# Constructing Galois representations with prescribed Iwasawa
$\lambda$-invariant

Anwesh Ray  Chennai Mathematical Institute, H1, SIPCOT IT Park, Kelambakkam,
Siruseri, Tamil Nadu 603103, India<EMAIL_ADDRESS>
###### Abstract.

Let $p\geq 5$ be a prime number. We consider the Iwasawa $\lambda$-invariants
associated to modular Bloch-Kato Selmer groups, considered over the cyclotomic
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$. Let $g$ be a $p$-ordinary cuspidal
newform of weight $2$ and trivial nebentype. We assume that the
$\mu$-invariant of $g$ vanishes, and that the image of the residual
representation associated to $g$ is suitably large. We show that for any
number $n$ greater than or equal to the $\lambda$-invariant of $g$, there are
infinitely many newforms $f$ that are $p$-congruent to $g$, with
$\lambda$-invariant equal to $n$. We also prove quantitative results regarding
the levels of such modular forms with prescribed $\lambda$-invariant.

###### Key words and phrases:

Iwasawa theory, modular forms, Selmer groups, Galois representations, Bloch-
Kato Selmer groups

###### 2020 Mathematics Subject Classification:

11R23, 11F80 (primary); 11F11, 11R18 (secondary)

##  1\. Introduction

The Iwasawa theory of Selmer groups associated to Galois representations
captures significant arithmetic information about motives. Let $p$ be a prime
number. Given an elliptic curve $E_{/\mathbb{Q}}$, and a number field
extension $F/\mathbb{Q}$, the $p$-primary Selmer group of $E$ over $F$ is of
fundamental importance and captures information about the Mordell-Weil group
$E(F)$ and the $p$-primary part of the Tate-Shafarevich group $\Sh(E/F)$. The
fundamental object of study in the Iwasawa theory of elliptic curves is the
$p$-primary Selmer group over the cyclotomic $\mathbb{Z}_{p}$-extension,
denoted $\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})$.
Mazur [Maz72] initiated the Iwasawa theory of elliptic curves $E$ with good
ordinary reduction at $p$, and associated structural invariants associated
with these Selmer groups.

###  1.1. Main results

In this paper we consider $\lambda$-invariants associated to Bloch-Kato Selmer
groups attached to modular Galois representations. We prove certain
qualitative and quantitative results about the levels of modular forms that
arise in natural families, for which the $\lambda$-invariant is prescribed to
be a fixed value. We fix a cuspidal Hecke newform $g$ of weight $2$ on
$\Gamma_{0}(N_{g})$. Associated with a fixed choice of embedding
$\iota_{p}\mathrel{\mathop{\mathchar
58\relax}}\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{p}$, let $\rho_{g}$
be the associated Galois representation. It is assumed that the image of the
residual representation $\bar{\rho}_{g}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\bar{\mathbb{F}}_{p})$
is up to conjugation, equal to $\operatorname{GL}_{2}(\mathbb{F}_{p})$. When
we assume that $p$ is odd and $g$ is $p$-ordinary, the Iwasawa $\mu$ and
$\lambda$-invariants are well defined, and denoted by $\mu_{p}(g)$ and
$\lambda_{p}(g)$ respectively (cf. section 3.2).

######  Theorem 1.1.

Let $p\geq 5$ be a prime and let $g$ be a normalized newform of weight $2$ on
$\Gamma_{0}(N_{g})$. We assume the following conditions

  1. (1)

We shall assume that the image of the residual representation $\bar{\rho}_{g}$
lies in $\operatorname{GL}_{2}(\mathbb{F}_{p})$. Moreover, the Galois
representation $\bar{\rho}_{g}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$
is surjective.

  2. (2)

The modular form $g$ is $p$-ordinary and $p\nmid N_{g}$,

  3. (3)

$g$ has optimal level, i.e., $N_{g}$ is the prime to $p$ part of the Artin
conductor of the residual representation,

  4. (4)

$\mu_{p}(g)=0$.

Associated with $g$, let $\Pi_{g}$ be the set of prime numbers $\ell$ such
that the following conditions are satisfied:

  1. (1)

$\ell\nmid N_{g}p$,

  2. (2)

$\ell\not\equiv\pm 1\mod{p}$ and $\ell^{p-1}\not\equiv 1\mod{p^{2}}$,

  3. (3)

$\bar{\rho}_{g}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell&0\\\
0&1\\\ \end{array}}\right)$ for a suitable choice of basis for the residual
representation.

Let $\Omega_{g}$ be the set of prime numbers such that

  1. (1)

$\ell\nmid N_{g}p$,

  2. (2)

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

  3. (3)

$\bar{\rho}_{g}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}-\ell&0\\\
0&-1\\\ \end{array}}\right)$ for a suitable choice of basis for the residual
representation.

The primes $\Pi_{g}$ and $\Omega_{g}$ have Dirichlet density
$\frac{(p-3)}{p(p-1)}$ and $\frac{(p-3)}{(p-1)^{2}}$ respectively.

Let $n,r\in\mathbb{Z}_{\geq 0}$ be such that $\operatorname{max}\\{n,r\\}>0$.
Let $n\geq 1$ be an integer. Then for any sets of primes
$\\{q_{1},\dots,q_{n}\\}\subset\Pi_{g}$ and
$\\{\ell_{1},\dots,\ell_{r}\\}\subset\Omega_{g}$, there exists a normalized
newform $f$ of weight $2$ of level
$N_{f}=N_{g}\times\prod_{i=1}^{n}q_{i}\times\prod_{j=1}^{r}\ell_{j}$ such that

  1. (1)

$f$ has good ordinary reduction at $p$,

  2. (2)

$\bar{\rho}_{g}\simeq\bar{\rho}_{f}$,

  3. (3)

$\mu_{p}(f)=0$,

  4. (4)

$\lambda_{p}(f)=\lambda_{p}(g)+n$.

In the special case when $\mu_{p}(g)=0$ and $\lambda_{p}(g)\leq 1$, one is
able to realize an infinite family of modular forms $f$ that are $p$-congruent
to $g$, for which the Bloch-Kato Selmer group over $\mathbb{Q}$ has prescribed
corank. We refer to Theorem 5.5 for the statement of the result.

The following result further illustrates Theorem 1.1.

######  Theorem 1.2.

There is a positive density set of primes $p$, such that for any $n\geq 0$,
there exist infinitely many normalized Hecke cuspdidal newforms $f$ of weight
$2$ such that

  1. (1)

$f$ has good ordinary reduction at $p$,

  2. (2)

$\bar{\rho}_{f}$ is surjective,

  3. (3)

$\mu_{p}(f)=0$,

  4. (4)

$\lambda_{p}(f)=n$.

###  1.2. Relationship with previous work

The existence of Galois representations for which the associated Selmer groups
have large $\lambda$-invariant has been studied by various authors, cf.
[Gre99, Ray23]. Our results are significantly stronger since we are able to
explicitly realize any large enough integer as a $\lambda$-invariant. If at a
given prime $p\geq 5$, there exists a newform $g$ satisfying the conditions of
Theorem 1.1, and such that $\lambda_{p}(g)=0$, then, every integer $n\geq 0$
is seen to arise from a modular form which is $p$-congruent to $g$. This is
indeed the case for a density $1$ set of primes $p$, as shown by the proof of
Theorem 1.2. Furthermore, not only is one able to construct infinitely many
modular Galois representations giving rise to a prescribed
$\lambda$-invariant, one also obtains an explicit and satisfactory
quantitative description of their levels. We contrast Theorem 1.1 to the
results of recent work by Hatley and Kundu [HK22], where it is shown that
there are infinitely many modular forms $f$ that are $p$-congruent to a fixed
modular form $g$, for which $\lambda_{p}(f)=\lambda_{p}(g)$. This
$\lambda$-stability result requires an additional assumption on $g$: that the
$\lambda$-invariant of $g$ is minimal in the family of all
$\lambda$-invariants for modular forms that are $p$-congruent to $g$. For
further details, we refer to p.15 of _loc. cit._ This assumption is clearly
satisfied when $\lambda_{p}(g)$ is $0$. This $\lambda$-stability result
follows from Theorem 1.1 in the special case when $n=0$, without the
additional hypothesis. The method used in proving our results does draw some
inspiration from [Ray23, HK22].

###  1.3. Organization

Including the introduction, the manuscript consists of 5 sections. In section
2, we set up basic notation and review the level raising theorems of Carayol
(cf. Theorem 2.1) and Diamond-Taylor (cf. Theorem 2.3). In section 3, we
describe the relationship between the Bloch-Kato and Greenberg Selmer groups
over the cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$. We describe
the Iwasawa invariants associated to these Selmer groups. In section 4, we
recall the results of Greenberg and Vatsal, and prove Theorem 4.9, which paves
the way to the proof of Theorem 1.1. In section 5, we compute the densities of
the sets $\Pi_{g}$ and $\Omega_{g}$, and prove Theorems 1.1 and 1.2.

### Acknowledgment

The project was initiated in 2022 when the author was a Simons postdoctoral
fellow at the Centre de recherches mathématiques in Montreal. From September
2022 to September 2023, the author’s research was supported by the CRM Simons
postdoctoral fellowship.

##  2\. Preliminaries

Fix an algebraic closure $\bar{\mathbb{Q}}/\mathbb{Q}$, and for each prime
$\ell$, let $\bar{\mathbb{Q}}_{\ell}$ be an algebraic closure and fix an
inclusion of $\iota_{\ell}\mathrel{\mathop{\mathchar
58\relax}}\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{\ell}$. Set
$\bar{\mathbb{Z}}$ to be the integral closure of $\mathbb{Z}$ in
$\bar{\mathbb{Q}}$. The choice of embedding $\iota_{\ell}$ corresponds to a
choice of prime $\mathfrak{l}|\ell$ of $\bar{\mathbb{Z}}$. The inclusion
$\iota_{\ell}$ induces an isomorphism of $\bar{\mathbb{Z}}_{\mathfrak{l}}$
with $\bar{\mathbb{Z}}_{\ell}$. Let $\operatorname{G}_{\ell}$ denote the
absolute Galois group
$\operatorname{Gal}(\bar{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$. The inclusion
$\iota_{\ell}$ induces an inclusion of $\operatorname{G}_{\ell}$ into
$\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$, whose image is the
decomposition group of $\mathfrak{l}$. Let $\operatorname{I}_{\ell}$ be the
inertia group of $\operatorname{G}_{\ell}$, and choose a Frobenius element
$\operatorname{fr}_{\ell}\in\operatorname{G}_{\ell}$. Given a Galois
representation $\rho\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\cdot)$,
let $\rho_{|\ell}$ be the restriction of $\rho$ to $\operatorname{G}_{\ell}$.
The representation $\rho$ is _unramified_ at $\ell$ if the restriction of
$\rho_{|\ell}$ to $\operatorname{I}_{\ell}$ is trivial. Fix a prime $p\geq 5$
and let $\mathfrak{p}|p$ be the prime above $p$, corresponding to the
inclusion $\iota_{p}$. Let $f=\sum_{n=1}^{\infty}a_{n}(f)q^{n}$ be a
normalized cuspidal newform of weight $k\geq 2$ and
$\rho_{f}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\bar{\mathbb{Q}}_{p})$
be the associated Galois representation. The Hodge-Tate weights for the Galois
representation are $\\{k-1,0\\}$. We note here that this is simply a matter of
convention (instead of the Hodge-Tate weights being $\\{0,1-k\\}$). There is a
finite extension $K$ over $\mathbb{Q}_{p}$ such that w.r.t a suitable choice
of basis, the image of $\rho_{f}$ lies in $\operatorname{GL}_{2}(K)$. In
greater detail, let $F$ be a number field containing the Fourier coefficients
of $f$, and $K$ is the completion $F_{\mathfrak{p}}$. We set $V_{f}\simeq
K^{2}$ to denote the underlying Galois module for the representation
$\rho_{f}$. Set $\mathcal{O}$ to denote the valuation ring of $K$, $\varpi$ be
a uniformizer of $\mathcal{O}$, and let $\kappa\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{O}/(\varpi)$ be the residue field of $\mathcal{O}$. There
exists a Galois stable $\mathcal{O}$-lattice $T_{f}\subset V_{f}$. We shall
also denote the integral representation on $T_{f}$ by
$\rho_{f}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathcal{O})$.
We denote its mod-$\varpi$ reduction by
$\bar{\rho}_{f}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\kappa)$.
We shall assume throughout the $\bar{\rho}_{f}$ is absolutely irreducible. In
this setting, it is easy to see that the Galois stable $\mathcal{O}$-lattice
$T_{f}$ is uniquely determined, and hence, there is no ambiguity in the
notation used.

###  2.1. The level raising results of Diamond-Taylor

Let $p$ be an odd prime number and $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\bar{\mathbb{F}}_{p})$
be an irreducible Galois representation. Let
$c\in\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ denote the complex
conjugation; $\bar{\rho}$ is _odd_ if $\operatorname{det}\bar{\rho}(c)=-1$.
Serre [Ser87] conjectured that any odd and irreducible representation is
modular. In greater detail, the strong form of the conjecture states that
$\bar{\rho}$ arises from a cuspidal newform $g$ of weight $k\geq 2$ on
$\Gamma_{1}(N_{\bar{\rho}})$, where $N_{\bar{\rho}}$ is the prime to $p$ part
of the Artin-conductor of $\bar{\rho}$ (cf. [RS99, p.11]). The weight
$k=k(\bar{\rho})$ is prescribed according to [Ser87, section 2]. Khare and
Wintenberger [KW09a, KW09b] proved Serre’s conjecture, building upon prior
work of Ribet [Rib90]. Suppose that $f$ is a newform of weight $k$ and level
$N_{f}$ coprime to $p$, such that the associated $p$-adic Galois
representation $\rho_{f}$ lifts $\bar{\rho}$. Then, the optimal level
$N_{\bar{\rho}}$ divides $N_{f}$. A theorem of Carayol [Car89] proves
necessary conditions for an integer $N_{f}$ to arise in this way from a
newform $f$.

######  Theorem 2.1 (Carayol).

Suppose there exists a modular form $f$ of weight $k$ of level $N_{f}$ such
that $\bar{\rho}=\bar{\rho}_{f}$ is absolutely irreducible. The level $N_{f}$
admits a factorization $N_{f}=N_{\bar{\rho}}\prod_{\ell}\ell^{\alpha(\ell)}$,
and for each $\ell$ with $\alpha(\ell)>0$, one of the following holds:

  1. (1)

$\ell\nmid N_{\bar{\rho}}$,
$\ell\left(\operatorname{trace}\bar{\rho}\left(\sigma_{\ell}\right)\right)^{2}=\left(1+\ell\right)^{2}\operatorname{det}\bar{\rho}\left(\sigma_{\ell}\right)$
in $\bar{\mathbb{F}}_{p}$, and $\alpha(\ell)=1$;

  2. (2)

$\ell\equiv-1\mod{p}$ and one of the following holds:

    1. (a)

$\ell\nmid N_{\bar{\rho}}$,
$\operatorname{trace}\bar{\rho}\left(\sigma_{\ell}\right)\equiv 0$ in
$\overline{\mathbb{F}}_{p}$, and $\alpha(\ell)=2$;

    2. (b)

$\ell\mid\\!\mid N_{\bar{\rho}}$, $\operatorname{det}\bar{\rho}$ is unramified
at $\ell$, and $\alpha(\ell)=1$;

  3. (3)

$\ell\equiv 1\mod{p}$ and one of the following holds:

    1. (a)

$\ell\nmid N_{\bar{\rho}}$ and $\alpha(\ell)=2$;

    2. (b)

$\ell\mid\\!\mid N_{\bar{\rho}}$ and $\alpha(\ell)=1$ or $\ell\nmid
N_{\bar{\rho}}$ and $\alpha(\ell)=1$.

######  Definition 2.2.

The set of levels satisfying the conditions outlined in Theorem 2.1 is denoted
by $\mathcal{S}(\bar{\rho})$.

######  Theorem 2.3 (Diamond-Taylor [DT94]).

Let $p\geq 5$ be a prime,

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

be an irreducible Galois representation. Assume that $\bar{\rho}$ arises from
a newform $g$ on $\Gamma_{1}(N_{\bar{\rho}})$ of weight $k$. Assume that $k$
lies in the range $2\leq k\leq p-2$, and let $M\in\mathcal{S}(\bar{\rho})$.
Then there exists a cuspidal newform $f$ of weight $k$ on $\Gamma_{1}(M)$,
such that $\bar{\rho}_{f}=\bar{\rho}$.

The result of Carayol and the above level-raising result of Diamond and Taylor
shows that $\mathcal{S}(\bar{\rho})$ is precisely the set of levels for the
cuspidal newforms $f$ such that $\bar{\rho}_{f}\simeq\bar{\rho}$.

##  3\. Selmer groups associated to modular forms

Let $f=\sum_{n=1}^{\infty}a_{n}(f)q^{n}$ be a normalized new cuspform on
$\Gamma_{1}(N_{f})$ and $p$ be an odd prime which is coprime to $N_{f}$. Let
$\rho_{f}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathcal{O})$
be the associated Galois representation and assume that the residual
representation is absolutely irreducible. Here, $K/\mathbb{Q}_{p}$ is a finite
extension and $\mathcal{O}$ is the valuation ring of $K$. Let $V_{f}$ and
$T_{f}$ be as defined in previous sections. We shall assume throughout that
$p$ is an ordinary prime, i.e., $p\nmid a_{p}(f)$. Set $\mu_{p^{n}}$ to denote
the $p^{n}$-th roots of unity in $\bar{\mathbb{Q}}$, and let
$\mathbb{Q}(\mu_{p^{n}})$ be the cyclotomic extension of $\mathbb{Q}$
generated by $\mu_{p^{n}}$. Note that the Galois group
$\operatorname{Gal}(\mathbb{Q}(\mu_{p^{n}})/\mathbb{Q})$ is isomorphic to
$\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)^{\times}$. Let
$\mathbb{Q}(\mu_{p^{\infty}})$ be the union of cyclotomic fields
$\mathbb{Q}(\mu_{p^{n}})$. The Galois group
$\operatorname{Gal}(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})$ is isomorphic to
$\mathbb{Z}_{p}^{\times}$ and decomposes into a product
$\Delta\times\mathbb{Z}_{p}$, where $\Delta$ is isomorphic to
$\mathbb{Z}/(p-1)\mathbb{Z}$. The cyclotomic $\mathbb{Z}_{p}$-extension
$\mathbb{Q}_{\operatorname{cyc}}$ is the field
$\mathbb{Q}(\mu_{p^{\infty}})^{\Delta}$ and the Galois group
$\operatorname{Gal}(\mathbb{Q}_{\operatorname{cyc}}/\mathbb{Q})$ is isomorphic
to $\mathbb{Z}_{p}$. For $n\geq 0$, the _$n$ -th layer_
$\mathbb{Q}_{n}/\mathbb{Q}$ is the extension contained in
$\mathbb{Q}_{\operatorname{cyc}}$ such that
$[\mathbb{Q}_{n}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]=p^{n}$.

###  3.1. Definition of Selmer groups

Let $S$ be a finite set of rational primes containing $\\{p,\infty\\}$, and
$\mathbb{Q}_{S}\subset\bar{\mathbb{Q}}$ be the maximal extension in which all
primes $\ell\notin S$ are unramified. Setting $A_{f}\mathrel{\mathop{\mathchar
58\relax}}=V_{f}/T_{f}$, we note that
$A_{f}\simeq\left(K/\mathcal{O}\right)^{2}$. Let $d^{\pm}$ be the dimension of
the $(\pm 1)$-eigenspaces for complex conjugation. We note that
$d^{+}=d^{-}=1$. Since we assume that $f$ is $p$-ordinary, there exists a
$1$-dimensional, $K$-subspace $W_{f}\subseteq V_{f}$ of dimension $1$, which
is stable under the action of $\operatorname{G}_{p}$. Let $C$ be the image of
$W_{f}$ in $A_{f}$, with respect to the mod-$T_{f}$ map $V_{f}\rightarrow
A_{f}$, and set $D=A_{f}/C$.

We assume that the set $S$ contains all primes dividing $N_{f}$. Let $L$ be an
extension of $\mathbb{Q}$ which is contained in
$\mathbb{Q}_{\operatorname{cyc}}$. Thus $L$ is the field $\mathbb{Q}_{n}$ for
some $n\geq 0$, or is the field $\mathbb{Q}_{\operatorname{cyc}}$. Given a
rational prime $\ell$, set $\ell(L)$ to be the set of primes of $L$ that lie
above $\ell$. We note that the set $\ell(\mathbb{Q}_{\operatorname{cyc}})$ is
finite. Given a prime $w$ of $\mathbb{Q}_{\operatorname{cyc}}$, let
$\mathbb{Q}_{\operatorname{cyc},w}$ be the union of all completions of
$p$-adic fields contained in $\mathbb{Q}_{\operatorname{cyc}}$ at $w$. Given a
prime $\ell\in S$, the Greenberg local Selmer condition
$\mathcal{H}_{\ell}(L,A_{f})$ is defined as follows. For $\ell\neq p$, the
local condition is defined by

$\mathcal{H}_{\ell}(L,A_{f})\mathrel{\mathop{\mathchar
58\relax}}=\prod_{w\in\ell(L)}H^{1}(L_{w},A_{f}).$

The local condition at $\ell=p$ is defined differently. Let $\eta_{p}$ be the
unique prime of $L$ that lies above $p$; let $\operatorname{I}_{\eta_{p}}$
denote the inertia subgroup of
$\operatorname{Gal}(\bar{L}_{\eta_{p}}/L_{\eta_{p}})$. Let

$H_{\eta_{p}}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(L_{\eta_{p}},A_{f})\longrightarrow
H^{1}(\operatorname{I}_{\eta_{p}},D)\right),$

and set

$\mathcal{H}_{p}(L,A_{f})\mathrel{\mathop{\mathchar
58\relax}}=H^{1}(L_{\eta_{p}},A_{f})/H_{\eta_{p}}.$

Following [G+89], the Greenberg Selmer group
$\operatorname{Sel}^{\operatorname{Gr}}(L,A_{f})$ is then defined as follows

$\operatorname{Sel}^{\operatorname{Gr}}(L,A_{f})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(\mathbb{Q}_{S}/L,A_{f})\longrightarrow\bigoplus_{\ell\in
S}\mathcal{H}_{\ell}(L,A_{f})\right).$

For $\ell\in S$, $\mathcal{H}_{\ell}(\mathbb{Q}_{\operatorname{cyc}},A_{f})$
is a cofinitely generated $\mathbb{Z}_{p}$-module (cf. [GV00, p.37]). We let
$\sigma_{\ell}(f)$ denote the $\mathcal{O}$-corank of
$\mathcal{H}_{\ell}(\mathbb{Q}_{\operatorname{cyc}},A_{f})$. For $\ell\neq p$,
we describe an algorithm to compute $\sigma_{\ell}(f)$. Let $V_{f}^{\prime}$
be the quotient $\left(V_{f}\right)_{\operatorname{I}_{\ell}}$ and
$\widetilde{P}_{\ell}(f;X)$ be the mod-$\varpi$ reduction of
$P_{\ell}(f;X)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{det}\left(\operatorname{Id}-\operatorname{fr}_{\ell}X\mid
V_{f}^{\prime}\right)$. Let $s_{\ell}$ be the number of primes of
$\mathbb{Q}_{\operatorname{cyc}}$ that lie above $\ell$. Note that $s_{\ell}$
is the maximal power of $p$ such that $\ell^{p-1}\equiv 1\mod{ps_{\ell}}$. Let
$d_{\ell}(f)$ be the multiplicity of $\ell^{-1}$ as a root of
$\widetilde{P}_{\ell}(f;X)$. By [GV00, Proposition 2.4], it follows that
$\sigma_{\ell}(f)$ is given by

(3.1) $\sigma_{\ell}(f)=s_{\ell}d_{\ell}(f).$

Given a number field $L/\mathbb{Q}$ contained in
$\mathbb{Q}_{\operatorname{cyc}}$, we let
$\operatorname{Sel}^{\operatorname{BK}}(L,A_{f})$ denote the Bloch-Kato Selmer
group associated to $A_{f}$, cf. [BK90] or [Och00, p.73, l.-1] for the precise
definition. The Bloch-Kato Selmer group over $\mathbb{Q}_{\operatorname{cyc}}$
is defined as the direct limit

$\operatorname{Sel}^{BK}(\mathbb{Q}_{\operatorname{cyc}},A_{f})\mathrel{\mathop{\mathchar
58\relax}}=\varinjlim_{n}\operatorname{Sel}^{BK}(\mathbb{Q}_{n},A_{f}).$

######  Proposition 3.1.

With respect to notation above, there is a natural map

$\operatorname{Sel}^{\operatorname{BK}}(\mathbb{Q}_{\operatorname{cyc}},A_{f})\rightarrow\operatorname{Sel}^{\operatorname{Gr}}(\mathbb{Q}_{\operatorname{cyc}},A_{f})$

with finite kernel and cokernel.

###### Proof.

The result above is [Och00, Corollary 4.3]. ∎

From the point of view of Iwasawa theory, we may work with either Selmer group
$\operatorname{Sel}^{\operatorname{BK}}(\mathbb{Q}_{\operatorname{cyc}},A_{f})$
or
$\operatorname{Sel}^{\operatorname{Gr}}(\mathbb{Q}_{\operatorname{cyc}},A_{f})$.
It conveniences us to work with the Greenberg Selmer group, and for ease of
notation, we simply set

$\operatorname{Sel}(\mathbb{Q}_{\operatorname{cyc}},A_{f})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Sel}^{\operatorname{Gr}}(\mathbb{Q}_{\operatorname{cyc}},A_{f}).$

###  3.2. Iwasawa invariants

We set $\Gamma\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\mathbb{Q}_{\operatorname{cyc}}/\mathbb{Q})$ and
$\Lambda$ denote the Iwasawa algebra
$\mathcal{O}\llbracket\Gamma\rrbracket\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{n}\mathcal{O}\left[\Gamma/\Gamma^{p^{n}}\right]$.
Given a module $M$ over $\Lambda$, we shall set
$M^{\vee}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Hom}_{\operatorname{cnts}}\left(M,\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)$.
A module $M$ over $\Lambda$ is said to be cofinitely generated (resp.
cotorsion) if $M^{\vee}$ is finitely generated (resp. torsion) as a module
over $\Lambda$. The Selmer group
$\operatorname{Sel}(\mathbb{Q}_{\operatorname{cyc}},A_{f})$ is a cofinitely
generated module over $\Lambda$. There are many cases of interest for which it
is known to be cotorsion over $\Lambda$, which we shall further discuss in the
next subsection. Given $\Lambda$-modules $M$ and $M^{\prime}$, a _pseudo-
isomorphism_ is a map of $\Lambda$-modules $M\rightarrow M^{\prime}$ whose
kernel and cokernel both have finite cardinality. Let $M$ be a cofinitely
generated and cotorsion $\Lambda$-module, then, as is well known, there is a
pseudo-isomorphism

$M^{\vee}\longrightarrow\left(\bigoplus_{i=1}^{s}\Lambda/(\varpi^{m_{i}})\right)\oplus\left(\bigoplus_{j=1}^{t}\Lambda/(f_{j}(T)^{n_{j}})\right).$

In the above sum, $s,t,m_{i},n_{i}\in\mathbb{Z}_{\geq 0}$, and $f_{j}(T)$ is
an irreducible distinguished polynomial in $\Lambda$, i.e., an irreducible
monic polynomial whose non-leading coefficients are divisible by $\varpi$. The
$\mu$-invariant is given by

$\mu_{p}(M)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\sum_{i}m_{i}&\text{ if }s>0;\\\ 0&\text{ if
}s=0.\end{cases}$

On the other hand, the $\lambda$-invariant is given by

$\lambda_{p}(M)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\sum_{j}n_{j}\operatorname{deg}(f_{j})&\text{ if
}t>0;\\\ 0&\text{ if }t=0.\end{cases}$

It follows from results of Kato [K+04] that
$\operatorname{Sel}(\mathbb{Q}_{\operatorname{cyc}}/A_{f})$ is a cotorsion
module over $\Lambda$. Then, the $\mu$ (resp. $\lambda$) invariant of
$\operatorname{Sel}(\mathbb{Q}_{\operatorname{cyc}},A_{f})$ is denoted by
$\mu_{p}(f)$ (resp. $\lambda_{p}(g)$). We note that $\mu_{p}(f)=0$ if and only
if $\operatorname{Sel}(\mathbb{Q}_{\operatorname{cyc}},A_{f})$ is cofinitely
generated as an $\mathcal{O}$-module, and in this case,

$\lambda_{p}(f)=\operatorname{corank}_{\mathcal{O}}\left(\operatorname{Sel}(\mathbb{Q}_{\operatorname{cyc}},A_{f})\right).$

It follows from Proposition 3.1 that the $\lambda$-invariant of
$\operatorname{Sel}(\mathbb{Q}_{\operatorname{cyc}},A_{f})$ is equal to the
$\lambda$-invariant of
$\operatorname{Sel}^{\operatorname{BK}}(\mathbb{Q}_{\operatorname{cyc}},A_{f})$.
We shall set $\operatorname{rank}^{\operatorname{BK}}(f)$ to denote the
$\mathcal{O}$-corank of the Bloch-Kato Selmer group
$\operatorname{Sel}^{\operatorname{BK}}(\mathbb{Q},A_{f})$.

######  Proposition 3.2.

Assume the $\mu_{p}(f)=0$. The following assertions hold:

  1. (1)

$\operatorname{rank}^{\operatorname{BK}}(f)\leq\lambda_{p}(f)$,

  2. (2)

$\operatorname{rank}^{\operatorname{BK}}(f)\equiv\lambda_{p}(f)\mod{2}$,

  3. (3)

suppose that $\lambda_{p}(f)\leq 1$, then,
$\operatorname{rank}^{\operatorname{BK}}(f)=\lambda_{p}(f)$.

###### Proof.

Part (1) follows as a direct consequence of [Och00, Theorem A]. Part (2)
follows from a standard argument due to Greenberg [Gre99, Proposition 3.10],
see [HL19, p.1288, proof of Theorem 5.7, ll. 9-15]. Part (3) is a direct
consequence of (1) and (2). ∎

######  Remark 3.3.

We note that the conventions used here are the same as those in [GV00, section
2]. Note that the ring $\mathcal{O}$ is chosen to be the valuation ring of
$F_{\mathfrak{p}}$. It is easy to see that the definition of the
$\lambda$-invariant $\lambda_{p}(f)$ is independent of the choice of field
$F$, and thus the valuation ring $\mathcal{O}$. Also, we note that the
$\mu$-invariant vanishes for the Selmer group over $\mathcal{O}$, if and only
if it vanishes after base-change by any valuation ring
$\mathcal{O}^{\prime}/\mathcal{O}$.

##  4\. Congruence relations between Selmer groups

We recall some results of Greenberg and Vatsal [GV00], and apply these results
to study the structure of Selmer groups of ordinary Galois representations.
Let $g$ be a Hecke newform of weight $2$ on $\Gamma_{0}(N_{g})$. Here, $N_{g}$
is the level of $g$. Assume that the following conditions are satisfied

  1. (1)

$p\nmid N_{g}$,

  2. (2)

$g$ is ordinary at $p$,

  3. (3)

$\bar{\rho}\mathrel{\mathop{\mathchar 58\relax}}=\bar{\rho}_{g}$ is absolutely
irreducible,

  4. (4)

$g$ has optimal level, i.e., $N_{g}$ is the prime to $p$ part of the Artin
conductor of $\bar{\rho}$.

Let $f$ be a Hecke new cuspform of weight $2$ on such that
$\bar{\rho}_{f}\simeq\bar{\rho}_{g}$. The understanding here is that the field
$F$ is chosen so that all the Fourier coefficients of both $f$ and $g$ are
contained in $F$. Remark 3.3 establishes that one may choose any large field
$F$ when studying the $\lambda$-invariant or the vanishing of the
$\mu$-invariant. Let $N_{f}$ be the level of $f$. We assume that $p\nmid
N_{f}$. Note that by Theorem 2.1, $N_{f}\in\mathcal{S}(\bar{\rho})$. We note
that $N_{g}$ divides $N_{f}$. We note that $f$ has ordinary reduction at $p$
(cf. [Ray23, Lemma 3.3]). Let $\epsilon_{f}$ be the nebentype of $f$, and
$\bar{\epsilon}_{f}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\kappa^{\times}$
its mod-$\varpi$ reduction. We find that
$\operatorname{det}\bar{\rho}_{g}=\bar{\chi}$, and
$\operatorname{det}\bar{\rho}_{g}=\bar{\chi}\bar{\epsilon}_{f}$ where
$\bar{\chi}$ is the mod-$p$ cyclotomic character. Since
$\bar{\rho}_{f}\simeq\bar{\rho}_{g}$, we find that $\bar{\epsilon}_{f}=1$. The
principal units in $\mathcal{O}^{\times}$ are the units that reduce to $1$
modulo $(\varpi)$. It is clear that if a principal unit is a root of unity,
then it is equal to $1$. Since $\epsilon_{f}$ is a finite order character,
$\bar{\epsilon}_{f}=1$ implies that $\epsilon_{f}=1$. Therefore, $f$ is a
weight $2$ cuspform on $\Gamma_{0}(N_{f})$.

######  Theorem 4.1 (Greenberg-Vatsal [GV00]).

With respect to above notation, assume that $\mu_{p}(g)=0$. Then,
$\mu_{p}(f)=0$, and

$\lambda_{p}(f)-\lambda_{p}(g)=\sum_{\ell\mid
N_{f}}\left(\sigma_{\ell}(g)-\sigma_{\ell}(f)\right),$

where the sum ranges over the primes $\ell$ dividing $N_{f}$.

We study the numbers $\left(\sigma_{\ell}(g)-\sigma_{\ell}(f)\right)$ for
primes $\ell|N_{f}$. Let $A$ be a local $\mathcal{O}$-algebra, with maximal
ideal $\mathfrak{m}_{A}$, and residue field $A/\mathfrak{m}_{A}$ isomorphic to
$\kappa$. Fix an $\mathcal{O}$-algebra isomorphism
$\varphi_{0}\mathrel{\mathop{\mathchar
58\relax}}A/\mathfrak{m}_{A}\rightarrow\kappa$. Let
$\varphi\mathrel{\mathop{\mathchar 58\relax}}A\rightarrow\kappa$ be the map
obtained upon composing the reduction modulo $\mathfrak{m}_{A}$ map, with
$\varphi_{0}$. We set

$\widehat{\operatorname{GL}}_{2}(A)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\left({\begin{array}[]{cc}a&b\\\ c&d\\\
\end{array}}\right)\in\operatorname{GL}_{2}(A)\mid\left({\begin{array}[]{cc}\varphi(a)&\varphi(b)\\\
\varphi(c)&\varphi(d)\\\ \end{array}}\right)=\left({\begin{array}[]{cc}1&0\\\
0&1\\\ \end{array}}\right)\right\\}.$

For $n\geq 1$, the map $\varphi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{O}/\varpi^{n}\rightarrow\kappa$ is the reduction modulo
$\varpi$ map.

######  Definition 4.2.

Associated with $g$, let $\Pi_{g}$ be the set of prime numbers $\ell$ such
that the following conditions are satisfied:

  1. (1)

$\ell\nmid N_{g}p$,

  2. (2)

$\ell\not\equiv\pm 1\mod{p}$ and $\ell^{p-1}\not\equiv 1\mod{p^{2}}$,

  3. (3)

$\bar{\rho}_{g}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell&0\\\
0&1\\\ \end{array}}\right)$ for a suitable choice of basis for the residual
representation.

Let $\Omega_{g}$ be the set of prime numbers such that

  1. (1)

$\ell\nmid N_{g}p$,

  2. (2)

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

  3. (3)

$\bar{\rho}_{g}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}-\ell&0\\\
0&-1\\\ \end{array}}\right)$ for a suitable choice of basis for the residual
representation.

In section 5, we shall introduce further assumptions on the image of
$\bar{\rho}_{g}$, so that the sets $\Pi_{g}$ and $\Omega_{g}$ have positive
density. For $\ell\in\Pi_{g}$ (resp. $\Omega_{g}$), we note that $\rho_{g}$ is
unramified at $\ell$. Since $\bar{\rho}_{f}\simeq\bar{\rho}_{g}$, we find that
$\bar{\rho}_{f}$ is unramified at $\ell$ as well. Thus,
$\rho_{f}(\operatorname{I}_{\ell})$ is contained in
$\widehat{\operatorname{GL}}_{2}(\mathcal{O})$, which is a pro-$p$ group.
Hence, $\rho_{f|\ell}$ factors through
$\operatorname{Gal}(\mathbb{Q}_{\ell}^{p-tr}/\mathbb{Q}_{\ell})$, where
$\mathbb{Q}_{\ell}^{p-tr}\subset\bar{\mathbb{Q}}_{\ell}$ is the maximal tamely
ramified extension of $\mathbb{Q}_{\ell}$ with pro-$p$ inertia. The Galois
group $\operatorname{Gal}(\mathbb{Q}_{\ell}^{p-tr}/\mathbb{Q}_{\ell})$ is
generated by a frobenius $\operatorname{fr}_{\ell}$, and the pro-$p$ tame
inertia generator $\tau_{\ell}$, subject to the relation
$\operatorname{fr}_{\ell}\tau_{\ell}\operatorname{fr}_{\ell}^{-1}=\tau_{\ell}^{\ell}$
(cf. for instance [Ram02, p.123, ll.15-16]).

Let $n\in\mathbb{Z}_{\geq 1}$, and consider the reduction mod-$\varpi^{n}$ map

$\pi_{n}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{GL}_{2}(\mathcal{O}/\varpi^{n+1})\rightarrow\operatorname{GL}_{2}(\mathcal{O}/\varpi^{n}).$

Given a ring $A$, denote by $\operatorname{M}_{2}(A)$ the group of $2\times 2$
matrices with entries in $A$. A matrix $M$ in the kernel of $\pi_{n}$ can be
written in the following form

$M=\operatorname{Id}+\varpi^{n}\left({\begin{array}[]{cc}a&b\\\ c&d\\\
\end{array}}\right).$

Here, $\left({\begin{array}[]{cc}a&b\\\ c&d\\\ \end{array}}\right)$ belongs to
$\operatorname{M}_{2}(\mathcal{O}/\varpi^{n+1})$. However, since the
coefficients of the matrix $M$ belong to $\mathcal{O}/\varpi^{n+1}$, it
follows that $a,b,c,d$ are determined modulo $\varpi$. Thus the matrix
$\left({\begin{array}[]{cc}a&b\\\ c&d\\\ \end{array}}\right)$ can be assumed
to be in $\operatorname{M}_{2}(\mathcal{O}/\varpi)$.

To illustrate this, consider an example in the case when $n=1$, and
$\left({\begin{array}[]{cc}a&b\\\ c&d\\\
\end{array}}\right)=\left({\begin{array}[]{cc}1&0\\\ 0&0\\\
\end{array}}\right)$. We find that

$M=\operatorname{Id}+\varpi\left({\begin{array}[]{cc}1&0\\\ 0&0\\\
\end{array}}\right)=\left({\begin{array}[]{cc}1+\varpi&0\\\ 0&1\\\
\end{array}}\right).$

On the other hand,

$\operatorname{Id}+\varpi\left({\begin{array}[]{cc}1+\varpi&0\\\ 0&0\\\
\end{array}}\right)=\left({\begin{array}[]{cc}1+\varpi+\varpi^{2}&0\\\ 0&1\\\
\end{array}}\right)=\left({\begin{array}[]{cc}1+\varpi&0\\\ 0&1\\\
\end{array}}\right)=M,$

since $\varpi^{2}=0$ when the coefficients are in $\mathcal{O}/\varpi^{2}$.

For $i=1,2$ and $M_{i}\in\operatorname{ker}\pi_{n}$, and let $B_{i}\in
M_{2}(\mathcal{O}/\varpi)$ be such that
$M_{i}=\operatorname{Id}+\varpi^{n}B_{i}$. Then, since $2n\geq n+1$, we have
$\varpi^{2n}=0$. Therefore, we find that

$M_{1}M_{2}=\left(\operatorname{Id}+\varpi^{n}B_{1}\right)\left(\operatorname{Id}+\varpi^{n}B_{2}\right)=\operatorname{Id}+\varpi^{n}(B_{1}+B_{2}).$

The inverse of $M=\operatorname{Id}+\varpi^{n}B$ is given by
$M^{-1}=\operatorname{Id}-\varpi^{n}B$. It is easy to see that the map

$\operatorname{M}_{2}(\mathcal{O}/\varpi)\rightarrow\operatorname{ker}\pi_{n},$

mapping $B$ to $\operatorname{Id}+\varpi^{n}B$ is an isomorphism of abelian
groups. Let $B\in\operatorname{ker}\pi_{n}$ and
$M\in\operatorname{GL}_{2}(\mathcal{O}/\varpi^{n+1})$. We write
$B=\operatorname{Id}+\varpi^{n}B^{\prime}$, where
$B^{\prime}=\left({\begin{array}[]{cc}a&b\\\ c&d\\\
\end{array}}\right)\in\operatorname{M}_{2}(\mathcal{O}/\varpi)$. We write

$[B^{\prime},M]\mathrel{\mathop{\mathchar
58\relax}}=B^{\prime}\bar{M}-\bar{M}B^{\prime}\in\operatorname{M}_{2}(\mathcal{O}/\varpi),$

where $\bar{M}$ is the reduction of $M$ modulo $\varpi$.

######  Lemma 4.3.

With respect to notation above,

$BMB^{-1}=M+\varpi^{n}[B^{\prime},M].$

###### Proof.

We find that since $\varpi^{n+1}=0$, in particular, $\varpi^{2n}=0$.
Therefore, we find that

$\begin{split}BMB^{-1}&=(\operatorname{Id}+\varpi^{n}B^{\prime})M(\operatorname{Id}-\varpi^{n}B^{\prime})\\\
&=M+\varpi^{n}[B^{\prime},M].\end{split}$

Note that $\varpi^{n}[B^{\prime},\bar{M}]$ is determined by the reduction of
$M$ modulo $\varpi$. ∎

######  Lemma 4.4.

Let $\ell$ be a prime such that

  1. (1)

$\ell\nmid N_{g}p$,

  2. (2)

$\ell\nmid\pm 1\mod{p}$,

  3. (3)

$\bar{\rho}_{g}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\eta\ell&\\\
&\eta\\\ \end{array}}\right)$, where $\eta=\pm 1$,

  4. (4)

$\rho_{f}$ is ramified at $\ell$.

Then, up to conjugation by a matrix in $\operatorname{GL}_{2}(\mathcal{O})$,

$\rho_{f}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell x&\\\ &x\\\
\end{array}}\right)\text{ and
}\rho_{f}(\tau_{\ell})=\left({\begin{array}[]{cc}1&y\\\ &1\\\
\end{array}}\right),$

where $x\equiv\eta\mod{\varpi}$ and $y\equiv 0\mod{\varpi}$.

###### Proof.

We choose an integral basis for $T_{f}$ so that
$\bar{\rho}_{f}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\eta\ell&0\\\
0&\eta\\\ \end{array}}\right)$. Since $\ell\nmid N_{g}p$, $\bar{\rho}_{g}$ is
unramified at $\ell$. Since $f$ and $g$ satisfy the congruence
$\bar{\rho}_{f}\simeq\bar{\rho}_{g}$, it follows that $\bar{\rho}_{f}$ is
unramified at $\ell$. Since $\tau_{\ell}$ is in the inertia group at $\ell$,
we find that $\bar{\rho}_{f}(\tau_{\ell})=\operatorname{Id}$. Thus, for $n=1$,
the upper triangular entry is $0$.

We show that there is a matrix $A\in\operatorname{GL}_{2}(\mathcal{O})$ so
that $\rho\mathrel{\mathop{\mathchar 58\relax}}=A\rho_{f}A^{-1}$ satisfies the
above conditions. In fact, the matrix $A$ can be chosen to be in
$\widehat{\operatorname{GL}}_{2}(\mathcal{O})$. For ease of notation, we set
$\rho^{\prime}\mathrel{\mathop{\mathchar 58\relax}}=\rho_{f}$ and
$\rho^{\prime}_{n}$ denote the $\mod{\varpi^{n}}$ reduction of
$\rho^{\prime}$. We shall inductively specify matrices
$A_{n}\in\widehat{\operatorname{GL}}_{2}(\mathcal{O}/\varpi^{n})$ such that
$A_{m}\equiv A_{n}\mod{\varpi^{n}}$ for all integers $m>n$. We shall then let
$A$ denote the matrix $(A_{n})$ in the inverse limit

$\widehat{\operatorname{GL}}_{2}(\mathcal{O})=\varprojlim_{n}\widehat{\operatorname{GL}}_{2}(\mathcal{O}/\varpi^{n}).$

Let us set $\rho_{n}\mathrel{\mathop{\mathchar
58\relax}}=A_{n}\rho_{n}^{\prime}A_{n}^{-1}$. The matrices $A_{n}$ shall have
the property that there exist $x_{n},y_{n},z_{n}\in\mathcal{O}/\varpi^{n}$
such that $\rho_{n}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell
x_{n}&\\\ &z_{n}\\\ \end{array}}\right)$ and
$\rho_{n}(\tau_{\ell})=\left({\begin{array}[]{cc}1&y_{n}\\\ &1\\\
\end{array}}\right)$, where $x_{n},z_{n}\equiv\eta\mod{\varpi}$ and
$y_{n}\equiv 0\mod{\varpi}$. Since $A_{m}\equiv A_{n}\mod{\varpi^{n}}$ for all
$m>n$, it shall then follow that $x_{m}\equiv x_{n}\mod{\varpi^{n}}$,
$y_{m}\equiv y_{n}\mod{\varpi^{n}}$ and $z_{m}\equiv z_{n}\mod{\varpi^{n}}$.
Setting $x,y,z$ to denote the inverse limits $(x_{n})$, $(y_{n})$ and
$(z_{n})$ respectively, we find that

$\rho(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell x&\\\ &z\\\
\end{array}}\right)\text{ and
}\rho(\tau_{\ell})=\left({\begin{array}[]{cc}1&y\\\ &1\\\
\end{array}}\right).$

The relation
$\operatorname{fr}_{\ell}\tau_{\ell}\operatorname{fr}_{\ell}^{-1}=\tau_{\ell}^{\ell}$
then implies that $x=z$.

Suppose that for some $n\geq 1$,

$\rho_{n}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell x_{n}&\\\
&z_{n}\\\ \end{array}}\right)\text{ and
}\rho_{n}(\tau_{\ell})=\left({\begin{array}[]{cc}1&y_{n}\\\ &1\\\
\end{array}}\right).$

It suffices to lift $A_{n}$ to
$A_{n+1}\in\widehat{\operatorname{GL}}_{2}(\mathcal{O}/\varpi^{n+1})$, so that

$\rho_{n+1}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell
x_{n+1}&\\\ &z_{n+1}\\\ \end{array}}\right)\text{ and
}\rho_{n+1}(\tau_{\ell})=\left({\begin{array}[]{cc}1&y_{n+1}\\\ &1\\\
\end{array}}\right),$

where $x_{n+1},y_{n+1},z_{n+1}$ lift $x_{n},y_{n}$ and $z_{n}$ respectively.
Let $B_{n+1}\in\widehat{\operatorname{GL}}_{2}(\mathcal{O}/\varpi^{n+1})$ be a
lift of $A_{n}$, and set $r_{n+1}\mathrel{\mathop{\mathchar
58\relax}}=B_{n+1}\rho_{n+1}^{\prime}B_{n+1}^{-1}$. Note that
$\rho_{n}=r_{n+1}\mod{\varpi^{n}}$. In particular,
$r_{n+1}(\operatorname{fr}_{\ell})$ lifts
$\rho_{n}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell x_{n}&\\\
&z_{n}\\\ \end{array}}\right)$ and $r_{n+1}(\tau_{\ell})$ lifts
$\rho_{n}(\tau_{\ell})=\left({\begin{array}[]{cc}1&y_{n}\\\ &1\\\
\end{array}}\right)$. Choose lifts $\tilde{x}_{n},\tilde{y}_{n},\tilde{z}_{n}$
of $x_{n},y_{n},z_{n}$ to $\mathcal{O}/\varpi^{n+1}$. Then,
$\rho_{n}(\operatorname{fr}_{\ell})$ is
$\left({\begin{array}[]{cc}\ell\tilde{x}_{n}&\\\ &\tilde{z}_{n}\\\
\end{array}}\right)\mod{\varpi^{n}}$, and this just means that there is
$\left({\begin{array}[]{cc}a&b\\\ c&d\\\ \end{array}}\right)$ with
coefficients determined mod-$\varpi$ such that

$r_{n+1}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell\tilde{x}_{n}&\\\
&\tilde{z}_{n}\\\
\end{array}}\right)+\varpi^{n}\left({\begin{array}[]{cc}a&b\\\ c&d\\\
\end{array}}\right).$

Similarly, there is $\left({\begin{array}[]{cc}e&f\\\ g&h\\\
\end{array}}\right)$ with coefficients determined mod-$\varpi$ such that

$r_{n+1}(\tau_{\ell})=\left({\begin{array}[]{cc}1&\widetilde{y}_{n}\\\ 0&1\\\
\end{array}}\right)+\varpi^{n}\left({\begin{array}[]{cc}e&f\\\ g&h\\\
\end{array}}\right).$

These coefficients $a,b,c,d,e,f,g,h\in\mathcal{O}/\varpi$ are determined by
$r_{n+1}$, i.e., they are determined by $\rho_{n+1}^{\prime}$ and the choice
of matrix $B_{n+1}$. Let

$B\mathrel{\mathop{\mathchar
58\relax}}=\left(\operatorname{Id}+\varpi^{n}B^{\prime}\right)\in\widehat{\operatorname{GL}}_{2}(\mathcal{O}/\varpi^{n+1}),$

and we set $r_{n+1}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=Br_{n+1}B^{-1}$. Write
$B^{\prime}=\left({\begin{array}[]{cc}a^{\prime}&b^{\prime}\\\
c^{\prime}&d^{\prime}\\\ \end{array}}\right)$, in due course, the values of
$a^{\prime},b^{\prime},c^{\prime}$ and $d^{\prime}$ shall be prescribed.

Let $M\in\operatorname{GL}_{2}(\mathcal{O}/\varpi^{n+1})$ and
$B\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Id}+\varpi^{n}B^{\prime}$ in
$\operatorname{ker}\pi_{n}$. It follows from Lemma 4.3 that

$BMB^{-1}=M+\varpi^{n}[B^{\prime},M].$

If $M\equiv\left({\begin{array}[]{cc}\eta\ell&\\\ &\eta\\\
\end{array}}\right)\mod{\varpi}$, then,

(4.1)
$BMB^{-1}=M+\varpi^{n}\left({\begin{array}[]{cc}&\eta(1-\ell)b^{\prime}\\\
\eta(\ell-1)c^{\prime}&\\\ \end{array}}\right).$

On the other hand, if $M\equiv\operatorname{Id}\mod{\varpi}$, then,
$[B^{\prime},M]=0$, and

(4.2) $BMB^{-1}=M.$

From the above computations, we find that

$\begin{split}&r_{n+1}^{\prime}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell\tilde{x}_{n}&\\\
&\tilde{z}_{n}\\\
\end{array}}\right)+\varpi^{n}\left({\begin{array}[]{cc}a&b+\eta(1-\ell)b^{\prime}\\\
c+\eta(\ell-1)c^{\prime}&d\\\ \end{array}}\right);\\\
&r_{n+1}^{\prime}(\tau_{\ell})=\left({\begin{array}[]{cc}1&\widetilde{y}_{n}\\\
0&1\\\ \end{array}}\right)+\varpi^{n}\left({\begin{array}[]{cc}e&f\\\ g&h\\\
\end{array}}\right).\end{split}$

The first of the above equations follows from (4.1) taking
$M\mathrel{\mathop{\mathchar 58\relax}}=r_{n+1}(\operatorname{fr}_{\ell})$ and
the second follows from (4.2) upon taking $M\mathrel{\mathop{\mathchar
58\relax}}=r_{n+1}(\tau_{\ell})$. Since $\ell\not\equiv 1\mod{p}$, we may
choose $B$ so that $r_{n+1}^{\prime}(\operatorname{fr}_{\ell})$ is diagonal.
In greater detail, $b^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\frac{b}{\eta(\ell-1)}$ and $c^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\frac{c}{\eta(1-\ell)}$, and set $a^{\prime}=0$ and $d^{\prime}=0$.
Thus, we write

$r_{n+1}^{\prime}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell
x_{n+1}&\\\ &z_{n+1}\\\ \end{array}}\right).$

We find that

$\begin{split}r_{n+1}^{\prime}(\operatorname{fr}_{\ell}\tau_{\ell}\operatorname{fr}_{\ell}^{-1})=&\left({\begin{array}[]{cc}\ell
x_{n+1}&\\\ &z_{n+1}\\\
\end{array}}\right)\left({\begin{array}[]{cc}1+\varpi^{n}e&\widetilde{y}_{n}+\varpi^{n}f\\\
\varpi^{n}g&1+\varpi^{n}h\\\ \end{array}}\right)\left({\begin{array}[]{cc}\ell
x_{n+1}&\\\ &z_{n+1}\\\ \end{array}}\right)^{-1},\\\
=&\left({\begin{array}[]{cc}1+\varpi^{n}e&\ell
x_{n+1}z_{n+1}^{-1}\left(\widetilde{y}_{n}+\varpi^{n}f\right)\\\
\varpi^{n}\ell^{-1}z_{n+1}x_{n+1}^{-1}g&1+\varpi^{n}h\\\
\end{array}}\right),\\\ =&\left({\begin{array}[]{cc}1+\varpi^{n}e&\ell
x_{n+1}z_{n+1}^{-1}\widetilde{y}_{n}+\varpi^{n}\ell f\\\
\varpi^{n}\ell^{-1}g&1+\varpi^{n}h\\\ \end{array}}\right),\\\
r_{n+1}^{\prime}(\tau_{\ell})^{\ell}=&\left({\begin{array}[]{cc}1+\varpi^{n}\ell
e&\ell\widetilde{y}_{n}+\varpi^{n}\ell f\\\ \varpi^{n}\ell g&1+\varpi^{n}\ell
h\\\ \end{array}}\right).\end{split}$

We note that $x_{n+1}\equiv z_{n+1}\equiv\eta\mod{\varpi}$, and thus
$x_{n+1}z_{n+1}^{-1}\equiv 1\mod{\varpi}$. The coefficients of the above
matrices are in $\mathcal{O}/\varpi^{n+1}$, and hence, the relations

$\varpi^{n}x_{n+1}z_{n+1}^{-1}=\varpi^{n}\text{ and
}\varpi^{n}z_{n+1}x_{n+1}^{-1}=\varpi^{n}$

are used in the above equations. Since $\ell\not\equiv\pm 1\mod{p}$, we find
that $e=g=h=0$. Hence, we find that

$\begin{split}&r_{n+1}^{\prime}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell
x_{n+1}&\\\ &z_{n+1}\\\ \end{array}}\right);\\\
&r_{n+1}^{\prime}(\tau_{\ell})=\left({\begin{array}[]{cc}1&y_{n+1}\\\ &1\\\
\end{array}}\right).\end{split}$

We set $A_{n+1}\mathrel{\mathop{\mathchar 58\relax}}=BB_{n+1}$, we note that
$A_{n}\equiv B_{n+1}\mod{\varpi^{n}}$, and
$B\equiv\operatorname{Id}\mod{\varpi^{n}}$. With respect to this choice,
$\rho_{n+1}\mathrel{\mathop{\mathchar 58\relax}}=r_{n+1}^{\prime}$, and
$A_{n}=A_{n+1}\mod{\varpi^{n}}$. This completes the inductive lifting
argument. By previous remarks in the proof, this is enough to establish the
result. ∎

######  Lemma 4.5.

Let $\ell\nmid N_{g}p$ be a prime that divides $N_{f}$. Then, the following
assertions hold.

  1. (1)

If $\ell\in\Pi_{g}$, then, $\sigma_{\ell}(g)=1$ and $\sigma_{\ell}(f)=0$.

  2. (2)

If $\ell\in\Omega_{g}$, then, $\sigma_{\ell}(g)=0$ and $\sigma_{\ell}(f)=0$.

###### Proof.

We begin by proving part (1). Since $\rho_{g}$ is unramified at $\ell$, we
find that $V^{\prime}_{g}=V_{g}$. Since
$\bar{\rho}_{g}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell&\\\
&1\\\ \end{array}}\right)$, it follows that
$\widetilde{P}_{\ell}(g;X)=(1-X)(1-\ell X)$. Since $\ell\not\equiv 1\mod{p}$,
it follows that $\ell^{-1}$ is a root of $\widetilde{P}_{\ell}(g;X)$ with
multiplicity $1$. Therefore, $d_{\ell}(g)=1$. On the other hand, it follows
from Lemma 4.4 that $V_{f}^{\prime}$ is the trivial
$\operatorname{G}_{\ell}$-module, we find that
$\widetilde{P}_{\ell}(f;X)=1-X$, and $\ell^{-1}$ is not a root of
$\widetilde{P}_{\ell}(f;X)$. Therefore, $d_{\ell}(f)=0$. Since
$\ell^{p-1}\not\equiv 1\mod{p^{2}}$, it follows that that $s_{\ell}=1$. Since
$\sigma_{\ell}(\cdot)=s_{\ell}d_{\ell}(\cdot)$, we find that
$\sigma_{\ell}(g)=1$ and $\sigma_{\ell}(f)=0$.

Next, we prove assertion (2). Since
$\bar{\rho}_{g}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}-\ell&\\\
&-1\\\ \end{array}}\right)$, it follows that
$\widetilde{P}_{\ell}(g;X)=(1+X)(1+\ell X)$. Thus, the roots of
$\widetilde{P}_{\ell}(g;X)$ are $\\{-1,-\ell^{-1}\\}$. Since
$\ell\not\equiv-1\mod{p}$, we find that $\ell^{-1}$ is not a root of
$\widetilde{P}_{\ell}(g;X)$, and therefore, $d_{\ell}(g)=0$. Hence, it follows
that $\sigma_{\ell}(g)=0$. As in the proof of part (1), it follows that
$\widetilde{P}_{\ell}(f;X)=(1+X)$. Therefore, $\ell^{-1}$ is not a root of
this polynomial, and it thus follows that $\sigma_{\ell}(f)=0$. ∎

Let $Q=\\{q_{1},\dots,q_{n}\\}$ be a set of primes contained in $\Pi_{g}$ and
$Q^{\prime}=\\{\ell_{1},\dots,\ell_{r}\\}$. For $(n,r)\in\mathbb{Z}_{\geq
0}^{2}$, we set $\mathfrak{T}(n,r)$ be the collection of all sets
$\Sigma=Q\cup Q^{\prime}$, where $Q=\\{q_{1},\dots,q_{n}\\}$ (resp.
$Q^{\prime}=\\{\ell_{1},\dots,\ell_{r}\\}$) is a subset of $\Pi_{g}$ (resp.
$\Omega_{g}$). The understanding is that when $n=0$ (resp. $r=0$), the set $Q$
(resp. $Q^{\prime}$) is empty. When we write
$\Sigma=\\{q_{1},\dots,q_{n},\ell_{1},\dots,\ell_{r}\\}$, we shall implicitly
mean that $\\{q_{1},\dots,q_{n}\\}$ is contained in $\Pi_{g}$ and
$\\{\ell_{1},\dots,\ell_{r}\\}$ is contained in $\Omega_{g}$.

######  Definition 4.6.

For $\Sigma\in\mathfrak{T}(n,r)$, set $N_{\Sigma}\mathrel{\mathop{\mathchar
58\relax}}=\prod_{i=1}^{n}q_{i}\times\prod_{j=1}^{r}\ell_{j}$. Let
$\mathcal{F}(\Sigma)$ be the set of newforms $f$ of weight $2$ such that

  1. (1)

$\bar{\rho}_{f}\simeq\bar{\rho}_{g}$,

  2. (2)

$N_{f}=N_{g}N_{\Sigma}$.

######  Proposition 4.7.

Let $g$ be a Hecke newform of optimal weight $k=2$ and optimal level. Then,
for $(n,r)$ such that $n>0$ or $r>0$, and $\Sigma\in\mathfrak{T}(n,r)$, the
set $\mathcal{F}(\Sigma)$ is nonempty.

###### Proof.

The result is a direct consequence of Theorem 2.3. ∎

######  Lemma 4.8.

Let $f\in\mathcal{F}(\Sigma)$, and let $\ell\neq p$ be a prime which divides
$N_{g}$. Then, $\sigma_{\ell}(f)=\sigma_{\ell}(g)$.

###### Proof.

It suffices for us to show that $d_{\ell}(f)=d_{\ell}(g)$. Let
$h\in\\{g,f\\}$, recall that $V_{h}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=(V_{h})_{\operatorname{I}_{\ell}}$. Note that as modules over the
inertia group $\operatorname{I}_{\ell}$, $V_{h}$ and $A_{h}[\varpi]$ are self
dual. Since
$\operatorname{ord}_{\ell}(N_{f})=\operatorname{ord}_{\ell}(N_{g})$, and
$N_{g}$ is the prime to $p$ part of the Artin-conductor of $\bar{\rho}$, it
follows that

(4.3)
$\operatorname{dim}A_{h}[\varpi]_{\operatorname{I}_{\ell}}=\operatorname{dim}(V_{h})_{\operatorname{I}_{\ell}},$

(cf. [EPW06, proof of Lemma 4.1.2]). Recall that
$P_{\ell}(h;X)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{det}\left(\operatorname{Id}-\operatorname{fr}_{\ell}X\mid
V_{h}^{\prime}\right)$ and that $\widetilde{P}_{\ell}(h;X)$ is the
mod-$\varpi$ reduction of $P_{\ell}(h;X)$. Set
$T_{h}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=(T_{h})_{\operatorname{I}_{\ell}}$. We identify $A_{h}[\varpi]$
with $T_{h}/\varpi T_{h}$, and thus $A_{h}[\varpi]_{\operatorname{I}_{\ell}}$
with $T_{h}^{\prime}/\varpi T_{h}^{\prime}$. The equality (4.3) implies that
$T_{h}^{\prime}$ is torsion free. We identify
$T_{h}^{\prime}\otimes_{\mathcal{O}}K$ with $V_{h}^{\prime}$, and since
$T_{h}^{\prime}$ is torsion free, we find that $T_{h}^{\prime}$ is an
$\mathcal{O}$-lattice in $V_{h}^{\prime}$, and that
$P_{\ell}(h;X)=\operatorname{det}\left(\operatorname{Id}-\operatorname{fr}_{\ell}X\mid
T_{h}^{\prime}\right)$. Therefore, the mod-$\varpi$ reduction of
$P_{\ell}(h;X)$ is given by

$\begin{split}\widetilde{P}_{\ell}(h;X)=&\operatorname{det}\left(\operatorname{Id}-\operatorname{fr}_{\ell}X\mid\left(T_{h}^{\prime}/\varpi
T_{h}^{\prime}\right)\right),\\\
=&\operatorname{det}\left(\operatorname{Id}-\operatorname{fr}_{\ell}X\mid
A_{h}[\varpi]_{\operatorname{I}_{\ell}}\right),\\\ \end{split}$

Since $A_{g}[\varpi]\simeq A_{f}[\varpi]$ as
$\operatorname{G}_{\ell}$-modules, we find that
$A_{g}[\varpi]_{\operatorname{I}_{\ell}}\simeq
A_{f}[\varpi]_{\operatorname{I}_{\ell}}$ as
$\operatorname{G}_{\ell}/\operatorname{I}_{\ell}$-modules. It thus follows
that $\widetilde{P}_{\ell}(f;X)=\widetilde{P}_{\ell}(g;X)$, from which we
deduce that $d_{\ell}(f)=d_{\ell}(g)$. ∎

######  Theorem 4.9.

Let $g$ be a Hecke newform of optimal level $N_{g}$, trivial nebentype and
weight $2$. Assume that the following conditions are satisfied

  1. (1)

$\bar{\rho}_{g}$ is absolutely irreducible,

  2. (2)

$g$ is $p$-ordinary and $p\nmid N_{g}$,

  3. (3)

$g$ has optimal level,

  4. (4)

$\mu_{p}(g)=0$.

Let $\Sigma\in\mathfrak{T}(n,r)$ and $f\in\mathcal{F}(\Sigma)$. Then, the
following assertions hold

  1. (1)

$f$ has good ordinary reduction at $p$,

  2. (2)

$\mu_{p}(f)=0$,

  3. (3)

$\lambda_{p}(f)=\lambda_{p}(g)+n$.

###### Proof.

As noted earlier, that $f$ has ordinary reduction at $p$ follows from [Ray23,
Lemma 3.3]. From Theorem 4.1, we find that $\mu_{p}(f)=0$ and

$\lambda_{p}(f)=\lambda_{p}(g)+\sum_{i=1}^{n}\left(\sigma_{q_{i}}(g)-\sigma_{q_{i}}(f)\right)+\sum_{j=1}^{r}\left(\sigma_{\ell_{j}}(g)-\sigma_{\ell_{j}}(f)\right)+\sum_{\ell|N_{g}}\left(\sigma_{\ell}(g)-\sigma_{\ell}(f)\right).$

For $\ell|N_{g}$, it follows from Lemma 4.8 that

$\sum_{\ell|N_{g}}\left(\sigma_{\ell}(g)-\sigma_{\ell}(f)\right)=0.$

It follows from (1) of Lemma 4.5 that

$\left(\sigma_{q_{i}}(g)-\sigma_{q_{i}}(f)\right)=1$

and it follows from (2) of Lemma 4.5 that

$\left(\sigma_{\ell_{j}}(g)-\sigma_{\ell_{j}}(f)\right)=0.$

It therefore follows that

$\lambda_{p}(f)=\lambda_{p}(g)+n.$

∎

##  5\. Constructing Galois representations with prescribed
$\lambda$-invariant

Throughout this section, $p\geq 5$. We introduce our assumptions. Let $g$ be a
normalized newform of weight $2$ on $\Gamma_{0}(N_{g})$. Throughout this
section, we assume the following conditions.

  1. (1)

The residue field $\kappa=\mathcal{O}/\varpi$ is $\mathbb{F}_{p}$, i.e.,
$f(\mathfrak{p}/p)=1$ where $\mathfrak{p}$ is the prime above $p$ prescribed
by $\iota_{p}$.

  2. (2)

The Galois representation $\bar{\rho}_{g}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$
is surjective.

  3. (3)

The modular form $g$ is $p$-ordinary and $p\nmid N_{g}$,

  4. (4)

$g$ has optimal level,

  5. (5)

$\mu_{p}(g)=0$.

We show that the sets $\Pi_{g}$ and $\Omega_{g}$ both have positive density.
Furthermore, we estimate these densities. We let $Y$ (resp. $Y^{\prime}$) be
the subset of $\operatorname{GL}_{2}(\mathbb{F}_{p})$ consisting of semisimple
matrices conjugate to $\left({\begin{array}[]{cc}a&\\\ &1\\\
\end{array}}\right)$ (resp. $\left({\begin{array}[]{cc}a&\\\ &-1\\\
\end{array}}\right)$), where $a\neq\pm 1$. It is easy to see that

$\\#Y=\\#Y^{\prime}=(p-3)\frac{\\#\operatorname{GL}_{2}(\mathbb{F}_{p})}{\\#\operatorname{T}(\mathbb{F}_{p})},$

where $\operatorname{T}$ denotes the diagonal torus. Therefore, we find that

(5.1)
$\frac{\\#Y}{\\#\operatorname{GL}_{2}(\mathbb{F}_{p})}=\frac{\\#Y^{\prime}}{\\#\operatorname{GL}_{2}(\mathbb{F}_{p})}=\frac{(p-3)}{(p-1)^{2}}.$

Let $\bar{\rho}$ denote the residual representation $\bar{\rho}_{g}$, and let
$\mathbb{Q}(\bar{\rho})$ be the Galois extension of $\mathbb{Q}$ which is the
fixed field of the kernel of $\bar{\rho}$. We refer to
$\mathbb{Q}(\bar{\rho})$ as the field _cut out by $\bar{\rho}$_. We set $G$ to
denote the Galois group
$\operatorname{Gal}(\mathbb{Q}(\bar{\rho})/\mathbb{Q})$. The residual
representation induces an isomorphism $\Phi\mathrel{\mathop{\mathchar
58\relax}}G\xrightarrow{\sim}\operatorname{GL}_{2}(\mathbb{F}_{p})$. Let $Z$
(resp. $Z^{\prime}$) denote $\Phi^{-1}(Y)$ (resp. $\Phi^{-1}(Y^{\prime})$). It
follows from (5.1) that

(5.2) $\frac{\\#Z}{\\#G}=\frac{\\#Z^{\prime}}{\\#G}=\frac{(p-3)}{(p-1)^{2}}.$

######  Lemma 5.1.

Let $\ell\nmid N_{g}p$ be a prime. The following assertions hold

  1. (1)

$\operatorname{fr}_{\ell}\in Z$ if and only if
$\bar{\rho}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell&\\\ &1\\\
\end{array}}\right)$ up to conjugation and $\ell\not\equiv\pm 1\mod{p}$;

  2. (2)

$\operatorname{fr}_{\ell}\in Z^{\prime}$ if and only if
$\bar{\rho}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}-\ell&\\\
&-1\\\ \end{array}}\right)$ up to conjugation and $\ell\not\equiv\pm
1\mod{p}$.

###### Proof.

Suppose that $\operatorname{fr}_{\ell}\in Z$ (resp.
$\operatorname{fr}_{\ell}\in Z^{\prime}$). Then, since the weight of $g$ is
$2$ and has trivial nebentype, we find that
$\operatorname{det}\bar{\rho}(\operatorname{fr}_{\ell})=\ell$. It follows that
$\bar{\rho}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}\ell&\\\ &1\\\
\end{array}}\right)$ (resp.
$\bar{\rho}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}-\ell&\\\
&-1\\\ \end{array}}\right)$) and $\ell\not\equiv\pm 1\mod{p}$. ∎

######  Proposition 5.2.

Let $\ell\nmid N_{g}p$ be a prime. Then, $\operatorname{fr}_{\ell}\in
Z^{\prime}$ if and only if $\ell\in\Omega_{g}$. As a consequence, it follows
that $\Omega_{g}$ has positive density equal to $\frac{(p-3)}{(p-1)^{2}}$.

###### Proof.

We find that $\operatorname{fr}_{\ell}\in Z^{\prime}$ if and only if
$\bar{\rho}(\operatorname{fr}_{\ell})=\left({\begin{array}[]{cc}-\ell&\\\
&-1\\\ \end{array}}\right)$ up to conjugation and $\ell\not\equiv\pm
1\mod{p}$. In other words, $\operatorname{fr}_{\ell}\in Z^{\prime}$ if and
only if $\ell\in\Omega_{g}$. By the Chebotarev density theorem, $\Omega_{g}$
has positive density equal to
$\frac{\\#Z^{\prime}}{\\#G}=\frac{(p-3)}{(p-1)^{2}}$. ∎

Recall that $\mathbb{Q}_{1}$ is the $\mathbb{Z}/p\mathbb{Z}$-extension of
$\mathbb{Q}$ contained in $\mathbb{Q}_{\operatorname{cyc}}$. Let
$\mathcal{L}/\mathbb{Q}$ be the compositum
$\mathbb{Q}(\bar{\rho})\cdot\mathbb{Q}_{1}$.

######  Lemma 5.3.

The extensions $\mathbb{Q}(\bar{\rho})$ and $\mathbb{Q}_{1}$ are linearly
disjoint.

###### Proof.

Let $E\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{Q}(\bar{\rho})\cap\mathbb{Q}_{1}$ and
$N\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\mathbb{Q}(\bar{\rho})/E)$. Note that $N$ is a
normal subgroup of $G\simeq\operatorname{GL}_{2}(\mathbb{F}_{p})$ and that
$[G\mathrel{\mathop{\mathchar 58\relax}}N]$ divides $p$. For $p\geq 5$, the
group $\operatorname{PSL}_{2}(\mathbb{F}_{p})$ is simple. It is easy to see
that $\operatorname{GL}_{2}(\mathbb{F}_{p})$ does not contain an index $p$
normal subgroup. Therefore, $N=G$ and thus, $\mathbb{Q}(\bar{\rho})$ and
$\mathbb{Q}_{1}$ are linearly disjoint. ∎

Let $\Gamma_{1}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\mathbb{Q}_{1}/\mathbb{Q})$ and we find that
$\operatorname{Gal}(\mathcal{L}/\mathbb{Q})\simeq G\times\Gamma_{1}$. Let $W$
be the product $Z\times(\Gamma_{1}\backslash\\{0\\})$.

######  Proposition 5.4.

Let $\ell\nmid N_{g}p$ be a prime. Then, $\operatorname{fr}_{\ell}\in W$ if
and only if $\ell\in\Pi_{g}$. As a result, $\Pi_{g}$ has density equal to
$\frac{(p-3)}{p(p-1)}$.

###### Proof.

Note that the prime $\ell$ is nonsplit in $\mathbb{Q}_{1}$ if and only if
$\ell^{p-1}\not\equiv 1\mod{p^{2}}$. The result is therefore a direct
consequence of Lemma 5.1. By the Chebotarev density theorem, $\Pi_{g}$ has
density equal to

$\frac{\\#Z\times(\\#\Gamma_{1}-1)}{\\#G\times\\#\Gamma_{1}}=\frac{(p-3)}{p(p-1)}.$

∎

###### Proof of Theorem 1.1.

The Theorem is a direct consequence of Proposition 5.2, Proposition 5.4 and
Theorem 4.9. ∎

######  Theorem 5.5.

Let $p\geq 5$ be a prime and let $g$ be a normalized newform of weight $2$ on
$\Gamma_{0}(N_{g})$ satisfying all of the conditions of Theorem 1.1.
Furthermore, assume that $\lambda_{p}(g)\leq 1$. Then, for any set of primes
$\\{\ell_{1},\dots,\ell_{r}\\}\subset\Omega_{g}$, there is a Hecke newform of
weight $2$ of level $N_{f}=N_{g}\ell_{1}\dots\ell_{r}$ such that

  1. (1)

$f$ has good ordinary reduction at $p$,

  2. (2)

$\bar{\rho}_{g}\simeq\bar{\rho}_{f}$,

  3. (3)

$\mu_{p}(f)=0$,

  4. (4)

$\operatorname{rank}^{\operatorname{BK}}(f)=\lambda_{p}(g)$.

###### Proof.

It follows from Theorem 1.1 that there exists $f$ satisfying the first three
assertions, and such that $\lambda_{p}(f)=\lambda_{p}(g)$. Proposition 3.2
implies that

$\operatorname{rank}^{\operatorname{BK}}(f)=\lambda_{p}(f)=\lambda_{p}(g).$

This proves the last assertion. ∎

###### Proof of Theorem 1.2.

It suffices to show that there is a non-CM Hecke newform $g$ of weight $2$ on
$\Gamma_{0}(N_{g})$ and a density $1$ set of primes $\Sigma$ such that for all
$p\in\Sigma$,

  1. (1)

$p\geq 5$,

  2. (2)

The residue field $\kappa=\mathcal{O}/\varpi$ is $\mathbb{F}_{p}$, i.e.,
$f(\mathfrak{p}/p)=1$.

  3. (3)

The Galois representation $\bar{\rho}_{g}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$
is surjective.

  4. (4)

The modular form $g$ is $p$-ordinary and $p\nmid N_{g}$,

  5. (5)

$g$ has optimal level, i.e., $N_{g}$ is the prime to $p$ part of the Artin
conductor of the residual representation,

  6. (6)

$\mu_{p}(g)=0$ and $\lambda_{p}(g)=0$.

It then follows from Theorem 1.1 that for $p\in\Sigma$, there are infinitely
many Hecke newforms of weight $2$ such that

  1. (1)

$f$ has good ordinary reduction at $p$,

  2. (2)

$\bar{\rho}_{f}$ is surjective,

  3. (3)

$\mu_{p}(f)=0$,

  4. (4)

$\lambda_{p}(f)=n$.

Let $E_{/\mathbb{Q}}$ be any non-CM elliptic curve with Mordell-Weil rank $0$,
and let $g$ be the associated Hecke newform. Consider the following
observations.

  * •

The field of Fourier coefficients of $g$ is $\mathbb{Q}$, since it is
associated to an elliptic curve over $\mathbb{Q}$. In particular, it follows
that the residue field $\kappa$ is isomorphic to $\mathbb{F}_{p}$ for all
primes $p$.

  * •

The set of primes at which $E$ has good ordinary reduction has density $1$, by
a result of Serre [Ser81].

  * •

Serre’s open image theorem [Ser72] shows that for all but finitely many primes
$p$, the residual representation $\bar{\rho}_{g}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$
is surjective.

  * •

If at a given prime $p$, $N_{g}$ is not optimal, then there must be a
$p$-congruence between $g$ and a newform $h$ of weight $2$ and strictly
smaller level. This follows from Ribet’s level lowering theorem. There are
only finitely many newforms of weight $2$ with level strictly less than
$N_{g}$. Also, there are only finitely many primes $p$ for which two newforms
are $p$-congruent. Therefore, for all but finitely many primes $p$, the level
$N_{g}$ is optimal.

  * •

Finally, it follows from a theorem of Greenberg [Gre99, Theorems 4.1, 5.1]
that for a density $1$ set of primes $p$, $\mu_{p}(g)=\lambda_{p}(g)=0$.

Therefore, there is a set of primes $\Sigma$ of density $1$ such that the
above conditions are satisfied. This completes the proof. ∎

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