Note that this definition and the proof that $W_k(\bpsi,1)$ specializes under a character $\psi$ of $A$ to $W_k(\psi_\fP, 1)$ uses the result of Lemma <ref>.  All the normalized Fourier coefficients of $W_k(\bpsi,1)$ lie in $A$, except for possibly the constant terms, which lie in $\Frac(A)$.

By <cit.>*Proposition 8.7, the constant term of $W_1(\psi_\fP, 1)$ at a cusp $[\cA] \in C_\infty(\fP, \fn) \setminus C_\infty(\fn)$ vanishes, and at a cusp $[\cA] \in C_\infty(\fn)$ equals

\[
\sgn(\N a) \psi^{-1}(a \fb_\cA^{-1})\frac{L_{\Sigma_\fp, \Sigma'}(\psi, 0)}{2^n}.
\]

The element of $A$ interpolating these specializations is the element $C^L_1(\cA)$ defined in (<ref>).  Since $C_1^L(\cA) \in A$, we obtain $W_1(\bpsi, 1) \in M_1(\fn,A, \bpsi)$.

We have an analogous construction of group ring forms over $G$.  We write
\[ \overline{A} = \overline{R}_\chi = \cO[G_p]_\chi. \]

\[ \overline{\bpsi} \colon \fg \longrightarrow \overline{A}^* \]

denote the reduction of $\bpsi$.  Note that $\overline{\bpsi}$ factors through $\fg \longrightarrow G$.
As in (<ref>), we let

\[ \fm_0 = \cond(H/F), \qquad \fm = \lcm(\fm_0, \ \prod_{\mathclap{\fq \in \Sigma \cup \Sigma'}} \fq \, ). \]

Note that here, $\fp$ does not divide the level $\fm$.  Since we have $\fg' = G'$, however, the levels $\fn$ and $\fm$ agree away from $p$, i.e. if $\fP_0$ denotes the $p$-part of $\fm$, then $\fn/\fP = \fm/\fP_0$.
(This fact implies that Propositions <ref> and <ref> below, which a priori would apply to the cusps in $C_\infty(\fP_0, \fm)$, also apply to the cusps in $C_\infty(\fP, \fn)$.)

We use  <cit.>*Proposition 8.14, eqn. (102) again to define, for odd $k \ge 1$,  a group ring valued modular form

\begin{equation} \label{e:fraca}
W_k(\overline{\bpsi}, 1) \in  M_k(\fm, \Frac(\overline{A}), \overline{\bpsi}) \subset M_k(\fn, \Frac(\overline{A}), \overline{\bpsi}).
\end{equation}

Again, the $q$-expansion coefficients of $W_k(\overline{\bpsi}, 1)$ other than possibly the constant terms lie in $\overline{A}$.
The specialization of $W_k(\overline{\bpsi}, 1)$ at a character $\psi$ of $\overline{A}$ is ${W}_k(\psi_{\fP_0}, 1)$, defined as in (<ref>) with $\fP$ replaced by $\fP_0$.

We now discuss the constant terms of $W_k(\overline{\bpsi}, 1)$ for odd $k \ge 1$.
The following result  follows directly from  <cit.>*Proposition 8.7.

If $\Sigma \setminus S_\infty$ is nonempty, the normalized constant term of $W_1(\overline{\bpsi},1)$ at a cusp $\cA$ such that $[\cA] \in C_\infty(\fP, \fn)$ is

\[
C^H_1(\cA) =
\begin{cases}
\sgn(\N a) {\overline{\bpsi}}^{-1}(a \fb_{\cA}^{-1})\Theta_H^\#/2^n & \text{if } [\cA] \in C_\infty(\fn) \\
0 &  \text{if } [\cA] \in C_\infty(\fP, \fn) \setminus C_\infty(\fn).
\end{cases}
\]

In particular, if $\Sigma \setminus S_\infty$ is nonempty we have $W_1(\overline{\bpsi},1) \in M_1(\fn, \overline{A}, \overline{\bpsi})$.

If $\Sigma = S_\infty$ (in which case $\fP_0 = 1$) the situation is more complicated for $W_1(\overline{\bpsi},1)$.
Furthermore in this case we  require the constant terms of the forms $W_k(\overline{\bpsi},1)$ for odd $k\ge 3$.

For each character $\psi$ of $\overline{A}$, let $\fb_0 = \cond(\psi)$, $\fb = \lcm(\fb_0, \fP_0)$, and $\fl = \fm/\fb$.
If $[\cA]$ is a cusp, we follow <cit.>*Definition 8.3 and define two sets of primes:

\[ J_\fl = \{ \fq \mid \fl \colon [\cA] \in C_0(\fq, \fn) \}, \qquad  J_\fl^c =  \{ \fq \mid \fl \colon [\cA] \in C_\infty(\fq, \fn) \}. \]

Define $C_k'(\cA)$ to be the unique element of
such that for each character $\psi$ of $\overline{A}$, we have

\begin{equation} \label{e:cpdef}
\psi(C_k'(\cA)) =  \frac{\tau(\psi)}{\N\fb_0^k} \psi(\fc_{\cA}) \sgn(\N(-c)) \frac{L_{S_{\infty}, \emptyset}(\psi^{-1}, 1-k)}{2^n}
\prod_{\fq \in J_{\fl}^c}(1 - \N \fq^k) \prod_{\fq \in J_{\fl}}(1- \psi(\fq))
\end{equation}

if $[\cA] \in C_0(\fb_0, \fn)$, and
\[  \psi(C_k'(\cA)) = 0 \]
if $[\cA] \in C_\infty(\fP, \fn) \setminus C_0(\fb_0, \fn)$.  Here $\tau(\psi)$ denotes the Gauss sum defined in <cit.>*Definition 4.1.  The following proposition follows directly from <cit.>*Proposition 8.7.

Suppose that $\Sigma = S_\infty$. The normalized constant term of $W_k(\overline{\bpsi}, 1)$ at a cusp $[\cA] \in C_\infty(\fP, \fn)$ is equal to $C^H_1(\cA) + C_1'(\cA)$ if $k = 1$, and is equal to $C_k'(\cA)$ if $k > 1$.

§.§ Construction of a cusp form

In this section we construct the cusp form required in our proof. The construction will need to be split into several cases.
Recall that $\fP$ is the $p$-part of $\fn$.  The ideal $\fP$ is divisible precisely by the primes in $\Sigma_\fp =\Sigma \cup \{ \fp \}$.
Define $\fP'$ to be the product of all other primes above $p$, i.e.

\[
\fP' = \prod_{\mathclap{\fq \mid p, \ \fq \nmid \fP}} \fq.
\]

* Case 1: the set $\Sigma \setminus S_\infty$ is non-empty, i.e. $\fp$ is not the only prime dividing $\fP$.

In Case 2, we have $\Sigma = S_\infty$.
Note that the eigenvalues of the form $W_1(\bpsi, 1)$ for the operator $U_\fq$, $\fq \mid \fP'$, are $\bpsi(\fq)$ and $1$.  If $\chi(\fq) \neq 1$, then these are not congruent modulo the maximal ideal of $A$.  To ensure that the Hecke algebras we work with are local, it will be convenient to project to the eigenspace with eigenvalue $1$.

\[
\fP'_0  =\prod_{\mathclap{\substack{\fq \mid \fP' \\ \chi(\fq) \neq 1}}} \fq, \qquad
\fP'_1  =\prod_{\mathclap{\substack{\fq \mid \fP' \\ \chi(\fq) = 1}}} \fq.
\]

We subdivide Case 2 into three cases:

* Case 2(a): $\fP'_0 \neq 1$.

* Case 2(b): $\fP'_0 = 1, \fP'_1 \neq 1$.

* Case 2(c): $\fP'_0 = \fP'_1 = 1$, i.e. $\fp$ is the only prime above $p$ in $F$.

We consider the module
\[ \Lambda = A \oplus \overline{A},\]

endowed with the canonical diagonal $A$-action.
We view $A = A \oplus 0$ as a submodule of $\Lambda$ and denote the vector $(0,1) \in \Lambda$ by $\lambda$, so elements of $\Lambda$ will be written $a + \lambda b$ with $a \in A, b \in \overline{A}.$

As in the previous section, we have the canonical characters
\[
\bpsi\colon \fg \longrightarrow A^*, \qquad \overline{\bpsi}  \colon \fg \longrightarrow G \longrightarrow \overline{A}^*.
\]

We denote the image of $I$ in $A$ (i.e. the kernel of the canonical projection $A \longrightarrow \overline{A}$) by $I_A$.
Recall the following result of Silliman <cit.>*Theorem 10.7, a  generalization of a result of Hida.
Fix a positive integer $m$.  For positive integers $k \equiv 0 \pmod{(p-1)p^{N}}$ with $N$ sufficiently large depending on $m$,
there is a modular form $V_{k} \in M_k(1, \Z_p, 1)$ such that
$c(\fm, V_{k})  \equiv 0 \pmod{p^m}$ for all integral ideals $\fm$, and such that the normalized constant term  $c_{\cA}(0,V_k)$
for each cusp $[\cA]$ is congruent to $1 \pmod{p^m}$.

To apply this result, we hereafter assume that $m$ is fixed and $k \equiv 1 \pmod{(p-1)p^{N})}$ with $N$ sufficiently large that the conclusion of Theorem <ref> holds.  In  subsequent arguments we will make $N$ larger still if necessary to obtain other properties of our modular forms.

In Case 1, we define

\[
f = (W_1( \bpsi,1) - \lambda W_1(\overline{\bpsi},1)) V_{k-1} \in M_k(\fn, \Lambda, \bpsi).
\]

Note here that $W_1( \bpsi,1) \in M_k(\fn, A, \bpsi)$ and $W_1(\overline{\bpsi},1) \in  M_k(\fn, \overline{A}, \overline{\bpsi})$ by
Propositions <ref> and <ref>, respectively.

In Case 2 we
recall the non-zerodivisor

\begin{equation} \label{e:xdef}
x = x(k) = \frac{\Theta^{H/F}_{S_{\infty}, \emptyset}(1-k)}{\Theta^{H/F}_{S_{\infty}, \emptyset}} \in \Frac(\overline{A}) \end{equation}

considered in <cit.>*Theorem 8.16.
The elements $\Theta^{H/F}_{S_{\infty}, \emptyset}(1-k)$ and $\Theta^{H/F}_{S_{\infty}, \emptyset}$ belong to $\Frac(\overline{A})$ and interpolate the nonzero algebraic numbers
$L_{S_\infty, \emptyset}(\psi^{-1}, 1-k)$ and $L_{S_{\infty}, \emptyset}(\psi^{-1}, 0)$, respectively, as $\psi$ ranges over the characters of $A$. For $k>1$ and $k \equiv 1 \pmod{(p-1)p^N)}$ with $N$ sufficiently large, the ratio $x$ belongs to $\overline{A}$ by loc. cit.  We hereafter assume that $N$ is chosen so that this holds.
\begin{equation} \label{e:tx}
\tilde{x} = \text{ any non-zerodivisor lift of }x \text{ to }A.
\end{equation}

We define in Case 2:

\begin{equation} \label{e:fdef2ab}
f = \tilde{x}  \left(W_1( \bpsi,1) - \lambda W_1(\overline{\bpsi},1)\right) V_{k-1} + \lambda W_k(\overline{\bpsi}, 1).
\end{equation}

A priori, in Case 2 the form $f$ lies in $M_k(\fn, \Lambda \otimes_{\Z_p} \Q_p, \bpsi)$ since we have only shown that the constant terms of the forms $W_1(\overline{\bpsi}, 1)$ and $W_k(\overline{\bpsi},1)$ lie in $\Frac(\overline{A}) = \overline{A} \otimes_{\Z_p} \Q_p$.  However, the proposition below shows that in fact $f \in M_k(\fn, \Lambda, \bpsi)$.
In order to state results in all cases simultaneously, we set $\tilde{x} = 1$ in Case 1 for the remainder of the paper.

With $N$ and $k$ as above, we have in all cases $f \in M_k(\fn, \Lambda, \bpsi)$.  Furthermore the form $f$ has constant terms at cusps $[\cA] \in C_\infty(\fP, \fn)$
lying in the submodule $ J \subset \Lambda$, where

\begin{equation}  \label{e:jdef}
J =   \tilde{x}J_0, \qquad J_0 = \left(I_A^2, \Theta_L^\# - \Theta_H^\# \lambda, \  p^m\Lambda\right).
\end{equation}

We give the proof in Case 2, as Case 1 is similar and in fact easier.

We first note that the non-constant term $q$-expansion coefficients of $f$ lie in $\Lambda$.
Indeed, all the forms appearing in the definition of $f$ have non-constant term $q$-expansion coefficients lying in $\Lambda$,
so any failure of integrality of non-constant terms can  arise only from multiplying the constant terms of $W_1(1, \overline{\bpsi})$ by the non-constant terms of $V_{k-1}$. For $k \equiv 1 \pmod{(p-1)p^{N}}$ and $N$ sufficiently large, the non-constant terms of $V_{k-1}$ are divisible by any desired power of $p$, and this product will be integral.

We will prove the integrality of the constant terms of $f$ and the statement about constant terms simultaneously.
By Propositions <ref> and <ref>, the constant term of $f$ at a cusp $[\cA] \in C_\infty(\fP, \fn)$ is equal to

\begin{align*}
& \left( C^L_1(\cA) - \lambda C^H_1(\cA)- \lambda C_1'(\cA)\right) \tilde{x} y + \lambda  C_k'(\cA)  \\
\equiv \ &  \left( \tilde{x}C_1^L(\cA) - \tilde{x} \lambda C_1^H(\cA)\right) -
\lambda \left(C_1'(\cA) \tilde{x}y - C_k'(\cA)\right),
\end{align*}

where $y = c_{\cA}(0, V_{k-1}) \equiv 1 \pmod{p^m}$ and the congruence is modulo $\tilde{x} p^m \Lambda \subset J$. For the first term, we have

\[
\left( \tilde{x} C_1^L(\cA) - \tilde{x} \lambda C_1^H(\cA) \right) =  \frac{\sgn(\N a) \bpsi^{-1}(a \fb_{\cA}^{-1})}{2^n}\left(\tilde{x}\Theta_L^\# - \tilde{x}\Theta_H^\# \lambda \right) \in J. \]

It remains to show that $x y C_1'(\cA) -   C_k'(\cA)$ lies in $\overline{A}$ and  is divisible by $x p^m$. Now
$C_1'(\cA)$ is some fixed element of $\Frac(\overline{A})$ and $x \in \overline{A}$, so for $y$ sufficiently close to 1 $p$-adically, $(y - 1) C_1'(\cA)$ will lie in $p^m \overline{A}$.  Therefore it suffices to show that $x C_1'(\cA) -  C_k'(\cA)$ lies in $\overline{A}$ and  is divisible by $x p^m$, i.e. that $C_1'(\cA) -  x^{-1} C_k'(\cA)$ lies in $p^m\overline{A}$.

For this, we note that by the definition (<ref>), multiplying by the factor $x^{-1}$ exactly replaces the $L$-value in the definition (<ref>) of $C_k'(\cA)$ with that appearing in $C_1'(\cA)$, hence

\[
C_1'(\cA) -  x^{-1}\Theta_H C_k'(\cA) = C_1'(\cA) \left( 1 - \N\fb_0^{1-k} \prod_{\fq \in J_{\fl}^c}\frac{1-\N\fl_i^k}{1 - \N\fl_i} \right).
\]

For  $k \equiv 1 \pmod{(p-1)p^{N}}$ and $N$ sufficiently large, the term in parentheses can be made divisible by arbitrarily large powers of $p$. (Note we are in Case 2, whence $\fb_0$ is prime to $p$.)
The result follows.

There exists a $p$-ordinary cusp form $g \in S_k(\fn\fP', \Lambda, \bpsi)^{p\text{-ord}}$ such that in Cases 1 and 2(a) we have

\[
g \equiv W_1( \bpsi_{\fP_0'},1) - \lambda W_1(\overline{\bpsi}_{\fP_0'},1) \pmod{J_0},
\]

while in Cases 2(b) and 2(c) we have

\[
g \equiv \tilde{x} \big( W_1(\bpsi, 1) -\lambda W_1( \overline{\bpsi}, 1) \big) + \lambda W_k(1, \overline{\bpsi}_p) \pmod{J}.
\]

These congruences are understood to mean that the $q$-expansion coefficients $c(\fa, g)$ for nonzero ideals $\fa \subset \cO_F$ are congruent modulo $J_0$ or $J$ to the coefficients of the expressions on the right.

Silliman's result <cit.>*Theorem 10.8
implies that there is an element $f' \in M_k(\fn, J, \bpsi)$ whose constant terms at $C_\infty(\fP, \fn)$ agree with those of $f$.  Therefore $f - f'$ has constant terms that vanish on $C_\infty(\fP, \fn)$.
Let $e_\fP^{\ord}, e_p^{\ord}$ denote the ordinary operators of Hida (see <cit.>*7.2.9).
<cit.>*Theorem 5.1, the form $g_0 = e_\fP^{\ord}(f - f')$ is cuspidal,
\[ g = e_p^{\ord}(f -f') = e_{\fP'}^{\ord} g_0\]
is cuspidal as well. Now the ordinary operator $e_p^{\ord}$ fixes the forms $W_1(\bpsi,1)$ and $W_1(\overline{\bpsi}, 1)$, whereas it sends $W_k(\overline{\bpsi}, 1)$ to its ordinary $p$-stabilization $W_k(\overline{\bpsi}, 1_p)$.  Furthermore, $V_{k-1} \equiv 1 \pmod{p^m}$ and $p^m \in J_0$. The result follows in Cases 2(b) and 2(c).

To complete the proof in Cases 1 and 2(a), we apply the operator $\prod_{\fq \mid \fP'_0} (U_\fq - \bpsi(\fq))$.  This operator sends $W_1(\bpsi, 1)$ and $W_1(\overline{\bpsi}, 1)$ to $W_1(\bpsi_{\fP'_0}, 1)$ and $W_1(\overline{\bpsi}_{\fP'_0}, 1)$, respectively, while it annihilates $W_k(\overline{\bpsi}, 1_p).$  This gives the result in Case 1.  In Case 2(a), we obtain a cusp form $g$ satisfying

\[ g \equiv  \tilde{x}  \left(W_1( \bpsi_{\fP'_0},1) - \lambda W_1(\overline{\bpsi}_{\fP_0'},1)\right)
\pmod{\tilde{x}J_0}. \]

This congruence in particular implies that the Fourier coefficients of $g$ are divisible by $\tilde{x}$.  Since $\tilde{x}$ is a non-zerodivisor, we may divide by it and obtain a cusp form satsifying the same congruence as in Case 1.

§.§ Homomorphism from the Hecke algebra

We now define certain Hecke algebras acting on $S_k(\fn\fP', A, \bpsi)$.
We define $\T$ to be the $A$-subalgebra of $\End_A(S_k(\fn\fP', A, \bpsi))$ generated over $A$ by the following operators:

* $T_\fl$ for $\fl \nmid \fn\fP'$,

* $S(\fa)$ for $(\fa, \fn\fP') = 1$,

* $U_\fq$ for $\fq \in \Sigma \setminus S_\infty$ (i.e. for prime $\fq \mid \fP$, $\fq \neq \fp$),

* $(U_\fp - 1)^2$, and

* $(U_\fp - 1)t$ for $t \in I$.

Let $\tilde{\T}= \T[U_\fp]$.
Finally let $\T^\dagger = \tilde{\T}[U_\fq, \fq \mid \fP']$.

Due to the presence of the involution $\#$ in Proposition <ref>, we  consider an “involuted" version of the ring $A_\sL$.  Define

\[
A_{\sL}^\# = A[\sL]/(\sL (\Theta_H)^\# - (\Theta_L)^\#, \sL I_A, \sL^2, I_A^2).
\]

At this point, we must momentarily abandon case 2(c), when $\fp$ is the only prime of $F$ above $p$.  In this case, the last bullet point of the theorem below—which in some sense is the most important—does not hold.  In case 2(c), $\fP' = 1$, and this last bullet point then reads $\Ann_{A_\sL^\#}(W) \subset (p^m)$, which does not hold in this case.  The algebra $W$ constructed in Theorem <ref> using the form $g$ is not large enough for our purposes.  In <ref> we will therefore provide a different approach to proving the congruence (<ref>) that only applies when there is exactly one prime of $F$ above $p$.
For the remainder of <ref> we assume that there is at least one prime of $F$ above $p$ other than $\fp$.

Suppose we are in Cases 1, 2(a), or 2(b), i.e. there exists a prime of $F$ above $p$ other than $\fp$.  There exists an $A_{\sL}^\#$-algebra $W$
and a surjective $A$-algebra homomorphism
\[ \varphi \colon \T^\dagger \longrightarrow W \]
such that:

* $T_\fl \mapsto \N\fl^{k-1} + \bpsi(\fl)$ for prime $\fl \nmid \fn\fP'$.

* $S(\fa) \mapsto \bpsi(\fa)$ for $(\fa, \fn\fP') = 1$.

* $U_\fq \mapsto 1$ for prime $\fq \in \Sigma \setminus S_\infty$.

* $U_\fp \mapsto 1 - \sL$.

* $U_\fq \mapsto 1$ for prime $\fq \mid \fP_0'$.

For prime $\fq \mid \fP'$, write $\epsilon_\fq = \varphi(U_\fq - \bpsi(\fq)) \in W$.  Then:

* $\epsilon_\fq(\epsilon_\fq -1 + \bpsi(\fq)) = 0$.

* $\Ann_{A_{\sL}^\#}\Big(\prod_{\fq \mid \fP'} \epsilon_\fq\Big) \subset (p^m)$.

We recall from (<ref>) the definition

\[
J_0 = \left(I_A^2, \Theta_L^\# - \Theta_H^\# \lambda, \  p^m\Lambda\right).
\]

To streamline the notation for all cases, write

\[ X = \begin{cases} 1 & \text{Cases 1/2(a)} \\
\tilde{x} & \text{Case 2(b),}
\end{cases} \qquad K = XJ_0, \]

and recall $\fP_0' = 1$ in Case 2(b).

\[
\cC = \prod_{\fa \subset \cO_F} \Lambda/{K},
\]
with the product indexed by the nonzero ideals $\fa \subset \cO_F$. There is an $A$-module map

\[
c: S_k(\fn\fP', \Lambda, \bpsi) \longrightarrow \cC
\]

that associates to each cusp form $h$ its collection of normalized $q$-expansion coefficients $c(\fa, h)$ reduced modulo ${K}$. Note that if we let the operators $T_\fl$, $U_\fl$ act by the usual formulae on Fourier coefficients (see <cit.>*eqn. (97)) and we let $S(\fa)$ act by $\bpsi(\fa)$, then the map $c$ is equivariant for these operators.

Let $\cF$ denote the image of the $\T^{\dagger}$-span of the cusp form $g$ defined in Corollary <ref> under the map $c$. This is an $A$-module of finite type. Define $W$ be the image of the canonical $A$-algebra homomorphism

\[
\T^{\dagger} \longrightarrow \End_{A}(\cF).
\]

This construction yields a canonical surjective $A$-algebra map

\[
\varphi: \T^{\dagger} \longrightarrow W
\]

that sends a Hecke operator to its action on the Hecke span of $g$ under the map $c$.

We will show in a moment that $W$ has the structure of an $A_{\sL}^\#$-algebra.  First we calculate the action of Hecke operators on $g$ using the congruence of Corollary <ref>:

\begin{equation} \label{e:gcong}
g \equiv  \begin{cases}
W_1( \bpsi_{\fP_0'},1) - \lambda W_1(\overline{\bpsi}_{\fP_0'},1) & \text{Cases 1/2(a)} \\
X \left(W_1( \bpsi,1) - \lambda W_1(\overline{\bpsi},1)\right) + \lambda W_k(\overline{\bpsi}, 1_p)
& \text{Case 2(b)}
\end{cases} \pmod{K}.
\end{equation}

The fact that $\varphi(S(\fa)) = \bpsi(\fa)$ is clear since all our forms have nebentypus $\bpsi$.
The operator $T_{\fl}$ acts by multiplication by $1 + \bpsi(\fl)$ on $W_1(\bpsi, 1)$ and $W_1(\overline{\bpsi}, 1)$, and it acts by multiplication by $\N\fl^{k-1} + \bpsi(\fl)$ on $W_{k-1}(\bpsi, 1_p)$.  Since $\N\fl^{k-1} \equiv 1 \pmod{p^m}$ and $Xp^m \in K$, we have

\[ T_\fl(g) \equiv (\N\fl^{k-1} + \bpsi(\fl)) g \pmod{K}. \]

Similarly $\varphi(U_\fq) = 1$ for prime $\fq \in \Sigma \setminus S_\infty$, since all the forms $W_1(\bpsi, 1)$, $W_1(\overline{\bpsi}, 1), W_k(\overline{\bpsi}, 1_p)$ have $U_\fq$-eigenvalue 1.

The most interesting action is that of $U_\fp$.  This operator fixes $W_1(\bpsi, 1)$, since $\fp \mid \fP$.  It also acts as $\overline{\bpsi}(\fp) = 1$ on $W_k(\overline{\bpsi}, 1_p)$ because of the $p$-stabilization.  Yet $U_\fp$ does not act as a scalar on $W_1(\overline{\bpsi}, 1)$, as it is not stabilized.  Instead $U_\fp - 1$ sends $W_1(\overline{\bpsi}, 1)$ to its $\fp$-stabilization $W_1(\overline{\bpsi}, 1_\fp) = W_1(\overline{\bpsi}_\fp, 1)$, where the equality follows since $\overline{\bpsi}(\fp) = 1$ and we are in weight 1.  The conclusion of these considerations is that

\begin{equation} \label{e:cc}
(U_\fp - 1)g \equiv
- \lambda X W_1(\overline{\bpsi}_{\fp\fP_0'},1)
\pmod{K}. \end{equation}

We can now give $W$ the structure of an $A_\sL^\#$-algebra by letting $\sL$ act by $\varphi(1 - U_\fp)$.  To prove that this is well-defined, we must show that the relations in $A_{\sL}^\#$ are satisfied in $W$, namely $\sL(\Theta^H)^\#- (\Theta^L)^\# = \sL I_A = \sL^2 = I_A^2 = 0$.  The most interesting of these is the first.  From (<ref>), we find the following congruences mod

\begin{align*}
(\Theta^H)^\# (1 - U_\fp)g & \equiv  (\Theta^H)^\# X \lambda W_1(\overline{\bpsi}_{\fp\fP_0'},1)   \\
& \equiv  (\Theta^L)^\# XW_1(\overline{\bpsi}_{\fp  \fP_0'},1)  \\
& \equiv  (\Theta^L)^\# g.
\end{align*}

The second congruence follows from the definition of $K$ and the third from (<ref>), in view of the fact that $(\Theta_L)^\# \in I_A$
annihilates $\lambda$ while $W_1(\bpsi_{\fP_0'}, 1) \equiv W_1(\overline{\bpsi}_{\fp\fP_0'}, 1) \pmod{I_A}$ and
$X (\Theta^L)^\# I_A \in X I_A^2 \subset K$.  This proves that the first relation holds, while the others are similar but easier.

Next we study the action of $U_\fq - \bpsi(\fq)$ for $\fq \mid \fP'$.  This operator sends $W_1(\bpsi, 1)$ to $W_1(\bpsi_\fq, 1)$, sends $W_1(\overline{\bpsi}, 1)$ to $W_1(\overline{\bpsi}_\fq, 1)$, and annihilates
$W_k(\bpsi, 1_p)$.  For $\fq \mid \fP_1'$, we therefore have

\begin{equation}
\label{e:uq}
(U_\fq - \bpsi(\fq))g \equiv X\left(W_1(\bpsi_{\fq \fP_0'}, 1) - \lambda W_1(\overline{\bpsi}_{\fq \fP_0'}, 1)\right)
\pmod{K}. \end{equation}

Note that $U_\fq$ acts as 1 on the form on the right.  This yields the equation

\begin{equation} \label{e:eqeq}
\epsilon_\fq(\epsilon_\fq -1 + \bpsi(\fq)) = 0,
\end{equation}

where $\epsilon_\fq = \varphi(U_\fq - \bpsi(\fq))$.  Meanwhile if $\fq \mid \fP_0'$, then in particular we are in case 1/2(a) and $U_\fq$ acts as 1 on $g \pmod{K}$ by (<ref>).  For such $\fq$ we have

\begin{equation} \label{e:eqeq2}
\epsilon_\fq = 1 - \bpsi(\fq)
\end{equation}
and (<ref>) holds trivially.

To prove the last item note that by (<ref>) and (<ref>) we have

\begin{equation} \label{e:alleqs}
\prod_{\fq \mid \fP'} (U_{\fq} - \bpsi(\fq)) g \equiv  X \left( W_1(\bpsi_{\fP'}, 1) - \lambda W_1(\overline{\bpsi}_{\fP'},1) \right) \prod_{\fq \mid \fP_0'} (1 - \bpsi(\fq)) \pmod{K}.
\end{equation}

In case 2(c), this congruence does not hold as there is a contribution from the term $W_k(\overline{\bpsi}, 1_p)$.  This term does not appear in (<ref>) in cases 1/2(a), and is annihilated by the single application of an $\epsilon_\fq$ for $\fq \mid \fP'_1$ in case 2(b).  This is why case 2(c) must be removed from the present analysis.

Returning to (<ref>), suppose this form is annihilated by an element $a + b \sL \in A_{\sL}^{\#}$, with $a, b \in A$. By definition, such an element acts by $a + b (1- U_{\fp})$ and hence, noting that $\prod_{\fq \mid \fP_0'} (1 - \bpsi(\fq))$ is a unit, we obtain

\[
a X \left( W_1(\bpsi_{\fP'},1) - \lambda W(\overline{\bpsi}_{\fP'}, 1)\right) + b X \lambda W_1(\overline{\bpsi}_{\fP'}, 1_\fp) \equiv 0 \pmod{K}.
\]

Analyzing the $q$-expansion coefficients $c(1, *)$ and $c(\fp, *)$ of this congruence, respectively, we obtain

\begin{align*}
X(a - \lambda a + \lambda b)  & \equiv 0 \pmod{K} \\
X(a - 2 \lambda a + \lambda b) & \equiv 0 \pmod{K}.
\end{align*}

From these, we deduce $X(a+\lambda b) \in K$, and since $X$ is a non-zerodivisor it follows that $a + \lambda b \in J_0 = ( \Theta_H^{\#} \lambda - \Theta_L^{\#}, I_A^2 \Lambda, p^m \Lambda)$. It follows that in $A_{\sL}^{\#}$, we have

\[
a+ b \sL \in (\Theta_H^{\#} \sL - \Theta_L^{\#}, I_A^2, p^m) = (p^m)
\]

as desired.

§.§ Galois Representation

In this section we recall the formalism of <cit.>*9.1–9.2 regarding the Galois representation associated to the Hecke algebra $\T^\dagger$.
As explained in <cit.>*8.5 the Hecke algebra ${\T}^{\dagger}$ is reduced.
The kernel of $\varphi$ is contained in a unique maximal ideal $\fm \subset \T^\dagger$.  This maximal ideal is generated by the maximal ideal $(\fm_{\cO}, I)$ of $A$ together with the elements $T_\fl - (1+ \chi(\fl))$ for $\fl \nmid \fn\fP'$, $S(\fa) - \chi(\fa)$ for $(\fa, \fn\fP') = 1$, $U_\fq - 1$ for all $\fq \mid p$.

Let $\T_{\fm}^\dagger$ denote the completion of $\T^\dagger$ with respect to $\fm$ (and similarly $\T_{\fm}$, $\tilde{\T}_\fm$  the completions of $\T$ and $\tilde{\T}$ with respect to their maximal ideals $\fm \cap \T$, $\fm \cap \tilde{\T}$, respectively).  Set

\[
K = \Frac(\T_{\fm}^{\dagger}).
\]

As in <cit.>*9.2, there is a Galois representation

\[
\rho: G_F \longrightarrow \GL_2(K)
\]

such that

(1) $\rho$ is unramified outside $\fn p$.

(2) For all primes $\fl \nmid \fn p$, the characteristic polynomial of $\rho(\Frob_{\fl})$ is given by

\begin{equation} \label{e:chartl}
{\rm char}(\rho(\Frob_{\fl}))(x) = x^2 - T_{\fl} x + \bpsi(\fl) \N\fl^{k-1}.
\end{equation}

(3) For  $\fq \mid p$, let $G_{F,\fq} \subset G_F$ denote a decomposition group at $\fq$.  We have

\begin{equation} \label{e:repord}
\rho|_{G_{F,\fq}} \sim \begin{pmatrix} \bpsi \varepsilon_{\cyc}^{k-1}\eta_\fq^{-1} & * \\ 0 & \eta_{\fq} \end{pmatrix},
\end{equation}

where $\eta_{\fp}\colon G_{F,\fq} \longrightarrow (\T_\fm^\dagger)^*$ is the unramified character given by $\eta_{\fp}(\rec(\varpi^{-1})) = U_{\fp}$.  Here $\varpi$ denotes a uniformizer of $F_\fq^*$ and $\rec\colon F_\fq^* \longrightarrow G_{F,\fq}^{\ab}$ is the local Artin reciprocity map.  Also, $\varepsilon_{\cyc} \colon G_F \longrightarrow \Z_p^*$ denotes the $p$-adic cyclotomic character.

For each $\fq \mid p$, let $V_{\fq}$ be the eigenspace of $\rho|_{G_{\fq}}$ i.e. the span of the vector $\binom{1}{0}$ in the basis for which (<ref>) holds.
We choose an element $\tau \in G_F$ as in <cit.>*Proposition 9.3 such that its restriction to $\fg$ is complex conjugation and such that for all $\fq \mid p$, the subspace $V_{\fq}$ projected to each factor of $K$ is not stable under $\rho(\tau)$. As in loc. cit., fix a basis such that $\rho(\tau) =
\begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2\end{pmatrix}$ where $\lambda_1 \equiv 1 \pmod{\fm}$ and $\lambda_2 \equiv -1 \pmod{\fm}$.
For a general $\sigma$, we write

\[
\rho(\sigma) = \begin{pmatrix} a(\sigma) & b(\sigma) \\ c(\sigma) & d(\sigma) \end{pmatrix}.
\]

For each $\fq \mid p$, there is a change of basis matrix $M_\fq = \begin{pmatrix} A_\fq & B_\fq \\ C_\fq & D_\fq \end{pmatrix} \in \GL_2(K)$ such that

\begin{equation} \label{e:matrixeqn}
\begin{pmatrix} a(\sigma) & b(\sigma) \\ c(\sigma) & d(\sigma)\end{pmatrix} M_{\fq} = M_{\fq} \begin{pmatrix} \bpsi \varepsilon_{\cyc}^{k-1} \eta_{\fq}^{-1} & * \\ 0 & \eta_{\fq} \end{pmatrix}.
\end{equation}

The choice of $\tau$ ensures that $A_\fq$ and $C_\fq$ are invertible in $K$. Furthermore, equating the upper left hand entries in (<ref>) yields:

\begin{equation} \label{e:bequation}
b(\sigma) = x_\fq (a(\sigma) -  \bpsi \varepsilon_{\cyc}^{k-1} \eta_{\fq}^{-1}(\sigma))  \text{ for all }\sigma \in G_{F,\fq}, \text{ where }
x_\fq = -\frac{A_\fq}{C_\fq}.
\end{equation}

§.§ Cohomology Class

Let $\varphi_{\fm} \colon \T_\fm^\dagger \longrightarrow W$ denote the extension of the homomorphism $\varphi$ to the completion of $\T^\dagger$.
\[ \I^\dagger = \ker(\varphi_{\fm}), \quad \tilde{\I} = \ker({\varphi}_{\fm}|_{\tilde{\T}_{\fm}}), \quad \I = \ker(\varphi_{\fm}|_{\T_\fm}). \]

As in <cit.>*9.3, the choice of basis for $\rho$ implies that $a(\sigma), d(\sigma) \in \T_\fm$ with

\begin{equation} \label{e:adcong}
a(\sigma) \equiv \varepsilon_{\cyc}^{k-1}(\sigma) \pmod{\I}, \qquad d(\sigma) \equiv \bpsi(\sigma) \pmod{\I}.
\end{equation}

Recall that  $K=\Frac(\T_{\fm}^\dagger)$.
Recall the elements $x_\fq \in K$ for $\fq \mid p$ defined in (<ref>), including the distinguished prime $\fp$.
Define $x_{\fp}' = x_{\fp}(U_\fp - 1)$.  Let:

\begin{align}
\tilde{\B} & = \tilde{\T}_{\fm} \text{-submodule of } K \text{ generated by } b(\sigma)  \text{ for all } \sigma \in G_F \text{ along with } \label{e:btbdef} \\
& \ \ \ \ \  x_{\fp}, x_{\fp}', \text{ and } x_\fq \text{ for all } \fq \in \Sigma. \nonumber \\
\tilde{\B}' & = \tilde{\T}_{\fm} \text{-submodule of } \tilde{\B} \text{ generated by } \tilde{\I} \tilde{\B}, p^m \tilde{\B}, \text{ and}  \nonumber \\
& \ \ \ \ \  b(\sigma) \text { for all } \sigma \in I_{\fq}, \fq \mid \fP'. \nonumber \\
\tilde{B} & = \tilde{\B}/\tilde{\B}'. \label{e:tbdef}
\end{align}

By construction, $\tilde{B}$ naturally has the structure of an  $A_{\sL}^\#$-algebra in which $\sL$ acts by multiplication by $(1 - U_\fp)$.

Denote by $\overline{b}(\sigma)$ the image of $b(\sigma)$ in $\overline{B}$.
Since $\rho$ is a Galois representation, we have

\[ b(\sigma \sigma') = a(\sigma) b(\sigma') + b(\sigma)d(\sigma') \qquad \text{ for all } \sigma, \sigma' \in G_F. \]

The congruences (<ref>) therefore imply that the function

\[ \kappa(\sigma) =  \overline{b}(\sigma)\bpsi^{-1}(\sigma) \]

is a 1-cocycle defining a cohomology class $[\kappa] \in H^1(G_F, \tilde{B}(\bpsi^{-1}))$.  Note here that since $p^m \tilde{\B} = 0$ in $\tilde{B}$ and $\varepsilon_{\cyc}^{k-1} \equiv 1 \pmod{p^m}$, the character $\varepsilon_{\cyc}^{k-1}$ acts trivially on $\tilde{B}$.
\begin{align}
{B}_0 &=  \T_\fm\text{-submodule of } \tilde{B} \text{ generated by the image of } b(\sigma) \text{ for all } \sigma \in G_F.  \nonumber \\
\begin{split}
B & =  \T_\fm\text{-submodule of } \tilde{B} \text{ generated by } B_0 \text{ along with }  \label{e:bmoddef} \\
& \ \ \ \ \  x_{\fp}, x_{\fp}', \text{ and } x_\fq \text{ for all } \fq \in \Sigma.
\end{split}
\end{align}

Note that $\tilde{\I}\tilde{\B} = 0$ in $\tilde{B}$ and every element of $\T_\fm$ is equivalent to an element of $A$ modulo $\I$ (see the first three bullet points of Theorem <ref>). Therefore, in the definition of $B_0$ and $B$, it is equivalent to replace $\T_\fm$ by $A$.

The $A$-module structure of $\tilde{B}(\bpsi^{-1})$ is by definition the composition of the involution $\#$ with the natural $A$-module structure of $\tilde{B}$.  The canonical $A_{\sL}^\#$-module structure of $\tilde{B}$ can therefore be viewed as an $A_{\sL}$-module structure on $\tilde{B}(\bpsi^{-1})$.

We now verify that our construction verifies all the properties required to apply Theorems <ref> and <ref>.
First we describe the class $\kappa$ locally at primes in $\Sigma_\fp \setminus S_\infty$ using (<ref>).

For $\fq \in \Sigma \setminus S_\infty$, we have

\[ \kappa(\sigma) = ( \bpsi^{-1}(\sigma) -1 )x_{\fq}, \qquad \sigma \in G_{F, \fq}. \]

\[ \kappa(\sigma) = (\bpsi^{-1}(\sigma)-1)x_{\fp} + \ord_{\fp}(\sigma)x_{\fp}', \qquad \sigma \in G_{F, \fp},\]

where $\ord_\fp$ is as in (<ref>).

Note  that
\begin{equation} \label{e:aeq}
a(\sigma) \equiv \varepsilon_{\cyc}^{k-1}(\sigma) \equiv 1 \pmod{(\I, p^m)} \end{equation}

for $k \equiv 1 \pmod{(p-1)p^N}$ and $N \ge m$.
If $\fq \in \Sigma \setminus S_\infty,$ then $U_\fq \equiv 1 \pmod{\I}$, and hence by the definition of  $\eta_\fq$ following (<ref>),
$\eta_\fq$ acts trivially on $\tilde{B}(\bpsi^{-1})$.  Therefore (<ref>) becomes

\[ \kappa(\sigma) = \overline{b}(\sigma) \bpsi^{-1}(\sigma)= ( \bpsi^{-1}(\sigma) -1 )x_{\fq}, \qquad \sigma \in G_{F, \fq}. \]

For $\fq = \fp$, this applies except that $U_\fp \not\equiv 1 \pmod{\I}$.
Since $(U_\fp-1)^2 \in \I$, we have
\[ U_\fp^n \equiv 1 + n(U_\fp - 1) \pmod{\I}. \]
It follows that

\[ \kappa(\sigma) = (\bpsi^{-1}(\sigma)-1)x_{\fp} + \ord_{\fp}(\sigma)x_{\fp}', \qquad \sigma \in G_{F, \fp},\]

\[ x_{\fp}' = ( U_\fp-1) x_{\fp} = -\sL x_{\fp}.  \qedhere \]

The $G$-module $\tilde{B}(\bpsi^{-1})$ and the cohomology class $[\kappa] \in H^1(G_F, \tilde{B}(\bpsi^{-1}))$ satisfy the following properties.

* The $A$-module $B$ generates $\tilde{B}(\bpsi^{-1})$ as an $A_{\sL}$-module.

* The class $[\kappa]$ is unramified outside $\Sigma'$, locally trivial at  $\Sigma$, and tamely ramified at $\Sigma'$.

* The image of the restriction

\[ [\kappa]_{|G_L} \in H^1(G_L, \tilde{B}) = \Hom_{\cont}(G_L, \tilde{B}) \]

is equal to ${B}_0$.

* The quotient
${B}/{B}_0$ is generated over $A$ by $x_\fq$ for $\fq \in \Sigma$ and the elements $x_{\fp}, x_{\fp}'$.  The element $x_{\fp}'$ is fixed by the action of $G_{F,\fp}$.

* The 1-cocycle $\kappa$ satisfies the following.

* For $\fq \in \Sigma$ finite and $\sigma \in G_{F, \fq}$, we have
$\kappa(\sigma) = (\bpsi^{-1}(\sigma)-1)x_\fq.$

* For $\sigma \in \WD(\overline{F}_\fp/F_\fp)$, we have
\begin{equation} \label{e:ksp}
\kappa(\sigma) =     (\bpsi^{-1}(\sigma)-1)    x_{\fp} + \ord_\fp(\sigma) x_{\fp}'. \end{equation}

* With respect to the $A_{\sL}$-structure on $\tilde{B}(\bpsi^{-1})$, we have  $x_{\fp}' + \sL x_\fp = 0.$

Recall that $\tilde{\T}_\fm$ is generated over $\T_\fm$ by $1 - U_\fp$, which gives the action of $\sL$ on $\tilde{B}(\bpsi^{-1})$.
Comparing (<ref>) and (<ref>), it then follows that $B$ generates $\tilde{B}(\bpsi^{-1})$ as an $A_{\sL}$-module. This proves the first bullet point.

The Galois representation $\rho$, and hence the cohomology class $[\kappa]$, is unramified outside $\Sigma \cup \Sigma'$ and the primes dividing $p$.  There are three types of primes above $p$: those in $\Sigma$, the distinguished prime $\fp$, and the primes dividing $\fP'$.

Since in the definition of $\tilde{B}$ we have taken the quotient by the $\tilde{\T}_{\fm}$-module spanned by $b(\sigma)$  for $\sigma \in I_\fq, \fq \mid \fP'$, it follows that $[\kappa]$ is unramified at the primes dividing $\fP'$.
For $\fq \in \Sigma \setminus S_\infty$, Lemma <ref>
expresses $\kappa_{| G_{F, \fq}}$ as a coboundary and demonstrates that $[\kappa]$ is locally trivial at $\fq \in \Sigma$.  At $\fp$, note that since $ \ord_{\fp}(\sigma) = 0 $  for $\sigma \in I_{\fp}$, Lemma <ref> shows that $[\kappa]$ is unramified at $\fp$.  To conclude the proof of the second item, note that $\Sigma'$ contains no primes above $p$, and all our modules are pro-$p$.  Therefore $[\kappa]$ is tamely ramified at all primes in $\Sigma'$.

For the third bullet point, let $B_L \subset B_0$ denote the image of $[\kappa]_{| G_L}$.  Then of course the image of $[\kappa]_{| G_L}$ in $H^1(G_L, (B_0/B_L)(\bpsi^{-1}))$ vanishes, and hence by <cit.>*Lemma 6.3, the image of $[\kappa]$ in $H^1(G_F, (B_0/B_L)(\bpsi^{-1}))$ vanishes.  Writing $\overline{\kappa}$ for the image of $\kappa$ in $B_0/B_L$, we may then write $\overline{\kappa}(\sigma) = z(\bpsi^{-1}(\sigma) - 1)$ for some $z \in B_0/B_L$.  Yet by construction $\kappa(\tau) = 0$ and $\bpsi^{-1}(\tau) = -1$, where $\tau$ is the special element used in <ref> to fix the chosen basis of $\rho$.  We therefore obtain $z=0$.  Hence $\overline{\kappa} = 0$ as a cocycle.  But by definition of $\kappa$ and $B_0$, the image of the cocycle $\kappa$ generates $B_0$ and hence the image of $\overline{\kappa}$ generates $B_0/B_L$.  It follows that $B_0/B_L = 0$, i.e. $B_0 = B_L$ as desired.

The remaining bullet points follow directly from the definitions or have already been established.

In view of Theorems <ref> and <ref>, we deduce from Theorem <ref>:

We have an $A_{\sL}$-module surjection

\[ \nabla_{\!\sL, A} \longtwoheadrightarrow \tilde{B}(\bpsi^{-1}) \]
and hence an inclusion
$\Fitt_{A_{\sL}}(\nabla_{\!\sL, A}) \subset \Fitt_{A_{\sL}}(\tilde{B}(\bpsi^{-1}))$.

§.§ Calculation of the Fitting ideal

It remains to prove that
\[  \Fitt_{A_{\sL}}(\tilde{B}(\bpsi^{-1})) \subset (p^m). \]

Applying the involution $\#$, this is equivalent to

\[
\Fitt_{A_{\sL}^\#}(\tilde{B}) \subset (p^m) = p^m A_{\sL}^\#.
\]

Removal of the twist by $\bpsi^{-1}$ will be convenient so that the usual $\fg$-module structure on $\tilde{B}$ via $\bpsi$ is compatible with the $\tilde{\T}_\fm$-module structure on $\tilde{B}$ and the homomorphism $\varphi_\fm$, which satsifies $\varphi_\fm(S(\fa)) = \bpsi(\fa)$.

We first recall the following lemma from  <cit.>*Lemma 9.9.
Let $\B_0 \subset \tilde{\B}$ denote the $\tilde{\T}_{\fm}$-submodule generated by the elements $b(\sigma)$ for $\sigma \in G_F$.
There are finitely many elements $b_1, \ldots, b_m \in {\B}_0$ that are non-zerodivisors in $K$, which  generate $\B_0$ as an $A$-module.

With notation as in Theorem <ref>, we have
\[ \Fitt_{A_{\sL}^\#}(\tilde{B}) \subset
\bigg(p^m, \Ann_{A_{\sL}^\#}\Big( \prod_{\fq \mid \fP'} \epsilon_\fq \cdot W\Big) \bigg), \]

and hence by the last item of that theorem we have

\[  \Fitt_{A_{\sL}^\#}(\tilde{B}) \subset (p^m).
\]

The proof proceeds closely along the lines of that in <cit.>*Theorem 9.10. Let  $\fq_1, \dotsc, \fq_r$ denote the primes dividing $\fP'$. For each $\fq_i$, choose an element $\sigma_i \in G_{F,\fq_i} \subset G_F$ that lifts $\rec(\varpi_i) \in G_{F,\fq_i}^{\ab}$, where
$\varpi_i$ is a uniformizer for $F_{\fq_i}$.  Define

\[ c_i := -b(\sigma_i)\bpsi(\fq_i) \varepsilon_{\cyc}^{1-k}(\sigma_i) = x_{\fq_i}(U_{\fq_i} - \bpsi(\fq_i) + \I) \in \tilde{\B}. \]

Here and throughout this proof, we use the notation $a = b + \I$ to mean $a = b + z$ for some $z \in \I$ to avoid needing to add distinct variable names for each such $z$ that appears.

By choosing the elements $b_1, \dots, b_m$ from Lemma <ref> together with $x_\fq$ for all $\fq \in \Sigma_\fp$, we get elements $b_1, \dotsc, b_n$ ($n = m + \#\Sigma_\fp$) of $\tilde{\B}$ that are non-zerodivisors in $K=\Frac(\T_{\fm})$ and generate this module over $\tilde{\T}_\fm$.  The images of these elements in $\tilde{B}$ are therefore $A_{\sL}^\#$-module generators.

To calculate $\Fitt_{A_{\sL}^\#}(\tilde{B}) $ we use the generating set $c_1, \dotsc, c_r, b_1, \dotsc, b_n$ for $\tilde{B}$ over $A_{\sL}^\#$.  Of course, these first $r$ generators are not necessary, but including them will aid us in proving the theorem.  Suppose we have a matrix

\[
M \in M_{(n+r) \times (n+r)}(A_{\sL}^\#)
\]

such that each row of $M$ represents a relation amongst our generators, i.e. such that

\[
M(c_1, \dotsc, c_r, b_1, \dotsc, b_n)^T \equiv 0 \text{ in } \tilde{B}^{n+r}.
\]

We need to show that $\det(M) \in A_{\sL}^\#$ satisfies $(\det(M) + p^m z) \prod_{\fq \mid \fP'} \epsilon_\fq = 0$ in $W$ for some $z \in A_{\sL}^\#$.

Write $M   = (Y | Z)$ in block matrix form, where

\[
Y = (y_{ij}) \in M_{(n+r)\times r}(A_{\sL}^\#), \qquad
Z = (z_{ij}) \in M_{(n+r)\times n}(A_{\sL}^\#).
\]

Since $\bpsi$ and $\eta_{\fq_i}$ are unramified at $\fq_i$ and $a(\sigma) \equiv \varepsilon_{\cyc}^{k-1} \pmod{\I}$, we have by (<ref>):

\[ b(I_{\fq_i}) \subset x_{\fq_i} \I. \]

Also, since the $b_i$ generate $\tilde{\B}$ over $\tilde{\T}_\fm$, every element of $\tilde{\I} \tilde{\B}$ can be written as a sum of elements of the form   $ t_i b_i$ with $t_i \in \tilde{\I} \subset \I^\dagger$.
Therefore each relation
\[ \sum_{j=1}^r y_{ij} c_j + \sum_{j=1}^n z_{ij} b_j \equiv  0 \text{ in } \tilde{B} \]
can be expressed as in equality in $K$ as

\begin{equation} \label{e:xyzsum}
\sum_{j=1}^r  x_{\fq_j}(\tilde{y}_{ij}(U_{\fq_j} - \bpsi(\fq_j)) + {\I}^\dagger) +\sum_{j=1}^n (\tilde{z}_{ij} + {\I}^\dagger + p^m {\T}^\dagger_\fm) b_j  = 0. \end{equation}

We reiterate that here and in what follows, the symbols $\I^\dagger$ (twice) and ${\T}^\dagger_\fm$ represent elements of those sets for which we do not, for notational reasons, introduce separate variable names. We have denoted by $\tilde{y}_{ij}$ and $\tilde{z}_{ij}$ elements of ${\T}^\dagger_{\fm}$ such that $\varphi(\tilde{y}_{ij}) = y_{ij}$, $\varphi(\tilde{z}_{ij}) = z_{ij}$.
It  follows from (<ref>) that if we define a matrix $M' \in M_{(n+r) \times (n+r)}(K)$  in block form by

\[
M' = \left( x_{\fq_j}(\tilde{y}_{ij}( U_{\fq_j} - \bpsi(\fq_j) ) + {\I}^\dagger) \ \ \ \ |  \ \ \ \ (\tilde{z}_{ij} + {\I}^\dagger + p^m {\T}_\fm^\dagger) b_j \right),
\]

then $\det(M') = 0$ in $K$ since it has rows that sum to 0.  We can cancel the factors $x_{\fq_i}$ and $b_j$ scaling the columns of $M'$, since these are non-zerodivisors in $K$.  We obtain that $\det(M'') = 0$ where

\[
M'' = \left( (\tilde{y}_{ij}(U_{\fq_j} - \bpsi(\fq_j)) + {\I}^\dagger) \ \ \ \ |  \ \ \ \ (\tilde{z}_{ij} +{\I}^\dagger + p^m {\T}_\fm^\dagger)  \right) \in M_{(n+r)\times (n+r)}(\T^\dagger).
\]

Taking the determinant of $M''$ and applying $\varphi$, we obtain

\[
0 = \varphi( \det(M''))= (\det(M) + p^m A_{\sL}^\#) \prod_{\fq \mid \fP'} \epsilon_\fq \text{ in } W.
\]

\[ \det(M) \in \bigg(p^m, \Ann_{A_{\sL}^\#}\Big( \prod_{\fq \mid \fP'} \epsilon_\fq \cdot W\Big) \bigg), \]

yielding the first statement of the theorem.
The second statement then follows immediately from  the last bullet point in Theorem <ref>.

We immediately find:

Suppose we are in cases 1, 2(a), or 2(b).
We have

\[ \Fitt_{A_{\sL}}(\nabla_{\!\sL, A}) = 0.\]

Corollary <ref> and Theorem <ref> combine to yield

\[
\Fitt_{A_{\sL}}(\nabla_{\!\sL, A}) \subset \Fitt_{A_{\sL}}(\tilde{B}(\bpsi^{-1})) \subset (p^{m}).
\]

Since this holds for all $m$, we have $\Fitt_{A_{\sL}}(\nabla_{\!\sL, A}) = 0$ as desired.

Combining our results yields the $p$-part of Gross's conjecture.
For now we assume there is more than one prime above $p$ in $F$, leaving the case of one prime for the next section.
Theorem <ref> and Lemma <ref> imply that
$\Fitt_{R_{\sL}}(\nabla_{\!\sL}) = 0.$  Theorem <ref> then yields
the $p$-part of the modified Gross conjecture:

\[ \rec_G(u_\fp^{\Sigma, \Sigma'}) \equiv \Theta_L \pmod{I^2}. \]

Lemma <ref> now gives the $p$-part of Gross's conjecture for the Brumer–Stark unit $u_\fp$:

\[ \rec_G(u_\fp) \equiv  \Theta_{S_\fp,T}^{L/F} \pmod{I^2}. \qedhere \]

§ THE CASE OF ONE PRIME ABOVE $P$ IN $F$

In this section we handle Case 2(c), where $\fp$ is the only prime of $F$ lying above $p$.  We impose this condition for the remainder of the paper.
Rather than calculating the Fitting ideal of $\nabla_{\!\sL}$, we prove the $p$-part of the modified Gross Conjecture (the congruence (<ref>)) by taking advantage of two features that present themselves when there is only one prime above $\fp$: (1) the cyclotomic tower is ramified only at $\fp$, hence we may deform up this tower without altering the depletion set $\Sigma_\fp$; (2) The rank one rational Gross–Stark conjecture, proven in <cit.> and <cit.>, is known.  In essence, our argument  is to show that the rational conjecture (for the cyclotomic tower) implies the integral conjecture (for arbitrary $L/F$).  The key input in this reduction is the strong version of the Brumer–Stark conjecture giving the Fitting ideal of $\nabla_{\Sigma_\fp}^{\Sigma'}$, conjectured by Burns and Kurihara and proven in <cit.>.  This result was stated in Theorem <ref> above, and we apply it in this section to the compositum of $L$ with cyclotomic extensions of $F$.  Another important result applied is the nonvanishing of the derivative of $p$-adic $L$-functions at $s=0$, which follows by combining the result of the rational Gross–Stark conjecture with the spectacular transcendence theorem of Brumer–Baker on the linear independence of $p$-adic logarithms of algebraic numbers.  This again takes advantage of the fact that we are in a rank 1 situation since there is only one prime above $p$ in $F$.

We begin by recalling the necessary results and notation concerning $p$-adic $L$-functions.

§.§ $p$-adic $L$-functions

In our current setting we have
$\Sigma = S_\infty$, $\Sigma_\fp = S_\infty \cup \{\fp\}$, and $\fp$ is the only prime of $F$ above $p$.
For each odd character $\chi$ of $G$, Deligne–Ribet and Cassou-Nogues construct a $p$-adic meromorphic function

\[ L_p(\chi\omega, s) \colon \Z_p \longrightarrow \Q_p(\chi) \]

satisfying the interpolation property

\[ L_p(\chi\omega, n) = L_{\Sigma_\fp, \emptyset}(\chi \omega^n, n) \]

for all integers $n \le 0$. Here $\omega \colon G_F \longrightarrow \mu_{p-1}$ denotes the  Teichmüller character and $\Q_p(\chi)$ denotes the extension of $\Q_p$ obtained by adjoining the values of $\chi$.  Analyticity and integrality are achieved if we incorporate the smoothing set $\Sigma'$, i.e. we have a $p$-adic analytic function

\[ L_{p, \Sigma'}(\chi\omega, s) \colon \Z_p \longrightarrow \Z_p(\chi) \]

\[ L_{p, \Sigma'}(\chi\omega, n) = L_{\Sigma_\fp, \Sigma'}(\chi \omega^n, n) \]

for all integers $n \le 0$. Moreover, these $p$-adic $L$-functions interpolate to a group ring valued Stickelberger function.
There exists a $p$-adic analytic function

\[ \Theta_{p, \Sigma'}^H(s) \colon \Z_p \longrightarrow \Z_p[G]^- \]
satisfying the interpolation property

\[ \Theta_{p,\Sigma'}^H(1-k) = \Theta_{\Sigma_\fp, \Sigma'}(1-k)
\]

for all positive integers $k \equiv 1\pmod{p-1}$.  As we now describe, $\Theta_{p, \Sigma'}^H(s)$ can be constructed as a certain $p$-adic integral.
Let $F_\infty/F$ denote the cyclotomic $\Z_p$-extension of $F$ and let $\fh_\infty = \Gal(HF_\infty/F)$.
For each integer $m \ge 0$ we let $F_m \subset F_\infty$ denote the $m$th layer of the tower and let $\fh_m = \Gal(HF_\infty/HF_m)$.
Then $\fh_m$ is an open subgroup of $\fh_\infty$ and its cosets provide a cover of $\fh_\infty$ by disjoint opens.
For $\sigma \in G_m = \fh_\infty/\fh_m \cong \Gal(HF_m/F)$ we define

\[ \mu(\sigma + \fh_m) = \zeta_{\Sigma_\fp, \Sigma'}(\sigma, 0), \quad \text{where }
\Theta_{\Sigma_\fp, \Sigma'}^{HF_m/F}(0) = \sum_{\tau \in G_m} \zeta_{\Sigma_\fp, \Sigma'}(\tau, 0)\tau^{-1}. \]

For $\sigma \in G$ define

\begin{equation} \label{e:zetaint}
\zeta_{p, \Sigma'}(\sigma, s) = \int_{\sigma + \fh_0} \langle \varepsilon_{\cyc}(\tau) \rangle^{-s} d \mu(\tau).
\end{equation}

Here $\fh_0 = \Gal(HF_\infty/H)$.  We then have

\[ \Theta_{p, \Sigma'}^H(s) = \sum_{\sigma \in G} \zeta_{p, \Sigma'}(\sigma, s) \sigma^{-1}. \]

Taking the derivative of (<ref>) with respect to $s$ and evaluating at 0, we obtain

\begin{equation}
\zeta_{p, \Sigma'}'(\sigma, 0) = -\int_{\sigma + \fh_0} \log_p \varepsilon_{\cyc}(\tau) d \mu(\tau).
\end{equation}

To evaluate this modulo $p^m$, we may take the Riemann sum over the cosets of $\fh_m$.  We obtain:

For every integer $m \ge 0$, we have
\[
(\Theta_{p, \Sigma'}^H)'(0) \equiv - \sum_{\sigma \in G_m}  \zeta_{\Sigma_\fp, \Sigma'}(\sigma, 0) \log_p\varepsilon_{\cyc}(\sigma)\overline{\sigma}^{-1} \pmod{p^m}. \]

Here $\overline{\sigma}$ denotes the image of $\sigma$ in $G$.

For notational simplicity, we will simply write $\Theta_H'$ for $(\Theta_{p, \Sigma'}^H)'(0)$ in the sequel.

§.§ The rational Gross–Stark conjecture

The following result, which we refer to as the rank 1 rational Gross–Stark conjecture, was proven in <cit.> and <cit.>.  The latter paper removed two assumptions from the former, making the result unconditional.

Let $u \in U_\fp^-$ such that

\begin{equation} \label{e:ordnz}
\chi(\ord_G(u)) = \sum_{\sigma \in G}  \chi(\sigma)^{-1} \ord_{\fP}(\sigma(u_\fp)) \neq 0.
\end{equation}

\begin{equation} \label{e:lddp}
\frac{\sum_{\sigma \in G} \chi(\sigma)^{-1} \log_p(\Norm_{F_{\fp}/\Q_p}(\sigma(u)))}{ \chi(\ord_G(u)) } = - \frac{L_p'(\chi^{-1}\omega,0)}{L(\chi^{-1},0)}. \end{equation}

When applied to the Brumer–Stark unit $u_{\fp}^{\Sigma,\Sigma'}$, this result can be interpreted as follows:

We have

\begin{equation} \label{e:ddp}
\Theta_H' = - \sum_{\sigma \in G} \overline{\bpsi}(\sigma)^{-1}  \log_p(\Norm_{F_{\fp}/\Q_p}(\sigma(u_{\fp}^{\Sigma, \Sigma'}))).
\end{equation}

It is enough to show that the two sides agree after the application of $\chi$ for every odd character $\chi$ of $G$, i.e. that

\begin{equation} \label{e:lddp2}
L_{p, \Sigma'}'(\chi^{-1} \omega, 0) =
-\sum_{\sigma \in G} \chi(\sigma)^{-1} \log_p(\Norm_{F_{\fp}/\Q_p}(\sigma(u_{\fp}^{\Sigma, \Sigma'}))).
\end{equation}

Noting that

\[ \frac{L_p'(\chi^{-1}\omega,0)}{L(\chi^{-1},0)} = \frac{L_{p, \Sigma'}'(\chi^{-1}\omega,0)}{L_{\Sigma,\Sigma'}(\chi^{-1},0)}
\]

since the smoothing factors $\prod_{\fq \mid \Sigma'} (1 - \chi^{-1}(\sigma_\fq) \N\fq)$ cancel, and recalling that

\[ \chi(\ord_G(u_{\fp}^{\Sigma,\Sigma'})) = L_{\Sigma,\Sigma'}(\chi^{-1},0) \]
by the definition of $u_{\fp}^{\Sigma,\Sigma'},$ the desired result (<ref>) follows directly from (<ref>).

The following result is known to the experts and has at its heart two deep facts—Theorem <ref> above and the celebrated transcendence result of Brumer–Baker on the linear independence of logarithms of algebraic numbers over $\overline{\Q}$.

Let $\chi$ be an odd character of $G$.  We have $L_p'(\chi\omega, 0) \neq 0$.

In view of Theorem <ref>, it suffices to prove that

\[ \sum_{\sigma \in G} \chi(\sigma)^{-1} \log_p(\Norm_{F_{\fp}/\Q_p}(\sigma(u))) \neq 0 \]
for any $u \in U_\fp^-$ such that $\chi(\ord_G(u)) \neq 0$.  This follows from the theorem of Brumer–Baker <cit.> as explained by Gross in <cit.>*Proposition 2.13.

We interpret this result in terms of the group-ring element $\Theta_H'$ as follows.

The element $\Theta_H' \in \overline{R} = \Z_p[G]^-$ is a non-zerodivisor.

§.§ The ring $R_{X,m}$ and module $\nabla_{X,m}$

Recall $R = \Z_p[\fg]^-$. For any nonnegative integer $m$, define the ring

\begin{equation} R_{X,m} = R[X]/(\Theta_L - \Theta_H' X, X^2, I X, I^2, p^mX). \label{e:rxdef} \end{equation}

For $m$ sufficiently large, the canonical map $R/I^2 \longrightarrow R_{X,m}$ is injective.

The proof is easier than Theorem <ref> since $\Theta_H'$ is a non-zerodivisor in $\overline{R}$.
Let $a \in R$ have image in $R_{X,m}$ that vanishes.  Writing down the fact that $a$ lies in the ideal defining $R_{X,m}$ shows that $a \equiv r\Theta_L \pmod{I^2}$ for some $r \in R$ such that $\overline{r} \Theta_H' \equiv 0 \pmod{p^m}$ in $\overline{R}$.  Yet since $\Theta_H'$ is a non-zerodivisor,
there exists a nonnegative integer $h$ such that $\Theta_H'$ divides $p^h$ in $\overline{R}$. We therefore find $p^h \overline{r} \equiv 0 \pmod{p^m}$, whence $\overline{r} \equiv 0 \pmod{p^{m-h}}$ for $m \ge h$.  Therefore $r \in (I, p^{m-h})$ so $a \in (I^2, p^{m-h}I)$.  But $\#\Gamma$ annihilates $I/I^2$ (recall $\Gamma = \Gal(L/H)$) so for $m$ large enough $p^{m-h}I \subset I^2$, and we have $a \in I^2$ as desired.

Recall that for an integer $m \ge 0$, $F_m$ denotes the $m$th layer of the cyclotomic $\Z_p$-extension of $F$.
Let $\fg_m = \Gal(LF_m/F)$ and $\Gamma_m = \Gal(LF_m/H)$.  Let $R_m = \Z_p[\fg_m]^-$ and
\[ \bpsi_m \colon \fg_m \longrightarrow R_m^* \]
denote the canonical character.  We define a ring homomorphism

\begin{equation} \label{e:betadef} \beta_m \colon R_m \longrightarrow R_{X,m}, \qquad \sigma \mapsto \sigma|_L  + \sigma|_H \cdot \log_p(\varepsilon_{\cyc}(\sigma)) X.
\end{equation}

This is well-defined since the image of $\sigma \in \fg_m$ determines the value of $\log_p(\varepsilon_{\cyc}(\sigma))$ modulo $p^m$.

\begin{equation} \label{e:nablaxdef}
\nabla_{X, m} = {\nabla}_{\Sigma_\fp}^{\Sigma'}(L F_m)_{R_m} \otimes_{R_m} R_{X,m},
\end{equation}

where the $R_m$-action on the right factor is given by $\beta_m$.

The $R_{X,m}$-module $ \nabla_{X, m}$ is quadratically presented and

\[ \Fitt_{R_{X,m}}(\nabla_{X,m})= (\rec_G(u_\fp^{\Sigma,\Sigma'}) - \Theta_L). \]

To prepare for the proof of Lemma <ref>,
define $I_m = \ker(\Z_p[\fg_m]^- \longrightarrow \Z_p[G]^-)$. We have $I_m/I_m^2 \cong \Z_p[G]^- \otimes \Gamma_m$. We lift the map $\rec_G$ defined in (<ref>) by defining

\[
\rec_{G,m} \colon (\cO_{H, \Sigma, \Sigma'}^*)_{\overline{R}} \longrightarrow I_m/I_m^2, \qquad \epsilon \mapsto \sum_{\sigma \in G} (\rec_{\fP,m}(\sigma(\epsilon)) - 1)\tilde{\sigma}_m^{-1},
\]

where $\rec_{\fP, m}\colon H_{\fP}^* \longrightarrow \Gamma_m$ is the reciprocity map and $\tilde{\sigma}_m$ is a lift of $\sigma$ in $\fg_m$. Under the projection $\fg_m \longrightarrow \fg$, the element $\rec_{\fP, m}(\epsilon)$ is mapped to $\rec_{\fP}(\epsilon)$. Furthermore,
\[ \varepsilon_{\cyc}(\rec_{\fP,m}(\epsilon)) \equiv  \Norm_{H_{\fP}/\Q_p}(\epsilon) \text{ in } (\Z/p^m\Z)^*. \]
Therefore, under the map $\beta_m$ defined in (<ref>), we have

\begin{equation} \label{e:recn}
\beta_m(\rec_{G,m}(\epsilon)) = \rec_G(\epsilon) + X \sum_{\sigma \in G}\overline{\bpsi}(\sigma)^{-1} \log_p  \Norm_{H_{\fP}/\Q_p}(\sigma(\epsilon)).
\end{equation}

Lemmas <ref> and <ref> (applied to $LF_m$ in place of $L$) imply that
${\nabla}_{\Sigma_\fp}^{\Sigma'}(L F_m)_{R_m}$ is  quadratically presented over $R_m$ and that

\[ \Fitt_{R_m} ({\nabla}_{\Sigma_\fp}^{\Sigma'}(L F_m)_{R_m}) = ( \rec_{G,m}(u_\fp^{\Sigma, \Sigma'})). \]

Applying (<ref>), we then calculate

\begin{align}
\beta_m(\rec_{G,m}(u_\fp^{\Sigma, \Sigma'})) &= \rec_G(u_\fp^{\Sigma, \Sigma'}) + X \sum_{\sigma \in G}\overline{\bpsi}(\sigma)^{-1} \log_p  \Norm_{H_{\fP}/\Q_p}(\sigma(u_\fp^{\Sigma, \Sigma'}))  \nonumber \\
& =  \rec_G(u_\fp^{\Sigma, \Sigma'}) - \Theta_H' X  \label{e:applygs} \\
& =  \rec_G(u_\fp^{\Sigma, \Sigma'}) - \Theta_L. \label{e:applyref}
\end{align}

Equation (<ref>) follows from Corollary <ref> and equation (<ref>) is one of the defining relations of $R_{X, m}$ in (<ref>). The lemma follows.

On the other hand, in our previous work <cit.> we we calculated the Fitting ideal of ${\nabla}_{\Sigma_\fp}^{\Sigma'}(L F_m)$ exactly, yielding the following result:

We have $\Fitt_{R_{X,m}}(\nabla_{X,m}) = 0$.

Theorem 3.3 of <cit.>, recalled in Theorem <ref> above,
implies that

\[    \Fitt_{R_m}(\nabla_{\Sigma_\fp}^{\Sigma'}(LF_m)_{R_m}) = (\Theta^{LF_m}_{\Sigma_\fp, \Sigma'}). \]

Passing to $R_{X,m}$ by applying $\beta_m$, we find

\begin{align}
\beta_m(\Theta^{LF_m}_{\Sigma_\fp, \Sigma'}) &= \Theta_L + X \sum_{\sigma \in \fg_m} \zeta_{\Sigma_\fp, \Sigma'}(\sigma, 0) \log_p\varepsilon_{\cyc}(\sigma)\overline{\sigma}^{-1} \nonumber \\
&= \Theta_L + X \sum_{\sigma \in G_m} \zeta_{\Sigma_\fp, \Sigma'}(\sigma, 0) \log_p\varepsilon_{\cyc}(\sigma)\overline{\sigma}^{-1}  \label{e:gtog} \\
&= \Theta_L - X \Theta_H'  \label{e:applyth} \\
&= 0.  \label{e:final} \end{align}

Here $\overline{\sigma}$ denotes the image of $\sigma$ in  $G$.  Equation (<ref>) follows from the distribution property of partial zeta functions since $\log_p \varepsilon_{\cyc}(\sigma) \overline{\sigma}^{-1}$ depends only on the image of $\sigma$ in $G_m$. Equation (<ref>) follows from Lemma <ref>.  Equation (<ref>) is a defining relation of $R_{X,m}$.  The result follows.

Combining Lemma <ref> with Theorem <ref>, we obtain
\begin{equation} \label{e:recutrx}
\rec_G(u_\fp^{\Sigma, \Sigma'}) - \Theta_L =0 \text{ in } R_{X,m}. \end{equation}

By Lemma <ref>,  equation (<ref>) for $m$ large enough yields

\[ \rec_G(u_\fp^{\Sigma, \Sigma'}) \equiv \Theta_L \pmod{I^2} \]

in $R$.  This  completes the proof of the $p$-part of the modified Gross conjecture (<ref>) in Case 2(c).
Lemma <ref> then yields the $p$-part of Gross's conjecture for the Brumer–Stark unit $u_\fp$:

\[ \rec_G(u_\fp) \equiv  \Theta_{S_\fp,T}^{L/F} \pmod{I^2}. \]

This completes the proof of Theorem <ref>.

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