields
$\mathcal{F}^{(v_{1})}$ which amounts to specifying
$h^{(v_{2})}(\sigma_{v_{1}})$.

Then we are done. Note that $\mathfrak{l}_{2}$ is not a Chebotarev condition,
it has only been observed that $\mathfrak{l}_{2}$ has positive upper density.
Consider the case when there is no choice of $v_{2}\in\mathfrak{l}_{2}$ for
which the above conditions are satisfied. Let $\mathfrak{l}_{v_{1}}$ be the
subset of $\mathfrak{l}$ for which
$(R,E)\neq(2A_{2}-C_{2},\frac{(A_{2}+C_{1})}{2})$ for the choice of $v_{1}$.
We have thus assumed that $\mathfrak{l}_{2}\subseteq\mathfrak{l}_{v_{1}}$, it
follows that the upper density $\delta(\mathfrak{l}_{2})$ is less than or
equal to the upper density $\delta(\mathfrak{l}_{v_{1}})$.

Set $\mathcal{E}^{(v_{1})}$ to be the composite of the field $L_{h^{(v_{1})}}$
with the fields in $\mathcal{F}^{(v_{1})}$ and let
$\mathfrak{F}_{\mathfrak{l}}$ be the composite of fields in
$\mathcal{F}_{\mathfrak{l}}$. We show that there is an element
$x\in\operatorname{Gal}(\mathcal{E}^{(v_{1})}\cdot\mathfrak{F}_{\mathfrak{l}}/K)$
such that if $v_{2}$ is trivial prime such that the Frobenius at $v_{2}$ maps
to $x$, then $v_{2}\in\mathfrak{l}$ and the conditions on $v_{2}$ are
satisfied. Said differently, if $\sigma_{v_{2}}=x$, then
$v_{2}\in\mathfrak{l}\backslash\mathfrak{l}_{v_{1}}$. If $F_{1}$ is any of the
fields in $\mathcal{F}^{(v_{1})}$ and $F_{2}$ is the composite of the other
fields in $\mathcal{F}^{(v_{1})}\cup\mathcal{F}_{\mathfrak{l}}$, Lemma 6.6
asserts that $F_{1}\cap F_{2}\subseteq J_{F_{1}}$. Lemma 6.6 asserts that
$L_{h^{(v_{1})}}$ is linearly disjoint over $L$ from the composite of all
fields in $\mathcal{F}^{(v_{1})}\cup\mathcal{F}_{\mathfrak{l}}$. To construct
such an element $x$, enumerate the fields in
$\mathcal{F}^{(v_{1})}=\\{E_{1},\dots,E_{k-1}\\}$ and set
$E_{k}:=F_{h^{(v_{1})}}$. Set $E_{0}:=\mathfrak{F}_{\mathfrak{l}}$ and let
$\mathcal{E}_{j}$ be the composite $E_{0}\cdots E_{j}$, note that
$\mathcal{E}_{k}=\mathcal{E}^{(v_{1})}\cdot\mathfrak{F}_{\mathfrak{l}}$.
Consider the filtration

$\mathcal{E}_{k}\supset\mathcal{E}_{k-1}\supset\dots\supset\mathcal{E}_{1}\supset\mathcal{E}_{0}\supset
K.$

Let $x_{0}\in\operatorname{Gal}(\mathcal{E}_{0}/K)$ be an element defining
$\mathfrak{l}$. Note that
$\operatorname{Gal}(\mathcal{E}_{1}/\mathcal{E}_{0})\simeq\operatorname{Gal}(E_{1}/E_{1}\cap\mathcal{E}_{0})$
and the intersection $E_{1}\cap\mathcal{E}_{0}$ is contained in $J_{E_{1}}$.
The condition on $E_{1}/K$ is on the $\bar{\chi}\sigma_{2L_{1}}$-eigenspace
$\operatorname{Gal}(E_{1}/J_{E_{1}})$. Hence $x_{0}$ lifts to a suitable
$x_{1}\in\operatorname{Gal}(\mathcal{E}_{1}/K)$. Repeating the process, we see
that $x_{1}$ lifts to $x_{k-1}\in\operatorname{Gal}(\mathcal{E}_{k-1}/K)$ such
that if $\sigma_{v_{2}}=x_{k-1}$, then $v_{2}\in\mathfrak{l}$ and
$h^{(v_{2})}(\sigma_{v_{1}})=(A_{2}+C_{1})/2$. Since
$E_{k}\cap\mathcal{E}_{k-1}=K$, it follows that $x_{k-1}$ can be lifted to
$x_{k}\in\operatorname{Gal}(\mathcal{E}_{k}/K)$ such that if
$\sigma_{v_{2}}=x$, then all conditions on $v_{2}$ are satisfied.

As a result,
$\delta(\mathfrak{l}\backslash\mathfrak{l}_{v_{1}})\geq\frac{1}{[\mathcal{E}^{(v_{1})}\cdot\mathfrak{F}_{\mathfrak{l}}:K]}$,
and hence,

$\delta(\mathfrak{l}_{v_{1}})\leq\left(1-\frac{1}{[\mathcal{E}^{(v_{1})}\cdot\mathfrak{F}_{\mathfrak{l}}:K]}\right).$

For $F\in\mathcal{F}^{(v_{1})}$, the Galois group $\operatorname{Gal}(F/K)$
may be identified with a Galois submodule of
$\operatorname{Ad}^{0}\bar{\rho}^{*}$. Hence $[F:K]\leq
q^{\dim(\operatorname{Ad}^{0}\bar{\rho})}$ for $F\in\mathcal{F}^{(v_{1})}$ is
a uniform bound independent of $v_{1}$. Similar reasoning shows that
$[L^{h^{(v_{1})}}:L]\leq q^{\dim(\operatorname{Ad}^{0}\bar{\rho})}$. Setting
$N:=(\\#\Phi+n+1)\cdot\dim\operatorname{Ad}^{0}\bar{\rho}$, deduce that

$\delta(\mathfrak{l}_{v_{1}})\leq
1-q^{-N}[\mathfrak{F}_{\mathfrak{l}}:K]^{-1}.$

Suppose that there is a sequence of $m$ primes
$v_{1}^{(1)},\dots,v_{1}^{(m)}\in\mathfrak{l}_{2}$, such that it is not
possible to find a second prime $v_{2}$ for any of the primes $v_{1}^{(j)}$.
In other words,
$\mathfrak{l}_{2}\subseteq\cap_{j=1}^{m}\mathfrak{l}_{v_{1}^{(j)}}$. We show
that the density of $\cap_{j=1}^{m}\mathfrak{l}_{v_{1}^{(j)}}$ approaches zero
as $m$ approaches infinity. Since the upper density of $\mathfrak{l}_{2}$ is
positive, we will eventually find a pair $(v_{1},v_{2})$. For convenience of
notation, set $w_{j}:=v_{1}^{(j)}$ and set $A=\\{w_{1},\dots,w_{m}\\}$. Fix
$1\leq j\leq m$ and enumerate the fields
$\mathcal{F}^{(w_{j})}=\\{E_{1},\dots,E_{k-1}\\}$ and set
$E_{k}=F_{h^{(w_{j})}}$. Denote by
$\mathfrak{E}_{j}:=\mathfrak{F}_{\mathfrak{l}}\cdot\mathcal{E}^{(w_{1})}\cdots\mathcal{E}^{(w_{j})}$
and let $C_{j}$ be the subset of $\operatorname{Gal}(\mathfrak{E}_{j}/K)$
defining the set $\cap_{i=1}^{j}\mathfrak{l}_{w_{i}}$. This means that
$v_{2}\in\cap_{i=1}^{j}\mathfrak{l}_{w_{i}}$ if and only if $\sigma_{v_{2}}\in
C_{j}$. We show that any element
$w\in\operatorname{Gal}(\mathfrak{E}_{j-1}/K)$ lifts to an element
$\tilde{w}\in\operatorname{Gal}(\mathfrak{E}_{j}/K)$ which is not in $C_{j}$.
This is shown by filtering $\mathfrak{E}_{j}/\mathfrak{E}_{j-1}$ by

$\mathfrak{E}_{j}=\mathcal{E}_{k}\supset\mathcal{E}_{k-1}\supset\cdots\mathcal{E}_{1}\supset\mathcal{E}_{0}=\mathfrak{E}_{j-1},$

where $\mathcal{E}_{l}:=\mathfrak{E}_{j-1}E_{1}\cdots E_{l}$. The argument is
identical to that provided before.

As a result,

$\\#C_{j}\leq([\mathfrak{E}_{j}:\mathfrak{E}_{j-1}]-1)\\#C_{j-1}.$

Therefore,

$\begin{split}\delta(\cap_{i=1}^{j}\mathfrak{l}_{w_{i}})=\frac{\\#C_{j}}{[\mathfrak{E}_{j}:K]}\leq&\left(1-\frac{1}{[\mathfrak{E}_{j}:\mathfrak{E}_{j-1}]}\right)\frac{\\#C_{j-1}}{[\mathfrak{E}_{j-1}:K]},\\\
\leq&\left(1-\frac{1}{[\mathcal{E}^{(w_{j})}:K]}\right)\frac{\\#C_{j-1}}{[\mathfrak{E}_{j-1}:K]},\\\
\leq&(1-q^{-N})\delta(\cap_{i=1}^{j-1}\mathfrak{l}_{w_{i}}).\end{split}$

Therefore,
$\delta(\cap_{i=1}^{m}\mathfrak{l}_{w_{i}})\leq(1-q^{-N})^{m-1}(1-q^{-N}[\mathfrak{F}_{\mathfrak{l}}:K]^{-1})$.
Since $\mathfrak{l}_{2}$ has positive upper density there is a large value of
$m$ such that $\mathfrak{l}_{2}$ is not contained in
$\cap_{i=1}^{m}\mathfrak{l}_{w_{i}}$. This shows that a pair $(v_{1},v_{2})$
satisfying the required conditions does exist. ∎

######  Proposition 6.9.

Let $\rho_{2}$ be as in Proposition 6.8. The image of $\rho_{2}$ is the
principal congruence subgroup of
$\operatorname{GSp}_{2n}(\text{W}(\mathbb{F}_{q})/p^{2})$ of similitude
character $1$.

###### Proof.

Recall that the similitude character $\kappa$ is prescribed to equal
$\kappa_{0}\chi^{k}$ where $\kappa_{0}$ is the Teichmüller lift of
$\bar{\kappa}$ and $k$ is divisible by $p(p-1)$. Therefore, we have that
$\kappa\equiv\kappa_{0}\mod{p^{2}}$, and as a result, elements in the
principal congruence subgroup in the image of $\rho_{2}$ necessarily have
similitude character $1$. Therefore, $\rho_{2}(\operatorname{G}_{L})$ may be
identified with a subspace of $\operatorname{Ad}^{0}\bar{\rho}$. In greater
detail, $\rho_{2}(g)$ is identified with
$\frac{1}{p}(\rho_{2}(g)-\operatorname{Id})$, for
$g\in\operatorname{G}_{L}=\operatorname{ker}\bar{\rho}$. It may be checked
that $\rho_{2}(\operatorname{G}_{L})$ is a $\operatorname{G}$-submodule of
$\operatorname{Ad}^{0}\bar{\rho}$ and that the natural
$\operatorname{G}$-action on $\operatorname{Gal}(\mathbb{Q}(\rho_{2})/L)$
(induced by conjugation) coincides with the $\operatorname{G}$-action on
$\rho_{2}(\operatorname{G}_{L})$ viewed as a submodule of
$\operatorname{Ad}^{0}\bar{\rho}$. Recall that
$\rho_{2}=(\operatorname{Id}+ph)\zeta_{2}$, where $h$ is the cohomology class
given by $-h^{(v_{1})}+2h^{(v_{2})}$. Since $\bar{\rho}$ is unramified at
$v_{1}$, we have that $\tau_{v_{1}}\in\operatorname{G}_{L}$. The cohomology
class $h^{(v_{2})}$ is unramified at $v_{1}$, as is $\zeta_{2}$. Therefore, we
have that

$\rho_{2}(\tau_{v_{1}})=(\operatorname{Id}+ph(\tau_{v_{1}}))\zeta_{2}(\tau_{v_{1}})=(\operatorname{Id}-ph^{(v_{1})}(\tau_{v_{1}})).$

Recall that by (6.4), we have that

$h^{(v_{1})}(\tau_{v_{1}})\in(\operatorname{Ad}^{0}\bar{\rho})_{-2L_{1}}\backslash\\{0\\}.$

Therefore, $\rho_{2}(\operatorname{G}_{L})$ is identified with a Galois-
submodule of $\operatorname{Ad}^{0}\bar{\rho}$ which contains an element with
non-zero $-2L_{1}$-component. From Lemma 4.4, it is deduced that this module
must be all of $\operatorname{Ad}^{0}\bar{\rho}$. This completes the proof. ∎

##  7\. Annihiliating the dual-Selmer Group

Let
$\rho_{3}:\operatorname{G}_{\mathbb{Q},T\cup\\{v_{1},v_{2}\\}}\rightarrow\operatorname{GSp}_{2n}(\text{W}(\mathbb{F}_{q})/p^{3})$
be the lift of $\bar{\rho}$ obtained from the application of Propositions 6.8
and 6.9. Recall that the Galois group $\operatorname{Gal}(K(\rho_{2})/K)$ is
identified with $\operatorname{Ad}^{0}\bar{\rho}$. As a result, once it is
shown that $\rho_{3}$ lifts to a characteristic zero representation $\rho$, it
shall follow that $\rho$ is irreducible. In showing that $\rho_{3}$ can be
lifted to characteristic zero, we enlarge the set of primes
$Z=T\cup\\{v_{1},v_{2}\\}$ to a finite set of primes $Y$ such that
$X:=Y\backslash S$ consists only of trivial primes. For $i=1,2$ set
$\mathcal{C}_{v_{i}}=\mathcal{C}_{v_{i}}^{ram}$ and for primes $v\in
X\backslash\\{v_{1},v_{2}\\}$, set $\mathcal{C}_{v}=\mathcal{C}_{v}^{nr}$. We
show that the dual-Selmer group
$H^{1}_{\mathcal{N}^{\perp}}(\operatorname{G}_{\mathbb{Q},Y},\operatorname{Ad}^{0}\bar{\rho}^{*})$
vanishes for a suitably chosen set of primes $Y$. For convenience of notation,
denote by $\mathscr{W}$ the Galois submodule
$(\operatorname{Ad}^{0}\bar{\rho})_{-2n+2}$ of
$\operatorname{Ad}^{0}\bar{\rho}$ spanned by root spaces
$(\operatorname{Ad}^{0}\bar{\rho})_{\beta}$ for $\beta\neq-2L_{1}$.

######  Proposition 7.1.

Let
$\rho_{3}:\operatorname{G}_{\mathbb{Q},T\cup\\{v_{1},v_{2}\\}}\rightarrow\operatorname{GSp}_{2n}(\text{W}(\mathbb{F}_{q})/p^{3})$
be the lift of $\bar{\rho}$ obtained from the application of Propositions 6.8
and 6.9. Let $Y$ be a finite set of primes which contains
$Z=T\cup\\{v_{1},v_{2}\\}$ such that $Y\backslash S$ consists of trivial
primes. Suppose $f\in
H^{1}_{\mathcal{N}}(\operatorname{G}_{\mathbb{Q},Y},\operatorname{Ad}^{0}\bar{\rho})$
and $\psi\in
H^{1}_{\mathcal{N}^{\perp}}(\operatorname{G}_{\mathbb{Q},Y},\operatorname{Ad}^{0}\bar{\rho}^{*})$
are nonzero classes. Then there exists a prime $v\notin Y$ such that

  1. (1)

$v$ is a trivial prime,

  2. (2)

$\rho_{3\restriction\operatorname{G}_{v}}$ satisfies
$\mathcal{C}_{v}=\mathcal{C}_{v}^{nr}$,

  3. (3)

$f$ does not satisfy $\mathcal{N}_{v}=\mathcal{N}_{v}^{nr}$,

  4. (4)

$\beta_{\restriction\operatorname{G}_{v}}=0$ for all $\beta\in
H^{1}(\operatorname{G}_{\mathbb{Q},Y},\mathscr{W}^{*})$,

  5. (5)

$\psi_{\restriction\operatorname{G}_{v}}\neq 0$ and one can extend
$\\{\psi\\}$ to a basis $\psi_{1}=\psi,\psi_{2},\dots,\psi_{k}$ of
$H^{1}(\operatorname{G}_{\mathbb{Q},Y},\operatorname{Ad}^{0}\bar{\rho}^{*})$
such that $\psi_{i\restriction\operatorname{G}_{v}}=0$ for $i>1$.

###### Proof.

Each condition is a union of Chebotarev conditions on a number of finite
extensions $J$ of $K$. Each of the extensions $J$ are Galois over $\mathbb{Q}$
with $\operatorname{Gal}(J/K)$ an $\mathbb{F}_{p}$-vector space. Let
$g\in\operatorname{G}^{\prime}$ and $x\in\operatorname{Gal}(J/K)$, define,
$g\cdot x:=\tilde{g}x\tilde{g}^{-1}$ where $\tilde{g}$ is a lift of $g$ to
$\operatorname{Gal}(J/\mathbb{Q})$. This gives $\operatorname{Gal}(J/K)$ the
structure of an $\mathbb{F}_{p}[\operatorname{G}^{\prime}]$-module. For each
condition, we list the choices for $J$ below as well as characters for the
$\mathbb{T}$-action on $\operatorname{Gal}(J/K)$:

Condition | $J$ | $\text{Eigenspaces of }\operatorname{Gal}(J/K)$
---|---|---
$(1)$ | $K(\mu_{p^{2}})$ | $1$
$(2)$ | $K(\rho_{2})$ | $1,\\{\sigma_{\lambda}\\}_{\lambda\in\Phi}$
$(3)$ | $K_{f}$ | $1,\\{\sigma_{\lambda}\\}_{\lambda\in\Phi}$