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\title[Erd\H{o}s--Kac theorem for signed Iwasawa invariants]
{An analogue of the Erd\H{o}s--Kac theorem for signed Iwasawa invariants of modular forms}

\author[A.~Ray]{Anwesh Ray\, \orcidlink{0000-0001-6946-1559}}
\address{Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Kelambakkam 603103, Tamil Nadu, India.}
\email{anwesh@cmi.ac.in}

\subjclass[2020]{Primary 11R23; Secondary 11N60, 11F33, 11F80}
\keywords{Signed Iwasawa invariants, non-ordinary modular forms,
signed Selmer groups, level raising, Erd\H{o}s--Kac theorem}

\begin{document}

\begin{abstract}
Erd\H{o}s and Kac proved that the number
of distinct prime factors of an integer has a Gaussian limiting
distribution. We prove an analogue for the signed
Iwasawa invariants of non-ordinary weight-two modular forms. Under
the hypotheses of this paper, we construct infinitely many forms
congruent to a given form $g$ that realize every common nonnegative
increase of its two signed $\lambda$-invariants. We count the levels realizing each increase.
In the resulting family, the common increase is asymptotically
Gaussian, with mean and variance $\delta_\Pi\log\log X+O_g(1)$ for a
positive constant $\delta_\Pi$.
\end{abstract}


\maketitle

\section{Introduction}
\label{sec:introduction}

 \subsection{Background and motivation} A central theme in arithmetic geometry is to understand how arithmetic
invariants vary in infinite families. Iwasawa theory encodes the
growth of class groups, Selmer groups and related arithmetic objects in
$p$-adic towers in terms of a small collection of algebraic invariants.

The theory originated in Iwasawa's study of ideal class groups in
infinite towers of number fields. Let $p$ be a prime number and $K_\infty/K$ be the cyclotomic
$\Zp$-extension. Let $K_n$ be the unique intermediate field
satisfying $[K_n:K]=p^n$. If $A_n$ denotes the $p$-primary part
of the ideal class group of $K_n$, then the inverse limit of the
groups $A_n$, with respect to the norm maps, is a module over the
Iwasawa algebra
$\Lambda=\Zp[[T]]$. Writing
$\#A_n=p^{e_n}$, Iwasawa proved that there are integers $\mu,\lambda\geq0$ and $\nu$
such that
\begin{equation}\label{eq:intro-iwasawa-growth}
 e_n=\mu p^n+\lambda n+\nu
\end{equation}
for all sufficiently large $n$; see \cite{Iwasawa1959}. This essentially leverages the structure theory of finitely generated and torsion modules over $\Lambda$. The integers
$\mu$ and $\lambda$ are the Iwasawa invariants of the tower.

Mazur extended this point of view to elliptic curves
\cite{Mazur1972}.  Let $E/\Q$ be an elliptic curve and let
$\Q_\infty/\Q$ be the cyclotomic $\Zp$-extension.  The role of the
class group is played by the $p$-primary Selmer group
$\Sel_{p^\infty}(E/\Q_\infty)$. For every finite extension $F/\Q$ contained in $\Q_\infty$, the Selmer
group fits into the fundamental exact sequence
\[
 0\longrightarrow
 E(F)\otimes_{\Z}\Q_p/\Z_p
 \longrightarrow
 \Sel_{p^\infty}(E/F)
 \longrightarrow
 \Sh(E/F)[p^\infty]
 \longrightarrow 0.
\] When its Pontryagin dual is torsion
over the cyclotomic Iwasawa algebra, it has well defined invariants
$\mu_p(E)$ and $\lambda_p(E)$ which arise from the structure theory of $\Lambda$-modules. These invariants describe the growth
of the Selmer groups in the finite layers of $\Q_\infty/\Q$ and are
closely related to the growth of Mordell--Weil groups and
Tate--Shafarevich groups.

The same construction applies to the Galois representation attached
to a modular form. Let $h$ be a normalized newform and let $V_h$ be
its $p$-adic Galois representation.  A choice of stable lattice
$T_h\subset V_h$ gives a discrete representation
$A_h=V_h/T_h$, and one defines a Selmer group over $\Q_\infty$ by
imposing suitable local conditions on $A_h$. In the ordinary case,
the local condition at $p$ is obtained from the ordinary filtration of
$V_h$. Kato's
Euler-system shows that the Pontryagin dual of the resulting
Selmer group is a finitely generated torsion module over the Iwasawa
algebra \cite{Kato2004}.  Its characteristic power series therefore
defines the associated $\mu$- and $\lambda$-invariants.

At a non-ordinary prime there is no ordinary filtration giving the
required local condition.  For elliptic curves with supersingular
reduction, Kobayashi introduced the plus and minus Selmer groups
\cite{Kobayashi2003}, while Pollack constructed the corresponding
bounded $p$-adic $L$-functions when $a_p=0$ \cite{Pollack2003}.
Signed Coleman maps and signed Selmer groups for modular forms were
subsequently developed by Lei, Lei--Loeffler--Zerbes and Sprung
\cite{Lei2011,LLZ2010,LLZ2011,Sprung2012,Sprung2017}.  In the
weight-two setting considered here, these constructions give two
signed Selmer groups, indexed by
$\natural\in\{\sharp,\flat\}$, and hence two pairs of Iwasawa
invariants $\bigl(\mu^\sharp(h),\lambda^\sharp(h)\bigr)$ and $\bigl(\mu^\flat(h),\lambda^\flat(h)\bigr)$.

A fundamental question is to understand how Iwasawa invariants vary
with respect to congruences of Galois representations. Greenberg and Vatsal proved that, under suitable hypotheses, the
vanishing of the Iwasawa $\mu$-invariant is preserved under congruences
between ordinary elliptic curves.  When the $\mu$-invariants vanish,
they expressed the difference of the corresponding
$\lambda$-invariants as a sum of explicit local correction terms
\cite{GreenbergVatsal2000}. Emerton--Pollack--Weston extended this study to ordinary modular forms
in Hida families \cite{EmertonPollackWeston2006}.  They showed that if
the algebraic or analytic $\mu$-invariant vanishes for one form, it
vanishes throughout the family. They also described how the
$\lambda$-invariants vary between branches and proved that they are
constant along each branch.  The algebraic and analytic variation
formulae agree; consequently, the main conjecture for one modular form implies
it for the other modular forms that are congruent to it. Kim extended the Greenberg--Vatsal comparison to the plus and minus
Selmer groups of congruent supersingular elliptic curves
\cite{Kim2009,Kim2013}.  Pollack--Weston used congruences to prove
Kida-type formulas, including one for supersingular elliptic curves
\cite{PollackWeston2006}.  Hatley and Lei then treated congruent
non-ordinary modular forms of the same weight. For each sign, they
showed that the $\mu$-invariant vanishes for one form if and only if it
vanishes for the other; when it vanishes, the difference of the
$\lambda$-invariants is determined by local terms at primes away from
$p$ \cite{HatleyLei2019}.

In \cite{Ray2024}, we combined the ordinary congruence formula with
level raising results of Diamond and Taylor \cite{DiamondTaylor1994}.  Starting from an
ordinary weight-two newform $g$ with $\mu_p(g)=0$, we showed
that, for every $n\geq0$, there are infinitely many newforms $f$
congruent to $g$ with $\mu_p(f)=0$ and
$\lambda_p(f)=\lambda_p(g)+n$. The present paper extends this construction to non-ordinary modular
forms and develops a probabilistic model for the variation of their
signed $\lambda$-invariants as the tame level varies. In the non-ordinary setting there are two signed
$\lambda$-invariants.  We show that level raising increases them by
the same amount, so their difference remains unchanged. We then study the distribution of the common increment as the tame
level varies.
\subsection{Main results}
Let $p\geq 5$ be a prime number. Let
\[
 g=\sum_{n\geq1}a_n(g)q^n\in S_2(\Gamma_0(N_g))
\]
be a normalized newform of trivial nebentypus, where $p\nmid N_g$,
and suppose that $g$ is non-ordinary at $p$.  Write
$G_\Q=\Gal(\overline{\Q}/\Q)$ and let
\[
 \bar\rho_g:G_\Q\longrightarrow\operatorname{GL}_2(\mathbf{k})
\]
be the residual Galois representation attached to $g$ at a fixed
$p$-adic place of its coefficient field, where $\mathbf{k}$ is the
corresponding residue field.  Let $\overline\chi_p$ denote the mod-$p$
cyclotomic character, and let $\mu_{p^r}$ denote the group of
$p^r$-th roots of unity.  Put
\[
 H_g=\bar\rho_g(G_\Q)\subseteq\operatorname{GL}_2(\mathbf{k}).
\]
Let $L_g$ be the fixed field of $\ker(\bar\rho_g)$.  Since $g$ has
weight two and trivial nebentypus,
$\det\bar\rho_g=\overline\chi_p$.  In particular,
$\det(H_g)=\F_p^\times$ and $\Q(\mu_p)\subseteq L_g$. For a prime $q\nmid N_gp$, we write $\Frob_q$ for an arithmetic
Frobenius element at $q$. 

For $h\in H_g$, put $d(h)=\det(h)$ and define the conjugacy-stable
subsets
\begin{align*}
 \mathcal C_g^+
 &=
 \{h\in H_g:d(h)\notin\{\pm1\},\ \Tr(h)=1+d(h)\},\\
 \mathcal C_g^-
 &=
 \{h\in H_g:d(h)\notin\{\pm1\},\ \Tr(h)=-(1+d(h))\}.
\end{align*}
Thus the characteristic polynomial of an element of
$\mathcal C_g^+$ is $(X-1)(X-d(h))$, whereas that of an element of
$\mathcal C_g^-$ is $(X+1)(X+d(h))$. 

For a non-ordinary modular form $h$ and
$\natural\in\{\sharp,\flat\}$, let $X^\natural(h)$ denote the
Pontryagin dual of its $\natural$-signed Selmer group over the
cyclotomic $\Z_p$-extension of $\Q$.  When $X^\natural(h)$ is torsion,
we denote its Iwasawa invariants by
$\mu^\natural(h)$ and $\lambda^\natural(h)$.
We impose the following
hypotheses.
\begin{enumerate}[label=(H\arabic*)]
\item\label{H:image}
The representation $\bar\rho_g$ is absolutely irreducible, both
$\mathcal C_g^+$ and $\mathcal C_g^-$ are nonempty, and
\[
 L_g\cap\Q(\mu_{p^2})=\Q(\mu_p).
\]
\item\label{H:optimal}
The integer $N_g$ is the prime-to-$p$ Artin conductor of
$\bar\rho_g$.
\item\label{H:base}
For each $\natural\in\{\sharp,\flat\}$, the signed Selmer dual
$X^\natural(g)$ is torsion over the cyclotomic Iwasawa algebra and
$\mu^\natural(g)=0$.
\end{enumerate}
The residual-image hypothesis \ref{H:image} is weaker than requiring
$\mathbf{k}=\Fp$ and
$H_g=\operatorname{GL}_2(\Fp)$.  In particular,
Proposition~\ref{prop:large-image-criterion} shows that it holds
whenever $\operatorname{SL}_2(\mathbf{k})\subseteq H_g$.

Let $\Pi_g$ be the set of primes $q\nmid N_gp$ such that
$\bar\rho_g(\Frob_q)\in\mathcal C_g^+$ and
$q^{p-1}\not\equiv1\pmod{p^2}$.  Let $\Omega_g$ be the set of primes
$\ell\nmid N_gp$ such that
$\bar\rho_g(\Frob_\ell)\in\mathcal C_g^-$.Write $\delta_\Pi$ and $\delta_\Omega$ for the natural densities of
$\Pi_g$ and $\Omega_g$, respectively.  These densities exist by the
Chebotarev density theorem, and Proposition~\ref{prop:aux-density}
gives
\begin{equation}\label{eq:intro-auxiliary-densities}
 \delta_\Pi
 =
 \frac{p-1}{p}\frac{|\mathcal C_g^+|}{|H_g|}\quad \text{and}\quad
 \delta_\Omega
 =
 \frac{|\mathcal C_g^-|}{|H_g|}.
\end{equation}
Both are positive by \ref{H:image}. When
$\mathbf{k}=\Fp$ and $H_g=\operatorname{GL}_2(\Fp)$, these become
\[
 \delta_\Pi=\frac{p-3}{p(p-1)}\quad \text{and}\quad
 \delta_\Omega=\frac{p-3}{(p-1)^2}.
\]

The first main result is the following signed analogue of the
prescription theorem of \cite{Ray2024}.

\begin{theorem}\label{thm:main}
Assume \ref{H:image}--\ref{H:base}.  Let $Q\subset\Pi_g$ and
$R\subset\Omega_g$ be finite sets.  There exists a normalized
weight-two newform $f$ of trivial nebentypus and exact level
\[
 N_f=N_g\prod_{q\in Q}q\prod_{\ell\in R}\ell
\]
such that $\bar\rho_f\simeq\bar\rho_g$ and $f$ is non-ordinary at $p$.
For each $\natural\in\{\sharp,\flat\}$, the signed Selmer dual
$X^\natural(f)$ is torsion over the cyclotomic Iwasawa algebra and
\[
 \mu^\natural(f)=0\quad \text{and}\quad
 \lambda^\natural(f)
 =
 \lambda^\natural(g)+|Q|.
\]
\end{theorem}

The distinction between the two sets of auxiliary primes is local.  A
prime in $\Pi_g$ contributes one to each signed $\lambda$-invariant,
whereas a prime in $\Omega_g$ contributes zero.  At the primes
dividing $N_g$, the optimal-level hypothesis implies that the local
correction terms for $f$ and $g$ agree.  Since these tame corrections
do not depend on the signed label, Theorem~\ref{thm:main} also gives
\[
 \lambda^\sharp(f)-\lambda^\flat(f)
 =
 \lambda^\sharp(g)-\lambda^\flat(g).
\]
As both auxiliary sets are infinite, we obtain the following
consequence.

\begin{corollary}\label{cor:realization}
Assume \ref{H:image}--\ref{H:base}.  For every $n\geq0$, there are
infinitely many normalized newforms $f$, of pairwise distinct exact
levels, such that $\bar\rho_f\simeq\bar\rho_g$ and the following assertions hold:
\begin{enumerate}
    \item $\mu^\sharp(f)=\mu^\flat(f)=0$,
    \item $\bigl(\lambda^\sharp(f),\lambda^\flat(f)\bigr)
 =
 \bigl(\lambda^\sharp(g)+n,\lambda^\flat(g)+n\bigr)$.
\end{enumerate}
\end{corollary}

Theorem~\ref{thm:main} determines the signed $\lambda$-invariants for
each choice of level-raising primes.  We now let these choices vary
and bound the product of the primes.  If the level is raised at
$Q\subset\Pi_g$ and $R\subset\Omega_g$, both signed
$\lambda$-invariants increase by $|Q|$.  Their common increment is
therefore the number of prime factors of the new level belonging to
$\Pi_g$.

For comparison, let $\omega(m)$ denote the number of distinct prime
divisors of a positive integer $m$. Hardy and Ramanujan proved that the
normal order of $\omega(m)$ is $\log\log m$
\cite{HardyRamanujan1917}, and Tur\'an gave a proof using the second
moment \cite{Turan1934}.  Erd\H{o}s and Kac showed that the
fluctuations about this normal order have a Gaussian limiting
distribution \cite{ErdosKac1940}.  More precisely, for every
$x\in\mathbb R$,
\[
 \lim_{Y\to\infty}
 \frac{1}{Y}
 \#
 \left\{
 m\leq Y:
 \frac{\omega(m)-\log\log Y}{\sqrt{\log\log Y}}\leq x
 \right\}
 =
 \frac{1}{\sqrt{2\pi}}
 \int_{-\infty}^{x}e^{-t^2/2}\,dt.
\]
We prove the corresponding limit law for the common signed
$\lambda$-increment, with $\delta_\Pi\log\log Y$ in place of
$\log\log Y$.

Call a positive integer $D$ \emph{admissible} if it is squarefree and
every prime divisor of $D$ belongs to $\Pi_g\cup\Omega_g$.  Since the
two sets are disjoint, we may write $D=D_\Pi D_\Omega$, where
$D_\Pi$ is supported on $\Pi_g$ and $D_\Omega$ is supported on
$\Omega_g$.  Put
\[
 \omega_\Pi(D)=\#\{q:q\mid D_\Pi\}.
\]
For every admissible $D$, choose one form $f_D$ supplied by
Theorem~\ref{thm:main}.  It has exact level $N_gD$, and
\begin{equation}\label{eq:intro-common-increment}
 \lambda^\sharp(f_D)-\lambda^\sharp(g)
 =
 \lambda^\flat(f_D)-\lambda^\flat(g)
 =
 \omega_\Pi(D).
\end{equation}
Thus the variation of the signed Iwasawa invariants is governed by an
additive arithmetic function supported on the Frobenian set $\Pi_g$.

Let
\[
 \mathscr D(Y)
 =
 \{D\leq Y:D\text{ is admissible}\}\quad \text{and}\quad
 \mathscr F(Y)
 =
 \{f_D:D\in\mathscr D(Y)\}.
\]
Since $f_D$ has exact level $N_gD$, different admissible integers give
different members of $\mathscr F(Y)$. For $\natural\in\{\sharp,\flat\}$ and $f\in\mathscr F(Y)$, set
\[
 \Delta^\natural(f)
 =
 \lambda^\natural(f)-\lambda^\natural(g).
\]
If $X$ is a real-valued function on $\mathscr F(Y)$, we write
$\mathbb E_Y[X]$ and $\operatorname{Var}_Y(X)$ for its expectation
and variance with respect to the uniform probability measure on
$\mathscr F(Y)$.



The second main result is the following.

\begin{theorem}[Analogue of Erd\H{o}s--Kac theorem]
\label{thm:erdos-kac}
Let $\natural\in\{\sharp,\flat\}$. Then for every real number $x$,
\begin{equation}\label{eq:EK-limit}
 \begin{aligned}
 &\lim_{Y\to\infty}
 \frac{1}{\#\mathscr F(Y)}
 \#\Biggl\{
 f\in\mathscr F(Y):\\[-2pt]
 &\hspace{5em}
 \frac{
 \lambda^\natural(f)-\lambda^\natural(g)
 -\delta_\Pi\log\log Y}
 {\sqrt{\delta_\Pi\log\log Y}}
 \leq x
 \Biggr\}
 =
 \frac{1}{\sqrt{2\pi}}
 \int_{-\infty}^x e^{-t^2/2}\,dt.
 \end{aligned}
\end{equation}
Moreover,
\begin{align}
 \mathbb E_Y[\Delta^\natural]
 &=
 \delta_\Pi\log\log Y+O_g(1),
 \label{eq:EK-mean}\\
 \operatorname{Var}_Y(\Delta^\natural)
 &=
 \delta_\Pi\log\log Y+O_g(1).
 \label{eq:EK-variance}
\end{align}
The two signed increments agree for every member of the family.
Consequently,
\[
 \lambda^\sharp(f)-\lambda^\flat(f)
 =
 \lambda^\sharp(g)-\lambda^\flat(g)
\]
for every $f\in\mathscr F(Y)$.
\end{theorem}

The Gaussian law describes the typical size of the common increase in
the two signed $\lambda$-invariants.  We also count the forms for which
this increase has a prescribed value.  For $n\geq0$, let
$\mathscr F_n(X)$ consist of the chosen forms $f_D$ of level at most
$X$ for which
\[\lambda^\sharp(f_D)-\lambda^\sharp(g)
=\lambda^\flat(f_D)-\lambda^\flat(g)=n.\]
For each fixed $n$, we prove that
\[
 \#\mathscr F_n(X)
 \sim
 \frac{C_{g,n}}{N_g}
 X(\log X)^{\delta_\Omega-1}(\log\log X)^n,
\]
where $C_{g,n}>0$.

\subsection{Organization of the paper}

Including the introduction, the article consists of five sections. Section~\ref{sec:signed} recalls the signed Selmer groups and the
congruence results we use.  In Section~\ref{sec:level-raising}, we
construct the auxiliary primes and the level-raised forms.
Section~\ref{sec:local} computes the local contributions to the signed
$\lambda$-invariants and proves Theorem~\ref{thm:main}.  Finally,
Section~\ref{sec:quantitative} counts the levels realizing each fixed
increase and proves the Erd\H{o}s--Kac theorem.

\section*{Statements and Declarations}

\noindent\textbf{Funding.}
The author received no funding for this work.

\noindent\textbf{Ethical approval.}
Not applicable.

\noindent\textbf{Consent to participate.}
Not applicable.

\noindent\textbf{Consent for publication.}
Not applicable.

\noindent\textbf{Conflict of interest.}
The author declares no conflict of interest.

\noindent\textbf{Data availability.}
No datasets were generated or analyzed during the current study.


\section{Signed Selmer groups and their congruences}
\label{sec:signed}
 Fix a prime number $p\geq 5$. Let $\chi_p$ be the $p$-adic cyclotomic character and $\bar{\chi}_p$ be its mod-$p$ reduction. Let $\Q(\mu_{p^\infty})$ be the infinite cyclotomic field generated by the $p$-power roots of unity $\mu_{p^\infty}$. Let $\Q_\infty/\Q$ be the cyclotomic $\Zp$-extension and set $\Gamma:=\Gal(\Q_\infty/\Q)$. Let $E/\Q_p$ be a finite extension with valuation ring $\cO$, uniformizer $\varpi$ and $\mathbf k=\cO/(\varpi)$. The Iwasawa algebra (with coefficients in $\cO$) is defined as the completed group algebra: 
 \[\Lambda_{\cO}:=\cO[[\Gamma]]=\varprojlim_n \cO[\Gamma/\Gamma^{p^n}].\]
 We choose a topological generator $\gamma$ of $\Gamma$ and identify  $\Lambda_{\cO}$ with $\cO[[T]]$ by setting $T:=\gamma-1$.
\subsection{Iwasawa modules}

We first describe the properties of modules over the Iwasawa algebra $\Lambda_{\cO}$. Here, if $M$ and $M'$ are two modules over $\Lambda_{\cO}$, then a pseudoisomorphism is a $\Lambda_{\cO}$-module homomorphism $\varphi: M\rightarrow M'$ whose kernel and cokernel both have finite cardinality. If $M$ is a finitely generated and torsion $\Lambda_{\cO}$-module, there is a pseudo-isomorphism
\begin{equation}\label{eq:structure-module}
 M\longrightarrow
 \bigoplus_{i=1}^{r}\Lambda_{\cO}/(\varpi^{\mu_i})
 \oplus
 \bigoplus_{j=1}^{s}\Lambda_{\cO}/(F_j(T)^{e_j}),
\end{equation}
where $\mu_i,e_j\geq1$ and the $F_j(T)$ are irreducible distinguished polynomials.  We then set
\[
 \mu(M)=\sum_{i=1}^{r}\mu_i\quad \text{and}\quad
 \lambda(M)=\sum_{j=1}^{s}e_j\deg F_j.
\]

\begin{lemma}\label{lem:modpi-criterion}
Let $M$ be a compact $\Lambda_{\cO}$-module, where
$\Lambda_{\cO}=\cO[[T]]$.  The following conditions are equivalent.
\begin{enumerate}[label=(\roman*)]
\item The quotient $M/\varpi M$ is finite-dimensional over $\mathbf{k}$.
\item The module $M$ is finitely generated and torsion over
$\Lambda_{\cO}$, and $\mu(M)=0$.
\end{enumerate}
If these conditions hold and $M$ has no nonzero finite
$\Lambda_{\cO}$-submodule, then
\[
 \lambda(M)=\dim_{\mathbf{k}}(M/\varpi M).
\]
\end{lemma}

\begin{proof}
We first assume that $M/\varpi M$ is finite-dimensional over
$\mathbf{k}$.  Let
\[
 \mathfrak n=(\varpi,T)
\]
be the maximal ideal of $\Lambda_{\cO}$.  Since
\[
 M/\mathfrak nM
 =
 (M/\varpi M)/T(M/\varpi M),
\]
the quotient $M/\mathfrak nM$ is finite-dimensional over
$\mathbf{k}$.  Choose elements $m_1,\ldots,m_r\in M$ whose images
generate $M/\mathfrak nM$, and let $N$ be the
$\Lambda_{\cO}$-submodule of $M$ generated by these elements.  Then
$M=N+\mathfrak nM$.  Since $M$ is compact and
$\Lambda_{\cO}$ is a complete local ring, the topological Nakayama
lemma gives $M=N$.  Thus $M$ is finitely generated over
$\Lambda_{\cO}$; see
\cite[Section~3, Corollary following the first theorem]{BalisterHowson1997}.

We may now apply the structure theorem for finitely generated
$\Lambda_{\cO}$-modules.  There is a homomorphism
\[
 \phi:M\longrightarrow M'
\]
with finite kernel and finite cokernel, where
\[
 M'=
 \Lambda_{\cO}^{\,r_0}
 \oplus
 \bigoplus_{i=1}^{u}
 \Lambda_{\cO}/(\varpi^{a_i})
 \oplus
 \bigoplus_{j=1}^{v}
 \Lambda_{\cO}/(F_j(T)^{e_j}).
\]
Here $a_i,e_j\geq1$, and each $F_j(T)$ is a distinguished
irreducible polynomial which is not divisible by $\varpi$.

It is easy to see that $M'/\varpi M'$ is finite-dimensional over $\mathbf{k}$. This rules out the first two types of elementary factors.
Indeed, a free summand $\Lambda_{\cO}$ contributes
$\mathbf{k}[[T]]$ to $M'/\varpi M'$, which is infinite-dimensional over
$\mathbf{k}$. Likewise,
\[
 \bigl(\Lambda_{\cO}/(\varpi^{a_i})\bigr)
 \big/
 \varpi\bigl(\Lambda_{\cO}/(\varpi^{a_i})\bigr)
 \simeq \mathbf{k}[[T]],
\]
so the presence of a factor $\Lambda_{\cO}/(\varpi^{a_i})$ would again
make $M'/\varpi M'$ infinite-dimensional.  It follows that
$r_0=u=0$. Therefore $M$ is torsion and $\mu(M)=0$.

We now prove the converse.  Suppose that $M$ is finitely generated
and torsion over $\Lambda_{\cO}$ and that $\mu(M)=0$.  The structure
theorem then gives a pseudo-isomorphism
\[
 \varphi: M\longrightarrow
 M':=\bigoplus_{j=1}^{v}
 \Lambda_{\cO}/(F_j(T)^{e_j}),
\]
with finite kernel and cokernel.  If $d_j=\deg F_j$, then, since
$F_j$ is distinguished,
\[
 F_j(T)\equiv T^{d_j}\pmod{\varpi}.
\] It follows
that $M'/\varpi M'$ is finite-dimensional. Since $M$ is pseudoisomorphic to $M'$, it is easy to see that $M/\varpi M$ is finite-dimensional.

It remains to prove the formula for the $\lambda$-invariant.  Assume
in addition that $M$ has no nonzero finite
$\Lambda_{\cO}$-submodule.  In the pseudo-isomorphism $\varphi$ above, the kernel
is finite and is a submodule of $M$.  Hence $K=0$.  We therefore
have an exact sequence
\[
 0\longrightarrow M\longrightarrow M'\longrightarrow C
 \longrightarrow0
\]
with $C$ finite.

Each module
$\Lambda_{\cO}/(F_j(T)^{e_j})$ is finite free over $\cO$ of rank
$e_jd_j$. Thus $M'$ is a finitely generated and
free $\cO$-module of rank
\[
 \sum_{j=1}^{v}e_jd_j=\lambda(M).
\]
Since $M$ is an $\cO$-submodule of the finite free module $M'$, it is
finitely generated and torsion-free over the discrete valuation ring
$\cO$, and hence is itself free over $\cO$.  The quotient $M'/M=C$ is
finite, so $M$ and $M'$ have the same $\cO$-rank.  Therefore
\[
 \operatorname{rank}_{\cO}M=\lambda(M).
\]
Finally, if a finite free $\cO$-module has rank $r$, then its reduction
modulo $\varpi$ has dimension $r$ over $\mathbf{k}$.  Hence
\[
 \dim_{\mathbf{k}}(M/\varpi M)
 =
 \operatorname{rank}_{\cO}M
 =
 \lambda(M),
\]
as required.
\end{proof}

\subsection{Signed local conditions}

Let $h=\sum_{n\geq1}a_n(h)q^n$ be a normalized newform of weight $2$, level prime to $p$, and trivial nebentypus. Choose an embedding $\overline{\Q}\hookrightarrow\overline{\Q}_p$ and let $E/\Q_p$ be a finite extension containing the image of the coefficient field of $h$. Associated to $h$ is a $2$-dimensional Galois representation over $E$; let $V_h$ denote the underlying vector space. Our convention is that the Hodge--Tate weights of $V_h$ are $\{0,-1\}$.  Let $T_h\subset V_h$ be Kato's Galois stable $\cO$-lattice, cf. \cite[Section~8.3]{Kato2004}. We then set
\[
 A_h=(V_h/T_h)(1)\quad \text{and}\quad
 \mathscr T_h=T_h(1).
\]
The lattice $\mathscr T_h$ is crystalline at $p$, with Hodge--Tate weights $\{0,1\}$. 

 For $n\geq 0$, put $K_n=\Qp(\mu_{p^{n+1}})$
and let $K_\infty=\bigcup_{n\geq0}K_n$. Thus
\[
 G_\infty:=\Gal(K_\infty/\Qp)
 \simeq \Delta\times\Gamma.
\]
The completion $\Q_{\infty,p}$ of $\Q_\infty$ at the prime above $p$ is $K_\infty^\Delta$. Thus passage to the trivial $\Delta$-component identifies the local conditions below with the corresponding local conditions over $\Q_{\infty,p}$.
If $T$ is a $\cO$-lattice equipped with a continuous action of
$G_{\Qp}$, its local Iwasawa cohomology is defined by
\begin{equation}\label{eq:iw-cohomology}
 H^1_{\Iw}(\Qp,T)
 :=
 \varprojlim_n H^1(K_n,T),
\end{equation}
where the inverse limit is taken with respect to the corestriction
maps.  It is naturally a module over
\[
 \cO[[G_\infty]]
 \simeq
 \Lambda_{\cO}(\Delta)
 :=
 \cO[\Delta]\widehat\otimes_{\cO}\Lambda_{\cO}.
\]

We apply this construction to the positive crystalline lattice
$\mathscr T_h$. The construction of Lei--Loeffler--Zerbes \cite{LLZ2010} gives, after fixing a
suitable basis, two Coleman maps
\[
 \operatorname{Col}_{h,1},\operatorname{Col}_{h,2}:
 H^1_{\Iw}(\Qp,\mathscr T_h)\longrightarrow
 \Lambda_{\cO}(\Delta).
\]
We use the notation $\sharp,\flat$ for the signed Coleman maps in the normalization customary in the weight-two non-ordinary setting; see \cite{Sprung2012,Sprung2017}. For a character $\eta:\Delta\rightarrow\cO^\times$, projection to the
$\eta$-isotypical component gives
\[
\operatorname{Col}_{h,\natural}^{\eta}:
 H^1_{\Iw}(\Qp,\mathscr T_h)^\eta
 \longrightarrow
 \Lambda_{\cO},
\]
where $\natural\in\{\sharp,\flat\}$. In this article we work throughout in the trivial
$\Delta$-component, so $\eta=\mathbf 1$.  We shall therefore suppress
$\eta$ from the notation and regard the signed Coleman maps as
$\Lambda_{\cO}$-valued maps.

We next recall the local Tate pairing used to define the signed local
condition.  The representation $A_h$ is the Cartier dual of
$\mathscr T_h$ in our normalization.  Thus there is a perfect
$G_{\Qp}$-equivariant pairing
\[
 \mathscr T_h\times A_h
 \longrightarrow E/\cO(1).
\]
For every finite extension $K/\Qp$, cup product followed by the local
invariant map gives Tate's local pairing
\[
 H^1(K,\mathscr T_h)\times H^1(K,A_h)
 \longrightarrow
 H^2(K,E/\cO(1))
 \xrightarrow{\operatorname{inv}_K}
 E/\cO.
\]
If $L/K$ is a finite extension, then
\[
 \langle\operatorname{cores}_{L/K}(x),y\rangle_K
 =
 \langle x,\operatorname{res}_{L/K}(y)\rangle_L.
\]
Passing to the inverse limit in the first variable and
the direct limit in the second gives a pairing
\begin{equation}\label{eq:iw-local-tate-pairing}
 \langle\ ,\ \rangle_p:
 H^1_{\Iw}(\Qp,\mathscr T_h)
 \times
 H^1(K_\infty,A_h)
 \longrightarrow E/\cO.
\end{equation}
If
$\natural\in\{\sharp,\flat\}$, the signed local condition at $p$ is
defined to be the orthogonal complement of the kernel of the
corresponding Coleman map:
\[
 H^1_{\natural}(K_\infty,A_h)
 =
 \bigl(\ker(\operatorname{Col}_{h}^{\natural})\bigr)^\perp.
\]
This is the local condition at $p$ used in the definition of the signed
Selmer group. For a prime $\ell\neq p$, the local condition is the unramified subgroup
\[
 H^1_{\ur}(\Q_{\infty,v},A_h)
 =\ker\bigl(H^1(\Q_{\infty,v},A_h)
 \longrightarrow H^1(I_v,A_h)\bigr),
\]
where $v$ is a prime of $\Q_\infty$ above $\ell$.  If $\Sigma$ contains $p$, $\infty$ and all primes at which $A_h$ is ramified, we define
\begin{equation}\label{eq:signed-selmer-definition}
 \Sel^\natural(A_h/\Q_\infty)
 =\ker\left(
 H^1(G_{\Q,\Sigma}/\Q_\infty,A_h)
 \longrightarrow
 \frac{H^1(\Q_{\infty,p},A_h)}{H^1_\natural(\Q_{\infty,p},A_h)}
 \oplus
\bigoplus_{\substack{ v\mid\ell\\ \ell\in\Sigma\setminus\{p\}}}
 \frac{H^1(\Q_{\infty,v},A_h)}{H^1_{\ur}(\Q_{\infty,v},A_h)}
 \right).
\end{equation}
Its Pontryagin dual is denoted
\[
 X^\natural(h)=\Sel^\natural(A_h/\Q_\infty)^\vee.
\]
Whenever $X^\natural(h)$ is torsion, its Iwasawa invariants are written $\mu^\natural(h)$ and $\lambda^\natural(h)$.

Let $\Sigma_0\subseteq\Sigma\setminus\{p,\infty\}$.  The non-primitive signed Selmer group $\Sel^{\natural,\Sigma_0}(A_h/\Q_\infty)$ is obtained from \eqref{eq:signed-selmer-definition} by imposing no local condition at the primes above $\Sigma_0$.  Set
\[
 Y_h^{\natural,\Sigma_0}
 =\Sel^{\natural,\Sigma_0}(A_h/\Q_\infty)^\vee.
\]
For $\ell\in\Sigma_0$, put
\[
 \mathcal H_{\ell}(A_h)
 =\prod_{v\mid\ell}
 \frac{H^1(\Q_{\infty,v},A_h)}{H^1_{\ur}(\Q_{\infty,v},A_h)}.
\]
Under the torsion hypothesis, the Poitou--Tate argument of \cite[Corollary~2.10]{HatleyLei2019} gives an exact sequence
\begin{equation}\label{eq:primitive-nonprimitive-exact}
 0\longrightarrow
 \Sel^\natural(A_h/\Q_\infty)
 \longrightarrow
 \Sel^{\natural,\Sigma_0}(A_h/\Q_\infty)
 \longrightarrow
 \bigoplus_{\ell\in\Sigma_0}\mathcal H_{\ell}(A_h)
 \longrightarrow0.
\end{equation}
The local modules on the right are cotorsion and have vanishing $\mu$-invariant; see \cite[Remark~2.11]{HatleyLei2019}.  If
\[
 \delta_{\ell,h}:=\lambda\bigl(\mathcal H_{\ell}(A_h)^\vee\bigr),
\]
then \eqref{eq:primitive-nonprimitive-exact} gives
\begin{equation}\label{eq:primitive-nonprimitive}
 \lambda^{\natural,\Sigma_0}(h)
 =\lambda^\natural(h)+\sum_{\ell\in\Sigma_0}\delta_{\ell,h}.
\end{equation}

\subsection{Residual comparison}

We next compare the residual signed Selmer groups for two congruent modular forms. 

Let $f$ and $g$ be $p$-non-ordinary modular forms of the same even
weight $k$, and let $T_f$ and $T_g$ be the corresponding
$G_{\Q}$-stable $\cO$-lattices.  We assume that the hypotheses
\textup{(irred)}, \textup{(inv)}, \textup{(BLZ)} and \textup{(Cong)}
of Hatley--Lei hold.  We recall these conditions explicitly; they are
summarized in \cite[p.~1289]{HatleyLei2019}.

The hypothesis \textup{(irred)} requires that $T_f/\varpi T_f$
be irreducible as a $G_{\Q}$-representation.  Since we assume that
$T_f/\varpi T_f\simeq T_g/\varpi T_g$, this is equivalent to the
corresponding assertion for $g$.
The hypothesis \textup{(Cong)} is precisely the residual congruence
\begin{equation}\label{eq:HL-cong}
 T_f/\varpi T_f
 \simeq
 T_g/\varpi T_g
\end{equation}
as $G_{\Q}$-representations.

The hypothesis
\textup{(inv)} is the vanishing
\begin{equation}\label{eq:HL-inv}
 A_h(m)^{G_{\Q_{\infty,p}}}=0
\end{equation}
for every $m\geq0$.

Finally, the hypothesis \textup{(BLZ)} requires that
\begin{equation}\label{eq:HL-BLZ}
 \ord_p(a_p(f)),
 \ord_p(a_p(g))
 >
 \left\lfloor\frac{k-2}{p-1}\right\rfloor;
\end{equation}
cf.
\cite[p.~1289]{HatleyLei2019} and
\cite[pp.~374--376]{BergerLiZhu2004}. When $k=2$, the right-hand side of \eqref{eq:HL-BLZ} is zero.
Consequently \textup{(BLZ)} becomes $\ord_p(a_p(f))>0$ and $\ord_p(a_p(g))>0$, i.e., both forms are non-ordinary at
$p$.


\begin{proposition}\label{prop:residual-comparison}
Let $\Sigma$ contain $p$, $\infty$, and all primes at which either
$f$ or $g$ is ramified, and set
$\Sigma_0=\Sigma\setminus\{p,\infty\}$. For either signed label
$\natural$, there is an isomorphism
\[
 \Sel^{\natural,\Sigma_0}(A_f/\Q_\infty)[\varpi]
 \simeq
 \Sel^{\natural,\Sigma_0}(A_g/\Q_\infty)[\varpi].
\]
\end{proposition}

\begin{proof}
This is a standard result and the method goes back to Greenberg--Vatsal
\cite[Section~2]{GreenbergVatsal2000}. The same
argument in the ordinary modular-form setting is recalled and used in
\cite[Section~4]{Ray2024}. In the non-ordinary setting, Hatley--Lei \cite[Propositions~4.3 and~4.5]{HatleyLei2019} identify the $\varpi$-torsion
of the non-primitive signed Selmer group with the residual Selmer group.
\end{proof}


\begin{proposition}
\label{prop:one-sided}
Let $f$ and $g$ be as in Proposition~\ref{prop:residual-comparison}.  Fix $\natural\in\{\sharp,\flat\}$.  If $X^\natural(g)$ is torsion and $\mu^\natural(g)=0$, then $X^\natural(f)$ is torsion and $\mu^\natural(f)=0$.
\end{proposition}

\begin{proof}
Take $\Sigma_0$ as in Proposition~\ref{prop:residual-comparison} and write $Y_h=Y_h^{\natural,\Sigma_0}$.  Since $X^\natural(g)$ is torsion, \eqref{eq:primitive-nonprimitive-exact} is an exact sequence for $g$. The local terms are torsion with vanishing $\mu$-invariant, so $Y_g$ is torsion with $\mu(Y_g)=0$.  Lemma~\ref{lem:modpi-criterion} gives $\dim_{\mathbf k}Y_g/\varpi Y_g<\infty$.

Pontryagin duality identifies
\[
 (\Sel^{\natural,\Sigma_0}(A_h/\Q_\infty)[\varpi])^\vee
 \simeq Y_h/\varpi Y_h
\]
for $h=g$ or $f$. Therefore
Proposition~\ref{prop:residual-comparison} gives
\[
 Y_f/\varpi Y_f\simeq Y_g/\varpi Y_g.
\]
It follows that $Y_f/\varpi Y_f$ is finite dimensional. Applying Lemma~\ref{lem:modpi-criterion} to $Y_f$ shows that $Y_f$ is finitely generated and torsion over $\Lambda_{\cO}$ with $\mu(Y_f)=0$.  The primitive signed Selmer group is a subgroup of the non-primitive group; after dualizing, $X^\natural(f)$ is a quotient of $Y_f$.  Hence $X^\natural(f)$ is torsion over $\Lambda_{\cO}$ with $\mu(X^\natural(f))=0$.
\end{proof}


\begin{lemma}
\label{lem:weight-two-finite-index}
Let $h$ be a weight-two $p$-non-ordinary newform of level prime to $p$ and trivial nebentypus. Assume \textup{(irred)}, \textup{(inv)}, \textup{(BLZ)} and \textup{(tor)} in the notation of Hatley--Lei. For each $\natural\in\{\sharp,\flat\}$, we have that $\Sel^\natural(A_h/\Q_\infty)$ has no proper $\Lambda_{\cO}$-submodule of finite index.
\end{lemma}

\begin{proof}
The result follows from \cite[Theorem~3.1]{HatleyLei2019}.
\end{proof}

\begin{proposition}[Signed congruence formula]
\label{prop:signed-congruence}
Let $f$ and $g$ satisfy the hypotheses of Proposition~\ref{prop:residual-comparison}.  If $X^\natural(g)$ is torsion and $\mu^\natural(g)=0$, then $X^\natural(f)$ is torsion, $\mu^\natural(f)=0$, and
\begin{equation}\label{eq:transition}
 \lambda^\natural(f)-\lambda^\natural(g)
 =\sum_{\ell\in\Sigma_0}
 \bigl(\delta_{\ell,g}-\delta_{\ell,f}\bigr).
\end{equation}
\end{proposition}

\begin{proof}
Proposition~\ref{prop:one-sided} first supplies the torsion and $\mu=0$ hypotheses for $f$. Thus the running hypothesis \textup{(tor)} of Hatley--Lei is satisfied for both $f$ and $g$.  By Lemma~\ref{lem:weight-two-finite-index}, both signed Selmer groups have no proper finite-index submodule.

By Proposition~\ref{prop:residual-comparison}, the residual non-primitive Selmer groups are isomorphic. It follows then from the \cite[Proof of Theorem~4.6]{HatleyLei2019} and \cite[Lemma~12]{Kim2009} that
\[
 \lambda^{\natural,\Sigma_0}(f)
 =\lambda^{\natural,\Sigma_0}(g).
\]
Applying \eqref{eq:primitive-nonprimitive} to $f$ and $g$ and subtracting gives \eqref{eq:transition}.
\end{proof}

\begin{theorem}
\label{thm:rigidity}
Let $f$ and $g$ be as in Proposition~\ref{prop:signed-congruence}.  Assume that $X^\sharp(g)$ and $X^\flat(g)$ are torsion and that $\mu^\sharp(g)=\mu^\flat(g)=0$.  Then
\[
 \lambda^\sharp(f)-\lambda^\flat(f)
 =\lambda^\sharp(g)-\lambda^\flat(g).
\]
More precisely, there is an integer $c(f,g)$ determined by the tame local correction terms such that
\[
 (\lambda^\sharp(f),\lambda^\flat(f))
 =(\lambda^\sharp(g)+c(f,g),\lambda^\flat(g)+c(f,g)).
\]
\end{theorem}

\begin{proof}
Apply Proposition~\ref{prop:signed-congruence} for $\sharp$ and $\flat$.  Every term on the right-hand side of \eqref{eq:transition} is attached to a prime $\ell\neq p$ and is therefore the same for the two signed Selmer structures. On subtracting the two identities, we obtain the result.
\end{proof}

\section{Residual images, non-optimal levels and auxiliary primes}
\label{sec:level-raising}

In this section, we carry out the global construction of the forms that
will be used in the proof of Theorem~\ref{thm:main}.  There are three
ingredients.  We first recall the part of the non-optimal-level theorem
of Carayol and Diamond--Taylor that allows one to raise the tame level at
several primes simultaneously.  We then isolate two conjugacy-stable
subsets of the residual image and use the Chebotarev density theorem to
produce two corresponding sets of auxiliary primes.  Finally, we show
that level raising at any finite collection of these primes preserves
the residual representation and forces the required local Steinberg
signs.  The argument is a refinement of the construction in
\cite[Sections~2, 4--5]{Ray2024}.

\subsection{Artin conductors and non-optimal levels}

Fix an algebraic closure $\overline\Q$ of $\Q$.  For every prime
$\ell$, choose an embedding $\overline\Q\hookrightarrow
\overline\Q_\ell$.  This identifies the absolute Galois group
$G_\ell=\Gal(\overline\Q_\ell/\Q_\ell)$ with a decomposition subgroup
of $G_\Q=\Gal(\overline\Q/\Q)$.  We write $I_\ell\subseteq G_\ell$
for the inertia subgroup and choose an arithmetic Frobenius element
$\Frob_\ell\in G_\ell/I_\ell$.

Let
\[
 \bar\rho:G_\Q\longrightarrow\operatorname{GL}_2(\mathbf k)
\]
be a continuous, odd and absolutely irreducible representation of
characteristic $p$.  If $\ell\neq p$, let
$a_\ell(\bar\rho)$ denote the Artin-conductor exponent of the local
representation $\bar\rho|_{G_\ell}$. The prime-to-$p$ Artin conductor of $\bar\rho$ is
\begin{equation}\label{eq:residual-artin-conductor}
 N(\bar\rho)=\prod_{\ell\neq p}\ell^{a_\ell(\bar\rho)}.
\end{equation}
If $h$ is a modular lift of $\bar\rho$ whose tame level is $N_h$, then
$N(\bar\rho)$ divides $N_h$; see
\cite{Ribet1994Report,Carayol1989}.  We say that $h$ has
\emph{optimal level} if $N_h=N(\bar\rho)$.  In this terminology, a
prime dividing $N_h/N(\bar\rho)$ records ramification that is present
in the characteristic-zero representation but disappears after
reduction modulo the maximal ideal.

Suppose that $\ell\nmid pN(\bar\rho)$.  The residual representation is
then unramified at $\ell$, and Carayol's condition for introducing
conductor exponent one at $\ell$ is
\begin{equation}\label{eq:carayol-condition}
 \ell\,\Tr\bigl(\bar\rho(\Frob_\ell)\bigr)^2
 =
 (1+\ell)^2\det\bigl(\bar\rho(\Frob_\ell)\bigr)
 \quad\text{in }\mathbf k.
\end{equation}
Since $\det(\bar\rho(\Frob_\ell))=\ell$, condition
\eqref{eq:carayol-condition} is equivalent to
\begin{equation}\label{eq:levelraising-trace}
 \Tr\bigl(\bar\rho(\Frob_\ell)\bigr)^2=(1+\ell)^2.
\end{equation}

Let $f$ be a weight-two newform with trivial nebentypus, and let
$\ell\neq p$ be a prime such that $\ell\Vert N_f$.  We say that $f$
is \emph{Steinberg at $\ell$} if its local automorphic representation
has the form $\pi_{f,\ell}\simeq \operatorname{St}\otimes\xi_\ell$,
where $\xi_\ell$ is an unramified character. Since the nebentypus of
$f$ is trivial, one has $\xi_\ell(\Frob_\ell)=\varepsilon_\ell$. The semisimplification of the local representation
at $\ell$ has unramified characters $\xi_\ell$ and $\xi_\ell\chi_p$.
Consequently, after reduction modulo $\varpi$, the Frobenius eigenvalues are $\varepsilon_\ell$ and $\varepsilon_\ell\ell$.


\begin{theorem}
\label{thm:carayol-DT}
Let $p\geq5$, and let
$\bar\rho:G_\Q\rightarrow\operatorname{GL}_2(\mathbf k)$ be odd and
absolutely irreducible, with determinant $\overline\chi_p$.  Suppose
that $\bar\rho$ arises from a normalized newform of weight two, trivial
nebentypus and optimal tame level $N(\bar\rho)$.  Let $S$ be a finite
set of primes such that every $\ell\in S$ satisfies
\begin{itemize}
    \item $\ell\nmid pN(\bar\rho)$,
    \item $\ell\not\equiv\pm1\pmod p$,
    \item $\Tr\bigl(\bar\rho(\Frob_\ell)\bigr)^2=(1+\ell)^2.$
\end{itemize}
Then there is a normalized weight-two newform $h$ of trivial
nebentypus, lifting $\bar\rho$, whose tame level is
\begin{equation}\label{eq:CDT-raised-level}
 N_h=N(\bar\rho)\prod_{\ell\in S}\ell.
\end{equation}
The form $h$ is new at every prime in $S$.  More precisely, its local
representation at such a prime is a Steinberg representation of sign
$\varepsilon_\ell\in\{\pm1\}$, and the residual Frobenius eigenvalues
are $\varepsilon_\ell$ and $\varepsilon_\ell\ell$.
\end{theorem}

\begin{proof}
This is the same non-optimal level argument as
\cite[Proposition~4.7]{Ray2024}.  Indeed, for every $\ell\in S$,
condition \eqref{eq:carayol-condition} is case~(1) of Carayol's
criterion, so adjoining $\ell$ increases the conductor exponent by
one.  Hence
\[
 M=N(\bar\rho)\prod_{\ell\in S}\ell
\]
is an admissible non-optimal level for $\bar\rho$.  Since $p\geq5$ and
the weight is $2$, the fixed-nebentypus form of the Diamond--Taylor
level-raising theorem applies in the trivial-character Hecke algebra and
produces a weight-two newform $f$ of trivial nebentypus lifting
$\bar\rho$ of level $M$; see \cite[Theorem~2.3]{Ray2024}, which recalls
\cite{DiamondTaylor1994}. Equivalently, one may apply the usual
level-raising theorem successively at the primes in $S$; at each step the
nebentypus remains trivial. This is precisely the argument used in
\cite[Proposition~4.7]{Ray2024}.

At every $\ell\in S$, the form $f$ is ramified at $\ell$ and
$\bar\rho_f(\Frob_\ell)$ has eigenvalues
$\epsilon_\ell\ell$ and $\epsilon_\ell$, with
$\epsilon_\ell\in\{\pm1\}$.  The local calculation of
\cite[Lemma~4.4]{Ray2024} therefore applies.  It shows that, after a
change of basis,
\[
 \rho_f(\Frob_\ell)=
 \begin{pmatrix}
  \ell x_\ell&0\\
  0&x_\ell
 \end{pmatrix}\quad \text{and}\quad
 \rho_f(\tau_\ell)=
 \begin{pmatrix}
  1&y_\ell\\
  0&1
 \end{pmatrix},
\]
where $x_\ell\equiv\epsilon_\ell\pmod{\varpi}$.
Since $f$ has weight two and trivial nebentypus,
$\det\rho_f=\chi_p$, and hence $x_\ell^2=1$.
As $p$ is odd, it follows that $x_\ell=\epsilon_\ell$.
Thus the local representation is of Steinberg type with sign
$\epsilon_\ell$. This proves
the final assertion.
\end{proof}

\subsection{The residual image}

We study the residual representation of the fixed form $g$ in more detail.
Set
\[
 H_g=\bar\rho_g(G_\Q)\subseteq\operatorname{GL}_2(\mathbf k),
\]
where $L_g=\overline\Q^{\ker\bar\rho_g}$ is the field cut out by $\bar \rho_g$. The representation $\bar\rho_g$ identifies $H_g$ with
$\Gal(L_g/\Q)$.  Since $g$ has weight two and trivial nebentypus, one
has $\det\bar\rho_g=\overline\chi_p$.  In particular,
$\det(H_g)=\Fp^\times$, where $\Fp^\times$ is viewed as a subgroup of
$\mathbf k^\times$.  Moreover, $\ker\bar\rho_g$ is contained in
$\ker\overline\chi_p$, and hence $\Q(\mu_p)\subseteq L_g$.

\begin{proposition}[Dickson]
\label{prop:dickson-zywina}
Let $G$ be a subgroup of $\operatorname{GL}_2(\mathbb F_{p^r})$, and
let $\overline G$ denote its image in
$\operatorname{PGL}_2(\mathbb F_{p^r})$.  Up to conjugacy, one of the
following possibilities occurs:
\begin{enumerate}[label=(\roman*)]
\item $G$ is contained in a Borel subgroup;
\item $G$ is contained in the normalizer of a Cartan subgroup;
\item $\overline G$ is isomorphic to $A_4$, $S_4$, or $A_5$;
\item there is a subfield $\mathbb F_{p^s}\subseteq\mathbb F_{p^r}$
such that $\overline G$ is conjugate to
$\operatorname{PSL}_2(\mathbb F_{p^s})$ or
$\operatorname{PGL}_2(\mathbb F_{p^s})$.
\end{enumerate}
\end{proposition}

\begin{proof}
This is Dickson's classification in the form recalled in
\cite[Lemma~5.1]{Zywina2023}.
\end{proof}
For $u\in H_g$, write
$d(u)=\det(u)$ and define
\begin{align}
 \mathcal C_g^+
 &=\bigl\{u\in H_g:d(u)\notin\{\pm1\},\
       \Tr(u)=1+d(u)\bigr\},\label{eq:Cplus}\\
 \mathcal C_g^-
 &=\bigl\{u\in H_g:d(u)\notin\{\pm1\},\
       \Tr(u)=-(1+d(u))\bigr\}.\label{eq:Cminus}
\end{align}
Both conditions depend only on the trace and determinant, so the two
sets are stable under conjugation in $H_g$.  If $u\in\mathcal C_g^+$,
then its characteristic polynomial factors as
\[
 X^2-(1+d(u))X+d(u)=(X-1)(X-d(u)).
\]
Similarly, if $u\in\mathcal C_g^-$, its characteristic polynomial is
$(X+1)(X+d(u))$.  Since $d(u)\notin\{\pm1\}$, the two roots in either
factorization are distinct.
The following proposition shows, in particular, that hypothesis
\ref{H:image} is automatic under a large-image condition. It
also gives explicit densities in that case.

\begin{proposition}
\label{prop:large-image-criterion}
Suppose that $\operatorname{SL}_2(\mathbf k)\subseteq H_g$, and put
$q_0=|\mathbf k|$.  Then $\mathcal C_g^+$ and $\mathcal C_g^-$ are
nonempty and
\begin{equation}\label{eq:large-image-disjointness}
 L_g\cap\Q(\mu_{p^2})=\Q(\mu_p).
\end{equation}
Moreover,
\begin{equation}\label{eq:large-image-C-density}
 \frac{|\mathcal C_g^+|}{|H_g|}
 =\frac{|\mathcal C_g^-|}{|H_g|}
 =\frac{p-3}{(p-1)(q_0-1)}.
\end{equation}
\end{proposition}

\begin{proof}
The determinant map on $H_g$ is surjective
onto $\Fp^\times$. It is easy to see that
\begin{equation}\label{eq:H-large}
 H_g=\bigl\{A\in\operatorname{GL}_2(\mathbf k):
               \det(A)\in\Fp^\times\bigr\}.
\end{equation}
For $a\in\Fp^\times\setminus\{\pm1\}$, the two matrices $\operatorname{diag}(a,1)$ and $\operatorname{diag}(-a,-1)$ both
belong to $H_g$. They belong to $\mathcal C_g^+$ and
$\mathcal C_g^-$, respectively.  Since $p\geq5$, there is at least one
such $a$, proving that both sets are nonempty.

We next prove \eqref{eq:large-image-disjointness}.  We already know
that $\Q(\mu_p)$ is contained in $L_g$.  Since
$[\Q(\mu_{p^2}):\Q(\mu_p)]=p$, there is no proper intermediate field
between these two cyclotomic fields.  Thus, if the intersection in
\eqref{eq:large-image-disjointness} were larger than $\Q(\mu_p)$, it
would contain all of $\Q(\mu_{p^2})$.  Restriction would then exhibit
the cyclic group
$\Gal(\Q(\mu_{p^2})/\Q(\mu_p))$, which has order $p$, as a quotient of
\[
 \Gal(L_g/\Q(\mu_p))
 =\ker\bigl(\det:H_g\longrightarrow\Fp^\times\bigr).
\]
By \eqref{eq:H-large}, this kernel is
$\operatorname{SL}_2(\mathbf k)$.  For $p\geq5$, the group
$\operatorname{SL}_2(\mathbf k)$ is perfect.  It has no nontrivial
abelian quotient and, in particular, no quotient of order $p$.

It remains to count the two conjugacy-stable subsets.  Fix
$a\in\Fp^\times\setminus\{\pm1\}$.  Every element of
$\mathcal C_g^+$ with determinant $a$ has distinct eigenvalues $1$ and
$a$, and is therefore conjugate in $\operatorname{GL}_2(\mathbf k)$
to $\operatorname{diag}(a,1)$.  Its centralizer is the split diagonal
torus, of order $(q_0-1)^2$.  Hence this conjugacy class has cardinality
\[
 \frac{|\operatorname{GL}_2(\mathbf k)|}{(q_0-1)^2}
 =q_0(q_0+1).
\]
The classes obtained from distinct values of $a$ are disjoint because
their determinants are distinct.  There are $p-3$ possible values of
$a$, and consequently
$|\mathcal C_g^+|=(p-3)q_0(q_0+1)$.  The same argument, using
$\operatorname{diag}(-a,-1)$, gives the identical formula for
$|\mathcal C_g^-|$. Finally, \eqref{eq:H-large} gives
$|H_g|=(p-1)|\operatorname{SL}_2(\mathbf k)|
=(p-1)q_0(q_0^2-1)$.  Dividing the preceding cardinalities by this
order proves \eqref{eq:large-image-C-density}.
\end{proof}

\subsection{The two Frobenian sets}

We now define the auxiliary primes.  Let $\Pi_g$ be the set of primes
$q\nmid N_gp$ such that $\bar\rho_g(\Frob_q)\in\mathcal C_g^+$ and $
q^{p-1}\not\equiv1\pmod{p^2}$. Let $\Omega_g$ be the set of primes $\ell\nmid N_gp$ such that $\bar\rho_g(\Frob_\ell)\in\mathcal C_g^-$. The determinant of $\bar\rho_g(\Frob_v)$ is the reduction of $v$
modulo $p$.  Since the determinant of an element of either
$\mathcal C_g^+$ or $\mathcal C_g^-$ is different from $\pm1$, every
prime in $\Pi_g\cup\Omega_g$ satisfies
$v\not\equiv\pm1\pmod p$.  Thus the congruence restriction required by
Theorem~\ref{thm:carayol-DT} is already built into the definitions.

\begin{proposition}[Densities of the auxiliary primes]
\label{prop:aux-density}
Assume \ref{H:image}.  The sets $\Pi_g$ and $\Omega_g$ are Frobenian,
and their densities are
\begin{equation}\label{eq:aux-densities}
 \delta(\Pi_g)=\frac{p-1}{p}\frac{|\mathcal C_g^+|}{|H_g|}
 \quad\text{and}\quad
 \delta(\Omega_g)=\frac{|\mathcal C_g^-|}{|H_g|}.
\end{equation}
In particular, both sets have positive density and are infinite.
\end{proposition}

\begin{proof}
The set $\Omega_g$ is the Chebotarev set in the finite Galois extension
$L_g/\Q$ associated with the union of conjugacy classes
$\mathcal C_g^-$.  Removing the finitely many primes that divide
$N_gp$ does not change its density.  The Chebotarev density theorem
therefore gives
$\delta(\Omega_g)=|\mathcal C_g^-|/|H_g|$.

The additional congruence $
q^{p-1}\not\equiv1\pmod{p^2}$ in the definition of $\Pi_g$ requires us to
adjoin the $p^2$-th roots of unity. Setting $M=\Q(\mu_{p^2})$, we find that both
$L_g$ and $M$ contain $\Q(\mu_p)$. Hypothesis \ref{H:image} states
that their intersection is $\Q(\mu_p)$. Consequently, the Galois group of the compositum
can be identified with the fibered product
\begin{equation}\label{eq:fiber-product}
 \Gal(L_gM/\Q)
 \simeq
 H_g\times_{\Fp^\times}(\Z/p^2\Z)^\times.
\end{equation}
Here the maps to $\Fp^\times$ are the determinant on $H_g$ and
reduction modulo $p$ on $(\Z/p^2\Z)^\times$.  Every element of
$\Fp^\times$ has exactly $p$ lifts to $(\Z/p^2\Z)^\times$, so the
group in \eqref{eq:fiber-product} has order $p|H_g|$.

Fix $u\in\mathcal C_g^+$ and set $a=d(u)$.  Among the $p$ elements of
$(\Z/p^2\Z)^\times$ which reduce to $a$, exactly one has
$(p-1)$-st power equal to $1$ modulo $p^2$.  Indeed, reduction modulo
$p$ induces an isomorphism from the subgroup of $(p-1)$-st roots of
unity in $(\Z/p^2\Z)^\times$ onto $\Fp^\times$, and the unique lift in
that subgroup is the Teichmuller lift of $a$.  The remaining $p-1$
lifts have $(p-1)$-st power different from $1$ modulo $p^2$.

It follows that the conjugacy-stable subset of
$\Gal(L_gM/\Q)$ defining $\Pi_g$ has cardinality
$(p-1)|\mathcal C_g^+|$.  The Chebotarev density theorem therefore gives
\[
 \delta(\Pi_g)
 =\frac{(p-1)|\mathcal C_g^+|}{p|H_g|}.
\]
This proves \eqref{eq:aux-densities}.  Positivity follows from the
nonemptiness of $\mathcal C_g^+$ and $\mathcal C_g^-$ in hypothesis
\ref{H:image}.
\end{proof}

If $v\nmid N_gp$, the characteristic polynomial of
$\bar\rho_g(\Frob_v)$ is the reduction of
$X^2-a_v(g)X+v$.  Hence, for $q\in\Pi_g$, the factorization attached to
$\mathcal C_g^+$ gives
$a_q(g)\equiv1+q\pmod\varpi$.  Likewise, if
$\ell\in\Omega_g$, then
$a_\ell(g)\equiv-(1+\ell)\pmod\varpi$.  In either case,
\eqref{eq:levelraising-trace} holds.  Every prime in
$\Pi_g\cup\Omega_g$ is therefore a level-raising prime. 

\subsection{Simultaneous level raising}

We may now apply the preceding construction to any finite collection
of auxiliary primes.

\begin{proposition}
\label{prop:level-raising}
Let $Q\subset\Pi_g$ and $R\subset\Omega_g$ be finite sets.  Under
hypotheses \ref{H:image} and \ref{H:optimal}, there is a normalized
newform $f$ of weight two and trivial nebentypus such that
$\bar\rho_f\simeq\bar\rho_g$ and
\begin{equation}\label{eq:simultaneous-raised-level}
 N_f=N_g\prod_{q\in Q}q\prod_{\ell\in R}\ell.
\end{equation}
Further, $f$ is
new at every prime in $Q\cup R$.  Its Steinberg sign is $+1$ at every
$q\in Q$ and $-1$ at every $\ell\in R$.
\end{proposition}

\begin{proof}
The two auxiliary sets are disjoint.  Indeed, an element of
$\mathcal C_g^+$ has trace $1+d$, while an element of
$\mathcal C_g^-$ has trace $-(1+d)$; these values cannot be equal when
$d\neq-1$.  Thus $Q\cap R=\varnothing$.  If both sets are empty, we
may simply take $f=g$.  Suppose otherwise and put $S=Q\cup R$.

Every prime in $S$ is prime to $pN_g$, is not congruent to $\pm1$
modulo $p$, and satisfies the trace condition
\eqref{eq:levelraising-trace}.  Hypothesis \ref{H:optimal} identifies
$N_g$ with $N(\bar\rho_g)$.  Theorem~\ref{thm:carayol-DT} therefore
produces a normalized newform $f$ of trivial nebentypus whose residual
representation is isomorphic to $\bar\rho_g$ and whose level is
the integer in \eqref{eq:simultaneous-raised-level}.  The same theorem
shows that $f$ is new at every prime in $S$.

It remains to determine the local signs.  Let $v\in S$, and let
$\varepsilon_v\in\{\pm1\}$ be the Steinberg sign of $f$ at $v$.  The
residual Frobenius eigenvalues of a Steinberg representation of sign
$\varepsilon_v$ are
$\{\varepsilon_v,\varepsilon_vv\}$, and hence its residual trace is
$\varepsilon_v(1+v)$.  If $q\in Q$, the definition of
$\mathcal C_g^+$ gives residual trace $1+q$.  Since
$q\not\equiv-1\pmod p$, the element $1+q$ is nonzero in $\mathbf k$,
and therefore $\varepsilon_q=+1$.  If $\ell\in R$, the definition of
$\mathcal C_g^-$ gives residual trace $-(1+\ell)$, and the same
argument gives $\varepsilon_\ell=-1$.
\end{proof}


\begin{lemma}[Preservation of non-ordinarity]
\label{lem:nonordinary-preserved}
The newform $f$ furnished by
Proposition~\ref{prop:level-raising} is non-ordinary at $p$.
\end{lemma}

\begin{proof}
After enlarging the coefficient field if necessary, choose stable
lattices for $f$ and $g$ whose reductions realize the fixed
isomorphism $\bar\rho_f\simeq\bar\rho_g$.  Restricting this isomorphism
to a decomposition group at $p$ gives
\begin{equation}\label{eq:local-residual-isomorphism-at-p}
 \bar\rho_f|_{G_{\Q_p}}\simeq\bar\rho_g|_{G_{\Q_p}}.
\end{equation}

Since $g$ has weight two, level prime to $p$, and is non-ordinary, its
$p$-th Fourier coefficient is zero modulo $\varpi$.  The local
description of the residual representation in the Fontaine--Laffaille
range shows that $\bar\rho_g|_{G_{\Q_p}}$ is of niveau two and is
irreducible; see \cite[Theorem~2.6]{Edixhoven1992}.  More explicitly,
after extending scalars to an algebraic closure of the residue field,
the semisimplification of its restriction to inertia has characters
$\omega_2$ and $\omega_2^p$, where $\omega_2$ is a fundamental
character of niveau two.  These two characters are interchanged by
Frobenius, so neither inertia eigenspace is stable under the full group
$G_{\Q_p}$.  This is precisely the irreducible niveau-two case.

Suppose, to the contrary, that $f$ were ordinary at $p$.  The
$p$-adic representation attached to an ordinary weight-two form has a
$G_{\Q_p}$-stable one-dimensional subspace and an unramified
one-dimensional quotient.  This gives a rank-one direct summand whose reduction
modulo $\varpi$ is a nonzero proper
$G_{\Q_p}$-stable subspace.  Thus the residual local representation of
$f$ would be reducible.  This contradicts
\eqref{eq:local-residual-isomorphism-at-p} and the irreducibility of the
residual local representation of $g$.  Hence $f$ is non-ordinary at
$p$.
\end{proof}
\section{The local correction terms and proof of the main theorem}
\label{sec:local}

Let $h$ be a weight-two newform and let $\ell\neq p$. With respect to the notation of \cite[Section~5]{HatleyLei2019}, set
\begin{equation}\label{eq:local-polynomial-def}
 P_{\ell,h}(X)
 =\det\left(1-\Frob_\ell X\mid V_h^{I_\ell}\right)\in\cO[X]
\end{equation}
and write $\widetilde P_{\ell,h}(X)$ for its reduction modulo $\varpi$.  Let $d_{\ell,h}$ be the multiplicity of $X=\ell^{-1}$ as a root of $\widetilde P_{\ell,h}(X)$.  If $r_\ell$ denotes the number of primes of $\Q_\infty$ above $\ell$, then Proposition~5.1 and Corollary~5.2 of \cite{HatleyLei2019} give
\begin{equation}\label{eq:delta-rd}
 \delta_{\ell,h}=r_\ell d_{\ell,h}.
\end{equation}
Indeed, the local $\lambda$-invariant at each prime $v\mid\ell$ is $d_{\ell,h}$, and there are $r_\ell$ such primes.

Let $q\in\Pi_g$.  Since $q\nmid N_gp$, the representation $V_g$ is unramified at $q$.  The residual Frobenius eigenvalues are $1$ and $q$, hence
\begin{equation}\label{eq:local-Pi-g}
 \widetilde P_{q,g}(X)=(1-X)(1-qX).
\end{equation}
Note that $X=q^{-1}$ is a root of the second factor.  It is not a root of the first factor because $q\not\equiv1\pmod p$, and thus
\begin{equation}\label{eq:d-Pi-g}
 d_{q,g}=1.
\end{equation}

Let $f$ be the form given by Proposition~\ref{prop:level-raising}.  Since $f$ is new of exponent one at $q$, its local automorphic representation is Steinberg up to an unramified twist.  Consequently $V_f^{I_q}$ is one-dimensional.  The positive sign in Proposition~\ref{prop:level-raising} gives
\begin{equation}\label{eq:local-Pi-f}
 \widetilde P_{q,f}(X)=1-X.
\end{equation}
At $X=q^{-1}$ this is nonzero because $q\not\equiv1\pmod p$, and hence
\begin{equation}\label{eq:d-Pi-f}
 d_{q,f}=0.
\end{equation}

The factor $r_q$ is controlled by the additional congruence modulo $p^2$.  We spell this out. The cyclotomic character gives an identification $\Gamma\simeq1+p\Zp$. The decomposition subgroup of a prime $q\neq p$ is the closed subgroup generated by the pro-$p$ part of $q$.  Equivalently it is generated by $q^{p-1}\in1+p\Zp$.  The condition
\[
 q^{p-1}\not\equiv1\pmod{p^2}
\]
is equivalent to $v_p(q^{p-1}-1)=1$, and an element of $1+p\Zp$ with this property is a topological generator.  Hence the decomposition subgroup is all of $\Gamma$, and therefore
\begin{equation}\label{eq:r-q-one}
 r_q=[\Gamma:\Gamma_q]=1.
\end{equation}
Combining \eqref{eq:delta-rd}, \eqref{eq:d-Pi-g}, \eqref{eq:d-Pi-f} and \eqref{eq:r-q-one} yields
\begin{equation}\label{eq:Pi-correction}
 \delta_{q,g}-\delta_{q,f}=1.
\end{equation}

Now let $\ell\in\Omega_g$.  The residual Frobenius eigenvalues of $g$ are $-1$ and $-\ell$, so
\begin{equation}\label{eq:local-Omega-g}
 \widetilde P_{\ell,g}(X)=(1+X)(1+\ell X).
\end{equation}
At $X=\ell^{-1}$ the two factors are $1+\ell^{-1}$ and $2$, both nonzero because $p\geq5$ and $\ell\not\equiv-1\pmod p$.  Therefore $d_{\ell,g}=0$.  The level-raised form has negative Steinberg sign and
\begin{equation}\label{eq:local-Omega-f}
 \widetilde P_{\ell,f}(X)=1+X,
\end{equation}
which is again nonzero at $X=\ell^{-1}$.  Hence $d_{\ell,f}=0$, and \eqref{eq:delta-rd} gives
\begin{equation}\label{eq:Omega-correction}
 \delta_{\ell,g}-\delta_{\ell,f}=0
\end{equation}
without any condition on the number of primes above $\ell$ in $\Q_\infty$.


\begin{lemma}
\label{lem:old-prime-lattice}
Assume that $h$ is congruent to $g$ and let $\ell\neq p$ be a prime such that the exponent of $\ell$ in the level of $h$ equals the exponent of $\ell$ in the Artin conductor of $\bar\rho_g$.  Setting $\bar T_h:=T_h/\varpi T_h$, we find that the natural inclusion map
\begin{equation}\label{eq:invariants-inclusion}
 T_h^{I_\ell}/\varpi T_h^{I_\ell}
 \longrightarrow
 (\bar T_h)^{I_\ell}
\end{equation}
is an isomorphism.  Consequently
\begin{equation}\label{eq:old-euler-reduction}
 \widetilde P_{\ell,h}(X)
 =\det\left(1-\Frob_\ell X\mid(\bar T_h)^{I_\ell}\right).
\end{equation}
\end{lemma}

\begin{proof}
This is precisely the argument of \cite[Lemma~4.8]{Ray2024}, applied to the inertia-invariant lattice and its Frobenius characteristic polynomial.
\end{proof}

\begin{lemma}\label{lem:old-prime-cancellation}
Let $f$ be as in Proposition~\ref{prop:level-raising} and let $\ell\mid N_g$. Then, we have that
\begin{enumerate}
    \item $\widetilde P_{\ell,f}(X)=\widetilde P_{\ell,g}(X)$,
    \item $d_{\ell,f}=d_{\ell,g}$,
    \item $\delta_{\ell,f}=\delta_{\ell,g}$.
\end{enumerate}
\end{lemma}

\begin{proof}
The forms $f$ and $g$ have the same $\ell$-adic level exponent, which equals the residual Artin-conductor exponent by Hypothesis~\ref{H:optimal}.  Lemma~\ref{lem:old-prime-lattice}, together with $\bar T_f\simeq\bar T_g$, therefore gives
$\widetilde P_{\ell,f}(X)=\widetilde P_{\ell,g}(X)$.  Equality of the multiplicities at $X=\widetilde\ell^{-1}$ gives $d_{\ell,f}=d_{\ell,g}$, and then \eqref{eq:delta-rd} gives $\delta_{\ell,f}=\delta_{\ell,g}$.
\end{proof}
\begin{proposition}
\label{prop:local-variation}
Let $f$ be as in Proposition~\ref{prop:level-raising}. Then the following assertions hold:
\begin{enumerate}[label=(\roman*)]
\item if $q\in Q\subset\Pi_g$, then $\delta_{q,g}-\delta_{q,f}=1$;
\item if $\ell\in R\subset\Omega_g$, then $\delta_{\ell,g}-\delta_{\ell,f}=0$;
\item if $\ell\mid N_g$, then $\delta_{\ell,g}=\delta_{\ell,f}$;
\item at every other tame prime the two reduced local Euler polynomials agree, and hence there is no contribution to the transition formula.
\end{enumerate}
\end{proposition}

\begin{proof}
Assertions (i) and (ii) are \eqref{eq:Pi-correction} and \eqref{eq:Omega-correction}.  Assertion (iii) is Lemma~\ref{lem:old-prime-cancellation}.  Finally, if $\ell\nmid pN_f$, both characteristic-zero representations are unramified at $\ell$ and the congruence $\bar\rho_f\simeq\bar\rho_g$ identifies the reduced Frobenius polynomials.  Thus $d_{\ell,f}=d_{\ell,g}$ and the corresponding term is zero.
\end{proof}

We can now prove the main theorem.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Choose finite disjoint sets $Q\subset\Pi_g$ and
$R\subset\Omega_g$.  Proposition~\ref{prop:level-raising} produces a
newform $f$ of the asserted level with
$\bar\rho_f\simeq\bar\rho_g$.  By Lemma~\ref{lem:nonordinary-preserved}, the
form $f$ is again non-ordinary at $p$. The hypotheses needed for the
signed congruence results hold for $f$ and $g$: \textup{(irred)} and
\textup{(Cong)} follow from \ref{H:image} and
$\bar\rho_f\simeq\bar\rho_g$, \textup{(BLZ)} follows from
non-ordinarity in weight two, and \textup{(inv)} follows from
\cite[Lemma~4.4]{Lei2011} since $2\leq p$. Fix a signed label
$\natural\in\{\sharp,\flat\}$.  Proposition~\ref{prop:one-sided}
implies that $X^\natural(f)$ is torsion and
$\mu^\natural(f)=0$.

Let $\Sigma$ contain $p$, $\infty$, and every prime dividing $N_f$, and
put $\Sigma_0=\Sigma\setminus\{p,\infty\}$. 
Proposition~\ref{prop:signed-congruence} then gives
\[
 \lambda^\natural(f)-\lambda^\natural(g)
 =
 \sum_{\ell\in\Sigma_0}
 \bigl(\delta_{\ell,g}-\delta_{\ell,f}\bigr).
\]
By Proposition~\ref{prop:local-variation}, every prime of $Q$
contributes one, every prime of $R$ contributes zero, and the
contributions at the old primes cancel.  The right-hand side is
therefore $|Q|$.  The calculation is independent of $\natural$, and
hence it holds simultaneously for $\sharp$ and $\flat$.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor:realization}]
Fix $n\geq0$.  Choose a set $Q\subset\Pi_g$ with $|Q|=n$.
If $n>0$, there are already infinitely many possible choices for $Q$. If $n=0$, take $Q=\varnothing$ and vary finite nonempty
sets $R\subset\Omega_g$. Since both auxiliary sets are infinite, this
gives infinitely many distinct tame levels.  The result then follows from Theorem~\ref{thm:main}.
\end{proof}


\section{Quantitative and probabilistic realization}
\label{sec:quantitative}

Throughout this section we retain the hypotheses and notation of
Theorem~\ref{thm:main}.  Thus $g$ is a weight-two newform which is
non-ordinary at $p$, both $X^\sharp(g)$ and $X^\flat(g)$ are torsion over
the cyclotomic Iwasawa algebra, and
$\mu^\sharp(g)=\mu^\flat(g)=0$.  Recall also that a prime in $\Pi_g$
contributes one to each signed $\lambda$-invariant, whereas a prime in
$\Omega_g$ changes the exact tame level without changing either signed
$\lambda$-invariant.  Write $\delta_\Pi$ and $\delta_\Omega$ for the positive densities of
$\Pi_g$ and $\Omega_g$, respectively, as in
Proposition~\ref{prop:aux-density}.  Thus
\[
 \delta_\Pi
 =
 \frac{p-1}{p}\frac{|\mathcal C_g^+|}{|H_g|}
 \qquad\text{and}\qquad
 \delta_\Omega
 =
 \frac{|\mathcal C_g^-|}{|H_g|}.
\]
When $\mathbf{k}=\mathbf F_p$ and
$H_g=\operatorname{GL}_2(\mathbf F_p)$, these specialize to
\[
 \delta_\Pi=\frac{p-3}{p(p-1)},
 \qquad
 \delta_\Omega=\frac{p-3}{(p-1)^2}.
\]  Since the two
sets of primes are disjoint, an admissible integer has a unique
factorization $D=D_\Pi D_\Omega$, where $D_\Pi$ is supported on $\Pi_g$
and $D_\Omega$ is supported on $\Omega_g$.  Put
\[
 \omega_\Pi(D)=\#\{q:q\mid D_\Pi\}.
\]
For every admissible $D$, choose one newform $f_D$ supplied by
Theorem~\ref{thm:main}, using the prime divisors of $D_\Pi$ as the set
$Q$ and the prime divisors of $D_\Omega$ as the set $R$.  The next
proposition records the arithmetic content which will be counted below.

\begin{proposition}
\label{prop:iwasawa-family}
For every admissible $D$, the form $f_D$ is a normalized weight-two
newform of trivial nebentypus and exact level $N_gD$.  It is
non-ordinary at $p$ and satisfies
$\bar\rho_{f_D}\simeq\bar\rho_g$.  Moreover, for each
$\natural\in\{\sharp,\flat\}$, the signed Selmer dual
$X^\natural(f_D)$ is torsion over the cyclotomic Iwasawa algebra,
\begin{equation}\label{eq:iwasawa-family}
 \mu^\natural(f_D)=0
 \quad\text{and}\quad
 \lambda^\natural(f_D)
 =\lambda^\natural(g)+\omega_\Pi(D).
\end{equation}
In particular,
\begin{equation}\label{eq:signed-difference-rigid-family}
 \lambda^\sharp(f_D)-\lambda^\flat(f_D)
 =
 \lambda^\sharp(g)-\lambda^\flat(g)
\end{equation}
for every admissible $D$.
\end{proposition}

\begin{proof}
Apply Proposition~\ref{prop:level-raising} with $Q$ and $R$ equal to the sets of prime divisors of $D_\Pi$ and $D_\Omega$, respectively.  This gives all the assertions concerning $f_D$.  Theorem~\ref{thm:main} then gives, for each $\natural\in\{\sharp,\flat\}$, \[\mu^\natural(f_D)=0\] and
\[
 \lambda^\natural(f_D)
 =\lambda^\natural(g)+|Q|
 =\lambda^\natural(g)+\omega_\Pi(D).
\]
Subtracting the formulas for $\sharp$ and $\flat$ proves
\eqref{eq:signed-difference-rigid-family}.
\end{proof}

We next recall the analytic input in the form needed here. Recall that a set of rational primes $\mathcal P$ is called Frobenian if
there exists a finite Galois extension $L/\mathbb Q$ and a union $C$ of
conjugacy classes in $\operatorname{Gal}(L/\mathbb Q)$ such that, apart
from finitely many primes, a prime $p$ belongs to $\mathcal P$ precisely
when the Frobenius conjugacy class $\operatorname{Frob}_p$ lies in $C$. 
\begin{lemma}
\label{lem:frobenian-prime-sum}
Let $\mathcal P$ be a Frobenian set of density $\delta$.  There is a
constant $c>0$ and a function $G_{\mathcal P}(s)$, holomorphic in the
dented zero-free region
\[
 \mathcal R_c=
 \left\{s=\sigma+it:
 \sigma\geq1-\frac{c}{1+\log(2+|t|)}\right\}\setminus(-\infty,1],
\]
and continuous at $1$, such that
\begin{equation}\label{eq:frobenian-prime-sum}
 \sum_{\ell\in\mathcal P}\ell^{-s}
 =
 \delta\log\frac{1}{s-1}+G_{\mathcal P}(s).
\end{equation}
The constant $c$ may be chosen so that
$\exp(G_{\mathcal P}(s))$ has at most polynomial growth in $|t|$
on $\mathcal R_c$. In particular,
$\sum_{\ell\leq Y,\,\ell\in\mathcal P}\ell^{-1}
=\delta\log\log Y+O_{\mathcal P}(1)$.
\end{lemma}

\begin{proof}
This is the Frobenian Dirichlet-series theorem of Serre
\cite{Serre1981}, together with the standard zero-free region for the
finite collection of Artin--Hecke $L$-functions occurring in its proof.
Brauer induction reduces the required continuation and growth bound to
Hecke $L$-functions. Since the extension defining $\mathcal P$ is fixed,
the region may be chosen uniformly for these finitely many factors.
\end{proof}

We define for $z\in \mathbb C$ and $\Re(s)>1$,
\begin{equation}\label{eq:marked-euler-product}
 F(s,z)
 =
 \prod_{\ell\in\Omega_g}(1+\ell^{-s})
 \prod_{q\in\Pi_g}(1+zq^{-s}).
\end{equation}
 Expanding the product gives
\begin{equation}\label{eq:marked-dirichlet-series}
 F(s,z)
 =
 \sum_{D\ \mathrm{admissible}}
 \frac{z^{\omega_\Pi(D)}}{D^s}.
\end{equation}
We set $\alpha(z):=\delta_\Omega+z\delta_\Pi$.

\begin{lemma}
\label{lem:SD-factorization}
There is an open set of $z$-values containing the closed interval
$[0,1]$ and a constant $c>0$ such that
\begin{equation}\label{eq:SD-factorization}
 F(s,z)=(s-1)^{-\alpha(z)}H(s,z),
\end{equation}
where $H(s,z)$ is holomorphic in both variables for $s\in\mathcal R_c$.
For every compact set $K$ of $z$-values in this open set, $H(s,z)$ has
at most polynomial growth in $|\Im(s)|$, uniformly for $z\in K$.
Moreover the open set can be chosen so that $H(1,z)$ is nonzero there.
Moreover, $H(1,x)>0$ for $0\leq x\leq1$.
\end{lemma}

\begin{proof}
For $\Re(s)>1$, the expansion $\log(1+w)=w+O(w^2)$ gives
\[
 \log F(s,z)
 =
 \sum_{\ell\in\Omega_g}\ell^{-s}
 +z\sum_{q\in\Pi_g}q^{-s}+R(s,z),
\]
where $R(s,z)$ is holomorphic for $\Re(s)>1/2$ and locally uniformly
holomorphic in $z$.  Lemma~\ref{lem:frobenian-prime-sum} identifies the
singular part with
$\alpha(z)\log(1/(s-1))$.  After shrinking $c$ if necessary,
exponentiating the remaining holomorphic terms gives
\eqref{eq:SD-factorization} throughout $\mathcal R_c$, with the stated
polynomial growth uniformly on compact sets of $z$-values.  For real
$s>1$ and real $x\geq0$, all Euler factors in $F(s,x)$ are positive.
Taking $s\to1^+$ proves positivity of the resulting nonzero limiting
factor.
\end{proof}

For $Y\geq2$, write
\[
 S(Y,z)=
 \sum_{\substack{D\leq Y\\D\ \mathrm{admissible}}}
 z^{\omega_\Pi(D)}.
\]
The Selberg--Delange theorem allows us to deduce an asymptotic formula for $S(Y,z)$ from the behavior of $F(s,z)$ near $s=1$.

\begin{proposition}
\label{prop:uniform-SD}
Let $K$ be a compact set of $z$-values contained in the open set from
Lemma~\ref{lem:SD-factorization}, and suppose that
$\inf_{z\in K}\Re\alpha(z)>0$.  Then, uniformly for $z\in K$,
\begin{equation}\label{eq:uniform-SD}
 S(Y,z)
 =
 Y(\log Y)^{\alpha(z)-1}
 \left(
 \frac{H(1,z)}{\Gamma(\alpha(z))}
 +O_K\left(\frac{1}{\log Y}\right)
 \right).
\end{equation}
On a slightly smaller compact set, this estimate may be differentiated
a fixed number of times with respect to $z$.
\end{proposition}

\begin{proof}
Put
\[
 G(s,z)=F(s,z)\zeta(s)^{-\alpha(z)}.
\]
Since $(s-1)\zeta(s)$ is holomorphic and nonzero in a sufficiently
small region of the form $\mathcal R_c$, Lemma~\ref{lem:SD-factorization}
shows that $G(s,z)$ is holomorphic there and has at most polynomial
growth, uniformly for $z\in K$.  If
$A=\max(1,\sup_{z\in K}|z|)$, then the coefficients of $F(s,z)$ are
bounded in absolute value by the nonnegative coefficients of $F(s,A)$.
Thus the type hypotheses in the uniform Selberg--Delange theorem are
satisfied uniformly for $z\in K$.  Theorem~II.5.2 of
\cite[Chapter~II.5]{Tenenbaum2015} gives \eqref{eq:uniform-SD}.
Applying the same uniform estimate on a slightly larger compact set and
using Cauchy's integral formula gives the asserted fixed number of
$z$-derivatives.
\end{proof}

For a fixed integer $n\geq0$, let
\[
 \mathscr D_n(Y)
 =
 \{D\leq Y:D\text{ is admissible and }\omega_\Pi(D)=n\},
\]
and let $\mathscr F_n(X)$ be the family consisting of the chosen forms
$f_D$ with $D\in\mathscr D_n(X/N_g)$.  Thus every member of
$\mathscr F_n(X)$ has level at most $X$.

\begin{theorem}
\label{thm:quantitative}
For every fixed $n\geq0$,
\begin{equation}\label{eq:Dn-asymptotic}
 \#\mathscr D_n(Y)
 \sim
 C_{g,n}Y(\log Y)^{\delta_\Omega-1}(\log\log Y)^n,
\end{equation}
where
\begin{equation}\label{eq:Cgn}
 C_{g,n}
 =
 \frac{H(1,0)}{\Gamma(\delta_\Omega)}
 \frac{\delta_\Pi^n}{n!}>0.
\end{equation}
Consequently,
\begin{equation}\label{eq:Fn-asymptotic}
 \#\mathscr F_n(X)
 \sim
 \frac{C_{g,n}}{N_g}
 X(\log X)^{\delta_\Omega-1}(\log\log X)^n.
\end{equation}
Every $f\in\mathscr F_n(X)$ is non-ordinary at $p$, satisfies
$\bar\rho_f\simeq\bar\rho_g$, and has torsion signed Selmer duals with
\begin{equation}\label{eq:quantitative-iwasawa-conclusion}
 \mu^\sharp(f)=\mu^\flat(f)=0
 \quad \text{and}\quad
 (\lambda^\sharp(f),\lambda^\flat(f))
 =
 (\lambda^\sharp(g)+n,\lambda^\flat(g)+n).
\end{equation}
\end{theorem}

\begin{proof}
For a power series $F(z)$, we write $[z^m]F(z)$ for the coefficient of $z^m$ in its expansion. By definition, the coefficient of $z^n$ in $S(Y,z)$ counts precisely
those admissible integers having exactly $n$ prime divisors in
$\Pi_g$. Thus
\[
 [z^n]S(Y,z)=\#\mathscr D_n(Y).
\]
We may therefore recover the required asymptotic by extracting the
coefficient of $z^n$ from the uniform Selberg--Delange estimate \eqref{eq:uniform-SD}.

Choose $\rho>0$ sufficiently small that
$\rho\delta_\Pi<\min\{\delta_\Omega,1\}$. The first inequality ensures
that $\Re(\delta_\Omega+z\delta_\Pi)>0$ throughout the closed disc
$|z|\leq\rho$, so Proposition~\ref{prop:uniform-SD} applies uniformly
on this disc. Put
\[
 B(z)=\frac{H(1,z)}
 {\Gamma(\delta_\Omega+z\delta_\Pi)}.
\]
This is holomorphic near the origin, and the Selberg--Delange estimate
takes the form
\[
 S(Y,z)
 =
 Y(\log Y)^{\delta_\Omega-1}
 e^{z\delta_\Pi\log\log Y}B(z)+E(Y,z),
\]
where, uniformly for $|z|=\rho$,
\[
 E(Y,z)
 \ll_\rho
 Y(\log Y)^{\delta_\Omega-2+\rho\delta_\Pi}.
\]
The additional factor $(\log Y)^{\rho\delta_\Pi}$ in this bound comes
from taking the maximum of
$|(\log Y)^{z\delta_\Pi}|$ on the circle $|z|=\rho$.

Let $L=\log\log Y$ and write
$B(z)=\sum_{j\geq0}b_jz^j$. Since
\[
 e^{z\delta_\Pi L}
 =
 \sum_{m\geq0}\frac{(\delta_\Pi L)^m}{m!}z^m,
\]
the coefficient of $z^n$ in
$B(z)e^{z\delta_\Pi L}$ is
\[
 \sum_{j=0}^n
 b_j\frac{(\delta_\Pi L)^{n-j}}{(n-j)!}.
\]
Here $n$ is fixed. The term with $j=0$ is the only term containing
$L^n$; all the remaining terms contain at most $L^{n-1}$. Consequently,
\[
 [z^n]\bigl(B(z)e^{z\delta_\Pi L}\bigr)
 =
 B(0)\frac{\delta_\Pi^n}{n!}L^n
 +O_{g,n}(L^{n-1}).
\]

It remains to verify that the error term remains negligible after
coefficient extraction. By Cauchy's coefficient formula,
\[
 |[z^n]E(Y,z)|
 \leq
 \rho^{-n}\max_{|z|=\rho}|E(Y,z)|
 \ll_{\rho,n}
 Y(\log Y)^{\delta_\Omega-2+\rho\delta_\Pi}.
\]
Relative to
$Y(\log Y)^{\delta_\Omega-1}(\log\log Y)^n$, this is bounded by a
constant multiple of
\[
 \frac{(\log Y)^{-1+\rho\delta_\Pi}}
 {(\log\log Y)^n},
\]
which tends to zero because $\rho\delta_\Pi<1$. We therefore obtain
\[
 \#\mathscr D_n(Y)
 \sim
 B(0)\frac{\delta_\Pi^n}{n!}
 Y(\log Y)^{\delta_\Omega-1}(\log\log Y)^n.
\]
Finally,
\[
 B(0)=\frac{H(1,0)}{\Gamma(\delta_\Omega)}>0,
\]
so this is precisely \eqref{eq:Dn-asymptotic}, with the constant given
in \eqref{eq:Cgn}.

To pass from admissible integers to modular forms, take
$Y=X/N_g$. The form $f_D$ has exact level $N_gD$, so two different
admissible integers give forms of different exact levels. Hence
\[
 \#\mathscr F_n(X)=\#\mathscr D_n(X/N_g).
\]
Since $N_g$ is fixed, replacing $Y$ by $X/N_g$ does not change the
powers of $\log X$ or $\log\log X$ in the asymptotic. This proves
\eqref{eq:Fn-asymptotic}. The assertion
\eqref{eq:quantitative-iwasawa-conclusion} now follows directly from
Proposition~\ref{prop:iwasawa-family}.
\end{proof}


We now allow the number of primes from $\Pi_g$ to vary.  Let
$\mathscr D(Y)$ denote the set of all admissible integers $D\leq Y$,
and consider the corresponding family
\[
 \mathscr F(Y)=\{f_D:D\in\mathscr D(Y)\}.
\]
Since $f_D$ has exact level $N_gD$, different values of $D$ give
different members of $\mathscr F(Y)$.  In particular,
$\#\mathscr F(Y)=\#\mathscr D(Y)$.

We regard $\mathscr F(Y)$ as a finite probability space by giving each
of its members equal weight.  Thus, if
$\mathscr A\subseteq\mathscr F(Y)$, then
\[
 \mathbb P_Y(\mathscr A)
 =
 \frac{\#\mathscr A}{\#\mathscr F(Y)}.
\]
If $X:\mathscr F(Y)\rightarrow\mathbb R$ is a real-valued
function, we define its expectation and variance by
\[
 \mathbb E_Y[X]
 =
 \frac{1}{\#\mathscr F(Y)}
 \sum_{f\in\mathscr F(Y)}X(f)
\]
and
\[
 \operatorname{Var}_Y(X)
 =
 \frac{1}{\#\mathscr F(Y)}
 \sum_{f\in\mathscr F(Y)}
 \bigl(X(f)-\mathbb E_Y[X]\bigr)^2.
\]
Equivalently,
$\operatorname{Var}_Y(X)=\mathbb E_Y[X^2]-\mathbb E_Y[X]^2$.

For $\natural\in\{\sharp,\flat\}$, define the signed increment by
\[
 \Delta^\natural(f_D)
 =
 \lambda^\natural(f_D)-\lambda^\natural(g).
\]
Proposition~\ref{prop:iwasawa-family} gives the pointwise identity
\begin{equation}\label{eq:increment-identity}
 \Delta^\sharp(f_D)
 =
 \Delta^\flat(f_D)
 =
 \omega_\Pi(D).
\end{equation}
Thus the variation of either signed $\lambda$-invariant is governed by
the elementary arithmetic function $\omega_\Pi(D)$.  The probabilistic
statements below are consequently distribution theorems for the signed
Iwasawa invariants in the family $\mathscr F(Y)$.

We first record the size of the underlying family.  Setting $z=1$ in
Proposition~\ref{prop:uniform-SD} removes the weight which records the
number of prime divisors in $\Pi_g$.  We obtain
\begin{equation}\label{eq:total-admissible}
 \#\mathscr F(Y)
 =
 \#\mathscr D(Y)
 \sim
 \frac{H(1,1)}
 {\Gamma(\delta_\Pi+\delta_\Omega)}
 Y(\log Y)^{\delta_\Pi+\delta_\Omega-1}.
\end{equation}

\begin{proposition}
\label{prop:mod-poisson}
There is a function
\[
 \Phi_g(z)
 =
 \frac{H(1,z)}{H(1,1)}
 \frac{\Gamma(\delta_\Pi+\delta_\Omega)}
 {\Gamma(\delta_\Omega+z\delta_\Pi)},
\]
which is holomorphic and nonzero in a neighborhood of $z=1$ and
satisfies $\Phi_g(1)=1$, such that, locally uniformly for $z$ near
$1$ and for either signed label $\natural$,
\begin{equation}\label{eq:mod-poisson}
 \frac{1}{\#\mathscr F(Y)}
 \sum_{f\in\mathscr F(Y)}z^{\Delta^\natural(f)}
 =
 \Phi_g(z)
 \exp\!\bigl(\delta_\Pi(z-1)\log\log Y\bigr)
 \left(
 1+O_g\left(\frac{1}{\log Y}\right)
 \right).
\end{equation}
The error term may be differentiated twice with respect to $z$ on a
neighborhood of $1$.
\end{proposition}

\begin{proof}
By \eqref{eq:increment-identity}, the numerator on the left-hand side
of \eqref{eq:mod-poisson} is
\[
 \sum_{D\in\mathscr D(Y)}z^{\omega_\Pi(D)}
 =
 S(Y,z).
\]
The denominator is $S(Y,1)$.

Choose a closed disc centered at $1$ on which
$\Re(\delta_\Omega+z\delta_\Pi)>0$.  Proposition~\ref{prop:uniform-SD}
applies uniformly on this disc and gives
\[
 S(Y,z)
 =
 Y(\log Y)^{\delta_\Omega+z\delta_\Pi-1}
 \left(
 \frac{H(1,z)}
 {\Gamma(\delta_\Omega+z\delta_\Pi)}
 +O_g\left(\frac{1}{\log Y}\right)
 \right).
\]
At $z=1$, the same formula gives \eqref{eq:total-admissible}.  Dividing
the two asymptotic formulae, and using the positivity of $H(1,1)$,
we find that
\[
 \frac{S(Y,z)}{S(Y,1)}
 =
 \frac{H(1,z)}{H(1,1)}
 \frac{\Gamma(\delta_\Pi+\delta_\Omega)}
 {\Gamma(\delta_\Omega+z\delta_\Pi)}
 (\log Y)^{\delta_\Pi(z-1)}
 \left(
 1+O_g\left(\frac{1}{\log Y}\right)
 \right).
\]
Since
\[
 (\log Y)^{\delta_\Pi(z-1)}
 =
 \exp\!\bigl(\delta_\Pi(z-1)\log\log Y\bigr),
\]
this is precisely \eqref{eq:mod-poisson}.

The function $\Phi_g$ is holomorphic near $1$.  It is also nonzero
after the neighborhood is made sufficiently small, since
$H(1,1)>0$ and the Gamma function is finite and nonzero at
$\delta_\Pi+\delta_\Omega>0$.  Finally, the Selberg--Delange estimate
is uniform and holomorphic in $z$.  Cauchy's integral formula therefore
allows its error term to be differentiated twice on a slightly
smaller disc.
\end{proof}

\begin{theorem}
\label{thm:levy-continuity}
Let $(Y_m)_{m\geq1}$ be a sequence tending to infinity, and let
\[
 X_m:\mathscr F(Y_m)\longrightarrow\mathbb R
\]
be a sequence of real-valued functions.  Suppose that, for every
$t\in\mathbb R$,
\[
 \frac{1}{\#\mathscr F(Y_m)}
 \sum_{f\in\mathscr F(Y_m)}e^{itX_m(f)}
 \longrightarrow e^{-t^2/2}.
\]
Then, for every $x\in\mathbb R$,
\[
 \frac{1}{\#\mathscr F(Y_m)}
 \#
 \{f\in\mathscr F(Y_m):X_m(f)\leq x\}
 \longrightarrow
 \frac{1}{\sqrt{2\pi}}
 \int_{-\infty}^{x}e^{-u^2/2}\,du.
\]
\end{theorem}

\begin{proof}
Let
\[
 \nu_m
 =
 \frac{1}{\#\mathscr F(Y_m)}
 \sum_{f\in\mathscr F(Y_m)}\delta_{X_m(f)}
\]
be the probability distribution of $X_m$.  Its characteristic function
is the expression on the left-hand side of the hypothesis.  Since this
converges pointwise to $e^{-t^2/2}$, L\'evy's continuity theorem
\cite[Theorem~26.3, p.~349]{Billingsley1995} shows that $\nu_m$
converges weakly to the standard normal distribution and the asserted
formula follows.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm:erdos-kac}]
We set $L_Y=\delta_\Pi\log\log Y$ and define \[Z_Y
 =
 \frac{\Delta^\natural-L_Y}{\sqrt{L_Y}}.
\]
Since $\delta_\Pi>0$, we have $L_Y\to\infty$.  We shall prove the
limiting distribution by showing that the characteristic functions of
the variables $Z_Y$ converge to that of a standard Gaussian.

Fix $t\in\mathbb R$ and set
\[
 z_Y=\exp\!\left(\frac{it}{\sqrt{L_Y}}\right).
\]
Then $z_Y\to1$, and hence Proposition~\ref{prop:mod-poisson} applies
for all sufficiently large $Y$. We find that
$z_Y^{\Delta^\natural}
=\exp(it\Delta^\natural/\sqrt{L_Y})$. We arrive at 
\[
 \mathbb E_Y[e^{itZ_Y}]
 =
 \Phi_g(z_Y)
 \exp\!\left(
 L_Y\left(
 z_Y-1-\frac{it}{\sqrt{L_Y}}
 \right)
 \right)
 (1+o(1)).
\]
The first factor tends to one because $\Phi_g$ is continuous at $1$
and $\Phi_g(1)=1$.  On the other hand, Taylor's formula gives
\[
 z_Y
 =
 1+\frac{it}{\sqrt{L_Y}}
 -\frac{t^2}{2L_Y}
 +O_t(L_Y^{-3/2}).
\]
It follows that
\[
 L_Y\left(
 z_Y-1-\frac{it}{\sqrt{L_Y}}
 \right)
 =
 -\frac{t^2}{2}+O_t(L_Y^{-1/2}).
\]
Consequently, we find that
\[
 \lim_{Y\rightarrow \infty} \mathbb E_Y[e^{itZ_Y}]
=
 e^{-t^2/2}.
\]
The function $e^{-t^2/2}$ is the characteristic function of the
standard normal distribution. Theorem \ref{thm:levy-continuity} now gives
\eqref{eq:EK-limit}.

It remains to prove \eqref{eq:EK-mean} and \eqref{eq:EK-variance}. We do this by computing the first two moments. Let
\[
 G_Y(z)=\mathbb E_Y[z^{\Delta^\natural}]
\]
be the probability generating function of the signed increment.
Taking logarithms in \eqref{eq:mod-poisson} gives, locally uniformly
near $z=1$,
\[
 \log G_Y(z)
 =
 L_Y(z-1)+\log\Phi_g(z)
 +O_g\left(\frac{1}{\log Y}\right).
\]
The differentiated assertion in Proposition~\ref{prop:mod-poisson}
allows us to differentiate this identity twice at $z=1$.

Since $G_Y(1)=1$, we have
\[
 \mathbb E_Y[\Delta^\natural]
 =
 G_Y'(1)
 =
 (\log G_Y)'(1)
 =
 L_Y+(\log\Phi_g)'(1)
 +O_g\left(\frac{1}{\log Y}\right).
\]
This proves \eqref{eq:EK-mean}.  For the variance, the usual identities
for a probability generating function give
\[
 \operatorname{Var}_Y(\Delta^\natural)
 =
 G_Y''(1)+G_Y'(1)-G_Y'(1)^2
 =
 (\log G_Y)''(1)+(\log G_Y)'(1).
\]
Differentiating the logarithmic asymptotic once more, we obtain
\[
 \operatorname{Var}_Y(\Delta^\natural)
 =
 L_Y
 +(\log\Phi_g)'(1)
 +(\log\Phi_g)''(1)
 +O_g\left(\frac{1}{\log Y}\right),
\]
which proves \eqref{eq:EK-variance}.

Finally, \eqref{eq:increment-identity} shows that the two signed
increments agree for every $D$, rather than merely having the same
limiting distribution.  Subtracting the two identities gives
\[
 \lambda^\sharp(f_D)-\lambda^\flat(f_D)
 =
 \lambda^\sharp(g)-\lambda^\flat(g);
\]
see \eqref{eq:signed-difference-rigid-family}. This completes the proof.
\end{proof}

\printbibliography

\end{document}