# A Refined Lifting Theorem for Supersingular Galois Representations

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

Let $p$ be an odd prime, $\mathbb{F}$ a finite field of characteristic $p$ and
let $\bar{\rho}:\text{G}_{\mathbb{Q}}\rightarrow\text{GL}_{2}(\mathbb{F})$ be
a continuous Galois representation. Denote by $\bar{\chi}$ the mod $p$
cyclotomic character. We assume that the local representation
$\bar{\rho}_{\restriction\text{G}_{\mathbb{Q}_{p}}}$ is flat and irreducible
and $\text{det}\bar{\rho}=\bar{\chi}$. The celebrated theorem of Khare and
Wintenberger asserts that if $\bar{\rho}$ satisfies some natural conditions,
there exists a normalized Hecke-eigenform $f=\sum_{n\geq 1}a_{n}q^{n}$ and a
prime $\mathfrak{p}|p$ in its field of Fourier coefficients such that the
associated $\mathfrak{p}$-adic representation ${\rho}_{f,\mathfrak{p}}$ lifts
$\bar{\rho}$. In this manuscript we prove a refined lifting theorem. Let
$\iota_{\mathfrak{p}}$ be the inclusion of the field of Fourier coefficients
$\mathbb{Q}(f)$ of $f$ into $\mathbb{Q}(f)_{\mathfrak{p}}$ and let $v_{p}$ be
a $p$-adic valuation on $\bar{\mathbb{Q}}_{p}$ normalized by $v_{p}(p)=1$. We
show that for any choice of $\lambda\in\mathbb{Z}_{\geq 1}$, there exists a
cuspidal Hecke eigenform $f$ of weight $2$ over $\Gamma_{1}(N)$ for $(N,p)=1$
and a prime $\mathfrak{p}|p$ such that
$v_{p}(\iota_{\mathfrak{p}}(a_{p}))=\lambda$ and the residual representation
$\bar{\rho}$ lifts to $\rho_{f,\mathfrak{p}}$. Thus all slopes
$\mathbb{Z}_{\geq 1}\subset\mathbb{Q}_{>0}$ are realized as we vary over all
Galois representations associated to cuspidal Hecke eigenforms which lift
$\bar{\rho}$.

##  1\. Introduction

###  1.1. Statement of the Main Result

Let $p$ be an odd rational prime and $\mathbb{F}$ a finite field of
characteristic $p$. Let $v_{p}$ be a $p$-adic valuation on the algebraic
closure $\bar{\mathbb{Q}}_{p}$ normalized by $v_{p}(p)=1$. Let $f$ be a
normalized Hecke eigenform and let $\mathbb{Q}(f)$ denote the number field
generated by the Fourier coefficients of $f$ and
$\mathcal{O}_{f}\subset\mathbb{Q}(f)$ its ring of integers. Choose a prime
$\mathfrak{p}|p$ of $\mathbb{Q}(f)$ and denote by
$\iota_{\mathfrak{p}}:\mathbb{Q}(f)\rightarrow\mathbb{Q}(f)_{\mathfrak{p}}$
the corresponding inclusion. Letting $\mathcal{O}_{f,\mathfrak{p}}$ denote the
$\mathfrak{p}$-adic completion of the ring of integers $\mathcal{O}_{f}$, the
$\mathfrak{p}$-adic Galois representation attached to $f$ is denoted by

$\rho_{f,\mathfrak{p}}:\text{G}_{\mathbb{Q}}\rightarrow\text{GL}_{2}(\mathcal{O}_{f,\mathfrak{p}}).$

Let $\bar{\rho}:\text{G}_{\mathbb{Q}}\rightarrow\text{GL}_{2}(\mathbb{F})$ be
an odd absolutely irreducible Galois representation which is unramified at all
but finitely many primes. The theorem of Khare and Wintenberger (previously
referred to as Serre’s conjecture) asserts that $\bar{\rho}$ lifts to a
modular $\mathfrak{p}$-adic Galois representation $\rho_{f,\mathfrak{p}}$ for
a normalized Hecke eigenform $f$ and a prime $\mathfrak{p}|p$ of
$\mathbb{Q}(f)$. We examine the case when
$\bar{\rho}_{\restriction\text{G}_{\mathbb{Q}_{p}}}$ is flat and irreducible.
In this case, an eigenform $f$ for which $\rho_{f,\mathfrak{p}}$ lifts
$\bar{\rho}$ is supersingular at $p$, i.e.
$v_{p}(\iota_{\mathfrak{p}}(a_{p}))\in\mathbb{Q}_{>0}$.

Let $\chi$ denote the $p$-adic cyclotomic character and $\bar{\chi}$ denote
its reduction and let $\omega_{2}$ be a choice of fundamental character of
level $2$.

######  Theorem 1.1.

Let $\bar{\rho}:\text{G}_{\mathbb{Q}}\rightarrow\text{GL}_{2}(\mathbb{F})$ be
a Galois representation for which

  1. (1)

$\bar{\rho}$ is unramified outside a set of finitely many primes $S$,

  2. (2)

the local representation $\bar{\rho}_{\restriction\text{G}_{\mathbb{Q}_{p}}}$
is flat and irreducible,

  3. (3)

$\det\bar{\rho}=\bar{\chi}$,

and let $\lambda\in\mathbb{Z}_{\geq 1}$.

There exists a eigenform $f=\sum_{n\geq 1}a_{n}q^{n}\in S_{2}(\Gamma_{1}(N))$
such that $(N,p)=1$ and for a certain $\mathfrak{p}|p$ in $\mathcal{O}_{f}$

  1. (1)

the residue field $k(\mathfrak{p})\subseteq\mathbb{F}$ and the ramification
index $e(\mathfrak{p}|p)=1$ and consequently,

$\mathcal{O}_{f,\mathfrak{p}}\subseteq W(\mathbb{F}).$

  2. (2)

With respect to the embedding
$\iota_{\mathfrak{p}}:\mathbb{Q}(f)\hookrightarrow\mathbb{Q}(f)_{\mathfrak{p}}$,

$v_{p}(\iota_{\mathfrak{p}}(a_{p}))=\lambda.$

  3. (3)

There is a finite set of primes $X$ disjoint from $S$ such that the Galois
representation $\rho_{f,\mathfrak{p}}$ is unramified outside $S\cup X$.

  4. (4)

The characteristic zero representation $\rho_{f,\mathfrak{p}}$ lifts
$\bar{\rho}$

$\text{G}_{\mathbb{Q},S\cup
X}$$\text{G}_{\mathbb{Q},S}$$\text{GL}_{2}(\mathbb{F}).$$\text{GL}_{2}(W(\mathbb{F}))$$\bar{\rho}$$\rho_{f,\mathfrak{p}}$

We are led to define the notion of the slope of any flat deformation of
$\bar{\rho}_{\restriction\text{G}_{\mathbb{Q}_{p}}}$. This notion for local
Galois representations is an adaptation of the global notion of the slope of
the $p$-th Fourier coefficient of a normalized Hecke eigencuspform giving rise
to a deformation of $\bar{\rho}$. We show that for any fixed
$\lambda\in\mathbb{Z}_{\geq 1}$, the slope-$\lambda$ flat deformations of
$\bar{\rho}_{\restriction\text{G}_{\mathbb{Q}_{p}}}$ are parametrized by a
closed subscheme of the deformation space of all flat deformations of
$\bar{\rho}_{\restriction\text{G}_{\mathbb{Q}_{p}}}$. Using the method of
[KLR05], we may lift $\bar{\rho}$ a number of steps so that when restricted to
$\text{G}_{\mathbb{Q}_{p}}$ it has slope-$\lambda$. The slope-$\lambda$
deformation condition has the property that at this stage it is preserved by
action of the entire tangent space of the functor of flat-deformations.
Consequently, at this stage the global deformation problem becomes balanced
and it becomes possible to lift it to characteristic zero by applying the
classical method due to Ramakrishna which involves annihiliating the
associated dual Selmer group of the deformation problem. The global
deformation comes from a normalized cuspidal Hecke eigenform $f$. The field of
Fourier coefficients of $f$ and the set of primes of ramification can be
expected to be really large for any choice of $\lambda$.

There has been some interest in understanding the variation in the slope of
the Fourier coefficient $a_{p}(f)$ for cuspidal Hecke eigenforms lifting a
fixed residual Galois representation from the point of view of the $p$-adic
Langlands program. The reader may contrast the global lifting theorem proved
in this manuscript with the local results of Berger-Li-Zhu (see [BLZ04]),
Buzzard-Gee ( see [BG09] and [BG13]) and Ganguli-Ghate (see [GG15]) who
classify the local representation $\bar{\rho}_{f\restriction
G_{\mathbb{Q}_{p}}}$ as $f$ ranges over all cuspidal Hecke eigenforms $f$ of
weight $k\geq 2$ such that the slope of the Fourier coefficient $a_{p}(f)$
lies in various sub-intervals of $(0,\infty)$.

###  1.2. A Sketch of the Approach via Deformation Theory

We now provide a slightly more detailed sketch of the strategy involved. We
lift our residual Galois representation to
$\rho:\text{G}_{\mathbb{Q}}\rightarrow\text{GL}_{2}(W(\mathbb{F}))$ such that

  1. (1)

$\rho$ is unramified at all but finitely many primes,

  2. (2)

$\text{det}\rho=\chi$,

  3. (3)

$\rho_{\restriction\text{G}_{\mathbb{Q}_{p}}}$ is flat and irreducible,

  4. (4)

$\rho_{\restriction\text{G}_{\mathbb{Q}_{p}}}$ satisfies a certain deformation
condition which will have the effect of prescribing the valuation of the
$p$-th Fourier coefficient of a normalized cuspidal eigenform which whose
associated Galois representation may satisfy this condition.

Fix $\lambda\in\mathbb{Z}_{\geq 1}$, at the prime $p$, we use a variation of
the Ramakrishna’s flat deformation functor from [Ram93]. If $f=\sum_{n\geq
1}a_{n}q^{n}$ is a normalized eigenform and $\mathfrak{p}|p$ is a prime of
$\mathbb{Q}(f)$ such that $\bar{\rho}_{f,\mathfrak{p}}=\bar{\rho}$ and
$\rho_{f,\mathfrak{p}}$ satisfies this local deformation condition at $p$,
then $v_{p}(\iota_{\mathfrak{p}}(a_{p}))=\lambda$. Let $\mathcal{C}$ be the
category of Noetherian coefficient rings over $W(\mathbb{F})$, for
$R\in\mathcal{C}$, its maximal ideal is denoted by $\mathfrak{m}_{R}$. Let
$\mathcal{F}$ denote Ramakrishna’s flat deformation functor defined so that
$\mathcal{F}(R)$ consists the deformations $\varrho$ of
$\bar{\rho}_{\restriction\text{G}_{\mathbb{Q}_{p}}}$ to $R$

$\text{G}_{\mathbb{Q}_{p}}$$\text{GL}_{2}(\mathbb{F})$$\text{GL}_{2}(R)$$\bar{\rho}_{\restriction\text{G}_{\mathbb{Q}_{p}}}$$\varrho$

such that $\varrho$ is flat and $\det\varrho=\chi$. It is a consequence of a
theorem of Ramakrishna that $\mathcal{F}$ is represented by
$\mathcal{R}=W(\mathbb{F})[|X|]$ (see [FM95, Theorem B2]).

The functor of points of $\hat{\mathbb{A}}^{1}=W(\mathbb{F})[|Y|]$ when
evaluated at $R\in\mathcal{C}$ with maximal ideal $\mathfrak{m}_{R}$ is

$\hat{\mathbb{A}}^{1}(R)={\rm{Hom}}_{\mathcal{C}}(W(\mathbb{F})[|Y|],R)\simeq\mathfrak{m}_{R}.$

The strategy employed in this manuscript involves a number of steps. There is
a subfunctor $\mathcal{F}^{\lambda}$ of
$\mathcal{F}\times_{\text{Spec}W(\mathbb{F})}\hat{\mathbb{A}}^{1}$ for which
$\mathcal{F}^{\lambda}(R)$ consists of pairs
$(\rho,U)\in\mathcal{F}(R)\times\mathfrak{m}_{R}$ with a prescribed slope
relation. Let $\pi_{1}$ and $\pi_{2}$ be the first and second projections of
$\text{Spec}\mathcal{R}\times_{\text{Spec}W(\mathbb{F})}\hat{\mathbb{A}}^{1}$,
we show that $\mathcal{F}^{\lambda}$ is representable by a closed subscheme of
$X_{\mathcal{F}^{\lambda}}$ and that $\pi_{2}$ induces an isomorphism
$\pi_{2}:X_{\mathcal{F}^{\lambda}}\xrightarrow{\sim}\hat{\mathbb{A}}^{1}$. At
a prime $v\in S\backslash\\{p\\}$ the local deformation condition
$\mathcal{C}_{v}$ is the condition described in Proposition 1 of [Ram02]. The
projection $\pi_{1}$ induces a natural transformation
$\pi_{1}:\mathcal{F}^{\lambda}\rightarrow\mathcal{F}$. Let
$\mathcal{C}_{p}^{\lambda}$ be the local condition at $p$ prescribed by

$\mathcal{C}_{p}^{\lambda}(R):=\text{image}\\{\pi_{1}(R):\mathcal{F}^{\lambda}(R)\rightarrow\mathcal{F}(R)\\}.$

We show that $\mathcal{C}_{p}^{\lambda}$ defines a liftable subfunctor of
$\mathcal{F}$. Moreover, the slope relation implies that if $f=\sum_{n\geq
1}a_{n}q^{n}$ is a normalized eigenform and $\mathfrak{p}|p$ is a prime of
$\mathbb{Q}(f)$ such that $\bar{\rho}_{f,\mathfrak{p}}=\bar{\rho}$ and
satisfies $\mathcal{F}^{\lambda}$ then
$v_{p}(\iota_{\mathfrak{p}}(a_{p}))=\lambda$. For a finite length ring
$R\in\mathcal{C}$ the precise definition of the slope of a deformation of
$\bar{\rho}_{\restriction\text{G}_{\mathbb{Q}_{p}}}$ involves the analysis of
the Fontaine-Laffaille module associated to it.

We apply the method in [KLR05] to lift $\bar{\rho}$ up to

$\rho_{\lambda+2}:\text{G}_{\mathbb{Q}}\rightarrow\text{GL}_{2}(W(\mathbb{F})/p^{\lambda+2})$

after which the Ramakrishna method is employed to produce a finite set of
auxiliary primes and a characteristic zero weight $2$ deformation

$\rho:\text{G}_{\mathbb{Q},S\cup X}\rightarrow\text{GL}_{2}(W(\mathbb{F}))$

subject to the local conditions $\mathcal{C}_{v}$. Since
$\rho_{\restriction\text{G}_{\mathbb{Q}_{p}}}\in\mathcal{C}_{p}^{\lambda}$
this deformation is in particular flat and supersingular.

The Fontaine-Mazur conjecture for absolutely irreducible
$\text{G}_{\mathbb{Q}}$-representations is fully established in [Kis09] for
all but one exception in the ordinary case. In particular, rank 2 super
singular representations which satisfy the conditions of the Fontaine-Mazur
conjecture are modular and consequently $\rho$ coincides with a family of
representations with slope $\lambda$ and varying weight whose points of
classical weight coincide with representations arising from cusp forms of
slope $\lambda$.

The methods in this manuscript do not as of yet carry over to non-integral
$\lambda\in\mathbb{Q}_{>0}$. The Ramakrishna method as originally conceived
applies to produce deformations of $\bar{\rho}$ to
$\rho:\text{G}_{\mathbb{Q}}\rightarrow\text{GL}_{2}(\mathcal{O})$ where
$\mathcal{O}$ is the valuation ring of an unramified extension of
$\mathbb{Q}_{p}$.

###  1.3. Acknowledgements

I am very grateful to my advisor Ravi Ramakrishna for introducing me to his
Galois-theoretic lifting method and for some helpful suggestions. I would also
like to thank Brian Hwang, Aftab Pande and Stefan Patrikis for some fruitful
conversations.

##  2\. Notation

In this section we summarize some basic notation in this manuscript.

  * •

The prime $p$ is an odd prime and $\mathbb{F}$ is a finite field of
characteristic $p$.

  * •

The $p$-adic valuation
$v_{p}:\bar{\mathbb{Q}}_{p}\rightarrow\mathbb{Q}\cup\\{\infty\\}$ is
normalized by $v_{p}(p)=1$.

  * •

The completion of $\bar{\mathbb{Q}}_{p}$ w.r.t the valuation $v_{p}$ is
denoted by $\mathbb{C}_{p}$. We simply denote the extension of the valuation
to $\mathbb{C}_{p}$ by $v_{p}$ which takes values in
$\mathbb{R}\cup\\{\infty\\}$. The valuation ring is denoted by
$\mathcal{O}_{\mathbb{C}_{p}}$.

  * •

The ring of Witt vectors with residue field $\mathbb{F}$ is denoted by
$W(\mathbb{F})$ and $K:=W(\mathbb{F})[1/p]$. The field $K$ is the unramified
extension of $\mathbb{Q}_{p}$ with residue field $\mathbb{F}$.

  * •

Let $f$ be a normalized cuspidal Hecke eigenform, the field of Fourier
coefficients of $f$ is the extension of $\mathbb{Q}$ generated by the Fourier
coefficients of the $q$-expansion of $f$ and is denoted by $\mathbb{Q}(f)$.
This is a number field. The ring of integers of $\mathbb{Q}(f)$ is denoted by
$\mathcal{O}_{f}$ and its completion at a finite prime $v$ of $\mathbb{Q}(f)$
is denoted by $\mathcal{O}_{f,v}$.

  * •

The residual representation $\bar{\rho}$ satisfies the hypotheses of Theorem
1.1. The adjoint representation $\text{Ad}^{0}\bar{\rho}$ is the
$\mathbb{F}[\text{G}_{\mathbb{Q}}]$ module of trace zero matrices

$\text{Ad}^{0}\bar{\rho}=\left\\{\left({\begin{array}[]{cc}a&b\\\ c&-a\\\
\end{array}}\right)\mid a,b,c\in\mathbb{F}\right\\}$

where $g\in\text{G}_{\mathbb{Q}}$ acts through conjugation via $\bar{\rho}$

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

  * •

Associated to a normalized Hecke-eigenform $f$ and a prime $\mathfrak{p}|p$ of
the field of Fourier coefficients $\mathbb{Q}(f)$ is the associated
$\mathfrak{p}$-adic Galois representation

$\rho_{f,\mathfrak{p}}:\text{G}_{\mathbb{Q}}\rightarrow\text{GL}_{2}(\mathcal{O}_{f,\mathfrak{p}}).$

  * •

Let $K=W(\mathbb{F})[1/p]$ and let $\sigma\in\text{Gal}(K/\mathbb{Q})$ denote
the Frobenius automorphism of $K$.

  * •

$\text{Rep}_{K}^{f}(\text{G}_{\mathbb{Q}_{p}})$ is the category of finite
dimensional continuous $G_{\mathbb{Q}_{p}}$ representations over $K$.

  * •

$\text{Rep}^{f}(\text{G}_{\mathbb{Q}_{p}})$ is the category of finite length
$W(\mathbb{F})[G_{\mathbb{Q}_{p}}]$-modules.

  * •

Let $\mathcal{C}$ be the category of finitely generated noetherian local
$W(\mathbb{F})$ algebras $R$ with maximal ideal $\mathfrak{m}$ and a
prescribed mod $\mathfrak{m}$ isomorphism
$R/\mathfrak{m}\xrightarrow{\sim}\mathbb{F}$. We shall refer to $\mathcal{C}$
as the category of coefficient rings over $W(\mathbb{F})$. A map
$f:(R_{1},\mathfrak{m}_{1})\rightarrow(R_{2},\mathfrak{m}_{2})$ in
$\mathcal{C}$ is a map of local rings compatible with reduction isomorphisms.
Let $\mathcal{C}^{f}$ be the full subcategory of finite length algebras.

  * •

Let $Z$ be a finite set of primes containing $S$ and $M$ an
$\mathbb{F}[\text{G}_{\mathbb{Q},Z}]$-module, for $i=0,1,2$, the i-group is
the kernel of the restriction map

${}_{Z}^{i}(M):=\text{ker}\\{H^{1}(G_{\mathbb{Q},Z},M)\rightarrow\bigoplus_{l\in
Z}H^{1}(G_{\mathbb{Q}_{l}},M)\\}.$

  * •

The functor of crystalline weight $2$ deformations of $\bar{\rho}$ is a
functor

$\mathcal{F}:\mathcal{C}\rightarrow\text{Sets}$

on $\mathcal{C}$ by $\mathcal{F}(R)$ consists of all deformations
$\rho:\text{G}_{\mathbb{Q}_{p}}\rightarrow\text{GL}_{2}(R)$ of $\bar{\rho}$
that are crystalline and with determinant $\det\rho=\chi$.

##  3\. Recollections on the Fontaine-Laffaille Functor

In this section, we review some properties of the Fontaine-Laffaille functor
and make preparations to describe a fixed slope subfunctor of the functor of
flat deformations of $\bar{\rho}_{\restriction\text{G}_{\mathbb{Q}_{p}}}$. The
standard references for this section are [FL82] and [Pat06].

Let $K=W(\mathbb{F})[1/p]$ and $\sigma\in\operatorname{Gal}(K/\mathbb{Q}_{p})$
the Frobenius element and let $\mathbb{C}_{p}$ denote the completion of
$\bar{\mathbb{Q}}_{p}$ w.r.t the valuation $v_{p}$. We choose a distinguished
element

$\varepsilon=(\varepsilon_{i})\in\mathcal{O}^{\flat}_{\mathbb{C}_{p}}:=\varprojlim_{x\mapsto
x^{p}}\mathcal{O}_{\mathbb{C}_{p}}/p$

for which $\varepsilon_{0}\neq 1$ is a $p$-th root of unity. For
$x\in\mathcal{O}^{\flat}_{\mathbb{C}_{p}}$ let $[x]$ denote the Teichmüller
lift of $x$ in $W(\mathcal{O}^{\flat}_{\mathbb{C}_{p}})$. The continuous
Galois-equivariant ring homomorphism
$\theta:W(\mathcal{O}^{\flat}_{\mathbb{C}_{p}})\rightarrow\mathcal{O}^{\flat}_{\mathbb{C}_{p}}$,
defined by
$\theta\left(\sum_{i}[x^{(i)}]p^{i}\right):=\sum_{i}x^{(i)}_{0}p^{i}$ is open
and surjective, the kernel of which is a principal ideal generated by a
special element $\xi\in W(\mathcal{O}^{\flat}_{\mathbb{C}_{p}})$ with some key
properties stated in [BC09, Proposition 4.4.3]. Let $\text{A}_{cris}$ denote
the divided power envelope of $W(\mathcal{O}^{\flat}_{\mathbb{C}_{p}})$ with
respect to the ideal $\ker\theta$, in other words,
$\text{A}_{cris}=W(\mathcal{O}^{\flat}_{\mathbb{C}_{p}})[\xi^{m}/m!]_{m\geq
1}$. The $p$-adic completion $\text{B}_{cris}^{+}$ is a local ring with
residue field $\mathbb{C}_{p}$ and its valuation and the crystalline period
ring $\text{B}_{cris}$ is then defined to be the ring obtained on inverting
the period $t:=\log[\varepsilon]\in\text{A}_{cris}$. The period ring
$\text{B}_{cris}:=\text{B}_{cris}^{+}[1/t]$ has an induced Galois stable
filtration of $\mathbb{Q}_{p}$ vector spaces given by
$F^{i}\text{B}_{cris}:=t^{i}\text{A}_{cris}$.

######  Definition 3.1.

A filtered $\varphi$-module $M$ is a $K$ vector space equipped with a
semilinear bijective map $\varphi$ with respect to $\sigma$ and a decreasing
filtration $\\{F^{i}M\\}$. For this filtration, $F^{i}M=M$ for $i\ll 0$ and
$F^{i}M=0$ for $i\gg 0$.

Fontaine’s crystalline period functor $\text{D}_{cris}$ from
$\text{Rep}_{K}^{f}(\text{G}_{\mathbb{Q}_{p}})$ to the category of finitely
filtered $\varphi$-modules is defined as follows, for
$V\in\text{Rep}_{K}^{f}(\text{G}_{\mathbb{Q}_{p}})$

$\text{D}_{cris}(V):=\left(V\otimes_{\mathbb{Q}_{p}}\text{B}_{cris}\right)^{G_{\mathbb{Q}_{p}}}$

with filtration

$F^{i}\text{D}_{cris}(V):=\left(V\otimes_{\mathbb{Q}_{p}}F^{i}\text{B}_{cris}\right)^{G_{\mathbb{Q}_{p}}}.$

The representation $V$ is crystalline if

$\dim_{K}V=\dim_{K}\text{D}_{\text{cris}}(V).$

A filtered $\varphi$-module $V$ is weakly-admissible if and only if there
exists a $W(\mathbb{F})$-lattice $M\subset V$ with induced filtration
$F^{i}M=M\cap F^{i}V$ for which

  1. (1)

$\varphi(F^{i}M)\subseteq p^{i}M$

  2. (2)

$\sum_{i}p^{-i}\varphi(F^{i}M)=M$.

Theorem A of [CF00] asserts that to a crystalline representation $\rho$, the
$\varphi$-module $\text{D}_{cris}(V_{\rho})$ is weakly-admissible and all
weakly-admissible $\varphi$-modules arise this way.

######  Definition 3.2.

A filtered Dieudonné $W(\mathbb{F})$-module also known as a Fontaine-Laffaille
Module it is furnished with a decreasing, exhaustive, separated filtration of
$W(\mathbb{F})$-submodules $\\{F^{i}M\\}$ and for each integer $i$ a
$\sigma$-semilinear map

$\varphi^{i}=\varphi_{M}^{i}:F^{i}M\rightarrow M$

such that

  1. (1)

the following compatibility relation is satisfied
$\varphi^{i+1}=p\varphi^{i}$,

  2. (2)

$\sum_{i}\varphi^{i}(F^{i}M)=M$.

Denote by $\varphi=\varphi^{0}:M\rightarrow M$.

A map $f:(M,\varphi^{i}_{M})\rightarrow(N,\varphi^{i}_{N})$ of Fontaine-
Laffaille modules is a $W(\mathbb{F})$-module map such that
$f(F^{i}M)\subseteq F^{i}N$ and $\varphi^{i}_{N}\circ f_{\restriction
F^{i}M}=f\circ\varphi^{i}_{M}$.

######  Definition 3.3.

Let $\text{MF}_{tor}^{f}$ denote the category of finite length Fontaine-
Laffaille modules $M$ with morphisms satisfying the conditions alluded to
above. For $a<b$ let $\text{MF}_{tor}^{f,[a,b]}$ let the full subcategory of
$\text{MF}_{tor}^{f}$ whose underlying modules $M$ satisfy $F^{a}M=M$ and
$F^{b}M=0$.

Let $\text{Rep}^{f}(\text{G}_{\mathbb{Q}_{p}})$ is the category of continuous
$W(\mathbb{F})[G_{\mathbb{Q}_{p}}]$ modules that are finite-length
$\mathbb{Z}_{p}$-modules. Fontaine and Laffaille in [FL82] showed that
$\text{MF}_{tor}^{f}$ and $\text{MF}_{tor}^{f,[a,b]}$ are abelian categories
and defined a contravariant functor

$\text{U}_{S}:\text{MF}_{tor}^{f,[0,p]}\rightarrow\text{Rep}^{f}(\text{G}_{\mathbb{Q}_{p}}).$

######  Definition 3.4.

Let $M\in\text{MF}_{tor}^{[a,b]}$, its dual
$M^{*}\in\text{MF}_{tor}^{[1-b,1-a]}$ is the module

$M^{*}={\rm{Hom}}_{W(\mathbb{F})}(M,K/W(\mathbb{F}))$

with filtration

$F^{i}(M^{*})={\rm{Hom}}_{W(\mathbb{F})}(M/F^{1-i}M,K/W(\mathbb{F})).$

We proceed to describe the maps $\varphi^{i}_{M^{*}}:F^{i}(M^{*})\rightarrow
M^{*}$. Let $f\in F^{i}(M^{*})$ and $m\in M$. Since
$M=\sum_{j}\varphi^{j}_{M}(F^{j}M)$, the element $m$ may be represented as a
sum $m=\sum_{j}\varphi^{j}_{M}(m_{j})$. To prescribe
$\varphi^{i}_{M^{*}}(f)(m)$ it suffices to define
$\varphi^{i}_{M^{*}}(f)(\varphi_{M}^{j}(m_{j}))$. These are taken as follows

$\varphi^{i}_{M^{*}}(f)(\varphi^{j}_{M}(m_{j}))=\begin{cases}0\text{ for
}j>-i;\\\ f(p^{-i-j}m_{j})\text{ for }j\leq-i.\\\ \end{cases}$

The reader may check that the maps $\varphi_{M^{*}}^{i}$ are well defined. We
now describe Tate-twists.

######  Definition 3.5.

For an object $M\in\text{MF}_{tor}^{f,[a,b]}$ and $m\in\mathbb{Z}$, the
$m$-fold Tate twist of $M$ is the module
$M(m)\in\text{MF}_{tor}^{f,[a+m,b+m]}$ whose underlying module is $M$ and
$F^{i}(M(m))=F^{i-m}(M)$ and $\varphi^{i}_{M}=\varphi^{i-m}_{M(m)}$.

We proceed to describe the covariant adaptations of the Fontaine-Laffaille
functor which satisfy a suitable compatibility relation with the tensor
product. This makes these functors more suitable to work with.

######  Definition 3.6.

We let

$\text{T}:\text{MF}_{tor}^{[2-p,1]}\rightarrow\text{Rep}^{f}(\text{G}_{\mathbb{Q}_{p}})$

defined by $\text{T}(M)=\text{U}_{S}(M^{*})$. We define

$\text{T}_{1}:\text{MF}_{tor}^{[3-p,2]}\rightarrow\text{Rep}^{f}(\text{G}_{\mathbb{Q}_{p}})$

by $\text{T}_{1}(M):=\text{U}_{S}(M^{*}(1))$.

We now collect a few facts about the functors T and $\text{T}_{1}$.

######  Fact 3.7.

The functor T is full and faithful and as a consequence, so is $\text{T}_{1}$.

######  Fact 3.8.

For each object $M$ of $\text{MF}_{tor}^{[2-p,1]}$,

$\text{length}_{W(\mathbb{F})}M=\text{length}_{W(\mathbb{F})}\text{T}(M)=\text{length}_{W(\mathbb{F})}\text{T}_{1}(M).$

Let $\mathcal{C}$ be the category of finitely generated noetherian local
$W(\mathbb{F})$-algebras $R$ with maximal ideal $\mathfrak{m}$ and a
prescribed mod $\mathfrak{m}$ isomorphism
$R/\mathfrak{m}\xrightarrow{\sim}\mathbb{F}$ and refer to $\mathcal{C}$ as the
category of coefficient rings over $W(\mathbb{F})$. A map
$f:(R_{1},\mathfrak{m}_{1})\rightarrow(R_{2},\mathfrak{m}_{2})$ in
$\mathcal{C}$ is a map of local rings compatible with reduction isomorphisms.
Let $\mathcal{C}^{f}$ be the full subcategory of finite-length algebras.

######  Definition 3.9.

For $R\in\mathcal{C}$, a deformation
$\rho:\text{G}_{\mathbb{Q}}\rightarrow\text{GL}_{2}(R)$ of $\bar{\rho}$ is
said to be crystalline if for every finite-length quotient $R^{\prime}$ of $R$
in $\mathcal{C}$ the pushforward $\rho\otimes_{R}R^{\prime}$ lies in the
essential image of $T$.

The residual representation
$\bar{\rho}:\text{G}_{\mathbb{Q}_{p}}\rightarrow\text{Aut}_{\mathbb{F}_{p}}(V_{\bar{\rho}})$
is flat, and consequently, there exists $M_{0}\in\text{MF}_{tor}^{[0,2]}$ such
that $\text{T}_{1}(M_{0})\simeq V_{\bar{\rho}}$. This is shown for instance in
[Ram93].

######  Definition 3.10.

Define the functor

$\mathcal{F}:\mathcal{C}\rightarrow\text{Sets}$

on $\mathcal{C}$ by $\mathcal{F}(R)$ consists of all deformations
$\rho:\text{G}_{\mathbb{Q}}\rightarrow\text{GL}_{2}(R)$ of $\bar{\rho}$ that
are flat and with determinant $\det\rho=\chi$.

The restriction of this functor to $\mathcal{C}^{f}$ has the following
description: an object of $\mathcal{F}(R)$ for $R\in\mathcal{C}^{f}$ can be
identified with $\text{T}_{1}(M)$ for $M\in\text{MF}_{tor}^{f}$. In fact,
$M\in\text{MF}_{tor}^{f,[0,2]}$ and as $\text{T}_{1}$ is full and faithful,
the $R$ action on the free $R$-module $\text{T}_{1}(M)$ carries over to a
faithful $R$ action on $M$. In greater detail, if $r\in R$, the endomorphism
induced by multiplication by $r$ on $\text{T}_{1}(M)$ is an
$R[\text{G}_{\mathbb{Q}_{p}}]$-module endomorphism. Also since $\text{T}_{1}$
is full and faithful, this multiplication by $r$ endomorphism concides with an
endomorphism

$\text{T}_{1}^{-1}(r):M\rightarrow M$

in $\text{MF}_{tor}^{[2-p,1]}$, i.e.

  * •

$\text{T}_{1}^{-1}(r):M\rightarrow M$ is a $W(\mathbb{F})$-module endomorphism

  * •

$\text{T}_{1}^{-1}(r)(M^{i})\subseteq M^{i}$

  * •

$\varphi^{i}\circ\text{T}_{1}^{-1}(r)_{\restriction
M^{i}}=\text{T}_{1}^{-1}(r)\circ\varphi^{i}.$

Denote $\varphi^{0}$ by $\varphi$ and $W(\mathbb{F})$-algebra of $\varphi$
equivariant $W(\mathbb{F})$-module homomorphisms of $M$ by
$\text{End}_{\varphi}(M)$. As a consequence of the faithfulness of $T_{1}$, we
obtain a ring homomorphism

$\text{T}_{1}^{-1}:R\rightarrow\text{End}_{\varphi}(M)$

taking $r\in R$ to the endomorphism $\text{T}_{1}^{-1}(r)$. We summarize these
observations by saying that $M$ is an $R$-module in the category
$\text{MF}_{tor}^{f}$. In particular, it is an $R[\varphi]$-module and that
the $R$ action on $M$ is uniquely determined by that on $\text{T}_{1}(M)$ and
preserves the filtration on $M$. In fact the module $M$ can be canonically
identified with a free $R$-module of rank $2$. The following result follows
from Lemma 4.2 in [Pat06] where the analogous result for the functor T is
stated. That the same implication is true for $\text{T}_{1}$ is because
$\text{T}_{1}(M)=\text{T}(M(-1))$.

######  Proposition 3.11.

Let $R\in\mathcal{C}^{f}$ and $M\in\text{MF}_{tor}^{[0,2]}$ with
$\text{T}_{1}(M)\in\mathcal{F}(R)$. With respect to the $R$-module structure
on $M$ induced from the $R$-module structure on the rank $2$ free $R$-module
$\text{T}_{1}(M)$, $M$ is also a free $R$-module of rank $2$ and
$\varphi:M\rightarrow M$ is an $R$ linear endomorphism.

######  Definition 3.12.

Let $R\in\mathcal{C}_{f}$ and $M\in\text{MF}_{tor}^{[0,2]}$ with
$\text{T}_{1}(M)\in\mathcal{F}(R)$. Note that by Proposition 3.11, the
$R$-module $M$ is free of rank $2$. The $R$-linear operator
$\varphi:=\varphi^{0}$ thus may be described by a $2\times 2$ matrix with
entries in $R$. The trace of $\varphi$ is the sum of the diagonal elements of
this matrix. We denote the trace by $\text{tr}\varphi_{\restriction M}$, or
sometimes to ease notation when working with Galois representations we use
$\text{tr}\varphi_{\restriction\text{T}_{1}(M)}$ interchangeably.

######  Fact 3.13.

Let $R\in\mathcal{C}^{f}$, we use $R$ to refer to an object of
$\text{MF}_{tor}^{f}$ which is concentrated in degree zero and $\varphi^{0}$
is the identity. Then $\text{T}(R)=R$ with trivial Galois action. Let
$\alpha\in\text{Aut}_{W(\mathbb{F})}(R)$, we denote by $R(\alpha)$ the object
concentrated at $0$ and with $\varphi^{0}=\alpha$. Then $\text{T}(R(\alpha))$
is the $R$-module of rank $1$ with unramified
$\text{G}_{\mathbb{Q}_{p}}$-action by $\alpha^{-1}$ (see Lemma 4.6 [Pat06]).

######  Definition 3.14.

Let $R\in\mathcal{C}^{f}$ and $A,B\in\text{MF}_{tor}^{f}$ be $R$-modules such
that the $R$-module action is compatible with the $W(\mathbb{F})$-action, the
filtrations and $\varphi^{i}$-maps. The tensor product $A\otimes_{R}B$ has a
natural filtration
$F^{m}(A\otimes_{R}B)=\bigoplus_{i+j=m}F^{i}(A)\otimes_{R}F^{j}(B)$ and
$\varphi^{m}_{A\otimes_{R}B}=\bigoplus_{i+j=m}\varphi^{i}_{A}\otimes_{R}\varphi^{j}_{B}$.

######  Lemma 3.15.

Let $A\in\text{MF}_{tor}^{f,[0,2]}$ and $B\in\text{MF}_{tor}^{f,[0,1]}$ and
$R\in\mathcal{C}^{f}$ such that $T_{1}(A)$ and $T(B)$ have the structure of
$R[\text{G}_{\mathbb{Q}_{p}}]$-modules compatible with their
$W(\mathbb{F})[\text{G}_{\mathbb{Q}_{p}}]$-module structure. Then, there is an
isomorphism of $R[\text{G}_{\mathbb{Q}_{p}}]$-modules
$\text{T}_{1}(A)\otimes_{R}\text{T}(B)\xrightarrow{\sim}\text{T}_{1}(A\otimes_{R}B)$.

###### Proof.

This statement follows from Lemma 4.3 of [Pat06]. ∎

Let $g:R_{1}\rightarrow R_{2}$ be a map of coefficient rings in
$\mathcal{C}^{f}$ and $\text{T}_{1}(M_{1})\in\mathcal{F}(R_{1})$. The push-
forward of $\text{T}_{1}(M_{1})$ to $\mathcal{F}(R_{2})$ is
$g_{*}\text{T}_{1}(M_{1}):=\text{T}_{1}(M_{1})\otimes_{R_{1}}R_{2}$. It
follows from Lemma 3.15 that there is an
$R_{2}[\text{G}_{\mathbb{Q}_{p}}]$-module isomorphism
$\text{T}_{1}(M_{1})\otimes_{R_{1}}R_{2}\simeq\text{T}_{1}(M_{1}\otimes_{R_{1}}R_{2})$
and as a result, we may identify $g_{*}\text{T}_{1}(M_{1})$ with
$\text{T}_{1}(M_{2})$ where $M_{2}:=M_{1}\otimes_{R_{1}}R_{2}$. The following
relation comes as a consequence of this identification.

######  Proposition 3.16.

With respect to notation introduced above, for $\varrho\in\mathcal{F}(R_{1})$,
the following relation holds

(3.1) $\text{tr}\varphi_{\restriction
g_{*}\varrho}=g\left(\text{tr}\varphi_{\restriction\varrho}\right).$

##  4\. The Fixed-Slope Functor

We recall that $K=W(\mathbb{F})[1/p]$ with Frobenius $\sigma$ and $N>0$
coprime to $p$. Let $f\in S_{2}(\Gamma_{1}(N))$ be a normalized eigenform with
nebentype character $\psi$. Let $\mathfrak{p}|p$ in the number field generated
by the Fourier coefficients $\mathbb{Q}(f)$ of $f$. The $\mathfrak{p}$-adic
Galois representation
$\rho_{f,\mathfrak{p}}:\text{G}_{\mathbb{Q}}\rightarrow\text{GL}_{2}(\mathcal{O}_{f,\mathfrak{p}})$
where $\mathcal{O}_{f,\mathfrak{p}}$ is the valuation ring of
$\mathbb{Q}(f)_{\mathfrak{p}}$ with residue field $k(\mathfrak{p})$. Assume
that the following conditions are satisfied

  1. (1)

$\rho_{f,\mathfrak{p}}$ is supersingular, i.e $f$ is supersingular at
$\mathfrak{p}$,

  2. (2)

$\mathbb{Q}(f)_{\mathfrak{p}}$ embeds in $K$, or in other words, the residue
field $k(\mathfrak{p})\subseteq\mathbb{F}$ and the ramification index of
$e(\mathfrak{p}|p)=1$,

  3. (3)

$\rho_{f,\mathfrak{p}}$ lifts $\bar{\rho}$.

In addition to these conditions, since $M$ is coprime to $p$ it follows that
$\rho_{f,\mathfrak{p}}$ satisifes the flat deformation condition. We have not
made any assumptions on the slope of $\iota_{\mathfrak{p}}(a_{p}(f))$ except
that it is positive. Let $V_{f}$ denote the $2$-dimensional
$\mathfrak{p}$-adic $G_{\mathbb{Q}_{p}}$-representation induced by $f$. This
representation is flat and in particular, crystalline and hence,

$\dim_{K}\text{D}_{cris}(V_{f})=\dim_{K}V_{f}=2.$

The following theorem is alluded to in [BG09] and follows from the main
theorem of [Sai97].

######  Theorem 4.1.

The characteristic polynomial of $\varphi$ on the $2$ dimensional $K$ vector
space $\text{D}_{cris}(V_{f})$ is

$ch(X)=X^{2}-\iota_{\mathfrak{p}}(a_{p})X+\psi(p)p^{k-1}.$

In order to describe a  fixed slope deformation condition we proceed to
describe the notion of the slope of any flat deformation of
$\bar{\rho}_{\restriction\text{G}_{\mathbb{Q}_{p}}}$ to any Artinian
coefficient ring $R$

$\text{G}_{\mathbb{Q}_{p}}$$\text{GL}_{2}(\mathbb{F})$.$\text{GL}_{2}(R)$$\bar{\rho}_{\restriction\text{G}_{\mathbb{Q}_{p}}}$$\varrho$

The following result gives us control on the $p$-adic valuation of
$\iota_{\mathfrak{p}}(a_{p}(f))$ via torsion Fontaine-Laffaille theory.

######  Proposition 4.2.

Let $\rho:G_{\mathbb{Q}}\rightarrow\text{GL}_{2}(W(\mathbb{F})/p^{m})$ be a
flat deformation of $\bar{\rho}$ such that there exists a cuspidal Hecke
eigenform $f\in S_{2}(\Gamma_{1}(M))$ of level $M$ coprime to $p$ such that
$\rho_{f,\mathfrak{p}}$ lifts $\rho$

$\text{G}_{\mathbb{Q}_{p}}$$\text{GL}_{2}(W(\mathbb{F})/p^{m})$.$\text{GL}_{2}(W(\mathbb{F}))$$\rho$$\rho_{f,\mathfrak{p}}$

Then
$\text{tr}\varphi_{\restriction\rho}=\iota_{\mathfrak{p}}(a_{p}(f))\mod{p^{m}}$.

###### Proof.

Let $V_{f}=V_{f,\mathfrak{p}}$ be the underlying vector space on which
$\text{G}_{\mathbb{Q}_{p}}$ acts via $\rho_{f,\mathfrak{p}}$ and let $L\simeq
W(\mathbb{F})\oplus W(\mathbb{F})$ be the Galois stable submodule of $V_{f}$.
The module $\mathcal{M}:=\varprojlim_{n}\text{T}_{1}^{-1}(L/p^{n}L)$ is
equipped with a $\varphi$-action which is by definition the inverse limit of
the $\varphi^{0}$-actions on $\text{T}_{1}^{-1}(L/p^{n}L)$ for $n\geq 1$. It
follows from Proposition 3.11 that $\mathcal{M}$ is a free module of rank $2$
over $W(\mathbb{F})$. Following [Hat18] we denote by
$\hat{\text{T}}^{*}_{cris}(\mathcal{M}):={\rm{Hom}}_{MF}(\mathcal{M},\text{A}_{cris})$
with its induced $\text{G}_{\mathbb{Q}_{p}}$-action. If we show that there is
a $\text{G}_{\mathbb{Q}_{p}}$-equivariant isomorphism

(4.1)
$L\otimes_{W(\mathbb{F})}K\simeq{\rm{Hom}}_{W(\mathbb{F})}(\hat{\text{T}}^{*}_{cris}(\mathcal{M}),W(\mathbb{F})(1))\otimes_{W(\mathbb{F})}K$

it follows from Theorem 2.11 of [Hat18] that there is a $\varphi$-equivariant
isomorphism $\text{D}_{cris}(V_{f})\simeq\mathcal{M}\otimes_{W(\mathbb{F})}K$.
In greater detail, letting
$V:=\hat{\text{T}}_{cris}^{*}(\mathcal{M})\otimes_{W(\mathbb{F})}K$, Theorem
2.11 of [Hat18] states that
$\text{D}_{cris}(V^{*})\simeq\mathcal{M}\otimes_{W(\mathbb{F})}K$ as
$\varphi$-modules. On the other hand, it would follow from 4.1 that

$\begin{split}V^{*}=&{\rm{Hom}}_{K}(V,K(1))\\\
\simeq&{\rm{Hom}}_{W(\mathbb{F})}(\hat{\text{T}}_{cris}^{*}(\mathcal{M}),W(\mathbb{F})(1))\otimes_{W(\mathbb{F})}K\\\
\simeq&L\otimes_{W(\mathbb{F})}K\simeq V_{f}.\end{split}$

The statement of the proposition follows from this and Theorem 4.1, and all
that remains to be proved is the isomorphism 4.1.

Following [Hat18] $\text{U}_{S}\simeq\text{T}_{cris}^{*}$ where
$\text{T}_{cris}^{*}$ is prescribed by

$\text{T}_{cris}^{*}(M)={\rm{Hom}}_{MF}(M,\varinjlim_{n}\text{A}_{cris}/p^{n}),$

and thus,

$\text{T}_{1}(M)={\rm{Hom}}_{W(\mathbb{F})}(\text{T}_{cris}^{*}(M),K/W(\mathbb{F})(1)).$

We now proceed to show 4.1. We observe that

$\begin{split}&{\rm{Hom}}_{W(\mathbb{F})}(\hat{\text{T}}^{*}_{cris}(\mathcal{M}),W(\mathbb{F})(1))\\\
=&{\rm{Hom}}_{W(\mathbb{F})}({\rm{Hom}}_{MF}(\mathcal{M},\text{A}_{cris}),W(\mathbb{F})(1))\\\
\simeq&{\rm{Hom}}_{W(\mathbb{F})}({\rm{Hom}}_{MF}(\mathcal{M},\text{A}_{cris})\otimes_{W(\mathbb{F})}K/W(\mathbb{F}),K/W(\mathbb{F})(1))\\\
\simeq&{\rm{Hom}}_{W(\mathbb{F})}(\varinjlim_{n}{\rm{Hom}}_{MF}(\mathcal{M},\text{A}_{cris})/p^{n},K/W(\mathbb{F})(1)).\\\
\end{split}$

On the other hand,

$\begin{split}L\simeq&\varprojlim_{m}\text{T}_{1}(\text{T}_{1}^{-1}(L_{m}))\\\
\simeq&\varprojlim_{m}{\rm{Hom}}_{W(\mathbb{F})}({\rm{Hom}}_{MF}(\text{T}_{1}^{-1}(L_{m}),\varinjlim_{n}\text{A}_{cris}/p^{n}),K/W(\mathbb{F})(1))\\\
\simeq&{\rm{Hom}}_{W(\mathbb{F})}(\varinjlim_{m}{\rm{Hom}}_{MF}(\text{T}_{1}^{-1}(L_{m}),\varinjlim_{n}\text{A}_{cris}/p^{n}),K/W(\mathbb{F})(1))\\\
\simeq&{\rm{Hom}}_{W(\mathbb{F})}(\varinjlim_{m}{\rm{Hom}}_{MF}(\mathcal{M}/p^{m},\varinjlim_{n}\text{A}_{cris}/p^{n}),K/W(\mathbb{F})(1))\\\
\simeq&{\rm{Hom}}_{W(\mathbb{F})}(\varinjlim_{m}{\rm{Hom}}_{MF}(\mathcal{M},\text{A}_{cris}/p^{m}),K/W(\mathbb{F})(1)).\end{split}$

The $\text{G}_{\mathbb{Q}_{p}}$-equivariant inclusion

$\varinjlim_{n}{\rm{Hom}}_{MF}(\mathcal{M},\text{A}_{cris})/p^{n}\hookrightarrow\varinjlim_{n}{\rm{Hom}}_{MF}(\mathcal{M},\text{A}_{cris}/p^{n})$

induces a $\text{G}_{\mathbb{Q}_{p}}$-equivariant surjection

$L\otimes_{W(\mathbb{F})}K\twoheadrightarrow{\rm{Hom}}_{W(\mathbb{F})}(\hat{\text{T}}^{*}_{cris}(\mathcal{M}),W(\mathbb{F})(1))\otimes_{W(\mathbb{F})}K\simeq
V^{*}.$

On the other hand, we observe that

$\dim_{K}L\otimes_{W(\mathbb{F})}K=\text{rank}_{W(\mathbb{F})}L=2$

and

$\dim_{K}V^{*}\geq\dim_{K}\text{D}_{cris}(V^{*})=\dim_{K}\mathcal{M}\otimes_{W(\mathbb{F})}K=2,$

and thus this surjection must be an isomorphism. ∎

######  Definition 4.3.

For $R\in\mathcal{C}_{f}$ with maximal ideal $\mathfrak{m}_{R}$ and
$\lambda\in\mathbb{Z}_{\geq 1}$ we define

$\mathcal{F}^{\lambda}(R)\subseteq(\mathcal{F}\times\hat{\mathbb{A}}^{1})(R)=\mathcal{F}(R)\times\mathfrak{m}_{R}$

the set of pairs $(\rho,U)$ such that $\rho$ lifts
$\bar{\rho}_{|G_{\mathbb{Q}_{p}}}$

$\text{tr}\varphi_{\restriction\rho}=p^{\lambda}(1+U).$

That $\mathcal{F}^{\lambda}$ is a functor follows from equation 3.1.

We recall that it is a consequence of a theorem of Ramakrishna that
$\mathcal{F}$ is pro-represented by $\mathcal{R}=W(\mathbb{F})[|X|]$ (see
[FM95, Theorem B2]). The functor
$\mathcal{F}\times_{\text{Spec}W(\mathbb{F})}\hat{\mathbb{A}}^{1}$ is
represented by the fibered product

$\text{Spec}\mathcal{R}\times_{\text{Spec}W(\mathbb{F})}\hat{\mathbb{A}}^{1}\simeq\text{Spec}W(\mathbb{F})[|X,Y|].$

Let $\mathcal{I}$ be an ideal in $\mathcal{R}=W(\mathbb{F})[|X|]$ such that
$\mathcal{R}/\mathcal{I}$ is Artinian. Let $\rho_{\mathcal{R}}$ be the
universal representation representing $\mathcal{F}$ and let
$\rho_{\mathcal{R}/\mathcal{I}}$ be the associated mod $\mathcal{I}$. Let
$\mathcal{I}^{\prime}$ be an ideal which contains $\mathcal{I}$, then by
equation 3.1,

$\text{tr}\varphi_{|\rho_{\mathcal{R}/\mathcal{I}^{\prime}}}=\text{tr}\varphi_{|\rho_{\mathcal{R}/\mathcal{I}}}\mod\mathcal{I^{\prime}},$

let $\Phi(X)\in\mathcal{R}=W(\mathbb{F})[|X|]$ be the limit over ideals
$\mathcal{I}$ for which $\mathcal{R}/\mathcal{I}$ is Artinian

$\Phi(X):=\varprojlim_{\mathcal{I}}\text{tr}\varphi_{|\rho_{\mathcal{R}/\mathcal{I}}}.$

The functor $\mathcal{F}^{\lambda}$ is represented by

$X_{\mathcal{F}^{\lambda}}=\text{Spec}\tilde{\mathcal{R}}^{\lambda}=\text{Spec}\left(\frac{W(\mathbb{F})[|X,Y|]}{(\Phi(X)-p^{\lambda}(1+Y))}\right).$

Let $\mathfrak{m}_{\mathcal{R}}$ be the maximal ideal of
$\mathcal{R}=W(\mathbb{F})[|X|]$, we observe that

$\text{tr}\varphi_{|\bar{\rho}}=\Phi(X)\>\text{mod}\>\mathfrak{m}_{\mathcal{R}}.$

It follows from the theorem of Khare and Wintenberger that
$\bar{\rho}=\bar{\rho}_{g,\mathfrak{q}}$ where $g$ is a cuspidal Hecke
eigenform and $\mathfrak{q}|p$ is a prime in the field of Fourier coefficients
$\mathbb{Q}(g)$. Since $\bar{\rho}_{\restriction G_{\mathbb{Q}_{p}}}$ is
irreducible, $g$ is supersingular at $\mathfrak{q}$, i.e.
$p|\iota_{\mathfrak{q}}(a_{p}(g))$. It follows from Proposition 4.2 the trace
of the residual representation $\text{tr}\varphi_{|\bar{\rho}}=0$ and as a
consequence $\Phi(X)\in\mathfrak{m}_{\mathcal{R}}$. The projection to the
second factor is a natural transformation
$\pi_{2}:\mathcal{F}\times_{\text{Spec}W(\mathbb{F})}\hat{\mathbb{A}}^{1}\rightarrow\hat{\mathbb{A}}^{1}$
and restricts to a natural transformation
$\pi_{2}:\mathcal{F}^{\lambda}\rightarrow\hat{\mathbb{A}}^{1}$ which induces
the map of $W(\mathbb{F})$-algebras

$\pi_{2}^{*}:W(\mathbb{F})[|Y|]\rightarrow\tilde{\mathcal{R}}^{\lambda}=\frac{W(\mathbb{F})[|X,Y|]}{(\Phi(X)-p^{\lambda}(1+Y))}$

for which $\pi_{2}^{*}(Y)=Y\>\text{mod}(\Phi(X)-p^{\lambda}(1+Y))$.

######  Proposition 4.4.

The map $\pi_{2}^{*}$ is an isomorphism of $W(\mathbb{F})$-algebras and thus
$\mathcal{F}^{\lambda}$ is representable by a power series ring in one
variable over $W(\mathbb{F})$.

###### Proof.

It follows from Proposition 4.2 the trace of the residual representation
$\text{tr}\varphi_{|\bar{\rho}}=0$ and as a consequence
$\Phi(X)\in\mathfrak{m}_{\mathcal{R}}$. The map $\pi_{2}^{*}$ is an
isomorphism provided the coefficient of $X$ in the power series expansion of
$\Phi(X)$ is a unit. In greater detail, suppose that the coefficient of $X$ in
the power series expansion of $\Phi(X)$ is a unit, then, $\pi_{2}^{*}$ is
injective since $\Phi(X)\neq 0$. On the other hand since the coefficient of
$X$ is a unit, we may write $X$ as a power series with coefficients in $Y$. In
other words, $X$ will lie in the image of $\pi_{2}^{*}$ and therefore the
$\pi_{2}^{*}$ will be surjective as well.

We proceed to prove that the coefficient of $X$ is a unit. Let
$M_{0}\in\text{MF}_{tor}^{f,[0,2]}$ be such that
$\text{T}_{1}(M_{0})\simeq\bar{\rho}_{|G_{\mathbb{Q}_{p}}}$. Choose a basis
$\\{e,k\\}$ of $M_{0}$ such that $k$ spans $F^{1}M_{0}$. The Fontaine-
Laffaille module $M_{0}$ is characterized by a single matrix
$\left({\begin{array}[]{c|c}a&c\\\ b&d\\\
\end{array}}\right)\in\text{M}_{2}(\mathbb{F})$ where

$\varphi^{0}(e)=ae+bk\text{ and }\varphi^{1}(k)=ce+dk.$

The relation $\varphi^{0}_{\restriction F^{1}M_{0}}=p\varphi^{1}=0$ implies
that $\varphi^{0}(f)=0$. The trace
$a=\text{tr}\varphi_{|\bar{\rho}_{|G_{\mathbb{Q}_{p}}}}=0$ since
$\bar{\rho}_{|G_{\mathbb{Q}_{p}}}$ is supersingular. Let $\tilde{M}_{0}$ be a
Fontaine-Laffaille module of dimension $4$ over $\mathbb{F}$ provided with the
structure of a free $\mathbb{F}[\epsilon]$-module which we proceed to
describe. We let
$\tilde{M}_{0}=\mathbb{F}[\epsilon]\cdot\tilde{e}\oplus\mathbb{F}[\epsilon]\cdot\tilde{k}$
with filtration and $\varphi^{i}$ defined as follows:

  * •

$F^{0}\tilde{M}_{0}:=\tilde{M}_{0}$,
$F^{1}\tilde{M}_{0}:=\mathbb{F}[\epsilon]\cdot\tilde{k}$ and
$F^{2}\tilde{M}_{0}:=0$,

  * •

the maps $\varphi^{j}$ for $j=0,1$ are $\mathbb{F}[\epsilon]$-module maps,
which similar to $M_{0}$ is characterized by a matrix
$\left({\begin{array}[]{c|c}\tilde{a}&\tilde{c}\\\ \tilde{b}&\tilde{d}\\\
\end{array}}\right)\in\text{M}_{2}(\mathbb{F}[\epsilon])$ where

$\varphi^{0}(\tilde{e})=\tilde{a}\cdot\tilde{e}+\tilde{b}\cdot\tilde{k}\text{
and }\varphi^{1}(\tilde{k})=\tilde{c}\cdot\tilde{e}+\tilde{d}\cdot\tilde{k}.$

We further assume that $\tilde{r}$ lifts $r$ for $r\in\\{a,b,c,d\\}$, then the
map $\mathcal{Q}:\tilde{M}_{0}\rightarrow M_{0}$ mapping $\tilde{e}\mapsto e$
and $\tilde{f}\mapsto f$ is a map of Fontaine-Laffaille modules since
$\varphi^{j}\circ\mathcal{Q}=\mathcal{Q}\circ\varphi^{j}$ for $j=0,1$.

It follows from previous discussions that $\text{T}_{1}(\tilde{M}_{0})$ is a
rank $2$ Galois representation $\varrho$ which lifts
$\bar{\rho}_{|G_{\mathbb{Q}_{p}}}$

$\text{G}_{\mathbb{Q}_{p}}$$\text{GL}_{2}(\mathbb{F})$.$\text{GL}_{2}(\mathbb{F}[\epsilon])$$\bar{\rho}_{\restriction\text{G}_{p}}$$\varrho$

We observe that $\text{tr}\varphi_{|\varrho}=\tilde{a}\in\mathbb{F}[\epsilon]$
lifts $a=0$. The element $\tilde{a}$ is thus in fact invariant in the
isomorphism class of $\tilde{M}_{0}$. We simply specify $\tilde{M}_{0}$ by
choosing the matrix $\left({\begin{array}[]{c|c}\tilde{a}&\tilde{c}\\\
\tilde{b}&\tilde{d}\\\ \end{array}}\right)$ such that $\tilde{a}\neq 0$,
clearly such a choice can be made. The Galois representation $\varrho$
coincides with a $W(\mathbb{F})$-algebra homomorphism
$H_{\varrho}:\mathcal{R}\rightarrow\mathbb{F}[\epsilon]$ such that

$H_{\varrho}(\Phi(X))=\Phi(H_{\varrho}(X))=\tilde{a}\neq 0.$

This implies that the coefficient of $X$ in the expansion of $\Phi(X)$ is a
unit from which the statement of the Proposition follows. ∎

##  5\. The Local Deformation Condition at $p$

Throughout this section we fix a choice of $\lambda\in\mathbb{Z}_{\geq 1}$.
Let $\mathcal{C}_{p}^{\lambda}$ be the sub-functor of $\mathcal{F}$ prescribed
by

(5.1)
$\mathcal{C}_{p}^{\lambda}:=\text{image}\\{\mathcal{F}^{\lambda}\rightarrow\mathcal{F}\\},$

and thus

$\mathcal{C}_{p}^{\lambda}(R)=\\{\rho\in\mathcal{F}(R)\mid\text{ there exists
}U\in\mathfrak{m}_{R}\text{ such that
}\text{tr}\varphi_{|\rho}=p^{\lambda}(1+U)\\}.$

We shall prescribe the deformation condition $\mathcal{C}_{p}^{\lambda}$ in
applying the Ramakrishna method of lifting the global representation
$\bar{\rho}$. We observe that if $f=\sum_{n\geq 1}a_{n}q^{n}$ is an eigenform
with field of Fourier coefficients $K_{f}$ and $\mathfrak{p}|p$ is a prime of
$K_{f}$ such that the ramification index $e(\mathfrak{p}|p)=1$, then if the
associated representation $\varrho=\rho_{f,\mathfrak{p}}$ is a characteristic
zero deformation of $\bar{\rho}$

$\text{G}_{\mathbb{Q}}$$\text{GL}_{2}(\mathbb{F})$.$\text{GL}_{2}(W(\mathbb{F}))$$\bar{\rho}$$\varrho$

such that ${\varrho}_{\restriction
G_{\mathbb{Q}_{p}}}\in\mathcal{C}_{p}^{\lambda}$, then it follows that
$\iota_{\mathfrak{p}}(a_{p})=\text{tr}\varphi_{|\varrho}=p^{\lambda}(1+U)$ for
some $U\in\mathbb{Z}_{p}$ for which $p|U$ and consequently, the slope of
$a_{p}$ is $\lambda$.

We shall show that $\bar{\rho}$ lifts to a geometric representation $\rho$, it
follows from the results of Kisin [Kis09] that such a representation $\rho$ is
modular, i.e. there exists an eigenform $f$ and a prime $\mathfrak{p}|p$ in
$K_{f}$ for which $e(\mathfrak{p}|p)=1$ such that
$\rho=\rho_{f,\mathfrak{p}}$. We proceed to provide a brief outline of the
lifting strategy which is an adaptation of that in [Ram02].

There are several steps in the lifting construction:

  1. (1)

we show that the local deformation condition $\mathcal{C}_{p}^{\lambda}$ is
liftable and that there is a one dimensional space
$\mathcal{N}_{p}^{\lambda}\subset H^{1}(G_{p},\text{Ad}^{0}\bar{\rho})$ such
that for any $N>\lambda$, deformation $\varrho$ of
$\bar{\rho}_{|G_{\mathbb{Q}_{p}}}$,

$\text{G}_{\mathbb{Q}_{p}}$$\text{GL}_{2}(\mathbb{F})$.$\text{GL}_{2}(W(\mathbb{F})/p^{N})$$\bar{\rho}_{|G_{\mathbb{Q}_{p}}}$$\varrho$

and $X\in\mathcal{N}_{p}^{\lambda}$, the twist

$(\text{Id}+p^{N-1}X)\varrho\in\mathcal{C}_{p}^{\lambda}(W(\mathbb{F})/p^{N}).$

  2. (2)

At primes $l\in S/\\{p\\}$ we specify local liftable deformation conditions
(see [Ram02]) $\mathcal{C}_{l}$ with tangent spaces $\mathcal{N}_{l}\subseteq
H^{1}(G_{\mathbb{Q}_{l}},Ad^{0}\bar{\rho})$ for which

$\dim\mathcal{N}_{l}=\dim H^{0}(G_{\mathbb{Q}_{l}},\text{Ad}^{0}\bar{\rho}).$

At a prime $l\notin S$ which is a nice prime, we may choose the local
deformation condition for $\bar{\rho}_{|G_{\mathbb{Q}_{l}}}$ denoted by
$\mathcal{C}_{l}$ and a tangent space $\mathcal{N}_{l}$ specified in [Ram02],
we recall that

$\dim\mathcal{N}_{l}=\dim H^{0}(G_{\mathbb{Q}_{l}},\text{Ad}^{0}\bar{\rho}).$

  3. (3)

We show that for a certain finite set of primes $X$ containing $S$, such that
$X/S$ consists of nice primes defined in loc. cit., one may exhibit a lift of
$\bar{\rho}$ to a mod $p^{\lambda+2}$ representation $\rho_{\lambda+2}$

$\text{G}_{\mathbb{Q},X}$$\text{GL}_{2}(\mathbb{F})$.$\text{GL}_{2}(W(\mathbb{F})/p^{\lambda+2})$$\bar{\rho}$$\rho_{\lambda+2}$

such that for $l\in X$, ${\rho_{\lambda+2}}_{\restriction
G_{\mathbb{Q}_{l}}}\in\mathcal{C}_{l}$.

  4. (4)

We lift $\rho_{\lambda+2}$ to a geometric representation one step at a time.
Suppose that $\rho_{m}$ is a lift of $\rho_{\lambda+2}$ for $m\geq\lambda+2$
which

     * •

is unramified at all primes $l\notin X$,

     * •

for $l\in X$, the local representation
${\rho_{m}}_{|G_{\mathbb{Q}_{l}}}\in\mathcal{C}_{l}$,

we lift $\rho_{m}$ to $\rho_{m+1}$ such that $\rho_{m+1}$ satisifes the same
conditions

$\text{G}_{\mathbb{Q}}$$\text{GL}_{2}(W(\mathbb{F})/p^{\lambda+2})$.$\text{GL}_{2}(W(\mathbb{F})/p^{m})$$\text{GL}_{2}(W(\mathbb{F})/p^{m+1})$$\rho_{\lambda+2}$$\rho_{m}$$\rho_{m+1}$

This can be done for any $m$ for an enlarged choice of primes $X$, the
argument goes back to Ramakrishna. The key step involves showing that a set of
nice primes $X/S$ may be chosen so that the associated dual Selmer group

$H^{1}_{\mathcal{N}^{\perp}}(G_{\mathbb{Q},X},\text{Ad}^{0}\bar{\rho}^{*}):=\text{ker}\\{H^{1}(G_{\mathbb{Q},X},\text{Ad}^{0}\bar{\rho}^{*})\xrightarrow{\text{res}}\bigoplus_{l}H^{1}(G_{\mathbb{Q}_{l}},\text{Ad}^{0}\bar{\rho}^{*})/\mathcal{N}_{l}^{\perp}\\}$

is zero. As a result of the local deformation conditions being large enough,
i.e. the dimensions of $\mathcal{N}_{l}$ for $l\in X$ being as specified, it
follows that we are in a balanced setting, i.e. the dimensions of the Selmer
and dual Selmer groups match

$\dim H^{1}_{\mathcal{N}}(G_{\mathbb{Q},X},\text{Ad}^{0}\bar{\rho})=\dim
H^{1}_{\mathcal{N}^{\perp}}(G_{\mathbb{Q},X},\text{Ad}^{0}\bar{\rho}^{*}).$

We show that the Selmer group can be made to vanish on choosing the
appropriate auxiliary primes $X/S$.

To complete the proof of the main theorem only steps 1 and 3 require some
argument, we refer to [Ram02] for a complete description of the steps 2 and 4.
In this section we lay out some of the properties which
$\mathcal{C}_{p}^{\lambda}$ satisfies and fully establish step 1. In the next
section we shall do so for step 3.

######  Definition 5.1.

We say that a sub-functor $\mathcal{F}^{\prime}$ of $\mathcal{F}$ is a
deformation condition if the following conditions 1 to 3 are satisfied, and
that $\mathcal{F}^{\prime}$ is a liftable deformation condition if condition 4
is also satisfied:

  1. (1)

$\mathcal{F}^{\prime}(\mathbb{F})=\bar{\rho}_{\restriction
G_{\mathbb{Q}_{p}}}.$

  2. (2)

Let $R_{1}$ and $R_{2}$ be in $\mathcal{C}$,
$\rho_{1}\in\mathcal{F}^{\prime}(R_{1})$ and
$\rho_{2}\in\mathcal{F}^{\prime}(R_{2})$. 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 $\rho_{1}\oplus\rho_{2}$ the induced $R_{3}$-representation, then
$\rho_{1}\oplus\rho_{2}\in\mathcal{F}^{\prime}(R_{3})$.

  3. (3)

Let $R\in\mathcal{C}$ with maximal ideal $\mathfrak{m}_{R}$. If
$\rho\in\mathcal{F}(R)$ and
$\rho\in\mathcal{F}^{\prime}(R/\mathfrak{m}_{R}^{n})$ for all $n>0$ it follows
that $\rho\in\mathcal{F}^{\prime}(R)$. In other words, the functor
$\mathcal{F}^{\prime}$ is continuous.

  4. (4)

Let $R\in\mathcal{C}$ and $I$ an ideal such that $I.\mathfrak{m}_{R}=0$. Then
for $\rho\in\mathcal{F}^{\prime}(R/I)$ there exists
$\rho^{\prime}\in\mathcal{F}^{\prime}(R)$ such that
$\rho=\rho^{\prime}\;\text{mod}\;I$.

######  Proposition 5.2.

The subfunctor $\mathcal{C}_{p}^{\lambda}$ of $\mathcal{F}$ is a liftable
deformation condition.

###### Proof.

We proceed to check the conditions of Definition 5.1.

Condition 1 requires that
$\mathcal{C}_{p}^{\lambda}(\mathbb{F})=\bar{\rho}_{|G_{\mathbb{Q}_{p}}}$, is
shown by only observing that $\mathcal{C}_{p}^{\lambda}(\mathbb{F})$ is
nonempty. The condition requiring
$\text{tr}\varphi_{|\bar{\rho}}=p^{\lambda}(1+U)$ is clearly satisfied since
$\text{tr}\varphi_{|\bar{\rho}}=0$ and $\lambda\geq 1$ and hence
$p^{\lambda}(1+U)=0$ as well.

We next check condition 2. Let $R_{j}$ and $I_{j}$ for $j=1,2$ and
$\alpha:R_{1}/I_{2}\xrightarrow{\sim}R_{2}/I_{2}$ be as in condition 2.
Suppose that $\rho_{j}\in\mathcal{C}_{p}^{\lambda}(R_{j})$ for $j=1,2$ such
that

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

By assumption there exist $U_{j}\in\mathfrak{m}_{R_{j}}$ for $j=1,2$ such that

$\text{tr}\varphi_{|\rho_{j}}=p^{\lambda}(1+U_{j}).$

Let $\bar{U_{j}}$ denote the reduction of $U_{j}$ modulo
$\text{Ann}_{R_{j}}(p^{\lambda})$ in $R_{j}/\text{Ann}_{R_{j}}(p^{\lambda})$.
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 induced $R_{3}$-representation. We have that
$\text{tr}\varphi_{|\rho_{3}}=p^{\lambda}(1+U_{1},1+U_{2})\in R_{3}$. Since
$p^{\lambda}(1+U_{1},1+U_{2})\in R_{3}$ see that $(\bar{U_{1}},\bar{U_{2}})\in
R_{3}/\text{Ann}_{R_{3}}(p^{\lambda})$ where we identify

$\begin{split}R_{3}/\text{Ann}_{R_{3}}(p^{\lambda})=&\\{(\bar{r_{1}},\bar{r_{2}})\in
R_{1}/\text{Ann}_{R_{1}}(p^{\lambda})\times
R_{2}/\text{Ann}_{R_{2}}(p^{\lambda})\\\
&\mid\alpha(\bar{r_{1}}\>\text{mod}(pR_{1}+I_{1}))=\bar{r_{2}}\>\text{mod}(pR_{2}+I_{2})\\}.\end{split}$

We choose a lift $(\bar{U_{1}},\bar{U_{2}})\in R_{3}/p^{\lambda}R_{3}$ to
$V=(V_{1},V_{2})\in R_{3}$. We observe that for $j=1,2$ since
$V_{j}-U_{j}\in\text{Ann}_{R_{j}}(p^{\lambda})$,

$p^{\lambda}(1+V_{j})=p^{\lambda}(1+U_{j})$

and we conclude that

$\text{tr}\varphi_{|\rho_{3}}=p^{\lambda}(1+V)$

and consequently $\rho_{3}\in\mathcal{C}_{p}^{\lambda}(R_{3})$.

Condition 3 is an immediate consequence of $\mathcal{F}$ being a deformation
condition. By Proposition 4.4 it follows that the functor
$\mathcal{F}^{\lambda}\simeq\hat{\mathbb{A}}^{1}$ and thus it follows that
condition 4 is satisfied. ∎

######  Proposition 5.3.

Let $\lambda\in\mathbb{Z}_{\geq 1}$ and $\mathcal{N}_{p}^{\lambda}\subseteq
H^{1}(G_{\mathbb{Q}_{p}},\text{Ad}^{0}\bar{\rho})$ be the one-dimensional
tangent space to the flat deformation functor (which is independent of
$\lambda$). The space $\mathcal{N}_{p}^{\lambda}$ stabilizes mod $p^{m}$ lifts
of $\bar{\rho}_{\restriction G_{\mathbb{Q}_{p}}}$ in
$\mathcal{C}_{p}^{\lambda}$ for $m>\lambda+1$. In other words, for
$m>\lambda+1$, $X\in\mathcal{N}_{p}^{\lambda}$ and
$\varrho_{m}\in\mathcal{C}_{p}^{\lambda}(W(\mathbb{F})/p^{m})$, the twist

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

###### Proof.

Recall that $\mathcal{N}_{p}^{\lambda}$ is the subspace of
$H^{1}(G_{\mathbb{Q}_{p}},\text{Ad}^{0}\bar{\rho})$ that constitutes the
tangent space to the functor of flat deformations (cf. [Ram02]) and that
$\mathcal{N}_{p}^{\lambda}$ is one-dimensional and is independent of
$\lambda$. The $p$ deformations
$\varrho^{(i)}=(\text{Id}+p^{m-1}X_{i})\varrho_{m}$ as $X_{i}$ ranges over
$\mathcal{N}_{p}^{\lambda}$ as $i=1,\dots,p$ are flat deformations of
$\varrho_{m-1}=\varrho_{m}\mod{p^{m-1}}$. For ease of notation, we denote by
$\varrho^{\prime}:=\varrho_{m}\>\text{mod}\>p^{m-\lambda-1}$ as is depicted
below

$\text{G}_{\mathbb{Q}_{p}}$$\text{GL}_{2}(W(\mathbb{F})/p^{m-\lambda-1})$.$\text{GL}_{2}(W(\mathbb{F})/p^{m})$$\varrho^{\prime}$$\varrho_{m}$

Since $\varrho_{m}\in\mathcal{C}_{p}^{\lambda}$ there exists $U\in
W(\mathbb{F})/p^{m}$ such that
$\text{tr}\varphi_{|\varrho_{m}}=p^{\lambda}(1+U)$. Setting
$U^{\prime}=U\mod{p^{m-\lambda-1}}$ we observe that $(\varrho_{m},U)$ is a
lift of $(\varrho^{\prime},U^{\prime})$.

We choose $U_{1},\dots,U_{p}\in W(\mathbb{F})/p^{m}$ such that $U_{j}\equiv
U\mod{p^{m-\lambda-1}}$ and such that they are all distinct modulo
$p^{m-\lambda}$. Since $U_{j}\equiv U\mod{p^{m-\lambda-1}}$ it follows that

$p^{\lambda}(1+U_{j})\equiv p^{\lambda}(1+U)\mod{p^{m-1}}.$

On the other hand, since $U_{j}$ are distinct modulo $p^{m-\lambda}$ it
follows that $p^{\lambda}(1+U_{j})$ are distinct mod $p^{m}$. Let
$\bar{U_{j}}=U_{j}\mod{p^{m-1}}$, the pair
$(\varrho_{m-1},\bar{U_{j}})\in\mathcal{F}^{\lambda}(W(\mathbb{F})/p^{m-1})$
for $j=1,\dots,p$. The functor
$\mathcal{F}^{\lambda}\simeq\hat{\mathbb{A}}^{1}$ and thus in particular,
there exists a lift
$(\tilde{\varrho}^{(j)},Q_{j})\in\mathcal{F}^{\lambda}(W(\mathbb{F})/p^{m})$
of $(\varrho_{m-1},\bar{U_{j}})$.

We observe that
$\tilde{\varrho}^{(j)}\in\mathcal{C}_{p}^{\lambda}(W(\mathbb{F})/p^{m})$ since
$(\tilde{\varrho}^{(j)},Q_{j})\in\mathcal{F}^{\lambda}(W(\mathbb{F})/p^{m})$
and the set of deformations $\tilde{\varrho}^{(j)}$ are deformations of
$\varrho_{m-1}$ for $j=1,\dots,p$. We show that $\tilde{\varrho}^{(j)}$ are
distinct deformations. The result will follow from this since it shall follow
that

$\\{\varrho^{(j)}\mid j=1,\dots,p\\}=\\{\tilde{\varrho}^{(j)}\mid
j=1,\dots,p\\}\subseteq\mathcal{C}_{p}^{\lambda}(W(\mathbb{F})/p^{m})$

and therefore,
$\varrho^{(j)}\in\mathcal{C}_{p}^{\lambda}(W(\mathbb{F})/p^{m})$. For
$X\in\mathcal{N}_{p}^{\lambda}=\\{X_{1},\dots,X_{p}\\}$ we have that $X=X_{k}$
for some choice of $k$ and the twist
$(\text{Id}+p^{m-1}X)\varrho_{m}=\varrho^{(k)}\in\mathcal{C}_{p}^{\lambda}(W(\mathbb{F})/p^{m})$.

Therefore, to conclude the proof, we proceed to show that for $i\neq j$, the
deformations of $\varrho_{m-1}$ are not strictly equivalent deformations,
$\tilde{\varrho}^{(i)}\not\simeq\tilde{\varrho}^{(j)}$. Since $\lambda\geq 1$
and $Q_{j}\equiv U_{j}\mod{p^{m-1}}$ (they are both lifts of $\bar{U_{j}}$) it
follows that

$\text{tr}\varphi_{|\tilde{\varrho}^{(j)}}=p^{\lambda}(1+Q_{j})=p^{\lambda}(1+U_{j})$

and likewise,

$\text{tr}\varphi_{|\tilde{\varrho}^{(i)}}=p^{\lambda}(1+Q_{i})=p^{\lambda}(1+U_{i}).$

Since $U_{i}\not\equiv U_{j}\mod{p^{m-\lambda}}$ it follows that

$\text{tr}\varphi_{|\tilde{\varrho}^{(i)}}=p^{\lambda}(1+U_{i})\neq
p^{\lambda}(1+U_{j})=\text{tr}\varphi_{|\tilde{\varrho}^{(j)}}$

and therefore, $\tilde{\varrho}^{(i)}\not\simeq\tilde{\varrho}^{(j)}$. The
proof is now complete. ∎

##  6\. Lifting to mod $p^{\lambda+2}$

In this section we show that $\bar{\rho}$ may be lifted a mod $p^{\lambda+2}$
representation $\rho_{\lambda+2}$

$\text{G}_{\mathbb{Q}}$$\text{GL}_{2}(\mathbb{F})$.$\text{GL}_{2}(W(\mathbb{F})/p^{\lambda+2})$$\bar{\rho}$$\rho_{\lambda+2}$

######  Definition 6.1.

A prime $l$ is said to be a nice prime if

  1. (1)

$l\not\equiv\pm 1\mod{p}$,

  2. (2)

$\bar{\rho}$ is unramified at $l$,

  3. (3)

$\bar{\rho}(\sigma_{l})$ has eigenvalues whose ratio equals $l$.

At a nice prime $l$, there is a smooth deformation condition $\mathcal{C}_{l}$
for which we refer to as Ramakrishna deformations. They were introduced in
[Ram02]. A deformation of $\bar{\rho}_{|G_{\mathbb{Q}_{l}}}$ is said to be
nice if it lies in $\mathcal{C}_{l}$. The tangent space of $\mathcal{C}_{l}$
at a nice prime $l$ is denoted by $\mathcal{N}_{l}$.

We enlarge the set of primes $S$ so that the -groups
${}^{1}_{S}(\text{Ad}^{0}\bar{\rho})$ and
${}^{2}_{S}(\text{Ad}^{0}\bar{\rho})$ are both trivial. For further details,
the reader may refer to loc. cit. At every prime $l\in S$ we prescribe a
smooth local deformation condition $\mathcal{C}_{l}$ with tangent space
$\mathcal{N}_{l}$ such that $\dim\mathcal{N}_{l}=\dim
H^{0}(G_{\mathbb{Q}_{l}},\text{Ad}^{0}\bar{\rho})$ as in loc. cit. At the
prime $p$ we let $\mathcal{C}_{p}^{\lambda}$ be the deformation condition 5.1
and $\mathcal{N}_{p}^{\lambda}\subseteq
H^{1}(G_{\mathbb{Q}_{p}},\text{Ad}^{0}\bar{\rho})$ the one-dimensional space
of Proposition 5.3.

Suppose that $m\leq\lambda$ and $\rho_{m}$ is a mod $p^{m}$ lift of
$\bar{\rho}$ which is unramified outside a finite set. We seek to replace
$\rho_{m}$ by a twist $\rho_{m}^{\prime}=(\text{Id}+p^{m-1}Z)\rho_{m}$ by a
global cohomology class $Z$ which does lift to some $\rho_{m+1}$ as depicted

$\text{G}_{\mathbb{Q}}$$\text{GL}_{2}(W(\mathbb{F})/p^{m})$.$\text{GL}_{2}(W(\mathbb{F})/p^{m+1})$$\rho_{m}^{\prime}$$\rho_{m+1}$

Furthermore we seek to do so so that $\rho_{m+1}$ like $\rho_{m}$ is ramified
outside finitely many primes.

Assume that $m\geq 2$. Suppose that $X_{m}$ is a finite set of primes
containing the primes above $p$, the set of primes $S$ and the set of primes
at which $\rho_{m}$ is ramified. Let $Z_{m}$ be a finite set of nice primes
disjoint from $X_{m}$, chosen so that a twist of $\rho_{m}$ lifts to
$\rho_{m+1}$ such that $\rho_{m+1}$ is unramified outside $X_{m+1}:=Z_{m}\cup
X_{m}$. For ease of notation we replace $\rho_{m}$ by this twist,

$\text{G}_{\mathbb{Q},X_{m+1}}$$\text{GL}_{2}(W(\mathbb{F})/p^{m})$.$\text{GL}_{2}(W(\mathbb{F})/p^{m+1})$$\rho_{m}$$\rho_{m+1}$

We may arrive at a collection of nice primes $Z_{m}$ by applying the method of
Khare, Larsen and Ramakrishna from [KLR05].

At each prime $l\in X_{n}$ there is a choice of a local cohomology class
$z_{l}\in H^{1}(G_{\mathbb{Q}_{l}},\text{Ad}^{0}\bar{\rho})$ such that the
twist $(\text{Id}+p^{m-1}z_{l})\rho_{m}\in\mathcal{C}_{l}$. Thus we arrive at
a tuple

$(z_{l})_{l\in X_{m}}\in\bigoplus_{l\in
X_{m}}H^{1}(G_{\mathbb{Q}_{l}},\text{Ad}^{0}\bar{\rho}).$

If $(z_{l})_{l}$ is the image of a global cohomology class $Z\in
H^{1}(G_{\mathbb{Q},X_{m}},\text{Ad}^{0}\bar{\rho})$ then we may simply
replace $\rho_{m}$ by its twist by $Z$,

$(\text{Id}+p^{m-1}Z)\rho_{m}.$

Since the local representations

$(\text{Id}+p^{m-1}Z){\rho_{m}}_{\restriction
G_{\mathbb{Q}_{l}}}\in\mathcal{C}_{l}$

and $\mathcal{C}_{l}$ is a liftable deformation condition, the obstruction to
lifting $\rho_{m}$ lies in ${}^{2}_{X_{m}}(\text{Ad}^{0}\bar{\rho})$ which we
have arranged to be trivial. Therefore, if such a global cohomology class
exists, a twist of $\rho_{m}$ by this global cohomology class lifts to
$\rho_{m+1}$.

We make preparations to complete the argument in the case when there does not
exist such a global cohomology class $Z$.

######  Proposition 6.2.

There exists a finite choice of primes $X_{m+1}$ containing $X_{m}$ such that
on replacing $\rho_{m}$ by its twist w.r.t a cohomology class $f\in
H^{1}(G_{\mathbb{Q},X_{m+1}},\text{Ad}^{0}\bar{\rho})$, the twist
$(\text{Id}+p^{m-1}\rho_{m})$ lifts to $\rho_{m+1}$ which is unramified at all
primes outside $X_{m+1}$. As a consequence, $\bar{\rho}$ lifts to
$\rho_{\lambda+2}$ which is unramified outside a finite set of primes
$X=X_{\lambda+2}$.

###### Proof.

Let $Q$ be a finite set of primes as in Lemma 8 of [KLR05] for which the
restriction map

$H^{1}(G_{\mathbb{Q},X_{m}\cup
Q},\text{Ad}^{0}\bar{\rho})\rightarrow\left(\oplus_{l\in
X_{m}}H^{1}(G_{\mathbb{Q}_{l}},\text{Ad}^{0}\bar{\rho})\right)\oplus\left(\oplus_{q\in
Q}H^{1}_{nr}(G_{\mathbb{Q}_{q}},\text{Ad}^{0}\bar{\rho})\right)$

is an isomorphism. Therefore, there exists a cohomology class $h\in
H^{1}(G_{\mathbb{Q},X_{m}\cup Q},\text{Ad}^{0}\bar{\rho})$ such that
$h_{\restriction G_{\mathbb{Q}_{l}}}=z_{l}$ and such that
$(\text{Id}+p^{m-1}h)\rho_{m}$ is nice at all primes $q\in Q$. By Corollory 11
of [KLR05] there exists a finite set of $\rho_{m}$-nice primes $V$ disjoint
from $X_{m}\cup Q$ such that there exists a cohomology class $g$ such that

  * •

$(\text{Id}+p^{m-1}(g+h))\rho_{m}$ is unramified outside $X_{m}\cup Q\cup V$

  * •

is unobstructed at all primes in $X_{m}\cup Q\cup V$.

We set $X_{m+1}=X_{m}\cup Q\cup V$, since
${}^{2}_{X_{m+1}}(\text{Ad}^{0}\bar{\rho})=0$, and there are no local
obstructions to lifting $\rho_{m}$ the statement of the Proposition follows. ∎

###### Proof.

(of Theorem 1.1)

Let $X$ be a finite set of primes for which $\bar{\rho}$ lifts to
$\rho_{\lambda+2}$ such that the ${\rho_{\lambda+2}}_{\restriction
G_{\mathbb{Q}_{l}}}\in\mathcal{C}_{l}$ for $l\in X$. The Selmer condition
$\\{\mathcal{N}_{l}\\}_{l\in X}$ is balanced, i.e.

$\dim H^{1}_{\mathcal{N}}(G_{\mathbb{Q},X},\text{Ad}^{0}\bar{\rho})=\dim
H^{1}_{\mathcal{N}^{\perp}}(G_{\mathbb{Q},X},\text{Ad}^{0}\bar{\rho}^{*}).$

It is shown for instance in [Ram02] that in this situation, on enlarging $X$
by a finite set of nice primes, the Selmer group and hence the dual Selmer
group

$H^{1}_{\mathcal{N}^{\perp}}(G_{\mathbb{Q},X},\text{Ad}^{0}\bar{\rho}^{*})=0.$

Consequently, from the Poitou Tate long exact sequence, we arrive at an
isomorphism

(6.1) $H^{1}(G_{\mathbb{Q},S\cup
X},\text{Ad}^{0}\bar{\rho})\xrightarrow{\sim}\bigoplus_{l\in
X}H^{1}(G_{\mathbb{Q}_{l}},\text{Ad}^{0}\bar{\rho})/\mathcal{N}_{l}.$

Let $\rho_{m}$ be a lift of $\rho_{\lambda+2}$ for $m\geq\lambda+2$. We lift
$\rho_{m}$ to $\rho_{m+1}$ by twisting $\rho_{m}$ by a global cohomology class
$f\in H^{1}(G_{\mathbb{Q},X},\text{Ad}^{0}\bar{\rho})$. In greater detail, for
$l\in X$, there exists a class $z_{l}+\mathcal{N}_{l}\in
H^{1}(G_{\mathbb{Q}_{l}},\text{Ad}^{0}\bar{\rho})/\mathcal{N}_{l}$ such that
for any representative $z_{l}\in
H^{1}(G_{\mathbb{Q}_{l}},\text{Ad}^{0}\bar{\rho})$ of this class, the twist
$(\text{Id}+p^{m-1}z_{l})({\rho_{m}}_{|G_{\mathbb{Q}_{l}}})\in\mathcal{C}_{l}$.
By the isomorphism 6.1 there is a global cohomology class $Z\in
H^{1}(G_{\mathbb{Q},X},\text{Ad}^{0}\bar{\rho})$ such that for all $l\in X$,
$Z_{\restriction G_{\mathbb{Q}_{l}}}=z_{l}$. We replace $\rho_{m}$ by its
twist by $Z$, we observe that

$\left((\text{Id}+p^{m-1}Z)\rho_{m}\right)_{\restriction
G_{\mathbb{Q}_{l}}}\in\mathcal{C}_{l}$

for all $l\in X$ and hence is unobstructed at all primes $l\in X$. Since
${}^{2}_{X}(\text{Ad}^{0}\bar{\rho})=0$ it follows that $\rho_{m}$ does lift
to $\rho_{m+1}$ where $\rho_{m+1}$ is unramified outside $X$. Once again
replacing $\rho_{m+1}$ by its twist by a suitable global cohomology class, we
may assume further that ${\rho_{m+1}}_{\restriction
G_{\mathbb{Q}_{l}}}\in\mathcal{C}_{l}$ at all primes $l\in X$. The main
theorem therefore follows as $\rho_{\lambda+2}$ therefore lifts to a
characterisitic zero representation $\rho$ for which $\rho_{\restriction
G_{\mathbb{Q}_{p}}}\in\mathcal{C}_{p}^{\lambda}$ and thus in particular,
$\text{tr}\varphi_{|\rho}\in W(\mathbb{F})$ such that
$v_{p}(\text{tr}\varphi_{|\rho})=\lambda$.

The representation $\rho$ satisfies the conditions of Fontaine and Mazur and
thus by [Kis09] there is a supersingular normalized eigenform $f\in
S_{2}(\Gamma_{1}(N))$ and a prime $\mathfrak{p}|p$ of the field of Fourier
coefficients $\mathbb{Q}(f)$ of $f$ for which $e(\mathfrak{p}|p)=1$, such that
the associated Galois representation $\rho_{f,\mathfrak{p}}\simeq\rho$. In
particular, $\iota_{\mathfrak{p}}(a_{p})=\text{tr}\varphi_{|\rho}$ and as a
consequence, $v_{p}(\iota_{\mathfrak{p}}(a_{p}))=\lambda$. Since
$\rho_{f,\mathfrak{p}}$ is flat when restricted to $G_{\mathbb{Q}_{p}}$ it
follows that $(N,p)=1$. Thus the proof of the theorem is complete. ∎

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11institutetext: Max-Planck-Institut für Extraterrestrische Physik (MPE),
Giessenbachstrasse 1, 85748 Garching bei München, Germany 22institutetext:
INAF - Osservatorio Astronomico di Brera, via E. Bianchi 46, 23807 Merate
(LC), Italy 33institutetext: Dipartimento di Matematica e Fisica, Universitá
degli Studi Roma Tre, via della Vasca Navale 84, I-00146 Roma Italy

# disnht: modeling X-ray absorption from distributed column densities

Nicola Locatelli 11 Gabriele Ponti 2211 Stefano Bianchi 33

(Accepted ???. Received ???; in original form ???)

Collecting and analysing X-ray photons over either spatial or temporal scales
encompassing varying optical depth values requires knowledge about the optical
depth distribution. In the case of sufficiently broad optical depth
distribution, assuming a single column density value leads to a misleading
interpretation of the source emission properties, nominally its spectral
model. We present a model description for the interstellar medium absorption
in X-ray spectra at moderate energy resolution, extracted over spatial or
temporal regions encompassing a set of independent column densities. The
absorption model (named disnht) approximates the distribution with a lognormal
one and is presented in table format. The solution table and source code are
made available and can be further generalized or tailored for arbitrary
optical depth distributions encompassed by the extraction region. The disnht
absorption model presented and its generalized solution are expected to be
relevant for present and upcoming large angular scale analyses of diffuse
X-ray emission, such as the ones from the extended ROentgen Survey with an
Imaging Telescope Array (eROSITA) and the future Athena missions.

††offprints: E-mail<EMAIL_ADDRESS>
ISM: abundances; X-rays: diffuse background; X-rays: ISM; Galaxy: halo

##  1 Introduction

Neutral and partially ionized material can efficiently absorb ultra-violet
light and X-rays (Wilms et al., 2000). Absorption is observed as an
exponential decrease in the received intensity of the light $I\propto
e^{-\tau}$, quantified through the optical depth $\tau=\sigma N_{H}$, where
$N_{H}$ is the integral of its neutral hydrogen along the line of sight in
cm-2. The abundances of all the other metals contributing to the absorption in
the intervening medium are also taken into account through the absorption
cross section $\sigma(X(Z),E)$, in cm2, as a function of the photon energy
$E=h\nu$, where $X(Z)$ maps the relative abundance of the element Z with
respect to the hydrogen column density $N_{H}$ (Rybicky et al., 1982). The
interstellar medium (ISM) hosts large amounts of such an absorbing material in
the form of a complex multiphase collection of ions, atoms, molecules and dust
grains diffused all across the thin stellar disk of our Galaxy, encompassing a
wide range of densities ($10^{-4}<n_{H,{\rm ISM}}<10^{6}\,\rm cm^{-3}$) and
temperatures ($T_{\rm ISM}<10^{6}K$) and contributing differently to the X-ray
absorption (e.g. Wolfire et al., 2003; Ferrière, 2001, for a review). When
analysing the light spectrum along one line of sight the absorption can be
easily detected and accounted for, thanks to its known characteristic energy
dependence (namely well known cross sections, Balucinska-Church & McCammon
1992). However, if the light collected into an instrument and summed into a
spectrum encompasses several independent $N_{H}$ elements (in space or time),
an appropriate description of the absorption necessarily requires a
description of the $N_{H}$ distribution of values, as a single column density
value describing the absorption within a given (and satisfactory) accuracy may
simply not exist. This fact follows from the mathematical evidence that a sum
over exponential functions can be approximated by a single exponential
function only in the case of a very peaked distribution of exponents around
one value. An incomplete but instructive list of cases in which this
assumption is broken include the spectral analysis of: faint diffuse emission
for which collection of photons over a large angular scale is performed to
increased the significance of the spectral features (e.g. Muno et al., 2009;
Masui et al., 2009; Yoshino et al., 2009; Gupta et al., 2012; Ponti et al.,
2015; Cheng et al., 2021; Wang et al., 2021); extraction regions holding high
($\sim 10^{24}$ cm-2) and very high ($\sim 10^{25}$ cm-2) absorption such as
toward dense clouds in the Galactic disc, for which collecting the photons
over even a few resolution elements can significantly affect the X-ray
spectrum (e.g. Molinari et al., 2011); pointlike sources holding column
densities variable over the observation timescale, such as for dipping low
mass X-ray binaries binaries (Díaz Trigo et al., 2006; Ponti et al., 2016;
Bianchi et al., 2017), changing-look active galactic nuclei (Matt et al.,
2003; Puccetti et al., 2007; Bianchi et al., 2009; Risaliti et al., 2009;
Marchese et al., 2012; LaMassa et al., 2015) and magnetic accreting white
dwarfs (Done & Magdziarz, 1998). Modeling of the spectral absorption through
distribution of column densities has already successfully performed in the
past using a power law column density distribution (pwab model, presented in
Done & Magdziarz, 1998) and more recently using a lognormal distribution
(Cheng et al., 2021; Wang et al., 2021). These examples show the increasing
need in the X-ray community of careful and systematic treatment of the
absorption when its properties change in a non-negligible way across space or
time. For these and all the other cases mentioned above, in this work we aim
to provide the community with a method and precomputed tables of absorption
factors $-\ln\,A_{\Theta}(E)$ able to properly reproduce the absorption curve.
The model produced will be useful in general to any X-ray spectral analysis
that involves a set of independent column densities broadly distributed around
an expected value. The DIStributed NH Table model (disnht) is written into an
ascii readable table and is also already implemented in .fits format readable
from the X-ray SPECtral fitting software XSPEC as an exponential table model
(etable). All the tables and materials used to create them, including the
source code are publicly available
111http://www.brera.inaf.it/hotmilk/models.html. The code can be used to
compute the absorption spectrum for any arbitrary column density distribution.
In Section 2 we describe the model creation and output table format and usage.
We present different realistic scenarios in Sec. 3 and we draw our conclusions
in Sec. 4.

##  2 Model

For any given set of M column densities $\Theta_{M}(N_{H})$, the exact overall
absorption spectrum can be computed as

$A_{\Theta}=\sum_{i}^{M}\exp\left\\{-\sigma(X(Z),E)\,N_{H,i}\right\\}$ (1)

where the cross section $\sigma$ is derived from Balucinska-Church & McCammon
(1992) and assuming Lodders 2003 as our benchmark model for the metals in the
ISM. Cross sections from different abundance sets commonly used in the XSPEC
software are also provided for completeness (Anders & Grevesse, 1989; Asplund
et al., 2009).

In the lack of an exact model for the absorption by a distribution of column
densities, it is common to simply assume an average value $\langle
N_{H}\rangle$ or to fit for a single value, so that the absorption spectrum is

$\tilde{A}_{\Theta}=\exp\left\\{-\sigma(X(Z),E)\,\langle
N_{H}\rangle\right\\}$ (2)

The discrepancy introduced in $\tilde{A}_{\Theta}$ with respect to the exact
absorption spectrum $A_{\Theta}$ can be significant. We show an example in
Fig. 1, where we accumulate the emission from a sample of absorbed spectra
$p_{m=1...M}$ with $F_{\nu,i}\propto\nu^{-2}$ and fixed (arbitrary)
normalization.

Figure 1:  Power law spectrum $F_{\nu}\propto\nu^{-2}$ absorbed by a
distribution of column densities. The distribution $\Theta$ is drawn from a
lognormal with avg${}_{\Theta}=21.0$ (upper left panel), 22.0 (upper right),
23.0 (lower left) and 24.0 (lower right) dex [cm-2]. Distributions for
different stdΘ are also shown in each panel: std${}_{\Theta}=0.5$ (green
dotted dashed line), 0.3 (red solid), 0.2 (yellow dotted), 0.01 (cyan solid).
The blue dashed line show an absorbed flat spectrum $\tilde{A}_{\Theta}$ with
same normalization, absorbed by $N_{H}=\rm avg_{\Theta}$ in each plot.

We assume the distribution of column density $\Theta_{M}(N_{H})$ to be
lognormal with expected value avg${}_{\Theta}=21,\,22,\,23,\,24$ dex and
standard deviation of std${}_{\Theta}=0.01,\,0.2,\,0.3,\,0.5$ dex. Every
spectrum $p_{m}$ is absorbed by a column density in the set $\Theta_{M}$
opacity equal to the one used by the tbabs model (Wilms et al., 2000). The
absorbed spectra are then summed together. The sum spectrum $A_{\Theta}$ (Eq.
1) is then compared with a single spectrum absorbed by the average value of
$\Theta_{M}$, $\tilde{A}_{\Theta}$ (Eq. 2). We note that assuming a shape for
the column density distribution $\Theta_{M}$ is necessary to make the data fit
to converge. An arbitrary distribution can instead be modeled whenever
precisely know, regardless of its shape. We find that a lognormal distribution
for the column density is a reasonable assumption in real cases where the
discrepancy between the exact and average-value solutions for the absorbed
spectrum becomes relevant (see Sec. 3 for further details). The lognormal
distribution is parametrized by its expected value avgΘ and standard deviation
stdΘ.

The discrepancy between the exact and average column density (blue dashed line
in Fig. 1) solutions can be computed in terms of the ratio
$A_{\Theta}/\tilde{A}_{\Theta}$. In the example shown in the upper left panel
$A_{\Theta}/\tilde{A}_{\Theta}$ is in excess of $1\%$ at $E\leq 1$ keV,
amounting to $10\%$ at 0.3 keV. The discrepancy increases exponentially at
lower energies, so that it is already a few orders of magnitudes at 0.1 keV.
This situation worsens for all values of avg${}_{\Theta}>21.0$ and
std${}_{\Theta}>>0.01$ dex.

We compute and tabulate the correct absorption $A_{\Theta}$ for all values of
avg${}_{\Theta}\in[19.5,25.5]$ and std${}_{\Theta}\in[0,1]$. The table is
multiplicative so that its entries, as a function of energy, are to be
multiplied to any unabsorbed spectrum. The XSPEC table provides instead the
quantity $-\ln A_{\Theta}$ so that, for instance, for $A_{\Theta}=e^{-2}$ the
XSPEC entry provides the value 2. The XSPEC class of this table is an etable
and it also can be used as a multiplicative model when loaded through the
command etable{<TABLE_NAME>}.

##  3 Discussion

Figure 2: Simulated spectrum absorbed by a lognormal distribution of column
densities. The emission model is an APEC at $T=1$ keV and solar abundances
(Lodders, 2003). The mock data are the sum of 100 simulated APEC absorbed by
different column density values of the neutral material extracted from a
lognormal distribution. The solid lines show disnht models with avgΘ values:
23.25 (best-fit) and different stdΘ 0.50 (green), 0.33 (red, best-fit), 0.20
(yellow) and 0.01 (cyan). The blue dashed line show the same emission model
absorbed by a single column density equal to avg${}_{\Theta}=23.25$ (with
best-fit kT=1.5 keV). Residuals $\chi\equiv\rm(data-model)/error$ and ratios
$\rm data/model$ are shown in the central and lower panel respectively.

We demonstrate the efficacy of the disnht model in retrieving the correct
values by simulating a set of 100 lines of sight encompassing a distribution
of column densities $N_{H}$. The $N_{H}$ distribution $\Theta$ is drawn from a
lognormal with mean value avg${}_{\Theta}=23.30$ and standard deviation
std${}_{\Theta}=0.35$. The avgΘ and stdΘ recomputed after the extraction are
23.25 and 0.34 dex respectively. For each $N_{H}$, we generate a mock spectrum
using the fakeit tool of XSPEC. We build the spectrum from a collisionally
ionized plasma model (APEC hereafter, Smith et al. 2001) at a temperature
$T_{{\small APEC}}=1$ keV and ISM metal abundances (taken from Lodders, 2003),
absorbed by neutral material, aiming to emulate the observation of a large
region of diffuse emission from the hot phase of the ISM in the midplane of
the Milky Way. We set the exposure time of each observation to 1 ks. The
instrumental response matrix and ancillary files have been taken from XMM-
Newton observations of the Galactic center region with the EPIC-PN camera
(Ponti et al., 2015). After simulating each spectrum individually, we sum
them. The resulting spectrum is shown in Fig. 2.

comp | param | tbabs | disnht
---|---|---|---
absorption |  $N_{H}$, avgΘ | $3.29\pm 0.06\times 10^{22}$ | $23.27\pm 0.03$
absorption | stdΘ | - | $0.340\pm 0.002$
APEC | $kT$ | $1.77\pm 0.04$ | $1.04\pm 0.02$
APEC | $Z/Z_{\odot}$ | $1.7\pm 0.2$ | $1.13\pm 0.15$
APEC | norm | $8.7\pm 0.7\cdot 10^{-4}$ | $8.1\pm 1.1\cdot 10^{-3}$
| C stat | 1594.4 | 964.33
| d.o.f. ‡ | 976 | 975
Table 1:  Best fit parameters from the spectral fit of the mock data.
$\ddagger$ degrees of freedom.

The best-fit values of the disnht$\times$APEC model are reported in Tab. 1.
Both the simulated distribution of column densities and the emission model are
reproduced by the disnht model best-fit parameters within the statistical
uncertainties, holding a C statistics of $\sim 964$ over 975 degrees of
freedom. The tbabs$\times$ APEC best-fit model instead provides a column
density of the absorbing material (Wilms et al., 2000) of $3.29\times
10^{22}\,\rm cm^{-2}$, about an order of magnitude lower than the mean of the
simulated (log-)distribution. Furthermore, when the disnht model is used, the
retrieved temperature and metallicity of the emission component are consistent
with a $kT=1$ keV collisionally ionized plasma with solar abundances, whereas
by using the tbabs model, the plasma characterization ($kT=1.56\pm 0.03$ keV,
$Z=1.7\pm 0.2\,Z_{\odot}$) significantly differs from the one originally set.
The C statistics held by the best-fit tbabs*APEC model is 1594.4 over 976
degrees of freedom. There is thus an overall improvement of the disnht model
with respect to tbabs, both in terms of describing the absorption and
(consequently) the emission properties of the spectrum, and of the statistical
goodness of the fit, at the expense of an additional degree of freedom.

###  3.1 Lognormal distribution approximation

The disnht model is also already provided and precomputed as an interpolation
between the discrete entries of a two dimensional table. Each dimension of
this table corresponds to one degree of freedom of the model. The physical
distribution of column density in space or time can be arbitrarily complex to
model, in terms of number of parameters required to describe it analytically.
The (log-)normal distribution, if applicable, offers the advantage of
providing physical insight rather than just a mathematical description and
also to require a minimal amount of free parameters: a mean and a dispersion
value. Is a (log-)normal distribution a sufficiently good representation of
the column density distribution? We consider the assumption as sufficiently
good by testing different observed column density distributions against a
lognormal with the same mean, standard deviation and number of elements as the
real distribution, through a Kolmogorov-Smirnov test. The approximation is
valid for rejection probabilities of the null hypothesis (of the two
distributions to have been drawn by the same one) larger than 0.05.

Figure 3: Goodness of the approximation with a lognormal distribution in terms
of Kolmogorov-Smirnov p-value distributions ($\log_{10}$ scale). The vertical
red line shows $p_{\rm KS}=0.05$. The Kolmogorov-Smirnov test is run over
column density sets extracted from circular region of different areas in the
$N_{H}$ all sky maps of HI4PI Collaboration et al. (2016).

We draw real distributions from the all sky $N_{H}$ maps of HI4PI
Collaboration et al. (2016). We consider this map representative of the
physical column density at high Galactic latitudes $b>10\deg$. We build column
density sets extracted over circular regions of different dimension to test
our assumption over different angular scales. For each extraction radius we
draw 1000 sets centered at random positions in the sky at $b>10\deg$. From
every set we obtain a probability of the set to be described by a lognormal
distribution. We reject the hypothesis for p-values smaller than 0.05. The
distributions of p-values obtained over sets at different angular scales are
shown in Fig. 3, for all p-values larger than $10^{-5}$. The original $N_{H}$
map has a resolution of $\simeq 3.4^{\prime}$. Larger angular scales then
encompass a larger number of set elements. For instance, the $\sim
0.1\,(1.0)\,\pi\deg^{2}$ extraction region encompass 30 (300) pixels. As shown
in Fig. 3, the column density distributions are generally consistent with a
lognormal one up to $\sim 1\deg$ radius, with the corresponding distribution
of p-values peaking at $p>0.05$. For larger regions, the real distribution
starts to significantly deviate from lognormal. From a deeper inspection, this
is due in most cases to a $N_{H}$ distribution resembling a sum of a few
lognormal ones corresponding to portions at small scales encompassed within a
single region. The non-negligible width of such distributions makes a solution
for the absorption computed from a single lognormal distribution still
preferable to a correction by a single (average) value of column density.
Similarly, the same test performed over the Galactic center column density map
shows consistency of the lognormal assumption up to a $6^{\prime}$ radius
angular scale, with larger regions deviating from a single lognormal. The
scale that limits the validity of the lognormal approximation has to depend in
general on both the angular power spectrum of the column density fluctuations
(and thus on the medium properties) as well as on the highest angular
resolution used to sample the column density across the sky.

###  3.2 Deviations from single column density absorption models

So far we have demonstrated that: i) there are several cases of significantly
broad column density distributions, produced by averaging over a large enough
sky area; ii) in many of such cases, the distribution of column densities can
be approximated with a lognormal one; iii) such an approximation leads to a
more precise characterization of the absorption spectrum in terms of
statistical power and accuracy of the best-fit values. To further demonstrate
the importance of adopting a more complex model than only a single exponential
function of the column density and also to assess when the latter is still a
good enough approximation to the spectrum, we computed the discrepancy between
the two methods, in terms of the quantities $A_{\Theta}$ and
$\tilde{A}_{\Theta}$ already introduced in Sec. 2. Provided that the limit for
small stdΘ of the disnht model gives the tbabs solution, the fractional
(percent) deviation of $\tilde{A}_{\Theta}$ with respect to $A_{\Theta}$ is

$\left(\frac{A_{\Theta}}{\tilde{A}_{\Theta}}(E)-1\right)\times 100$ (3)

and provides a mean to distinguish when the two models actually describe a
significantly different absorption spectrum and to what extent.

Figure 4: Percent deviation $(A_{\Theta}/\tilde{A}_{\Theta}-1)\times 100$ (log
scale) between the absorption at 0.3 keV produced by a lognormal distribution
of column densities and a single-parameter absorption model (i.e. tbabs
$(N_{H}=\rm avg_{\Theta})$), as a function of the free parameters avgΘ and
stdΘ. Blue and red points show the scatter plot for distributions extracted
respectively from $6^{\prime}\times 6^{\prime}$ random regions in the Galactic
center column density map (Molinari et al., 2011) and $1\deg\times 1\deg$
random region at Galactic latitudes $b>10\deg$ from the HI column densities of
HI4PI Collaboration et al. (2016).

We quantify the fractional deviation at energy 0.3 keV. At lower (higher)
energies the models will deviate more (less). We study the deviation for a
grid of parameters in the ranges avg${}_{\Theta}\in[19.5;23.0]$ using a 0.25
dex step and std${}_{\Theta}\in[0.02;0.22]$ using a 0.02 dex step. The results
are shown in Fig 4. Because of the exponential form, the difference between a
negligible correction and an important correction is very large, even in log
scale. We thus compress the color scale to highlight corrections between 1 and
30% ($+100\%$ corresponding to a factor 2). A $\sim 1\%$ correction may in
fact be statistically significant in a high signal-to-noise spectrum. White
color regions of the parameter space indicate a negligible difference between
the disnht and tbabs models. For such parameters the latter model is
statistically favourable in the absence on priors on the distribution, since
it requires only one (less) free parameter. We then plot in Fig. 4 the points
describing the column density distributions in the random regions extracted
from the maps already considered in this work, for the upper bound angular
scales where the lognormal approximation is verified, namely $1\deg\times
1\deg$ for the HI4PI Collaboration et al. (2016) map (red dots) and
$6^{\prime}\times 6^{\prime}$ for the Molinari et al. (2011) Galactic center
map (blue dots). As we can see from Fig 4, for the majority of the high
Galactic latitude regions the tbabs model provides a good solution to the
absorption. This is due to the column densities of absorbing material being on
average lower than for lines of sight passing through the disk. Furthermore,
the smoother absorbing medium at high Galactic latitudes distributes close
around the mean value. In fact, we can see how the importance of the
correction provided by the disnht model increases toward the upper right
corner of the plot (i.e. for increasing average column density, but also for
increasing scatter around the mean). However, even at high latitude, there are
regions that require a non-negligible correction provided by the disnht model.
The situation is drastically different for Galactic center observations and in
general throughout the Galactic disk. In these regions both the average column
density and the scatter around the mean increase of orders of magnitude. In
these context, the large average value makes the absorption different than the
tbabs model even for narrower distributions and the correction factor is very
large (i.e. $A_{\Theta}/\tilde{A}_{\Theta}>2$) already at 0.3 keV.

##  4 Conclusions

We have presented a model for the absorption of soft X-ray spectra in the case
that photons are collected from different lines of sight holding a broad
distribution of column density values. After showing the efficacy of the model
in retrieving the absorption properties of the spectrum, we proved that
several cases exist for which it is relevant to consider it. In such cases the
assumption of an absorption spectrum described by a single column density
value breaks. We have proven that in such cases the lognormal distribution is
a fair approximation to the physical one. The absorption model proposed is
computed by assuming a constant layer of emission to be absorbed by a
distribution of column densities. This is a rather simple assumption compared
to a general case in which emission from a plasma is expected to deviate from
homogeneity regardless of absorption. However, our goal is to focus on the
modeling of the absorption of the spectrum, rather than on its emission. In
general, whenever the source emission can be considered smooth with respect
the angular scale of the absorption medium fluctuations, the disnht model is
valid and provides a better description of the absorbing medium with respect
to a single-valued column density model. However, in case of a non smooth
underlying emission, a correction for the column density distribution is still
to be preferred to a simple average-value analysis of the absorption, in terms
of result accuracy, for large enough column density distributions. We expect
the proposed model to be relevant for present and future generation of X-ray
imaging telescopes furnished with survey capabilities and large fields of
view, such as eROSITA (Merloni et al., 2012; Predehl et al., 2021) and Athena
(Nandra et al., 2013).

The model is made available either through a python code to compute the
absorption spectrum given an arbitrary column density distribution, or
precomputed under the assumption of a lognormal distribution both in ascii and
.fits tables (etable, XSPEC readable)
formats222http://www.brera.inaf.it/hotmilk/models.html.

## Acknowledgements

We thank our anonymous reviewer for the helpful comments on the manuscript. NL
and GP acknowledge financial support from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program
HotMilk (grant agreement No. [865637]). SB acknowledges financial support from
the Italian Space Agency (grant 2017-12-H.0).

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