# Arithmetic Statistics for Galois deformation rings

Anwesh Ray  Department of Mathematics
University of British Columbia
Vancouver BC, Canada<EMAIL_ADDRESS>and  Tom Weston  Department of
Mathematics, University of Massachusetts, Amherst, MA, USA.
<EMAIL_ADDRESS>
###### Abstract.

Given an elliptic curve $E$ defined over the rational numbers and a prime $p$
at which $E$ has good reduction, we consider the Galois deformation ring
parametrizing lifts of the residual representation on the $p$-torsion group
$E[p]$. For a fixed elliptic curve without complex multiplication, it is shown
that these deformation rings are unobstructed for all but finitely many
primes. For a fixed prime $p$ and varying elliptic curve $E$, we relate the
problem to the question of how often $p$ does not divide the modular degree.
Heuristics due to M.Watkins based on those of Cohen and Lenstra indicate that
this proportion should be $\prod_{i\geq
1}\left(1-\frac{1}{p^{i}}\right)\approx 1-\frac{1}{p}-\frac{1}{p^{2}}$. This
heuristic is supported by computations which indicate that most elliptic
curves (satisfying further conditions) have smooth deformation rings at a
given prime $p\geq 5$, and this proportion comes close to $100\%$ as $p$ gets
larger.

##  1\. Introduction

Given an elliptic curve $E$ defined over the rationals, and a prime number
$p$, denote by $E[p]$ the $p$-torsion subgroup of $E(\bar{\mathbb{Q}})$.
B.Mazur introduced deformation functors, parametrizing lifts of the residual
representation

$\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$

on $E[p]$. These functors are represented by universal deformation rings and
their study gained considerable momentum in [33, 29, 4], where the modularity
of an elliptic curve $E_{/\mathbb{Q}}$ is established. The approach involved
showing that a certain deformation ring associated to $\bar{\rho}$ is in fact
isomorphic to a localized Hecke algebra associated to a space of modular
forms. A result of this flavor which establishes an isomorphism between a
deformation ring and a localized Hecke algebra is known as an
$\operatorname{R}=\mathbb{T}$ theorem. Here, $\operatorname{R}$ refers to the
deformation ring associated to a residual representation $\bar{\rho}$ and
$\mathbb{T}$ is the associated localized Hecke algebra. The Galois
representations which arise from elliptic curves and modular forms satisfy
additional local conditions, and hence, the deformations functors of interest
are subject to a local constraint at $p$, which is defined using $p$-adic
Hodge theory. There has been considerable interest in extending such
$\operatorname{R}=\mathbb{T}$ results to more general Galois representations
arising from cuspidal automorphic representations of higher rank, see for
instance [12, 30, 3]. For instance, a brilliant application of such automorphy
results led to the resolution of the Sato-Tate conjecture, see [2].

In this paper, we shall consider deformations with fixed determinant equal to
the cyclotomic character. Given the importance of deformation rings in proving
modularity results, it is of natural interest to explicitly characterize
Galois deformation rings. Such investigations were carried out by N.Boston and
B.Mazur, see [6, 7, 8]. Presentations for Galois deformation rings with
prescribed local conditions are discussed in the work of G. Böckle [5]. The
deformation ring is presented by generators and relations. When there are no
relations, the ring is smooth. If this is the case, the deformation
ring/functor is said to be unobstructed and it is in fact isomorphic to a
power series ring over $\mathbb{Z}_{p}$. On the other hand, there may be
relations, in which case it’s structure is more complicated. Such relations
arise from local and global obstructions to lifting. The local obstructions
are characterized by local cohomology classes, while in the other hand, the
global obstructions are characterized by global cohomology classes subject to
local constraints (see [5, Theorem 5.2] for further details).

The deformation rings studied in this paper are equipped with a local
condition at $p$, which we shall refer to as geometric deformation rings.
There is one such ring $\mathcal{R}_{E,p}$ associated to each pair $(E,p)$,
where $E$ is a rational elliptic curve and $p$ an odd prime at which $E$ has
good reduction. We study the structure of these rings on average, more
precisely, how often they are unobstructed. Since the determinant is fixed
throughout, the geometric deformation ring is smooth if and only of it is
isomorphic to $\mathbb{Z}_{p}$. In this case, there is a unique
characteristic-zero lift. We study the following questions.

######  Question 1.1.

  1. (1)

For a fixed elliptic curve $E$, how often is the deformation ring
$\mathcal{R}_{E,p}$ unobstructed?

  2. (2)

For a fixed odd prime $p$ and $E$ varying over all rational elliptic curves
with good reduction at $p$, how often is the deformation ring
$\mathcal{R}_{E,p}$ unobstructed?

The first question was raised by Mazur in [23, section 11], and studied by the
second named author in [34]. In loc. cit. one considers a Hecke eigencuspform
$f$ of weight $k\geq 2$ and squarefree level. At each prime $p$, denote by
$\bar{\rho}_{f,p}$ the residual representation at $p$. Then it is shown that
for $100\%$ of the primes $p$, the full deformation ring associated to
$\bar{\rho}_{f,p}$ (with no constraints on the determinant or local
conditions) is unobstructed (and isomorphic to $\mathbb{Z}_{p}\llbracket
X_{1},X_{2},X_{3}\rrbracket$). Furthermore, when $k>2$, the deformation ring
is unobstructed for all but finitely many primes. These results were extended
by J.Hatley to the case when the level is not squarefree, see [19]. When $f$
is the modular form associated to an elliptic curve, the weight $k=2$ and it
is still expected (but not proven except in a few cases; see [27]) that the
full deformation ring is unobstructed for all by finitely many primes. These
results have been generalized to Hilbert modular forms by A.Gamzon [16], and
subsequently to regular algebraic conjugate selfdual cuspidal (RACSDC)
automorphic representations by D.Guiraud [17].

We show that for a fixed elliptic curve $E_{/\mathbb{Q}}$ without complex
multiplication, the geometric deformation ring $\mathcal{R}_{E,p}$ is
isomorphic to $\mathbb{Z}_{p}$ for all but an explicit finite set of primes
$p$, see Theorem 4.6 and Corollary 4.7. Next, we consider the problem where
$p$ is fixed and $E$ varies over elliptic curves ordered according to height.
This is certainly a more ambitious problem, and we are only able to combine
our results with heuristics. We restrict ourselves to elliptic curves with
good ordinary reduction at $p$, squarefree conductor and for which the
residual representation $\bar{\rho}_{E,p}$ on $E[p]$ is irreducible.
Furthermore, we insist that all primes $\ell$ dividing the conductor of $E$
are not $\pm 1\mod{p}$, see Definition 5.1. For this set of elliptic curves,
it is shown that the deformation ring $\mathcal{R}_{E,p}$ is unobstructed
provided the degree of the modular parametrization $X_{0}(N)\rightarrow E$ is
coprime to $p$. This condition has been studied in detail by M.Watkins in
[31]. Cohen-Lenstra heuristics indicate that the probability that $p$ divides
the modular degree of an elliptic curve is given by the product

$1-\prod_{i\geq
1}\left(1-\frac{1}{p^{i}}\right)=\frac{1}{p}+\frac{1}{p^{2}}-\frac{1}{p^{3}}+\dots.$

The heuristics indicate that the proportion of elliptic curves subject to the
above conditions for which $\mathcal{R}_{E,p}$ is not isomorphic to
$\mathbb{Z}_{p}$ becomes smaller as $p$ increases. These heurists are
supported by computation which provide further insight into the problem.

_Organization:_ Including the introduction, the article consists of five
sections. In section 2, we discuss the essential objects of study in the
paper, namely the deformation rings associated to elliptic curves. In section
3, we discuss presentations for deformation rings and establish a criterion
for unobstructedness, in the setting in which there is a local condition at
$p$. In section 4, we study the first of the two aforementioned questions. In
this section, it is shown that given a non-CM elliptic curve, the deformation
rings considered are unobstructed for all but a finite explicit set of primes.
In section 5, the second question is studied, when $p\geq 5$ is a fixed prime
and $E$ varies over a certain collection of elliptic curves. In this section
the question of unobstructedness of the deformation ring is related to the
modular degree. Cohen-Lenstra heuristics are supported by computations in this
section.

### Acknowledgements

AR would like to thank Ravi Ramakrishna for helpful conversations.

##  2\. Deformation theory of Galois representations

Throughout this section, we fix an elliptic curve $E$ defined over the
rationals and a prime number $p\geq 3$ such that the following simplifying
assumptions are in place

  1. (1)

$E$ has good reduction at $p$,

  2. (2)

the residual representation $\bar{\rho}_{E,p}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$
on the $p$-torsion points $E[p]$ is absolutely irreducible.

Fix an algebraic closure $\bar{\mathbb{Q}}$ of $\mathbb{Q}$. For a finite set
of primes $\Sigma$, let $\mathbb{Q}_{\Sigma}\subset\bar{\mathbb{Q}}$ be the
maximal extension of $\mathbb{Q}$ which is unramified at all primes
$v\notin\Sigma$ and $\operatorname{G}_{\Sigma}$ denote the Galois group
$\operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q})$.

###  2.1. Galois deformations

Associated to the residual representation $\bar{\rho}=\bar{\rho}_{E,p}$ are
various Galois deformation rings, which parametrize the Galois representations
which lift $\bar{\rho}$. We recall how these rings are defined and the
properties that characterize them. Throughout, $S$ will denote the set of
primes $\ell$ at which $E$ has bad reduction. Note that by assumption, $S$
does not contain $p$. The global representation gives rise to a local one at
each prime $\ell$. For every prime $\ell$, choose an algebraic closure
$\bar{\mathbb{Q}}_{\ell}$ of $\mathbb{Q}_{\ell}$ and an embedding
$\iota_{\ell}\mathrel{\mathop{\mathchar
58\relax}}\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{\ell}$. Set
$\operatorname{G}_{\ell}$ to denote the absolute Galois group
$\operatorname{Gal}(\bar{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ and note that
$\iota_{\ell}$ induces an inclusion
$\operatorname{G}_{\ell}\hookrightarrow\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$.
With respect to this inclusion, we let $\bar{\rho}_{|\ell}$ be the restriction
of $\bar{\rho}$ to $\operatorname{G}_{\ell}$. We introduce various functors of
Galois deformations associated to the residual representation $\bar{\rho}$,
some of which shall involve conditions on the local representations. First, we
introduce the functor of deformations with no local constraints. Let
$\mathcal{C}$ be the category of commutative complete local noetherian
$\mathbb{Z}_{p}$-algebras with residue field isomorphic to $\mathbb{F}_{p}$.
Such rings are referred to as coefficient rings and are inverse limits of
finite length local rings. A coefficient ring $A$ is equipped with the inverse
limit topology and admits a presentation of the form

$A\simeq\frac{\mathbb{Z}_{p}\llbracket
X_{1},\dots,X_{n}\rrbracket}{(f_{1},\dots,f_{m})},$

where $f_{1},\dots,f_{m}$ are in the maximal ideal of $A$. Here,
$\mathbb{Z}_{p}\llbracket X_{1},\dots,X_{n}\rrbracket$ is the formal power
series ring over $\mathbb{Z}_{p}$ in indeterminates $X_{1},\dots,X_{n}$.
Important examples of such coefficient rings include $\mathbb{Z}_{p}$,
$\mathbb{Z}/p^{N}$, the ring of dual numbers
$\mathbb{F}_{p}[\epsilon]/(\epsilon^{2})$ and the Iwasawa algebra
$\mathbb{Z}_{p}\llbracket T\rrbracket$. A coefficient ring $A$ comes equipped
with a unique residue map $A\rightarrow\mathbb{F}_{p}$ which is a
$\mathbb{Z}_{p}$-algebra homomorphism. A map between coefficient rings is a
$\mathbb{Z}_{p}$-algebra homomorphism which commutes with residue maps. Denote
by $\widehat{\operatorname{GL}_{2}}(A)$ the subgroup of
$\operatorname{GL}_{2}(A)$ consisting of matrices which reduce to the identity
in $\operatorname{GL}_{2}(\mathbb{F}_{p})$. Two continuous $A$-valued
representations

$\rho_{1},\rho_{2}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{S\cup\\{p\\}}\rightarrow\operatorname{GL}_{2}(A)$

lifting $\bar{\rho}$ are strictly equivalent if they are conjugate by some
matrix in $\widehat{\operatorname{GL}_{2}}(A)$.

######  Definition 2.1.

A deformation $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{S\cup\\{p\\}}\rightarrow\operatorname{GL}_{2}(A)$
of $\bar{\rho}$ is a strict equivalence class of continuous lifts of
$\bar{\rho}$.

Let $\chi$ denote the cyclotomic character. Throughout, all deformations
$\rho$ of $\bar{\rho}$ are stipulated to have determinant equal to $\chi$.
When we refer to a Galois deformation $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{S\cup\\{p\\}}\rightarrow\operatorname{GL}_{2}(A)$,
it is to be understood that $\rho$ is an actual representation whose strict
equivalence class is the deformation in question.

###  2.2. Deformation rings

The functor of deformations

$\operatorname{D}_{\bar{\rho}}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{C}\rightarrow\operatorname{Sets}$

sends a ring $A$ to the set of deformations of $\bar{\rho}$ of the form
$\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{S\cup\\{p\\}}\rightarrow\operatorname{GL}_{2}(A)$.
Note that $S\cup\\{p\\}$ is the set of primes at which the deformations are
allowed to be ramified, and this is suppressed in the notation. In fact,
throughout the paper, we will only be considering deformations with minimal
ramification in this sense. Since it is assumed that $\bar{\rho}$ is
irreducible, it follows from [21, Proposition 1] that there is a universal
Galois deformation

$\rho^{\operatorname{univ}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{S\cup\\{p\\}}\rightarrow\operatorname{GL}_{2}(\operatorname{R}_{\bar{\rho}})$

representing $\operatorname{D}_{\bar{\rho}}$. This means that if $A$ is a
coefficient ring and $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{S\cup\\{p\\}}\rightarrow\operatorname{GL}_{2}(A)$
is a Galois deformation of $\bar{\rho}$, then there is a unique map of
coefficient rings $\operatorname{R}_{\bar{\rho}}\rightarrow A$ such that the
following diagram commutes

$\operatorname{G}_{S\cup\\{p\\}}$$\operatorname{GL}_{2}(A)$.$\operatorname{GL}_{2}(\operatorname{R}_{\bar{\rho}})$$\rho$$\rho^{\operatorname{univ}}$

Here, $\operatorname{R}_{\bar{\rho}}$ is referred to as the universal
deformation ring associated to $\operatorname{D}_{\bar{\rho}}$.

Let $\operatorname{I}_{p}$ be the inertia group at the prime $p$. A character
$\psi\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{1}(\mathbb{Z}_{p})$
is geometric if
$\operatorname{\psi}_{\restriction\operatorname{I}_{p}}=\chi^{k-1}_{\restriction\operatorname{I}_{p}}$,
for $k\in\mathbb{Z}_{\geq 2}$. Fontaine and Mazur introduce the notion of a
geometric Galois representation.

######  Definition 2.2.

Let $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(\bar{\mathbb{Q}}_{p})$
be a continuous Galois representation, $\rho$ is said to be geometric if it
satisfies the following conditions:

  1. (1)

$\rho$ is irreducible,

  2. (2)

$\operatorname{det}\rho$ is an odd geometric character,

  3. (3)

$\rho$ is unramified away from a finite set of primes,

  4. (4)

the local representation $\rho_{\restriction p}$ is deRham (see [9, section 6]
for the definition and basic properties of deRham representations).

When $\rho$ arises from a Hecke eigencuspform, it is necessarily geometric.
Fontaine and Mazur conjectured that $2$-dimensional geometric Galois
representations arise from Hecke eigencuspforms. This conjecture has been
settled in all but one exceptional case (see [32] and [20]). One is primarily
interested in Galois representations that are geometric in the above sense.
Throughout, we shall impose an additional deformation condition at $p$, which
will guarantee that the deformation rings parametrize geometric lifts. We will
further elaborate on this in the next subsection.

Note that if $\rho$ is a deformation of $\bar{\rho}$ then
$\rho_{\restriction\ell}$ is a deformation of $\bar{\rho}_{\restriction\ell}$.
Therefore, a global deformation gives rise to a local deformation at each
prime $\ell$. At each prime $\ell$, set $\operatorname{Def}_{\ell}(A)$ to be
the set of $A$-deformations of $\bar{\rho}_{\restriction\ell}$. The
association $A\mapsto\operatorname{Def}_{\ell}(A)$ defines a functor
$\operatorname{Def}_{\ell}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{C}\rightarrow\operatorname{Sets}$. Let $A$ be a coefficient
ring with maximal ideal $\mathfrak{m}$. Let $I$ be an ideal in $A$, the
quotient map $A\rightarrow A/I$ is said to be a small extension if
$\mathfrak{m}$ is principal and $I\cdot\mathfrak{m}=0$.

######  Definition 2.3.

A functor of deformations $\mathcal{C}_{\ell}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{C}\rightarrow\operatorname{Sets}$ is a subfunctor of
$\operatorname{Def}_{\ell}$. Say that $\mathcal{C}_{\ell}$ is deformation
condition if the conditions $\eqref{dc1}-\eqref{dc3}$ stated below are
satisfied:

  1. (1)

$\mathcal{C}_{\ell}(\mathbb{F}_{p})=\\{\bar{\rho}_{\restriction\ell}\\}.$

  2. (2)

For $i=1,2$, let $R_{i}\in\mathcal{C}$ and
$\rho_{i}\in\mathcal{C}_{\ell}(R_{i})$. Let $I_{1}$ be an ideal in $R_{1}$ and
$I_{2}$ an ideal in $R_{2}$ such that there is an isomorphism
$\alpha\mathrel{\mathop{\mathchar
58\relax}}R_{1}/I_{1}\xrightarrow{\sim}R_{2}/I_{2}$ 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 $\rho_{1}\times_{\alpha}\rho_{2}$ the induced $R_{3}$-representation, then
$\rho_{1}\times_{\alpha}\rho_{2}\in\mathcal{C}_{\ell}(R_{3})$.

  3. (3)

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

A deformation functor $\mathcal{C}_{\ell}$ is said to be liftable if for every
small extension $A\rightarrow A/I$ the induced map
$\mathcal{C}_{\ell}(A)\rightarrow\mathcal{C}_{\ell}(A/I)$ is surjective.

Condition $\eqref{dc2}$ is referred to as the Mayer-Vietoris property. By a
well-known result of Grothendieck [22, section 18], conditions $\eqref{dc1}$
to $\eqref{dc3}$ guarantee that $\mathcal{C}_{\ell}$ is representable.

######  Definition 2.4.

Set $\operatorname{Ad}^{0}\bar{\rho}$ to denote the Galois module whose
underlying vector space consists of $2\times 2$ matrices with entries in
$\mathbb{F}_{p}$, and with trace zero. The Galois action is as follows: for
$g\in\operatorname{G}_{\mathbb{Q}}$ and $v\in\operatorname{Ad}^{0}\bar{\rho}$,
set $g\cdot v\mathrel{\mathop{\mathchar
58\relax}}=\bar{\rho}(g)v\bar{\rho}(g)^{-1}$.

Note that $\operatorname{Ad}^{0}\bar{\rho}^{*}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Hom}(\operatorname{Ad}^{0}\bar{\rho},\mu_{p})$ can be
identified with $\operatorname{Ad}^{0}\bar{\rho}(1)$, i.e.,
$\operatorname{Ad}^{0}\bar{\rho}$ twisted by the action of the mod-$p$
cyclotomic character $\bar{\chi}$. Let $\ell$ be a prime number, the functor
of local deformations $\operatorname{Def}_{\ell}$ is equipped with a tangent
space. As a set, this is defined to be
$\operatorname{Def}_{\ell}\left(\mathbb{F}_{p}[\epsilon]/(\epsilon^{2})\right)$.
It has the structure of a vector space over $\mathbb{F}_{p}$ and is in
bijection with
$H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})$. The
bijection identifies a cohomology class $f$ with the deformation

$(\operatorname{Id}+\epsilon
f)\bar{\rho}_{\restriction\ell}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\ell}\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p}[\epsilon]/(\epsilon^{2})).$

The tangent space $\mathcal{N}_{\ell}$ of a deformation condition
$\mathcal{C}_{\ell}$ consists of the cohomology classes $f\in
H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})$, such that
$(\operatorname{Id}+\epsilon
f)\bar{\rho}_{\restriction\ell}\in\mathcal{C}_{\ell}\left(\mathbb{F}_{p}[\epsilon]/(\epsilon^{2})\right)$.
For $\ell\in S$, the deformation functor $\operatorname{Def}_{\ell}$ is
unobstructed if it is liftable in the sense of Definition 2.3. The following
is a criterion for local unobstructedness.

######  Lemma 2.5.

The functor $\operatorname{Def}_{\ell}$ is unobstructed if
$H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*})=0$.

###### Proof.

By the local Euler characteristic formula,
$H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*})$ is dual
to $H^{2}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})$. Let
$A\rightarrow A/I$ be a small extension and $t$ a generator of the principal
ideal $\mathfrak{m}_{A}$. Identify $\operatorname{Ad}^{0}\bar{\rho}$ with the
kernel of the mod-$I$ reduction map

$\operatorname{SL}_{2}(A)\rightarrow\operatorname{SL}_{2}(A/I)$

by identifying $\left({\begin{array}[]{cc}a&b\\\ c&-a\\\ \end{array}}\right)$
with

$\operatorname{Id}+t\left({\begin{array}[]{cc}a&b\\\ c&-a\\\
\end{array}}\right)=\left({\begin{array}[]{cc}1+ta&tb\\\ tc&1-ta\\\
\end{array}}\right).$

We show that

$\varrho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\ell}\rightarrow\operatorname{GL}_{2}(A/I)$

lifts to

$\tilde{\varrho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\ell}\rightarrow\operatorname{GL}_{2}(A).$

Let $\tau\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\ell}\rightarrow\operatorname{GL}_{2}(A)$ be a
continuous lift of $\varrho$ (not necessarily a homomorphism), for which
$\det\tau=\chi$. Such a lift does always exist. Note that for
$g_{1},g_{2}\in\operatorname{G}_{\mathbb{Q}}$,

$\mathcal{O}(\varrho)(g_{1},g_{2})\mathrel{\mathop{\mathchar
58\relax}}=\tau(g_{1}g_{2})\tau(g_{2})^{-1}\tau(g_{1})^{-1}$

is an element of

$\operatorname{Ad}^{0}\bar{\rho}=\operatorname{ker}\left\\{\operatorname{SL}_{2}(A)\rightarrow\operatorname{SL}_{2}(A/I)\right\\}.$

This defines a cohomology class $\mathcal{O}(\varrho)\in
H^{2}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})$, which is
trivial precisely when $\varrho$ lifts to an actual representation

$\tilde{\varrho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\ell}\rightarrow\operatorname{GL}_{2}(A).$

Since $H^{2}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})=0$, a
lift $\tilde{\varrho}$ must exist and the result follows. ∎

###  2.3. The flat deformation condition

The deformation rings we consider shall parametrize geometric Galois
representations $\rho$ lifting $\bar{\rho}$ for which
$\operatorname{det}\rho=\chi$. This is ensured by choosing an appropriate
local deformation condition $\mathcal{C}_{p}$ for $\bar{\rho}_{|p}$. A
deformation $\varrho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{p}\rightarrow\operatorname{GL}_{2}(A)$ of
$\bar{\rho}_{|p}$ is said to be flat if for each finite length quotient $A/I$
of $A$, the mod-$I$ representation $\varrho_{I}\mathrel{\mathop{\mathchar
58\relax}}=\varrho\mod{I}$ arises from the Galois action on the generic fibre
of a a finite flat group scheme over $\mathbb{Z}_{p}$. Such flat deformations
are characterized by certain filtered Dieudonné modules.

######  Definition 2.6.

A filtered Dieudonné module $M$ is a $\mathbb{Z}_{p}$-module furnished with a
decreasing, exhaustive, separated filtration $(M^{i})$ of
sub-$\mathbb{Z}_{p}$-modules. For each integer $i$, we are given a
$\mathbb{Z}_{p}$-linear map $\varphi^{i}\mathrel{\mathop{\mathchar
58\relax}}M^{i}\rightarrow M$ such that, for $x\in M^{i+1}$,
$\varphi^{i+1}(x)=p\varphi^{i}(x)$. Denote by
$\operatorname{MF}_{\operatorname{tor}}^{f}$ the category of filtered
Dieudonné-modules $M$ that are of finite length and for which
$\sum_{i}\varphi^{i}(M)=M$. For $j>0$, set
$\operatorname{MF}_{\operatorname{tor}}^{f,j}$ to be the full subcategory of
$\operatorname{MF}_{\operatorname{tor}}^{f}$ consisting of modules $M$ for
which $M^{0}=M$ and $M^{j}=0$.

Let $\operatorname{Rep}_{\mathbb{Z}_{p}}^{f}$ be the category of finite length
$\mathbb{Z}_{p}[\operatorname{G}_{p}]$-modules. For $j<p$, Fontaine and
Laffaille show in [18] that there is a faithful, exact contravariant functor

$\operatorname{U}_{S}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{MF}_{\operatorname{tor}}^{f,j}\rightarrow\operatorname{Rep}_{\mathbb{Z}_{p}}^{f}.$

Since we fix the determinant of our deformations to be equal to $\chi$, we
need only focus our attention to the category
$\operatorname{MF}_{\operatorname{tor}}^{f,2}$. Fontaine and Laffaille show
that $\operatorname{U}_{S}$ induces an anti-equivalence between this category
and the category of $\operatorname{G}_{p}$-modules arising from finite flat
group schemes over $\mathbb{Z}_{p}$. Since the elliptic curve $E$ is assumed
to have good reduction at $p$, the residual representation $\bar{\rho}_{|p}$
is flat, i.e., arises from a finite flat group scheme over $\mathbb{Z}_{p}$.
The result of Fontaine and Laffaille implies that there exists a uniquely
determined module $M_{0}\in\operatorname{MF}_{\operatorname{tor}}^{f,2}$, such
that $\operatorname{U}_{S}(M_{0})=V_{\bar{\rho}}$, where $V_{\bar{\rho}}$ is
the underlying $\mathbb{F}_{p}$-vector space on which $\operatorname{G}_{p}$
acts via $\bar{\rho}_{|p}$. Let $R\in\mathcal{C}$, and $R/I$ be a finite
length quotient of $R$. The mod-$I$ reduction of
$\varrho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{p}\rightarrow\operatorname{GL}_{2}(R)$ is denoted
$\varrho_{I}$ and its underlying module is denoted $V_{\varrho_{I}}$. We take
note of this result.

######  Theorem 2.7 (Fontaine-Laffaille).

Let $R\in\mathcal{C}$, a deformation $\varrho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{p}\rightarrow\operatorname{GL}_{2}(R)$ is flat if
for each finite length quotient $R/I$, there is a Fontaine Laffaille module
$M_{\varrho_{I}}\in\operatorname{MF}_{\operatorname{tor}}^{f,2}$, such that
$\operatorname{U}_{S}(M_{\varrho_{I}})=V_{\varrho_{I}}$.

###### Proof.

Please refer to [18, section 9], where the result is proved. ∎

Let $\operatorname{F}_{2}$ be the subfunctor of $\operatorname{Def}_{p}$
consisting of flat deformations. Ramakrishna in [25] showed that under certain
hypotheses, $\operatorname{F}_{2}$ is representable by a power series ring
over $\mathbb{Z}_{p}$. Let us recall these results. Set
$\operatorname{End}_{p}(\bar{\rho})$ to denote the ring of
$\operatorname{G}_{p}$-module endomorphisms of $V_{\bar{\rho}}$.

######  Theorem 2.8 (Ramakrishna).

Assume that $\operatorname{End}_{p}(\bar{\rho})=\mathbb{F}_{p}$, then,
$\operatorname{F}_{2}$ is pro-representable by a smooth ring
$R_{2}\simeq\mathbb{Z}_{p}\llbracket X\rrbracket$.

###### Proof.

The result of Ramakrishna is proved in the setting where the determinants of
the deformations $\varrho$ of $\bar{\rho}_{|p}$ are not fixed. Since $\varrho$
is flat, $\operatorname{det}\varrho_{|\operatorname{I}_{p}}=\chi$. Note
however, that we fix the entire determinant $\operatorname{det}\varrho=\chi$.
Ramakrishna shows that the flat deformation functor without fixed determinant
is representable by $\mathbb{Z}_{p}\llbracket X_{1},X_{2}\rrbracket$. This is
proven by showing that the full-adjoint tangent space corresponding to flat
deformations contained in
$H^{1}(\operatorname{G}_{p},\operatorname{Ad}\bar{\rho})$ is $2$-dimensional,
and hence, the ring is of the form $\mathbb{Z}_{p}\llbracket
X_{1},X_{2}\rrbracket/\mathcal{I}$. It is also shown that this functor is
representable by a smooth ring over $\mathbb{Z}_{p}$, hence, the ideal
$\mathcal{I}=0$. The same argument in the case when the determinant is fixed,
shows that when $\det=\chi$ is fixed, the functor is represented by
$\mathbb{Z}_{p}\llbracket X\rrbracket/\mathcal{J}$, and since the functor is
liftable, it must be isomorphic to $\mathbb{Z}_{p}\llbracket X\rrbracket$. ∎

When the condition $\operatorname{End}_{p}(\bar{\rho})=\mathbb{F}_{p}$ is
satisfied, set $\mathcal{C}_{p}$ to denote the flat deformation functor with
$\operatorname{det}=\chi$. Let $\mathcal{N}_{p}\subseteq
H^{1}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho})$ denote its
tangent space, note that by the above result, $\mathcal{N}_{p}$ is
$1$-dimensional. Also note that in this setting when
$\operatorname{End}_{p}(\bar{\rho})=\mathbb{F}_{p}$, it follows that
$H^{0}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho})=0$. Therefore,
the formula

$\dim\mathcal{N}_{p}=1+\dim
H^{0}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho})$

is automatically satisfied. When $E$ has supersingular reduction at $p$, the
condition $\operatorname{End}_{p}(\bar{\rho})=\mathbb{F}_{p}$ follows from
[28]. On the other hand, when $E$ is ordinary at $p$, there is an unramified
character $\psi$ such that
$\bar{\rho}_{|p}=\left({\begin{array}[]{cc}\psi\bar{\chi}&\ast\\\
0&\psi^{-1}\\\ \end{array}}\right)$. In this setting,
$\operatorname{End}_{p}(\bar{\rho})=\mathbb{F}_{p}$ if and only if the
representation $\bar{\rho}_{|p}$ is non-split. Therefore, consider the case
when $\bar{\rho}_{|p}$ is split. In this case, Ramakrishna shows in [26, Table
2, p. 128] that there is a pair $(\mathcal{C}_{p},\mathcal{N}_{p})$, where
$\mathcal{C}_{p}$ is a liftable deformation functor with determinant equal to
$\chi$ and $\mathcal{N}_{p}$ is the tangent space of $\mathcal{C}_{p}$.
Moreover, these deformations are all ordinary, and are defined as follows.

  * •

When $\bar{\rho}_{|p}$ is twist equivalent to
$\left({\begin{array}[]{cc}\bar{\chi}&0\\\ 0&1\\\ \end{array}}\right)$, it is
shown that there is subfunctor $\mathcal{C}_{p}$ of $\operatorname{Def}_{p}$
consisting of flat deformations represented by $\mathbb{Z}_{p}\llbracket
X_{1},X_{2}\rrbracket$.

  * •

When $\bar{\rho}_{|p}$ is not twist equivalent to
$\left({\begin{array}[]{cc}\bar{\chi}&0\\\ 0&1\\\ \end{array}}\right)$, it is
shown that there is subfunctor $\mathcal{C}_{p}$ of $\operatorname{Def}_{p}$
consisting of ordinary deformations represented by $\mathbb{Z}_{p}\llbracket
X_{1},X_{2}\rrbracket$. In this setting, we check that all infinitesimal lifts
$\mathcal{C}_{p}\left(\mathbb{F}_{p}[\epsilon]/(\epsilon^{2})\right)$ are
flat.

######  Proposition 2.9.

Let $E$ be an elliptic curve with good reduction at $p\geq 5$, and
$\bar{\rho}$ the residual representation. Then, $\bar{\rho}_{|p}$ comes
equipped with a liftable deformation condition $\mathcal{C}_{p}$ along with
tangent space $\mathcal{N}_{p}$, such that

  1. (1)

all deformations $\varrho$ satisfying $\mathcal{C}_{p}$ are ordinary (resp.
crystalline) if $E$ has ordinary (resp. supersingular) reduction at $p$,

  2. (2)

$\dim\mathcal{N}_{p}=1+\dim
H^{0}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho})$,

  3. (3)

the deformations
$\varrho\in\mathcal{C}_{p}\left(\mathbb{F}_{p}[\epsilon]/(\epsilon^{2})\right)$
are flat, i.e., arise via $\operatorname{U}_{S}$.

###### Proof.

The above discussion shows that the only case that needs to be considered is
when $\bar{\rho}_{|p}=\left({\begin{array}[]{cc}\psi\bar{\chi}&0\\\
0&\psi^{-1}\\\ \end{array}}\right)$ for an unramified character $\psi$. Let
$M_{0}$ be the Fontaine-Laffaille module associated to $V_{\bar{\rho}}$. Then
$M_{0}$ is a direct sum $M_{0}=N_{1}\oplus N_{2}$, where $N_{1}$ and $N_{2}$
are $1$-dimensional Fontaine-Laffaille modules corresponding to the Galois
modules $\mathbb{F}_{p}(\psi\bar{\chi})$ and $\mathbb{F}_{p}(\psi^{-1})$
respectively. 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
dashed matrix

$\left({\begin{array}[]{c|c}a&c\\\ b&d\\\
\end{array}}\right)\in\text{M}_{2}(\mathbb{F}),$

where $\varphi^{0}(e)=ae+bk$ and $\varphi^{1}(k)=ce+dk$. The relation
$\varphi^{0}_{\restriction M_{0}^{1}}=p\varphi^{1}=0$ implies that
$\varphi^{0}(k)=0$. Therefore, the matrices corresponding to $\varphi^{0}$ and
$\varphi^{1}$ are as follows

$\varphi^{0}=\left({\begin{array}[]{cc}a&0\\\ b&0\\\
\end{array}}\right)\text{, and }\varphi^{1}=\left({\begin{array}[]{c}c\\\ d\\\
\end{array}}\right).$

On the other hand, according to loc. cit., $\mathcal{C}_{p}$ is the ordinary
deformation condition and $\mathcal{N}_{p}$ is the ordinary tangent space and
has dimension $2$. Explicitly, $\mathcal{N}_{p}\subset
H^{1}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho})$ has a basis of
two cohomology classes $f_{1}$ and $f_{2}$, where $f_{1}$ is ramified and has
image in $\left({\begin{array}[]{cc}0&\ast\\\ 0&0\\\ \end{array}}\right)$, and
on the other hand, $f_{2}$ is unramified and has image in the diagonal in
$\operatorname{Ad}^{0}\bar{\rho}$. Both cohomology classes are seen to arise
from Fontaine Laffaille modules $\widetilde{M}_{0}$ which fit into a short
exact sequence

$0\rightarrow M_{0}\rightarrow\widetilde{M}_{0}\rightarrow M_{0}\rightarrow
0.$

The definition of $\widetilde{M}_{0}$ involves a choice of a dashed matrix
with entries in $\mathbb{F}_{p}[\epsilon]/\epsilon^{2}$ lifting that
corresponding $M_{0}$. We leave the details of this simple matrix calculation
to the reader. ∎

##  3\. Presentations for Geometric Deformation Rings

Assume throughout that $p\geq 5$ and that

$\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{S\cup\\{p\\}}\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$

is as in the previous section. In this section, we consider Galois
representations which are ordinary/crystalline when localized at $p$, and thus
in particular, are deRham. Such representations will be geometric and under a
mild additional hypothesis, are known to arise from Hecke eigencuspforms.
Following Böckle [5], we discuss presentations for the associated Galois
deformation rings. We do not impose any local conditions at the primes
$\ell\in S$. However, at $p$, we impose the condition $\mathcal{C}_{p}$
defined in the previous section. Recall that throughout, the determinant of
all deformations considered is fixed and equal to $\chi$, the cyclotomic
character.

We introduce an additional hypothesis which will play a role in simplifying
some results in arithmetic statistics, however, the results in this section
shall not assume this hypothesis.

######  Hypothesis 3.1.

We say that condition $(\star)$ is satisfied for $\bar{\rho}$ if for all
$\ell\in S$,

$H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*})=0.$

We introduce Selmer and dual Selmer groups which will play an important role
in describing presentations for Galois deformation rings. First, we introduce
the notion of Selmer data for the set of primes $S\cup\\{p\\}$. A Selmer datum
$\mathcal{L}$ consists of a choice of subspace $\mathcal{L}_{\ell}\subseteq
H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})$ for each prime
$\ell\in S\cup\\{p\\}$; we set $\mathcal{L}_{\infty}=0$. Let
$\mathcal{L}_{\ell}^{\perp}\subset
H^{1}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho}^{*})$ be the
orthogonal complement of $\mathcal{L}_{\ell}$ with respect to the local Tate-
duality pairing. Then, the Selmer and dual Selmer groups associated to the
datum $\mathcal{L}$ are defined as follows

(3.1)
$\begin{split}&H^{1}_{\mathcal{L}}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})\mathrel{\mathop{\mathchar
58\relax}}=\text{ker}\left\\{H^{1}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})\longrightarrow\bigoplus_{\ell\in
S\cup\\{p\\}}\frac{H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})}{\mathcal{L}_{\ell}}\right\\}\\\
&H^{1}_{\mathcal{L}^{\perp}}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})\mathrel{\mathop{\mathchar
58\relax}}=\text{ker}\left\\{H^{1}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})\longrightarrow\bigoplus_{\ell\in
S\cup\\{p\\}}\frac{H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*})}{\mathcal{L}_{\ell}^{\perp}}\right\\}\\\
\end{split}$

respectively.

In this paper, we work in a special case, where $\mathcal{L}_{\ell}$ is the
full space $H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})$
for $\ell\in S$ and $\mathcal{L}_{p}=\mathcal{N}_{p}$. The following notation
we use is not standard and there are no conditions at the primes $\ell\in S$.
The Selmer and dual Selmer groups are as follows

(3.2) $\begin{split}&H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})\mathrel{\mathop{\mathchar
58\relax}}=\text{ker}\left\\{H^{1}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})\longrightarrow\frac{H^{1}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho})}{\mathcal{N}_{p}}\right\\}\\\
&H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})\mathrel{\mathop{\mathchar
58\relax}}=\text{ker}\left\\{H^{1}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})\longrightarrow\frac{H^{1}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho}^{*})}{\mathcal{N}_{p}^{\perp}}\right\\}\\\
\end{split}$

respectively.

######  Proposition 3.2.

With respect to notation introduced above, we have that

$\dim H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})-\dim
H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})=\sum_{\ell\in
S}H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*}).$

In particular, $(\star)$ is satisfied if and only if

$\dim H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})=\dim
H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*}).$

###### Proof.

Suppose for each prime $\ell\in S\cup\\{p,\infty\\}$, there is a choice of
subspace $\mathcal{L}_{\ell}\subseteq
H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})$. The Selmer
condition $\mathcal{L}$ is the data $\\{\mathcal{L}_{\ell}\\}$ and the
dimensions of the Selmer and dual Selmer groups are related as follows

$\begin{split}&\dim
H^{1}_{\mathcal{L}}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})-\dim
H^{1}_{\mathcal{L}^{\perp}}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})\\\
=&\dim
H^{0}(\operatorname{G}_{\mathbb{Q}},\operatorname{Ad}^{0}\bar{\rho})-\dim
H^{0}(\operatorname{G}_{\mathbb{Q}},\operatorname{Ad}^{0}\bar{\rho}^{*})\\\
+&\sum_{\ell\in S\cup\\{p,\infty\\}}\left(\dim\mathcal{L}_{\ell}-\dim
H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})\right),\\\
\end{split}$

according to Wiles’ formula [24, Theorem 8.7.9]. The representation
$\bar{\rho}$ associated to an elliptic curve is odd, i.e.,
$\det\bar{\rho}(c)=-1$, where $c$ denotes complex conjugation. It is therefore
easy to show that $\dim
H^{0}(\operatorname{G}_{\infty},\operatorname{Ad}^{0}\bar{\rho})=1$. Now,
specify the Selmer data as follows

$\mathcal{L}_{\ell}=\begin{cases}0\text{ if }\ell=\infty,\\\
\mathcal{N}_{p}\text{ if }\ell=p,\\\
H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})\text{ if
}\ell\in S.\\\ \end{cases}$

In this case, the Selmer group
$H^{1}_{\mathcal{L}}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})$
coincides with the Selmer group $H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})$.
Since $\operatorname{Ad}^{0}\bar{\rho}$ is assumed to be irreducible, we have
that

$\dim
H^{0}(\operatorname{G}_{\mathbb{Q}},\operatorname{Ad}^{0}\bar{\rho})=\dim
H^{0}(\operatorname{G}_{\mathbb{Q}},\operatorname{Ad}^{0}\bar{\rho}^{*})=0.$

Also, note that for $\ell\neq p$ by the local Euler characteristic formula,

$\dim H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})-\dim
H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})=\dim
H^{2}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}).$

On the other hand, according to local duality,

$\dim H^{2}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})=\dim
H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*}).$

Finally, note that

$\dim\mathcal{N}_{p}=1+\dim
H^{0}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho}).$

Putting it all together, we obtain the result.

∎

For the next part of the discussion, a good reference is [5]. For
$R\in\mathcal{C}$, let $\operatorname{D}_{\bar{\rho}}^{\langle p\rangle}(R)$
be the subset of $\operatorname{D}_{\bar{\rho}}(R)$ consisting of deformations
$\rho$ such that the local representation $\rho_{|p}$ satisfies
$\mathcal{C}_{p}$. The functor $\operatorname{D}_{\bar{\rho}}^{\langle
p\rangle}$ is represented by a universal deformation

$\rho^{\operatorname{univ},\langle p\rangle}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{S}\rightarrow\operatorname{GL}_{2}(\operatorname{R}_{\bar{\rho}}^{\langle
p\rangle}),$

and we shall refer to $\operatorname{R}_{\bar{\rho}}^{\langle p\rangle}$ as
the universal geometric deformation ring. This is abuse of terminology, since
it does not capture all geometric deformations, since there may indeed be
characteristic zero deformations $\varrho$ of $\bar{\rho}_{|p}$ that are
deRham yet do not satisfy $\mathcal{C}_{p}$. However, since our constructions
require choosing a suitable local condition that is smooth and satisfies other
suitable properties, we work in this setting.

Let $\ell\in S$, and $\rho_{\ell}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\ell}\rightarrow\operatorname{GL}_{2}(R_{\ell})$
be the universal deformation of $\bar{\rho}_{|\ell}$ representing the functor
$\operatorname{Def}_{\ell}$. This local deformation ring has a presentation

$R_{\ell}\simeq\frac{\mathbb{Z}_{p}\llbracket
X_{1},\dots,X_{u}\rrbracket}{\left(g_{1},\dots,g_{v}\right)}$

where

$u=\dim H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})\textit{
and }v=\dim H^{2}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}).$

Denote the ideal of relations by $\mathcal{I}_{\ell}\mathrel{\mathop{\mathchar
58\relax}}=\left(g_{1},\dots,g_{v}\right)$. Let
$\rho_{p}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{p}\rightarrow\operatorname{GL}_{2}(R_{p})$ be the
universal deformation of $\bar{\rho}_{|p}$ representing the functor
$\mathcal{C}_{p}$. Note that since the deformation functor
$\mathcal{C}_{p}^{\psi}$ is liftable, the local ring $R_{p}$ is a smooth power
series ring

$R_{p}\simeq\mathbb{Z}_{p}\llbracket X_{1},\dots X_{u}\rrbracket$

with $u=\dim\mathcal{N}_{p}$ generators.

By universality, the local representation
$\rho_{|\ell}^{\operatorname{univ},\langle p\rangle}$ arises by composing

$\rho_{\ell}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\ell}\rightarrow\operatorname{GL}_{2}(R_{\ell})$

by the map
$\operatorname{GL}_{2}(R_{\ell})\rightarrow\operatorname{GL}_{2}(\operatorname{R}_{\bar{\rho}}^{\langle
p\rangle})$ induced by a uniquely determined map of local rings
$R_{\ell}\rightarrow\operatorname{R}_{\bar{\rho}}^{\langle p\rangle}$. Let
$\mathcal{J}_{\ell}$ denote the ideal generated by the image of
$\mathcal{I}_{\ell}$ under the image of this map. The following result is a
consequence of [5, Theorem 5.2].

######  Theorem 3.3.

There is a presentation

$\operatorname{R}_{\bar{\rho}}^{\langle
p\rangle}\simeq\frac{\mathbb{Z}_{p}\llbracket
X_{1},\dots,X_{t}\rrbracket}{\mathcal{J}},$

with

$t\mathrel{\mathop{\mathchar 58\relax}}=\dim H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})$

and where $\mathcal{J}$ is generated by the ideals $\mathcal{J}_{\ell}$, for
$\ell\in S$, and $s\mathrel{\mathop{\mathchar 58\relax}}=\dim H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})$
globally defined relations $f_{1},\dots,f_{s}\in(p,X_{1},\dots,X_{t})^{2}$.
Each ideal $\mathcal{J}_{\ell}$ is generated by at most $\dim
H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*})$ local
relations.

######  Corollary 3.4.

Suppose that $(\star)$ is satisfied and

$H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})=0.$

Then, $\operatorname{R}_{\bar{\rho}}^{\langle p\rangle}$ is isomorphic
$\mathbb{Z}_{p}$.

###### Proof.

It follows from Proposition 3.2 that both the Selmer group $H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})$
and the dual Selmer group $H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})$
are zero, hence, $s=t=0$. Furthermore, since
$H^{2}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})=0$ for
$\ell\in S$, it follows that $\mathcal{J}_{\ell}=0$ are all $\ell\in S$. Since
$s=0$, it follows that there no globally defined relations, and hence,
$\mathcal{J}=0$. Therefore, it follows from Theorem 3.3 that
$\operatorname{R}_{\bar{\rho}}^{\langle p\rangle}$ is isomorphic
$\mathbb{Z}_{p}$. ∎

##  4\. Statistics for a fixed Elliptic Curve $E_{/\mathbb{Q}}$ and varying
prime $p$

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with squarefree
conductor $N$ and $p\geq 5$ a prime. Then, the Galois action on the
$p$-torsion points gives rise to the $2$-dimensional mod-$p$ Galois
representation $\bar{\rho}=\bar{\rho}_{E,p}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q},S\cup\\{p\\}}\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$.
Associated to each pair $(E,p)$ such that:

  1. (1)

$E$ has good reduction at $p$,

  2. (2)

the Galois representation $\bar{\rho}$ is absolutely irreducible,

let $\mathcal{R}_{E,p}=\operatorname{R}_{\bar{\rho}}^{\langle p\rangle}$ be
the geometric deformation ring introduced in the previous section. Note that
for a fixed elliptic curve $E$ without complex multiplication, all but
finitely many primes $p$ satisfy the above conditions. In this section, we
show that $\mathcal{R}_{E,p}\simeq\mathbb{Z}_{p}$ for all but finitely many
primes $p$ (that satisfy the conditions above). This result may be contrasted
with the main result of [34], where it is shown that
$\operatorname{R}_{\bar{\rho}}$ is unobstructed for all but a density zero set
of primes.

There are 2 steps to the arguments in this section.

  1. (1)

It is shown that condition $(\star)$ is satisfied for all but finitely many
primes $p$.

  2. (2)

It is shown that for all but finitely many $p$, $H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})=0$
for the residual representation $\bar{\rho}=\bar{\rho}_{E,p}$.

###  4.1. Bloch-Kato Selmer groups

In order to prove our results about the vanishing of the Selmer group
$H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})$,
we relate it to the Bloch-Kato Selmer group, which is associated with the
characteristic zero representation. Let us recall some standard definitions.
In the next discussion, assume that the pair $(E,p)$ is fixed, and recall that
$S$ is the set of primes $\ell\neq p$ at which $E$ has bad reduction. Let
$\operatorname{T}_{p}(E)$ denote the $p$-adic Tate-module associated to $E$
and $\operatorname{V}_{p}(E)$ be the $p$-adic vector space
$\operatorname{T}_{p}(E)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$, equipped with
the action of $\operatorname{G}_{S\cup\\{p\\}}$, and let
$\operatorname{Ad}_{p}^{0}(E)$ be the adjoint representation associated to
$\operatorname{V}_{p}(E)$. For a $\operatorname{G}_{S\cup\\{p\\}}$-module $M$,
set $H^{i}(\mathbb{Q}_{S\cup\\{p\\}}/\mathbb{Q},M)$ to denote the group of
continuous classes
$H^{i}_{\operatorname{cnts}}(\operatorname{G}_{S\cup\\{p\\}},M)$. For each
place $\ell$ of $\mathbb{Q}$, Bloch and Kato define a subspace

$H^{1}_{\bf{f}}\left(\mathbb{Q}_{\ell},\operatorname{Ad}_{p}^{0}(E)\right)\subseteq
H^{1}\left(\mathbb{Q}_{\ell},\operatorname{Ad}_{p}^{0}(E)\right)$

as follows

$H^{1}_{\bf{f}}\left(\mathbb{Q}_{\ell},\operatorname{Ad}_{p}^{0}(E)\right)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}H^{1}_{nr}\left(\mathbb{Q}_{\ell},\operatorname{Ad}_{p}^{0}(E)\right)&\text{
if }\ell\neq p,\infty,\\\
\operatorname{ker}\left\\{H^{1}(\mathbb{Q}_{\ell},\operatorname{Ad}_{p}^{0}(E))\longrightarrow
H^{1}(\mathbb{Q}_{\ell},\operatorname{B}_{\operatorname{crys}}\otimes\operatorname{Ad}_{p}^{0}(E))\right\\}&\text{
if }\ell=p,\\\ 0&\text{ if }\ell=\infty.\end{cases}$

Let $\operatorname{W}_{p}(E)$ be the quotient
$\operatorname{Ad}_{p}^{0}(E)\otimes_{\mathbb{Q}_{p}}\mathbb{Q}_{p}/\mathbb{Z}_{p}$;
the natural quotient map
$\operatorname{Ad}_{p}^{0}(E)\rightarrow\operatorname{W}_{p}(E)$ induces a map
on passing to cohomology

$H^{1}\left(\mathbb{Q}_{\ell},\operatorname{Ad}_{p}^{0}(E)\right)\rightarrow
H^{1}\left(\mathbb{Q}_{\ell},\operatorname{W}_{p}(E)\right).$

Let $\ell$ be any prime, set

$H^{1}_{\bf{f}}\left(\mathbb{Q}_{\ell},\operatorname{W}_{p}(E)\right)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{im}\left\\{H^{1}_{\bf{f}}\left(\mathbb{Q}_{\ell},\operatorname{Ad}_{p}^{0}(E)\right)\longrightarrow
H^{1}(\mathbb{Q}_{\ell},\operatorname{W}_{p}(E))\right\\}.$

For $M$ denoting either $\operatorname{Ad}_{p}^{0}(E)$ or
$\operatorname{W}_{p}(E)$, the Bloch-Kato Selmer group associated to $M$ is
defined as follows

$H^{1}_{\bf{f}}(\mathbb{Q}_{S\cup\\{p\\}}/\mathbb{Q},M)\mathrel{\mathop{\mathchar
58\relax}}=\ker\left\\{H^{1}(\mathbb{Q}_{S\cup\\{p\\}}/\mathbb{Q},M)\longrightarrow\bigoplus_{\ell}\frac{H^{1}(\mathbb{Q}_{\ell},M)}{H^{1}_{\bf{f}}(\mathbb{Q}_{\ell},M)}\right\\}.$

For $\ell\notin\\{p,\infty\\}$, the local cohomology group
$H^{1}_{\bf{f}}\left(\mathbb{Q},\operatorname{W}_{p}(E)\right)$ coincides with
the maximal divisible subgroup of
$H^{1}_{\operatorname{nr}}\left(\mathbb{Q}_{\ell},\operatorname{W}_{p}(E)\right)$.
For any finite set of primes $\Sigma$ not containing $p,\infty$, define a
larger Selmer group as follows

$H^{1}_{\Sigma}(\mathbb{Q},M)\longrightarrow\left(\bigoplus_{\ell\notin\Sigma\cup\\{p,\infty\\}}\frac{H^{1}(\mathbb{Q}_{\ell},M)}{H^{1}_{\operatorname{nr}}(\mathbb{Q}_{\ell},M)}\right)\oplus\left(\frac{H^{1}(\mathbb{Q}_{p},M)}{H^{1}_{\bf{f}}(\mathbb{Q}_{p},M)}\right).$

Note that the inclusion
$H^{1}_{\bf{f}}\left(\mathbb{Q},\operatorname{W}_{p}(E)\right)\subseteq
H^{1}_{\emptyset}\left(\mathbb{Q},\operatorname{W}_{p}(E)\right)$ is an
equality if
$H^{0}\left(\operatorname{I}_{\ell},\operatorname{W}_{p}(E)\right)$ is
divisible for all $\ell\neq p$.

###  4.2. Vanishing of the dual Selmer group

We now study the relationship between the Bloch-Kato Selmer group and the
smoothness of the geometric deformation ring. In order to arrive at such a
relationship, we establish a criterion for the vanishing of the dual Selmer
group $H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})$.

######  Lemma 4.1.

Let $\mathcal{N}_{p}\subseteq
H^{1}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho})$ be the tangent
space of $\mathcal{C}_{p}$. Then, the image of $\mathcal{N}_{p}$ under the
natural map

$H^{1}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho})\rightarrow
H^{1}\left(\operatorname{G}_{p},\operatorname{W}_{p}(E)\right)[p]$

is contained in
$H^{1}_{\bf{f}}\left(\operatorname{G}_{p},\operatorname{W}_{p}(E)\right)[p]$.

###### Proof.

It is well known that elements of
$H^{1}_{\bf{f}}\left(\operatorname{G}_{p},\operatorname{W}_{p}(E)\right)$ are
precisely those classes whose corresponding extensions lie in the image of the
Fontaine-Laffaille functor, see the discussion in [14, section 1.1.2 and
p.697]. On the other hand, it follows from part (3) of Proposition 2.9 that
all cohomology classes $f\in\mathcal{N}_{p}$ have this property. Hence, any
class $f\in\mathcal{N}_{p}$ must map to
$H^{1}_{\bf{f}}\left(\operatorname{G}_{p},\operatorname{W}_{p}(E)\right)$. ∎

######  Proposition 4.2.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and $p\geq 5$ a prime
at which $E$ has good reduction. Assume that the following conditions hold:

  1. (1)

the mod-$p$ Galois representation $\bar{\rho}_{E,p}$ is absolutely
irreducible,

  2. (2)

$H^{1}_{\emptyset}(\mathbb{Q},\operatorname{W}_{p}(E))=0$.

Then the following assertions hold:

  1. (1)

$\dim H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})=\sum_{\ell\in
S}\dim H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*})$,

  2. (2)

$H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})=0$.

###### Proof.

Throughout this proof, we set $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}=\bar{\rho}_{E,p}$. Identify $\operatorname{W}_{p}(E)[p]$ with
$\operatorname{Ad}^{0}\bar{\rho}$, and note that since $\bar{\rho}$ is assumed
to be irreducible, it follows that
$H^{0}(\mathbb{Q},\operatorname{Ad}^{0}\bar{\rho})=0$, and as a result, the
natural map

(4.1) $H^{1}(\mathbb{Q},\operatorname{Ad}^{0}\bar{\rho})\longrightarrow
H^{1}(\mathbb{Q},\operatorname{W}_{p}(E))$

is injective. Let $\mathcal{T}\subset H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})$
consist of cohomology classes $f$ that are unramified at all primes $\ell\neq
p$. In other words,

$\mathcal{T}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left\\{H^{1}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})\longrightarrow\left(\bigoplus_{\ell\in
S}\frac{H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})}{H^{1}(\operatorname{G}_{\ell}/\operatorname{I}_{\ell},\operatorname{Ad}^{0}\bar{\rho})}\right)\oplus\left(\frac{H^{1}(\operatorname{G}_{p},\operatorname{Ad}^{0}\bar{\rho})}{\mathcal{N}_{p}}\right)\right\\}.$

We show that the above injection (4.1) restricts to a map

$\mathcal{T}\longrightarrow
H^{1}_{\emptyset}\left(\mathbb{Q},\operatorname{W}_{p}(E)\right).$

It suffices to observe that the local conditions defining $\mathcal{T}$ as a
subspace of $H^{1}(\mathbb{Q},\operatorname{Ad}^{0}\bar{\rho})$ are compatible
with those defining
$H^{1}_{\emptyset}\left(\mathbb{Q},\operatorname{W}_{p}(E)\right)$. Clearly,
this is the case for $\ell\neq p$, and for $\ell=p$, the assertion follows
from Lemma 4.1. As a result, the assumption that
$H^{1}_{\emptyset}\left(\mathbb{Q},\operatorname{W}_{p}(E)\right)=0$ implies
that $\mathcal{T}=0$. On the other hand, consider the short exact sequence

$0\rightarrow\mathcal{T}\rightarrow H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})\rightarrow\bigoplus_{\ell\in
S}\frac{H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})}{H^{1}_{\operatorname{nr}}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})},$

from which we obtain the inequality

$\begin{split}\dim H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})&\leq\dim\mathcal{T}+\sum_{\ell\in
S}\left(\dim
H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})-\dim
H^{1}_{nr}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})\right)\\\
&\leq\dim\mathcal{T}+\sum_{\ell\in S}\left(\dim
H^{1}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})-\dim
H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho})\right)\\\
&=\dim\mathcal{T}+\sum_{\ell\in S}\dim
H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*}),\\\
&=\sum_{\ell\in S}\dim
H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*}),\end{split}$

where in the last step, we invoke $\dim\mathcal{T}=0$.

Proposition 3.2 asserts that

$\dim H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})=\dim
H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})+\sum_{\ell\in
S}\dim H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*}).$

Therefore, $H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})=0$
and

$\dim H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho})=\sum_{\ell\in
S}\dim H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*}).$

∎

###  4.3. Congruence primes and vanishing of the Bloch-Kato Selmer group

Next, we discuss conditions for the vanishing of
$H^{1}_{\emptyset}(\mathbb{Q},\operatorname{W}_{p}(E))$. Let $N$ denote the
conductor of $E$, and $f$ the Hecke eigencuspform of weight $2$ associated
with $E$. We introduce the notion of a congruence prime, which will be key in
our analysis of the vanishing of the Selmer group
$H^{1}_{\emptyset}\left(\mathbb{Q},\operatorname{W}_{p}(E)\right)$.

######  Definition 4.3.

We say that $p$ is a congruence prime for $f$ if there exists a newform
$f_{0}$ of weight $2$ and level $d|N$ such that:

  1. (1)

$f_{0}$ has character lifting the trivial one,

  2. (2)

$f_{0}$ is not Galois conjugate to $f$,

  3. (3)

$\bar{\rho}_{f_{0},\lambda}\simeq\bar{\rho}_{f,\lambda}$ for some prime
$\lambda|p$ of $\bar{\mathbb{Q}}$.

Let $\operatorname{Cong}(f)$ denote the set of congruence primes. A prime $p$
is a strict congruence prime if there is a newform satisfying the above
conditions with level $N$. Denote by
$\operatorname{Cong}_{N}(f)\subseteq\operatorname{Cong}(f)$ the subset of
strict congruence primes.

The following result gives a relationship between congruence primes and the
vanishing of the Selmer group
$H^{1}_{\emptyset}\left(\mathbb{Q},\operatorname{W}_{p}(E)\right)$, it is
largely based on results in [14]

######  Proposition 4.4.

Let $p$ be a prime, assume that that:

  1. (1)

$\bar{\rho}_{f,p}$ is absolutely irreducible,

  2. (2)

$p>2$,

  3. (3)

either $N>1$ or $p>3$,

  4. (4)

$p\nmid N$,

  5. (5)

$\ell\not\equiv 1\mod{p}$ for all primes $\ell|N$,

  6. (6)

$\bar{\rho}_{f,p}$ is ramified at all primes $\ell|N$.

Then $H^{1}_{\emptyset}(\mathbb{Q},\operatorname{W}_{p}(E))\neq 0$ if and only
if $p\in\operatorname{Cong}_{N}(f)$.

###### Proof.

This is a special case of [35, Proposition 17]. ∎

######  Proposition 4.5.

Suppose that $\ell\in S$, $p\geq 3$ and $\ell\not\equiv\pm 1\mod{p}$. Then,
$H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*})=0$ if and
only if $\bar{\rho}$ is ramified at $\ell$.

###### Proof.

We refer the reader to [35, Lemma 11]. ∎

######  Theorem 4.6.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with squarefree
conductor $N$. Let $\Sigma(E)$ be the set of primes $p$ such that one of the
following conditions is satisfied:

  1. (1)

$p\leq 3$,

  2. (2)

$\bar{\rho}_{f,p}$ is not absolutely irreducible

  3. (3)

$p\mid N$,

  4. (4)

there is a prime $\ell|N$, such that $\ell\equiv\pm 1\mod{p}$,

  5. (5)

$p\in\operatorname{Cong}(f)$.

Then, for $p\notin\Sigma(E)$, the geometric deformation ring
$\mathcal{R}_{E,p}$ is isomorphic to $\mathbb{Z}_{p}$.

###### Proof.

Let $p\notin\Sigma(E)$ and assume by way of contradiction that
$\mathcal{R}_{E,p}\not\simeq\mathbb{Z}_{p}$. According to Corollary 3.4, there
are two possibilities:

  1. (1)

$H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*})\neq 0$ for
some prime $\ell\in S$,

  2. (2)

$H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})\neq
0$.

Suppose that the first of the above possibilities does hold. According to
Proposition 4.5, it the residual representation $\bar{\rho}$ must be
unramified at $\ell$. However, according to [35, Lemma 12], it follows that
$p\in\operatorname{Cong}(f)$. However, since $p\notin\Sigma(E)$, it follows
therefore that
$H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*})=0$ for all
primes $\ell\in S$. Therefore, the only possibility is that $H^{1}_{\langle
p\rangle}(\operatorname{G}_{S\cup\\{p\\}},\operatorname{Ad}^{0}\bar{\rho}^{*})\neq
0$. Note that since $p\notin\Sigma(E)$, the residual representation
$\bar{\rho}$ is absolutely irreducible. Hence, it follows from Proposition 4.2
that $H^{1}_{\emptyset}(\mathbb{Q},\operatorname{W}_{p}(E))\neq 0$. Since
$H^{0}(\operatorname{G}_{\ell},\operatorname{Ad}^{0}\bar{\rho}^{*})=0$ for all
primes $\ell\in S$, it follows from Proposition 4.5 that $\bar{\rho}$ is
ramified at all primes $\ell\in S$. Proposition 4.4 then implies that
$p\in\operatorname{Cong}_{N}(f)$, which is not possible since
$p\notin\Sigma(E)$. The contradiction implies that
$\mathcal{R}_{E,p}\simeq\mathbb{Z}_{p}$ for all primes $p\notin\Sigma(E)$. ∎

######  Corollary 4.7.

Let $E$ be an elliptic curve over $\mathbb{Q}$ with squarefree conductor $N$
and without complex multiplication. Then, for all but finitely many primes
$p$, the Galois deformation ring $\mathcal{R}_{E,p}$ is isomorphic to
$\mathbb{Z}_{p}$.

###### Proof.

It follows from Serre’s open image theorem that the set of primes $p$ at which
the residual representation $\bar{\rho}_{E,p}$ is absolutely irreducible is
finite. It is easy to see that the set of primes $\Sigma(E)$ in the statement
of Theorem 4.6 is finite and the result follows from the theorem. ∎

##  5\. Statistics for a fixed prime $p$ and varying elliptic curve
$E_{/\mathbb{Q}}$

In this section, we are able to provide some partial answers to the dual
problem. Namely, we fix a prime $p$ throughout and let $E$ vary over all non-
CM elliptic curves over $\mathbb{Q}$. We will need to assume that $p\geq 5$.
We note now that our results apply to a sparse set of elliptic curves
$E_{/\mathbb{Q}}$.

###  5.1. The congruence number and modular degree

######  Definition 5.1.

Let $\mathscr{E}_{p}$ be the set of elliptic curves $E_{/\mathbb{Q}}$ for
which the following conditions are satisfied:

  1. (1)

$E$ has good reduction at $p$,

  2. (2)

the conductor $N$ of $E$ is squarefree,

  3. (3)

the residual representation $\bar{\rho}=\bar{\rho}_{E,p}$ is absolutely
irreducible,

  4. (4)

$\ell\not\equiv\pm 1\mod{p}$ for all primes $\ell\mid N$.

######  Remark 5.2.

It is shown by J. Cremona and M. Sadek in [13] that the proportion of elliptic
curves over $\mathbb{Q}$ ordered by height satisfying (1) (resp. (2)) is
$(1-\frac{1}{p})$ (resp. $\zeta(10)/\zeta(2)\approx$ 60.85%). On the other
hand, it is a result of W.Duke that $\bar{\rho}_{E,p}$ is irreducible for
$100\%$ of elliptic curves, see [15].

######  Remark 5.3.

We observed computationally for elliptic curves with conductor at most 4000
that the conditions (1), (2) and (4) appear to imply (3). We are not
immediately aware of any obvious reason for this.

As $E$ varies in the set $\mathscr{E}_{p}$, one would like to understand how
often is the geometric deformation ring $\mathcal{R}_{E,p}$ isomorphic to
$\mathbb{Z}_{p}$? Let $f$ be the modular form corresponding to $E$. Note that
according to Theorem 4.6, this is the case when
$p\notin\operatorname{Cong}(f)$. Let $\mathscr{E}_{p}^{\prime}$ be the subset
of $\mathscr{E}_{p}$ for which this additional condition is satisfied.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, there is a unique
minimal Weierstrass equation $y^{2}=x^{3}+ax+b$, where
$(a,b)\in\mathbb{Z}^{2}$ is such that $\operatorname{gcd}(a^{3},b^{2})$ is not
divisible by any twelfth power. The height of $E$ is given by
$H(E)\mathrel{\mathop{\mathchar 58\relax}}=\max\left(\mathinner{\\!\left\lvert
a\right\rvert}^{3},b^{2}\right)$. Ordering elliptic curves over $\mathbb{Q}$
according to height, for any subset $\mathscr{S}\subseteq\mathscr{E}_{p}$ and
$x>0$, let $\mathscr{S}(x)$ consist of all elliptic curves $E\in\mathscr{S}$
with height $\leq x$. The lower density of $\mathscr{S}$ is given by

$\mathfrak{d}(\mathscr{S})\mathrel{\mathop{\mathchar
58\relax}}=\liminf_{x\rightarrow\infty}\frac{\\#\mathscr{S}(x)}{\\#\mathscr{E}_{p}(x)}.$

From a statistical point of view, one would like to characterize
$\mathfrak{d}(\mathscr{E}_{p}^{\prime})$, i.e., the lower density for the
proportion of curves $E\in\mathscr{E}_{p}$ such that
$\mathcal{R}_{E,p}\simeq\mathbb{Z}_{p}$.

We are not able to prove unconditional results, instead, we resort to
heuristics and explicit calculations. We recall the notions of modular degree
and congruence number associated to an elliptic curve $E_{/\mathbb{Q}}$.

######  Definition 5.4.

Let $E_{/\mathbb{Q}}$ be an elliptic curve of conductor $N$ and
$\phi_{E}\mathrel{\mathop{\mathchar 58\relax}}X_{0}(N)\rightarrow E$ the
modular homomorphism associated to $E$. The modular degree $m_{E}$ is the
degree of $\phi_{E}$.

Let $f$ be the normalized newform on $\Gamma_{0}(N)$ associated to $E$. The
congruence number is a number which is precisely divisible by the primes in
$\operatorname{Cong}(f)$, see [1] for the definition. Let $r_{E}$ the
congruence number.

Given $r\in\mathbb{Z}_{\geq 1}$, set $r^{(p)}$ to denote $|r|_{p}^{-1}$, i.e.,
the $p$-part of $r$. Since $p\nmid N$, it follows from [1, Theorem 2.1] that
$r_{E}^{(p)}=m_{E}^{(p)}$. Note that an elliptic curve $E\in\mathscr{E}_{p}$
belongs to $\mathscr{E}_{p}^{\prime}$, precisely when $r_{E}^{(p)}=1$.
Therefore, we may formulate the problem in terms of the distribution of the
$p$-primary part of the modular degree

$\liminf_{x\rightarrow\infty}\frac{\\#\mathscr{E}_{p}^{\prime}(x)}{\\#\mathscr{E}_{p}(x)}=\liminf_{x\rightarrow\infty}\frac{\\#\\{E\in\mathscr{E}_{p}(x)\mid
m_{E}^{(p)}=1\\}}{\\#\mathscr{E}_{p}(x)}.$

The probablity that $p\mid m_{E}^{(p)}$ for odd primes $p$ is given by Cohen-
Lenstra heuristics, see [31, p.499]. It follows from such heuristics that the
proportion of curves $m_{E}^{(p)}=1$ is

$\prod_{i}\left(1-\frac{1}{p^{i}}\right)=1-\frac{1}{p}-\frac{1}{p^{2}}+\frac{1}{p^{3}}+\dots.$

Watkins obtains some evidence for these heuristics in loc. cit. One can only
expect that the stipulation that $m_{E}^{(p)}=1$ be largely independent from
the other conditions (1)-(4).

In the rest of this paper, we show through explicit calculation that one may
expect that $m_{E}^{(p)}=1$ holds for a large proportion of curves, especially
as $p\rightarrow\infty$. This leads us to expect the following,

  1. (1)

for any odd prime $p$,

$\liminf_{x\rightarrow\infty}\frac{\\#\mathscr{E}_{p}^{\prime}(x)}{\\#\mathscr{E}_{p}(x)}>0.$

  2. (2)

As $p\rightarrow\infty$,

$\lim_{p\rightarrow\infty}\left(\liminf_{x\rightarrow\infty}\frac{\\#\mathscr{E}_{p}^{\prime}(x)}{\\#\mathscr{E}_{p}(x)}\right)\rightarrow
1.$

###  5.2. Statistics for congruence primes

For newforms $f,g$ of levels $M,N$, we say that a rational prime $p$ is a
congruence prime for $f$ and $g$ if there is a prime $\mathfrak{p}$ of
$\bar{\mathbb{Q}}$ containing $p$ such that there is a congruence of Fourier
coefficients

$a_{\ell}(f)\equiv a_{\ell}(g)\pmod{\mathfrak{p}}$

for all primes $\ell$ not dividing $pMN$.

Let $E$ be an elliptic curve with conductor $N$ and associated newform $f\in
S_{2}(\Gamma_{0}(N))$. Recall that a prime $p$ is a congruence prime for $E$
if there is a weight $2$ newform $g$ of level dividing $N$ such that $p$ is a
congruence prime for $f_{E}$ and $g$. We say that a congruence prime is strict
(resp. proper) if the level of $g$ can be taken equal to $N$ (resp. strictly
less than $N$). In deformation theory, strict congruence primes correspond to
global obstructions while proper congruence primes correspond to local
obstructions.

The behavior of $2$ as a congruence prime was studied in [11], where they
proved that it is a congruence prime for the vast majority of eigenforms. The
behavior for odd primes is less clear. We computed all congruence primes for
the 13,352 isogeny classes of elliptic curves of conductor at most $4000$.
Computations were done in Magma using the package of Montes to deal with the
large number fields which arise as coefficient fields. (Attempts to extend the
computations to larger conductor were unsuccessful due to the extremely large
coefficient fields which begin to arise at that stage.)

$p$ | % proper | % strict | % cong
---|---|---|---
2 | 82.9 | 99.1 | 99.5
3 | 49.1 | 61.7 | 75.4
5 | 16.3 | 23.8 | 37.3
7 | 7.8 | 13.2 | 20.0
11 | 2.5 | 4.8 | 7.2
13 | 1.6 | 3.4 | 4.9
17 | 0.7 | 2.1 | 2.8
19 | 0.5 | 1.3 | 1.8
23 | 0.3 | 0.9 | 1.1
29 | 0 | 0.6 | 0.6
31 | 0.1 | 0.4 | 0.4
37 | 0 | 0.3 | 0.4
41 | 0 | 0.3 | 0.3
43 | 0 | 0.1 | 0.1
47 | 0 | 0.1 | 0.1

The proportions for all primes between 53 and 97 were never more than $0.1\%$.

It is difficult to see any immediate pattern in this data beyond the not
surprising observation that small congruence primes are relatively common and
large congruence primes are rare. The data for the 179 isogency classes of
elliptic curves of prime conductor in our sample (in which case all
congruences are necessarily strict) was perhaps more revealing.

$p$ | % cong | % Watkins $S_{3}$ | Watkins CL
---|---|---|---
3 | 40.2 | 44.8 | 44.0
5 | 22.9 | 24.2 | 24.0
7 | 11.2 | 16.3 | 16.3
11 | 9.5 | 9.9 | 9.9
13 | 7.3 | 8.1 | 8.3
17 | 3.9 | 6.2 | 6.2
19 | 5.6 | 5.3 | 5.5
23 | 3.4 | 4.7 | 4.5
29 | 3.4 | 3.4 | 3.6
31 | 2.8 | 3.4 | 3.3
37 | 1.1 | 2.7 | 2.8

Here we list also the proportion of elliptic curves $E$ in Watkin’s set
$S_{3}$ (consisting of 52878 non-Setzer–Neumann elliptic curves of prime
discriminant of absolute value less than $10^{7}$, as computed by [10]) with
$p$ dividing the modular degree, as well as the Cohen–Lenstra prediction
Watkins developed for that case. Given that our data set is quite small, the
fit is certainly respectably close.

Since our results are concerned with elliptic curves of squarefree conductor,
we also report the proportions in that case, which are notably higher than the
overall average, at least for larger $p$. The data here is for the 4931
isogeny classes of elliptic curves of squarefree conductor $\leq 4000$.

$p$ | % cong
---|---
2 | 98.8
3 | 63.7
5 | 37.9
7 | 21.6
11 | 9.3
13 | 6.8
17 | 4.1
19 | 2.8
23 | 1.9
29 | 1.2

Finally, and perhaps most relevantly, we note that individual elliptic curves
tend to have very few congruence primes.

$n$ | % $E$ with $n$ proper cp | % $E$ with $n$ strict cp | % $E$ with $n$ cp
---|---|---|---
0 | 4.3% | 0.3% | 0.1%
1 | 40.8% | 16.6% | 8.2%
2 | 44.1% | 55.7% | 42.2%
3 | 10.2% | 25.0% | 38.8%
4 | 0.6% | 2.3% | 9.7%
5 | 0.0% | 0.1% | 0.9%
6 | 0 | 0 | 0.0%

There were a total of six elliptic curves in the sample with six congruence
primes.

Putting everything together, we computed for small primes $p$ the proportion
of isogeny classes of conductor $\leq 4000$ lying in our set
$\mathcal{E}_{p}$, and then also what proportion of those further satisfy
$p\nmid m_{E}$.

$p$ | % in ${\mathcal{E}}_{p}$ | % in ${\mathcal{E}}_{p}$ with $p\nmid m_{E}$
---|---|---
5 | 10.3% | 6.6%
7 | 17.2% | 13.5%
11 | 25.1% | 22.3%
13 | 28.6% | 26.7%
17 | 30.4% | 29.2%
19 | 30.9% | 30.0%
23 | 32.1% | 31.5%
29 | 33.3% | 33.0%

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