# On the Iwasawa invariants of BDP Selmer groups and BDP $p$-adic
$L$-functions

Antonio Lei  Department of Mathematics and Statistics
University of Ottawa
150 Louis-Pasteur Pvt
Ottawa, ON
Canada K1N 6N5<EMAIL_ADDRESS>,  Katharina Müller  Département de
Mathématiques et de Statistique
Université Laval, Pavillion Alexandre-Vachon
1045 Avenue de la Médecine
Québec, QC
Canada G1V 0A6<EMAIL_ADDRESS>and  Jiacheng Xia  Département
de Mathématiques et de Statistique
Université Laval, Pavillion Alexandre-Vachon
1045 Avenue de la Médecine
Québec, QC
Canada G1V 0A6<EMAIL_ADDRESS>
###### Abstract.

Let $p$ be an odd prime. Let $f_{1}$ and $f_{2}$ be weight $2$ Hecke eigen
cusp forms with isomorphic residual Galois representations at $p$.
Greenberg–Vatsal and Emerton–Pollack–Weston showed that if $p$ is a good
ordinary prime for the two forms, the Iwasawa invariants of their $p$-primary
Selmer groups and $p$-adic $L$-functions over the cyclotomic
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ are closely related. The goal of
this article is to generalize these results to the anticyclotomic setting.
More precisely, let $K$ be an imaginary quadratic field where $p$ splits.
Suppose that the generalized Heegner hypothesis holds with respect to both
$(f_{1},K)$ and $(f_{2},K)$. We study relations between the Iwasawa invariants
of the BDP Selmer groups and the BDP $p$-adic $L$-functions of $f_{1}$ and
$f_{2}$.

###### Key words and phrases:

Anticyclotomic Iwasawa theory, congruences of modular forms, Heegner points

###### 2020 Mathematics Subject Classification:

11R23 (primary); 11G40 (secondary)

##  1\. Introduction

Let $K$ be an imaginary quadratic field with discriminant $d_{K}$. Throughout
this article, $p\geq 3$ is a fixed prime number that is split in $K$. We fix
once and for all an embedding
$\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}_{p}}$ and let
$\mathfrak{p}$ be the prime of $K$ above $p$ determined by this embedding
given by the pre-image of the maximal ideal of the ring of integers of
$\overline{\mathbb{Q}_{p}}$. Let $f$ be a normalized newform (by which we
always mean a Hecke eigenform) of $S_{2}(\Gamma_{0}(N),\mathcal{O})$, where
$\mathcal{O}$ is the ring of integers of a number field $L$. Let $K_{\infty}$
be the anticyclotomic $\mathbb{Z}_{p}$-extension of $K$. We assume that the
class number of $K$ is coprime to $p$. In particular, this implies that the
two primes above $p$ are totally ramified in $K_{\infty}$.

We assume that $N$ is coprime to $pd_{K}$ and we factorize $N$ into a product
$N^{+}N^{-}$, where $N^{+}$ (resp. $N^{-}$) is the product of the prime
factors of $N$ that are split (resp. inert) in $K$. We say that $f$ satisfies
the generalized Heegner hypothesis (with respect to $K$) if:

  * (GHH)

$N^{-}$ is the square-free product of an even number of primes.

In [BDP13, Bra11], under the assumption that $N^{-}=1$,
Bertolini–Darmon–Prasanna and Brakočević constructed a $p$-adic $L$-function
$L_{p}^{\mathrm{BDP}}(f)\in\Lambda^{\mathrm{ac}}$ whose square interpolates
the $L$-values of $f_{/K}$ twisted by Hecke characters over $K$ of certain
infinity type (this element is commonly known as the BDP $p$-adic $L$-function
attached to $f$; For the definition of $\Lambda^{\mathrm{ac}}$, see §2). The
construction of $L_{p}^{\mathrm{BDP}}(f)$ has been generalized by Hunter
Brooks [HB15] to the setting $N^{-}>1$. The Iwasawa main conjecture predicts
that the Pontryagin dual of the so-called BDP Selmer group
$\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{f})$ attached to $f$ over
$K_{\infty}$ is torsion over $\Lambda^{\mathrm{ac}}$. Furthermore, its
characteristic ideal (see Definition 2.1) satisfies

(IMC)
$\left(L_{p}^{\mathrm{BDP}}(f)^{2}\right)\stackrel{{\scriptstyle?}}{{=}}\mathrm{Char}_{\Lambda^{\mathrm{ac}}}\left(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{f})^{\vee}\otimes\Lambda^{\mathrm{ac}}\right)$

as ideals in $\Lambda^{\mathrm{ac}}$.

The conjecture (IMC) is equivalent to Perrin-Riou's Heegner point main
conjecture formulated in [PR87] when $f$ is $p$-ordinary (see for example
[BCK21, Theorem 5.2]). For similar results in the non-ordinary case, see
[CÇSS18, Theorem 3.7] and [CW22b, Theorem 6.8]. This equivalence, together
with the works of Howard [How04b, How04a] (in the ordinary case) and the work
of Kobayashi–Ota [KO20] combined with [LZ22] (in the non-ordinary case), show
that the inclusion $\subseteq$ in (IMC) holds under certain hypotheses. Under
different hypotheses, the other inclusion or even the full equality of (IMC)
has been studied in [CW22a, Theorem 5.2], [BCK21, Theorem B], [CW22b, Theorem
6.8] and [CÇSS18, Theorem 5.8]. In a recent work of Sweeting [Swe22, Theorem
E], (IMC) has been shown to hold in the $p$-ordinary case under very mild
hypotheses.

The goal of this article is to study how Iwasawa invariants of the objects
featured in (IMC) behave under congruences, generalizing results of
Greenberg–Vatsal [GV00] on objects defined over the cyclotomic
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ in the $p$-ordinary case (see
[Kim09, HL19a] for similar results in the $p$-non-ordinary case) and covering
both the ordinary and non-ordinary cases simultaneously. Note that analogous
results in the anticyclotomic setting have been studied in [Kim17, CKL17]
under the hypothesis that $N^{-}$ is a squarefree product of an odd number of
primes. In particular, (GHH) is violated and the results therein concern
different Selmer groups and $p$-adic $L$-functions from those featured in
(IMC). Under mild technical hypothesis one implication of the correpsonding
main conjecture is proved in [BD05].

In order to explain our results, we first introduce certain notation that will
be utilized throughout the article. For $i\in\\{1,2\\}$, let $f_{i}\in
S_{2}(\Gamma_{0}(N_{i}),\mathcal{O})$ be a newform. Here, $\mathcal{O}$ is the
ring of integers of a number field $L$. We factor $N_{i}=N_{i}^{+}N_{i}^{-}$
as above and assume that (GHH) holds for $f_{i}$.

Suppose that $p\nmid N_{1}N_{2}$ and fix a place $v$ of $\mathcal{O}$ above
$p$. We denote the completion of $\mathcal{O}$ at $v$ by $\mathcal{O}_{v}$ and
fix a uniformizer $\varpi$ of $\mathcal{O}_{v}$. Let

$\rho_{i}\colon G_{\mathbb{Q}}\to\textup{Aut}(V_{i})=\textup{GL}_{2}(L_{v})$

be the $p$-adic representation attached to $f_{i}$ of [Del69], such that the
identity $\textup{Tr}(\rho_{i}(\textup{Frob}_{l}))=a_{l}(f_{i})$ holds for
every prime $l$ coprime to $N_{i}p$, where $\textup{Frob}_{l}$ is the
arithmetic Frobenius at $l$. Note that the Hodge–Tate weights of $V_{i}$ are
$0$ and $1$ (our convention is that the cyclotomic character has Hodge–Tate
weight $1$). The $p$-adic Tate module of the abelian variety attached to
$f_{i}$ determines a $G_{\mathbb{Q}}$-stable lattice $T_{i}\subset V_{i}$. We
define $A_{i}=V_{i}/T_{i}$. By an abuse of notation, we denote the
representation

$G_{\mathbb{Q}}\to\textup{Aut}(T_{i})=\textup{GL}_{2}(\mathcal{O}_{v})$

by the same symbol $\rho_{i}$. The residual representation of $\rho_{i}$ is
denoted by $\bar{\rho}_{i}$. We assume throughout that

(Cong) $\bar{\rho}_{1}\simeq\bar{\rho}_{2}$

as $G_{\mathbb{Q}}$-representations.

For $i\in\\{1,2\\}$, we consider the following conditions:

  * (FG)

$\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{i})^{\vee}$ is a finitely
generated $\mathcal{O}_{v}$-module.

  * ($H^{0}$)

$H^{0}(K_{w},A_{i})=\\{0\\}$ for all $w\mid pN^{-}_{1}N^{-}_{2}$ and
$i\in\\{1,2\\}$.

Our main results are the following theorems.

######  Theorem A (Corollary 3.8).

Suppose that hypotheses ($H^{0}$) holds for both $f_{1}$ and $f_{2}$. Then
(FG) holds for $i=1$ if and only if it holds for $i=2$. Furthermore, when
these equivalent conditions hold, we have

$\operatorname{rank}_{\mathcal{O}_{v}}(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{1})^{\vee})+\sum_{\ell\in\Sigma_{0}}c_{\ell,1}=\operatorname{rank}_{\mathcal{O}_{v}}(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{2})^{\vee})+\sum_{\ell\in\Sigma_{0}}c_{\ell,2},$

where $\Sigma_{0}$ is the set of primes dividing $N^{+}_{1}N^{+}_{2}$ and
$c_{\ell,i}$ are explicit local terms defined in Lemma 3.5.

This generalizes similar results in [HL19b, HL22] where $N_{1}^{-}$ and
$N_{2}^{-}$ are assumed to be $1$. Under appropriate hypotheses, [LZ22,
Theorem A] and [HL22, Theorem 5.1] show that (FG) holds. Theorem A allows us
to illustrate new instances where (FG) holds that are not covered by the
aforementioned results (see Lemma 3.10).

On the analytic results, we prove a similar result under different hypotheses.
In particular, we consider for $i\in\\{1,2\\}$:

  * (mu)

$L_{p}^{\mathrm{BDP}}(f_{i})$ is an Iwasawa function that is not divisible by
$\varpi$.

  * (div)

If $N_{i}^{-}\neq 1$, then $p\nmid\prod_{q\mid N_{i}}(q-1)q(q+1)$.

  * (sq-fr)

If $N_{i}^{-}\neq 1$, then $N_{i}$ is square-free.

######  Remark 1.1.

Assume that $\overline{\rho}$ is irreducible and let $M$ be the conductor of
$\overline{\rho}$. We can write $N=M\prod_{i=1}^{s}q_{i}^{e_{i}}$ for some
primes $q_{i}$ and exponents $e_{i}$. If (div) is satisfied, then [DT94,
Theorem 1] implies that $(q_{i},M)=1$ and $e_{i}=1$ for $1\leq i\leq s$. Thus,
(sq-fr) is satisfied if $M$ is square-free.

######  Theorem B (Theorem 4.20).

Suppose that the hypotheses (div), (sq-fr) and (unit)111The hypothesis (unit)
concerns the $p$-divisibility of certain explicit factors. We refer the reader
to the statement of Theorem 4.10 in the main body of the article for further
details. hold for both $f_{1}$ and $f_{2}$. Let $\alpha(f_{i})$ be the factor
introduced in Theorem 4.2.222The constant $\alpha(f_{i})$ is a quotient of
Petersson products when $N_{i}^{-}>1$. If $N_{i}^{-}=1$, then
$\alpha(f_{i})=1$. Then, $\alpha(f_{i})$ is a $p$-adic unit and (mu) holds for
$i=1$ if and only if the same is true for $i=2$. Furthermore, when these
equivalent conditions hold, the $\lambda$-invariants (see Definition 2.1) of
the BDP $p$-adic $L$-functions of $f_{1}$ and $f_{2}$ satisfy

$2\lambda(L_{p}^{\mathrm{BDP}}(f_{1}))+\sum_{q|\tilde{N}_{3}^{(1)}}d_{q,1}=2\lambda(L_{p}^{\mathrm{BDP}}(f_{2}))+\sum_{q|\tilde{N}_{3}^{(2)}}d_{q,2},$

where $\tilde{N}_{3}^{(1)}$, $\tilde{N}_{3}^{(2)}$, $d_{q,1}$ and $d_{q,2}$
are explicitly defined in Theorem 4.10.

Note that under certain hypotheses, Hsieh [Hsi14, Theorem B] and Burungale
[Bur17, Theorem B] have shown that (mu) holds for a large family of modular
forms. Analogous to the algebraic side, Theorem B allows us to find new
examples of modular forms for which (mu) holds (see Lemma 4.22). The residual
representations of these modular forms at $p$ are not absolutely irreducible.
In particular, they are not covered by the previous results of Hsieh and
Burungale.

Our proof of Theorem B allows us to deduce that, up to certain explicit Euler
factors, the logarithms of the Heegner points on two abelian varieties of
$\operatorname{GL}_{2}$-type satisfying (Cong) are congruent modulo $\varpi$.
In particular, this allows us to generalize [KL19, Theorem 1.16] (with $m=1$),
relaxing the strong Heegner hypothesis to (GHH). See Theorem 4.25 for the
precise statement.

Our proof of Theorem B relies on a comparison between
$L_{p}^{\mathrm{BDP}}(f_{i})$ and the anticyclotomic specialization of the
2-variable Rankin–Selberg $p$-adic $L$-function multiplied by Katz' $p$-adic
$L$-function. This is the origin of the factor $\alpha(f_{1})$. Such
comparison was already explored extensively in several previous works
including [CR18, Cas13, CW22b]. The crux of the method we employ dwells on
congruence properties enjoyed by Rankin–Selberg $p$-adic $L$-functions. As far
as the authors are aware, such congruence properties are currently not proven
directly for the BDP $p$-adic $L$-functions, which are constructed using
quaternionic modular forms via the Jacquet–Langlands correspondence. Under
certain hypotheses, results of Emerton [Eme02] imply that these quaternionic
modular forms are congruent when (Cong) holds. However, it seems to be an open
problem in the generality we currently work with (we refer the reader to the
discussion in [Mar17, P.1780] for more details).

Finally, when the hypotheses of Theorems A and B hold simultaneously, we can
compare the local terms to deduce:

######  Theorem C (Theorem 5.1).

Let $f_{1}$ and $f_{2}$ be newforms of level $N_{1}$ and $N_{2}$ respectively,
satisfying (GHH), (Cong), ($H^{0}$), (div), (sq-fr) and (unit). Assume that
(mu) holds for $f_{1}$ and that $\alpha(f_{1})$ is a $p$-adic unit. Suppose
that one inclusion in (IMC) holds for $f_{2}$ and that the full equality of
(IMC) holds for $f_{1}$. Then the full equality of (IMC) holds for $f_{2}$.

## Acknowledgement

Parts of these works were carried out during AL's visit at University College
Dublin in fall 2022 supported by a Distinguished Visiting Professorship and
the Seed Funding Scheme. He thanks UCD for the financial support and the warm
hospitality. He also thanks Kazim Buyukboduk and Daniele Casazza for
interesting discussions on subjects related to topics studied in this paper
during his visit. The authors would like to thank Tobias Berger, Francesc
Castella, Antonio Cauchi, Daniel Delbourgo, Neil Dummigan, Jeffrey Hatley,
Ernest Hunter Brooks, David Loeffler, Kimball Martin, Ariel Pacetti and Jan
Vonk for answering their questions during the preparation of the article. The
authors would also like to thank Chan-Ho Kim, Chao Li and Luochen Zhao for
their helpful suggestions and comments. Finally, we thank the anonymous
referee for carefully reading an earlier version of the article as well as
constructive feedback. All three authors’ research is supported by the NSERC
Discovery Grants Program RGPIN-2020-04259 and RGPAS-2020-00096.

##  2\. Notation

Let $\mathfrak{c}$ be an integral ideal of $K$. We denote the ray class field
of conductor $\mathfrak{c}$ by $K_{\mathfrak{c}}$. Let
$\mathcal{K}=\bigcup_{n\geq 1}K_{p^{n}}$ and denote
$\operatorname{Gal}(\mathcal{K}/K)$ by $H_{p^{\infty}}$. Analogously, we
define $H_{\mathfrak{p}^{\infty}}$ as the Galois group of $\bigcup_{n\geq
1}K_{\mathfrak{p}^{n}}/K$. Note that
$H_{p^{\infty}}\cong\mathbb{Z}_{p}^{2}\times\Delta$ for a finite abelian group
$\Delta$. Our assumption that $p$ is coprime to the class number of $K$
implies that $\Delta$ is of order coprime to $p$.

The group $\operatorname{Gal}(K/\mathbb{Q})$ acts on the abelian group
$H_{p^{\infty}}$ via the formula $\sigma\mapsto\tilde{c}\sigma\tilde{c}^{-1}$,
where $\tilde{c}$ is a lift of $c\in\operatorname{Gal}(K/\mathbb{Q})$ to
$\operatorname{Gal}(\mathcal{K}/\mathbb{Q})$. We will frequently use the
notation $\sigma^{c}$ to denote $\tilde{c}\sigma\tilde{c}^{-1}$. Let
$H^{+}_{p^{\infty}}$ be the subgroup on which
$\operatorname{Gal}(K/\mathbb{Q})$ acts trivially and define
$H^{-}_{p^{\infty}}=H_{p^{\infty}}/H^{+}_{p^{\infty}}$. Let
$K^{\textup{a}}_{\infty}\subset\mathcal{K}$ be the unique subextension with
Galois group $H^{-}_{p^{\infty}}$ (note that $K^{\textup{a}}_{\infty}$
contains the anticyclotomic $\mathbb{Z}_{p}$-extension $K_{\infty}$ of $K$).
We denote $\operatorname{Gal}(K_{\infty}/K)$ by $\Gamma^{\mathrm{ac}}$.

Let $L_{v}$ and $\mathcal{O}_{v}$ be defined as in the introduction. Let $R$
be the completion of the integral closure of $\mathcal{O}_{v}$ in the maximal
unramified extension of $L_{v}$.

We define the following Iwasawa algebras

$\displaystyle\Lambda_{p}=R[[H_{p^{\infty}}]],\quad\Lambda_{\mathfrak{p}}=R[[H_{\mathfrak{p}^{\infty}}]],\quad\Lambda^{\textup{a}}=R[[H^{-}_{p^{\infty}}]],$
$\displaystyle\Lambda^{\textup{ac}}=R[[\Gamma^{\mathrm{ac}}]],\quad\Lambda^{\mathrm{ac}}_{\mathcal{O}_{v}}=\mathcal{O}_{v}[[\Gamma^{\mathrm{ac}}]].$

There are natural projections

$\pi^{\textup{a}}\colon\Lambda_{p}\to\Lambda^{\textup{a}},\quad\pi^{\textup{ac}}\colon\Lambda_{p}\to\Lambda^{\textup{ac}}.$

These projections extend naturally to projections on the corresponding
fraction fields. By an abuse of notation, we denote these projections by the
same symbols.

Given an element $\sigma$ in $H^{-}_{p^{\infty}}$, we denote any lift of
$\sigma$ to $H_{p^{\infty}}$ by $\tilde{\sigma}$. Note that the element
$\tilde{\sigma}^{1-c}\in\Lambda_{p}$ does not depend on the lift
$\tilde{\sigma}$ we chose by the definition of $H^{+}_{p^{\infty}}$. Thus, we
obtain a well-defined map

$\mathcal{V}\colon\Lambda^{\textup{a}}\to\Lambda_{p}.$

such that

$\pi^{\textup{a}}\circ\mathcal{V}(\sigma)=\sigma^{2}.$

In the remainder of the article, we frequently employ the notion of Iwasawa
invariants of elements in $\Lambda^{\mathrm{ac}}$ and
$\Lambda^{\mathrm{ac}}_{\mathcal{O}_{v}}$. Note that $\mathcal{O}_{v}$ and $R$
are local rings with maximal ideals $\mathfrak{m}_{\mathcal{O}_{v}}$ and
$\mathfrak{m}_{R}$, respectively. For each of the rings $R$ and
$\mathcal{O}_{v}$ we fix a generator of the maximal ideal, denoted by
$\varpi_{\mathcal{O}_{v}}$ and $\varpi_{R}$, respectively. If the ring is
clear from the context we omit the superscript.

######  Definition 2.1.

For every element $F\in\Lambda^{\mathrm{ac}}$, the Weierstrass preparation
theorem says that

$F=\varpi_{R}^{m}UG,$

for an integer $m\geq 0$, a unit $U$ in $\Lambda^{\mathrm{ac}}$ and a
distinguished polynomial $G$. We define the $\lambda$-invariant and
$\mu$-invariant of $F$ by $\lambda(F)=\deg(G)$ and $\mu(F)=m$, respectively.
If $F\in\Lambda^{\mathrm{ac}}_{\mathcal{O}_{v}}$, we define $\lambda(F)$ and
$\mu(F)$ in the same way.333Note that the absolute ramification indices in
$\mathcal{O}_{v}$ and in $R$ are the same. In particular, the $\mu$\- and
$\lambda$-invariants are independent of the choice of Iwasawa algebra.

Given a finitely generated torsion module $M$ over $\Lambda^{\mathrm{ac}}$, it
is pseudo-isomorphic to

$\bigoplus_{i=1}^{r}\Lambda^{\mathrm{ac}}/F_{i},$

for some integer $r\geq 1$ and nonzero elements
$F_{i}\in\Lambda^{\mathrm{ac}}$. We define the characteristic ideal of $M$ to
be

$\mathrm{Char}_{\Lambda^{\mathrm{ac}}}(M)=\left(\prod_{i=1}^{r}F_{i}\right)\subset\Lambda^{\mathrm{ac}}.$

We define $\mu(M)$ and $\lambda(M)$ to be $\mu(F)$ and $\lambda(F)$,
respectively, where $F$ is any generator of
$\mathrm{Char}_{\Lambda^{\mathrm{ac}}}(M)$. We define these objects in a
similar fashion if $\Lambda^{\mathrm{ac}}$ is replaced by
$\Lambda^{\mathrm{ac}}_{\mathcal{O}_{v}}$.

##  3\. Iwasawa invariants of BDP Selmer groups under congruences

We study how the Iwasawa invariants of BDP Selmer groups behave for congruent
modular forms, proving Theorem A stated in the introduction. Throughout, we
fix $f_{i}\in S_{2}(\Gamma_{0}(N_{i}),\mathcal{O})$ for $i\in\\{1,2\\}$
satisfying the conditions (GHH), (Cong) and ($H^{0}$). Note that the
hypothesis ($H^{0}$) implies that $H^{0}(K,A_{i})=H^{0}(K_{\infty},A_{i})=0$.

######  Lemma 3.1.

Let $w$ be a prime of $K$ dividing $N^{-}_{1}N^{-}_{2}$. Let $I_{w}$ be the
inertia subgroup inside $G_{K_{w}}$. Then
$H^{1}(G_{K_{w}}/I_{w},A_{i}^{I_{w}})$ and $H^{0}(K_{w},T_{i})$ are trivial.

###### Proof.

Let $K_{w}^{\mathrm{cyc}}$ be the cyclotomic $\mathbb{Z}_{p}$-extension of
$K_{w}$ and write
$G_{w}^{\mathrm{cyc}}=\operatorname{Gal}\left(\overline{K_{w}}/K_{w}^{\mathrm{cyc}}\right)$.
Since $w\nmid p$, we have the inclusion $I_{w}\subset G_{w}^{\mathrm{cyc}}$
and that

$G_{K_{w}}/G_{w}^{\mathrm{cyc}}\cong\operatorname{Gal}(K_{w}^{\textup{cyc}}/K_{w})\cong(G_{K_{w}}/I_{w})^{\textup{pro-p}}\cong\mathbb{Z}_{p}.$

Therefore, the triviality of $H^{1}(G_{K_{w}}/I_{w},A_{i}^{I_{w}})$ would
follow from showing that

$H^{1}\left(\operatorname{Gal}(K_{w}^{\mathrm{cyc}}/K_{w}),A_{i}^{G^{\mathrm{cyc}}_{w}}\right)=\\{0\\}.$

Indeed, under the hypothesis ($H^{0}$), we have $H^{0}(K_{w},A_{i})=\\{0\\}$,
which implies that $H^{0}(K_{w}^{\mathrm{cyc}},A_{i})=\\{0\\}$. Thus, the
first assertion of the lemma follows.

For the second assertion, note that $A_{i}\cong
T_{i}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$. Thus, $H^{0}(K_{w},A_{i})=\\{0\\}$
implies that the same holds for $H^{0}(K_{w},T_{i})$. ∎

######  Definition 3.2.

Let $\Sigma$ be the set of all primes dividing $N_{1}N_{2}p$ and let
$\Sigma_{0}$ be the set of primes dividing $N^{+}_{1}N^{+}_{2}$. We define the
BDP Selmer group of $f_{i}$ over $K_{n}$ by

$\operatorname{Sel}^{\mathrm{BDP}}(K_{n},A_{i})=\ker\left(H^{1}(K_{\Sigma}/K_{n},A_{i})\to\prod_{w|\Sigma\setminus\\{\mathfrak{p}^{c}\\}}H^{1}(K_{n,w},A_{i})\right)$

and the imprimitive BDP Selmer group (with respect to $\Sigma_{0}$)

$\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{n},A_{i})=\ker\left(H^{1}(K_{\Sigma}/K_{n},A_{i})\to\prod_{w|\Sigma\setminus(\Sigma_{0}\cup\\{\mathfrak{p}^{c}\\})}H^{1}(K_{n,w},A_{i})\right).$

(Here, given a set $S$ of primes, the product $\displaystyle\prod_{w|S}$ runs
through all the primes dividing an element of $S$.)

Note that
$\operatorname{Sel}^{\mathrm{BDP}}(K_{n},A_{i})\subset\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{n},A_{i})$
and that any prime in $\Sigma_{0}$ is finitely decomposed in $K_{\infty}/K$
(see [Bri07, Theorem 2]). In the standard definition of BDP Selmer groups, one
usually requires the cocycles to be unramified at primes away from $p$ instead
of trivial (see [KO20, (1.1)] or [castellaJLMS, Definition 2.2]). Our
assumption that $H^{0}(K_{w},A_{i})=\\{0\\}$ for all $w\mid
N^{-}_{1}N^{-}_{2}$ implies that the natural map

$H^{1}(K_{w},A_{i})\to H^{1}(I_{w},A_{i})$

is injective (see also Lemma 3.1). It then follows from [Kid18, the proof of
Proposition 3.1] that our definition of BDP Selmer groups coincides with the
standard one. We adopt the above definition to make the arguments on the
algebraic side more accessible to the readers.

We now prove several preliminary lemmas regarding these Selmer groups.

######  Lemma 3.3.

Assume that $\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{i})$ is
$\Lambda^{\textup{ac}}_{\mathcal{O}_{v}}$-cotorsion. Then the global-to-local
map

$H^{1}(K_{\Sigma}/K_{\infty},A_{i})\to\prod_{w|\Sigma\setminus\\{\mathfrak{p}^{c}\\}}H^{1}(K_{\infty,w},A_{i})$

is surjective.

###### Proof.

We follow the strategy of [PW11, proof of Proposition A.2]. As in loc. cit.,
since the local cohomology groups $H^{1}(K_{\infty,w},A_{i})$ do not contain
nontrivial $\Lambda^{\mathrm{ac}}_{\mathcal{O}_{v}}$-submodules of finite
index, it suffices to show that the cokernel of the global-to-local map is
finite.

Fix $i\in\\{1,2\\}$. For an integer $t$, let $B_{t}$ denote the tensor product
of $G_{K}$-representations $A_{i}\otimes\kappa^{t}$, where $\kappa$ is a fixed
character

$\kappa\colon\Gamma^{\mathrm{ac}}\to 1+p\mathbb{Z}_{p}$

that sends a topological generator of $\Gamma^{\mathrm{ac}}$ to $1+p$. Note
that $B_{t}\cong A$ as $\operatorname{Gal}(K_{\Sigma}/K_{\infty})$-modules and
thus we may replace $A_{i}$ by $B_{t}$ in the statement of the lemma.

It follows from Lemma 3.1 and [Kid18, Propositions 4.2 and 4.3] that
$H^{1}(K_{\infty,w},B_{t})$ is
$\Lambda_{\mathcal{O}_{v}}^{\mathrm{ac}}$-cotorsion for all
$w|\Sigma\setminus\\{p\\}$. Furthermore, by assumption,
$\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{i})$ is also
$\Lambda^{\textup{ac}}_{\mathcal{O}_{v}}$-cotorsion. Therefore, we may choose
$t$ such that

$\operatorname{rank}_{\mathcal{O}_{v}}(\operatorname{Sel}^{\mathrm{BDP}}(K_{n},B_{t})^{\vee})=\operatorname{rank}_{\mathcal{O}_{v}}(H^{1}(K_{n,w},B_{t})^{\vee})=0$

for all $n\geq 0$ and all $w|\Sigma\setminus\\{p\\}$. When we need to specify
at which level the place $w$ is defined, we will write $w_{n}$ and
$w_{\infty}$, respectively.

If $w|\mathfrak{p}$, [Gre89, Proposition 1] tells us that
$H^{1}(K_{\infty,v_{\infty}},B_{t})^{\vee}$ is of rank 2 over
$\Lambda^{\textup{ac}}_{\mathcal{O}_{v}}$ for all $t$. Consequently, for all
but finitely many choices of $t$, the $\mathcal{O}_{v}$-module
$H^{1}(K_{n,v_{n}},B_{t})^{\vee}$ is of rank $2p^{n}$ for all $n\geq 0$.
Furthermore, [Gre89, Proposition 3] implies that
$H^{1}(K_{\Sigma}/K_{n},B_{t})^{\vee}$ has $\mathcal{O}_{v}$-rank at least
$2p^{n}$.444While it is assumed that $\mathcal{O}_{v}=\mathbb{Z}_{p}$ in loc.
cit., the proof carries over to the more general case in a straightforward
manner. Thus, we may choose $t$ such that the cokernel of the global-to-local
map

$H^{1}(K_{\Sigma}/K_{n},B_{t})\to\prod_{w|\Sigma\setminus\\{\mathfrak{p}^{c}\\}}H^{1}(K_{n,w},B_{t})$

is finite. It remains to show that this finite cokernel is uniformly bounded
independently of $n$, which can be proved via the same argument given in
[GV00, proof of Proposition (2.1)]. We give an outline of the proof for the
convenience of the reader.

Let $T_{-t}=T_{i}\otimes\kappa^{-t}$. Note that
$\textup{Hom}(B_{t},L_{v}/\mathcal{O}_{v}(1))=T_{-t}$. Consider the following
global-to-local maps

$\displaystyle\alpha_{0}\colon$ $\displaystyle
H^{1}(K_{\Sigma}/K_{n},B_{t})\to\prod_{w|\Sigma\setminus\\{\mathfrak{p}^{c}\\}}H^{1}(K_{n,w},B_{t}),$
$\displaystyle\alpha\colon$ $\displaystyle
H^{1}(K_{\Sigma}/K_{n},B_{t})\to\prod_{w|\Sigma}H^{1}(K_{n,w},B_{t}),$
$\displaystyle\beta\colon$ $\displaystyle
H^{1}(K_{\Sigma}/K_{n},T_{-t})\to\prod_{w|\Sigma}H^{1}(K_{n,w},T_{-t}).$

By an abuse of notation, we denote the prime of $K_{n}$ lying above
$\mathfrak{p}^{c}$ by the same symbol. Let

$G_{B_{t}}=\langle\textup{im}(\alpha),H^{1}(K_{n,\mathfrak{p}^{c}},B_{t})\rangle$

be the smallest subgroup of $\prod_{w|\Sigma}H^{1}(K_{n,w},B_{t})$ containing
the image of $\alpha$ and $H^{1}(K_{n,\mathfrak{p}^{c}},B_{t})$. Let
$G_{M}\subset\prod_{w|\Sigma}H^{1}(K_{n,w},T_{-t})$ be the orthogonal
complement of $G_{B_{t}}$ under local Tate duality.

We have chosen $t$ such that the cokernel of $\alpha_{0}$ is finite. For such
$t$, the group $G_{M}$ is also finite. Note that $\ker(\beta)^{\vee}$ is a
subgroup of $H^{2}(K_{\infty}/K_{n},B_{t})$, which is trivial (by the
assumption that $p$ is odd). Let $S_{M}$ be the preimage of $G_{M}$ under
$\beta$. It follows that $S_{M}$ is finite and that its cardinality is bounded
by $H^{1}(K_{\Sigma}/K_{n},T_{-t})_{\textup{tors}}$. A standard argument (see
[GV00, page 19]) then shows that

$\left|H^{1}(K_{\Sigma}/K_{n},T_{-t})_{\textup{tors}}\right|\leq\left|H^{0}(K_{\infty},A_{i})\right|,$

which is trivial under ($H^{0}$) (see the discussion before Lemma 3.1). This
shows that $\alpha_{0}$ is surjective. In particular, the cardinality of the
cokernel of $\alpha_{0}$ is indeed bounded independently of $n$ as required. ∎

######  Lemma 3.4.

If the $\Lambda_{\mathcal{O}_{v}}^{\mathrm{ac}}$-module
$\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{\infty},A_{i})^{\vee}$ is
torsion, then it does not contain a non-trivial finite submodule.

###### Proof.

Let $B_{t}$ and $T_{-t}$ be defined as in the proof of Lemma 3.3. Once again,
we may replace $A_{i}$ by $B_{t}$. We choose $t$ such that

$\operatorname{rank}_{\mathcal{O}_{v}}(\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{n},B_{t})^{\vee})=0,\quad\text{and}\quad\operatorname{rank}_{\mathcal{O}_{v}}(H^{1}(K_{\Sigma}/K_{n},B_{t})^{\vee})=2p^{n}$

for all $n\geq 0$. Note that

$\operatorname{rank}_{\Lambda^{\textup{ac}}_{\mathcal{O}_{v}}}(H^{1}(K_{\Sigma}/K_{\infty},B_{t})^{\vee})=2.$

It follows from [Gre89, Proposition 3] that

$\operatorname{rank}_{\Lambda^{\textup{ac}}_{\mathcal{O}_{v}}}(H^{2}(K_{\Sigma}/K_{\infty},B_{t})^{\vee})=0.$

By [Gre89, Proposition 4 and 5], $H^{1}(K_{\Sigma}/K_{\infty},B_{t})^{\vee}$
has no nontrivial finite submodule. We can now conclude as in [Gre99, Proof of
proposition 4.14] that
$H^{1}(K_{\Sigma}/K_{\infty},B_{t})_{\Gamma^{\textup{ac}}}$ is trivial.

Take $n=0$ in the proof of Lemma 3.3. Recall that

$H^{1}(K_{\Sigma}/K,B_{t})\to\left(\prod_{w|\Sigma\setminus(\Sigma_{0}\cup\\{\mathfrak{p}^{c}\\})}H^{1}(K_{w},B_{t})\right)$

is surjective. As $H^{0}(K_{\infty},B_{t})=H^{0}(K_{\infty},A)=0$, it follows
from the inflation-restriction exact sequence that
$H^{1}(K_{\Sigma}/K_{\infty},B_{t})^{\Gamma^{\textup{ac}}}\cong
H^{1}(K_{\Sigma}/K,B_{t})$. We thus obtain a surjection

$H^{1}(K_{\Sigma}/K,B_{t})^{\Gamma^{\textup{ac}}}\to\left(\prod_{v\in\Sigma\setminus(\Sigma_{0}\cup\\{\mathfrak{p}^{c}\\})}H^{1}(K_{\infty,w},B_{t})\right)^{\Gamma^{\textup{ac}}}.$

The short exact sequence

$0\to\operatorname{Sel}_{\Sigma_{0}}^{\mathrm{BDP}}(K_{\infty},B_{t})\to
H^{1}(K_{\Sigma}/K_{\infty},B_{t})\to\left(\prod_{w|\Sigma\setminus(\Sigma_{0}\cup\\{\mathfrak{p}^{c}\\})}H^{1}(K_{\infty,w},B_{t})\right)\to
0$

gives the exact sequence

$\displaystyle H^{1}(K_{\Sigma}/K,B_{t})^{\Gamma^{\textup{ac}}}$
$\displaystyle\to\left(\prod_{w|\Sigma\setminus(\Sigma_{0}\cup\\{\mathfrak{p}^{c}\\})}H^{1}(K_{\infty,w},B_{t})\right)^{\Gamma^{\textup{ac}}}\to$
$\displaystyle\to\operatorname{Sel}_{\Sigma_{0}}^{\mathrm{BDP}}(K_{\infty},B_{t})_{\Gamma^{\textup{ac}}}\to
H^{1}(K_{\Sigma}/K_{\infty},B_{t})_{\Gamma^{\textup{ac}}}.$

As the first map is surjective and the last term is trivial, we deduce that

$\operatorname{Sel}_{\Sigma_{0}}^{\mathrm{BDP}}(K_{\infty},B_{t})_{\Gamma^{\textup{ac}}}=0,$

which implies that
$\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{\infty},B_{t})^{\vee}$ does
not contain a nontrivial finite submodule. ∎

We now relate the Iwasawa invariants of the standard BDP Selmer groups to
their imprimitive counterparts.

######  Lemma 3.5.

For any prime $l$ in $K$ that lies above a rational prime $q$ that is split in
$K/\mathbb{Q}$, we define the polynomial
$P_{l}^{f_{i}}(X)=\det(1-X\textup{Frob}_{l}|(V_{i})_{I_{l}})$, where $I_{l}$
is the inertia group at $l$. Let

$\mathcal{P}^{f_{i}}_{l}=P^{f_{i}}_{l}(N(l)^{-1}\gamma_{l})\in\Lambda^{\mathrm{ac}}_{\mathcal{O}_{v}},$

where $\gamma_{l}$ is the Frobenius of $l$ in $K_{\infty}/K$ and $N$ denotes
the norm map from $K$ to $\mathbb{Q}$.

Then $\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{\infty},A_{i})$ is
cotorsion over $\Lambda^{\mathrm{ac}}_{\mathcal{O}_{v}}$ if and only if
$\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{i})$ is cotorsion over
$\Lambda^{\mathrm{ac}}_{\mathcal{O}_{v}}$. Furthermore, when these Selmer
groups are indeed cotorsion over $\Lambda^{\mathrm{ac}}_{\mathcal{O}_{v}}$, we
have

$\displaystyle\mu(\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{\infty},A_{i})^{\vee})$
$\displaystyle=\mu(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{i})^{\vee}),$
$\displaystyle\lambda(\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{\infty},A_{i})^{\vee})$
$\displaystyle=\lambda(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{i})^{\vee})+\sum_{l|\Sigma_{0}}\lambda(\mathcal{P}^{f_{i}}_{l}).$

######  Remark 3.6.

It can be verified that $\mathcal{P}^{f_{i}}_{l}$ has trivial $\mu$-invariant
and that it is a unit if $l\mid q$ for a rational prime $q$ such that
$a_{q}(f_{i})=0$.

###### Proof.

By definition, we have an exact sequence

(3.1)
$0\to\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{i})\to\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{\infty},A_{i})\to\prod_{w|\Sigma_{0}}H^{1}(K_{\infty,w},A_{i}).$

Every prime dividing $\Sigma_{0}$ is finitely decomposed in $K_{\infty}$.
Thus, $H^{1}(K_{\infty,w},A_{i})$ is cotorsion over
$\Lambda_{\mathcal{O}_{v}}^{\mathrm{ac}}$ (see [GV00, proof of Proposition
(2.4)]). This proves the equivalence of the cotorsionness of the two Selmer
groups.

Suppose that these Selmer groups are indeed cotorsion. Then Lemma 3.3 tells us
that the last arrow in (3.1) is surjective. As discussed in [GV00, proof of
Proposition (2.4)], $H^{1}(K_{\infty,w},A_{i})^{\vee}$ is
$\Lambda_{\mathcal{O}_{v}}^{\mathrm{ac}}$-torsion, with characteristic ideal
generated by $\mathcal{P}^{f_{i}}$. Furthermore, its $\mu$-invariant is zero.
Therefore, the lemma follows from the additivity of Iwasawa invariants in
exact sequences. ∎

######  Lemma 3.7.

Let $\varpi$ be a uniformizer of $\mathcal{O}_{v}$. Assume that
$A_{1}[\varpi]\cong A_{2}[\varpi]$ as $G_{K}$-modules.555Note that this is a
weaker condition than (Cong). Then

$\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{\infty},A_{1})[\varpi]\cong\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{\infty},A_{2})[\varpi].$

###### Proof.

Fix $i\in\\{1,2\\}$. Let $w$ be a prime of $K_{\infty}$ dividing
$\Sigma\setminus\Sigma_{0}$. Recall that $H^{0}(K_{w},A_{i})=\\{0\\}$. In
particular, since $K_{\infty,w}/K_{w}$ is a pro-$p$ extension,
$H^{0}(K_{\infty,w},A_{i})=\\{0\\}$. On taking cohomology of the short exact
sequence

$0\to A_{i}[\varpi]\to A_{i}\to A_{i}\to 0,$

we deduce that

$H^{1}(K_{\infty,w},A_{i}[\varpi])\cong H^{1}(K_{\infty,w},A_{i})[\varpi].$

We also have

$H^{1}(K_{\infty},A_{i}[\varpi])\cong H^{1}(K_{\infty},A_{i})[\varpi].$

Thus,

$\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{\infty},A_{i})[\varpi]\cong\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{\infty},A_{i}[\varpi]),$

where
$\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{\infty},A_{i}[\varpi])$ is
defined in the obvious manner.

The assumption on the isomorphism between $A_{1}[\varpi]$ and $A_{2}[\varpi]$
implies that

$\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{\infty},A_{1}[\varpi])\cong\operatorname{Sel}^{\mathrm{BDP}}_{\Sigma_{0}}(K_{\infty},A_{2}[\varpi]).$

Hence the result follows. ∎

We are now ready to prove Theorem A.

######  Corollary 3.8.

Assume that $A_{1}[\varpi]\cong A_{2}[\varpi]$ as $G_{K}$-modules. Assume
further that $\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{1})$ is
$\Lambda^{\textup{ac}}_{\mathcal{O}_{v}}$-cotorsion and that
$\mu(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{1})^{\vee})=0$. Then
$\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{2})$ is cotorsion,
$\mu(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{2})^{\vee})=0$ and

$\lambda(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{1})^{\vee})+\sum_{l\in\Sigma_{0}}\lambda(\mathcal{P}^{f_{1}}_{l})=\lambda(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{2})^{\vee})+\sum_{l\in\Sigma_{0}}\lambda(\mathcal{P}^{f_{2}}_{l}).$

###### Proof.

Note that
$\operatorname{Sel}_{\Sigma_{0}}^{\mathrm{BDP}}(K_{\infty},A_{1})[\varpi]$ is
finite due to the assumption that
$\operatorname{Sel}_{\Sigma_{0}}^{\mathrm{BDP}}(K_{\infty},A_{1})$ is
$\Lambda^{\textup{ac}}_{\mathcal{O}_{v}}$-cotorsion and that its
$\mu$-invariant vanishes. By the proof of Lemma 3.7,

$\operatorname{Sel}_{\Sigma_{0}}^{\mathrm{BDP}}(K_{\infty},A_{1})[\varpi]\cong\operatorname{Sel}_{\Sigma_{0}}^{\mathrm{BDP}}(K_{\infty},A_{2})[\varpi].$

In particular,
$\operatorname{Sel}_{\Sigma_{0}}^{\mathrm{BDP}}(K_{\infty},A_{2})[\varpi]$ is
also finite. This implies that the
$\Lambda_{\mathcal{O}_{v}}^{\mathrm{ac}}$-module
$\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{2})^{\vee}$ is torsion with
vanishing $\mu$-invariant. Furthermore,

$\lambda(\operatorname{Sel}_{\Sigma_{0}}^{\mathrm{BDP}}(K_{\infty},A_{i})^{\vee})=\operatorname{ord}_{\varpi}(|\operatorname{Sel}_{\Sigma_{0}}^{\mathrm{BDP}}(K_{\infty},A_{i})[\varpi]|)$

by Lemma 3.4. Hence, the asserted equation involving $\lambda$-invariants
follows from Lemmas 3.5 and 3.7. ∎

###  3.1. An application

We conclude this section by showing that Theorem A implies the existence of an
infinite family of modular forms whose BDP Selmer groups over $K_{\infty}$
have vanishing $\mu$-invariant.

######  Definition 3.9.

Let $f=\sum a_{n}q^{n}\in S_{2}(\Gamma_{0}(N),\overline{\mathbb{Z}})$. Let
$N=N_{1}N_{2}N_{0}$ be a decomposition on $N$ into pairwise coprime integers,
such that $N_{1}N_{2}$ is square-free. Let $w$ be a place above $p$ in the
coefficient ring of $f$. Let $\phi$ and $\psi$ be Dirichlet characters. We say
that $f$ has partial Eisenstein decent by $(\phi,\psi,N_{1},N_{2},N_{0})$ if

  * •

$a_{l}\equiv\phi(l)+l\psi(l)\pmod{w}\quad\forall(l,N)=1$

  * •

$a_{l}\equiv\phi(l)\pmod{w}\quad\forall l\mid N_{1}$

  * •

$a_{l}\equiv l\psi(l)\pmod{w}\quad\forall l\mid N_{2}$

  * •

$a_{l}\equiv 0\pmod{w}\quad\forall l\mid N_{0}$.

######  Lemma 3.10.

Let $f$ be a newform of level $N$ and weight $2$ and let $\chi$ be a quadratic
character such that $\chi(p)=-1$. Assume that $p\in\\{3,5\\}$ and that $f$ has
partial Eisenstein decent by $(\chi,\chi,N_{1},N_{2},N_{0})$. Let $K$ be an
imaginary quadratic field satisfying (GHH) with respect to
$N\textup{cond}(\chi)$. We assume that $N^{-}=1$ and that $11$ and $19$ split
in $K$. Then

$\mu((\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A)^{\vee})=0.$

###### Proof.

Let $E/\mathbb{Q}$ be the elliptic curve with LMFDB label 11a.2 if $p=5$ and
the elliptic curve with LMFDB label 19.a2 if $p=3$. Then

$E[p]\cong\mathbb{F}_{p}\oplus\mathbb{F}_{p}(\omega)$

as $G_{\mathbb{Q}}$-representations. Let $g$ be the modular form associated to
the quadratic twist $E^{\chi}$. Recall that $v$ is a fixed place of $K$ above
$p$. Note that $H^{0}(K_{v},E^{\chi}[p^{\infty}])=\\{0\\}$ by construction. As
discussed in [Kri16, Remark 32], we have $a_{n}(f)\equiv
a_{n}(g)\pmod{\varpi}$ for all $n$ that are coprime to
$pN\textup{cond}(E^{\chi})$. By [Kri16, Theorem 34],

$\overline{\rho}_{f}\cong\mathbb{F}_{p}(\chi)\oplus\mathbb{F}_{p}(\chi\omega)$

as $G_{\mathbb{Q}}$-representations. We recall from [CGLS22, theorem 1.5.1]
that the $\mu$-invariant of
$\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},E^{\chi}[p^{\infty}])^{\vee}$
vanishes. Thus, the lemma follows from Corollary 3.8. ∎

######  Remark 3.11.

Let $M/\mathbb{Q}$ be the extension cut out by $\chi$. If $q$ is a rational
prime coprime to $p$ that ramifies in $M$, then $\chi\omega$ is ramified at
$q$ as well and $q^{2}\mid N$. Recall that $N_{1}N_{2}$ is square free. Thus,
$\chi$ is unramified at all primes dividing $N_{1}N_{2}$. Assume now that $q$
is inert in $K$ and let $w_{K}$ and $w_{M}$ be places above $q$ in $K$ and
$M$, respectively. Then $M_{w_{M}}\subset K_{w_{K}}$. Thus,
$H^{0}(K_{w_{K}},A)\neq\\{0\\}$. So, we can only apply Corollary 3.8 in the
case $N^{-}=1$.

##  4\. $\mathrm{BDP}$ $p$-adic $L$-functions of congruent modular forms

The main goal of this section is to prove Theorem B. As discussed in the
introduction, our strategy is to relate the BDP $p$-adic $L$-function to the
anticyclotomic specialization of the product of the Katz $p$-adic $L$-function
and a 2-variable Rankin–Selberg $p$-adic $L$-function. We review these objects
in the following subsection.

###  4.1. Review on $p$-adic $L$-functions

Let $\psi$ be a Hecke character of $K$ of conductor $\mathfrak{f}$. We say
that $\psi$ is of infinity type $(k,j)$ for integers $k$ and $j$ if

$\psi((\alpha))=\alpha^{k}\overline{\alpha}^{j}\quad\textup{for all
}\alpha\equiv 1\pmod{\mathfrak{f}}.$

If $\mathfrak{f}=(1)$, then $\psi((\alpha))=\alpha^{k}\overline{\alpha}^{j}$
for all principal ideals. Note that a Hecke character of trivial conductor and
infinity type $(k,j)$ can only exist if $|\mathcal{O}_{K}^{\times}|$ divides
$k-j$. We say that $\psi$ is anticyclotomic if it factors through
$H_{p^{\infty}}^{-}$. Note that such a character is of infinity type $(k,-k)$.
Let $\psi$ be an anticyclotomic character of trivial conductor. Let $q$ be a
rational prime that is is inert in $K/\mathbb{Q}$ and write
$\mathfrak{q}=(q)$. Then we have

$\psi(\mathfrak{q})=q^{k}\left(\overline{q}\right)^{-k}=1.$

The $L$-function associated to a Hecke character $\psi$ over $K$ is given by

$L(\psi,s)=\sum\frac{\psi(\mathfrak{a})}{N(\mathfrak{a})^{s}}$

for $\Re(s)\gg 0$, where the sum runs over all integral ideals of $K$ that are
coprime to $\mathfrak{f}$. We have an Euler product

$L(\psi,s)=\prod_{{\mathfrak{p}}}\left(1-\frac{\psi(\mathfrak{p})}{N(\mathfrak{p})^{s}}\right)^{-1},$

where the product runs over all prime ideals coprime to $\mathfrak{f}$.

######  Theorem 4.1.

There exist CM periods $\Omega\in\mathbb{C}^{\times}$ and $\Omega_{p}\in
R^{\times}$ and an element $L_{K,p}\in\Lambda$ such that

$L_{K,p}({\psi}^{-1})=\Gamma(k)\Omega^{j-k}\Omega_{p}^{k-j}(\sqrt{|D|})^{j}(2\pi)^{-j}(1-{\psi}(\overline{\mathfrak{p}}))(1-p^{-1}{{\psi}}(\mathfrak{p}))L({\psi}^{-1},0)$

for all Hecke characters of infinity type $(k,j)$ with $0\leq-j<k$ and trivial
conductor.

###### Proof.

This is proved in [Kat78, Section 5.3.0]. ∎

######  Theorem 4.2.

Let $\mathcal{N}^{+}$ be an ideal of $\mathcal{O}_{K}$ such that
$\mathcal{O}_{K}/\mathcal{N}^{+}\cong\mathbb{Z}/N^{+}\mathbb{Z}$. There exists
a $p$-adic L-function $L^{\mathrm{BDP}}_{p}(f)\in\Lambda^{\mathrm{ac}}$ such
that for each anticyclotomic Hecke character $\psi$ of trivial conductor and
infinity type $(-m,m)$, where $m\equiv 0\ \pmod{p-1}$, we have

$\displaystyle L^{\mathrm{BDP}}_{p}(f)(\psi)^{2}=$
$\displaystyle\left(\frac{\Omega_{p}}{\Omega_{k}}\right)^{4m}\frac{\Gamma(m)\Gamma(m+1)\psi({\mathfrak{N}^{+}})^{-1}}{4(2\pi)^{1-2m}\sqrt{D_{K}}^{2m-1}}$
$\displaystyle\times\alpha(f)(1-a_{p}\psi(\overline{\mathfrak{p}})p^{-1}-\psi(\overline{\mathfrak{p}})^{2}p^{-1})^{2}L(f,\chi,1),$

where $\alpha(f)\in\mathcal{O}_{v}$ is defined by

$\alpha(f)=\begin{cases}\frac{\langle f,f\rangle}{\langle
f^{\mathrm{JL}},f^{\mathrm{JL}}\rangle}&N^{-}>1,\\\
1&\text{otherwise.}\end{cases}$

Here, $\langle-,-\rangle$ denotes the Petersson inner product and
$f^{\mathrm{JL}}$ denotes Jacquet–Langlands transform of $f$, which are
normalized as in [Pra06, Sections 1-2].

###### Proof.

The construction of $L_{p}^{\mathrm{BDP}}(f)$ originates from [BDP13] and
[Bra11]. The normalization we employ here is the one given in [BCK21, Section
4]. See [CW22b, Proposition 2.1] for the interpolation formula. Note that in
loc. cit. it is assumed that $\mathcal{O}_{v}=\mathbb{Z}_{p}$. But the
$p$-adic Waldspurger formula used in the proof holds in our current setting.
The same formula can also be deduced from [Bur17, Theorem 5.6] if
$|\mathcal{O}_{K}^{\times}|=2$.666Note that [CW22b] and [Bur17] use opposite
conventions for infinity types of Hecke characters. ∎

Let $N$ be an integer that is coprime to $p$. Let
$S_{2}(\Gamma_{0}(N),\mathcal{O}_{v})$ be the set of cusp forms of level
$\Gamma_{0}(N)$ with coefficients in $\mathcal{O}_{v}$. Set

$\mathbf{g}=\sum_{(\mathfrak{a},\mathfrak{p})=1}[\mathfrak{a}]q^{N(\mathfrak{a})}\in\mathbb{Z}_{p}[[H_{\mathfrak{p}^{\infty}}]][[q]]$

to be a CM Hida family over $\mathbb{Z}_{p}[[H_{\mathfrak{p}^{\infty}}]]$.

Given a character of $H_{p^{\infty}}$, we may factorize it in a unique way
$\varphi\psi$, where $\varphi$ factors through $H_{\mathfrak{p}^{\infty}}$ and
$\psi$ factors through the norm map
$H_{p^{\infty}}\rightarrow\mathbb{Z}_{p}^{\times}$. Following [LLZ15, §6.3],
we define:

######  Definition 4.3.

Let ${\mathcal{L}}_{N,p}\mathrel{\mathop{\mathchar
58\relax}}S_{2}(\Gamma_{0}(N),\mathcal{O}_{v})\rightarrow\operatorname{Frac}\Lambda$
be the $\mathcal{O}_{v}$-linear map satisfying

${\mathcal{L}}_{N,p}(h)(\varphi\psi)=\frac{\langle\mathbf{g}(\varphi)^{*},\Xi_{1/N|d_{K}|}(h,\varphi\psi)^{\operatorname{ord},p}\rangle_{N|d_{K}|}}{\langle\mathbf{g}(\varphi),\mathbf{g}(\varphi)\rangle_{|d_{K}|}},$

where $\langle-,-\rangle_{M}$ denotes the Petersson product at level
$\Gamma_{1}(M)\cap\Gamma_{0}(p)$,

$\Xi_{1/N|d_{K}|}(h,\varphi\psi)^{\operatorname{ord},p}=e_{\operatorname{ord}}[{\mathcal{E}}_{1/N|d_{K}|}(\psi-1,-1-\varphi_{\mathbb{Q}}-\psi)\cdot
h].$

Here, $\mathbf{g}(\varphi)^{*}$ denotes the complex conjugate of the
specialization of $\mathbf{g}$ at $\varphi$, $e_{\operatorname{ord}}$ is
Hida's ordinary projector, ${\mathcal{E}}_{1/N|d_{K}|}(-,-)$ denotes the
$p$-depleted Eisenstein series over $\mathrm{Spec}\
\mathbb{Z}_{p}[[\mathbb{Z}_{p}^{\times}]]^{2}$ given in [LLZ14, Definition
5.3.1] and $\varphi_{\mathbb{Q}}$ denotes the composition of $\varphi$ with
$\mathbb{Z}_{p}^{\times}\hookrightarrow(\mathcal{\mathcal{O}}_{K}\times\mathbb{Z}_{p})^{\times}\rightarrow
H_{p^{\infty}}$. Note that we have written the characters additively.

When $h\in S_{2}(\Gamma_{0}(N),\mathcal{O}_{v})$ is a normalized Hecke
eigenform, ${\mathcal{L}}_{N,p}(h)$ is a 2-variable Rankin–Selberg $p$-adic
$L$-function attached to $h$, interpolating complex $L$-values of the base
change of $h$ to $K$ twisted by Hecke characters of certain infinity type (see
the discussion in [LLZ15, end of §6.3]).

######  Remark 4.4.

Recall that the Petersson inner product is linear in the first coordinate and
skew linear in the second. We have chosen $L$ to be a number field containing
the coefficients of the new forms $f_{1}$ and $f_{2}$. In particular, we can
assume that $L$ is totally real. Using the interpolating property of the
2-variable Rankinn–Selberg $p$-adic $L$-function we see that for $f_{1},f_{2}$
in $S_{2}(\Gamma_{0}(N),\mathcal{O}_{v})$ and $a\in\mathcal{O}_{v}$

$\mathcal{L}(h_{1}-h_{2})=\mathcal{L}(h_{1})-\mathcal{L}(h_{2}),\quad\mathcal{L}(ah)=a\mathcal{L}(h).$

###  4.2. Anticyclotomic specialization

In this subsection, we study the anticyclotomic specialization of $L_{K,p}$
and ${\mathcal{L}}_{N,p}$. Throughout, we fix a normalized Hecke eigenform
$f\in S_{2}(\Gamma_{0}(N),\mathcal{O}_{v})$, where $N$ is coprime to $p\cdot
d_{K}$. Note that $f$ may be an oldform at level $N$. We factorize
$N=N^{+}N^{-}$ and assume that (GHH) holds as before.

######  Definition 4.5.

Let $\iota\colon\Lambda^{\mathrm{ac}}\to\Lambda^{\mathrm{ac}}$ be the
$R$-linear map sending an element $\gamma\in\Gamma^{\mathrm{ac}}$ by
$\gamma\mapsto\gamma^{2}$. Let $L^{\textup{ac}}_{K,p}\in\Lambda^{\textup{ac}}$
be the image of $L_{K,p}$ under the composition $\iota\circ\pi^{\textup{ac}}$.

Let
$\mathcal{L}^{\textup{ac}}_{N,p}(f)\in\operatorname{Frac}{\Lambda^{\textup{ac}}}$
be the image of $\mathcal{L}_{N,p}(f)$ under $\pi^{\textup{ac}}$.

######  Theorem 4.6.

Assume that $f$ is a newform of level $N$ satisfying (div) and (sq-fr). Then
there exists a unit $u_{K}\in(\Lambda^{\mathrm{ac}})^{\times}$ (only depending
on $K$) such that

$\mathcal{L}_{p,N}^{\textup{ac}}(f)L^{\textup{ac}}_{K,p}=\alpha(f)Nu_{K}\sigma^{-1}_{\mathcal{N}^{+}}L_{p}^{\mathrm{BDP}}(f)^{2}\in\Lambda^{\textup{ac}},$

where $\sigma_{\mathcal{N}^{+}}$ is the image of $\mathcal{N}^{+}$ under the
Artin homomorphism and $\mathcal{N}^{+}$ is given as in the statement of
Theorem 4.2.

###### Proof.

Upon writing down the interpolation formulae alluded to in the proof of
[CW22b, Proposition 2.7], we deduce that
$\mathcal{L}_{p,N}^{\textup{ac}}(f)L^{\textup{ac}}_{K,p}$ is equal to, up to a
unit of $\Lambda^{\mathrm{ac}}$ of the form
$Nu_{K}\sigma^{-1}_{\mathcal{N}^{+}}$ (where $u_{K}$ only depends on $K$), the
$p$-adic $L$-function $\alpha(f)L_{p}^{\mathrm{BDP}}(f)^{2}$. By assumption
(div) and [Pra06, Theorem 2.4], $\alpha(f)$ is a $p$-adic integer. As
$L_{p}^{\mathrm{BDP}}(f)$ lies inside $\Lambda^{\mathrm{ac}}$, the theorem
follows. ∎

######  Remark 4.7.

If $f\in S_{2}(\Gamma_{0}(N),\mathbb{Z})$ is associated to a rational elliptic
curve, then $\alpha(f)$ is a $p$-adic unit whenever $p$ is non-Eisenstein and
the residual representation $\overline{\rho_{f}}\colon G_{\mathbb{Q}}\to
GL_{2}(\mathcal{O}_{v}/\varpi)$ attached to $f$ is ramified at each prime
$l\mid N_{i}^{-}$; see [Pra06, page 912].

######  Definition 4.8.

Let $f=\sum b_{n}q^{n}$ be a normalized newform of level $N^{\prime}$ and $N$
an integer such that $N^{\prime}\mid N$. Assume that $N^{\prime}$ satisfies
(sq-fr). If $q\mid(N/N^{\prime})$, we assume that $q^{2}\mid N$. Let
$\check{f}=\sum a_{n}q^{n}$ be the eigenform of level $N$ with Fourier
coefficients $a_{q}=0$ for $q\mid N/N^{\prime}$ and $a_{q}=b_{q}$ for all $q$
that are coprime to $N/N^{\prime}$.

While $\check{f}$ depends on $N$, we suppress it from the notation since it
will be clear from the context what $N$ is. Recall that we decompose any Hecke
character of $H_{p^{\infty}}$ as a product $\varphi\psi$, where $\varphi$
factors through $H_{\mathfrak{p^{\infty}}}$. We denote by
$\mathbf{g}(\varphi)$ the specialization of the Hida family $\mathbf{g}$ at
$\varphi$. Let $P_{q}(\mathbf{g}(\varphi),f,q^{-s})$ be the Euler factor of
the tensor product of the representations associated to $\mathbf{g}(\varphi)$
and $f$. The following lemma follows directly from definition.

######  Lemma 4.9.

Let $\tilde{N}_{3}$ be a product of rational primes that split in
$K/\mathbb{Q}$. Let $\varphi\psi$ be an anticyclotomic Hecke character of
infinity type $(b,-b)$ with $b\equiv 0\pmod{(p-1)}$ and let $\rho$ be the
character on $\Gamma^{ac}$ induced from $\varphi\psi$. Then

$\prod_{q\mid\tilde{N}_{3}}\prod_{\mathfrak{q}\mid
q}\mathcal{P}_{\mathfrak{q}}^{f}(\rho)=\prod_{q\mid\tilde{N}_{3}}P_{q}(\mathbf{g}(\varphi),f,q^{-(1+b)}).$

The following theorem can be regarded as an analytic analogue of Lemma 3.5.

######  Theorem 4.10.

Let $N^{\prime}|N$ be positive integers that are coprime to $p\cdot d_{K}$.
Assume that $N^{\prime}$ satisfies (sq-fr). If $q$ is a prime dividing
$N/N^{\prime}$, we assume further that $q^{2}\mid N$. Let $f=\sum
b_{q}q^{n}\in S_{2}(\Gamma_{0}(N^{\prime}),\mathcal{O}_{v})$ be a normalized
newform such that $(f,N^{\prime})$ satisfy the hypotheses (GHH) and (div). Let
$\check{f}$ be the eigenform of level $N$ defined in Definition 4.8. Then

  * (a)

$\mathcal{L}^{\textup{ac}}_{p,N}(\check{f})L^{\textup{ac}}_{K,p}\in\Lambda^{\textup{ac}}$.

  * (b)

Let $\tilde{N}_{1}$ be the product of all primes $q$ that divide $N$ but not
$N^{\prime}$ and are inert in $K/\mathbb{Q}$. Let $\tilde{N}_{2}$ be the
product of all common prime divisors of $N^{\prime}$ and $N/N^{\prime}$ that
are inert in $K/\mathbb{Q}$. Let $\tilde{N}_{3}$ be the product of all $q\mid
N/N^{\prime}$ that are split in $K/\mathbb{Q}$. If

(unit)
$\prod_{q\mid\tilde{N}_{1}}(1+q-b_{q})(1+q+b_{q})\prod_{q\mid\tilde{N}_{2}}(q-b_{q})(q+b_{q})\in\mathcal{O}_{v}^{\times},$

then

$\displaystyle\mu(\mathcal{L}^{\mathrm{ac}}_{p,N}(\check{f})L^{\textup{ac}}_{K,p})$
$\displaystyle=\mu(\mathcal{L}^{\mathrm{ac}}_{p,N^{\prime}}(f)L^{\textup{ac}}_{K,p}),$
$\displaystyle\lambda(\mathcal{L}_{p,N}^{\mathrm{ac}}(\check{f})L^{\textup{ac}}_{K,p})$
$\displaystyle=\lambda(\mathcal{L}^{\mathrm{ac}}_{p,N^{\prime}}(f)L^{\textup{ac}}_{K,p})+\sum_{\mathfrak{q}\mid\tilde{N}_{3}}\lambda(\mathcal{P}^{f}_{\mathfrak{q}}),$

where the sum runs over primes of $K$ dividing $\tilde{N}_{3}$ and
$\mathcal{P}^{f}_{\mathfrak{q}}$ is defined as in Lemma 3.5.

###### Proof.

Let $\varphi\psi$ be an anticyclotomic Hecke character of trivial conductor
and infinity type $(b,-b)$ with $b\equiv 0\pmod{(p-1)}$. Let
$h\in\\{f,\check{f}\\}$. Let $M=N^{\prime}|d_{k}|$ if $h=f$ and $M=N|d_{k}|$
if $h=\check{f}$. Let $\mathcal{D}({\mathbf{g}}(\varphi),h,1/M,s)$ be the
integral defined in [LLZ14, §4.2, P.55]. Here, $s$ is a complex variable. It
follows from [LLZ14, Proposition 5.4.2] that

(4.1)
$\mathcal{L}_{p,M}(h)(\varphi\psi)=(*)\mathcal{D}({\mathbf{g}}(\varphi),h,1/M,1+b),$

where $(*)$ is a factor only depending on $f$ but not on $h$ nor $M$.

We now compare $\mathcal{D}({\mathbf{g}}(\varphi),\check{f},1/N|d_{k}|,1+b)$
and $\mathcal{D}({\mathbf{g}}(\varphi),f,1/N^{\prime}|d_{k}|,1+b)$. Let

$C({\mathbf{g}}(\varphi),h,M,s)=\prod_{q\mid
M}P_{q}(\mathbf{g}(\varphi),h,q^{-s})\sum_{S(M)}a_{n}b_{n}n^{-s},$

where $S(M)$ is the set of integers whose prime factors divide $M$, and
$a_{n}$ and $b_{n}$ are the Fourier coefficients of ${\mathbf{g}}(\varphi)$
and $h$, respectively. We recall from [LLZ14, Theorem 4.2.3] that
$C({\mathbf{g}}(\varphi),h,M,s)$ is a polynomial in $q^{-s}$ for $q\mid M$. In
particular, it is an entire function in $s$.

Thanks to [LLZ14, Theorem 4.2.3], $\mathcal{D}({\mathbf{g}}(\varphi),h,1/M,s)$
and $C({\mathbf{g}}(\varphi),h,M,s)$ differ only by a $p$-adic unit and the
completed $L$-function $\Lambda({\mathbf{g}}(\varphi),f,s)$, which is
independent of the choice of $h$ (see [LLZ14, Proposition 4.1.5] for the
precise definition). In the remainder of the proof, we prove that up to a
$p$-adic integer, $C({\mathbf{g}}(\varphi),\check{f},N|d_{k}|,s)$ and
$C({\mathbf{g}}(\varphi),f,N^{\prime}|d_{k}|,s)$ differ by
$\prod_{q|\tilde{N}_{3}}P_{q}(\mathbf{g}(\varphi),f,q^{-s})$. It would then
follow from Lemma 4.9 and (4.1) that
$\mathcal{L}_{p,N}^{\mathrm{ac}}(\check{f})L^{\textup{ac}}_{K,p}$ and
$\mathcal{L}^{\mathrm{ac}}_{p,N^{\prime}}(f)L^{\textup{ac}}_{K,p}\prod_{\mathfrak{q}\mid\tilde{N}_{3}}\mathcal{P}^{f}_{\mathfrak{q}}$
have the same Iwasawa invariants. This would imply the assertions of the
theorem since
$\mathcal{L}_{p,N^{\prime}}^{\textup{ac}}(f)L^{\textup{ac}}_{K,p}\in\Lambda^{\mathrm{ac}}$
by Theorem 4.6 and
$\mathcal{P}_{\mathfrak{q}}^{f}\in\Lambda^{\mathrm{ac}}_{\mathcal{O}_{v}}$
with $\mu(\mathcal{P}_{\mathfrak{q}}^{f})=0$ for all
$\mathfrak{q}|\tilde{N}_{3}$.

Without loss of generality, we may assume that $N/N^{\prime}=q^{l}$ for some
prime $q$. We will distinguish three cases:

  * (1)

$q\not\mid N^{\prime}$

  * (2)

$q\mid N^{\prime}$ but $q^{2}\not\mid N^{\prime}$

  * (3)

$q^{2}\mid N^{\prime}$

In case (3), $\check{f}=f$ and $\mathcal{P}^{f}_{\mathfrak{q}}$ is a unit.

We consider case (1). By the definition of $\check{f}$, we have $b_{q}=0$.
Thus,

$\sum_{S(N|d_{k}|)}a_{n}b_{n}n^{-s}=\sum_{S(N^{\prime}|d_{k}|)}a_{n}b_{n}n^{-s},$

which implies that

$C({\mathbf{g}}(\varphi),\check{f},N|d_{k}|,s)=P_{q}(\mathbf{g}(\varphi),f,q^{-s})C({\mathbf{g}}(\varphi),f,N^{\prime}|d_{k}|,s).$

If $q$ is inert in $K$, then

$P_{q}(\mathbf{g}(\varphi),f,q^{-(1+b)})=(1-b_{q}q^{-1}+q^{-1})(1+b_{q}q^{-1}+q^{-1})$

does not depend on $\varphi$ (see [LLZ14, Proposition 4.1.2] for details). In
particular, it does not depend on $\varphi\psi$ and is a $p$-adic integer.

It remains to consider case (2). In this case,

$\prod_{q\mid N}P_{q}(\mathbf{g}(\varphi),f,q^{-s})=\prod_{q\mid
N^{\prime}}P_{q}(\mathbf{g}(\varphi),f,q^{-s})$

and only the sum $\sum_{S(M)}a_{n}b_{n}n^{-s}$ depends on $M$ (since the
Fourier coefficient at $q$ differs). Decomposing $\sum_{S(M)}a_{n}b_{n}n^{-s}$
into a finite Euler product, we see that only an Euler factor at $q$ is
altered when passing from $f$ to $\check{f}$.

If $q$ is inert in $K$, this factor again does not depend on $\varphi\psi$. If
$q$ splits in $K$, we can proceed as in case (1). ∎

######  Remark 4.11.

Keeping track of the units in the proof of Theorem 4.10 yields

$\displaystyle
L^{\mathrm{ac}}_{K,p}\mathcal{L}^{\mathrm{ac}}_{p,N}(\check{f})=(N/N^{\prime})\prod_{q\mid\tilde{N}_{1}}(1+q^{-1}-b_{q}q^{-1})(1+q^{-1}+b_{q}q^{-1})$
$\displaystyle\prod_{q\mid\tilde{N}_{2}}(1+b_{q}q^{-1})(1-b_{q}q^{-1})\prod_{\mathfrak{q}\mid\tilde{N}_{3}}\mathcal{P}^{f}_{\mathfrak{q}}\times
L^{\mathrm{ac}}_{K,p}\mathcal{L}^{\mathrm{ac}}_{p,N}(f)$

As an immediate consequence, we deduce:

######  Corollary 4.12.

Let $h\in S_{2}(\Gamma_{0}(N),\mathcal{O}_{v})$ be an $L_{v}$-linear
combination of Hecke eigenforms which satisfy the hypotheses of Theorem 4.10.
Then,

$L^{\textup{ac}}_{K,p}\mathcal{L}_{p,N}^{\textup{ac}}(h)\in\Lambda^{\mathrm{ac}}\otimes\mathbb{Q}_{p}.$

###  4.3. Integrality and anticyclotomic specialization

The goal of this subsection is to prove an integral version of Corollary 4.12.
Looking at the interpolation formula for $\mathcal{L}_{p,N}(h)$ one would
suspect that the denominator of $\mathcal{L}_{N,p}(h)$ is cleared by an ideal
interpolating the values
$\langle\mathbf{g}(\varphi),\mathbf{g}(\varphi)\rangle_{|d_{K}|}$, where
$\varphi$ is a Hecke character as above. It turns out that this is true in
view of the following result on congruence ideals due to Hida and Tilouine,
which allows us to study the denominator of ${\mathcal{L}}_{p,N}(h)$.

######  Proposition 4.13.

Let $H_{\mathbf{g}}$ be an annihilator in the congruence ideal associated to
the Hida family $\mathbf{g}$ as defined in [HT94]. For any $h\in
S_{2}(\Gamma_{0}(N),\mathcal{O}_{v})$, we have

$H_{\mathbf{g}}\mathcal{L}_{p,N}(h)\in\Lambda_{p}$

###### Proof.

Our strategy is similar to the one employed in [Col20, Sections 4-5]. Let
$\varphi\psi$ be a Hecke character of trivial conductor and infinity type
$(-l_{2},l_{1})$ with $l_{2}-l_{1}\geq 1$. Then $\mathbf{g}(\varphi)$ is a
newform of level $|d_{K}|$ and weight $l_{2}-l_{1}+1$. Analogously to [Hid88,
Section 7], we define the $\Lambda_{p}$-linear map

$\ell_{\mathbf{g}}\mathrel{\mathop{\mathchar
58\relax}}S(|d_{K}|,\Lambda_{p})\rightarrow\operatorname{Frac}\Lambda_{p}$

whose specialization at $\varphi\psi$ is given by

$\ell_{\mathbf{g}(\varphi)}(h(\varphi\psi))=a(1,(e_{\operatorname{ord}}h(\varphi\psi))\big{|}1_{\mathbf{g}(\varphi)}),$

where $h(\varphi\psi)$ is a form of level $|d_{K}|$ and weight
$l_{2}-l_{1}+1$, $1_{\mathbf{g}(\varphi)}$ is the projector induced by
$\mathbf{g}(\varphi)$ defined in [Hid85] and $a(1,f)$ denotes the first
Fourier coefficient of $f$. It follows from [Hid88, Proposition 7.8] that
$H_{\mathbf{g}}\ell_{\mathbf{g}}$ takes values in $\Lambda_{p}$. Note that
Hida defines the function $\ell_{\mathbf{g}}$ only on the space of ordinary
$\Lambda_{p}$-adic modular forms. But the ordinary projector
$e_{\operatorname{ord}}$ induces an injection of Hecke algebras
$T^{\operatorname{ord}}\to T$. Thus, we can extend $\ell_{\mathbf{g}}$ to the
space of all modular forms by applying $e_{\operatorname{ord}}$ to
$h(\varphi\psi)$.

Recall that
$\mathbf{g}(\varphi)\big{|}1_{\mathbf{g}(\varphi)}=\mathbf{g}(\varphi)$. If
$g^{\prime}$ is an eigenform that is not a multiple of $\mathbf{g}(\varphi)$,
then $g^{\prime}\big{|}1_{\mathbf{g}(\varphi)}=0$. It follows that

$\ell_{\mathbf{g}(\varphi)}(h(\varphi\psi))=\frac{\langle\mathbf{g}(\varphi)^{*},e_{\operatorname{ord}}(h(\varphi\psi))\rangle_{|d_{K}|}}{\langle\mathbf{g}(\varphi),\mathbf{g}(\varphi)\rangle_{|d_{K}|}}.$

We extend this linear functional to forms of level $N|d_{K}|$ as follows.
Consider the trace operator

$\displaystyle\textup{Tr}\colon S(\Gamma_{1}(N|d_{K}|),\Lambda_{p})$
$\displaystyle\to\ S(\Gamma_{1}(|d_{K}|,\Lambda_{p}),$ $\displaystyle f_{k}$
$\displaystyle\mapsto\sum_{\gamma}f|_{k}\gamma,$

where $f_{k}$ is the weight $k$ specialization of a Hida family $\mathbf{f}$,
$|_{k}\gamma$ denotes the slash action for modular forms and the sum runs over
a set of representatives for
$\big{(}\Gamma_{1}(|d_{K}|)\cap\Gamma_{0}(p)\big{)}/\big{(}\Gamma_{1}(|d_{K}|N)\cap\Gamma_{0}(p)\big{)}$.
Then

$\displaystyle\langle\mathbf{g}^{*}(\varphi),\textup{Tr}(f^{\prime})\rangle_{|d_{K}|}$
$\displaystyle=\ $
$\displaystyle\frac{1}{[\Gamma_{1}(|d_{K}|)\cap\Gamma_{0}(p)\mathrel{\mathop{\mathchar
58\relax}}\Gamma_{1}(N|d_{K}|)\cap\Gamma_{0}(p)]}\langle\mathbf{g}^{*}(\varphi),\textup{Tr}(f^{\prime})\rangle_{N|d_{K}|}$
$\displaystyle=\ $
$\displaystyle\langle\mathbf{g}^{*}(\varphi),f^{\prime}\rangle_{N|d_{K}|}.$

Let $\ell^{\prime}_{\mathbf{g}}=\ell_{\mathbf{g}}\circ\textup{Tr}$. By
definition $H_{\mathbf{g}}\ell^{\prime}_{\mathbf{g}}$ is integral and its
specialization at $\varphi\psi$ is given by

$\ell^{\prime}_{\mathbf{g}(\varphi)}(h(\varphi\psi))=\frac{\langle\mathbf{g}(\varphi)^{*},e_{\operatorname{ord}}(h(\varphi\psi))\rangle_{N|d_{K}|}}{\langle\mathbf{g}(\varphi),\mathbf{g}(\varphi)\rangle_{|d_{K}|}},$

which concludes the proof. ∎

We review the relation between the congruence ideal $H_{\mathbf{g}}$ and the
anticyclotomic specialization of $L_{K,p}$.

Let $M(\mathcal{K})/\mathcal{K}$ (resp,
$M(K^{\textup{a}}_{\infty})/K^{\textup{a}}_{\infty})$) be the maximal abelian
pro-$p$ extension unramified outside $\mathfrak{p}$. Then
$X(\mathcal{K})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(M(\mathcal{K})/\mathcal{K})$ (resp.
$X(K^{\textup{a}}_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(M(K^{\textup{a}}_{\infty})/K^{\textup{a}}_{\infty})$)
is a finitely generated $\Lambda$-torsion (resp.
$\Lambda^{\textup{a}}$-torsion) module.

Recall from §2 that $\Delta$ denotes the torsion subgroup of $H_{p^{\infty}}$.
Let $\chi$ be a character of $\Delta$. As $|\Delta|$ is coprime to $p$, each
$\Lambda_{p}$-module $M$ decomposes into $M=\oplus M_{\chi}$, where $M_{\chi}$
is a $\Lambda_{p}$-module on which $\Delta$ acts via $\chi$. In particular,
$\Lambda_{\chi}$ is the maximal subalgebra of $\Lambda_{p}$ on which $\Delta$
acts via $\chi$. Let $F_{\chi}$ be a generator of the characteristic ideal of
$X(\mathcal{K})_{\chi}$ as a $\Lambda_{\chi}$-module.

The following result on the Iwasawa main conjecture over $K$ is due to Rubin.

######  Theorem 4.14 (Rubin).

As ideals in $\Lambda_{\chi}$, we have the equality

$(F_{\chi})=(L_{K,p,\chi}),$

where $L_{K,p,\chi}$ denotes the $\chi$-isotypic component of $L_{K,p}$.

###### Proof.

This follows from [Rub91, Theorem 4.1(i)] combined with the remark after
Theorem 12.2 of op. cit. ∎

Similarly, every $\Lambda^{\textup{a}}$-module $M$ decomposes as $M=\oplus
M_{\chi}$, where the direct sum runs over all characters of $\Delta$ that
factor through $\operatorname{Gal}(K^{\textup{a}}_{\infty}/K)$. For any such
$\chi$, we denote by $f_{\chi}$ a generator of the characteristic ideal of
$X(K^{\textup{a}}_{\infty})_{\chi}$ as a $\Lambda_{\chi}^{\textup{a}}$-module.
The following theorem is due to Hida and Tilouine.

######  Theorem 4.15.

[HT94, Theorem 0.3] Let $\chi$ be a character of $\Delta$ factoring through
$\operatorname{Gal}(K^{a}_{\infty}/K)$ and let $H_{\mathbf{g},\chi}$ be the
$\chi$-isotypic component of $H_{\mathbf{g}}$. Then as ideals in
$\Lambda_{\chi}^{\textup{a}}$,

$H_{\mathbf{g},\chi}\mid h(K)\mathcal{V}(f_{\chi}),$

where $h(K)$ denotes the class number of $K$ and $\mathcal{V}$ is defined as
in §2.

######  Corollary 4.16.

We have

$\mu(\pi^{\textup{a}}(H_{\mathbf{g},\chi}))\leq\mu(f_{\chi}),$

for all characters of $\Delta$ that factor through
$\operatorname{Gal}(K^{\textup{a}}_{\infty}/K)$.

###### Proof.

By Theorem 4.15, $\pi^{\textup{a}}(H_{\mathbf{g},\chi})\mid
h(K)\pi^{\textup{a}}\mathcal{V}(f_{\chi})$ over $\Lambda^{\textup{a}}_{\chi}$.
As $p$ does not divide $h(K)$ by assumption and
$\mu(\pi^{\textup{a}}(\mathcal{V}(G)))=\mu(G)$ for every
$G\in\Lambda^{\textup{a}}$, the claim follows. ∎

Next we prove a relation between $f_{\chi}$ and $\pi^{\textup{a}}(F_{\chi})$.

######  Lemma 4.17.

Let $\chi$ be a character of $\Delta$ that factors through
$\operatorname{Gal}(K^{\textup{a}}_{\infty}/K)$. Then

$\mu(\pi^{\textup{a}}(F_{\chi}))=\mu(f_{\chi}).$

###### Proof.

For every $\Lambda_{p}$-module $M$ we denote by $\pi\colon M\to M/\ker(\pi)M$
the natural projection induced by
$\pi^{\textup{a}}\colon\Lambda_{p}\to\Lambda^{\textup{a}}$. Let
$M^{\prime}(\mathcal{K})\subset M(\mathcal{K})$ be the maximal extension that
is abelian over $K^{\textup{a}}_{\infty}$. Note that $\mathcal{K}\subset
M^{\prime}(\mathcal{K})$ and that $\pi(X(\mathcal{K}))$ is the maximal
quotient of $X(\mathcal{K})$ on which
$\operatorname{Gal}(\mathcal{K}/K^{\textup{a}}_{\infty})$ acts trivially. It
follows that
$\pi(X(\mathcal{K}))=\operatorname{Gal}(M^{\prime}(\mathcal{K})/\mathcal{K})$.
As $\mathcal{K}/K^{\textup{a}}_{\infty}$ is unramified we see that
$M^{\prime}(\mathcal{K})=M(K^{\textup{a}}_{\infty})$. Thus, we have an exact
sequence.

$0\to\pi(X(\mathcal{K}))_{\chi}\to
X(K^{\textup{a}}_{\infty})_{\chi}\to\operatorname{Gal}(K_{\infty}/K^{\textup{a}}_{\infty})^{\textup{pro-}p}.$

The lemma now follows from the fact that
$\operatorname{Gal}(K_{\infty}/K^{\textup{a}}_{\infty})^{\textup{pro-}p}$ is
finitely generated over $\mathbb{Z}_{p}$ (so its $\mu$-invariant vanishes). ∎

######  Corollary 4.18.

Let $\mathbbm{1}$ be the trivial character of $\Delta$. Then we have

$\mu(\pi^{\mathrm{ac}}(H_{\mathbf{g},\mathbbm{1}}))\leq\mu(L^{\textup{ac}}_{K,p}).$

###### Proof.

It follows from Theorem 4.14, Corollary 4.16 and Lemma 4.17 that

$\mu(\pi^{\mathrm{ac}}(H_{\mathbf{g},\mathbbm{1}}))\leq\mu(\pi^{\mathrm{ac}}(L_{K,p,\mathbbm{1}})).$

By the definition of $L^{\mathrm{ac}}_{p,K}$, we have
$\mu(\pi^{\mathrm{ac}}(L_{K,p,\mathbbm{1}}))\leq\mu(L^{\mathrm{ac}}_{p,K})$,
which concludes the proof. ∎

We can now deduce the following integral version of Corollary 4.12.

######  Corollary 4.19.

Let $h$ be as in Corollary 4.12. Then

$\mathcal{L}^{\textup{ac}}_{p,N}(h)L^{\textup{ac}}_{K,p}\in\Lambda^{ac}.$

###### Proof.

This follows from Proposition 4.13 combined with Corollaries 4.12 and 4.18. ∎

###  4.4. Proof of Theorem B

We are now ready to give the proof of Theorem B stated in the introduction.

######  Theorem 4.20.

Let $N_{1}$ and $N_{2}$ be two positive integers that are coprime to $p\cdot
d_{K}$ satisfying (GHH). Assume that $N_{i}$ satisfies (sq-fr). For
$i\in\\{1,2\\}$, let $f_{i}=\sum b^{(i)}_{q}q^{n}\in
S_{2}(\Gamma_{0}(N_{i}),\mathcal{O}_{v})$ be a newform satisfying (Cong),
(div), and (unit), with respect to $N^{\prime}=N_{i}$ and $N$ chosen to be a
common multiple of $N_{1}$ and $N_{2}$ that is also a square.

Then $\alpha(f_{1})$ is a $p$-adic unit and
$\mu(L_{p}^{\mathrm{BDP}}(f_{1}))=0$ if and only if $\alpha(f_{2})$ is a
$p$-adic unit and $\mu(L_{p}^{\mathrm{BDP}}(f_{2}))=0$. Furthermore, when this
holds,

$2\lambda(L_{p}^{\mathrm{BDP}}(f_{1}))+\sum_{\mathfrak{q}\mid\tilde{N}^{(1)}_{3}}\lambda(\mathcal{P}^{f_{1}}_{\mathfrak{q}})=2\lambda(L_{p}^{\mathrm{BDP}}(f_{2}))+\sum_{\mathfrak{q}\mid\tilde{N}^{(2)}_{3}}\lambda(\mathcal{P}^{f_{2}}_{\mathfrak{q}}).$

###### Proof.

The hypothesis (Cong) implies that $b_{q}^{(1)}\equiv
b_{q}^{(2)}\pmod{\varpi}$ for all $q$ that are coprime to $pN$. If $f_{i}$ are
non-ordinary at $p$, then $b^{(1)}_{p}\equiv b^{(2)}_{p}\equiv 0\
\pmod{\varpi}$. If $f_{i}$ are ordinary at $p$, the congruence
$b_{p}^{(1)}\equiv b_{p}^{(2)}\pmod{\varpi}$ follows from [Wil88, Theorem 2].
In other words, the Fourier coefficients of $f_{1}$ and $f_{2}$ are congruent
modulo $\varpi$ at primes outside $N$. Let $\check{f}_{1}$ and $\check{f}_{2}$
be the elements of $S_{2}(\Gamma_{0}(N),\mathcal{O}_{v})$ given by Definition
4.8. All their Fourier coefficients are congruent modulo $\varpi$. In
particular, there exists $h\in S_{2}(\Gamma_{0}(N),\mathcal{O}_{v})$ such that
$\check{f}_{1}-\check{f}_{2}=\varpi h$.

By Corollary 4.19 we see that
$L^{\textup{ac}}_{K,p}\mathcal{L}^{\textup{ac}}(h)\in\Lambda^{\textup{ac}}$.
As discussed in Remark 4.4, we have

(4.2)
$\left(\mathcal{L}^{\textup{ac}}_{p,N}(\check{f}_{1})-\mathcal{L}^{\textup{ac}}_{p,N}(\check{f}_{2})\right)L^{\textup{ac}}_{K,p}=\varpi
L^{\textup{ac}}_{K,p}\mathcal{L}^{\textup{ac}}_{p,N}(h)\in\varpi\Lambda^{\textup{ac}}.$

Thus,

$\mu(\mathcal{L}^{\textup{ac}}_{p,N}(\check{f}_{1})L^{\textup{ac}}_{K,p})=0\Leftrightarrow\mu(\mathcal{L}^{\textup{ac}}_{p,N}(\check{f}_{2})L^{\textup{ac}}_{K,p})=0.$

Consequently, Theorem 4.10(b) implies that

$\mu(\mathcal{L}^{\textup{ac}}_{p,N_{1}}(\check{f}_{1})L^{\textup{ac}}_{K,p})=0\Leftrightarrow\mu(\mathcal{L}^{\textup{ac}}_{p,N_{2}}(\check{f}_{2})L^{\textup{ac}}_{K,p})=0.$

Therefore, the first assertion of the theorem follows from Theorems 4.6.

When these equivalent conditions hold, equation (4.2) implies that

$\lambda(\mathcal{L}^{\textup{ac}}_{p,N}(\check{f}_{1})L^{\textup{ac}}_{K,p})=\lambda(\mathcal{L}^{\textup{ac}}_{p,N}(\check{f}_{2})L^{\textup{ac}}_{K,p}).$

Therefore, combining Theorems 4.6 and 4.10(b) gives rise to the equations

$\displaystyle
2\lambda({L}_{p}^{\mathrm{BDP}}(f_{1}))+\sum_{q\mid\tilde{N}^{(1)}_{3}}\sum_{\mathfrak{q}\mid
q}\lambda(\mathcal{P}^{f_{1}}_{\mathfrak{q}})=\ $
$\displaystyle\lambda(\mathcal{L}^{\mathrm{ac}}_{p,N_{1}}(f_{1})L^{\textup{ac}}_{K,p})+\sum_{q\mid\tilde{N}^{(1)}_{3}}\sum_{\mathfrak{q}\mid
q}\lambda(\mathcal{P}^{f_{1}}_{\mathfrak{q}})$ $\displaystyle=\ $
$\displaystyle\lambda(\mathcal{L}^{\textup{ac}}_{p,N}(\check{f}_{1})L^{\textup{ac}}_{K,p})$
$\displaystyle=\ $
$\displaystyle\lambda(\mathcal{L}^{\textup{ac}}_{p,N}(\check{f}_{2})L^{\textup{ac}}_{K,p})$
$\displaystyle=\ $
$\displaystyle\lambda(\mathcal{L}_{p,N_{2}}^{\mathrm{ac}}(f_{2})L^{\textup{ac}}_{K,p})+\sum_{q\mid\tilde{N}^{(2)}_{3}}\sum_{\mathfrak{q}\mid
q}\lambda(\mathcal{P}^{f_{2}}_{\mathfrak{q}})$ $\displaystyle=\ $
$\displaystyle
2\lambda(L_{p}^{\mathrm{BDP}}(f_{2}))+\sum_{q\mid\tilde{N}^{(2)}_{3}}\sum_{\mathfrak{q}\mid
q}\lambda(\mathcal{P}^{f_{2}}_{\mathfrak{q}}),$

which concludes the proof. ∎

######  Remark 4.21.

For $1\leq j\leq 3$ let $M^{(i)}_{j}\mid\tilde{N}_{j}^{(i)}$ be the product of
all primes dividing $\tilde{N}^{(i)}_{3}$ such that $b^{(1)}_{q}\not\equiv
b^{(2)}_{q}\pmod{\varpi}$ and redefine $N$ as the greatest common multiple of
$N_{1}$ and $N_{2}$ such that $q^{2}\mid N$ for each prime $q$ that divides
one of the numbers $M^{(i)}_{j}$ for $i\in\\{1,2\\}$ and $1\leq j\leq 3$.
These choices are enough to ensure that $\check{f_{1}}$ and $\check{f_{2}}$
have congruent Fourier-coefficients everywhere.

Analogous to Lemma 3.10, we deduce the following application of Theorem 4.20.

######  Lemma 4.22.

Let $f$ be a newform of level $N$ and weight $2$ and let $\chi$ be a quadratic
character such that $\chi(p)=-1$. Assume that $p\in\\{3,5\\}$ and that $f$ has
partial Eisenstein decent by $(\chi,\chi,N_{1},N_{2},N_{0})$. Let $K$ be an
imaginary quadratic field so that (GHH) holds with respect to
$N\textup{cond}(\chi)$. We assume that $N^{-}=1$ and that $11$ and $19$ split
in $K$. Then

$\mu(L_{p}^{\mathrm{BDP}}(f))=0.$

###### Proof.

Let $E$ be the elliptic curve and $E^{\chi}$ its quadratic twist, given as in
the proof of Lemma 3.10. Let $g$ be the weight-two modular form corresponding
to $E^{\chi}$. By [CGLS22, Theorem 2.2.2] $\mu(L^{\mathrm{BDP}}_{p}(g))=0$. As
$N^{-}=1$ we know that $\alpha(g)=1$ and that (div) and (unit) are satisfied.
Thus, we can apply Theorem 4.20 to $f$ and $g$. ∎

###  4.5. An application: congruence relations between Heegner points

For $i\in\\{1,2\\}$, let $f_{i}\in S_{2}(\Gamma_{0}(N_{i}),\mathcal{O}_{v})$
be a newform. Recall from Theorem 4.6 that up to an explicit unit $u$ of
$\Lambda^{\mathrm{ac}}$ only depending on $N$ and the imaginary quadratic
field $K$, we have

$\mathcal{L}_{p,N}^{\textup{ac}}(f_{i})L^{\textup{ac}}_{K,p}=u\alpha(f_{i})L_{p}^{\mathrm{BDP}}(f_{i})^{2}.$

We choose $N$ to be a common multiple of $N_{1}$ and $N_{2}$ that is a perfect
square. Let $\check{f_{i}}\in S_{2}(\Gamma_{0}(N),\mathcal{O}_{v})$ be the
oldform of level $N$ corresponding to the newform $f_{i}$ as in Definition 4.8
.

Recall from the proof of Theorem 4.10 that for an anticyclotomic Hecke
character $\varphi\psi$ of trivial conductor and infinity type $(b,-b)$, there
is an explicit integer $c(f_{i})\in\mathcal{O}_{v}$ independent of
$\varphi\psi$ given by a product of Euler factors at primes that are inert in
$K/\mathbb{Q}$ such that the equation

$C({\mathbf{g}}(\varphi),\check{f_{i}},N,b+1)=c(f_{i})\prod_{\mathfrak{q}\mid
M_{3}^{(i)}}\mathcal{P}^{f_{i}}_{\mathfrak{q}}C({\mathbf{g}}(\varphi),f_{i},N_{i},b+1)$

holds. The proof of Theorem 4.20 implies the following congruence result for
BDP $p$-adic $L$-functions.

######  Proposition 4.23.

Let $f_{1}$ and $f_{2}$ be two newforms satisfying the assumptions in Remark
4.21 and all other conditions of Theorem 4.20. Then we have the congruence
relation

$\displaystyle\alpha(f_{1})c(f_{1})\prod_{\mathfrak{q}\mid
M^{(1)}_{3}}\mathcal{P}^{f_{1}}_{\mathfrak{q}}\sigma^{-1}_{\mathcal{N}_{1}^{+}}L_{p}^{\mathrm{BDP}}(f_{1})^{2}(\varphi\psi)$
$\displaystyle\equiv\sigma^{-1}_{\mathcal{N}_{2}^{+}}\alpha(f_{2})c(f_{2})\prod_{\mathfrak{q}\mid\tilde{M}^{(2)}_{3}}\mathcal{P}^{f_{2}}_{\mathfrak{q}}L_{p}^{\mathrm{BDP}}(f_{2})^{2}(\varphi\psi)\pmod{\varpi}.$

Recall that $f_{i}$ was chosen to be a new form of level
$N_{i}=N^{+}_{i}N^{-}_{i}$. Let $B_{i}$ be the indefinite quaternion algebra
of conductor $N^{-}_{i}$ over $\mathbb{Q}$ and let $\textup{Sh}_{B_{i}}$ be
the Shimura curve of level $\Gamma_{1}(N_{i}^{+})$. We denote its Jacobian by
$J_{B_{i}}$. Let $f_{i}^{\textup{JL}}$ be the Jacquet-Langlands transfer of
$f_{i}$ to $\textup{Sh}_{B_{i}}$, normalized by requiring the corresponding
section of a certain canonical line bundle ${\mathcal{L}}$ over a canonical
proper smooth model of $\textup{Sh}_{B_{i}}$ over $\mathbf{Z}_{p}$ to be
integral and primitive with respect to the integral structure of
${\mathcal{L}}$ (see [Pra06, Section 2] for details). The modular form
$f_{i}^{\textup{JL}}$ induces a differential form $\omega_{f_{i}}$ on
$\textup{Sh}_{B_{i}}$ and $J_{B_{i}}$.

We fix an embedding $K\hookrightarrow B_{i}$. Let $\tau$ be its fixed point in
the upper half plane and we let $(A,t,\omega)$ be the triple associated to
$\tau$, where $A$ denotes a false elliptic curve, $t$ a level
$N^{+}$-structure and $\omega$ a global non-vanishing differential (see [HB15,
Section 2] for details). There is a well-defined action of the ideal class
group $\textup{Cl}_{K}$ on the equivalence classes of triples
$[(A,t,\omega)]$. For each ideal class $[\mathfrak{A}]$ coprime to $N^{+}$ we
define the CM point $P_{\mathfrak{A}}$ on $\textup{Sh}_{B_{i}}$ to be the
point corresponing to $[\mathfrak{A}(A,t,\omega)]$. We define the Heegner
point $\Delta_{f_{i}}$ associated to $f_{i}$ as the sum

$\sum_{[\mathfrak{A}]\in\textup{Cl}_{K}}(P_{\mathfrak{A}}-\infty),$

which is a priori defined over a finite extension of $K$ containing the
Fourier-coefficients of $f_{i}$ and a quadratic extension only depending on
$N_{i}$. As we are only working with weight-two forms, one can actually show
that it is defined over $K$ (see [HB15, discussion after Proposition 8.12]).
Translating [HB15, Proposition 8.12] into our setting, we obtain the following
relation.

######  Theorem 4.24.

Let $\mathbbm{1}$ be the trivial character on $\Gamma^{\mathrm{ac}}$. Then

$\displaystyle(L_{p}^{\mathrm{BDP}}(f)(\mathbbm{1}))^{2}=(1-a_{p}p^{-1}+p^{-1})^{2}\log_{\omega_{f_{i}}}(\Delta_{f_{i}})^{2}.$

We conclude this section with the following partial generalization of [KL19,
Theorem 3.9].

######  Theorem 4.25.

Let $f_{1}$ and $f_{2}$ be newforms of level $N_{1}$ and $N_{2}$ respectively.
Assume that the conditions of Remark 4.21 and all the conditions of Theorem
4.20 are satisfied. Then

$\displaystyle
c(f_{1})\alpha(f_{1})\prod_{q\mid{M}^{(1)}_{3}}\prod_{\mathfrak{q}\mid
q}\mathcal{P}^{f_{1}}_{\mathfrak{q}}(0)(1-b^{(1)}_{p}p^{-1}+p^{-1})^{2}\log_{\omega_{f_{1}}}(\Delta_{f_{1}})^{2}$
$\displaystyle\equiv
c(f_{2})\alpha(f_{2})\prod_{q\mid{M}^{(2)}_{3}}\prod_{\mathfrak{q}\mid
q}\mathcal{P}^{f_{2}}_{\mathfrak{q}}(0)(1-b^{(2)}_{p}p^{-1}+p^{-1})^{2}\log_{\omega_{f_{2}}}(\Delta_{f_{2}})^{2}\pmod{\varpi}.$

###### Proof.

This is an immediate consequence of Theorems 4.23 and 4.24. ∎

######  Remark 4.26.

Note that for $q$ such that $q\mid{M}^{(i)}_{3}$ and $q\not\mid N_{i}$ we have

$\prod_{\mathfrak{q}\mid
q}\mathcal{P}^{f_{i}}_{\mathfrak{q}}(0)=(1-b^{(i)}_{q}q^{-1}+q^{-1})^{2}.$

If $q\mid{M}^{(i)}_{3}$ and $q\mid N_{i}$, then we have

$\prod_{\mathfrak{q}\mid
q}\mathcal{P}^{f_{i}}_{\mathfrak{q}}(0)=(1-b^{(i)}_{q}q^{-1})^{2}.$

Thus, if $N^{-}=1$ we recover the congruence relation proved in [KL19, Theorem
3.9] for $m=1$ at the trivial character.

Unfortunately, this generalized congruence relation does not allow us to
improve their lower bound on the Goldfeld conjecture: Let $E$ be an elliptic
curve and let $d$ be a square free integer that is coprime to the conductor
$N$ of $E$. Then the conductor of the twisted elliptic curve $E^{d}$ is
$d^{2}N$. Let $K$ be an imaginary quadratic field simultaneously satisfying
(GHH) with respect to $N$ and $d^{2}N$. Then all prime factors of $d$ have to
split in $K$. Thus, we obtain the same lower bound as in [KL19, Theorem 1.12].

##  5\. The BDP main conjecture and congruent modular forms

We now prove Theorem C stated in the introduction.

######  Theorem 5.1.

Let $f_{1}$ and $f_{2}$ be two newforms of level $N_{1}$ and $N_{2}$,
respectively, satisfying (GHH), (Cong), ($H^{0}$), (div), (sq-fr) and (unit).
Assume that $\mu(L_{p}^{\mathrm{BDP}}(f_{1}))=0$ and that $\alpha(f_{1})$ is a
$p$-adic unit. If one inclusion in (IMC) holds for $f_{2}$ and that the full
equality of (IMC) holds for $f_{1}$, then the full equality of (IMC) holds for
$f_{2}$.

###### Proof.

The assumption that (IMC) holds for $f_{1}$ implies that
$\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{1})$ is
$\Lambda^{\mathrm{ac}}_{\mathcal{O}_{v}}$-cotorsion and

$2\lambda(L_{p}^{\mathrm{BDP}}(f_{1}))=\lambda(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{1})^{\vee}).$

Furthermore, we have

$\mu(L_{p}^{\mathrm{BDP}}(f_{1}))=\mu(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{1})^{\vee})=0.$

On the one hand, Corollary 3.8 tells us that

$\mu(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{2})^{\vee})=0$

and

(5.1)
$\lambda(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{1})^{\vee})+\sum_{l|\Sigma_{0}}\lambda(\mathcal{P}^{f_{1}}_{l})=\lambda(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{2})^{\vee})+\sum_{l|\Sigma_{0}}\lambda(\mathcal{P}^{f_{2}}_{l}).$

Theorem 4.20 tells us that $\mu(L_{p}^{\mathrm{BDP}}(f_{2}))=0$ and that

(5.2)
$2\lambda(L_{p}^{\mathrm{BDP}}(f_{1}))+\sum_{\mathfrak{q}\mid\tilde{N}^{(1)}_{3}}\lambda(\mathcal{P}^{f_{1}}_{\mathfrak{q}})=2\lambda(L_{p}^{\mathrm{BDP}}(f_{2}))+\sum_{\mathfrak{q}\mid\tilde{N}^{(2)}_{3}}\lambda(\mathcal{P}^{f_{2}}_{\mathfrak{q}}).$

As we assume that one inclusion of (IMC) holds for $f_{2}$ and
$\mu(L_{p}^{\mathrm{BDP}}(f_{2}))=\mu(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{2})^{\vee})=0$,
to show that the full equality of (IMC) holds, it is enough to show that
$2\lambda(L_{p}^{\mathrm{BDP}}(f_{2}))$ equals
$\lambda(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{2})^{\vee})$.

Recall that for $i\in\\{1,2\\}$, $\mathcal{P}_{\mathfrak{q}}^{f_{i}}=1$ for
all $v\in\Sigma_{0}$ such that $b^{(i)}_{q}=0$. Thus,
$\sum_{\mathfrak{q}\mid\tilde{N}^{(i)}_{3}}\lambda(\mathcal{P}^{f_{i}}_{\mathfrak{q}})=\sum_{l|\Sigma_{0}}\lambda(\mathcal{P}^{f_{i}}_{l})$.
Combining equations (5.1) and (5.2) gives

$\displaystyle\lambda(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{2})^{\vee})+\sum_{l|\Sigma_{0}}\lambda(\mathcal{P}^{f_{2}}_{l})=\
$
$\displaystyle\lambda(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{1})^{\vee})+\sum_{l|\Sigma_{0}}\lambda(\mathcal{P}^{f_{1}}_{l})$
$\displaystyle=\ $
$\displaystyle\lambda(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{1})^{\vee})+\sum_{\mathfrak{q}\mid\tilde{N}^{(1)}_{3}}\lambda(\mathcal{P}^{f_{1}}_{\mathfrak{q}})$
$\displaystyle=\ $ $\displaystyle
2\lambda(L_{p}^{\mathrm{BDP}}(f_{1}))+\sum_{\mathfrak{q}\mid\tilde{N}^{(1)}_{3}}\lambda(\mathcal{P}^{f_{1}}_{\mathfrak{q}})$
$\displaystyle=\ $ $\displaystyle
2\lambda(L_{p}^{\mathrm{BDP}}(f_{2}))+\sum_{\mathfrak{q}\mid\tilde{N}^{(2)}_{3}}\lambda(\mathcal{P}^{f_{2}}_{\mathfrak{q}})$
$\displaystyle=\ $ $\displaystyle
2\lambda(L_{p}^{\mathrm{BDP}}(f_{2}))+\sum_{l|\Sigma_{0}}\lambda(\mathcal{P}^{f_{2}}_{l}).$

Thus,

$\lambda(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{2})^{\vee})=2\lambda(L_{p}^{\mathrm{BDP}}(f_{2})),$

which concludes the proof. ∎

######  Remark 5.2.

Let $f_{1}$ and $f_{2}$ be newforms of level $N_{1}$ and $N_{2}$,
respectively, satisfying (GHH), (Cong), ($H^{0}$), (div), (sq-fr) and (unit).
Assume that the $\mu(L_{p}^{\mathrm{BDP}}(f_{1}))=0$ and that $\alpha(f_{1})$
is a $p$-adic unit. Assume that the full equality of (IMC) holds for $f_{1}$.
Below we give a list of (additional) sufficient conditions to make sure that
one inclusion of (IMC) holds for $f_{2}$. In particular, (IMC) holds for
$f_{2}$ in these cases.

  * •

Assume that $p$ is an ordinary prime for $f_{2}$ and that its residual
representation $\overline{\rho_{2}}$ is absolutely irreducible. By [BCK21,
Theorem 5.2], one implication of (IMC) is true if and only if one implication
of the Perrin-Riou Heegner point main conjecture is true. According to [Swe22,
Theorem E] there are non-negative integers $r$ and $s$ such that

$\Lambda^{\mathrm{ac}}\varpi^{r}\textup{Char}_{\Lambda^{ac}_{\mathcal{O}_{v}}}(\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{f_{2}})^{\vee})=\Lambda^{\mathrm{ac}}\varpi^{s}(L_{p}^{\mathrm{BDP}}(f_{2}))^{2}.$

Under the assumptions of Theorem 5.1 all $\mu$-invariants vanish and we obtain
$r=s=0$.

  * •

Assume that $f_{2}$ is non-ordinary at $p$ and that $N_{2}^{-}=1$. Then [KO20,
Theorem 1.5] implies

${L}_{p}^{\mathrm{BDP}}(f_{2})^{2}\in\Lambda^{\mathrm{ac}}\varpi^{l}\mathrm{Char}_{\Lambda_{\mathcal{O}_{v}}^{\mathrm{ac}}}\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{2})^{\vee},$

for some non-negative integer $l$. Using again that all $\mu$-invariants
vanish we can assume that $l=0$. Note that under additional assumptions on
$f_{2}$ [LZ22, Theorem A] proves one inclusion of (IMC) without the additional
factor $\varpi^{l}$.

  * •

Assume that $f_{2}$ is non-ordinary at $p\geq 5$ and that $f_{2}$ satisfies

    * –

$N_{2}^{-}\neq 1$

    * –

If $N_{2}$ is odd, $2$ splits in $K$

    * –

$N_{2}$ is square free.

    * –

$f_{2}$ is associated to an elliptic curve $E/\mathbb{Q}$.

Then [CW22b, Theorem 6.8 and Theorem C] imply that there is a non-negative
integer $r$ such that

$\varpi^{r}\Lambda^{\mathrm{ac}}\mathrm{Char}_{\Lambda_{\mathcal{O}_{v}}^{\mathrm{ac}}}\operatorname{Sel}^{\mathrm{BDP}}(K_{\infty},A_{2})^{\vee}\in{L}^{\mathrm{BDP}}_{p}(f_{2})^{2}\Lambda^{\mathrm{ac}}.$

As we have already seen in the proof of Theorem 5.1 that in our setting all
$\mu$-invariants vanish, we can choose $r=0$ and obtain that one inclusion of
(IMC) holds for $f_{2}$.

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