# Iwasawa invariants in residually reducible Hida families

Robert Pollack  Max Planck Institut für Mathematik, Bonn/Boston University
<EMAIL_ADDRESS>and  Preston Wake  Michigan State University
<EMAIL_ADDRESS>
###### Abstract.

We study the variation of $\mu$-invariants of modular forms in a cuspidal Hida
family in the case that the family intersects an Eisenstein family. We allow
for intersections that occur because of “trivial zeros” (that is, because $p$
divides an Euler factor) as in Mazur’s Eisenstein ideal paper, and pay special
attention to the case of the $5$-adic family passing through the elliptic
curve $X_{0}(11)$.

##  1\. Introduction

###  1.1. Congruences from $L$-values

In [BP19], the first author and Bellaïche studied the $\mu$-invariants of Hida
families of cuspidal eigenforms which admit congruences with Eisenstein
series. The congruences studied there were ones that arose from
$p$-divisibilities of $L$-values, in this case, Bernouli numbers. For
instance, consider the case of tame level $N=1$ where $p\mid B_{k}$, the
$k$-th Bernoulli number. In this situation, there is a cuspidal Hida family
which has non-trivial intersections with an Eisenstein Hida family with the
intersections occuring at the $p$-adic weights which are zeroes of the
$p$-adic $\zeta$-function (which is non-trivial as $p\mid B_{k}$).

Assuming certain Hecke algebras were Gorenstein, it was shown in [BP19] that
the $\mu$-invariants in the cuspidal family blew up as one approached these
intersection points. Moreover, precise formulas were conjectured about the
values of these $\mu$-invariants which in favorable situations (i.e. when the
cuspidal Hida family was rank 1 over weight space) were given simply as the
$p$-adic valuation of certain special values of the $p$-adic $\zeta$-function.
When $U_{p}-1$ generates the Eisenstein ideal, these conjectures were shown to
hold for the branch of the $\Lambda$-function with trivial tame character, and
the $\lambda$-invariants in these families were shown to be identically zero.

###  1.2. Congruences from Euler factors

In this paper, we aim to treat the analogous situation where congruences arise
because of divisibility of Euler factors. For example, consider the case of
$N$ a prime such that $N\equiv 1\pmod{p}$. By Mazur’s famous result [Maz77],
as long as $p>3$, there is always a cuspidal eigenform in
$S_{2}(\Gamma_{0}(N))$ congruent to the unique ordinary Eisenstein series of
weight 2 and level $N$. The most famous example of this congruence is when
$p=5$ and $N=11$. In this case, there is a unique such cuspidal eigenform and
it is exactly the modular form corresponding to the elliptic curve
$X_{0}(11)$.

The ordinary $p$-stabilization of this form to level $Np=55$ lives in a Hida
family that is rank 1 over weight space. However, there is a key difference in
this situation from the case treated in [BP19] where there was a unique
Eisenstein family to consider. In our situation, there are two Eisenstein
families; namely, one where $U_{N}=1$ for all weights and one where
$U_{N}=N^{k-1}$. Since $N\equiv 1\pmod{p}$, these two families are congruent
modulo $p$. Moreover, the localization of the Hida Hecke algebra which
corresponds to these two families together with the cuspidal family is not
Gorenstein and the methods of [BP19] do not apply.

To avoid this problem, we employ a method already used in [Oht14] and [WWE21]
where we remove $U_{N}$ from our Hecke algebra and replace it with $w_{N}$,
the Atkin-Lehner involution. While the $U_{N}$-eigenvalues of the two
Eisenstein families are congruent, the $w_{N}$-eigenvalues of these families
are $1$ and $-1$, and the congruence is broken. (It is verified in Appendix A
that Hida theory still works as expected for these modified Hida Hecke
algebras.)

We are interested in the Eisenstein family with $w_{N}$-eigenvalue $-1$ as our
cuspidal family has $w_{N}$-eigenvalue equal to $-1$. The localization of the
modified Hecke algebra corresponding to this Eisenstein family together with
our cuspidal family turns out to be Gorenstein and we are good shape to
generalize the methods of [BP19].

We state our results in the specific case of $p=5$ and $N=11$. To set up a
little bit of notation, let $L_{p}^{+}(k,s)$ denote the (plus) two-variable
$p$-adic $L$-function over $\mathbb{Q}_{\infty}$, the cyclotomic
$\mathbb{Z}_{p}$-extension, corresponding to the Hida family through
$X_{0}(11)$.

######  Theorem 1.1.

Let $N=11$, $p=5$, and let $k$ be an integer with $k\equiv 2\pmod{4}$. Let
$f_{k}$ denote the unique ordinary eigenform in $S_{k}(\Gamma_{0}(Np))$ whose
residual representation is $1\oplus\omega$, where $\omega$ is the mod-$p$
cyclotomic character. Then

$L_{p}^{+}(k,s)=(N^{k/2}-1)\cdot U(k,s)$

where $U(k,s)$ is a unit a power series in $k$ and $s$. In particular, the
Iwasawa invariants of $f_{k}$ are given by

  1. (1)

$\mu(f_{k})=\mathrm{val}_{p}(a_{p}(f_{k})-1)=1+\mathrm{val}_{p}(k)$ and

  2. (2)

$\lambda(f_{k})=0$.

Note that considering weight $2$ in the above theorem yields a $\mu$-invariant
of 1. This conclusion corresponds to the famous fact that $\mu(X_{0}(11))=1$
for $p=5$ (see [Maz72, Section 10]). Further, the above theorem implies that
as one moves closer and closer to weight $0$ in this Hida family, the
$\mu$-invariants blow up linearly. Theorem 1.1 is a special case of Theorem
1.2 below.

###  1.3. The more general case

We now discuss the key ingredients that go into Theorem 1.1 to unravel what is
special about this case with $(N,p)=(11,5)$ and weight $2$, and what can be
said in greater generality. To this end, we now let $N$ and $p$ be prime
numbers with $N\equiv 1\pmod{p}$, and let $k_{0}$ be an even integer with
$0<k_{0}<p-1$. Let $\mathbb{T}_{\mathfrak{m}}$ denote the completion of the
Hida Hecke algebra with tame level $N$ at the maximal ideal given by the
residual representation $\overline{\rho}=1\oplus\omega^{k_{0}}$ and with
Atkin–Lehner sign $w_{N}=-1$; let $\mathbb{T}^{0}_{\mathfrak{m}}$ denote the
quotient of $\mathbb{T}_{\mathfrak{m}}$ that acts faithfully on cuspforms. Let
$\mathcal{I}_{\mathfrak{m}}\subseteq\mathbb{T}_{\mathfrak{m}}$ denote the
Eisenstein ideal.

The following facts hold in the setting of Theorem 1.1, where $p=5$, $N=11$,
and $k_{0}=2$:

  1. (1)

$\mathbb{T}_{\mathfrak{m}}$ and $\mathbb{T}^{0}_{\mathfrak{m}}$ are Gorenstein

  2. (2)

$\mathcal{I}_{\mathfrak{m}}$ is generated by $U_{p}-1$

  3. (3)

$p\nmid\frac{B_{k_{0}}}{k_{0}}$

  4. (4)

$N\equiv 1\pmod{p}$ but $N\not\equiv 1\pmod{p^{2}}$

  5. (5)

the rank of $\mathbb{T}^{0}_{\mathfrak{m}}$ over the Iwasawa algebra $\Lambda$
equals 1

We will now examine each of these facts in turn and how (if at all) they are
used in the proof of Theorem 1.1; then we will state the general version
Theorem 1.2 of the theorem. We first note that (2) implies (1) (see Lemma
2.8). In fact, (1) holds whenever $\mathcal{I}_{\mathfrak{m}}$ is principal.

Condition (2) is crucial to obtain as precise a result as in Theorem 1.1.
However, we make use of several recent results [Deo23, Wak23, WWE21] about the
structure of these Hecke algebras which provide a numerical criterion that is
equivalent to (2), as we now explain. Let
$\log_{N}:\mathbb{F}_{N}^{\times}\to\mathbb{F}_{p}$ be a surjective
homomorphism (noting that $p\mid(N-1)$, this is a reduction modulo $p$ of a
choice of discrete logarithm). Further, let $\zeta$ denote a primitive $p$-th
root of unity in $\mathbb{F}_{N}^{\times}$. As recalled in Section 5.4 below,
the results of [Deo23, Wak23, WWE21] imply that if

$\sum_{i=1}^{p-1}i^{k_{0}-2}\log_{N}(1-\zeta^{i})\neq
0\text{~{}in~{}}\mathbb{F}_{p},$

then $\mathcal{I}_{\mathfrak{m}}$ is a principal ideal, and moreover, if
further

$p\mbox{ is not a }p\mbox{-th power modulo }N,$

then $U_{p}-1$ generates $\mathcal{I}_{\mathfrak{m}}$. Note that when
$k_{0}\equiv 2\pmod{p-1}$, the sum appearing in the first condition
$\sum_{i=1}^{p-1}i^{k_{0}-2}\log_{N}(1-\zeta^{i})$ simplifies to
$\log_{N}(p)$, so the first condition is equivalent to the second condition.
In particular, in the case $(N,p,k_{0})=(11,5,2)$, the fact that 5 is not a
5-th power modulo 11 implies (2) (and thus (1)) holds.

Moving on, condition (3) is necessary to ensure that only Euler-factor-type
congruences appear. If (3) fails, then there will be a mix of congruences from
$L$-values (as in [BP19]) and from Euler factors and the situation is more
complicated. Although it seems very interesting to understand this situation,
it is outside the scope of this paper.

Condition (4) is not a serious one. It is used to compute the valuation of
$N^{k/2}-1$. In general, we have

$\mathrm{val}_{p}(N^{k/2}-1)=\mathrm{val}_{p}(N-1)+\mathrm{val}_{p}(k).$

Condition (5) is important in that without it one cannot simply speak of
$f_{k}$ as there is no longer a unique cuspidal eigenform in our Hida family
in each weight. Further, without this condition, the crossing points of the
Eisenstein family and the cuspidal family cannot be as finely controlled.
Nonetheless, we still have a two-variable $p$-adic $L$-function which simply
equals $U_{p}-1$ up to a unit. Thus we can obtain a formula for the
$\mu$-invariants in terms of the $U_{p}$-eigenvalues of each form, but we can
only give a formula that only depends on $k$ and $N$ for the sum of the
$\mu$-invariants of all the forms in weight $k$ in our Hida family.

We note that the same results on the structure of Hecke algebras [Deo23,
Wak23, WWE21, WWE20] provide a numerical condition that forces condition (5)
to hold. Namely, $\mathbb{T}^{0}_{\mathfrak{m}}$ has rank 1 over $\Lambda$ if

$\prod_{i=1}^{N-1}i^{\left(\sum_{j=1}^{i-1}j^{k_{0}-1}\right)}\mbox{ is not a
}p\mbox{-th power modulo }N.$

Further, if there is a single Galois conjugacy class of modular forms in our
Hida family in a given weight, then all of their Iwasawa invariants are the
same and we can completely control the situation in this case. On the one
hand, we know of no numerical criterion to force there to be a single Galois
conjugacy class in a fixed weight. But on the other hand, examples where there
are more than one conjugacy class seem rare. When $p=5$, there are 163 primes
$N$ less than 5000 for which $N\equiv 1\pmod{5}$. Of these, 48 have
$\mathbb{T}^{0}_{\mathfrak{m}}$ with rank greater than 1, but only 6 have more
than one Galois conjugacy class of forms in weight 2.

We now state our main result in more general terms. Some notation: Let
$L^{+}_{p}(\mathfrak{m})$ denote the (plus) two-variable $p$-adic $L$-function
attached to our Hida family and write $L^{+}_{p}(\mathfrak{m},\omega^{0})$ for
its branch corresponding to $\mathbb{Q}_{\infty}$, the cyclotomic
$\mathbb{Z}_{p}$-extension—that is, the branch with trivial tame character.
For an eigenform $g$ in our Hida family, write $\varpi_{g}$ for a uniformizer
of the normalization of the ring generated by the Hecke-eigenvalue of $g$.

######  Theorem 1.2.

Assume that

  1. (1)

$p\nmid\frac{B_{k_{0}}}{k_{0}}$,

  2. (2)

$\displaystyle\sum_{i=1}^{p-1}i^{k_{0}-2}\log_{N}(1-\zeta^{i})\neq 0$ in
$\mathbb{F}_{p}$, and

  3. (3)

$p$ is not a $p$-th power modulo $N$.

Then $L^{+}_{p}(\mathfrak{m},\omega^{0})$ has the simple form

$L^{+}_{p}(\mathfrak{m},\omega^{0})=(U_{p}-1)\cdot U$

where $U$ is a unit. In particular, for every form $g$ in our Hida family,

$\mu(g)=\mathrm{ord}_{\varpi_{g}}(a_{p}(g)-1)\mbox{ and }\lambda(g)=0.$

Further, for every integer $k$ with $k\equiv k_{0}\pmod{p-1}$,

$\sum_{g}\mu(g)=\mathrm{val}_{p}(N-1)+\mathrm{val}_{p}(k)$

where the sum is over all Galois conjugacy classes of weight $k$ forms in our
Hida family.

Theorem 1.2 is stated, and proven, in a more precise form in Theorem 5.12
below.

###  1.4. Normalizations

Like the results in [BP19], the above theorem can be thought of as explaining
$\mu$-invariants through $p$-adic variation. For any individual form, a
positive $\mu$-invariant can almost be thought of as an error in
normalization. One can simply change the complex period defining the $p$-adic
$L$-function by a power of $p$ to simply force a $\mu$-invariant to be 0.
However, in the family one now sees the $\mu$-invariants as arising from
valuations of special values of $p$-adic analytic functions. In the setting of
[BP19], in the rank 1 case, the relevant analytic function was the $p$-adic
$\zeta$-function (which itself has $\mu$-invariant 0). In this paper, the
relevant function is simply $\langle N\rangle^{1/2}-1$ where $\langle
N\rangle$ is the element in $\Lambda$ which specializes to $N^{k}$ in weight
$k$. Note that $\langle N\rangle^{1/2}-1$ also has $\mu$-invariant 0.

With that said, one can wonder why these two-variable $p$-adic $L$-functions
are divisible by these analytic functions that depends only on $k$ and not on
the cyclotomic variable $s$. Indeed, the origin of this project was an attempt
to reconcile the results of [BP19] with results of the second author which
said that these $\mu$-invariants were identically 0 along the family!

The reconciliation of these results is that there are two natural
normalizations of the $p$-adic $L$-function and one leads to positive
$\mu$-invariants which blow up in the family and the other leads to
$\mu$-invariants that are 0 everywhere. We now explain.

In the construction of the two-variable $p$-adic $L$-function, one considers a
certain explicit class in

$\mathcal{L}^{+}_{p}(\mathfrak{m})\in
X{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}:=\varprojlim_{n}H^{1}(Y_{1}(Np^{r}),\mathbb{Z}_{p})^{-,\mathrm{ord}}_{\mathfrak{m}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$

built out of modular symbols where the upper “$\mathrm{ord}$” denotes the
ordinary projector and $\mathfrak{m}$ denotes the completion at the maximal
ideal corresponding to our Hida family (the change in sign in the notation is
due to Poincaré duality: we think of the negative-signed cohomology class
$\mathcal{L}^{+}_{p}(\mathfrak{m})$ as a _functional_ on modular symbols with
positive sign §3). We refer to this class as the two-variable $p$-adic
$L$-symbol.

Typically, when constructing $p$-adic $L$-functions, one extracts a function
from a symbol by choosing a basis for the space of symbols. In this case, that
would involve choosing a basis for the space $X$ as a
$\mathbb{T}_{\mathfrak{m}}$-module. However, the space $X$ is not necessarily
cyclic as a $\mathbb{T}_{\mathfrak{m}}$-module; in fact, by Ohta’s
Eichler–Shimura isomorphism for Hida families [Oht99], it is known that $X$ is
a dualizing module for $\mathbb{T}_{\mathfrak{m}}$. Hence $X$ is free of rank
$1$ over $\mathbb{T}_{\mathfrak{m}}$ if and only if
$\mathbb{T}_{\mathfrak{m}}$ is a Gorenstein ring. In fact, if
$\mathbb{T}_{\mathfrak{m}}$ is Gorenstein, then there is a canonical generator
$\\{0,\infty\\}\in X$ and there is a unique
$L_{p}(\mathfrak{m})\in\mathbb{T}_{\mathfrak{m}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$
such that

$\mathcal{L}^{+}_{p}(\mathfrak{m})=L^{+}_{p}(\mathfrak{m})\cdot\\{0,\infty\\}.$

The element $L^{+}_{p}(\mathfrak{m})$ can be viewed as a two-variable $p$-adic
$L$-function. Indeed, the cyclotomic variable comes from the fact that
$L^{+}_{p}(\mathfrak{m})$ lives in a group algebra over
$\mathbb{Z}_{p}^{\times}$ while the weight variable is parametrized by
$\mathrm{Spec}(\mathbb{T}_{\mathfrak{m}})$. If $f$ is a classical form in our
Hida family and $\mathfrak{p}_{f}$ is the corresponding height 1 prime ideal
of $\mathbb{T}_{\mathfrak{m}}$, then the image of $L^{+}_{p}(\mathfrak{m})$ in
$\mathbb{T}_{\mathfrak{m}}/\mathfrak{p}_{f}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$
is the one-variable $p$-adic $L$-function of $f$.

As discussed in length in [BP19], the specializations of
$L^{+}_{p}(\mathfrak{m})$ to Eisenstein forms in the family always vanish and
this is the reason why the $\mu$-invariants blow up in the family. We write
$L_{p}^{+,\rm mod}(\mathfrak{m})$ for the image of $L^{+}_{p}(\mathfrak{m})$
in $\mathbb{T}^{0}_{\mathfrak{m}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$ and we
refer to $L_{p}^{+,\rm mod}(\mathfrak{m})$ as the two-variable $p$-adic
$L$-function with the modular normalization. Here “modular” is referring to
the fact that this $L$-function first arose from a Hecke-algebra on the full
space of modular forms.

On the other hand there is another natural normalization. A closer examination
of the definition of the $p$-adic $L$-symbol shows that it actually lives in
the cohomology of the compact modular curve:

$\mathcal{L}^{+}_{p}(\mathfrak{m})\in
X^{0}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}:=\varprojlim_{n}H^{1}(X_{1}(Np^{r}),\mathbb{Z}_{p})^{-,\mathrm{ord}}_{\mathfrak{m}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}.$

If we instead assume that $\mathbb{T}^{0}_{\mathfrak{m}}$, the cuspidal Hecke
algebra localized at $\mathfrak{m}$, is Gorenstein, then we again have $X^{0}$
is free of rank 1 over $\mathbb{T}^{0}_{\mathfrak{m}}$; however, in this case
there is no natural generator of this space. Nonetheless, pick a generator $e$
of $X^{0}$, and write

$\mathcal{L}^{+}_{p}(\mathfrak{m})=L_{p}^{+,\mathrm{cusp}}(\mathfrak{m})\cdot
e$

for a unique
$L_{p}^{+,\mathrm{cusp}}(\mathfrak{m})\in\mathbb{T}^{0}_{\mathfrak{m}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$.
We call this a two-variable $p$-adic $L$-function with the cuspidal
normalization as it was defined directly from the cuspidal Hecke algebra. It
is well-defined up to a unit in $\mathbb{T}^{0}_{\mathfrak{m}}$.

When both $\mathbb{T}_{\mathfrak{m}}$ and $\mathbb{T}^{0}_{\mathfrak{m}}$ are
Gorenstein, then both definitions $L_{p}^{+,\rm mod}(\mathfrak{m})$ and
$L_{p}^{+,\mathrm{cusp}}(\mathfrak{m})$ make sense and one can ask how they
are related. Following ideas of Ohta [Oht05], we show that both
$\mathbb{T}_{\mathfrak{m}}$ and $\mathbb{T}^{0}_{\mathfrak{m}}$ are Gorenstein
if and only if the Eisenstein ideal $\mathcal{I}_{\mathfrak{m}}$ is principal.
Moreover, we show that $L_{p}^{+,\mathrm{cusp}}(\mathfrak{m})$ divides
$L_{p}^{+,\rm mod}(\mathfrak{m})$ and that the ratio is a generator of the
Eisenstein ideal. Note that in Theorem 1.2, $U_{p}-1$ generates the Eisenstein
ideal and this is exactly the analytic factor that causes the $\mu$-invariants
to grow in the family. In that theorem, the modular normalization is used. If
we instead used the cuspidal normalization, we would lose that factor of
$U_{p}-1$ and all of the $\mu$-invariants would be 0.

###  1.5. Further questions

In this paper, we only consider that analytic side of Iwasawa theory, in that
we only consider $p$-adic $L$-functions. There is a parallel algebraic side,
involving Selmer groups. This algebraic side is particularly interesting in
view of conjectures of Greenberg [Gre01] regarding algebraic $\mu$-invariants
of elliptic curves. We are currently working to prove the algebraic analog of
Theorem 1.2.

We also restrict our attention to the case where the tame level $N$ is prime
and where $p\nmid B_{k_{0}}$. This restriction on the level is natural in view
of our goal to understand the context for the example of $X_{0}(11)$. However,
this assumption has the drawback of limiting the types of examples we can
explore: for instance, $X_{0}(11)$ is the only elliptic curve we know of for
which Theorem 1.2 applies. Many of the techniques developed in this paper are
general and can be applied to other situations as soon as the relevant results
on the structure of Hecke algebras are known. Some interesting situations to
consider include:

  * •

$N$ is squarefree. (There are some results on the structure of Hecke algebras
proven in [WWE21].)

  * •

$N$ is the square of a prime. (This situation was considered in [LW22].)

  * •

$N$ is prime but $p\mid B_{k_{0}}$, so there are congruences of both
“$L$-value” and “Euler-factor” type.

In this greater generality, it will usually not be true that the Eisenstein
ideal is principal, so it is unlikely that one can hope for results as precise
as Theorem 1.2. We make some conjectures (see Conjectures 4.5 and 5.14)
regarding what is happening with Iwasawa invariants in some cases where the
Eisenstein ideal fails to be principal. We hope that these conjectures can
serve as a starting point for investigating these more general situations.

###  1.6. Structure of paper

In Section 2, we begin with an axiomatic approach to describing Hecke
algebras, Eisenstein ideals, and $p$-adic $L$-functions when there is a unique
Eisenstein series parametrized by our Hecke algebra. This axiomatic approach
applies equally to the case where the congruences arise by $p$-divisibility of
$L$-values or of Euler factors and studies the two possible normalizations
discussed in §1.4. In Section 3, we review the construction of the two-
variable $p$-adic $L$-symbol $\mathcal{L}^{+}_{p}(\mathfrak{m})$. In Section
4, we consider the setting of [BP19]. We reprove the main results of [BP19] as
an illustration of the axiomatics of Section 2 and also make conjectures about
what happens when the Eisenstein ideal is not principal. In Section 5, we move
to the case of trivial tame character and prime level. This section contains
our main results on analytic $\mu$-invariants. Finally, in Appendix A, we
verify that Hida theory works equally well for Hecke algebras in which $U_{N}$
has been replaced by $w_{N}$.

Acknowledgements: We dedicate this paper to the memory of Joël Bellaïche. The
impact of Joël’s mathematics on this paper, from his pioneering work on
pseudorepresentations to the paper [BP19], will be clear to the reader. Even
the genesis of this paper goes back to a meeting between the authors and Joël,
about [BP19] and generalizations, at the Glenn Stevens birthday conference in
2014. At the same conference, Joël first introduced the second author to Carl
Wang-Erickson, which led to a fruitful collaboration that provided many of the
results that make this paper possible, especially [WWE20, WWE17, WWE21].

We would like to thank Carl Wang-Erickson for many helpful discussions on the
topic of this paper.

The first author thanks the Max Planck Institute for Mathematics in Bonn and
the second author thanks the Institute for Advanced Study in Princeton for
wonderful environments during their stays there. The first author was
supported by NSF grants DMS-1702178 and DMS-2302285 and the second author by
NSF grant DMS-1901867.

##  2\. Axiomatics

In Section 4, we consider the same situation as [BP19], with a primitive tame
character, and in Section 5, we consider the non-primitive setting of Mazur’s
Eisenstein ideal. The two situations have a lot of commonalities, and to
emphasize this, in this section, we consider a purely abstract situation which
covers both of these at once.

The main point is that the standard construction of the two-variable $p$-adic
$L$-function gives not a _function_ but an element of a two-variable modular
symbols space. In both situations, the modular symbols space is known to be
isomorphic to the _dualizing module_ of the cuspidal Hecke algebra. If this
dualizing module is isomorphic to the Hecke algebra (i.e. if the cuspidal
Hecke algebra is Gorenstein), then the symbol gives a function. The $p$-adic
$L$-function can also be thought of as an element of the larger dualizing
module of the full Hecke algebra, and if that Hecke algebra is Gorenstein,
then the symbol again gives a function. If both Hecke algebras are Gorenstein,
this gives two different ways to produce a function from the symbol, and the
two functions are different by a normalizing factor.

Much of this discussion revolves around commutative algebra questions of
whether one or the other, or both, or these Hecke algebras is Gorenstein. In
this section, we discuss these commutative algebra issues in the abstract.

###  2.1. Set up

The first key property common to both of the situations we consider is that,
though there are congruences between cuspidal families and an Eisenstein
family, there are no congruences between different Eisenstein families. We
axiomatize this as follows.

######  Definition 2.1.

A _single-Eisenstein Hecke algebra set up_ is the data of
$(\mathcal{O},A,R,\mathcal{E},r_{0})$ where

  * •

$\mathcal{O}$ is a complete discrete valuation ring with uniformizer $\varpi$
and residue field $k$,

  * •

$A$ is a flat, local $\mathcal{O}$-algebra that is a complete commutative
Noetherian local complete intersection ring with maximal ideal
$\mathfrak{m}_{A}$,

  * •

$R$ is a commutative local $A$-algebra,

  * •

$\mathcal{E}:R\to A$ is an $A$-algebra homomorphism, and

  * •

$r_{0}\in R$ is an element the annihilates $\ker(\mathcal{E})$

satisfying the conditions

  * •

$R$ is a free $A$-module of finite rank $d$,

  * •

$R/\mathrm{Ann}_{R}(\ker(\mathcal{E}))$ is a free $A$-module of rank $d-1$,

  * •

$r_{0}$ generates $\mathrm{Ann}_{R}(\ker(\mathcal{E}))$ as an $A$-module, and

  * •

$\mathcal{E}(r_{0})\in A$ is a not a zero divisor.

Given this set up, let $I=\ker(\mathcal{E})$, let
$R^{0}=R/\mathrm{Ann}_{R}(I)$, and let $I^{0}\subset R^{0}$ be the image of
$I$ in $R^{0}$.

######  Remark 2.2.

In applications, $\mathcal{O}$ will be $\mathbb{Z}_{p}$ or a finite extension
of $\mathbb{Z}_{p}$, $A$ will be a ring of diamond operators, $R$ will be a
localization of a Hecke algebra at an Eisenstein maximal ideal, $\mathcal{E}$
will be the action of $R$ on an $A$-family of Eisenstein series, and $r_{0}$
will be a Hecke operator with the property, for a modular form $f$, the
coefficient $a_{1}(r_{0}f)$ of $q$ in the $q$-expansion of $r_{0}f$ equals the
residue of $f$ at a cusp. If $\mathcal{E}$ is the only family of Eisenstein
series supported by $R$, then $I$ is the Eisenstein ideal and $R^{0}$ is
maximal the quotient of $R$ that acts faithfully on cuspforms.

Let $(\mathcal{O},A,R,\mathcal{E},r_{0})$ be a single-Eisenstein Hecke algebra
set up. Note that the map $\mathcal{E}$ induces an isomorphism $R/I\cong A$,
making $A$ into an $R$-module. Moreover, the map $A\to\mathrm{Ann}_{R}(I)$
given by $1\mapsto r_{0}$ is an isomorphism of $R$-modules, so that for all
$r\in R$,

(2.1) $r\cdot r_{0}=\mathcal{E}(r)r_{0}.$

Note also that the action of $R$ on $I$ factors through $R^{0}$, so that if
$r\in R^{0}$ and $y\in I$, then $ry$ is a well-defined element of $I$.

The second key property is that the $p$-adic $L$-function is given as an
element of a module that is dual to the cuspidal Hecke algebra.

######  Definition 2.3.

Let $(\mathcal{O},A,R,\mathcal{E},r_{0})$ be a single-Eisenstein Hecke algebra
set up. Then an _$L$ -symbol set up_ is the data of a triple
$(X,\phi,x,\mathcal{L})$ where

  * •

$X$ is a $R$-module,

  * •

$\phi:X\to A$ is an $A$-linear functional on $X$, with kernel
$X^{0}=\ker(\phi)$,

  * •

$x\in X$ is an element such that $\phi(x)=1$, and

  * •

$\mathcal{L}\in\ker(\phi)\otimes_{\mathcal{O}}\mathcal{O}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$,

satisfying the condition that

  * •

there is an isomorphism $f:X\xrightarrow{\sim}\mathrm{Hom}_{A}(R,A)$ of
$R$-modules sending $X^{0}$ to $\mathrm{Hom}_{A}(R^{0},A)$.

Note that, for all $g\in\mathrm{Hom}_{A}(R,A)$ and $r\in R$, the element
$(r-\mathcal{E}(r))g$ is in $\mathrm{Hom}_{A}(R^{0},A)$ by (2.1). This implies
that, for all $y\in X$, the element $(r-\mathcal{E}(r))y$ is in $X^{0}$, so
that $\phi$ must be an isomorphism of $R$-modules:

$\phi(ry)=\mathcal{E}(r)\phi(y).$

Recall that a local ring is called _Gorenstein_ if it has a dualizing module
that is free of rank $1$. For our purposes, the important facts to remember
are that local complete intersection rings are Gorenstein, and that if $B\to
B^{\prime}$ is a finite flat local ring homomorphism and $M$ is a dualizing
module for $B$, then $\mathrm{Hom}_{B}(B^{\prime},M)$ is a dualizing module
for $B^{\prime}$. In particular, in an $L$-symbol set up, $X$ is a dualizing
module for $R$ and $X^{0}$ is a dualizing module for $R^{0}$.

###  2.2. Algebra around Gorensteinness

Let $(\mathcal{O},A,R,\mathcal{E},r_{0})$ be a single-Eisenstein Hecke algebra
set up. The composite map

$R\xrightarrow{\mathcal{E}}A\to A/\mathcal{E}(r_{0})A$

factors through $R^{0}$. Let $A^{0}=A/\mathcal{E}(r_{0})A$ and let
$\mathcal{E}^{0}:R^{0}\to A^{0}$ denote the induced map. Since the two
quotient maps $R\to R^{0}\xrightarrow{\mathcal{E}^{0}}A^{0}$ and
$R\xrightarrow{\mathcal{E}}A\to A^{0}$ are equal, there is a map to the fiber
product:

$R\to R^{0}\times_{A^{0}}A.$

######  Lemma 2.4.

The map $R\to R^{0}\times_{A^{0}}A$ is an isomorphism of $A$-algebras. In
particular, the natural maps $R^{0}/I^{0}\to A^{0}$ and $I\to I^{0}$ are
isomorphisms of $R$-modules.

###### Proof.

Consider the commutative diagram of $R$-modules with exact rows

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Ar_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{E}}$$\textstyle{R^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{E}^{0}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\mathcal{E}(r_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The leftmost vertical arrow is an isomorphism: it is surjective by definition
and injective since $\mathcal{E}(r_{0})$ is assumed to be a non-zero-divisor.
By the five-lemma, the map $\ker(\mathcal{E})\to\ker(\mathcal{E}^{0})$ is an
isomorphism. This is enough to imply that the map $R\to R^{0}\times_{A^{0}}A$
is an isomorphism. Indeed, the kernel is easily seen to equal $\ker(Ar_{0}\to
A\mathcal{E}(r_{0}))$, which is zero. To see it is surjective, let
$(t^{0},a)\in R^{0}\times_{A^{0}}A$, and choose a lift $t\in R$ of $t$. Since
$\mathcal{E}(t)\equiv\mathcal{E}^{0}(t^{0})\equiv a\mod{A\mathcal{E}(r_{0})}$,
it follows that $\mathcal{E}(t)-a\in A\mathcal{E}(r_{0})$, so
$\mathcal{E}(t)-a=a^{\prime}\mathcal{E}(r_{0})$ for some $a^{\prime}\in A$.
Then $t-a^{\prime}r_{0}$ maps to
$(t^{0},\mathcal{E}(t-a^{\prime}r_{0}))=(t^{0},a)$. ∎

Now let $(X,\phi,x,\mathcal{L})$ be an $L$-symbols set up. The following gives
criteria for $R$ to be Gorenstein.

######  Lemma 2.5.

The following are equivalent:

  1. (1)

$R$ is Gorenstein,

  2. (2)

$X$ is a free $R$-module of rank 1,

  3. (3)

$X$ is cyclic as an $R$-module,

  4. (4)

$X$ is generated by $x$ as an $R$-module,

  5. (5)

The map $I\to X^{0}$ given by $t\mapsto t\cdot x$ is an isomorphism of
$R$-modules.

###### Proof.

The equivalence of (1) and (2) is the definition of Gorenstein, and clearly
(2) implies (3). Assume (3), and let $y\in X$ be a generator of $X$ as an
$R$-module. Let $r\in R$ be such that $ry=x$. Then

$1=\phi(x)=\phi(ry)=\mathcal{E}(r)\phi(y).$

This implies $\mathcal{E}(r)\in A^{\times}$, so, since $R$ is local, $r\in
R^{\times}$. Since $x=ry$, this implies (4). Since $X$ is a faithful
$R$-module, (4) implies (2). Lastly, the equivalence of (4) and (5) follows by
applying the five-lemma to the commutative diagram

(2.2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{E}}$$\scriptstyle{r\mapsto
r\cdot
x}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

∎

It follows from the lemma that if $R$ is Gorenstein, then $I$ is a dualizing
module for $R^{0}$. In particular, if $R$ and $R^{0}$ are both Gorenstein,
then $I$ is principal. We will see that the converse is true as well. First,
we require the following lemma on the structure of $R$ when $I$ is principal.

######  Lemma 2.6.

Assume that $I$ is a principal ideal with a generator $t\in I$. Let $F(X)\in
A[X]$ be the characteristic polynomial of the $A$-linear endomorphism

$R^{0}\to R^{0},\ r\mapsto tr.$

Then there are $A$-algebra isomorphisms

$A[X]/(F(X))\xrightarrow{\sim}R^{0},\ A[X]/(XF(X))\xrightarrow{\sim}R$

given by $X\mapsto t$. Moreover, $F(X)$ is a distinguished polynomial with
$F(0)A=\mathcal{E}(r_{0})A$.

###### Proof.

Let $\phi$ denote the $A$-algebra homomorphism

$A[X]\to R,\ X\mapsto t.$

First note that $\phi$ is surjective. To see this, it suffices, by Nakayama’s
lemma to prove that $\phi\otimes_{A}A/\mathfrak{m}_{A}$ is surjective, so we
may assume that $A$ is a field. In that case, $R$ is a $d$-dimensional algebra
over the field $A$ with maximal ideal $I$, so $I^{d}=0$. Fix $r\in R$ and note
that $r-\mathcal{E}(r)\in I$, so there is $r_{1}\in R$ such that
$r=\mathcal{E}(r)+r_{1}t$. Let $r_{0}=R$ and inductively choose $r_{i}\in R$
such that

$r_{i}=\mathcal{E}(r_{i})+r_{i+1}t.$

Then

$r=\mathcal{E}(r_{0})+\mathcal{E}(r_{1})t+\dots+\mathcal{E}(r_{d-1})t^{d-1},$

so $r$ is in the image of $\phi$.

Composing $\phi$ with the surjection $R\twoheadrightarrow R^{0}$ yields a
surjective map $\phi^{0}:A[X]\to R^{0}$. Then $\phi^{0}$ factors through
$A[X]/(F(X))$ by the Cayley-Hamilton Theorem, and, since $F(X)$ is monic,
$A[X]/(F(X))$ is a free $A$-module of rank equal to the degree of $F(X)$.
Since $R^{0}$ is a free $A$-module of rank equal to $\deg(F)$, this implies
that $\phi^{0}$ is an isomorphism. Since $R^{0}$ is local, it follows that
$F(X)$ is distinguished. The remaining parts follow from the isomorphism
$R\cong R^{0}\times_{A^{0}}A$ of Lemma 2.4. ∎

######  Example 2.7.

Suppose that $R^{0}=A$. Then Lemma 2.4 implies that
$I^{0}=\mathcal{E}(r_{0})A$ and that $I$ is generated by
$r_{0}-\mathcal{E}(r_{0})$. Then there is an isomorphism

$A[X]/(X^{2}-\mathcal{E}(r_{0})X)\xrightarrow{\sim}R$

given by $X\mapsto r_{0}-\mathcal{E}(r_{0})$.

######  Lemma 2.8.

The following are equivalent:

  1. (1)

$I^{0}$ is a principal ideal.

  2. (2)

$I$ is a principal ideal.

  3. (3)

$I$ is a free $R^{0}$-module of rank $1$.

  4. (4)

Both $R$ and $R^{0}$ are local complete intersection rings.

  5. (5)

Both $R$ and $R^{0}$ are Gorenstein.

  6. (6)

There is $t\in I$ such that $t\cdot x$ is a generator of $X^{0}$ as a
$R^{0}$-module.

Furthermore, if all of these statements are true, then (6) is true for all
generators $t$ of $I$ and every such $t$ is a generator.

###### Proof.

By Lemma 2.4, the map $I\to I^{0}$ is an isomorphism, so (1) implies (2).
Since $I$ is a faithful $R^{0}$-module, (2) implies (3), and clearly (3)
implies (1).

Now assume (2). By Lemma 2.6, there are isomorphisms

$R^{0}/\mathfrak{m}_{A}R^{0}\cong k[X]/(X^{d-1}),\ R/\mathfrak{m}_{A}R\cong
k[X]/(X^{d}),$

so $R^{0}/\mathfrak{m}_{A}R^{0}$ and $R/\mathfrak{m}_{A}R$ are visibly local
complete intersection rings. Since $A$ is a local complete intersection and
$A\to R^{0}$ and $A\to R$ are flat, this implies (4) (see [Sta, Tag 09Q7]).
Moreover, (4) implies (5) by general algebra (see [Sta, Tag 0DW6]).

Now assume (5). Since $R^{0}$ is Gorenstein, there is a generator $x^{0}\in
X^{0}$ of $X^{0}$ as an $R^{0}$-module. By Lemma 2.5 (part (5)) implies that
$x^{0}$ is of the form $x^{0}=tx$ for some $t\in I$, proving (6).

Now assume (6). Then the leftmost vertical map in (2.2) is surjective. By the
snake lemma, the center vertical map is also surjective, and, since $X$ is a
faithful $R$-module, this implies that the center vertical map (and hence all
the vertical maps by the 5 lemma) in (2.2) are isomorphisms. By (6), the
composite map

$R^{0}\xrightarrow{r\mapsto rt}I\xrightarrow{a\mapsto ax}X^{0}$

is surjective. Since the second map is an isomorphism, this implies that $t$
generates $I$, proving (2). ∎

###  2.3. Algebra around $L$-functions

Let $(\mathcal{O},A,R,\mathcal{E},r_{0})$ be a single-Eisenstein Hecke algebra
set up and let $(X,\phi,x,\mathcal{L})$ be an $L$-symbol set up for it. In
this section, we define different elements of
$R{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$, under different Gorenstein
hypotheses, which we think of as normalizations of the $L$-function (as in
Section 1.4) associated to the $L$-symbol set up. When both $R$ and $R^{0}$
are Gorenstein, then there are two distinct normalizations and we compare
them.

######  Lemma 2.9.

Suppose that $R$ is Gorenstein. Then there is a unique $L\in
R{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$ such that $\mathcal{L}=L\cdot x$.
Moreover, $L$ is in the subset $I{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}\subset
R{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$.

###### Proof.

By Lemma 2.5, the map

$R{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}\xrightarrow{r\mapsto
rx}X\otimes_{\mathcal{O}}\mathcal{O}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$

is an isomorphism and it induces an isomorphism
$I{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}\to
X^{0}\otimes_{\mathcal{O}}\mathcal{O}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$.
Then $L$ is the preimage of $\mathcal{L}$ under the second isomorphism. ∎

######  Definition 2.10.

If $R$ is Gorenstein, the element $L\in
R{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$ of Lemma 2.9 is called the modular
normalization of the $p$-adic $L$-function.

######  Lemma 2.11.

Suppose that $R^{0}$ is Gorenstein, so that $X^{0}$ is a free $R^{0}$-module
of rank $1$. Then, for each generator $e$ of $X^{0}$, there is a unique
$L^{0}_{e}\in R^{0}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$ such that
$\mathcal{L}=L^{0}_{e}\cdot e$. Moreover, the class of $L^{0}_{e}$ in the
quotient multiplicitive monoid
$R^{0}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}/(R^{0})^{\times}$ is independent
of the choice of generator $e$.

###### Proof.

The first part is clear since
$X^{0}\otimes_{\mathcal{O}}\mathcal{O}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$
is a free $R^{0}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$-module with generator
$e$. If $f$ is another generator for $X^{0}$, then $f=Ue$ for a unique
$U\in(R^{0})^{\times}$, and

$L^{0}_{e}\cdot e=\mathcal{L}=L^{0}_{f}\cdot f=L^{0}_{f}\cdot Ue$

so $L^{0}_{e}=L^{0}_{f}\cdot U$ and the second part follows. ∎

######  Definition 2.12.

If $R^{0}$ is Gorenstein, let $L^{0}$ denote the class of $L^{0}_{e}$ in
$R^{0}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}/(R^{0})^{\times}$ for some choice
of $e$ (Lemma 2.11 says that $L^{0}$ is independent of the choice). The class
$L^{0}$ is called the _cuspidal normalization of the $p$-adic $L$-function_.

If both $R$ and $R^{0}$ are Gorenstein, then there are two different
normalizations $L$ and $L^{0}$. By Lemma 2.8, both $R$ and $R^{0}$ are
Gorenstein if and only if $I$ is principal; the following lemma essentially
says that the two normalizations $L$ and $L^{0}$ differ by a generator of $I$.

######  Lemma 2.13.

Suppose that $I$ is principal.

  1. (1)

There is a unique element $L\in I{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$ such
that $\mathcal{L}=L\cdot x$.

  2. (2)

For each generator $t\in I$, the element $tx\in X^{0}$ is a generator for
$X^{0}$ as a $R^{0}$-module, and there is a unique element $L^{0}_{tx}\in
R^{0}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$ such that
$\mathcal{L}=L^{0}_{tx}\cdot tx$.

  3. (3)

For each generator $t\in I$, there is an equality $L=L^{0}_{tx}\cdot t$ of
elements of $I{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$.

###### Proof.

  1. (1)

By Lemma 2.8, $R$ is Gorenstein, so (1) follows from Lemma 2.9.

  2. (2)

By Lemma 2.8, $tx\in X^{0}$ is a generator for $X^{0}$ as a $T^{0}$-module,
and (2) follows from Lemma 2.11.

  3. (3)

By Lemmas 2.4 and 2.8, the map $R\to I$ given by $1\mapsto t$ factors through
an isomorphism $R^{0}\xrightarrow{\sim}I$. Since $L\in
I{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$, there is a unique element $L_{t}\in
T^{0}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$ such that $L=L_{t}\cdot t$. Then,
by (1), $\mathcal{L}=L_{t}\cdot tx$. On the other hand, by (2), $L^{0}_{tx}$
is the unique element of $T^{0}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$
satisfying $\mathcal{L}=L^{0}_{tx}\cdot tx$. Hence $L_{t}=L^{0}_{tx}$, and (3)
follows.

∎

Note that $R{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$ (and similarly
$R^{0}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$) is a semi-local ring with
components labeled by the characters of $(\mathbb{Z}/p\mathbb{Z})^{\times}$;
choosing a generator of $1+p\mathbb{Z}_{p}$, each component is isomorphic to a
power series ring $R{[\\![}u{]\\!]}$. For $j\in\\{0,\dots,p-2\\}$ and $f\in
R{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$, let $f(\omega^{j})\in
R{[\\![}u{]\\!]}$ denote the image of $f$ in the $\omega^{j}$-component. If
$c:R{[\\![}u{]\\!]}\to R$ and is a $R$-algebra homomorphism, we often write
$f(\omega^{j},c)\in R$ instead of $c(f(\omega^{j}))$, and think of this “as
evaluation at $c$”.

######  Lemma 2.14.

Suppose that $I$ is principal and that there is a $j\in\\{0,\dots,p-2\\}$ and
an $R$-algebra homomorphism $c:R{[\\![}u{]\\!]}\twoheadrightarrow R$ such that
$L(\omega^{j},c)\in I$ is a generator. Let $t=L(\omega^{j},c)$ and let
$L_{tx}^{0}\in R^{0}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$ be as in Lemma
2.13(2). Then:

  1. (1)

$L(\omega^{j})=L(\omega^{j},c)\cdot L_{tx}^{0}(\omega^{j})$, and

  2. (2)

$L_{tx}^{0}(\omega^{j})\in(R^{0}{[\\![}u{]\\!]})^{\times}$.

In particular,
$L^{0}(\omega^{j})\in(R^{0}{[\\![}u{]\\!]})^{\times}/(R^{0})^{\times}$ and
$L(\omega^{j})\equiv L(\omega^{j},c)\pmod{(R^{0}{[\\![}u{]\\!]})^{\times}}$.

###### Proof.

By Lemma 2.13 (3), there is an equality
$L(\omega^{j})=L^{0}_{tx}(\omega^{j})t$., which proves (1). Applying the
homomorphism $c$ yields

$t=L(\omega^{j},c)=c(L^{0}_{tx}(\omega^{j}))t.$

Since $I$ is a free $R^{0}$-module, this implies that
$c(L^{0}_{tx}(\omega^{j}))=1$ and hence that
$L^{0}_{tx}(\omega^{j})\in(R^{0}{[\\![}u{]\\!]})^{\times}$ as
$R^{0}{[\\![}u{]\\!]}$ is local. This proves (2). ∎

###  2.4. Content and $\mu$-invariant

These results about the relationships between the modular and cuspidal
normalizations have implications about their content and $\mu$-invariants. We
first recall the definitions of these concepts.

######  Definition 2.15.

Let $B$ be a commutative ring $B$. The _content_ of a power series $f=\sum
a_{i}(f)u^{i}\in B{[\\![}u{]\\!]}$ is the ideal
$\mathrm{content}_{B}(f)\subset B$ generated by all the coefficients
$a_{i}(f)$; the series $f$ _has unit content_ if $\mathrm{content}_{B}(f)=B$.
Note that, for every $b\in B$, there is an equality

(2.3) $\mathrm{content}_{B}(bf)=b\cdot\mathrm{content}_{B}(f),$

and, in particular, that the content of $f$ depends only on the image of $f$
in $B{[\\![}u{]\\!]}/B^{\times}$.

If $B$ is local and $f$ has unit content, then $a_{i}(f)\in R^{\times}$ for
some $i$, and the _$\lambda$ -invariant_ $\lambda(f)\in\mathbb{Z}$ is defined
as the minimal $i$ such that $a_{i}(f)\in R^{\times}$. Note that $f$ is a unit
if and only if it has unit content and $\lambda(f)=0$.

Finally, if $B$ is a DVR with uniformizer $\varpi$, then the _$\mu$
-invariant_ $\mu(f)\in\mathbb{Z}$ is defined to the unique integer $n$ such
that $\mathrm{content}_{R}(f)=\varpi^{n}R$. In this case, $f$ has unit content
if and only if $\mu(f)=0$.

Note that the values of a power series function are in the content ideal. In
other words, if $c:B{[\\![}u{]\\!]}\to B$ is $B$-algebra homomorphism, then
$c(f)\in\mathrm{content}_{B}(f)$. Indeed, $c(f)=\sum a_{i}(f)c(u)^{i}$ and
each $a_{i}(f)$ is in $\mathrm{content}_{B}(f)$.

Now let $(\mathcal{O},A,R,\mathcal{E},r_{0})$ be a single-Eisenstein Hecke
algebra set up and let $(X,\phi,x,\mathcal{L})$ be an $L$-symbol set up for
it.

######  Lemma 2.16.

Assume that $I$ is principal, fix $j\in\\{0,\dots,p-1\\}$, and let $c\colon
R{[\\![}u{]\\!]}\to~{}R$ be an $R$-algebra homomorphism.

  1. (1)

$L(\omega^{j},c)\in\mathrm{content}_{R}(L(\omega^{j}))$.

  2. (2)

There is an equality
$\mathrm{content}_{R}(L(\omega^{j}))=\mathrm{content}_{R}(L^{0}(\omega^{j}))I$
of ideals in $R$.

  3. (3)

If $L(\omega^{j},c)$ is a generator of $I$, then
$\mathrm{content}_{R}(L(\omega^{j}))=I$ and $L^{0}(\omega^{j})$ is a unit.

###### Proof.

(1) is the general fact that the values of a power series function are in the
content ideal. (2) and (3) follow from Lemmas 2.13 and 2.14, respectively. ∎

To discuss $\mu$-invariants, we must work over a DVR. We now fix an
$\mathcal{O}$-algebra homomorphism

$w:A\to\mathcal{O},$

which, in our applications, will correspond to fixing a weight. Let
$R_{w}=R\otimes_{A,w}\mathcal{O}$ and $I_{w}$ be the image of $I$ in $R_{w}$,
and similarly for $R_{w}^{0}$ and $I_{w}^{0}$.

Assume that $I$ is principal and fix $j\in\\{0,\dots,p-2\\}$. Let $t\in I$ be
a generator and let $F(X)\in A[X]$ be the characteristic polynomial of $t$, as
in Lemma 2.6. Let $F_{w}(X)\in\mathcal{O}[X]$ denote the image of $F(X)$ under
$w$. Let $F_{w}(X)=\prod_{i=1}^{r}F_{w,i}(X)$ be the factorization of
$F_{w}(X)$ in $\mathcal{O}[X]$ into irreducible polynomials, and let
$\mathcal{O}_{w,i}$ be the normalization of $\mathcal{O}[X]/(F_{w,i}(X))$, and
let $\varpi_{w,i}$ be a uniformizer in $\mathcal{O}_{w,i}$ and let $X_{w,i}$
be the image of $X$ in $\mathcal{O}_{w,i}$. Then, by Lemma 2.6, the
normalization of $R^{0}_{w}$ is isomorphic to
$\prod_{i=1}^{r}\mathcal{O}_{w,i}$. For $i=1,\dots,r$, let
$L_{w,i}(\omega^{j})\in\mathcal{O}_{w,i}{[\\![}u{]\\!]}$ be the image of
$L(\omega^{j})$ under the map

$R{[\\![}u{]\\!]}\to R_{w}{[\\![}u{]\\!]}\to\mathcal{O}_{w,i}{[\\![}u{]\\!]}.$

######  Remark 2.17.

In our applications, the irreducible factors $F_{w,i}(X)$ correspond to the
cuspidal eigenforms of weight $w$ that are congruent to the given Eisenstein
series. The rings $\mathcal{O}_{w,i}$ are the valuation ring in their
corresponding Hecke fields, and the elements $L_{w,i}(\omega^{j})$ are their
$p$-adic $L$-functions.

######  Lemma 2.18.

With the notation as in the previous paragraph, suppose there is a surjective
$R$-algebra homomorphism $c:R{[\\![}u{]\\!]}\to R$ such that
$L(\omega^{j},c)=t$.

  1. (1)

There is an equality

$\mathrm{val}_{\varpi}(F_{w}(0))=\sum_{i=1}^{r}\mu(L_{w,i}(\omega^{j})).$

  2. (2)

Suppose that there is an integer $M>0$ such that
$\mathrm{val}_{\varpi}(F_{w}(0))>Mr$. Then there exists an $i$ such that
$\mu(L_{w,i}(\omega^{j}))>M$.

###### Proof.

By Lemma 2.14, there is an equality

$L(\omega^{j})=L(\omega^{j},c)L_{tx}^{0}(\omega^{j})=tL_{tx}^{0}(\omega^{j})$

and $L_{tx}^{0}(\omega^{j})\in R^{0}{[\\![}u{]\\!]}^{\times}$. Since, for each
$i=1,\dots,r$, the map

$R\to R_{w}\to\mathcal{O}_{w,i}$

sends $t$ to $X_{w,i}$, it follows that $L_{w,i}(\omega^{j})=X_{w,i}U_{i}$ for
a unit $U_{i}\in\mathcal{O}_{w,i}{[\\![}u{]\\!]}^{\times}$. In particular, the
$\mu$-invariant of $L_{w,i}(\omega^{j})$ is the valuation of $X_{w,i}$:

$\mu(L_{w,i}(\omega^{j}))=\mathrm{val}_{\varpi_{w,i}}(X_{w,i}).$

Since $\mathcal{O}_{w,i}$ is the normalization of
$\mathcal{O}[X]/(F_{w,i}(X))$, the valuation of $X_{w,i}$ is given by

$\mathrm{val}_{\varpi_{w,i}}(X_{w,i})=\mathrm{val}_{\varpi}(F_{w,i}(0)).$

Combining the last two equalities with the fact that
$F_{w}(0)=\prod_{i}F_{w,i}(0)$ gives

$\mathrm{val}_{\varpi}(F_{w}(0))=\sum_{i=1}^{r}\mathrm{val}_{\varpi}(F_{w,i}(0))=\sum_{i=1}^{r}\mu(L_{w,i}(\omega^{j})),$

which proves (1). Part (2) is clear from (1). ∎

######  Example 2.19.

Suppose that $R^{0}=A$. Then, by Example 2.7, $F(X)=X+\mathcal{E}(r_{0})$, so
$F_{w}(X)=X-w(\mathcal{E}(r_{0}))$ for every choice of $w$, and
$\mu(L_{w}(\omega^{j}))=\mathrm{val}_{\varpi}(w(\mathcal{E}(r_{0})))$.

##  3\. Two-variable $p$-adic $L$-symbols

The singular cohomology groups

$H^{1}(Y_{1}(Np^{r}),\mathbb{Z}_{p})\mbox{~{}~{}and~{}~{}}H^{1}(X_{1}(Np^{r}),\mathbb{Z}_{p})$

can be respectively identified with the homology groups

$H_{1}(Y_{1}(Np^{r}),\\{{\rm
cusps}\\},\mathbb{Z}_{p})\mbox{~{}~{}and~{}~{}}H_{1}(X_{1}(Np^{r}),\mathbb{Z}_{p})$

as in [Sha11, Proposition 3.5]. We will make this identification implicitly
and write $\\{\alpha,\beta\\}_{r}$ for the cohomology class corresponding to
the geodesic connecting $\alpha$ to $\beta$ for $\alpha$ and $\beta$ in
$\mathbb{P}^{1}(\mathbb{Q})$. These groups can also be identified with étale
(co)homology groups, but one has to be careful about Galois actions, since
Poincaré duality has a one-Tate-twist in it—this is all discussed in [Sha11,
Section 3.5].

Write $H^{1}(Y_{1}(Np^{r}),\mathbb{Z}_{p})^{\pm,\mathrm{ord}}$ for the
ordinary subspace of this cohomology group with sign $\pm$ and set
$\\{\alpha,\beta\\}_{r}^{\pm,\mathrm{ord}}$ to be the projection of
$\\{\alpha,\beta\\}_{r}$ to this subspace. Set
$H^{1}_{\Lambda}(Y_{1}(N)):=\displaystyle\varprojlim_{r}H^{1}(Y_{1}(Np^{r}),\mathbb{Z}_{p})^{\mathrm{ord}}$
and analogously define $H^{1}_{\Lambda}(X_{1}(N))$. We can then write down the
two-variable $p$-adic $L$-symbol explictly as follows:

$\mathcal{L}_{p}^{\pm}:=\varprojlim_{r}\left(\sum_{a\in\left(\mathbb{Z}/p^{r}\mathbb{Z}\right)^{\times}}U_{p}^{-r}\\{\infty,a/p^{r}\\}^{\mp,\mathrm{ord}}_{r}\otimes[a]\right)_{r}\in
H^{1}_{\Lambda}(Y_{1}(N))^{\mp}\otimes\mathbb{Z}_{p}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}.$

(The sign change here is intentional: we want to consider $\mathcal{L}_{p}$ as
a _functional_ on cohomology classes (i.e. a homology class), and the sign
change appears because of the one-twist mentioned above in Poincaré duality.
With this convention, $\mathcal{L}_{p}^{\pm}$ is a functional on cohomology
classes of the same sign). By [Oht99, Proposition 4.3.4], we have that
$\\{\infty,a/p^{r}\\}_{r}^{\mathrm{ord}}$ is in
$H^{1}(X_{1}(Np^{r}),\mathbb{Z}_{p})^{\mathrm{ord}}$ and thus
$\mathcal{L}_{p}^{\pm}$ actually lives in
$H^{1}_{\Lambda}(X_{1}(N))^{\mp}\otimes\mathbb{Z}_{p}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$.

Lastly, let $\mathbb{T}$ denote the Hida Hecke algebra (to be defined more
carefully in the following section). Then $H^{1}_{\Lambda}(Y_{1}(N))^{\pm}$ is
a $\mathbb{T}$-module and for any maximal ideal
$\mathfrak{m}\subseteq\mathbb{T}$, we have that
$H^{1}_{\Lambda}(Y_{1}(N))^{\pm}_{\mathfrak{m}}$ is a direct summand of
$H^{1}_{\Lambda}(Y_{1}(N))^{\pm}$. We write
$\mathcal{L}_{p}^{\pm}(\mathfrak{m})$ for the projection of
$\mathcal{L}_{p}^{\pm}$ to
$H^{1}_{\Lambda}(Y_{1}(N))^{\pm}_{\mathfrak{m}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$.

##  4\. The primitive case

In this section, we consider the case Eisenstein families with a _primitive_
tame character. In this case, congruences modulo $p$ between Eisenstein series
and cuspforms arise because $p$ divides an $L$-value, as in [Rib76]. In the
next section, we will consider the case of _trivial_ tame character and
congruences that occur because $p$ divides an Euler-factor, as in [Maz77].
This primitive case is the same setting that was considered in [BP19] and we
obtain similar results. The main novelty is that we highlight the role played
by the two possible normalizations, modular and cuspidal, which allows us to
obtain results and conjectures when the Eisenstein ideal is not assumed to be
principal.

###  4.1. Setup

Let $p\geq 5$ and let $N$ be an integer with $p\nmid N\varphi(N)$. Let
$\mathfrak{H}$ denote the $p$-adic Hida Hecke algebra of tame level
$\Gamma_{1}(N)$. It is an algebra over
$\mathbb{Z}_{p}{[\\![}(\mathbb{Z}/N\mathbb{Z})^{\times}\times\mathbb{Z}_{p}^{\times}{]\\!]}$
generated by $T_{q}$ for primes $q\nmid Np$ and $U_{\ell}$ for $\ell\mid Np$.
The Eisenstein ideal $\mathcal{I}\subset\mathfrak{H}$ is the ideal generated
by $T_{q}-(1+\langle q\rangle q^{-1})$ for $q\nmid Np$ and by $U_{\ell}-1$ for
$\ell\mid Np$.

Let $\mathfrak{m}\subset\mathfrak{H}$ be a (Good Eisen) maximal ideal, in the
sense of [BP19, §3.1], and let $\mathbb{T}_{\mathfrak{m}}$ be the completion
of $\mathfrak{H}$ at $\mathfrak{m}$. This determines a character
$\theta_{\mathfrak{m}}:(\mathbb{Z}/Np\mathbb{Z})^{\times}\to\overline{\mathbb{Q}}_{p}^{\times}$,
as explained in _loc. cit._ , and the (Good Eisen) condition implies that the
tame part $\theta_{\mathfrak{m}}|_{(\mathbb{Z}/N\mathbb{Z})^{\times}}$ of
$\theta_{\mathfrak{m}}$ is primitive. Let $\mathcal{O}_{\mathfrak{m}}$ be the
valuation ring in the $p$-adic field generated by the values of
$\theta_{\mathfrak{m}}$ and let $\varpi\in\mathcal{O}_{\mathfrak{m}}$ be a
uniformizer. Let
$\Lambda_{\mathfrak{m}}=\mathcal{O}_{\mathfrak{m}}{[\\![}T{]\\!]}$; it is a
regular local flat $\mathcal{O}_{\mathfrak{m}}$-algebra, and
$\mathbb{T}_{\mathfrak{m}}$ is a finite flat local $\Lambda_{\mathfrak{m}}$.
By [BP19, Lemma 3.1], there is a $\Lambda_{\mathfrak{m}}$-algebra homomorphism
$\mathrm{Eis}_{\mathfrak{m}}:\mathbb{T}_{\mathfrak{m}}\to\Lambda_{\mathfrak{m}}$
with kernel $\mathcal{I}_{\mathfrak{m}}$ that satisfies
$\mathrm{Ann}_{\mathbb{T}_{\mathfrak{m}}}(\mathcal{I}_{\mathfrak{m}})\cong\Lambda_{\mathfrak{m}}$
and is generated by the Hecke operator $T_{0}$ determined by
$a_{1}(T_{0}f)=a_{0}(f)$.

It is straightforward to see that
$(\mathcal{O},A,R,\mathcal{E},r_{0})=(\mathcal{O}_{\mathfrak{m}},\Lambda_{\mathfrak{m}},\mathbb{T}_{\mathfrak{m}},\mathrm{Eis}_{\mathfrak{m}},T_{0})$
is a single-Eisenstein Hecke algebra set up in the sense of Definition 2.1. We
note that $\mathcal{E}(r_{0})=\mathrm{Eis}_{\mathfrak{m}}(T_{0})$ is the
constant term of the Eisenstein family, which in this case is the Kubota-
Leopoldt series
$L_{p}(\psi_{\mathfrak{m}}^{-1},\kappa)\in\Lambda_{\mathfrak{m}}$ of [BP19,
§3.8].

We seek now to form our $L$-symbol set up as in Definition 2.3. Let
$C_{1}(Np^{r})$ denote the cusps of $X_{1}(Np^{r})$; following [FK12, Section
1.3.2], cusps lying over the cusp $0$ of $X_{0}(Np^{r})$ will be called
_0-cusps_. Let
$\mathcal{C}_{r}=\ker(\mathbb{Z}_{p}[C_{1}(Np^{r})(\mathbb{C})]\stackrel{{\scriptstyle\Sigma}}{{\to}}\mathbb{Z}_{p})$
where $\Sigma$ is the augmentation map. There is an exact sequence

(4.1) $0\to H^{1}(X_{1}(Np^{r},\mathbb{Z}_{p})\to
H^{1}(Y_{1}(Np^{r},\mathbb{Z}_{p})\xrightarrow{\partial}\mathcal{C}_{r}\to 0,$

where $\partial$ is the boundary map at the cusps, which satisfies
$\partial(\\{\alpha,\beta\\}_{r})=\alpha-\beta$. By [Oht03, Proposition
(3.1.2)], the localized inverse limit
$\mathcal{C}_{\mathfrak{m}}=(\varprojlim_{r}(\mathcal{C}_{r}))_{\mathfrak{m}}$
is free of rank one as a $\Lambda$-module, generated by the projection of the
class of $0-\infty$ to the $\mathfrak{m}$-part. Taking inverse limit and
localization of the sequences (4.1) then yields an exact sequence

(4.2) $0\to H^{1}_{\Lambda}(X_{1}(N))_{\mathfrak{m}}\to
H^{1}_{\Lambda}(Y_{1}(N))_{\mathfrak{m}}\xrightarrow{\phi}\Lambda_{\mathfrak{m}}\to
0$

where $\phi(\\{0,\infty\\})=1$. This is the exact sequence of [Oht03, Theorem
(1.5.5) (III)] (see also [FK12, Section 6.2.5] for the description of $\phi$
as the boundary at 0-cusps).

Since $\\{0,\infty\\}\in H^{1}_{\Lambda}(Y_{1}(N))_{\mathfrak{m}}^{-}$, taking
minus-parts of (4.2) yields an exact sequence

(4.3) $0\to H^{1}_{\Lambda}(X_{1}(N))^{-}_{\mathfrak{m}}\to
H^{1}_{\Lambda}(Y_{1}(N))^{-}_{\mathfrak{m}}\xrightarrow{\phi}\Lambda_{\mathfrak{m}}\to
0.$

######  Proposition 4.1.

Taking
$(X,\phi,x,\mathcal{L})=(H^{1}_{\Lambda}(Y_{1}(N))^{-}_{\mathfrak{m}},\phi,\\{0,\infty\\},\mathcal{L}^{+}_{p}(\mathfrak{m}))$
gives an $L$-symbol set up.

###### Proof.

The exact sequence (4.3) implies that
$\ker(\phi)=H^{1}_{\Lambda}(X_{1}(N))^{-}_{\mathfrak{m}}$. By [FK12, §1.7.13
and Proposition 6.3.5], $H^{1}_{\Lambda}(Y_{1}(N))^{-}_{\mathfrak{m}}$ is a
dualizing module for $\mathbb{T}_{\mathfrak{m}}$ and
$H^{1}_{\Lambda}(X_{1}(N))^{-}_{\mathfrak{m}}$ is a dualizing module for
$\mathbb{T}_{\mathfrak{m}}^{0}$. Moreover, these isomorphism are compatible in
the sense that the isomorphism
$H^{1}_{\Lambda}(Y_{1}(N))^{-}_{\mathfrak{m}}\cong\mathrm{Hom}_{\Lambda_{\mathfrak{m}}}(\mathbb{T}_{\mathfrak{m}},\Lambda_{\mathfrak{m}})$
sends $H^{1}_{\Lambda}(X_{1}(N))^{-}_{\mathfrak{m}}$ isomorphically to
$\mathrm{Hom}_{\Lambda_{\mathfrak{m}}}(\mathbb{T}_{\mathfrak{m}}^{0},\Lambda_{\mathfrak{m}})$
(see [Oht03, Diagram (3.5.3)]). Since $\phi(\\{0,\infty\\})=1$ and
$\mathcal{L}^{+}_{p}(\mathfrak{m})\in
H^{1}_{\Lambda}(X_{1}(N))^{-}_{\mathfrak{m}}\otimes_{\mathbb{Z}_{p}}\mathbb{Z}_{p}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$,
this completes the verification. ∎

###  4.2. Results

Applying the results of §2 yields the following theorem.

######  Theorem 4.2.

  1. (1)

If $\mathbb{T}_{\mathfrak{m}}$ is Gorenstein, then there is a unique
$L^{+}_{p}(\mathfrak{m})\in\mathbb{T}_{\mathfrak{m}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$
such that
$\mathcal{L}^{+}_{p}(\mathfrak{m})=L^{+}_{p}(\mathfrak{m})\cdot\\{0,\infty\\}$.
Moreover,
$\mathrm{content}(L^{+}_{p}(\mathfrak{m}))\subseteq\mathcal{I}_{\mathfrak{m}}$.

  2. (2)

Suppose that $\mathbb{T}^{0}_{\mathfrak{m}}$ is Gorenstein, so that
$H^{1}_{\Lambda}(X_{1}(N))^{+}_{\mathfrak{m}}$ is free of rank 1 over
$\mathbb{T}^{0}_{\mathfrak{m}}$. Then for every generator $e$ of
$H^{1}_{\Lambda}(X_{1}(N))^{+}_{\mathfrak{m}}$, there is a unique
$L^{+}_{p}(\mathfrak{m})^{0}_{e}\in\mathbb{T}_{\mathfrak{m}}^{0}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$
such that
$\mathcal{L}^{+}_{p}(\mathfrak{m})=L^{+}_{p}(\mathfrak{m})^{0}_{e}\cdot e$.

  3. (3)

If $\mathcal{I}_{\mathfrak{m}}$ is a principal ideal with generator $t$, then
$\mathbb{T}_{\mathfrak{m}}$ and $\mathbb{T}^{0}_{\mathfrak{m}}$ are Gorenstein
and $H^{1}_{\Lambda}(X_{1}(N))^{+}_{\mathfrak{m}}$ is generated by
$t\\{0,\infty\\}$. Moreover, there is an equality

$L^{+}_{p}(\mathfrak{m})=L^{+}_{p}(\mathfrak{m})^{0}_{t\\{0,\infty\\}}t$

in $\mathcal{I}_{\mathfrak{m}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$.

###### Proof.

Part (1) is Lemma 2.9. Part (2) is Lemma 2.11. Part (3) is Lemma 2.13. ∎

######  Remark 4.3.

Part (3) is a mild refinement of [BP19, Theorem 3.14].

For an integer $k\geq 2$, let
$k:\Lambda_{\mathfrak{m}}\to\mathcal{O}_{\mathfrak{m}}$ be the weight-$k$
specialization, so that
$\mathbb{T}_{\mathfrak{m}}\otimes_{\Lambda_{\mathfrak{m}},k}\mathcal{O}_{\mathfrak{m}}=\mathbb{T}_{\mathfrak{m},k}$,
the Hecke algebra acting on weight-$k$ forms. The normalization of
$\mathbb{T}_{\mathfrak{m},k}$ is
$\mathbb{T}_{\mathfrak{m},k}\to\prod_{i=1}^{r_{k}}\mathcal{O}_{f_{k,i}}$,
where $\\{f_{k,i}~{}|~{}i=1,\dots r_{k}\\}$ is a complete list of Galois-
orbits of modular eigenforms with coefficients in $\overline{\mathbb{Q}}_{p}$
that are residually Eisenstein, and $\mathcal{O}_{f_{k,i}}$ are the valuation
rings in their respective $p$-adic Hecke eigenfields. Let
$\varpi_{f_{k,i}}\in\mathcal{O}_{f_{k,i}}$ be a uniformizer. Recall that the
image of $L^{+}_{p}(\mathfrak{m})$ in
$\mathcal{O}_{f_{k,i}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$ is
$L^{+}_{p}(f_{k,i})$, the one-variable $p$-adic $L$-function of $f_{k,i}$.

We write $L^{+}_{p}(\mathfrak{m},\omega^{j})$ for the projection of
$L^{+}_{p}(\mathfrak{m})$ to the $\omega^{j}$-component of
$\mathbb{T}_{\mathfrak{m}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$ and likewise
write $L^{+}_{p}(\mathfrak{m},\omega^{j})^{0}_{e}$ for the projection of
$L^{+}_{p}(\mathfrak{m})^{0}_{e}$ to the $\omega^{j}$-component of
$\mathbb{T}^{0}_{\mathfrak{m}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$. The
following analyzes the $\omega^{0}$-components of these $p$-adic $L$-functions
and is a mild refinement of the results of [BP19, §3.7].

######  Theorem 4.4.

Suppose that $U_{p}-1$ generates $\mathcal{I}_{\mathfrak{m}}$ and let
$e=(U_{p}-1)\\{0,\infty\\}$. Then

  1. (1)

$L^{+}_{p}(\mathfrak{m},\omega^{0})=L^{+}_{p}(\mathfrak{m},\omega^{0})^{0}_{e}\cdot(U_{p}-1)$
and
$L^{+}_{p}(\mathfrak{m},\omega^{0})^{0}_{e}\in(\mathbb{T}_{\mathfrak{m}}^{0}{[\\![}u{]\\!]})^{\times}$.

  2. (2)

For every weight $k$ and index $i$, the $\mu$\- and $\lambda$-invariants of
$f_{k,i}$ are

$\mu(L^{+}_{p}(f_{k,i},\omega^{0}))=\mathrm{val}_{\varpi_{f_{k,i}}}(a_{p}(f_{k,i})-1)\mbox{
and }\lambda(L^{+}_{p}(f_{k,i},\omega^{0}))=0.$

  3. (3)

For every weight $k$, the sum of the $\mu$-invariants is given by

$\sum_{i=1}^{r_{k}}\mu(L^{+}_{p}(f_{k,i},\omega^{0}))=\mathrm{val}_{\varpi}(L_{p}(\psi_{\mathfrak{m}}^{-1},k)).$

  4. (4)

For every integer $M$, there is a weight $k$ and index $i$ such that

$\mu(L_{p}^{+}(f_{k,i},\omega^{0}))>M.$

###### Proof.

Let $\mathbf{1}:\Lambda_{\mathfrak{m}}\to\mathcal{O}_{\mathfrak{m}}$ be
evaluation at the trivial character. The argument of [BP19, Theorem 3.15]
shows that, if $\mathbb{T}_{\mathfrak{m}}$ is Gorenstein, then

$L^{+}_{p}(\mathfrak{m},\mathbf{1})=L^{+}_{p}(\mathfrak{m},\omega^{0},\mathbf{1})=(U_{p}-1)v$

for $v\in(\mathbb{T}_{\mathfrak{m}})^{\times}$. Thus part (1) follows from
Lemma 2.14 and immediately implies (2), whereas part (3) follows from Lemma
2.18(1). Finally let $M>0$ be an integer, and choose an integer $k$ close
enough to a zero of $L_{p}(\psi_{\mathfrak{m}}^{-1},\kappa)$ that

$\mathrm{val}_{\varpi}(L_{p}(\psi_{\mathfrak{m}}^{-1},k)))>M\mathrm{rank}_{\Lambda}(\mathbb{T}^{0}_{\mathfrak{m}}).$

Since $\mathrm{rank}_{\Lambda}(\mathbb{T}^{0}_{\mathfrak{m}})\geq r_{k}$, part
(4) follows from Lemma 2.18(2). ∎

###  4.3. Conjectures

We make the following conjecture.

######  Conjecture 4.5.

  1. (1)

Suppose that $\mathbb{T}_{\mathfrak{m}}$ is Gorenstein. For each $j$, there is
an equality

$\mathrm{content}_{\Lambda_{\mathfrak{m}}}(L^{+}_{p}(\mathfrak{m},\omega^{j}))=\mathcal{I}_{\mathfrak{m}}.$

  2. (2)

Suppose that $\mathbb{T}^{0}_{\mathfrak{m}}$ is Gorenstein and let $e$ be a
generator of $H^{1}_{\Lambda}(X_{1}(N))^{-}_{\mathfrak{m}}$ as a
$\mathbb{T}^{0}_{\mathfrak{m}}$-module. Then, for each $j$,
$L^{+}_{p}(\mathfrak{m},\omega^{j})^{0}_{e}$ has unit content.

We wish to compare this conjecture with [BP19, Conjecture 3.16]. For a height-
one prime $\mathfrak{p}\subset\mathbb{T}_{\mathfrak{m}}^{0}$, let
$\mathcal{O}_{\mathfrak{p}}$ be the normalization of
$\mathbb{T}^{0}_{\mathfrak{m}}/\mathfrak{p}$ (which is a DVR) and let
$\varpi_{\mathfrak{p}}$ denote a uniformizer in $\mathcal{O}_{\mathfrak{p}}$.
Let
$L^{+}_{p}(\mathfrak{p})^{0}_{e},L^{+}_{p}(\mathfrak{p})\in\mathcal{O}_{\mathfrak{p}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$
denote the images of $L^{+}_{p}(\mathfrak{m})^{0}_{e}$ and
$L^{+}_{p}(\mathfrak{m})$ (supposing they exist). If
$\mathcal{I}_{\mathfrak{m}}$ is generated by $\mathfrak{t}$, let
$\mathfrak{t}_{\mathfrak{p}}$ denote the image of $\mathfrak{t}$ in
$\mathcal{O}_{\mathfrak{p}}$.

######  Conjecture 4.6 (Bellaïche-Pollack).

Suppose that $\mathcal{I}_{\mathfrak{m}}$ is generated by $\mathfrak{t}$.
Then, for every height-one prime
$\mathfrak{p}\subset\mathbb{T}_{\mathfrak{m}}^{0}$, there is an equality
$\mu(L^{+}_{p}(\mathfrak{p},\omega^{j}))=\mathrm{val}_{\varpi_{\mathfrak{p}}}(\mathfrak{t}_{\mathfrak{p}})$
for each $j$.

######  Lemma 4.7.

Suppose that $H^{1}_{\Lambda}(X_{1}(N))^{-}_{\mathfrak{m}}$ is generated by an
element $e$ as a $\mathbb{T}^{0}_{\mathfrak{m}}$-module. The following are
equivalent:

  1. (1)

$L^{+}_{p}(\mathfrak{m},\omega^{j})^{0}_{e}$ has unit content.

  2. (2)

for some height-one prime $\mathfrak{p}\subset\mathbb{T}_{\mathfrak{m}}^{0}$,
the $\mu$-invariant $\mu(L^{+}_{p}(\mathfrak{p},\omega^{j})^{0}_{e})$
vanishes.

  3. (3)

for all height-one primes $\mathfrak{p}\subset\mathbb{T}_{\mathfrak{m}}^{0}$,
the $\mu$-invariant $\mu(L^{+}_{p}(\mathfrak{p},\omega^{j})^{0}_{e})$
vanishes.

###### Proof.

Clear since $\mathbb{T}^{0}_{\mathfrak{m}}$ is a local ring. ∎

######  Lemma 4.8.

Suppose that $\mathcal{I}_{\mathfrak{m}}$ is generated by $\mathfrak{t}$ and
let $e=\mathfrak{t}\\{0,\infty\\}$. Then for every height-one prime
$\mathfrak{p}\subset\mathbb{T}^{0}_{\mathfrak{m}}$, there is an equality

$\mu(L^{+}_{p}(\mathfrak{p},\omega^{j}))=\mu(L^{+}_{p}(\mathfrak{p},\omega^{j})^{0}_{e})+\mathrm{val}_{\varpi_{\mathfrak{p}}}(\mathfrak{t}_{\mathfrak{p}}).$

In particular,
$\mu(L^{+}_{p}(\mathfrak{p},\omega^{j}))\geq\mathrm{val}_{\varpi_{\mathfrak{p}}}(\mathfrak{t}_{\mathfrak{p}})$,
with equality if and only if $\mu(L^{+}_{p}(\mathfrak{p},\omega^{j})^{0}_{e})$
is zero.

###### Proof.

Clear from Lemma 2.13. ∎

######  Proposition 4.9.

Suppose that $\mathcal{I}_{\mathfrak{m}}$ is principal and let
$j_{0}\in\mathbb{Z}/(p-1)\mathbb{Z}$. Then the following are equivalent:

  1. (1)

Conjecture 4.5 (1) for $j=j_{0}$.

  2. (2)

Conjecture 4.5 (2) for $j=j_{0}$.

  3. (3)

Conjecture 4.6 for $j=j_{0}$.

Moreover, if $U_{p}-1$ generates $\mathcal{I}_{\mathfrak{m}}$, then these
conjectures are all true for $j=0$.

###### Proof.

Let $\mathfrak{t}$ be a generator of $\mathcal{I}_{\mathfrak{m}}$ and let
$e=\mathfrak{t}\\{0,\infty\\}$. Theorem 4.2(3) implies
$L^{+}_{p}(\mathfrak{m},\omega^{j_{0}})=\mathfrak{t}\cdot
L^{+}_{p}(\mathfrak{m},\omega^{j_{0}})^{0}_{e}$, so by the multiplicative
property (2.3) of content, it follows that

$\mathrm{content}_{\Lambda_{\mathfrak{m}}}(L^{+}_{p}(\mathfrak{m},\omega^{j_{0}}))=\mathfrak{t}\cdot\mathrm{content}_{\Lambda_{\mathfrak{m}}}(L^{+}_{p}(\mathfrak{m},\omega^{j_{0}})^{0}_{e})=\mathcal{I}_{\mathfrak{m}}\cdot\mathrm{content}_{\Lambda_{\mathfrak{m}}}(L^{+}_{p}(\mathfrak{m},\omega^{j_{0}})^{0}_{e}).$

This makes the equivalence of (1) and (2) clear.

Now assume (2), so that $L^{+}_{p}(\mathfrak{m},\omega^{j_{0}})^{0}_{e}$ has
unit content, and let $\mathfrak{p}\subset\mathbb{T}^{0}_{\mathfrak{m}}$ be a
height-one prime. Then $\mu(L^{+}_{p}(\mathfrak{p},\omega^{j_{0}})^{0}_{e})=0$
by Lemma 4.7. It then follows from Lemma 4.8 that
$\mu(L^{+}_{p}(\mathfrak{m},\omega^{j_{0}}))=\mathrm{val}_{\varpi_{\mathfrak{p}}}(\mathfrak{t}_{\mathfrak{p}})$,
proving (3).

Now assume (3), so that
$\mu(L^{+}_{p}(\mathfrak{m},\omega^{j_{0}}))=\mathrm{val}_{\varpi_{\mathfrak{p}}}(\mathfrak{t}_{\mathfrak{p}})$
for all $\mathfrak{p}$. Then
$\mu(L^{+}_{p}(\mathfrak{p},\omega^{j_{0}})^{0}_{e})$ vanishes by Lemma 4.8,
and so $L^{+}_{p}(\mathfrak{m},\omega^{j_{0}})^{0}_{e}$ has unit content by
Lemma 4.7, proving (2).

For the last claim, if $U_{p}-1$ generates $\mathcal{I}_{\mathfrak{m}}$, then
Theorem 4.4 implies that $L^{+}_{p}(\mathfrak{m},\omega^{0})^{0}_{e}$ is a
unit and Conjecture 4.5 (2) is clear for $j=0$. ∎

##  5\. Mazur case

In this section, we consider the case of tame level $\Gamma_{0}(N)$ for a
prime $N$. This is, in some sense, the opposite of the previous section:
whereas in Section 4 we considered forms with _primitive_ tame character, in
this section we consider _trivial_ tame character. The main difference is that
the constant term of the relevant Eisenstein series are multiplied by an Euler
factor at $N$ and congruences can occur because $p$ divides that Euler factor.
We focus on these kinds of congruences in the most interesting case: when
$N\equiv 1\pmod{p}$. This includes the case considered by Mazur in his
original article [Maz77] on the Eisenstein ideal, and so we refer to this set
up as the “Mazur case”. We make use of recent advances [WWE20, Wak23, WWE21,
Deo23, Lec18, Lec21] about the Gorenstein property for the relevant Hecke
algebras. This allows us to replace the Gorenstein assumptions in the results
of Section 4 with some numerical criteria.

Another difference between this setup and the primitive case considered in
Section 4 is in the way we specify a unique Eisenstein family. In tame level
$\Gamma_{0}(N)$, the space of ordinary Eisenstein families has rank 2. A
generic basis of the space is given by $U_{N}$-eigenforms: one where $U_{N}$
acts by 1 and one where $U_{N}$ acts by $N^{k-1}$. Since $N\equiv 1\pmod{p}$,
these two families are congruent and cannot be separated by localizing a
maximal ideal of the Hecke algebra. To resolve this issue, following an idea
of Ohta [Oht14] as in [WWE21], we replace the $U_{N}$ operator in the Hecke
algebra by the Atkin-Lehner involution $w_{N}$. The two $w_{N}$-eigenvector
ordinary Eisenstein families are not congruent, so there is a unique
Eisenstein family after localizing at a maximal ideal in this new Hecke
algebra.

The following subsection summarizes the results of Appendix A where it is
verified that Hida theory works as expected for these modified Hecke algebras.
After that, we verify that there is a unique ordinary Eisenstein family where
$w_{N}$ acts by $-1$ and we compute the constant term of its $q$-expansion. We
explain how to use Ohta’s results on $\Lambda$-adic Eichler-Shimura for tame
level $\Gamma_{1}(N)$ to prove the same results for tame level
$\Gamma_{0}(N)$. We then establish various numerical criteria that guarantee
that our Hecke algebras are Gorenstein and further ones that guarantee that
$U_{p}-1$ generates the Eisenstein ideal. With these results in hand, we apply
the axiomatic setup of §2 to deduce our main analytic results.

###  5.1. Hida theory with Atkin-Lehner operators

Fix an even integer $k_{0}$ with $0<k_{0}<p-1$. For integers $r\geq 0,k\geq 2$
with $k\equiv k_{0}\bmod{p-1}$, let $\mathfrak{H}_{k,Np^{r}}$ denote
subalgebra of
$\mathrm{End}_{\mathbb{Z}_{p}}(M_{k}(\Gamma_{0}(N)\cap\Gamma_{1}(p^{r}),\mathbb{Z}_{p}))$
generated by operators $T_{q}$ for primes $q\nmid Np$, together with $w_{N}$
and $U_{p}$. Let $\mathfrak{h}_{k,Np^{r}}$ denote the image of
$\mathfrak{H}_{k,Np^{r}}$ in
$\mathrm{End}_{\mathbb{Z}_{p}}(S_{k}(\Gamma_{0}(N)\cap\Gamma_{1}(p^{r}),\mathbb{Z}_{p}))$.
Let $\mathfrak{H}_{k}^{\mathrm{ord}}$ denote the inverse limit over $r$ of the
ordinary part $\mathfrak{H}_{k,Np^{r}}^{\mathrm{ord}}$ of
$\mathfrak{H}_{k,Np^{r}}$, and similarly for
$\mathfrak{h}_{k}^{\mathrm{ord}}$. The main result proven in Appendix A is
that these algebras satisfy the main theorems of Hida theory, just as for the
Hecke algebras with $U_{N}$-operators:

  * •

(independence of weight) $\mathfrak{H}_{k}^{\mathrm{ord}}$ and
$\mathfrak{h}_{k}^{\mathrm{ord}}$ are independent of $k$ (and depend only on
$k_{0}$), and so can be denoted simply by $\mathfrak{H}^{\mathrm{ord}}$ and
$\mathfrak{h}^{\mathrm{ord}}$,

  * •

(freeness over $\Lambda$) the algebras $\mathfrak{H}^{\mathrm{ord}}$ and
$\mathfrak{h}^{\mathrm{ord}}$ are free $\Lambda$-modules of finite rank,

  * •

(duality) the pairing $(f,T)\mapsto a_{1}(Tf)$ is a perfect duality between
$\mathfrak{H}^{\mathrm{ord}}$ and $\Lambda$-adic modular forms (and similarly
for $\mathfrak{h}^{\mathrm{ord}}$ and cuspforms),

  * •

(control) the natural maps
$\mathfrak{H}^{\mathrm{ord}}\to\mathfrak{H}^{\mathrm{ord}}_{k,Np^{r}}$ induce
isomorphisms
$\mathfrak{H}^{\mathrm{ord}}/\omega_{r,k}\to\mathfrak{H}^{\mathrm{ord}}_{k,Np^{r}}$
for a particular $\omega_{r,k}\in\Lambda$ (and similarly for
$\mathfrak{h}^{\mathrm{ord}}$).

###  5.2. Eisenstein series

Let $\mathfrak{m}\subseteq\mathfrak{H}^{\mathrm{ord}}$ denote the maximal
ideal corresponding to the residual representation $1\oplus\omega^{k_{0}-1}$
and which contains $w_{N}+1$. Write $\mathbb{T}_{\mathfrak{m}}$ for the
completion of $\mathfrak{H}^{\mathrm{ord}}$ at $\mathfrak{m}$ and
$\mathbb{T}^{0}_{\mathfrak{m}}$ for the completion of
$\mathfrak{h}^{\mathrm{ord}}$ at $\mathfrak{m}$. Then both
$\mathbb{T}_{\mathfrak{m}}$ and $\mathbb{T}^{0}_{\mathfrak{m}}$ are modules
over $\Lambda=\mathbb{Z}_{p}{[\\![}T{]\\!]}$ the Iwasawa algebra.

Let $E^{\mathrm{ord}}$ denote the family of Eisenstein series whose
specialization to a weight $k\equiv k_{0}\pmod{p-1}$ is
$E^{\mathrm{ord}}_{k}$, the unique ordinary Eisenstein series of weight $k$
and level $p$ whose constant term is $-(1-p^{k-1})\frac{B_{k}}{2k}$. We wish
to promote this family to an eigenfamily of level $Np$ where $w_{N}$ acts by
$-1$. To this end, note that it is easy to compute the action of $w_{N}$ on
$E^{\mathrm{ord}}_{k}$ thought of as a form of level $Np$ as this form is old
at $N$. Indeed, for any form $f$ of level $Np$, we have $w_{N}f=N^{1-k/2}\cdot
f\big{|}_{k}\left(\begin{smallmatrix}Na&b\\\ Np&Nd\end{smallmatrix}\right)$
where this matrix has determinant $N$ and thus

$\displaystyle w_{N}E^{\mathrm{ord}}_{k}(z)$
$\displaystyle=N^{1-k/2}\cdot\left(E^{\mathrm{ord}}_{k}\big{|}_{k}\left(\begin{smallmatrix}Na&b\\\
Np&Nd\end{smallmatrix}\right)\right)(z)$
$\displaystyle=N^{1-k/2}\cdot\left(E^{\mathrm{ord}}_{k}\big{|}_{k}\left(\begin{smallmatrix}a&b\\\
p&Nd\end{smallmatrix}\right)\left(\begin{smallmatrix}N&0\\\
0&1\end{smallmatrix}\right)\right)(z)$
$\displaystyle=N^{1-k/2}\cdot\left(E^{\mathrm{ord}}_{k}\big{|}_{k}\left(\begin{smallmatrix}N&0\\\
0&1\end{smallmatrix}\right)\right)(z)$ $\displaystyle=N^{k/2}\cdot
E^{\mathrm{ord}}_{k}(Nz).$

With this formula in hand, an easy computation shows that
$\mathcal{E}^{\pm}_{k}(q):=E^{\mathrm{ord}}_{k}(q)\pm
N^{k/2}E^{\mathrm{ord}}_{k}(q^{N})$ is a $w_{N}$-eigenform with eigenvalue
$\pm 1$. In particular, there is a unique Eisenstein series in
$M_{k}(\Gamma_{0}(Np))$ where $w_{N}$ acts with sign $\pm 1$. Further note
that the constant term of $\mathcal{E}^{\pm}_{k}(q)$ is given by $-(1\pm
N^{k/2})(1-p^{k-1})\frac{B_{k}}{2k}$.

Let $\mathcal{E}^{-}$ denote the family of Eisenstein series that in weight
$k\equiv k_{0}\pmod{p-1}$ specializes to $\mathcal{E}^{-}_{k}$. Let
$\mathrm{Eis}:\mathbb{T}_{\mathfrak{m}}\to\Lambda$ be the homomorphism
corresponding to $\mathcal{E}^{-}$ and let $\mathcal{I}_{\mathfrak{m}}$ denote
the kernel of this map. As $\mathcal{E}^{-}$ is the unique Eisenstein family
parametrized by $\mathbb{T}_{\mathfrak{m}}$, the ideal
$\mathrm{Ann}_{\mathbb{T}_{\mathfrak{m}}}(\mathcal{I}_{\mathfrak{m}})$ is free
of rank one as a $\Lambda$-module, is generated by the Hecke operator $T_{0}$
determined by $a_{1}(T_{0}f)=a_{0}(f)$, and
$\mathbb{T}^{0}_{\mathfrak{m}}=\mathbb{T}_{\mathfrak{m}}/T_{0}\mathbb{T}_{\mathfrak{m}}$.

Taken together, all this implies that
$(\mathcal{O},A,R,\mathcal{E},r_{0})=(\mathbb{Z}_{p},\Lambda,\mathbb{T}_{\mathfrak{m}},\mathrm{Eis},T_{0})$
is a single-Eisenstein Hecke algebra set up. Further, the element
$\xi=\mathrm{Eis}(T_{0})$ corresponds to the constant term of
$\mathcal{E}^{-}$ which is $\zeta_{p,k_{0}}\cdot(1-\langle N\rangle^{1/2})$
where $\zeta_{p,k_{0}}\in\Lambda$ is the $\omega^{k_{0}}$-branch of the
$p$-adic $\zeta$-function and $\langle N\rangle\in\Lambda$ is the element
which specializes to $N^{k}$ in weight $k$.

###  5.3. $\Lambda$-adic Eichler-Shimura for tame level $\Gamma_{0}(N)$

Consider the $\Lambda$-adic étale cohomology groups
$H^{1}_{\Lambda}(Y_{0}(N))$ and $H^{1}_{\Lambda}(X_{0}(N))$ with tame level
$\Gamma_{0}(N)$, defined as

$H^{1}_{\Lambda}(Y_{0}(N))=\varprojlim
H^{1}(Y(\Gamma_{0}(N)\cap\Gamma_{1}(p^{r})),\mathbb{Z}_{p})^{\mathrm{ord},(k_{0})},$

and similarly for $X_{0}(N)$, where the superscript ${(k_{0})}$ means the
$\omega^{k_{0}}$-eigenspace for the diamond-operator action of
$(\mathbb{Z}/p\mathbb{Z})^{\times}$. In this section, we use Ohta’s results on
the structure of $H^{1}_{\Lambda}(Y_{1}(N))$ and $H^{1}_{\Lambda}(X_{1}(N))$
to deduce analogous results for $H^{1}_{\Lambda}(Y_{0}(N))$ and
$H^{1}_{\Lambda}(X_{0}(N))$. The main input is the following result about the
structure of $H^{1}_{\Lambda}(Y_{1}(N))$ as a $\mathbb{Z}_{p}[\Delta]$-module,
where
$\Delta=\Gamma_{0}(N)/\Gamma_{1}(N)\cong(\mathbb{Z}/N\mathbb{Z})^{\times}$ is
the group of diamond operators of level $N$.

######  Lemma 5.1.

The $\Lambda$-adic cohomology group $H^{1}_{\Lambda}(Y_{1}(N))$ is a
projective $\mathbb{Z}_{p}[\Delta]$-module, and the natural maps

$H^{1}_{\Lambda}(Y_{1}(N))_{\Delta}\to H^{1}_{\Lambda}(Y_{0}(N))\to
H^{1}_{\Lambda}(Y_{1}(N))^{\Delta}$

are isomorphisms.

###### Proof.

It is enough to prove the result at level $\Gamma_{1}(Np^{r})$ for fixed $r$;
taking inverse limits gives the result for $\Lambda$-adic cohomology. Let
$Y_{1}=Y_{1}(Np^{r})$ and $Y_{0}=Y(\Gamma_{0}(N)\cap\Gamma_{1}(p^{r}))$.

First note that $H^{0}(Y_{1},\mathbb{Z}_{p})^{\mathrm{ord}}$ is zero because
$U_{p}$ acts by $p$ on $H^{0}(Y_{1},\mathbb{Z}_{p})$. Then the following are
evident:

  1. (1)

$Y_{1}\to Y_{0}$ is an étale covering with Galois group $\Delta$,

  2. (2)

the cohomology $H^{i}(Y_{1},\mathbb{Z}_{p})^{\mathrm{ord}}$ is only supported
in degree $i=1$, and

  3. (3)

$H^{1}(Y_{1},\mathbb{Z}_{p})^{\mathrm{ord}}$ is $p$-torsion-free.

These three facts are enough to imply that
$H^{1}(Y_{1},\mathbb{Z}_{p})^{\mathrm{ord}}$ is a projective
$\mathbb{Z}_{p}[\Delta]$-module, as follows. By the comparison isomorphism
between étale and Betti cohomology, it is enough to show that the singular
cohomology is projective. Let $C^{\bullet}(Y_{1},\mathbb{Z}_{p})$ be the
complex of singular cochains on $Y_{1}$; by (1), it is a bounded complex of
flat $\mathbb{Z}_{p}[\Delta]$-modules. Hecke operators act on
$C^{\bullet}(Y_{1},\mathbb{Z}_{p})$, and, since the ordinary projector is an
idempotent, the complex $C^{\bullet}(Y_{1},\mathbb{Z}_{p})^{\mathrm{ord}}$ is
still a bounded complex of flat $\mathbb{Z}_{p}[\Delta]$-modules and its
cohomology groups are $H^{i}(Y_{1},\mathbb{Z}_{p})^{\mathrm{ord}}$. Hence,
there is a perfect complex $C^{\bullet}$ of $\mathbb{Z}_{p}[\Delta]$-modules
and a quasi-isomorphism

$C^{\bullet}\simeq C^{\bullet}(Y_{1},\mathbb{Z}_{p})^{\mathrm{ord}}$

(see [Mum08, Lemma II.5.1, pg. 47]). By (2), this implies that
$H^{1}(Y_{1},\mathbb{Z}_{p})^{\mathrm{ord}}$ has finite projective dimension
over $\mathbb{Z}_{p}[\Delta]$, which, together with (3), implies that it is
projective (see [Bro94, Theorems 8.10 and 8.12, pg. 152], for instance, or use
the Auslander-Buchsbaum formula).

The composition of the two maps

$H^{1}(Y_{1},\mathbb{Z}_{p})^{\mathrm{ord}}_{\Delta}\to
H^{1}(Y_{0},\mathbb{Z}_{p})^{\mathrm{ord}}\to(H^{1}(Y_{1},\mathbb{Z}_{p})^{\mathrm{ord}})^{\Delta}$

is equal to the map induced by multiplication by the norm element of
$\mathbb{Z}_{p}[\Delta]$ on $H^{1}(Y_{1},\mathbb{Z}_{p})^{\mathrm{ord}}$,
which is an isomorphism since $H^{1}(Y_{1},\mathbb{Z}_{p})^{\mathrm{ord}}$ is
$\mathbb{Z}_{p}[\Delta]$-projective. The map
$H^{1}(Y_{0},\mathbb{Z}_{p})^{\mathrm{ord}}\to(H^{1}(Y_{1},\mathbb{Z}_{p})^{\mathrm{ord}})^{\Delta}$
is clearly injective, so this completes the proof. ∎

This lemma allows us to to prove the following analog of Ohta’s $\Lambda$-adic
Eichler-Shimura isomorphisms.

######  Proposition 5.2.

There are split-exact sequences of $\mathbb{T}_{\mathfrak{m}}$-modules

$\displaystyle 0\to\mathbb{T}_{\mathfrak{m}}^{0}\to
H^{1}_{\Lambda}(Y_{0}(N))_{\mathfrak{m}}\to\mathrm{Hom}_{\Lambda}(\mathbb{T}_{\mathfrak{m}},\Lambda)\to
0$ $\displaystyle 0\to\mathbb{T}_{\mathfrak{m}}^{0}\to
H^{1}_{\Lambda}(X_{0}(N))_{\mathfrak{m}}\to\mathrm{Hom}_{\Lambda}(\mathbb{T}_{\mathfrak{m}}^{0},\Lambda)\to
0.$

###### Proof.

Let $\mathfrak{H}^{\mathrm{ord}}_{1}$ be the $p$-adic Hida Hecke algebra of
tame level $\Gamma_{1}(N)$ and let $\mathfrak{h}^{\mathrm{ord}}_{1}$ be its
cuspidal quotient. Let $m_{\Lambda,1}$ and $S_{\Lambda,1}$ be the spaces of
$\Lambda$-adic modular forms and cuspforms, respectively, of tame level
$\Gamma_{1}(N)$, and let $m_{\Lambda}$ and $S_{\Lambda}$ be the corresponding
spaces of tame level $\Gamma_{0}(N)$.

Ohta’s $\Lambda$-adic Eichler-Shimura isomorphisms [Oht99, Oht00] imply that
there are exact sequences of $\mathfrak{H}_{1}^{\mathrm{ord}}$-modules

(5.1) $\displaystyle 0\to\mathfrak{h}_{1}^{\mathrm{ord}}\to
H^{1}_{\Lambda}(Y_{1}(N))\to m_{\Lambda,1}\to 0$ (5.2) $\displaystyle
0\to\mathfrak{h}_{1}^{\mathrm{ord}}\to H^{1}_{\Lambda}(X_{1}(N))\to
S_{\Lambda,1}\to 0.$

These sequences are defined using the action of $G_{\mathbb{Q}_{p}}$ and since
$k_{0}\neq 1$ (because $k_{0}$ is even) the actions on the sub and quotient
are distinguished, so the sequences split as
$\mathfrak{H}_{1}^{\mathrm{ord}}$-modules (see [Oht03, Section 3.4] and [FK12,
Section 6.3.12]); a fortiori, they split as $\mathbb{Z}_{p}[\Delta]$-modules
(since the diamond operators are in $\mathfrak{H}_{1}^{\mathrm{ord}}$). Lemma
5.1 and the splitting of (5.1) imply that $\mathfrak{h}_{1}^{\mathrm{ord}}$
and $m_{\Lambda,1}$ are $\mathbb{Z}_{p}[\Delta]$-projective. Since the dual of
a projective $\mathbb{Z}_{p}[\Delta]$-module is projective, this implies that
$\mathfrak{H}_{1}^{\mathrm{ord}}$ and $S_{\Lambda,1}$ are projective
$\mathbb{Z}_{p}[\Delta]$-modules. Then the sequence (5.2) implies that
$H^{1}_{\Lambda}(X_{1}(N))$ is also $\mathbb{Z}_{p}[\Delta]$-projective. Just
as in the proof of Lemma 5.1, this implies that the natural map

$H^{1}_{\Lambda}(X_{0}(N))\to H^{1}_{\Lambda}(X_{1}(N))^{\Delta}$

is an isomorphism. Hence taking $\Delta$-invariants of the split-exact
sequences (5.1) and (5.2) yields exact sequences

(5.3) $\displaystyle 0\to(\mathfrak{h}_{1}^{\mathrm{ord}})^{\Delta}\to
H^{1}_{\Lambda}(Y_{0}(N))\to(m_{\Lambda,1})^{\Delta}\to 0$ (5.4)
$\displaystyle 0\to(\mathfrak{h}_{1}^{\mathrm{ord}})^{\Delta}\to
H^{1}_{\Lambda}(X_{0}(N))\to(S_{\Lambda,1})^{\Delta}\to 0.$

Now note that the natural maps $m_{\Lambda}\to(m_{\Lambda,1})^{\Delta}$ and
$S_{\Lambda}\to(S_{\Lambda,1})^{\Delta}$ are isomorphisms because a modular
form of level $\Gamma_{1}(N)$ that is invariant under the diamond operators is
also a form of level $\Gamma_{0}(N)$. This implies that the dual map

(5.5)
$\mathrm{Hom}_{\Lambda}(S_{\Lambda,1},\Lambda)_{\Delta}\to\mathrm{Hom}_{\Lambda}(S_{\Lambda},\Lambda),$

is an isomorphism. But, by duality (see Theorem A.3 and Theorem A.9), there
are isomorphisms
$\mathrm{Hom}_{\Lambda}(S_{\Lambda,1},\Lambda)\cong\mathfrak{h}_{1}^{\mathrm{ord}}$
and
$\mathrm{Hom}_{\Lambda}(S_{\Lambda},\Lambda)\cong\mathfrak{h}^{\mathrm{ord}}$.
Moreover, since $\mathfrak{h}_{1}^{\mathrm{ord}}$ is
$\mathbb{Z}_{p}[\Delta]$-free, the norm map induces an isomorphism

$(\mathfrak{h}_{1}^{\mathrm{ord}})_{\Delta}\cong(\mathfrak{h}_{1}^{\mathrm{ord}})^{\Delta}.$

Combining this isomorphism with the duality isomorphisms and (5.5) gives a
string of isomorphisms

$(\mathfrak{h}_{1}^{\mathrm{ord}})^{\Delta}\cong(\mathfrak{h}_{1}^{\mathrm{ord}})_{\Delta}\cong\mathrm{Hom}_{\Lambda}(S_{\Lambda,1},\Lambda)_{\Delta}\cong\mathrm{Hom}_{\Lambda}(S_{\Lambda},\Lambda)\cong\mathfrak{h}^{\mathrm{ord}}.$

Hence (5.3) and (5.4) are isomorphic to

(5.6) $\displaystyle 0\to\mathfrak{h}^{\mathrm{ord}}\to
H^{1}_{\Lambda}(Y_{0}(N))\to m_{\Lambda}\to 0$ (5.7) $\displaystyle
0\to\mathfrak{h}^{\mathrm{ord}}\to H^{1}_{\Lambda}(X_{0}(N))\to S_{\Lambda}\to
0.$

Localizing at $\mathfrak{m}$ then completes the proof. ∎

Just as in Section 4.1, there is an exact sequence

$0\to H^{1}_{\Lambda}(X_{0}(N))\to
H^{1}_{\Lambda}(Y_{0}(N))\xrightarrow{\phi}\Lambda\to 0.$

where $\phi(\\{0,\infty\\})=1$. Since $\\{0,\infty\\}\in
H^{1}_{\Lambda}(Y_{0}(N))^{-}$, this implies that there is an exact sequence

$0\to H^{1}_{\Lambda}(X_{0}(N))^{-}\to
H^{1}_{\Lambda}(Y_{0}(N))^{-}\xrightarrow{\phi}\Lambda\to 0.$

######  Proposition 5.3.

Taking
$(X,\phi,x,\mathcal{L})=(H^{1}_{\Lambda}(Y_{0}(N))^{-},\phi,\\{0,\infty\\},\mathcal{L}_{p}^{+}(\mathfrak{m}))$
gives an $L$-symbol set up.

###### Proof.

Just as in the proof of Proposition 4.1, given Proposition 5.2. ∎

###  5.4. Gorenstein results and generators of the Eisenstein ideal

In this section, we give some numerical criteria for when the Eisenstein ideal
$\mathcal{I}_{\mathfrak{m}}$ is principal (and thus
$\mathbb{T}_{\mathfrak{m}}$ and $\mathbb{T}^{0}_{\mathfrak{m}}$ are
Gorenstein), when $\mathcal{I}_{\mathfrak{m}}$ is generated by $U_{p}-1$, and
when $\mathbb{T}^{0}_{\mathfrak{m}}$ has rank 1 over $\Lambda$.

Let $\mathbb{T}^{\rm tame}_{\mathfrak{m},k}$ and $\mathbb{T}_{\mathfrak{m},k}$
denote the respective Hecke algebras of $M_{k}(\Gamma_{0}(N))^{\mathrm{ord}}$
and $M_{k}(\Gamma_{0}(Np))^{\mathrm{ord}}$ and let $\mathbb{T}^{\rm
0,tame}_{\mathfrak{m},k}$ and $\mathbb{T}^{0}_{\mathfrak{m},k}$ denote the
respective Hecke algebras of $S_{k}(\Gamma_{0}(N))^{\mathrm{ord}}$ and
$S_{k}(\Gamma_{0}(Np))^{\mathrm{ord}}$. Here all of these Hecke algebras are
defined to contain $w_{N}$ rather than $U_{N}$.

######  Theorem 5.4.

If $k>2$, there are isomorphisms
$\mathbb{T}_{\mathfrak{m},k}\cong\mathbb{T}^{\rm tame}_{\mathfrak{m},k}$ and
$\mathbb{T}^{0}_{\mathfrak{m},k}\cong\mathbb{T}^{\rm
0,tame}_{\mathfrak{m},k}$. For $k=2$, the same conclusions hold when $p$ is
not a $p$-th power modulo $N$.

###### Proof.

This result is easy is $k\neq 2$. Indeed, an eigenform $f$ of weight $k$ which
is $p$-new has $a_{p}(f)=\pm p^{\frac{k-2}{2}}$ and thus is ordinary if and
only if $k=2$. In particular, when $k\neq 2$, none of these ordinary Hecke
algebras grow from level $N$ to level $Np$.

The case of $k=2$ is much deeper and follows from the results of the second
author and Wang-Erickson, namely [WWE21, Theorem 1.4.5 and Proposition A.3.1]
and rely on the hypothesis that $p$ is not a $p$-th power modulo $N$. ∎

Returning to the setting of §5.2, we have $\mathbb{T}_{\mathfrak{m}}$ and
$\mathbb{T}^{0}_{\mathfrak{m}}$ are the localized Hida Hecke algebras
corresponding to the residual representation $1\oplus\omega^{k_{0}-1}$ and we
now give criteria for when these rings are Gorenstein. To this end, let $g$
denote a generator of $\mathbb{F}_{N}^{\times}$ and define
$\log_{N}:\mathbb{F}_{N}^{\times}\to\mathbb{F}_{p}$ by
$\log_{N}(g^{a})=a\pmod{p}$. Note that this map is well-defined as $a$ is
defined modulo $N-1$ and $p\mid N-1$. Further, fix $\zeta_{p}$ a $p$-th root
of unity in $\mathbb{F}_{N}^{\times}$.

######  Theorem 5.5.

Assume that both of the following two conditions hold:

  1. (1)

$p\nmid B_{k_{0}}$, and

  2. (2)

$\displaystyle\sum_{i=1}^{p-1}i^{k_{0}-2}\log_{N}(1-\zeta_{p}^{i})\neq
0\text{~{}in~{}}\mathbb{F}_{p}$.

Then $\mathcal{I}_{\mathfrak{m}}$ is a principal ideal. In particular,
$\mathbb{T}_{\mathfrak{m}}$ and $\mathbb{T}^{0}_{\mathfrak{m}}$ are Gorenstein
and $\mathbb{T}_{\mathfrak{m},k}$ and $\mathbb{T}^{0}_{\mathfrak{m},k}$ are
Gorenstein for all $k\equiv k_{0}\pmod{p-1}$.

###### Proof.

The results of [Deo23] imply that $\mathcal{I}_{\mathfrak{m}}$ is principal
(see, in particular, [Deo23, Remark 5.7]). Once we know
$\mathcal{I}_{\mathfrak{m}}$ is principal, then Lemma 2.8 implies that
$\mathbb{T}_{\mathfrak{m}}$ and $\mathbb{T}^{0}_{\mathfrak{m}}$ are Gorenstein
and Theorem A.9 implies that $\mathbb{T}_{\mathfrak{m},k}$ and
$\mathbb{T}^{0}_{\mathfrak{m},k}$ are Gorenstein for all $k\equiv
k_{0}\pmod{p-1}$. ∎

######  Remark 5.6.

We note that if $k_{0}\equiv 2\pmod{p-1}$, then

$\sum_{i=1}^{p-1}i^{k_{0}-2}\log_{N}(1-\zeta_{p}^{i})=\sum_{i=1}^{p-1}\log_{N}(1-\zeta_{p}^{i})=\log_{N}\left(\prod_{i=1}^{p=1}1-\zeta_{p}^{i}\right)=\log_{N}(p).$

In particular, when $k_{0}\equiv 2\pmod{p-1}$, the hypotheses of Theorem 5.5
reduce simply to asking that $p$ is not a $p$-th power modulo $N$

We now give criteria for when $U_{p}-1$ generates the Eisenstein ideal.

######  Theorem 5.7.

Assume the hypotheses of Theorem 5.5. Then $p$ is not a $p$-th power modulo
$N$ if and only if $U_{p}-1$ generates the Eisenstein ideal.

###### Proof.

First note that $U_{p}-1$ generates the Eisenstein ideal
$\mathcal{I}_{\mathfrak{m}}$ of $\mathbb{T}_{\mathfrak{m}}$ if and only if
$U_{p}-1$ generates the Eisenstein ideal $\mathcal{I}_{\mathfrak{m},k}$ of
$\mathbb{T}_{\mathfrak{m},k}$ for one (equivalently any) $k\equiv
k_{0}\pmod{p-1}$. Thus it suffices to work in $\mathbb{T}_{\mathfrak{m},k}$ to
establish the above theorem.

To this end, choose $k\equiv k_{0}\pmod{p-1}$ with $k>2$. By Theorem 5.5,
$\mathcal{I}_{\mathfrak{m},k}$ is principal, and hence there is a surjective
homomorphism $\mathbb{Z}_{p}[x]\to\mathbb{T}_{\mathfrak{m},k}$ sending $x$ to
a generator of $\mathcal{I}_{\mathfrak{m},k}$. This map induces an isomorphism
$\mathbb{T}_{\mathfrak{m},k}/p\mathbb{T}_{\mathfrak{m},k}\cong\mathbb{F}_{p}[x]/(x^{r})$
for some $r\geq 1$. On the other hand, [Wak23, Proposition 4.4.2] gives a
surjective homomorphism
$\phi:\mathbb{T}_{\mathfrak{m},k}\to\mathbb{F}_{p}[\epsilon]/(\epsilon^{2})$
such that

$\displaystyle T_{\ell}$ $\displaystyle\mapsto
1+\ell^{k_{0}-1}+\epsilon(\ell^{k_{0}-1}-1)\log_{N}(\ell),$ $\displaystyle
U_{p}$ $\displaystyle\mapsto 1-\epsilon\log_{N}(p).$

Hence, an element $t\in\mathcal{I}_{\mathfrak{m},k}$ is a generator if and
only if $\phi(t)\neq 0$. It follows that $U_{p}-1$ generates
$\mathcal{I}_{\mathfrak{m},k}$ if only if $p$ is not a $p$-th power modulo
$N$. ∎

######  Remark 5.8.

We note that when $k_{0}\equiv 2\pmod{p-1}$, all of the hypotheses of Theorem
5.7 again reduce to simply assuming that $p$ is not a $p$-th power modulo $N$.

Lastly, we give a criteria for when $\mathbb{T}^{0}_{\mathfrak{m}}$ has rank 1
over $\Lambda$.

######  Theorem 5.9.

Assume the hypotheses of Theorem 5.7 and that

(5.8) $\prod_{i=1}^{N-1}i^{\left(\sum_{j=1}^{i-1}j^{k_{0}-1}\right)}$

is not a $p$-th power modulo $N$. Then $\mathbb{T}^{0}_{\mathfrak{m}}$ has
rank 1 over $\Lambda$.

###### Proof.

By Theorem A.9, to compute the rank of $\mathbb{T}^{0}_{\mathfrak{m}}$ over
$\Lambda$, it suffices to compute the rank of
$\mathbb{T}^{0}_{\mathfrak{m},k}$ over $\mathbb{Z}_{p}$ for any $k\equiv
k_{0}\pmod{p-1}$. The theorem then follows from [Deo23, Corollary B]. ∎

######  Remark 5.10.

In [Lec18, pg. 36], it is verified that when $k_{0}\equiv 2\pmod{p-1}$ the
quantity in (5.8) is a $p$-th power modulo $N$ if and only if Merel’s number
$\prod_{i=1}^{\frac{N-1}{2}}i^{i}$ is a $p$-th power modulo $N$.

###  5.5. Results

By Proposition 5.3,
$(H^{1}_{\Lambda}(Y_{0}(N))^{-},\phi,\\{0,\infty\\},\mathcal{L}_{p}^{+}(\mathfrak{m}))$
is an $L$-symbol set up for the single-Eisenstein Hecke algebra set up
$(\mathbb{Z}_{p},\Lambda,\mathbb{T}_{\mathfrak{m}},\mathcal{E}^{-},T_{0})$.
Applying the results of Section 2 to this set up, we obtain a theorem which is
essentially identical to Theorem 4.2:

######  Theorem 5.11.

  1. (1)

If $\mathbb{T}_{\mathfrak{m}}$ is Gorenstein, then there is a unique
$L^{+}_{p}(\mathfrak{m})\in\mathbb{T}_{\mathfrak{m}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$
such that
$\mathcal{L}^{+}_{p}(\mathfrak{m})=L^{+}_{p}(\mathfrak{m})\cdot\\{0,\infty\\}$.
Moreover,
$\mathrm{content}(L^{+}_{p}(\mathfrak{m}))\subseteq\mathcal{I}_{\mathfrak{m}}$.

  2. (2)

Suppose that $\mathbb{T}^{0}_{\mathfrak{m}}$ is Gorenstein, so that
$H^{1}_{\Lambda}(X_{0}(N))^{+}_{\mathfrak{m}}$ is free of rank 1 over
$\mathbb{T}^{0}_{\mathfrak{m}}$. Then, for every generator $e$ of
$H^{1}_{\Lambda}(X_{0}(N))^{+}_{\mathfrak{m}}$, there is a unique
$L^{+}_{p}(\mathfrak{m})^{0}_{e}\in\mathbb{T}_{\mathfrak{m}}^{0}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$
such that
$\mathcal{L}^{+}_{p}(\mathfrak{m})=L^{+}_{p}(\mathfrak{m})^{0}_{e}\cdot e$.

  3. (3)

If $\mathcal{I}_{\mathfrak{m}}$ is a principal ideal with generator $t$, then
$\mathbb{T}_{\mathfrak{m}}$ and $\mathbb{T}^{0}_{\mathfrak{m}}$ are Gorenstein
and $H^{1}_{\Lambda}(X_{0}(N))^{+}_{\mathfrak{m}}$ is generated by
$t\\{0,\infty\\}$. Moreover, there is an equality

$L^{+}_{p}(\mathfrak{m})=L^{+}_{p}(\mathfrak{m})^{0}_{t\\{0,\infty\\}}t$

in $\mathcal{I}_{\mathfrak{m}}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$.

Note that Theorem 5.5 gives criteria for when the hypothesis of part (3)
holds. We now move on to the case where $U_{p}-1$ generates the Eisenstein
ideal. We use notation analogous to that used in Section 4.2. In particular,
for a fixed weight $k$, the forms $f_{k,1},\dots,f_{k,r_{k}}$ are a complete
list of Galois-conjugacy-classes of eigenforms for
$\mathbb{T}_{\mathfrak{m},k}$ and $\varpi_{f_{k,i}}$ denotes a uniformizer in
the $p$-adic Hecke field of $f_{k,i}$.

######  Theorem 5.12.

Let $e=(U_{p}-1)\\{0,\infty\\}$. Assume that all of the following three
conditions hold:

  1. (a)

$p\nmid B_{k_{0}}$,

  2. (b)

$\displaystyle\sum_{i=1}^{p-1}i^{k_{0}-2}\log_{N}(1-\zeta_{p}^{i})\neq 0$ in
$\mathbb{F}_{p}$, and

  3. (c)

$p$ is not a $p$-th power modulo $N$.

Then the following four statements are all true:

  1. (1)

$L^{+}_{p}(\mathfrak{m},\omega^{0})=L^{+}_{p}(\mathfrak{m},\omega^{0})^{0}_{e}\cdot(U_{p}-1)$
and
$L^{+}_{p}(\mathfrak{m},\omega^{0})^{0}_{e}\in(\mathbb{T}_{\mathfrak{m}}^{0}{[\\![}u{]\\!]})^{\times}$.

  2. (2)

For every weight $k$ and index $i$, the $\mu$\- and $\lambda$-invariants of
$f_{k,i}$ are

$\mu(L^{+}_{p}(f_{k,i},\omega^{0}))=\mathrm{val}_{\varpi_{f_{k,i}}}(a_{p}(f_{k,i})-1)\mbox{
and }\lambda(L^{+}_{p}(f_{k,i},\omega^{0}))=0.$

  3. (3)

For every weight $k$, the sum of the $\mu$-invariants is given by

$\sum_{i=1}^{r_{k}}\mu(L^{+}_{p}(f_{k,i},\omega^{0}))=\mathrm{val}_{p}(N-1)+\mathrm{val}_{p}(k).$

  4. (4)

For every integer $M$ and every weight $k$ such that
$\mathrm{val}_{p}(k)>M\mathrm{rank}_{\Lambda}(\mathbb{T}^{0}_{\mathfrak{m}})$,
there exists an index $i$ such that

$\mu(L^{+}_{p}(f_{k,i},\omega^{0}))>M.$

If, in addition,

$\prod_{i=1}^{N-1}i^{\left(\sum_{j=1}^{i-1}j^{k_{0}-1}\right)}\mbox{ is not a
}p\mbox{-th power},$

then $\operatorname{rank}_{\Lambda}\mathbb{T}^{0}_{\mathfrak{m}}=1$ and

$\mu(L^{+}_{p}(f_{k},\omega^{0}))=\mathrm{val}_{p}(N-1)+\mathrm{val}_{p}(k),$

where $f_{k}$ is the unique form of weight $k$ in the Hida family
corresponding to the isomorphism $\mathbb{T}^{0}_{\mathfrak{m}}\cong\Lambda$.

###### Proof.

By Theorem 5.7, the assumptions (a)-(c) imply that $U_{p}-1$ generates
$\mathcal{I}_{\mathfrak{m}}$. The proof of parts (1)-(4) works verbatim as in
the proof of Theorem 4.4, except that the role of the Kubota–Leopold series
$L_{p}(\psi^{-1}_{\mathfrak{m}},\kappa)$ is played by
$\mathcal{E}^{-}(T_{0})$, which equals
$\zeta_{p,k_{0}}(\kappa)(1-N^{\kappa/2})$. In particular, for every weight
$k$,

$\displaystyle\mathrm{val}_{\varpi}(\mathcal{E}^{-}(T_{0})|_{\kappa=k})$
$\displaystyle=\mathrm{val}_{\varpi}(\zeta_{p,k_{0}}(k))+\mathrm{val}_{\varpi}(1-N^{k/2})$
$\displaystyle=\mathrm{val}_{\varpi}(N-1)+\mathrm{val}_{\varpi}(k),$

because $\mathrm{val}_{\varpi}(\zeta_{p,k_{0}}(k))=0$ by (a). The last claim
follows from Theorem 5.9 and (3). ∎

######  Remark 5.13.

We note that the relative advantage of Theorem 5.12 over Theorem 4.4 is that
all of the hypotheses are simple numerical criteria that can be verified to be
true in any given case.

###  5.6. Conjectures and evidence

The below is a re-statement of Conjecture 4.5 but now in the setting of this
section.

######  Conjecture 5.14.

  1. (1)

Suppose that $\mathbb{T}_{\mathfrak{m}}$ is Gorenstein. For each $j$, there is
an equality

$\mathrm{content}(L^{+}_{p}(\mathfrak{m},\omega^{j}))=\mathcal{I}_{\mathfrak{m}}.$

  2. (2)

Suppose that $\mathbb{T}^{0}_{\mathfrak{m}}$ is Gorenstein with
$H^{1}_{\Lambda}(X_{1}(N))^{+}_{\mathfrak{m}}$ generated by an element $e$ as
a $\mathbb{T}^{0}_{\mathfrak{m}}$-module. Then
$L^{+}_{p}(\mathfrak{m},\omega^{j})^{0}_{e}$ has unit content.

If $\mathcal{I}_{\mathfrak{m}}$ is principal then both
$\mathbb{T}_{\mathfrak{m}}$ and $\mathbb{T}^{0}_{\mathfrak{m}}$ are Gorenstein
and the hypotheses of both parts of the above conjecture are satisfied. When
$U_{p}-1$ generates $\mathcal{I}_{\mathfrak{m}}$, then the $j=0$ case of this
conjecture follows from Theorem 5.12.

We ran numerical tests of this conjecture analogous to what was done in [BP19,
Section 3.9]. Namely, for all primes $N<80$, we ran through all primes $p\geq
5$ such that $N\equiv 1\pmod{p}$. For each such pair $(N,p)$, we considered
$2\leq k_{0}\leq p-3$ and when the corresponding
$\mathbb{T}^{0}_{\mathfrak{m}}$ had rank 1, we verified the above conjecture
for all values of $j$. The assumption of the rank 1 of
$\mathbb{T}^{0}_{\mathfrak{m}}$ is needed because the method of computation
involved iterating $U_{p}$ on overconvergent modular symbols which converges
when there is a unique form in the Hida family in each weight $k$. This
assumption on the rank of $\mathbb{T}^{0}_{\mathfrak{m}}$ also implies that
both $\mathbb{T}^{0}_{\mathfrak{m}}$ and $\mathbb{T}_{\mathfrak{m}}$ are
Gorenstein and so $\mathcal{I}_{\mathfrak{m}}$ is principal in this case. In
all, we verified the conjecture (for all allowable values of $j$) for 35
eigenforms.

##  Appendix A Hida theory with tame Atkin-Lehner involutions

We consider a variant of the Hida Hecke algebra with operators $w_{\ell}$ at
primes $\ell$ dividing the tame level rather than $U_{\ell}$. Let
$N=p\ell_{1}\dots\ell_{r}$ be squarefree, and assume $p>3$.

###  A.1. Review of Hida theory

In this section, we review Hida theory. All of these results are well-known,
see, for example, [Hid86a], [Hid86b], [Hid93, Chapter 7], [Wil88], [FK12,
Section 1.5].

####  A.1.1. Classical modular forms

For $r\geq 0$ let $\Gamma_{r}$ denote the congruence subgroup
$\Gamma_{0}(N)\cap\Gamma_{1}(p^{r})$. For a ring $R$, let
$M_{k}(\Gamma_{r},R)$ and $S_{k}(\Gamma_{r},R)$ denote the spaces of modular
forms and cusp forms, respectively, of weight $k$ and level $\Gamma_{r}$, with
coefficients in $R$. There is an injective $q$-expansion map

$\mathrm{expand}_{q}:M_{k}(\Gamma_{r},R)\to R{[\\![}q{]\\!]}$

which we denote by $f\mapsto\sum_{n}a_{n}(f)q^{n}$. For $f\in
S_{k}(\Gamma_{r},R)$ we have $a_{0}(f)=0$. We define

$m_{k}(\Gamma_{r},R)=\\{f\in M_{k}(\Gamma_{r},Q(R))\ :\ a_{n}(f)\in R\text{
for all }n>0\\}$

where $Q(R)$ is the localization $S^{-1}R$ where $S$ is the set of non-zero-
divisors of $R$.

There are Hecke operators $T_{n}$ for $(n,N)=1$ and $U_{\ell}$ for $\ell\mid
N$ as well as the Atkin-Lehner involutions $w_{\ell}$ for $\ell\mid N$ and
diamond operators $\langle n\rangle$ for $(n,N)=1$. These all act on
$m_{k}(\Gamma_{r},R)$ and preserve the submodules $M_{k}(\Gamma_{r},R)$ and
$S_{k}(\Gamma_{r},R)$. We let

$\mathfrak{H}^{\prime}_{r,k}\subset\mathrm{End}_{\mathbb{Z}_{p}}(m_{k}(\Gamma_{r},\mathbb{Z}_{p})),\
\mathfrak{h}^{\prime}_{r,k}\subset\mathrm{End}_{\mathbb{Z}_{p}}(S_{k}(\Gamma_{r},\mathbb{Z}_{p}))$

be the $\mathbb{Z}_{p}$-subalgebras generated by the diamond operators and
$T_{n}$ for $(n,N)=1$ as well as $U_{\ell}$ for $\ell|N$. These are
commutative algebras, and the subject of Hida theory.

We let

$\mathfrak{H}_{r,k}\subset\mathrm{End}_{\mathbb{Z}_{p}}(m_{k}(\Gamma_{r},\mathbb{Z}_{p})),\
\mathfrak{h}_{r,k}\subset\mathrm{End}_{\mathbb{Z}_{p}}(S_{k}(\Gamma_{r},\mathbb{Z}_{p}))$

be the $\mathbb{Z}_{p}$-subalgebras generated by the diamond operators and
$T_{n}$ for $(n,N)=1$ as well as $U_{p}$ and $w_{\ell}$ for
$\ell|\frac{N}{p}$. These are commutative algebras, and are the main focus in
this paper.

######  Lemma A.1.

Let $M$ be $m_{k}(\Gamma_{r},\mathbb{Z}_{p})$ or
$S_{k}(\Gamma_{r},\mathbb{Z}_{p})$, and let $H$ be
$\mathfrak{H}^{\prime}_{r,k}$ or $\mathfrak{h}^{\prime}_{r,k}$, respectively.
Then $M$ and $H$ are free $\mathbb{Z}_{p}$-modules of finite rank and the
pairing

$M\times H\to\mathbb{Z}_{p}$

given by $(f,T)\mapsto a_{1}(Tf)$ is perfect.

####  A.1.2. Cohomology

Let $Y(\Gamma_{r})$ be the modular curve with level $\Gamma_{r}$ and let
$X(\Gamma_{r})$ be its compactification. For $k\geq 2$, let $\mathcal{F}_{k}$
denote the twisted constant $p$-adic étale sheaf on $Y(\Gamma_{r})$ associated
to the representation $\mathrm{Sym}^{k-2}\mathrm{Std}$ of $\mathrm{GL}_{2}$,
and denote by the same letter its pushforward to $X(\Gamma_{r})$. Consider the
cohomology groups $H^{1}(r,k),H^{1}_{c}(r,k),H^{1}_{P}(r,k)$ defined as
follows

$H^{1}(r,k):=H^{1}(Y(\Gamma_{r}),\mathcal{F}_{k}),\
H^{1}_{c}(r,k):=H^{1}_{c}(Y(\Gamma_{r}),\mathcal{F}_{k}),\
H^{1}_{P}(r,k):=H^{1}(X(\Gamma_{r}),\mathcal{F}_{k}).$

By Eichler-Shimura theory, the algebras $\mathfrak{H}_{r,k}^{\prime}$ and
$\mathfrak{H}_{r,k}$ act on $H^{1}_{\dagger}(r,k)$ for any
$\dagger\in\\{\emptyset,c,P\\}$, faithfully if $\dagger\in\\{\emptyset,c\\}$,
and factoring exactly through $\mathfrak{h}_{r,k}^{\prime}$ and
$\mathfrak{h}_{r,k}$, respectively, if $\dagger=P$.

######  Lemma A.2.

For any $k\geq 2$, any $r^{\prime}\geq r\geq 0$, and any
$\dagger\in\\{\emptyset,c,P\\}$, the trace maps

$H^{1}_{\dagger}(r^{\prime},k)\to H^{1}_{\dagger}(r,k)$

commute with the actions of $T_{\ell}$ and $\langle\ell\rangle$ for any
$\ell\nmid N$, $U_{\ell}$ for any $\ell\mid N$, and $w_{\ell}$ for any
$\ell\mid\mid N$.

###### Proof.

For $T_{\ell}$ and $U_{\ell}$, this is proven in [Oht93, Lemma 7.4.1], and
essentially the same proof works for $w_{\ell}$. Indeed, just as in that
proof, it is enough to that $w_{\ell}$ commutes with the natural map

$H^{1}_{\dagger}(r,k)\to H^{1}_{\dagger}(r^{\prime},k)$

for $\dagger\in\\{\emptyset,c\\}$. This commutivity is clear because on both
spaces $w_{\ell}$ is given by the same double coset operator. Explicitly, at
any level $M$ with $\ell||M$, $w_{\ell}$ is the double coset operator
associated to any matrix $W_{\ell,M}$ of the form

$W_{\ell,M}=\left(\begin{array}[]{cc}\ell x&y\\\ Mz&\ell w\end{array}\right)$

such that $\det(W_{\ell,M})=\ell$ (the operator is independent of the choice
of $W_{\ell,M}$). We see that any choice of $W_{\ell,Np^{r^{\prime}}}$ is also
a valid choice for $W_{\ell,Np^{r}}$. ∎

####  A.1.3. Hida theory

For a $\mathfrak{H}_{r,k}$-module or $\mathfrak{H}_{r,k}^{\prime}$-module $M$,
let $M^{\mathrm{ord}}$ denote the largest direct summand on which $U_{p}$ acts
invertibly. For $k$ fixed, let

$\mathfrak{H}^{\prime\mathrm{ord}}_{k}=\varprojlim_{r\geq
0}\mathfrak{H}^{\prime\mathrm{ord}}_{r,k},\
\mathfrak{h}^{\prime\mathrm{ord}}_{k}=\varprojlim_{r\geq
0}\mathfrak{h}^{\prime\mathrm{ord}}_{r,k}$

where the transition maps send $T_{q}$ and $U_{\ell}$ to the operator with the
same name (this is well-defined by Lemma A.2). These are algebras over the
Iwasawa algebra $\Lambda=\mathbb{Z}_{p}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$,
via the diamond operator action. For $a=1,\dots,p-1$, and any
$\Lambda$-module, $M$, let $M^{(a)}$ denote the direct summand where the
torsion subgroup of $\mathbb{Z}_{p}^{\times}$ acts by the $a$-th power of the
Teichemuller character. Identify $\Lambda^{(a)}$ with
$\mathbb{Z}_{p}{[\\![}1+p\mathbb{Z}_{p}{]\\!]}$, and, for any $k\equiv
a\pmod{p}$, let
$\omega_{r,k}=[1+p]^{p^{r}}-(1+p)^{(k-2)p^{r}}\in\Lambda^{(a)}$.

Let

$m_{k,\Lambda}^{\mathrm{ord}}=\varprojlim_{r\geq
0}m_{k}(\Gamma_{r},\mathbb{Z}_{p})^{\mathrm{ord}},\
S_{k,\Lambda}^{\mathrm{ord}}=\varprojlim_{r\geq
0}S_{k}(\Gamma_{r},\mathbb{Z}_{p})^{\mathrm{ord}}$

where the transition maps are the trace maps. These are faithful modules for
$\mathfrak{H}^{\prime\mathrm{ord}}_{k}$ and
$\mathfrak{h}^{\prime\mathrm{ord}}_{k}$, respectively. The maps

$m_{k}(\Gamma_{r},\mathbb{Z}_{p})^{\mathrm{ord}}\to\mathbb{Q}_{p}[(\mathbb{Z}/p^{r}\mathbb{Z})^{\times}]{[\\![}q{]\\!]}$

given by

$f\mapsto\sum_{a\in(\mathbb{Z}/p^{r}\mathbb{Z})^{\times}}(\mathrm{expand}_{q}(\langle
a\rangle^{-1}U_{p}^{r}w_{Np^{r}}(f))[a]$

induce injective $\Lambda$-module homomorphisms

$m_{k,\Lambda}^{\mathrm{ord}}\to Q(\Lambda)+q\Lambda{[\\![}q{]\\!]},\
S_{k,\Lambda}^{\mathrm{ord}}\to q\Lambda{[\\![}q{]\\!]},$

and we identify these modules with there images in
$Q(\Lambda){[\\![}q{]\\!]}$. The following is known as Hida’s control theorem.

######  Theorem A.3.

The $\Lambda$-modules
$m_{k,\Lambda}^{\mathrm{ord}},S_{k,\Lambda}^{\mathrm{ord}},\mathfrak{H}^{\prime\mathrm{ord}}_{k}$
and $\mathfrak{h}^{\prime\mathrm{ord}}_{k}$ are independent of the integer
$k\geq 2$. We may and do drop the subscript $k$ from the notation, and denote
these objects simply by
$m_{\Lambda}^{\mathrm{ord}},S_{\Lambda}^{\mathrm{ord}},\mathfrak{H}^{\prime\mathrm{ord}}$
and $\mathfrak{h}^{\prime\mathrm{ord}}$.

Let $M$ be $m_{\Lambda}^{\mathrm{ord}}$ or $S_{\Lambda}^{\mathrm{ord}}$ and
let $H$ be $\mathfrak{H}^{\prime\mathrm{ord}}$ or
$\mathfrak{h}^{\prime\mathrm{ord}}$, respectively. Let $M_{r,k}$ and $H_{r,k}$
be the fixed weight and level versions. We have:

  1. (1)

$M$ and $H$ are free $\Lambda$-modules of finite rank.

  2. (2)

The pairing

$M\times H\to\Lambda,$

given by $(f,T)\mapsto a_{1}(Tf)$, is perfect.

  3. (3)

For any $k\geq 2$ and $r\geq 0$, the natural maps

$M/\omega_{r,k}M\to M_{r,k},\ H/\omega_{r,k}H\to H_{r,k}$

are isomorphisms.

###  A.2. Hida theory with Atkin-Lehner operators

In this section, we prove the analog of the control theorem for the algebras
$\mathfrak{H}_{r,k}^{\mathrm{ord}}$ and $\mathfrak{h}_{r,k}^{\mathrm{ord}}$.

For an element $\epsilon=(\epsilon_{1},\dots,\epsilon_{r})\in\\{\pm 1\\}^{r}$
and an $\mathfrak{H}_{r,k}$-module $M$, let $M^{\epsilon}$ denote the summand
of $M$ on which $w_{\ell_{i}}$ acts by $\epsilon_{i}$ for $i=1,\dots,r$.

####  A.2.1. Atkin-Lehner theory, and reducing from level $N$ to level $N/p$

We will frequently make use of the following result, which is a variant of
Atkin-Lehner theory (c.f. [AL70, Theorem 1]) that allows for more-general
coefficient rings. It was first proven by Mazur [Maz77, Lemma II.5.9, pg. 83]
in the case $M>5$ is prime, and in the general case by Ohta [Oht14, Corollary
(2.1.4)].

######  Lemma A.4.

Let $M$ be an integer and let $f\in M_{k}(\Gamma_{0}(M),R)$ with $R$ a
$\mathbb{Z}[1/2M]$-algebra, and suppose that $f$ is an eigenform for all
$w_{\ell}$ with $\ell|M$ and that $a_{n}(f)=0$ for all $(n,M)=1$. Then $f$ is
constant.

Note that the lemma requires that the level be invertible in the coefficient
ring. We want to apply the lemma to rings $R$ where $p\notin R^{\times}$. The
following result, which essentially says that ordinary forms are old-at-$p$
when the weight is at least 3, is useful to remove $p$ from the level. It was
proven by Gouvêa [Gou92, Lemma 3] for cusp forms, and the same proof works for
modular forms (c.f. [Oht05, Proposition 1.3.2]).

######  Lemma A.5.

Let $M$ be any integer that is prime to $p$ and let $k>2$. Then the map

$M_{k}(\Gamma_{1}(M),\mathbb{Z}_{p})^{\mathrm{ord}}\to
M_{k}(\Gamma_{1}(M)\cap\Gamma_{0}(p),\mathbb{Z}_{p})^{\mathrm{ord}}$

is an isomorphism.

####  A.2.2. Duality

The key to the proof of the control theorem will be the following duality
result, which is the analog of Lemma A.1.

######  Proposition A.6.

For any $r\geq 0$ and $k\geq 2$, consider the pairings

$(-,-)_{r,k}:m_{k}(\Gamma_{r},\mathbb{Z}_{p})^{\mathrm{ord}}\times\mathfrak{H}_{r,k}^{\mathrm{ord}}\to\mathbb{Z}_{p},\
(-,-)^{0}_{r,k}:S_{k}(\Gamma_{r},\mathbb{Z}_{p})^{\mathrm{ord}}\times\mathfrak{h}_{r,k}^{\mathrm{ord}}\to\mathbb{Z}_{p}$

given by $(f,T)\mapsto a_{1}(Tf)$.

  1. (1)

For any $r,k$, the resulting maps

$\mathfrak{H}_{r,k}^{\mathrm{ord}}\to\mathrm{Hom}_{\mathbb{Z}_{p}}(m_{k}(\Gamma_{r},\mathbb{Z}_{p})^{\mathrm{ord}},\mathbb{Z}_{p}),\
\mathfrak{h}_{r,k}^{\mathrm{ord}}\to\mathrm{Hom}_{\mathbb{Z}_{p}}(S_{k}(\Gamma_{r},\mathbb{Z}_{p})^{\mathrm{ord}},\mathbb{Z}_{p})$

are injective.

  2. (2)

For $r=0$ and any $k>2$, the pairings are perfect.

For the proof of the proposition, we need a lemma comparing
$m_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$ to
$M_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$.

######  Lemma A.7.

For any $\epsilon\in\\{\pm 1\\}^{r}$, there is an exact sequence

$0\to M_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}\to
m_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}\xrightarrow{a_{0}}\left(\frac{1}{p^{m_{k,\epsilon}}}\mathbb{Z}_{p}\right)/\mathbb{Z}_{p}\to
0$

for some integer $m_{k,\epsilon}\geq 0$. Moreover, if we define $m^{\prime}$
as the maximal integer $m$ such that there exists $h\in
M_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$ with $h\equiv
1\pmod{p^{m}}$ and $a_{0}(h)=1$, then $m_{k,\epsilon}=m^{\prime}$.

###### Proof.

We have an exact sequence

$0\to M_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}\to
m_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}\xrightarrow{a_{0}}\mathbb{Q}_{p}/\mathbb{Z}_{p}.$

Since $m_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$ is finitely
generated, the image of the $a_{0}$ map is a finite cyclic $p$-group. We
define $m_{k,\epsilon}\geq 0$ so that the order of this group is
$p^{m_{k,\epsilon}}$, and we get the desired exact sequence.

If $h^{\prime}\in m_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$
satisfies $a_{0}(h^{\prime})=\frac{1}{p^{m_{k,\epsilon}}}$, then
$h:=p^{m_{k,\epsilon}}h^{\prime}\in
M_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$ satisfies $h\equiv
1\pmod{p^{m_{k,\epsilon}}}$ and $a_{0}(h)=1$, so $m_{k,\epsilon}\leq
m^{\prime}$. Conversely, if $h\equiv 1\pmod{p^{m^{\prime}}}$ and $a_{0}(h)=1$,
then $h^{\prime}:=p^{-m^{\prime}}h\in
m_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$ and
$a_{0}(h^{\prime})=p^{-m^{\prime}}$ so $m_{k,\epsilon}\geq m^{\prime}$. ∎

######  Remark A.8.

It is known that $m_{k,\epsilon}=0$ if $(p-1)\nmid k$, but we shall not use
this.

###### Proof of Proposition A.6.

We give the proof only for the modular case, the cuspidal case being similar
but easier. We can decompose $m_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord}}$
and $\mathfrak{H}_{0,k}^{\mathrm{ord}}$ into eigenspaces for the Aktin-Lehner
operators. It is enough to show that, for any choice of $\epsilon$, the
induced pairing

$(-,-)_{r,k}:m_{k}(\Gamma_{r},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}\times\mathfrak{H}_{r,k}^{\mathrm{ord},\epsilon}\to\mathbb{Z}_{p},$

has the desired properties.

We first show that the pairing is perfect with $\mathbb{Q}_{p}$-coefficients,
which will imply (1). Suppose that $f\in
M_{k}(\Gamma_{r},\mathbb{Q}_{p})^{\mathrm{ord},\epsilon}$ satisfies
$(f,T)_{r,k}=0$ for all $T\in\mathfrak{H}_{r,k}^{\mathrm{ord},\epsilon}[1/p]$.
This implies that $a_{n}(f)=0$ for all $(n,N/p)=1$. By Lemma A.4, this implies
that $f$ is constant, and hence $0$. Conversely, suppose that
$T\in\mathfrak{H}_{r,k}^{\mathrm{ord},\epsilon}[1/p]$ satisfies
$(f,T)_{r,k}=0$ for all $f\in
M_{k}(\Gamma_{r},\mathbb{Q}_{p})^{\mathrm{ord},\epsilon}$. Then for any $g\in
M_{k}(\Gamma_{r},\mathbb{Q}_{p})^{\mathrm{ord},\epsilon}$, we have
$a_{n}(Tg)=(T_{n}g,T)_{r,k}=0$ for all $(n,N/p)=1$. The same lemma then
implies that $Tg=0$, which implies that $T=0$ since
$\mathfrak{H}_{r,k}^{\mathrm{ord},\epsilon}[1/p]$ acts faithfully on
$M_{k}(\Gamma_{r},\mathbb{Q}_{p})^{\mathrm{ord},\epsilon}$.

Now we consider (2). By the perfectness of the $\mathbb{Q}_{p}$-pairing, we
have that

$m_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}\to\mathrm{Hom}_{\mathbb{Z}_{p}}(\mathfrak{H}_{0,k}^{\mathrm{ord},\epsilon},\mathbb{Z}_{p})$

is injective. We have to show it is surjective. Let
$\phi:\mathfrak{H}_{0,k}^{\mathrm{ord},\epsilon}\to\mathbb{Z}_{p}$. We see
that there is $f^{\prime}\in
M_{k}(\Gamma_{0},\mathbb{Q}_{p})^{\mathrm{ord},\epsilon}$ such that
$\phi(T)=(T,f^{\prime})_{k}$ for all
$T\in\mathfrak{H}_{0,k}^{\mathrm{ord},\epsilon}$. This implies that
$a_{n}(f^{\prime})=\phi(T_{n}f^{\prime})\in\mathbb{Z}_{p}$ for all
$(n,N/p)=1$, where $T_{n}:=U_{p^{r}}T_{n/p^{r}}$ if $p^{r}||n$. If
$f^{\prime}\in m_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$, we
are done, so assume for a contradiction that it is not. Then there is a
minimal $r\geq 1$ such that $f=p^{r}f^{\prime}\in
m_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$. By the minimality
of $r$, we must have $a_{n}(f)\in\mathbb{Z}_{p}^{\times}$ for some $n>0$
(clearly this $n$ must have $(n,N/p)>1$).

Assume that $f\in M_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$.
By Lemma A.5, we may consider $f\in
M_{k}(\Gamma_{0}(N/p),\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$. Consider the
image $\bar{f}$ in
$M_{k}(\Gamma_{0}(N/p),\mathbb{F}_{p})^{\mathrm{ord},\epsilon}$. Since
$a_{n}(f^{\prime})\in\mathbb{Z}_{p}$ for all $(n,N)=1$ and since $r\geq 1$, we
have $a_{n}(\bar{f})=0$ for all $(n,N)=1$. As $N/p\in\mathbb{F}_{p}^{\times}$,
Lemma A.4 applies and we get that $\bar{f}$ is constant. This implies that
$a_{n}(f)\equiv 0\pmod{p}$ for all $n>0$, a contradiction.

Finally, assume that $f\not\in
M_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$, so
$s:=-\mathrm{val}_{p}(a_{0}(f))>0$. Then $g:=a_{0}(f)^{-1}f\in
M_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$ satisfies
$a_{0}(g)=1$. By Lemma A.5, we may consider $g\in
M_{k}(\Gamma_{0}(N/p),\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$. Since
$a_{n}(f^{\prime})\in\mathbb{Z}_{p}$ for all $(n,N)=1$, we have
$a_{n}(g)=0\pmod{p^{r+s}}$ for all $(n,N)=1$. As
$N/p\in(\mathbb{Z}/p^{r+s}\mathbb{Z})^{\times}$, Lemma A.4 implies that
$g\pmod{p^{r+s}}$ is the constant $a_{0}(g)=1$. By Lemma A.7, and since $r>0$,
this implies that $s<m_{k,\epsilon}$.

Now fix $h\in M_{k}(\Gamma_{0},\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$ such
that $h\equiv 1\pmod{p^{m_{k,\epsilon}}}$ and such that $a_{0}(h)=1$. Since
$\mathrm{val}_{p}(a_{0}(f))=-s$ and $s<m_{k,\epsilon}$, we see that
$a_{n}(a_{0}(f)h)\in p\mathbb{Z}_{p}$ for all $n>0$. Then letting
$f^{\prime\prime}=f-a_{0}(f)h$, we see that $f^{\prime\prime}\in
M_{k}(\Gamma_{0}(N/p),\mathbb{Z}_{p})^{\mathrm{ord},\epsilon}$. Moreover, we
see that
$a_{n}(f^{\prime\prime})=p^{r}a_{n}(f^{\prime})-a_{n}(a_{0}(f)h)\equiv
0\pmod{p}$, for all $(n,N/p)=1$. By Lemma A.4 this implies that
$f^{\prime\prime}\pmod{p}$ is constant. In particular, for all $n>0$ we have

$0\equiv a_{n}(f^{\prime\prime})\equiv a_{n}(f)\pmod{p},$

a contradiction. ∎

####  A.2.3. Control theorem

For $k$ fixed, let

$\mathfrak{H}^{\mathrm{ord}}_{k}=\varprojlim_{r\geq
0}\mathfrak{H}^{\mathrm{ord}}_{r,k},\
\mathfrak{h}^{\mathrm{ord}}_{k}=\varprojlim_{r\geq
0}\mathfrak{h}^{\mathrm{ord}}_{r,k}$

where the transition maps send $T_{q}$ and $w_{\ell}$ to the operator with the
same name (this is well-defined by Lemma A.2). These are algebras over the
Iwasawa algebra $\Lambda=\mathbb{Z}_{p}{[\\![}\mathbb{Z}_{p}^{\times}{]\\!]}$,
via the diamond operator action. We can now prove the main theorem, which is a
control theorem for the algebras $\mathfrak{H}^{\mathrm{ord}}_{k}$ and
$\mathfrak{h}^{\mathrm{ord}}_{k}$.

######  Theorem A.9.

The algebras $\mathfrak{H}^{\mathrm{ord}}_{k}$ and
$\mathfrak{h}^{\mathrm{ord}}_{k}$ for various $k$ are canonically identified
with each other. We can and do drop the subscript $k$ from the notation and
simply refer to these algebras as $\mathfrak{H}^{\mathrm{ord}}$ and
$\mathfrak{h}^{\mathrm{ord}}$. Moreover:

  1. (1)

There are canonical isomorphisms of $\Lambda$-modules
$\mathfrak{H}^{\mathrm{ord}}\to\mathfrak{H}^{\prime\mathrm{ord}}$ and
$\mathfrak{h}^{\mathrm{ord}}\to\mathfrak{h}^{\prime\mathrm{ord}}$. In
particular, $\mathfrak{H}^{\mathrm{ord}}$ and $\mathfrak{h}^{\mathrm{ord}}$
are free $\Lambda$-modules of finite rank.

  2. (2)

The pairings

$m_{\Lambda}^{\mathrm{ord}}\times\mathfrak{H}^{\mathrm{ord}}\to\Lambda,\
S_{\Lambda}^{\mathrm{ord}}\times\mathfrak{h}^{\mathrm{ord}}\to\Lambda,$

given by $(f,T)\mapsto a_{1}(Tf)$, are perfect.

  3. (3)

The natural maps

$\mathfrak{H}^{\mathrm{ord}}/\omega_{r,k}\mathfrak{H}^{\mathrm{ord}}\to\mathfrak{H}^{\mathrm{ord}}_{r,k},\
\mathfrak{h}^{\mathrm{ord}}/\omega_{r,k}\mathfrak{h}^{\mathrm{ord}}\to\mathfrak{h}^{\mathrm{ord}}_{r,k}$

are isomorphisms for all $r\geq 0$ and $k\geq 2$.

  4. (4)

The pairings $(-,-)_{r,k}$ and $(-,-)_{r,k}^{0}$ of Proposition A.6 are
perfect for all $r\geq 0$ and $k\geq 2$.

###### Proof.

We give the proofs only for modular case, the cuspidal cases being similar.
First, we see that $\mathfrak{H}^{\mathrm{ord}}_{k}$ is the subalgebra of
$\mathrm{End}_{\Lambda}(m^{\mathrm{ord}}_{k,\Lambda})$ generated by the
operators $T_{q}$, $U_{p}$ and $w_{\ell}$. But by Theorem A.3,
$m^{\mathrm{ord}}_{k,\Lambda}$ is independent of $k$. This shows that these
algebras are independent of $k$.

The map $\mathfrak{H}^{\mathrm{ord}}\to\mathfrak{H}^{\prime\mathrm{ord}}$ in
(1) is given as the composite

$\mathfrak{H}^{\mathrm{ord}}\to\mathrm{Hom}_{\Lambda}(m^{\mathrm{ord}}_{\Lambda},\Lambda)\xrightarrow{\sim}\mathfrak{H}^{\prime\mathrm{ord}}$

where the first map is induced by the pairing in (2), and the second map is
the isomorphism of Theorem A.3 (2). Let
$X=\mathrm{coker}(\mathfrak{H}^{\mathrm{ord}}\to\mathfrak{H}^{\prime\mathrm{ord}})$,
which is a finitely generated $\Lambda$-module.

Similarly, for any fixed $k$ and $r$, we have a map
$\mathfrak{H}^{\mathrm{ord}}_{r,k}\to\mathfrak{H}^{\prime\mathrm{ord}}_{r,k}$
defined as the composite

$\mathfrak{H}^{\mathrm{ord}}_{r,k}\to\mathrm{Hom}_{\mathbb{Z}_{p}}(m_{k}(\Gamma_{r},\mathbb{Z}_{p})^{\mathrm{ord}},\mathbb{Z}_{p})\xrightarrow{\sim}\mathfrak{H}^{\prime\mathrm{ord}}_{r,k},$

where the first map is given by $(-,-)_{r,k}$ and the second map is the
isomorphism given by Lemma A.1. This map is an isomorphism for every $r$ and
$k$ if and only if (4) holds.

We have a commutative diagram to compare these maps:

(A.1)
$\textstyle{\mathfrak{H}^{\mathrm{ord}}/\omega_{r,k}\mathfrak{H}^{\mathrm{ord}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{H}^{\prime\mathrm{ord}}/\omega_{r,k}\mathfrak{H}^{\prime\mathrm{ord}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{\mathfrak{H}^{\mathrm{ord}}_{r,k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{H}^{\prime\mathrm{ord}}_{r,k}.}$

The leftmost vertical arrow is surjective because the operators $T_{q}$,
$U_{p}$ and $w_{\ell}$ map to the operators of the same name, the lower
horizontal arrow is injective by Proposition A.6(1), and the rightmost
vertical arrow is an isomorphism by Theorem A.3 (3). For $r=0$ and $k>2$,
Proposition A.6(2) implies that the lower horizontal arrow is an isomorphism.
This implies that the map

$\mathfrak{H}^{\mathrm{ord}}/\omega_{0,k}\mathfrak{H}^{\mathrm{ord}}\to\mathfrak{H}^{\prime\mathrm{ord}}/\omega_{0,k}\mathfrak{H}^{\prime\mathrm{ord}}$

is surjective for all $k>2$. In other words, we have $X/\omega_{0,k}X=0$ for
all $k>2$. The elements $\omega_{0,3},\omega_{0,4},\dots,\omega_{0,p+1}$ are
in the $p-1$ different maximal ideals of $\Lambda$, so $X=0$ by Nakayama’s
lemma. This implies that
$\mathfrak{H}^{\mathrm{ord}}\to\mathfrak{H}^{\prime\mathrm{ord}}$ is
surjective.

Returning to the diagram (A.1) for arbitrary $r\geq 0$ and $k\geq 2$, we see
that the lower horizontal arrow is also surjective, proving (4). Since the map
$\mathfrak{H}^{\mathrm{ord}}\to\mathfrak{H}^{\prime\mathrm{ord}}$ is the
inverse limit (for $k$ fixed and $r$ increasing) of these maps, it is also an
isomorphism proving (1). By the definition of the map
$\mathfrak{H}^{\mathrm{ord}}\to\mathfrak{H}^{\prime\mathrm{ord}}$, this also
proves (2). This shows that all the arrows except the leftmost vertical in
(A.1) are isomorphisms, so it is too, proving (3). ∎

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