# On the anticyclotomic Iwasawa theory
of newforms at Eisenstein primes
of semistable reduction

Timo Keller  (T. Keller) Rijksuniveriteit Groningen, Bernoulli Institute,
Bernoulliborg, Nijenborgh 9, 9747 AG Groningen, The Netherlands  Leibniz
Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete
Mathematik, Welfengarten 1, 30167 Hannover, Germany  Department of
Mathematics, Chair of Computer Algebra, Universität Bayreuth, Germany
<EMAIL_ADDRESS>https://www.timo-keller.de and  Mulun Yin  (M. Yin) University
of California Santa Barbara, South Hall, Santa Barbara, CA 93106, USA
<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

Let $f$ be a newform of weight $k$ and level $N$ with trivial nebentypus. Let
$\mathfrak{p}\nmid 2N$ be a maximal ideal of the ring of integers of the
coefficient field of $f$ such that the self-dual twist of the
mod-$\mathfrak{p}$ Galois representation of $f$ is reducible with constituents
$\varphi,\psi$. Denote a decomposition group over the rational prime $p$ below
$\mathfrak{p}$ by $G_{p}$. We remove the condition
$\varphi|_{G_{p}}\neq\mathbf{1},\omega$ from [CGLS22], and generalize their
results to newforms of arbitrary weights. As a consequence, we prove some
Iwasawa Main Conjectures and get the $p$-part of the strong BSD Conjecture for
elliptic curves of analytic rank $0$ or $1$ over $\mathbf{Q}$ in this setting.
In particular, non-trivial $p$-torsion is allowed in the Mordell–Weil group.
Using Hida families, we also prove an Iwasawa Main Conjecture for newforms of
weight $2$ of multiplicative reduction at Eisenstein primes. In the above
situations, we also get $p$-converse to the theorems of
Gross–Zagier–Kolyvagin. The $p$-converse theorems have applications to
Goldfeld’s conjecture in certain quadratic twist families of elliptic curves
having a $3$-isogeny.

###### 2020 Mathematics Subject Classification:

11G40 (Primary) 11G05, 11G10, 14G10 (Secondary)

TK was supported by the Deutsche Forschungsgemeinschaft (DFG),
Geschäftszeichen STO 299/18-1, AOBJ: 667349 and by the 2021 MSCA Postdoctoral
Fellowship 01064790 – ExplicitRatPoints while working on this article.

## Introduction

###  0.1. Statement of the main results.

Let $f\in S_{k}(\Gamma_{0}(N))$ with $k=2r$ be a newform with coefficient
field $\mathbf{Q}(f)$ and coefficient ring $\mathbf{Z}[f]$. Let $F$ be a
finite extension of the completion of $\mathbf{Q}(f)$ at a chosen prime above
$p>2$, and denote by $\mathcal{O}$ the ring of integers in $F$. Let
$\mathfrak{p}$ be the maximal ideal of $\mathcal{O}$ and let $\pi$ be a
generator of $\mathfrak{p}$. Assume that $\mathfrak{p}$ is an _Eisenstein
prime_ for $f$, i.e., that the mod-$\mathfrak{p}$ residual Galois
representation $\overline{\rho}_{f}$ of $G_{\mathbf{Q}}$ associated with $f$
is reducible. This means that its semisimplification is the direct sum of two
characters. After replacing $\overline{\rho}_{f}$ by its self-dual twist
$\overline{\rho}_{f}(1-r)$, these characters are
$\varphi,\psi=\varphi^{-1}\omega$. (In the case $k=2$, $\overline{\rho}_{f}$
is the Galois representation of $G_{\mathbf{Q}}$ acting on
$A_{f}[\mathfrak{p}](\overline{\mathbf{Q}})$ where $A_{f}/\mathbf{Q}$ is the
modular abelian variety attached to $f$). Note that by Serre’s Modularity
Conjecture, a theorem of Khare–Kisin–Wintenberger, simple RM abelian varieties
over $\mathbf{Q}$ are attached to newforms of weight $2$, and vice versa.

Let $K/\mathbf{Q}$ be a Heegner field for $N$, i.e., an imaginary quadratic
field such that all primes dividing $N$ split completely in $K$. We further
assume that $D_{K}$ is odd and $\neq-3$, and that $p=v\overline{v}$ splits in
$K$.

In [CGLS22], it is proved for $k=2$ and $\mathbf{Z}[f]=\mathbf{Z}$ that, under
the assumption $\varphi|_{G_{v}}\neq\mathbf{1},\omega$ (where $\mathbf{1}$ is
the trivial and $\omega$ the mod-$p$ cyclotomic character), the anticyclotomic
Main Conjecture and Perrin-Riou’s Heegner point Main Conjecture hold. From
this, the $p$-converse to Gross–Zagier–Kolyvagin’s theorem and the $p$-part of
BSD in the analytic rank $1$ case (under the assumption that $\varphi$ is
ramified at $p$ and odd or unramified at $p$ and even, which was later removed
by [CGS23]) easily follow.

However, this excludes in particular the case where
$A_{f}(\mathbf{Q})[\mathfrak{p}]\neq 0$ in the case $k=2$. In this article, we
close this gap and generalize the results to all weights and all coefficient
rings.

Our main results are the Iwasawa Main Conjectures Theorem A and Theorem B, a
$p$-converse theorem Theorem E to the theorem of
Gross–Zagier–Kolyvagin–Logachëv, and the $p$-part of the BSD formula Theorem C
in our situation when $p$ does not divide the level of $f$. For newforms of
weight $2$, we obtain Iwasawa Main Conjectures at Eisenstein primes of
semistable reduction (i.e., good reduction or bad multiplicative reduction).
See Theorem D.

###  0.2. Method of proof and outline of the paper.

First, we assume $p$ does not divide the level of $f$ from section 1 through
section 4. Following the strategies in [CGLS22], the proof of our Iwasawa Main
Conjectures for $f$ is divided into two steps. We first compare the Iwasawa
$\lambda$\- and $\mu$-invariants of the algebraic side of and analytic side,
then prove a one side divisibility. The starting point is again the
observation that a Main Conjecture should be equivalent to an imprimitive one.
More specifically, for an appropriate Selmer group $\mathfrak{X}_{f}$ for $f$
and a corresponding Bertolini–Darmon–Prasanna $p$-adic $L$-function
$\mathcal{L}_{f}$, there is an equality for their ‘$S$-imprimitive’ versions

${\operatorname{Char}}(\mathfrak{X}^{S}_{f})\Lambda^{{\operatorname{nr}}}=(\mathcal{L}^{S}_{f})$

as ideals in $\Lambda^{{\operatorname{nr}}}$, where the $p$-adic $L$-function
for $f$ lives. Here we consider the primitive and imprimitive unramified
Selmer groups (which are identified with the corresponding Greenberg Selmer
groups in [CGLS22] in their setting) of the characters $\varphi$ and $\psi$
and of $f$ over the anticyclotomic Iwasawa algebra
$\Lambda\colonequals\mathcal{O}\llbracket\Gamma\rrbracket$, where $\Gamma$ is
the Galois group of the anticyclotomic $\mathbf{Z}_{p}$-extension $K_{\infty}$
of $K$. Here
$\Lambda^{{\operatorname{nr}}}\coloneq\Lambda\hat{\otimes}_{\mathbf{Z}_{p}}\mathbf{Z}_{p}^{{\operatorname{nr}}}$,
for $\mathbf{Z}_{p}^{{\operatorname{nr}}}$ the completion of the ring of
integers of the maximal unramified extension of $\mathbf{Q}_{p}$. Denote by
$\mathbf{F}$ the residue field of $F$.

In Section 1, we show that the primitive unramified Selmer group of $f$ and
the characters $\varphi,\psi$ appearing in the semisimplification of
$\overline{\rho}_{f}$ are $\Lambda$-torsion with $\mu$-invariant $0$. In most
cases, we get a simple relation between the $\lambda$-invariants

(0.1)
$\lambda(\mathfrak{X}^{S}_{f})=\lambda(\mathfrak{X}^{S}_{\varphi})+\lambda(\mathfrak{X}^{S}_{\psi})$

as in [CGLS22], but there is a special case when one of the characters is
trivial over $G_{K}$, where we get

(0.2)
$\lambda(\mathfrak{X}^{S}_{f})+1=\lambda(\mathfrak{X}^{S}_{\varphi})+\lambda(\mathfrak{X}^{S}_{\psi}).$

From here, we can easily get the same comparisons between the corresponding
primitive Selmer groups. We mention that this difference in
$\lambda$-invariants agrees with the Iwasawa Main Conjecture for the trivial
character, which takes on a different form from others

${\operatorname{Char}}(\mathfrak{X}_{\mathbf{1}})=(\mathcal{L}_{\mathbf{1}}\cdot
T).$

Since we allow one of the characters to be trivial on $G_{p}$, the proofs of
(0.1) and (0.2) requires more general ideas than those in op. cit. in that the
Selmer groups $\mathfrak{X}^{S}_{?}$ ($?=f,\vartheta$) may not be almost
divisible in the sense that they might contain non-trivial finite
$\Lambda$-submodules. Indeed, the vanishing results of some $\text{{H}}^{0}$
and $\text{{H}}^{2}$ cohomology groups in op. cit. heavily rely on this
condition and no longer hold in our situation. We will compare these groups
for $f$ to those of the characters directly and obtain an expected relation
between their Selmer groups.

By Ribet’s Lemma (see [Bel09]), we can find a sub-lattice of the lattice for
the $\mathfrak{p}$-adic Galois representation of $f$ such that the associated
mod-$\mathfrak{p}$ Galois representation of $f$ has no nonzero trivial
subrepresentation over $G_{K}$. More precisely, we can fix a non-split
extension

$0\to\mathbf{F}(\varphi)\to\overline{\rho}_{f}\to\mathbf{F}(\psi)\to 0$

where the first character restricts to $\omega$ on $G_{p}$ so in particular
it’s non-trivial over $G_{K}$. We can assume this since both the Iwasawa Main
Conjectures and the BSD Conjecture is invariant under isogenies. This choice
of the lattice is crucial for our computations.

In Section 2, we show an analogous formula for the $\mu$\- and
$\lambda$-invariants for the Katz $p$-adic $L$-functions of the characters and
the BDP $L$-function of $f/K$. That is,

$\lambda(\mathcal{L}^{S}_{f})=\lambda(\mathcal{L}^{S}_{\varphi})+\lambda(\mathcal{L}^{S}_{\psi}),$

where $\mathcal{L}^{S}_{?}\coloneq\mathcal{L}_{?}\cdot\prod_{w\in
S}\mathcal{P}_{w}(?)$ for certain elements $\mathcal{P}_{w}(?)$ in $\Lambda$
(see Lemma 1.1.1, section 1.5). Here $?=f,\ \varphi\text{ or }\psi$. Using the
known results on the Iwasawa Main Conjecture for the characters proved by
Rubin [Rub91], we get the equality between algebraic and analytic Iwasawa
invariants for $f$.

Hence it suffices to show that the characteristic ideal of the unramified
Selmer group divides the $p$-adic $L$-function. For this, we can refer to
[CGLS22, §§ 3, 4] because we chose a lattice $T_{f}$ such that
$\overline{\rho}_{f}$ has no nonzero trivial subrepresentation. The Main
Conjectures can then be applied to yield the main results of this paper for
the good reduction part.

The first anticyclotomic Iwasawa Main Conjecture we obtain is the following,
which can be seen as a special case of Greenberg’s Main Conjectures.

######  Theorem A (Theorem 3.0.7 (IMC2)).

Let $f\in S_{k}(\Gamma_{0}(N))$ be a newform of arbitrary weight $k$ and $p>2$
an ordinary Eisenstein prime not divining $N$, and let $K$ be an imaginary
quadratic field satisfying the hypotheses in section 0.1. Let
$\mathfrak{X}_{f}$ be the Pontryagin dual of the unramified Selmer groups for
$f$ (see section 1.4 for precise definitions) and let $\mathcal{L}_{f}$ be the
corresponding BDP $p$-adic $L$-function. Then $\mathfrak{X}_{f}$ is
$\Lambda$-torsion, and

${\operatorname{Char}}(\mathfrak{X}_{f})\Lambda^{{\operatorname{nr}}}=(\mathcal{L}_{f})$

as ideals in $\Lambda^{{\operatorname{nr}}}$.

A second Iwasawa Main Conjecture of Perrin-Riou type, which is equivalent to
the one above, is also obtained.

######  Theorem B (Theorem 3.0.7 (IMC1)).

Let $f\in S_{k}(\Gamma_{0}(N))$ be a newform of arbitrary weight $k$ and $p>2$
an ordinary Eisenstein prime not dividing $N$, and let $K$ be an imaginary
quadratic field satisfying the hypotheses in section 0.1. Then both
$\text{{H}}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})$ and
$\mathcal{X}=\text{{H}}^{1}_{\mathcal{F}_{\Lambda}}(K,M_{f})^{\vee}$ have
$\Lambda$-rank one (see [CGLS22, section 3] for general definitions), and the
equality

${\operatorname{Char}}_{\Lambda}(\mathcal{X}_{{\operatorname{tors}}})={\operatorname{Char}}_{\Lambda}(\text{{H}}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})/\Lambda\kappa_{\infty})^{2}$

holds in $\Lambda$.

We now formulate our main results on the $p$-part of strong BSD for elliptic
curves of good reduction at Eisenstein primes.

######  Theorem C (Theorem 4.2.1).

Let $E/\mathbf{Q}$ be an elliptic curve and $p>2$ a prime of good reduction.
Assume that $E$ admits a cyclic $p$-isogeny with kernel
$C=\mathbf{F}_{p}(\varphi)$ for some character $\varphi\colon
G_{\mathbf{Q}}\to\mathbf{F}_{p}^{\times}$ (equivalently, $E[p]$ is reducible).
Assume that $r_{\operatorname{an}}(E)\in\\{0,1\\}$.

Then the $p$-part of the BSD formula holds for $E/\mathbf{Q}$, i.e.,

${\operatorname{ord}}_{p}\Big{(}\frac{L^{*}(E/\mathbf{Q},1)}{\Omega_{E}\cdot{\operatorname{Reg}}_{E/\mathbf{Q}}}\Big{)}={\operatorname{ord}}_{p}\Big{(}\frac{{\operatorname{Tam}}(E/\mathbf{Q})\\#\Sha(E/\mathbf{Q})[p^{\infty}]}{(\\#E(\mathbf{Q})_{\operatorname{tors}})^{2}}\Big{)}.$

Here, $L^{*}(E/\mathbf{Q},1)$ denotes the leading Taylor coefficient of
$L(E/\mathbf{Q},s)$ at $s=1$.

Finally, we modify Skinner’s use of Hida families in [Ski14] to prove a Main
Conjecture in the multiplicative reduction case in section 5. The obstruction
is again that our imprimitive Selmer groups $\mathfrak{X}^{S}_{?}$ may have
finite $\Lambda$-submodules. In particular, the characteristic ideals are no
longer identified with the Fitting ideals. The formal argument can be carried
out by replacing the Selmer groups by their free parts. We would also need to
modify several pieces that go into the proof into the residually reducible
setting.

######  Theorem D (Theorem 5.0.3).

Let $f$ be a newform of weight $2$ and $p\|N$ an odd Eisenstein prime of
multiplicative reduction. Let $K$ be an imaginary quadratic field satisfying
the hypotheses in section 0.1. Then the following equation holds in
$\Lambda^{{\operatorname{nr}}}$:

${\operatorname{Char}}(\mathfrak{X}_{f})\Lambda^{{\operatorname{nr}}}=(\mathcal{L}_{f}).$

We remark that our Main Conjecture would yield a rank $1$ BSD formula provided
the results in [CGS23] are extended to higher weight modular forms.

###  0.3. Examples

Here are the elliptic curves $E$ over $\mathbf{Q}$ in the [LMFDB] with
$E(\mathbf{Q})_{\operatorname{tors}}\cong\mathbf{Z}/p$, rank $0$ or $1$, and
$p$ a prime of good reduction: $p=3$, $p=5$, and $p=7$.

Here are concrete examples for each of these three infinite families, an
example of smallest conductor in each case. Our main result implies the
$p$-part of strong BSD for all their twists that have analytic rank $0$ or
$1$, of which there exist infinitely many. Case $p=3$: 19a3 has torsion
subgroup $\mathbf{Z}/3$ and rank $1$. Case $p=5$: 11a3 ($=X_{1}(11)$) has
torsion subgroup $\mathbf{Z}/5$ and rank $0$. Case $p=7$: 26b2 has torsion
subgroup $\mathbf{Z}/7$ and rank $1$.

There are also examples of higher dimension: The descent computations for
absolutely simple modular abelian surfaces in [KS22] and [KS23, § 6] are not
necessary anymore for odd primes of good reduction. For example, the genus $2$
curve $X\colon y^{2}=5x^{6}+10x^{5}-13x^{4}-30x^{3}+9x^{2}+20x-8$ has
absolutely simple semistable RM Jacobian $J$ of squarefree level $5\cdot 13$
and rank $0$ associated to the newform 65.2.a.c. Note that this curves is not
covered by _op. cit._ The algorithms from _op. cit._ prove the prime-to-$3$
part of strong BSD for $J/\mathbf{Q}$ with
$\\#\Sha(J/\mathbf{Q})_{\operatorname{an}}=2$. Our results allow us to also
prove the $3$-part: Let the good ordinary prime $3$ split as
$\mathfrak{p}\mathfrak{p}^{\prime}$ in $\mathbf{Z}[f]=\mathbf{Z}[\sqrt{3}]$
such that $\rho_{\mathfrak{p}}$ is irreducible and
$\rho_{\mathfrak{p}^{\prime}}$ is reducible with a trivial $1$-dimensional
subrepresentation coming from the $3$-torsion. The results and algorithms of
_op. cit._ prove that the Heegner index $I_{K}$ for $D_{K}=-131$ (so $3$ is
split in $\mathcal{O}_{K}$) equals $\mathfrak{p}^{\prime}$ times some ideal
lying above $2$, and that the Tamagawa product considered as an ideal in
$\mathbf{Z}[f]$ equals $\mathfrak{p}^{\prime}$. Hence [KS23, Theorem 5.2.2]
proves that $\Sha(J/\mathbf{Q})[\mathfrak{p}]=0$. Our main theorem implies
[CGLS22, eq. (5.5)] with $p$-torsion replaced by $\mathfrak{p}$-torsion
throughout, showing that
$v_{\mathfrak{p}^{\prime}}(\\#\Sha(J/K)[\mathfrak{p}^{\prime\infty}])=2v_{\mathfrak{p}^{\prime}}(I_{K})-2v_{\mathfrak{p}^{\prime}}({\operatorname{Tam}}(J/\mathbf{Q}))=2-2=0$.
Since $\mathfrak{p}^{\prime}\nmid 2$, we have
$\Sha(J/K)[\mathfrak{p}^{\prime}]\simeq\Sha(J/\mathbf{Q})[\mathfrak{p}^{\prime}]\oplus\Sha(J^{K}/\mathbf{Q})[\mathfrak{p}^{\prime}]$,
hence we get $\Sha(J/\mathbf{Q})[\mathfrak{p}^{\prime}]=0$, so strong BSD
holds for $J/\mathbf{Q}$ with $\Sha(J/\mathbf{Q})\cong\mathbf{Z}/2$. (Note
that the argument with [CGLS22, eq. (5.5)] also provides evidence for the
conjecture that the Tamagawa product divides the Heegner index.)

###  0.4. A $p$-converse theorem and applications to Goldfeld’s conjecture

######  Theorem E (Theorem 4.1.1, Corollary 5.0.4).

Let $A/\mathbf{Q}$ be a simple RM abelian variety associated to a newform $f$
and $\mathfrak{P}\mid p>2$ a prime ideal of its endomorphism ring
$\mathcal{O}$ of good or bad multiplicative reduction such that
$\rho_{A,\mathfrak{P}}$ is reducible.

Then

${\operatorname{corank}}_{\mathcal{O}}{\operatorname{Sel}}_{\mathfrak{P}^{\infty}}(A/\mathbf{Q})=r\in\\{0,1\\}\implies{\operatorname{rk}}_{\mathcal{O}}A(\mathbf{Q})=r_{\operatorname{an}}(f)=r,$

and $\Sha(A/\mathbf{Q})[\mathfrak{P}^{\infty}]$ is finite.

Our $3$-converse theorem has several applications to Goldfeld’s conjecture,
improving the bounds for the proportion of rank $1$ twists when being combined
with results [BKOS19, Shn21, BFS23] on the average $3$-Selmer rank. See Remark
4.1.2.

###  0.5. Relation to previous works.

Results in this paper in the good reduction case can be seen as direct
generalizations of those in [CGLS22] to higher weight modular forms while
removing the technical condition $\vartheta|_{G_{p}}\neq\mathbf{1},\omega$.
Thus it would automatically generalize the results which require the main
theorem in op. cit. as an input, for example [CGS23, Theorem A]. We also
obtain generalizations of the structure theorems in [How04] as well as the
anticyclotomic control theorem in [JSW17] in the residually reducible case
which allows torsion. Since our characters are more general than those in
[CGLS22], the results on the proportion of ranks of quadratic twists of an
elliptic curve over $\mathbf{Q}$ with a rational $3$-isogeny are also
improved, following [BKOS19].

Finally, we obtain new results for newforms with multiplicative reduction at
Eisenstein primes which include a proof of an Iwasawa Main Conjecture. These
results are analogous to those in [Ski14], but the residually reducible case
were untouched.

###  0.6. Ordering of the characters.

Recall the short exact sequence relating the residual representation
$\overline{\rho}_{f}$ to the characters

$0\to\mathbf{F}(\varphi)\to\overline{\rho}_{f}\to\mathbf{F}(\psi)\to 0,$

where we assumed that the first character $\varphi$ restricts to $\omega$ on
$G_{p}$ and that the sequence is non-split. We could as well fix another
ordering where the first character restricts to $\mathbf{1}$ on $G_{p}$. In
fact, our original approach assumed that locally, the trivial character is a
subrepresentation, regardless of whether the sequence is split or not. This
made our computations in Section 1 easier. However, we had to heavily extend
the results on the Kolyvagin system from [CGLS22, § 3], or rather [How04], to
allow non-trivial torsion group $\text{{H}}^{0}(K,\overline{\rho}_{f})$. As
our results might be useful for the future, we included them in the Appendix
A. We also deduce a version of the anticyclotomic control theorem which allows
non-trivial torsion in Appendix B. It should be mentioned that a direct proof
of everything with torsion is possible, but we do not include it in this paper
for simplicity.

###  0.7. Notation.

Pontryagin duals are denoted by $(-)^{\vee}$. The mod-$p$ cyclotomic character
is denoted by $\omega$ and the $p$-adic cyclotomic character by $\chi$. For
any subextension $L/K$ of $\bar{K}/K$, denote by $G_{L}$ the absolute Galois
group of $L$.

###  0.8. Acknowledgements.

We thank Debanjana Kundu and Otmar Venjakob for helpful discussions and Ari
Shnidman for pointing out the application of our $p$-converse theorem to
Goldfeld’s conjecture. We also thank MY’s advisor Francesc Castella for his
guidance throughout this project. This work is part of MY’s forthcoming Ph.D.
thesis.

##  1\. Algebraic side

Let $f\in S_{k}(\Gamma_{0}(N))^{\operatorname{new}}$ be a newform of weight
$k=2r\geq 2$ and level $N$ with trivial nebentypus. Fix an odd prime $p\nmid
N$. Let $\mathbf{Z}[f]$ be the coefficient ring of $f$ with field of fractions
$\mathbf{Q}(f)$. Let $F$ be a finite extension of a completion of
$\mathbf{Q}(f)$ at a chosen prime above $p$ with ring of integers
$\mathcal{O}$. Let $\mathfrak{p}=(\pi)$ be the maximal ideal of $\mathcal{O}$
and let $\mathbf{F}/\mathfrak{p}$ be the residue field of $\mathcal{O}$. We
use the Pontryagin dual
$(-)^{\vee}={\operatorname{Hom}}_{\mathcal{O},{\operatorname{cts}}}(-,F/\mathcal{O})$
for locally compact $\mathcal{O}$-modules.

Recall that $\omega$ is the mod-$p$ cyclotomic character. Denote by

$\rho_{f}\colon{\operatorname{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})\to{\operatorname{Aut}}_{F}(V_{f}(1-r))\cong{\operatorname{GL}}_{2}(F)$

the _self-dual Tate twist_ of the $p$-adic Galois representation attached to
$f$. It has determinant $\chi$.

Let $K\subset\overline{\mathbf{Q}}$ be an imaginary quadratic field in which
$p=v\overline{v}$ splits, with $v$ the prime of $K$ above $p$ induced by
$\iota_{p}$. We also fix an embedding
$\iota_{\infty}\colon\overline{\mathbf{Q}}\hookrightarrow\mathbf{C}$.

We assume that $K$ satisfies the following Heegner hypothesis:

$\text{every prime }\ell\mid N\text{ splits in }K$

Let $G_{K}\colonequals{\operatorname{Gal}}(\overline{\mathbf{Q}}/K)\subset
G_{\mathbf{Q}}\colonequals{\operatorname{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})$
be the absolute Galois group of $K$, and for each place $w$ of $K$ let
$I_{w}\subset G_{w}\subset G_{K}$ be the corresponding inertia and
decomposition groups, unique up to conjugation. Let
${\operatorname{Frob}}_{w}\in G_{w}/I_{w}$ be the arithmetic Frobenius. For
the prime $v\mid p$, we assume $G_{v}$ is chosen so that it is identified with
${\operatorname{Gal}}(\overline{\mathbf{Q}}_{p}/\mathbf{Q}_{p})$ via
$\iota_{p}$.

Let $\Gamma\colonequals{\operatorname{Gal}}(K_{\infty}/K)$ be the Galois group
of the anticyclotomic $\mathbf{Z}_{p}$-extension $K_{\infty}$ of $K$, and let
$\Lambda\colonequals\mathcal{O}\llbracket\Gamma\rrbracket$ be the
anticyclotomic Iwasawa algebra over $\mathcal{O}$. We shall often identify
$\Lambda$ with the power series ring $\mathcal{O}\llbracket T\rrbracket$ by
setting $T=\gamma-1$ for a fixed topological generator $\gamma\in\Gamma$.

We record the following well-known properties of the anticyclotomic
$\mathbf{Z}_{p}$-extension $K_{\infty}/K$ of an imaginary quadratic field $K$,
i.e., the $\mathbf{Z}_{p}$-extension on which complex conjugation acts as
inversion. Readers who are only interested in the main results of this paper
can directly go to section 1.1.

######  Lemma 1.0.1 (splitting of primes in $K_{\infty}$).

Let $w\mid\ell\neq p$ be a place of $K$. If $\ell$ is unramified and split in
$K$, there are only finitely many primes of $K_{\infty}$ lying above $\ell$.
If $\ell$ is ramified or inert in $K$, $\ell$ splits completely in
$K_{\infty}$.

Note that this is in contrast to the _cyclotomic_ $\mathbf{Z}_{p}$-extension,
in which no primes are infinitely split.

######  Lemma 1.0.2 (local behavior of $K_{\infty}$ above $p$).

Let $K$ be an imaginary quadratic field and $p=v\overline{v}$ a rational prime
completely split in $K$. Let $K_{\infty}/K$ be the anticyclotomic
$\mathbf{Z}_{p}$-extension.

  1. (i)

$\Gamma$ is the $\mathbf{Z}_{p}$-quotient of
$\varprojlim_{n}{\operatorname{Gal}}(K_{n}/K)$, where $K_{n}$ is the ring
class field of the order
$\mathcal{O}_{p^{n}}\colonequals\mathbf{Z}+p^{n}\mathcal{O}_{K}$. The
unramified part of $\Gamma$ at $v\mid p$ is cyclic of order dividing
$h_{K}=\\#{\operatorname{Gal}}(K_{0}/K)$, and $K_{\infty,v}/K_{1,v}$ is
totally ramified at $v$.

  2. (ii)

The completion $K_{\infty,v}$ of $K_{\infty}$ at $v\mid p$ is a ramified
$\mathbf{Z}_{p}$-extension of $K_{v}=\mathbf{Q}_{p}$ different from
$K_{v,cyc}$, the cyclotomic $\mathbf{Z}_{p}$-extension of $K_{v}$. In
particular, $K_{\infty,v}\cap K_{v,cyc}$ is a finite extension of $K_{v}$.

  3. (iii)

Let $v\mid p$ be a place of $K$. The image of the compositions
$G_{v}\hookrightarrow G_{K}\twoheadrightarrow\Gamma$ and $I_{v}\hookrightarrow
G_{v}\twoheadrightarrow\Gamma$ are open. If $p\nmid h_{K}$, the composition
$I_{v}\hookrightarrow G_{v}\twoheadrightarrow\Gamma$ is surjective.

###### Proof.

  1. (i)

Note that $K_{0}$ is the Hilbert class field of $K$, so by class field theory
${\operatorname{Frob}}_{v}\in{\operatorname{Gal}}(K_{0}/K)$ has order equal to
the order of $v$ in the class group of $K$. All primes above $p$ are totally
ramified in $K_{n}/K_{0}$, and one has canonical isomorphisms
${\operatorname{Gal}}(K_{n}/K)\simeq{\operatorname{Pic}}(\mathcal{O}_{n})\simeq(\mathcal{O}_{K}/p^{n})^{\times}/(\mathbf{Z}/p^{n})^{\times}$
from [Cox13, equation (7.27)].

  2. (ii)

The first claim follows from [LNQD00, Theorem 3.2]; condition (ii) there is
satisfied for $K$. Thus

${\operatorname{Gal}}((K_{\infty,v}\cap
K_{v,{\operatorname{cyc}}})/K_{v})\cong{\operatorname{Gal}}(K_{\infty,v}/K_{v})/{\operatorname{Gal}}(K_{\infty,v}/(K_{\infty,v}\cap
K_{v,{\operatorname{cyc}}}))$

must be a non-trivial quotient of
${\operatorname{Gal}}(K_{\infty,v}/K_{v})\cong\mathbf{Z}_{p}$ by an open
subgroup, and hence it must be finite.

  3. (iii)

By the (ii), $G_{v}\hookrightarrow G_{K_{v}}$ is open. The remaining
statements follow from the (i).∎

######  Corollary 1.0.3 (the localization is an isomorphism away from $p$).

Let $A$ be a $p^{\infty}$-torsion $G_{K_{w}}$-module. If $\ell\neq p$ is
unramified and split with $w\mid\ell$ in $K$, then the restriction morphism
$\text{{H}}^{1}(K_{\infty,w}/K_{w},A)\to\text{{H}}^{1}(I_{w},A)^{\Gamma/I_{w}}$
is an isomorphism.

###### Proof.

See [PW11, Remark 3.1 and Lemma 3.2]. Note that the kernel and cokernel of the
restriction morphism are the first and second Galois cohomology groups of a
$p$-primary Galois module and that the profinite degree of the absolute Galois
group $G_{\infty,w}$ of $K_{\infty,w}$ is prime to $p$. ∎

######  Lemma 1.0.4 (the dual of the Iwasawa algebra).

The Pontryagin dual $\Lambda^{\vee}$ of the Iwasawa algebra is isomorphic to
$(F/\mathcal{O})[\varinjlim_{n}(\Gamma/\Gamma_{n})^{\vee}]\simeq(\varinjlim_{n}F/\mathcal{O})[(\Gamma/\Gamma_{n})^{\vee}]$.

###### Proof.

This follows from the exactness property of the Pontryagin dual, which
interchanges projective and injective limits, and the definition of the
Iwasawa algebra as a projective limit. ∎

######  Corollary 1.0.5 (Galois cohomology of Iwasawa algebra equals Galois
cohomology over the $\mathbf{Z}_{p}$-extension).

One has
$\text{{H}}^{i}(K,\Lambda^{\vee})\simeq\text{{H}}^{i}(K_{\infty},F/\mathcal{O})$.
For the twist of $\Lambda^{\vee}$ by a character $\vartheta\colon
G_{K}\to\mathbf{F}^{\times}\hookrightarrow\mathcal{O}^{\times}$, one has
$\text{{H}}^{i}(K,\mathcal{O}(\vartheta)\otimes_{\mathcal{O}}\Lambda^{\vee})\simeq\text{{H}}^{i}(K_{\infty},F/\mathcal{O}(\vartheta))$.
If $K_{\vartheta}$ is obtained from $K$ by adjoining the values of
$\vartheta$, then $K_{\vartheta}$ is linearly disjoint from $K_{\infty}$ and
one has
$\text{{H}}^{i}(K_{\infty},F/\mathcal{O}(\vartheta))\simeq\text{{H}}^{i}(K_{\infty}K_{\vartheta},F/\mathcal{O})^{\Delta_{\vartheta}}$.

###### Proof.

See [CGLS22, proof of Theorem 1.2.2]. Note that their proof works for any
$\vartheta$. ∎

###  1.1. Local cohomology groups of characters.

Recall that $\mathbf{F}=\mathcal{O}/\mathfrak{p}$ is the residue field of
$\mathcal{O}$, a finite field of characteristic $p$. Let $\vartheta\colon
G_{K}\to\mathbf{F}^{\times}$ be a character with conductor divisible only by
primes that are split in $K$. Since the case where
$\vartheta|_{G_{p}}\neq\mathbf{1},\omega$ is already dealt with in [CGLS22],
we focus on the complementary cases. Via the Teichmüller lift
$\mathbf{F}^{\times}\hookrightarrow F^{\times}$, we shall also view
$\vartheta$ as taking values in $\mathcal{O}^{\times}$. Set

$M_{\vartheta}\colonequals\mathcal{O}(\vartheta)\otimes_{\mathcal{O}}\Lambda^{\vee}.$

The module $M_{\vartheta}$ is equipped with a $G_{K}$-action via
$\vartheta\otimes\Psi^{-1}$, where $\Psi\colon G_{K}\to\Lambda^{\times}$ is
the character arising from the projection $G_{K}\twoheadrightarrow\Gamma$. One
has a $G_{K}$-equivariant isomorphism
$M_{\vartheta}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\operatorname{Hom}}_{\mathcal{O},{\operatorname{cts}}}(\Lambda,\mathcal{O}(\vartheta))$.

Note that one has an exact sequence

(1.1) $0\to(F/\mathcal{O})(\vartheta)\to
M_{\vartheta}\stackrel{{\scriptstyle\cdot T}}{{\to}}M_{\vartheta}\to 0.$

In this section, we study the local cohomology of $M_{\vartheta}$ at various
primes $w$ of $K$.

####  1.1.1. $w\nmid p$ split in K

######  Lemma 1.1.1.

Let $\gamma_{w}$ be the image of ${\operatorname{Frob}}_{w}$ in the
decomposition group of $w\mid\ell$. Let
$\mathcal{P}_{w}(\vartheta)=P_{w}(\ell^{-1}\gamma_{w})\in\Lambda$ with
$P_{w}\colonequals\det(1-{\operatorname{Frob}}_{w}X\mid F(\vartheta)_{I_{w}})$
the Euler factor at $w$ of the $L$-function of $\vartheta$.

The module $\text{{H}}^{1}(K_{w},M_{\vartheta})^{\vee}$ is $\Lambda$-torsion
with characteristic ideal $(\mathcal{P}_{w}(\vartheta))$. In particular, it
has $\mu$-invariant $0$.

###### Proof.

See [CGLS22, Lemma 1.1.1]. Note that their proof works for any $\vartheta$. ∎

####  1.1.2. $w\mid p$ in $K$

######  Lemma 1.1.2.

Let $X$ be a finitely generated $\Lambda$-module.

  1. (i)

If

     * •

$X[T]=0$ and

     * •

$X/TX$ is a free $\mathcal{O}$-module of rank $r$,

then $X$ is a free $\Lambda$-module of rank $r$ (and conversely).

  2. (ii)

If

     * •

$X[T]=0$ and

     * •

${\operatorname{rk}}_{\mathcal{O}}X/TX=r$,

then ${\operatorname{rk}}_{\Lambda}X=r$.

###### Proof.

  1. (i)

This is [CGLS22, Lemma 1.1.2] (see also [NSW08, Proposition 5.3.19 (ii)]).

  2. (ii)

Consider the structure theorem

$X\sim\Lambda^{s}\oplus\bigoplus_{i}\Lambda/(f_{i})\oplus\bigoplus_{j}\Lambda/(\pi^{m_{j}})$

for some distinguished polynomials $f_{i}$. That $X[T]=0$ implies there is no
$f_{i}$. Then $X/TX$ would have $\mathcal{O}$-rank $s$ because
$(\Lambda/(\pi^{m_{j}}))/T$ is finite for any $j$. Hence $r=s$. ∎

We now study the case $\vartheta|_{G_{v}}\in\\{\mathbf{1},\omega\\}$ excluded
in [CGLS22]. Note that in each of these cases one property in [CGLS22] fails.

######  Proposition 1.1.3.

Assume that $\vartheta|_{G_{v}}\in\\{\mathbf{1},\omega\\}$ (all other cases
have been considered in the analogous [CGLS22, Proposition 1.1.3]). Then:

  1. (i)

The restriction morphism

$r_{w}\colon\text{{H}}^{1}(K_{w},M_{\vartheta})\to\text{{H}}^{1}(I_{w},M_{\vartheta})^{G_{w}/I_{w}}$

is surjective.

     * •

If $\vartheta|_{G_{w}}=\mathbf{1}$, its kernel is a cofree
$\mathcal{O}$-module of rank $1$. In particular, $ker(r_{w})^{\vee}$ has
$\Lambda$-rank $0$ and it has no non-trivial finite $\Lambda$-submodule
because all non-trivial subgroups of $\mathcal{O}$ are infinite.

     * •

If $\vartheta|_{G_{w}}=\omega$, it is an isomorphism.

  2. (ii)

     * •

If $\vartheta|_{G_{w}}=\mathbf{1}$, $\text{{H}}^{1}(K_{w},M_{\vartheta})$ is
$\Lambda$-cofree of rank $1$.

     * •

If $\vartheta|_{G_{w}}=\omega$,
$X\colonequals\text{{H}}^{1}(K_{w},M_{\vartheta})^{\vee}$ satisfies
${\operatorname{pd}}_{\Lambda}X\leq 1$, (hence) has no non-trivial finite
($\Lambda$-)submodule, and is generated by $2$ elements with $\Lambda$-rank
equal to $1$.

###### Proof.

Note that $K_{w}=\mathbf{Q}_{p}$ and that $K_{w,\infty}/K_{w}$ is an
infinitely ramified $\mathbf{Z}_{p}$-extension by Lemma 1.0.2 not containing
$\mu_{p}$ because $p>2$ and $([K_{w}(\mu_{p}):K_{w}],p)=1$.

(i): Since ${\operatorname{cd}}(K_{w})=1$ and
$G_{w}/I_{w}\simeq\hat{\mathbf{Z}}\cdot{\operatorname{Frob}}_{w}$, $r_{w}$ is
surjective and $\ker(r_{w})\simeq
M_{\vartheta}^{I_{w}}/({\operatorname{Frob}}_{w}-1)M_{\vartheta}^{I_{w}}$ by
[Rub00, Lemma I.3.2 (i)]. Let us now determine the $\ker(r_{w})$ by
determining its Pontryagin dual as a $\Lambda$-module. Dualizing

$0\to M_{\vartheta}^{G_{w}}\to
M_{\vartheta}^{I_{w}}\xrightarrow{{\operatorname{Frob}}_{w}-1}M_{\vartheta}^{I_{w}}\to
M_{\vartheta}^{I_{w}}/({\operatorname{Frob}}_{w}-1)M_{\vartheta}^{I_{w}}\to 0$

and noting that Pontryagin duality transforms $G_{w}$-invariants into the
quotient by $T$, $\ker(r_{w})^{\vee}$ sits in an exact sequence

(1.2)
$0\to\ker(r_{w})^{\vee}\to(M_{\vartheta}^{\vee})_{I_{w}}\to(M_{\vartheta}^{\vee})_{I_{w}}\to
M_{\vartheta}^{\vee}/T\to 0.$

If $\vartheta|_{G_{w}}\neq\mathbf{1}$, $M_{\vartheta}^{G_{w}}=0$, so
$(M_{\vartheta}^{\vee})_{I_{w}}\to(M_{\vartheta}^{\vee})_{I_{w}}$ is a
surjection of finitely generated modules over the Noetherian algebra
$\Lambda$, hence an isomorphism.

If $\vartheta|_{G_{w}}=\mathbf{1}$, the exact sequence (1.2) reads

$0\to\ker(r_{w})^{\vee}\to\Lambda_{I_{w}}\to\Lambda_{I_{w}}\to\Lambda/T\to 0,$

i.e.,

$0\to\ker(r_{w})^{\vee}\to\mathcal{O}[\Gamma/I_{w}]\to\mathcal{O}[\Gamma/I_{w}]\to\mathcal{O}\to
0.$

Since $I_{w}\subseteq G_{w}$ is of finite index by Lemma 1.0.2,
$\Gamma/I_{w}\cong\mathbf{Z}/p^{m}$ with $p^{m}\mid h_{K}$. Hence all modules
are $\mathcal{O}$ torsion-free and
${\operatorname{rk}}_{\mathcal{O}}\ker(r_{w})^{\vee}=1$. We now prove (ii).
Let $X$ be the $\Lambda$-module $\text{{H}}^{1}(K_{w},M_{\vartheta})^{\vee}$.
It is finitely generated because it is finite mod $(p,T)$. We determine $X[T]$
and $X/TX$ if $\vartheta|_{G_{w}}\in\\{\mathbf{1},\omega\\}$.

Claim 0:
$\text{{H}}^{i}(K_{w},M_{\vartheta})=\text{{H}}^{i}(K_{w,\infty},F/\mathcal{O}(\vartheta))$
and $\text{{H}}^{2}(K_{w},M_{\vartheta})=0$ independent of $\vartheta$.

The first statement is Shapiro’s lemma. Note that $K_{w,\infty}/K_{w}$ is a
(infinitely ramified) $\mathbf{Z}_{p}$-extension by Lemma 1.0.2. For the
second statement, use the first and that from [Nek06]

$\text{{H}}^{2}(K_{w},M_{\vartheta})=\text{{H}}^{2}(K_{w,\infty},F/\mathcal{O})=\text{{H}}^{0}_{\operatorname{Iw}}(K_{w,\infty}/K_{w},\mathcal{O}(\chi\vartheta^{-1}))=0,$

where the $0$-th Iwasawa cohomology is $0$ by [SV16, Remark 5.11, Lemma 5.12].

Claim 1:
$\text{{H}}^{1}(K_{w},M_{\mathbf{1}})/T=\text{{H}}^{1}(K_{w},M_{\omega})/T=0$.

From Claim 0, there is an isomorphism

$\text{{H}}^{1}(K_{w},M_{\vartheta})/T\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\text{{H}}^{2}(K_{w},F/\mathcal{O}(\vartheta)).$

From (1.4), one gets

$0\to\text{{H}}^{1}(K_{w},F/\mathcal{O}(\vartheta))/\mathfrak{p}\to\text{{H}}^{2}(K_{w},\mathbf{F}(\vartheta))\to\text{{H}}^{2}(K_{w},F/\mathcal{O}(\vartheta))[\mathfrak{p}]\to
0$

and

$0\to\text{{H}}^{2}(K_{w},F/\mathcal{O}(\vartheta))/\mathfrak{p}\to\text{{H}}^{3}(K_{w},\mathbf{F}(\vartheta))\to\text{{H}}^{3}(K_{w},F/\mathcal{O}(\vartheta))[\mathfrak{p}]\to
0.$

In the first sequence,
$\text{{H}}^{2}(K_{w},\mathbf{F}(\vartheta))\simeq\text{{H}}^{0}(K_{w},\mathbf{F}(\omega\vartheta^{-1}))^{\vee}$
by Tate’s local duality. In the second sequence, the $\text{{H}}^{3}$ terms
are $0$ since ${\operatorname{cd}}(K_{w})=2$. Hence
$\text{{H}}^{2}(K_{w},F/\mathcal{O}(\vartheta))$ is $p$-divisible and finitely
generated, hence isomorphic to $(F/\mathcal{O})^{r}$ for some $r$. If
$\vartheta=\mathbf{1}$, $r=0$, since the middle term in the first sequence is
$0$, and so $\text{{H}}^{1}(K_{w},M_{\mathbf{1}})/T=0$. If $\vartheta=\omega$,
$\text{{H}}^{2}(K_{w},F/\mathcal{O}(\omega))\cong\text{{H}}^{0}(K_{w},\mathbf{Z}_{p}(\omega))^{\vee}$
by Tate’s local duality (extended to $\mathcal{O}$-modules using that
projective limits are dual to injective ones) and the latter is $0$.

Claim 2a: $\text{{H}}^{1}(K_{w},M_{\mathbf{1}})[T]=F/\mathcal{O}$.

Equivalently, $(X/TX)^{\vee}=0$ for the Pontryagin dual. The long exact
cohomology sequence over $K_{w}$ for (1.1) gives an exact sequence

(1.3)
$0\to\text{{H}}^{0}(K_{w},M_{\vartheta})/T\to\text{{H}}^{1}(K_{w},F/\mathcal{O}(\vartheta))\to
X^{\vee}[T]\to 0.$

By Claim 0,
$\text{{H}}^{0}(K_{w},M_{\vartheta})/T=(F/\mathcal{O})/T=F/\mathcal{O}$, hence
$(\text{{H}}^{0}(K_{w},M_{\vartheta})/T)^{\vee}=\text{{H}}^{0}(K_{w},M_{\vartheta})^{\vee}[T]=\mathcal{O}$.

If $\vartheta|_{G_{w}}=\mathbf{1}$, the middle term
$\text{{H}}^{1}(K_{w},F/\mathcal{O}(\vartheta))$ is canonically isomorphic to
${\operatorname{Gal}}(K_{w}^{{\operatorname{ab}},p}/K_{w})^{\vee}=(K_{w}^{\times})^{\wedge,p}=\mathbf{Z}_{p}^{2}$
by local class field theory since $\mu_{p}\not\subset K_{w}^{\times}$. Hence
the Pontryagin dual of the exact sequence (1.3) reads

$0\to X/TX\to\mathcal{O}^{2}\to\mathcal{O}\to 0,$

hence ${\operatorname{rk}}_{\mathcal{O}}X/TX=1$ and $X/TX$ is
$\mathcal{O}$-free, hence $X/TX\cong\mathcal{O}$ and so $X^{\vee}[T]\cong
F/\mathcal{O}$.

Since $X[T]=0$ from Claim 1, apply Lemma 1.1.2 to
$X\colonequals\text{{H}}^{1}(K_{w},M_{\vartheta})^{\vee}$ to get (ii) in the
case $\vartheta|_{G_{w}}=\mathbf{1}$.

Claim 2b: $\text{{H}}^{1}(K_{w},M_{\omega})[T]\cong
F/\mathcal{O}\oplus\mathcal{O}/\mathfrak{p}^{n}$ for some $n>0$.

If $\vartheta|_{G_{w}}=\omega$, we look at the sequence (1.3) again:

$\text{{H}}^{0}(K_{w},M_{\vartheta})/T=(F/\mathcal{O}(\omega))^{G_{K_{w,\infty}}}/T=0/T=0$

because $\mu_{p}\not\subset K_{w,\infty}$:
${\operatorname{Gal}}(K_{w}(\mu_{p})/K_{w})\simeq(\mathbf{Z}/p)^{\times}$
cannot be a quotient of the $\mathbf{Z}_{p}$-extension $K_{w,\infty}/K_{w}$ if
$p>2$ by Lemma 1.0.2 (iii). Hence

$\text{{H}}^{1}(K_{w},F/\mathcal{O}(\omega))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}X^{\vee}[T]=(X/TX)^{\vee}.$

The long exact Galois cohomology sequence of

(1.4) $0\to\mathbf{F}(\omega)\to
F/\mathcal{O}(\omega)\stackrel{{\scriptstyle\pi}}{{\to}}F/\mathcal{O}(\omega)\to
0$

gives the short exact sequence

$0\to\text{{H}}^{0}(K_{w},F/\mathcal{O}(\omega))/\mathfrak{p}\to\text{{H}}^{1}(K_{w},\mathbf{F}(\omega))\to\text{{H}}^{1}(K_{w},F/\mathcal{O}(\omega))[\mathfrak{p}]\to
0.$

We have already seen $\text{{H}}^{0}(K_{w},F/\mathcal{O}(\omega))=0$, hence
one has an isomorphism

$\text{{H}}^{1}(K_{w},\mathbf{F}(\omega))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\text{{H}}^{1}(K_{w},F/\mathcal{O}(\omega))[\mathfrak{p}].$

One can compute the (finite) $\mathbf{F}$-dimension of
$\text{{H}}^{1}(K_{w},\mathbf{F}(\omega))$ using the local Euler–Poincaré
characteristic formula, or by noting that
$\text{{H}}^{1}(K_{w},\mathbf{F}(\omega))$ is dual to
$\text{{H}}^{1}(K_{w},\mathbf{F})$ by Tate’s local duality, and the latter is
an $\mathbf{F}$-vector space of dimension
${\operatorname{dim}}_{\mathbf{F}}(K_{w}^{\times})/p=2$ by local class field
theory, noting that $\mu_{p}\not\subset
K_{w}^{\times}=\mathbf{Q}_{p}^{\times}$ since $p>2$.

The Galois cohomology of (1.4) also gives an exact sequence

$0\to\text{{H}}^{1}(K_{w},F/\mathcal{O}(\omega))/\mathfrak{p}\to\text{{H}}^{2}(K_{w},\mathbf{F}(\omega))\to\text{{H}}^{2}(K_{w},F/\mathcal{O}(\omega))[\mathfrak{p}]\to
0.$

The last term evaluates by Tate’s local duality theorem to

$\text{{H}}^{2}(K_{w},F/\mathcal{O}(\omega))[\mathfrak{p}]\simeq(\text{{H}}^{0}(K_{w},\mathcal{O}(\chi\omega^{-1}))/\mathfrak{p})^{\vee}=0$

with $\chi\colon G_{K_{w}}\to\mathcal{O}^{\times}$ the $p$-adic cyclotomic
character and $\chi\omega^{-1}$ its projection $\chi^{1}$ to the principal
units $1+p\mathbf{Z}_{p}$ (note that $p>2$). Hence

(1.5)
$\text{{H}}^{1}(K_{w},F/\mathcal{O}(\omega))/\mathfrak{p}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\text{{H}}^{2}(K_{w},\mathbf{F}(\omega))=\text{{H}}^{0}(K_{w},\mathbf{F})^{\vee}=\mathbf{F}$

by Tate’s local duality.

Hence we know $\text{{H}}^{1}(K_{w},F/\mathcal{O}(\omega))$ mod $\mathfrak{p}$
(namely $\mathbf{F}$) and its $\mathfrak{p}$-torsion (namely
$\mathbf{F}^{2}$). As it is a cofinitely generated $\mathcal{O}$-module (since
its $\mathfrak{p}$-torsion is finite dimensional), writing it as
$(F/\mathcal{O})^{r}\oplus A$ with $A$ a finite $\mathcal{O}$-module,
$A/\mathfrak{p}\cong\mathbf{F}$ from (1.5), hence
$A\cong\mathcal{O}/\mathfrak{p}^{n}$ for some $n>0$, and so
$\mathbf{F}^{2}\cong\text{{H}}^{1}(K_{w},\mathbf{F}(\omega))=\text{{H}}^{1}(K_{w},F/\mathcal{O}(\omega))[\mathfrak{p}]=\mathbf{F}^{r+1}$.
Hence $r=1$ and $\text{{H}}^{1}(K_{w},F/\mathcal{O}(\omega))\cong
F/\mathcal{O}\oplus\mathcal{O}/\mathfrak{p}^{n}$ with $n>0$, agreeing with the
local Euler–Poincaré characteristic formula
$0-2+1=\chi(K_{w},\mathbf{F}(\omega))=-[K_{w}:\mathbf{Q}_{p}]=-1$.

Determination of $\text{{H}}^{1}(K_{w},M_{\omega})$ as a $\Lambda$-module.

Hence we know from (1.3) that
$X/TX\cong\mathcal{O}\oplus\mathcal{O}/\mathfrak{p}^{n}$ with $n>0$ in the
case $\vartheta|_{G_{w}}=\omega$, so by Lemma 1.1.2 $X$ is not $\Lambda$-free
and is generated by $\leq 2$ elements over $\Lambda$, and that $X[T]=0$ from
Claim 1. When $X[T]=0$, $X$ is $\Lambda$-torsion free, and there is a
presentation

${0}$${C}$${\Lambda^{2}}$${X}$${0}$${0}$${C/T}$${\Lambda^{2}/T}$${X/T}$${0}$${0}$${\mathfrak{p}^{n}\mathcal{O}}$${\mathcal{O}^{2}}$${\mathcal{O}\oplus\mathcal{O}/\mathfrak{p}^{n}}$${0}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

of $X$ as a $\Lambda$-module. Since $C\subset\Lambda^{2}$ is $\Lambda$-torsion
free and $C/T\cong\mathfrak{p}^{n}\mathcal{O}$, $C\cong\Lambda$ by Lemma
1.1.2. Hence the presentation implies that ${\operatorname{rk}}_{\Lambda}X=1$
and ${\operatorname{pd}}_{\Lambda}X\leq 1$, hence $X$ has no finite
$\Lambda$-submodule by [NSW08, Proposition (5.3.19) (i)]. ∎

###  1.2. Selmer groups of characters

Let $\vartheta\colon G_{K}\to\mathbf{F}_{\mathfrak{p}}^{\times}$ be a
character whose conductor is divisible only by primes split in $K$ (i.e.,
unramified over $\mathbf{Q}$ and have degree $1$).

Let $\Sigma$ be a finite set of places of $K$ containing $\infty$ and the
primes dividing $p$ or the conductor of $\vartheta$ and such that every finite
place in $\Sigma$ is split in $K$. Denote the maximal extension of $K$
unramified outside $\Sigma$ by $K^{\Sigma}$. Recall that $p$ splits as
$v\overline{v}$ in $K$.

######  Definition 1.2.1.

  1. (i)

The _Greenberg Selmer group_ of $\vartheta$ is

$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{Gr}}}(K,M_{\vartheta})\colonequals\ker\Big{\\{}\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta})\xrightarrow{{\operatorname{res}}_{\operatorname{Gr}}}\prod_{w\in\Sigma,w\nmid
p}\text{{H}}^{1}(K_{w},M_{\vartheta})\times\text{{H}}^{1}(K_{\overline{v}},M_{\vartheta})\Big{\\}}.$

  2. (ii)

For $S\colonequals\Sigma\setminus\\{v,\overline{v},\infty\\}$, let the _$S$
-imprimitive Selmer group_ of $\vartheta$ be

$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{Gr}}^{S}}(K,M_{\vartheta})\colonequals\ker\Big{\\{}\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta})\to\text{{H}}^{1}(K_{\overline{v}},M_{\vartheta})\Big{\\}},$

Replacing $M_{\vartheta}$ by $M_{\vartheta}[\mathfrak{p}]$ in the above
definitions, we obtain the _residual Greenberg Selmer groups_
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{Gr}}}(K,M_{\vartheta}[\mathfrak{p}])$
and
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{Gr}}^{S}}(K,M_{\vartheta}[\mathfrak{p}])$.

  3. (iii)

The _Selmer group with unramified local conditions away from $v$_, also called
unramified Selmer group for short, of $\vartheta$ is

$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}}(K,M_{\vartheta})\colonequals\ker\Big{\\{}\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta})\xrightarrow{{\operatorname{res}}_{{\operatorname{nr}},\vartheta}}\prod_{w\in\Sigma,w\nmid
p}\text{{H}}^{1}(I_{w},M_{\vartheta})^{G_{w}/I_{w}}\times\text{{H}}^{1}(I_{\overline{v}},M_{\vartheta})^{G_{\overline{v}}/I_{\overline{v}}}\Big{\\}}.$

  4. (iv)

For $S\colonequals\Sigma\setminus\\{v,\overline{v},\infty\\}$, let the _$S$
-imprimitive unramified Selmer group_ of $\vartheta$ be

$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\vartheta})\colonequals\ker\Big{\\{}\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta})\to\text{{H}}^{1}(I_{\overline{v}},M_{\vartheta})^{G_{\overline{v}}/I_{\overline{v}}}\Big{\\}},$

Replacing $M_{\vartheta}$ by $M_{\vartheta}[\mathfrak{p}]$ in the above
definitions, we obtain the _residual unramified Selmer groups_
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}}(K,M_{\vartheta}[\mathfrak{p}])$
and
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\vartheta}[\mathfrak{p}])$.

Note these Selmer groups are $\Lambda$-cofinitely generated (by finiteness of
the $\mathfrak{p}$-Selmer groups and Nakayama’s lemma), and the Selmer groups
are independent of the choice of $\Sigma$ as reflected in their notation.

The local conditions for these unramified Selmer groups are relaxed (namely,
the full group) at $v$ and $\mathcal{L}_{v}=\ker(r_{w})$ for
$w\in\Sigma\setminus\\{v\\}$ where
$r_{w}\colon\text{{H}}^{1}(K_{w},M_{\vartheta})\to\text{{H}}^{1}(I_{w},M_{\vartheta})^{G_{w}/I_{w}}$
is the restriction in Proposition 1.1.3. Note that
$\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta})={\operatorname{Sel}}^{\Sigma}(K,M_{\vartheta})$
with the $\Sigma$-relaxed Selmer group and
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}}(K,M_{\vartheta})={\operatorname{Sel}}^{\Sigma}_{\Sigma\setminus\\{v\\}}(K,M_{\vartheta})$
with the $\Sigma\setminus\\{v\\}$-strict Selmer group by [Rub00, Lemma I.5.3].
Analogously,
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\vartheta})={\operatorname{Sel}}^{\Sigma}_{\\{\overline{v}\\}}(K,M_{\vartheta})$
is the $\overline{v}$-strict Selmer group. The $S$-imprimitive Selmer group
has less local conditions than the Selmer group, omitting the local conditions
at $S$. Hence one has an inclusion
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}}(K,M_{\vartheta})\subseteq\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\vartheta})$.

Note that because $K_{w}=\mathbf{Q}_{p}$ and
$[\mathbf{Q}_{p}(\mu_{p}):\mathbf{Q}_{p}]=p-1>1$, $\omega|_{G_{w}}\neq 1$.
Hence also $\omega|_{G_{K}}\neq 1$. We need this because we use several times
that $\omega$ has no invariants over $G_{w}$ and $G_{K}$.

######  Theorem 1.2.2.

Assume that the conductor of $\vartheta$ is only divisible by primes that
split in $K$. Then
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}}(K,M_{\vartheta})^{\vee}$ is
a finitely generated $\Lambda$-torsion module with $\mu$-invariant $0$.

###### Proof.

The first part of this proposition is essentially proved in [Rub91] (with
[Rub94] to remove the assumption on $h_{K}$). As explained in [CGLS22, Theorem
1.2.2], the dual Selmer groups
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}}(K,M_{\vartheta})^{\vee}$ are
readily identified with the $\vartheta$-isotypic components of
$\mathcal{X}_{\infty}$ (which are shown to be $\Lambda$-torsion in [Rub91,
Remark (ii) after Theorem 4.1]) where $\mathcal{X}_{\infty}$ is the Galois
group of the maximal abelian pro-$p$ extension of $K_{\infty}K_{\vartheta}$
unramified outside $v$, $K_{\vartheta}$ being the fixed field of
$\ker(\vartheta)$.

That $\mu$-invariants is $0$ follows from the vanishing result of [Hid10] and
the Iwasawa Main Conjectures in [Rub91] together with [CW78] (or more
generally, [dS87, III.1.10] to remove the assumption on $p\neq 3$, that $p$ is
an anomalous prime and that $i\neq 0$ which corresponds to the case where
$\vartheta|_{G_{K}}=\mathbf{1}$). More precisely, if
$\vartheta|_{G_{K}}\neq\mathbf{1}$, then the Main Conjecture for $\vartheta$
identifies Char$(\mathfrak{X}_{\vartheta})$ with $(\mathcal{L}_{\vartheta})$
and hence the algebraic $\mu$-invariant equals the analytic one which
vanishes. If $\vartheta|_{G_{K}}=\mathbf{1}$, the Main Conjecture identifies
Char$(\mathfrak{X}_{\mathbf{1}})$ with $(\mathcal{L}_{\mathbf{1}}\cdot T)$ so
again both $\mu$-invariants vanish. ∎

######  Remark 1.2.3.

  1. (i)

Recall that from Corollary 1.0.3, the local conditions of our Selmer group at
$w\nmid p$ agree with the Greenberg’s local conditions. This is also discussed
in [CGLS22, Theorem 1.2.2]. As a consequence, our Selmer group can be
alternatively defined as

$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}}(K,M_{\vartheta})\colonequals\ker\Big{\\{}\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta})\xrightarrow{{\operatorname{res}}}\prod_{w\in\Sigma,w\nmid
p}\text{{H}}^{1}(K_{w},M_{\vartheta})\times\text{{H}}^{1}(I_{\overline{v}},M_{\vartheta})^{G_{\overline{v}}/I_{\overline{v}}}\Big{\\}}.$

  2. (ii)

We remark here that Theorem 1.2.2 is crucial in the proof of the fact that the
global-to-local map defining the above Selmer group (as well as the
imprimitive one) is surjective.

###### Proof.

Regarding (ii): We want to show that the conditions (1)–(4) of [PW11,
Proposition A.2] are satisfied for
$\text{{H}}^{1}(K,M_{\vartheta})\simeq\text{{H}}^{1}(K_{\infty},F/\mathcal{O}(\vartheta))$
by Shapiro’s lemma(see Corollary 1.0.5). The conditions are for $K/\mathbf{Q}$
an imaginary quadratic number field with $p=v\overline{v}$ and local
conditions $\mathcal{L}_{w}$:

  1. (1)

No place $w\mid p$ splits completely in $K_{\infty}$.

  2. (2)

The Selmer group is $\Lambda$-cotorsion.

  3. (3)

$\text{{H}}^{0}(K_{\infty},{\operatorname{Hom}}(\mathcal{O}(\vartheta),F/\mathcal{O}(1)))$
is finite.

  4. (4)

$r_{v}+r_{\overline{v}}\colonequals{\operatorname{corank}}_{\Lambda}\mathcal{L}_{v}+{\operatorname{corank}}_{\Lambda}\mathcal{L}_{\overline{v}}={\operatorname{rk}}_{\mathcal{O}}\mathcal{O}(\vartheta)\equalscolon[K:\mathbf{Q}]d-d\equalscolon\delta(K,V)$.

Condition (1) is satisfied for the anticyclotomic $\mathbf{Z}_{p}$-extension
by Lemma 1.0.2. By Theorem 1.2.2, $\mathfrak{X}_{\vartheta}$ is
$\Lambda$-torsion; this is condition (2). Denoting by $\chi$ the $p$-adic
cyclotomic character. Then the Cartier dual
${\operatorname{Hom}}(\mathcal{O}(\vartheta),F/\mathcal{O}(1))=F/\mathcal{O}(\chi\vartheta^{-1})$
has finite $G_{K_{\infty}}$-invariants for any finite order character
$\vartheta$. (This is equivalent to showing that
$\chi\vartheta^{-1}|_{G_{K_{\infty}}}\neq\mathbf{1}$. Note that
$\ker(\chi)=K_{{\operatorname{cyc}}}$, the cyclotomic
$\mathbf{Z}_{p}$-extension of $K$, is linearly disjoint from
$K_{{\operatorname{cyc}}}$, so
$\chi(G_{K_{\infty}})=\chi(G_{K_{\infty}}G_{K_{{\operatorname{cyc}}}})=\chi(G_{K})$.
Hence $\chi|_{G_{K_{\infty}}}\colon G_{K_{\infty}}\to\mathbf{Z}_{p}^{\times}$
has open image. But multiplying it by the character $\vartheta^{-1}$ with
finite image does not change this property.) This means that condition (3) is
satisfied. Condition (4) is satisfied since one has
$r_{v}=1,r_{\overline{v}}=0$ (from Proposition 1.1.3), $[K:\mathbf{Q}]=2$,
$d=1$ (we have characters) and $\delta(K,F(\vartheta))=1$ (there is exactly
one archimedean place up to equivalence, and it is complex, and $d=1$).

Surjectivity for the $S$-imprimitive Selmer group immediately follows. ∎

Consider the (finitely generated) $\Lambda$-modules

$\displaystyle\mathfrak{X}_{\vartheta}^{S}$
$\displaystyle\colonequals\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\vartheta})^{\vee},$
$\displaystyle\mathfrak{X}_{\vartheta}$
$\displaystyle\colonequals\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}}(K,M_{\vartheta})^{\vee}.$

We now determine $\lambda(\mathfrak{X}_{\vartheta}^{S})$ in terms of
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\vartheta}[\mathfrak{p}])$.

######  Lemma 1.2.4 (relation between the $p$-torsion of the $S$-imprimitive
Selmer group and the residual one).

One has an isomorphism

$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\vartheta}[\mathfrak{p}])\simeq\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\vartheta})[\mathfrak{p}].$

Note that one has a surjection
${\operatorname{Sel}}^{\Sigma}(K,M_{\vartheta}[\mathfrak{p}])\twoheadrightarrow{\operatorname{Sel}}^{\Sigma}(K,M_{\vartheta})[\mathfrak{p}]$
by [Rub00, Lemma I.5.4].

###### Proof.

The long exact cohomology sequence of $0\to M_{\vartheta}[\mathfrak{p}]\to
M_{\vartheta}\to M_{\vartheta}\to 0$ (note that $M_{\vartheta}$ is
$\pi$-divisible since $\Lambda^{\vee}$ is) yields a short exact sequence

$0\to\text{{H}}^{0}(K^{\Sigma}/K,M_{\vartheta})/\pi\to\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta}[\mathfrak{p}])\to\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta})[\mathfrak{p}]\to
0,$

with an analogous short exact sequence with $K^{\Sigma}/K$ replaced by
$K_{\overline{v}}$. This allows us to first study the $p$-torsion of the
Greenberg Selmer groups, which recovers [CGLS22, Lemma 1.2.4] for arbitrary
characters (their assumption $\vartheta|_{G_{\overline{v}}}\neq 1$ is not
essential). Indeed, by the snake lemma one has a commutative diagram with
exact rows

(Gr,$\vartheta$)
${0}$${\ker(r_{\overline{v}}^{{\operatorname{Gr}},0})}$${\text{{H}}^{1}_{\mathcal{F}_{\operatorname{Gr}}^{S}}(K,M_{\vartheta}[\mathfrak{p}])}$${\text{{H}}^{1}_{\mathcal{F}_{\operatorname{Gr}}^{S}}(K,M_{\vartheta})[\mathfrak{p}]}$${0}$${\text{{H}}^{0}(K^{\Sigma}/K,M_{\vartheta})/\mathfrak{p}}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta}[\mathfrak{p}])}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta})[\mathfrak{p}]}$${0}$${0}$${\text{{H}}^{0}(K_{\overline{v}},M_{\vartheta})/\mathfrak{p}}$${\text{{H}}^{1}(K_{\overline{v}},M_{\vartheta}[\mathfrak{p}])}$${\text{{H}}^{1}(K_{\overline{v}},M_{\vartheta})[\mathfrak{p}]}$${0}$$\scriptstyle{r_{\overline{v}}^{{\operatorname{Gr}},0}}$$\scriptstyle{r_{\overline{v}}^{{\operatorname{Gr}},1}}$

We have to study the kernel and the cokernel of

$r_{\overline{v}}^{{\operatorname{Gr}},0}\colon\text{{H}}^{0}(K^{\Sigma}/K,M_{\vartheta})/\mathfrak{p}\to\text{{H}}^{0}(K_{\overline{v}},M_{\vartheta})/\mathfrak{p}.$

But its domain and codomain are $0$:
$\text{{H}}^{0}(K^{\Sigma}/K,M_{\vartheta})\simeq\text{{H}}^{0}(K^{\Sigma}_{\infty}/K_{\infty},F/\mathcal{O}(\vartheta))$
and
$\text{{H}}^{0}(K_{\overline{v}},M_{\vartheta})\simeq\text{{H}}^{0}(K_{\overline{v},\infty},F/\mathcal{O}(\vartheta))$,
both isomorphisms by Shapiro’s lemma, which are both $\pi$-divisible: If
$\vartheta=\mathbf{1}$, they are $F/\mathcal{O}$; and if
$\vartheta\neq\mathbf{1}$ but $\vartheta|_{G_{\overline{v}}}=\mathbf{1}$, they
are $0$ and $F/\mathcal{O}$. Otherwise
$0=\text{{H}}^{0}(K^{\Sigma}_{\infty}/K_{\infty},\mathbf{F}(\vartheta))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\text{{H}}^{0}(K^{\Sigma}_{\infty}/K_{\infty},F/\mathcal{O})[\mathfrak{p}]$
(and similar for the local $\text{{H}}^{0}$) shows that they are torsion
$p$-groups with trivial $\pi$-torsion, hence $0$. In particular, the middle
row gives the identification
$\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta}[\mathfrak{p}])=\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta})[\mathfrak{p}]$.

We then study the imprimitive Selmer groups with unramified local conditions.
There is a similar diagram

(nr,$\vartheta$)
${0}$${\ker(r_{\overline{v}}^{{\operatorname{nr}},0})}$${\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}^{S}}(K,M_{\vartheta}[\mathfrak{p}])}$${\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}^{S}}(K,M_{\vartheta})[\mathfrak{p}]}$${0}$${\text{{H}}^{0}(K^{\Sigma}/K,M_{\vartheta})/\mathfrak{p}}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta}[\mathfrak{p}])}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta})[\mathfrak{p}]}$${0}$${0}$${\ker(f_{\vartheta})}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{\vartheta}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{\vartheta}[\mathfrak{p}]})}}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{\vartheta})}{\ker({\operatorname{res}}_{M_{\vartheta}})}[\mathfrak{p}]}$$\scriptstyle{r_{\overline{v}}^{{\operatorname{nr}},0}}$$\scriptstyle{r_{\overline{v}}^{{\operatorname{nr}},1}}$$\scriptstyle{f_{\vartheta}}$

where again we need to study the domain and codomain of
$r_{\overline{v}}^{{\operatorname{nr}},0}$. But the domain remains the same,
so still $\ker(r_{\overline{v}}^{{\operatorname{nr}},0})=0$. We claim that the
codomain $\ker(f_{\vartheta})=0$ as well.

From here and onward, we use $\tilde{\omega}$ (resp. $\tilde{\mathbf{1}}$) to
denote a character $\vartheta$ of $G_{K}$ whose restriction to
$G_{\overline{v}}$ is $\omega$ (resp. $\mathbf{1}$) which may or may not be
$\omega$ (resp. $\mathbf{1}$) on $G_{K}$.

When $\vartheta|_{G_{\overline{v}}}=\omega$, Proposition 1.1.3 (i) shows that
$\ker({\operatorname{res}}_{M_{\tilde{\omega}}})=0$. Actually, the same
argument shows that
$\ker({\operatorname{res}}_{M_{\tilde{\omega}}[\mathfrak{p}]})=0$ as well,
since both $M_{\tilde{\omega}}^{G_{\overline{v}}}$ and
$M_{\tilde{\omega}}[\mathfrak{p}]^{G_{\overline{v}}}$ are $0$. Therefore, if
$\vartheta|_{G_{\overline{v}}}=\omega$, the above two diagrams agree, so
$\ker(f_{\tilde{\omega}})=\text{{H}}^{0}(K_{\overline{v}},M_{\tilde{\omega}})/\mathfrak{p}=0$.

When $\vartheta|_{G_{\overline{v}}}=\mathbf{1}$, however, we need to study the
map $f_{\tilde{\mathbf{1}}}$ in the bottom row more carefully. Note that in
general for $C$ a $\mathfrak{p}$-divisible submodule of $A$ we have an
isomorphism

(1.6) $A/C[\mathfrak{p}]\simeq A[\mathfrak{p}]/C[\mathfrak{p}].$

Notice that from Proposition 1.1.3 (i),
$\ker({\operatorname{res}}_{M_{\tilde{\mathbf{1}}}})=F/\mathcal{O}$. A similar
argument shows that for $M_{\tilde{\mathbf{1}}}[\mathfrak{p}]$, the exact
sequence (1.2) reads

$0\to\ker({\operatorname{res}}_{M_{\tilde{\mathbf{1}}}[\mathfrak{p}]})^{\vee}\to\mathbf{F}[\Gamma/I_{w}]\to\mathbf{F}[\Gamma/I_{w}]\to\mathbf{F}\to
0,$

so
$\ker({\operatorname{res}}_{M_{\tilde{\mathbf{1}}}[\mathfrak{p}]})=\mathbf{F}$.

Note also that from the last row in the first diagram (Gr,$\vartheta$), since
$\text{{H}}^{0}(K_{\overline{v}},M_{\tilde{\mathbf{1}}})/\mathfrak{p}=0$,
there is an identification
$\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\mathbf{1}}}[\mathfrak{p}])=\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\mathbf{1}}})[\mathfrak{p}]$.
Therefore the domain of $f_{\tilde{\mathbf{1}}}$ is identified with
$\frac{\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\mathbf{1}}})[\mathfrak{p}]}{\ker({\operatorname{res}}_{M_{\tilde{\mathbf{1}}}[\mathfrak{p}]})}$.
Let $A=\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\mathbf{1}}})$ and
$C=\ker({\operatorname{res}}_{M_{\tilde{\mathbf{1}}}})=F/\mathcal{O}$. The map
$f_{\tilde{\mathbf{1}}}$ then looks like

$A[\mathfrak{p}]/\mathbf{F}\xrightarrow{f_{\tilde{\mathbf{1}}}}A/(F/\mathcal{O})[\mathfrak{p}]=A[\mathfrak{p}]/(F/\mathcal{O})[\mathfrak{p}]=A[\mathfrak{p}]/\mathbf{F},$

where the identification in the middle comes from (1.6). However,
$f_{\tilde{\mathbf{1}}}$ is induced by the natural map
$\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\mathbf{1}}}[\mathfrak{p}])\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\mathbf{1}}})[\mathfrak{p}]:=A[\mathfrak{p}]$,
so $f_{\tilde{\mathbf{1}}}$ must itself be an isomorphism on the quotients.
Hence $\ker(f_{\tilde{\mathbf{1}}})=0$ as well.

Hence in both cases $\vartheta|_{G_{\overline{v}}}=\omega$ and $\mathbf{1}$,
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\vartheta}[\mathfrak{p}])\to\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\vartheta})[\mathfrak{p}]$
is an isomorphism by the snake lemma applied to the second diagram. ∎

######  Proposition 1.2.5.

The $S$-imprimitive Selmer group $\mathfrak{X}_{\vartheta}^{S}$ is
$\Lambda$-torsion with $\mu$-invariant $0$ and $\lambda$-invariant

$\lambda(\mathfrak{X}_{\vartheta}^{S})=\lambda(\mathfrak{X}_{\vartheta})+\sum_{w\in\Sigma,w\nmid
p}\lambda(\mathcal{P}_{w}(\vartheta)).$

Moreover,
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\vartheta}[\mathfrak{p}])$
is finite and if $\vartheta|_{G_{\overline{v}}}\neq\omega$,
${\operatorname{dim}}_{\mathbf{F}}\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\vartheta}[\mathfrak{p}])=\lambda(\mathfrak{X}_{\vartheta}^{S})$.

###### Proof.

When $\vartheta|_{G_{\overline{v}}}\neq\mathbf{1},\omega$, this is [CGLS22,
Proposition 1.2.5]. Now we obtain from Remark 1.2.3 that the restriction map
defining $\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}}(K,M_{\vartheta})$
using unramified local conditions is surjective, so the sequence

(1.7)
$0\to\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}}(K,M_{\vartheta})\to\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta})\to\prod_{w\in\Sigma,w\nmid
p}\text{{H}}^{1}(K_{w},M_{\vartheta})\times\text{{H}}^{1}(I_{\overline{v}},M_{\vartheta})^{G_{\overline{v}}/I_{\overline{v}}}\to
0$

is exact. From the definitions, this readily yields the exact sequence

(1.8)
$0\to\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}}(K,M_{\vartheta})\to\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\vartheta})\to\prod_{w\in
S}\text{{H}}^{1}(K_{w},M_{\vartheta})\to 0,$

which combined with Theorem 1.2.2 and Lemma 1.1.1 gives the first part of the
proposition.

Essentially the same proof in loc. cit. shows
$\text{{H}}^{2}(K^{\Sigma}/K,M_{\vartheta})$ is 0. More precisely, we adopt
the arguments from Greenberg in [Gre89], as we now explain.

First note that because ${\operatorname{Gal}}(K^{\Sigma}/K)$ has
$p$-cohomological dimension $2$, $\text{{H}}^{2}(K^{\Sigma}/K,M_{\vartheta})$
is cofree. Proposition 3 in op. cit. yields the following equation of
$\Lambda$-coranks:

${\operatorname{corank}}_{\Lambda}(\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta}))-{\operatorname{corank}}_{\Lambda}(\text{{H}}^{2}(K^{\Sigma}/K,M_{\vartheta}))=1.$

Since
${\operatorname{corank}}_{\Lambda}(\text{{H}}^{1}(K^{\Sigma}/K,M_{\vartheta}))\leq
1$(it follows from the exact sequence (1.7), together with Theorem 1.2.2,
Lemma 1.1.1 and Proposition 1.1.3), this gives the cotorsionness of
$\text{{H}}^{2}(K^{\Sigma}/K,M_{\vartheta})$, so
$\text{{H}}^{2}(K^{\Sigma}/K,M_{\vartheta})=0$.

Then [Gre89, Proposition 5] shows that
$\text{{H}}^{1}(K^{\Sigma}/K,M_{\tilde{{\mathbf{1}}}})^{\vee}$ has no finite
$\Lambda$-submodules. From Proposition 1.1.3, we see that
$P^{S}:=(\frac{\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{{\mathbf{1}}}})}{\ker({\operatorname{res}}_{{M_{\tilde{\mathbf{1}}}}})})^{\vee}$
fits into an exact sequence

$0\to P^{S}\to\Lambda\to\mathcal{O}\to 0,$

so $P^{S}$ is a free $\Lambda$-module of rank $1$. Then (1.7) and (1.8)
readily yields

$\mathfrak{X}^{S}_{\tilde{\mathbf{1}}}\cong\frac{\text{{H}}^{1}(K^{\Sigma}/K,M_{\tilde{\mathbf{1}}})^{\vee}}{P^{S}}$

as $\Lambda$-modules, and by [GV00, Lemma 2.6] we conclude that
$\mathfrak{X}^{S}_{\tilde{\mathbf{1}}}$ has no finite $\Lambda$-submodules.

Since $\mathfrak{X}_{\tilde{\mathbf{1}}}^{S}$ is torsion with $\mu$-invariant
$0$ by Theorem 1.2.2, the finiteness of
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\tilde{\mathbf{1}}})[\mathfrak{p}]$
(and hence of
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\tilde{\mathbf{1}}}[\mathfrak{p}])$
by Lemma 1.2.4) follows from the structure theorem. It then follows that
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\tilde{\mathbf{1}}})$
is divisible, and in particular

$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\tilde{\mathbf{1}}})\cong(F/\mathcal{O})^{\lambda(\mathfrak{X}_{\tilde{\mathbf{1}}}^{S})},$

which together with Lemma 1.2.4 gives the formula for $\lambda$-invariant.

The same argument shows that
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\tilde{\omega}}[\mathfrak{p}])$
is finite, except that $\mathfrak{X}^{S}_{\tilde{\omega}}$ might have some
finite submodules (also from Proposition 1.1.3) so we don’t obtain a simple
formula about the $\lambda$-invariant. The computation will be postponed to
section 1.4. ∎

######  Corollary 1.2.6.

  1. (i)

Assume $\vartheta|_{G_{\overline{v}}}\neq\omega$. Then
$\text{{H}}^{2}(K^{\Sigma}/K,M_{\vartheta}[\mathfrak{p}])=0$.

  2. (ii)

When $\vartheta|_{G_{\overline{v}}}=\mathbf{1}$, one has an exact sequence

$0\to\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}^{S}}(K,M_{\mathbf{1}}[\mathfrak{p}])\to\text{{H}}^{1}(K^{\Sigma}/K,M_{\mathbf{1}}[\mathfrak{p}])\to\text{{H}}^{1}(K_{\overline{v}},M_{\mathbf{1}}[\mathfrak{p}])/\ker({\operatorname{res}}_{M_{\tilde{\mathbf{1}}}[\mathfrak{p}]})\to
0.$

###### Proof.

(i) When $\vartheta|_{G_{\overline{v}}}\neq\mathbf{1},\omega$, this is proved
in [CGLS22]. Assume $\vartheta|_{G_{\overline{v}}}=\mathbf{1}$.

The cohomology long exact sequence induced by multiplication by $p$ on
$M_{\tilde{\mathbf{1}}}$ yields an isomorphism

(1.9)
$\frac{\text{{H}}^{1}(K^{\Sigma}/K,M_{\tilde{\mathbf{1}}})}{p\text{{H}}^{1}(K^{\Sigma}/K,M_{\tilde{\mathbf{1}}})}\cong\text{{H}}^{2}(K^{\Sigma}/K,M_{\tilde{\mathbf{1}}}[\mathfrak{p}]).$

On the other hand, from the exactness of (1.7) we deduce the exact sequence

(1.10)
$0\to\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\tilde{\mathbf{1}}})\to\text{{H}}^{1}(K^{\Sigma}/K,M_{\tilde{\mathbf{1}}})\to\text{{H}}^{1}(I_{\overline{v}},M_{\tilde{\mathbf{1}}})^{G_{\overline{v}}/I_{\overline{v}}}\to
0.$

Since
$\text{{H}}^{1}(I_{\overline{v}},M_{\tilde{\mathbf{1}}})^{G_{\overline{v}}/I_{\overline{v}}}$
is a quotient object of
$\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\mathbf{1}}})$, the dual of
$\text{{H}}^{1}(I_{\overline{v}},M_{\tilde{\mathbf{1}}})^{G_{\overline{v}}/I_{\overline{v}}}$
is a subobject of
$\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\mathbf{1}}})^{\vee}$. By
Proposition 1.1.3 (ii) the latter is free of $\Lambda$-rank 1, so the dual of
$\text{{H}}^{1}(I_{\overline{v}},M_{\tilde{\mathbf{1}}})^{G_{\overline{v}}/I_{\overline{v}}}$
has no $p$-torsion, too. Since we showed in Proposition 1.2.5 that
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\tilde{\mathbf{1}}})$
is divisible, it follows from (1.10) that
$\text{{H}}^{1}(K^{\Sigma}/K,M_{\tilde{\mathbf{1}}})^{\vee}$ has no
$p$-torsion, and so

$\text{{H}}^{2}(K^{\Sigma}/K,M_{\tilde{\mathbf{1}}}[\mathfrak{p}])\cong 0$

by (1.9).

(ii) By Remark 1.2.3 (ii), the global to local map defining the imprimitive
unramified Selmer group
$\text{{H}}^{1}_{\mathcal{F}^{S}_{{\operatorname{nr}}}}(K,M_{\tilde{\mathbf{1}}})$
is surjective. By Proposition 1.2.5,
$\text{{H}}^{1}_{\mathcal{F}^{S}_{{\operatorname{nr}}}}(K,M_{\tilde{\mathbf{1}}})$
is divisible. Now with our Lemma 1.2.4, the proof of the second claim in
Corollary 1.2.6 in op. cit. applies almost verbatim to our case. This finishes
the proof. ∎

###  1.3. Local cohomology groups of $\rho_{f}$

Recall that $f\in S_{k}(\Gamma_{0}(N))^{{\operatorname{new}}}$ is a newform of
weight $k=2r$. Let $T\colonequals T_{f}$ be the self-dual twist of the
$\mathfrak{p}$-adic Galois representation of $f$, $V\colonequals
V_{f}\colonequals T_{f}\otimes_{\mathcal{O}}F$, and $W\colonequals
W_{f}\colonequals V_{f}/T_{f}$. Let $M_{f}$ be the $G_{K}$-module

$M_{f}\colonequals T\otimes_{\mathcal{O}}\Lambda^{\vee}.$

Note that one has an exact sequence

$0\to W_{f}\to M_{f}\stackrel{{\scriptstyle\cdot T}}{{\to}}M_{f}\to 0.$

Recall that by $\tilde{\mathbf{1}}$ (resp. $\tilde{\omega}$) we mean a
character $\vartheta$ of $G_{K}$ with $\vartheta|_{G_{p}}=\mathbf{1}$ (resp.
$\omega$) which may or may not be $\mathbf{1}$ (resp. $\omega$) on $G_{K}$.
Compared to [CGLS22], we then have two new cases to deal with:

  1. (i)

$0\to\mathbf{F}(\tilde{\mathbf{1}})\to\overline{\rho}_{f}\to\mathbf{F}(\tilde{\omega})\to
0$

  2. (ii)

$0\to\mathbf{F}(\tilde{\omega})\to\overline{\rho}_{f}\to\mathbf{F}(\tilde{\mathbf{1}})\to
0$

We will see in section 1.4 that the choice of the ordering of the characters
appearing in the decomposition in $\overline{\rho}_{f}$ will not matter. For
our convenience, assume we are in case (ii). This assumption is in effect
throughout section 1.3 and section 1.4.

######  Proposition 1.3.1.

Let $\rho$ be an irreducible $2$-dimensional $p$-adic Galois representation
whose residual representation $\overline{\rho}$ has semisimplification the
direct sum of two $1$-dimensional characters. Then there is a lattice such
that the associated Galois representation is a non-split extension of those
two characters in a given ordering.

###### Proof.

See [Bel09, Proposition 1.4] ∎

Note that $V_{f}$ is an irreducible $G_{K}$-module (this follows from the Tate
conjecture for abelian varieties if $k=2$ and in for newforms of general
weight from [Rib77, Theorem 2.3]), but becomes reducible when restricted to
$G_{K_{v}}$ because it is crystalline with distinct Hodge–Tate weights and
characteristic polynomial having a unit root [BC09, Theorem 8.3.6]. Hence by
isogeny invariance of the BSD Conjecture, we can assume
$\overline{\rho}_{f}|_{G_{K}}$ non-split with no trivial subrepresentation and
that there is a short exact sequence

$0\to\mathbf{F}(\widetilde{\varphi})\to\overline{\rho}_{f}|_{G_{K}}\to\mathbf{F}(\widetilde{\psi})\to
0$

with $\varphi=\widetilde{\varphi}|_{G_{K_{v}}}=\omega$.

######  Theorem 1.3.2.

The $p$-adic Galois representation $\rho_{f}|_{G_{K_{w}}}$ is crystalline with
Hodge–Tate weights $\\{r,1-r\\}$.

###### Proof.

This is because $p\nmid N$ [Sch90, Theorem 1.2.4 (ii)] (note that the
condition $p\geq k$ there can be removed by [Tsu99]). It has Hodge–Tate
weights $\\{r,1-r\\}$ since we have Hodge–Tate weights $\\{0,k-1\\}$ before
taking the self-dual twist by $1-r$ [Sch90, 4.2.0]. ∎

We first compute some $\text{{H}}^{0}$ cohomology groups.

######  Proposition 1.3.3.

Let $p>2$ be a prime. Let $K/\mathbf{Q}$ be an imaginary quadratic field with
$p=v\overline{v}$ split in $K$ and $K_{\infty}|K$ be the anticyclotomic
$\mathbf{Z}_{p}$-extension.

Assume the reduction $\overline{\rho}_{f}=V_{f}/T_{f}[\mathfrak{p}]$ of
$V_{f}$ a $2$-dimensional residual Galois representation of $G_{\mathbf{Q}}$
which is a non-split extension

$0\to\mathbf{F}(\varphi)\to\overline{\rho}_{f}\to\mathbf{F}(\psi)\to 0$

with $\varphi=\varepsilon\omega^{k-1}$ and $\psi=\varepsilon^{-1}$ and
$\varepsilon$ unramified at $p$. Then
$\text{{H}}^{0}(G_{K_{\infty}},\overline{\rho}_{f})=0$ (i.e.,
$\overline{\rho}_{f}$ remains non-split over $G_{K_{\infty}}$).

###### Proof.

Since $\overline{\rho}_{f}$ is non-split, it defines a nonzero extension class
in

${\operatorname{Ext}}_{\mathbf{F}[G_{K}]}^{1}(\mathbf{F}(\varepsilon^{-1}),\mathbf{F}(\varepsilon\omega^{k-1}))\simeq{\operatorname{Ext}}_{\mathbf{F}[G_{K}]}^{1}(\mathbf{F},\mathbf{F}(\varepsilon^{2}\omega^{k-1}))\simeq\text{{H}}^{1}(K,\mathbf{F}(\varepsilon^{2}\omega^{k-1}));$

here we use that twisting by $\varepsilon$ induces an isomorphism on
${\operatorname{Ext}}^{1}$. Thus it suffices to show that
${\operatorname{res}}\colon\text{{H}}^{1}(K,\mathbf{F}(\varepsilon^{2}\omega^{k-1}))\to\text{{H}}^{1}(K_{\infty},\mathbf{F}(\varepsilon^{2}\omega^{k-1}))$
is injective.

Let $L=K(\varepsilon^{2})$. It is an extension of $K$ of degree dividing
$(\\#\mathbf{F}-1)/2$, hence coprime to $p$. It is also unramified at $p$.
Since $K(\omega^{k-1})/K$ is totally ramified at $p$ as $p>2$ and
$K\neq\mathbf{Q}(\sqrt{-3}),\mathbf{Q}(\sqrt{-1})$, one has $K(\omega)\cap
L=K$. Note that $K(\omega^{k-1})\supsetneq K$ because otherwise $2\mid
p-1={\operatorname{ord}}(\omega)\mid(k-1)$, but $k$ is even.

Now consider the commutative diagram of inflation-restriction sequences

${0}$${\text{{H}}^{1}(K_{\infty}|K,\mathbf{F}(\varepsilon^{2}\omega^{k-1})^{G_{K_{\infty}}})}$${\text{{H}}^{1}(K,\mathbf{F}(\varepsilon^{2}\omega^{k-1}))}$${\text{{H}}^{1}(K_{\infty},\mathbf{F}(\varepsilon^{2}\omega^{k-1}))}$${0}$${\text{{H}}^{1}(LK_{\infty}|L,\mathbf{F}(\varepsilon^{2}\omega^{k-1})^{G_{LK_{\infty}}})}$${\text{{H}}^{1}(L,\mathbf{F}(\varepsilon^{2}\omega^{k-1}))}$${\text{{H}}^{1}(LK_{\infty},\mathbf{F}(\varepsilon^{2}\omega^{k-1}))}$$\scriptstyle{{\operatorname{inf}}}$$\scriptstyle{{\operatorname{res}}}$$\scriptstyle{{\operatorname{inf}}}$$\scriptstyle{{\operatorname{res}}}$

The middle vertical arrow is injective by restriction-corestriction because
$p\nmid[L:K]$. The lower left group is $0$ because
$\mathbf{F}(\varepsilon^{2}\omega^{k-1})^{G_{LK_{\infty}}}=\mathbf{F}(\omega^{k-1})^{G_{LK_{\infty}}}=0$
since no non-trivial $p$-th root of unity is contained in $K_{\infty}$ because
it is a $\mathbf{Z}_{p}$-extension of $K$ and $\mu_{p}(K)=\\{1\\}$ since $p>2$
and $K\neq\mathbf{Q}(\sqrt{-3}),\mathbf{Q}(\sqrt{-1})$; note that
$\omega^{k-1}\neq\mathbf{1}$. Hence the lower restriction morphism is
injective, hence by commutativity and exactness of the diagram the top
restriction is injective. ∎

For simplicity, in the local cohomology groups, we write the characters as
$\mathbf{1}$ and $\omega$ instead of $\tilde{\mathbf{1}}$ and $\tilde{\omega}$
since we do not care about their global behavior.

######  Proposition 1.3.4.

Let $w\mid p$. Assume the Galois representation $\overline{\rho}_{f}$ does not
have a trivial subrepresentation. Assume that for $w\mid p$,
$\overline{\rho}_{f}|_{G_{K_{w}}}$ is an extension of the trivial
representation $\mathbf{F}$ by $\mathbf{F}(\omega)$.

  1. (i)

$\text{{H}}^{0}(K,V_{f}/T_{f})=0$.

  2. (ii)

$\text{{H}}^{0}(K_{w},V_{f}/T_{f})$ is finite and cyclic, i.e., isomorphic to
$\mathcal{O}/\mathfrak{p}^{m}$ with $m\geq 0$.

  3. (iii)

$\text{{H}}^{0}(K_{w},M_{f})$ is finite and cyclic, i.e., isomorphic to
$\mathcal{O}/\mathfrak{p}^{m}$ with $m\geq 0$.

###### Proof.

  1. (i)

Follows by assumption on $T_{f}$ by Proposition 1.3.1.

  2. (ii)

Note that we have $V_{f}/T_{f}=M_{f}[T]$, hence
$\text{{H}}^{0}(?,V_{f}/T_{f})\subseteq\text{{H}}^{0}(?_{\infty},V_{f}/T_{f})[T]$
for $?\in\\{K,K_{w}\\}$, so (ii) follows from (iii).

  3. (iii)

Note that $\rho_{f}|_{G_{K_{w}}}$ is crystalline with Hodge–Tate weights
$\\{r,1-r\\}$ by Theorem 1.3.2. By [BC09, Theorem 8.3.6], $\rho_{f}|_{G_{w}}$
is reducible

$0\to F(\varphi)\to\rho_{f}|_{G_{w}}\to F(\psi)\to 0$

with characters $\varphi,\psi$ having Hodge–Tate weights $\\{r,1-r\\}$. By
[BC09, Proposition 8.3.4],
$\\{\varphi,\psi\\}=\\{\varepsilon\chi^{r},\varepsilon^{-1}\chi^{1-r}\\}$ with
$\varepsilon$ an unramified character of $G_{K_{w}}$. But for $m\neq 0$,
$\varepsilon\chi^{m}$ is non-trivial on $G_{K_{\infty,w}}$ by Lemma 1.0.2
(ii). Hence $\text{{H}}^{0}(K_{\infty,w},F(\varepsilon\chi^{m}))=0$, so
$\text{{H}}^{0}(K_{\infty,w},F/\mathcal{O}(\varepsilon\chi^{m}))\subseteq
F/\mathcal{O}$ is finite and cyclic. (Note that
$\text{{H}}^{0}(K_{\infty,w},F(\varepsilon\chi^{m}))=0$ implies
$\text{{H}}^{0}(K_{\infty,w},F/\mathcal{O}(\varepsilon\chi^{m}))$ finite by
[KO20, Lemma 2.7].) It remains to show that for one of $?=\varphi,\psi$, the
invariants $\text{{H}}^{0}(K_{\infty,w},F/\mathcal{O}(?))=0$.

Note that at least one
$\text{{H}}^{0}(K_{w},\mathbf{F}(\varphi)),\text{{H}}^{0}(K_{w},\mathbf{F}(\psi))$
is $0$ (since one of them is $\omega$ when restricted to $G_{w}$). Since
${\operatorname{ord}}(\omega|_{G_{K_{\infty}}})=p-1$ (since $p-1$ is coprime
to $p$), also at least one
$\text{{H}}^{0}(K_{\infty,w},F/\mathcal{O}(\varphi)),\text{{H}}^{0}(K_{\infty,w},F/\mathcal{O}(\psi))$
is $0$. Hence $\text{{H}}^{0}(K_{\infty,w},V_{f}/T_{f})\subseteq
F/\mathcal{O}$. It remains to show that we cannot have equality.

Case $r>1$: Note that in this case, both Hodge–Tate weights are $\neq 0$. Let
$K_{w}^{\varepsilon}$ be a finite extension of $K_{w}$ which trivializes
$\varepsilon$. Then the restriction of $\varphi,\psi\colon
G_{K_{w}^{\varepsilon}}\to\mathbf{Z}_{p}^{\times}$ have open image and are
equal to some non-zero powers $\chi^{a}$ of $\chi$. We claim that the
restriction of $\varphi,\psi$ to $K_{\infty,w}$ also have open image, from
which the proposition follows. It suffices to show the images of
$\varphi,\psi\colon
G_{K_{w}^{\varepsilon}K_{\infty,w}}\to\mathbf{Z}_{p}^{\times}$ are open.

Let $K_{w,{\operatorname{cyc}}}^{\varepsilon}$ be a finite extension of
$K_{w,{\operatorname{cyc}}}$ which trivializes $\varepsilon$. Then
$\varphi=\chi^{a}$ on $G_{K_{w,{\operatorname{cyc}}}^{\varepsilon}\cap
K_{w}^{\varepsilon}K_{\infty,w}}$. From Lemma 1.0.2 (ii) we know that

${\operatorname{Gal}}((K_{\infty,w}\cap
K_{w,{\operatorname{cyc}}})/K_{w})\cong{\operatorname{Gal}}(K_{\infty,w}/K_{w})/{\operatorname{Gal}}(K_{\infty,w}/(K_{\infty,w}\cap
K_{w,{\operatorname{cyc}}}))$

must be finite since it’s a non-trivial quotient of
${\operatorname{Gal}}(K_{\infty,w}/K_{w})\cong\mathbf{Z}_{p}$ by a proper open
subgroup. It follows that $K_{w,{\operatorname{cyc}}}^{\varepsilon}\cap
K_{w}^{\varepsilon}K_{\infty,w}$ is also a finite extension of $K_{w}$. Now
$\varphi(G_{K_{w}^{\varepsilon}K_{\infty,w}})=\chi^{a}(G_{K_{w}^{\varepsilon}K_{\infty,w}})=\chi^{a}(G_{K_{w,{\operatorname{cyc}}}^{\varepsilon}}G_{K_{w}^{\varepsilon}K_{\infty,w}})=\chi^{a}(G_{K_{w,{\operatorname{cyc}}}^{\varepsilon}\cap
K_{w}^{\varepsilon}K_{\infty,w}})$ must be open. The same is true for $\psi$.

Case $r=1$: Since $r=1$, $\rho_{f}$ is the $\mathfrak{p}$-adic Galois
representation attached to an abelian variety $A/K_{w}$ with good reduction.
Note that then $F/\mathcal{O}(\varphi)[\mathfrak{p}]=\mathbf{F}(\omega)$ and
we must show that $\text{{H}}^{0}(K_{\infty,w},F/\mathcal{O}(\psi))\subsetneq
F/\mathcal{O}$. Since $r=1$, $\psi$ is unramified. Hence
$\text{{H}}^{0}(K_{w}^{\operatorname{nr}},F/\mathcal{O}(\psi))=F/\mathcal{O}$.
Hence $G_{K_{w}^{\operatorname{nr}}}\subseteq\ker\psi$. We have to show that
$\text{{H}}^{0}(K_{\infty,w},F(\psi))\subsetneq F/\mathcal{O}$, i.e.,
$G_{K_{\infty,w}}\not\subseteq\ker\psi$. Suppose otherwise, then
$G_{K_{w}^{\operatorname{nr}}}G_{K_{\infty,w}}\subseteq\ker\psi$, i.e.,
$\text{{H}}^{0}(K_{\infty,w}\cap
K_{w}^{\operatorname{nr}},F/\mathcal{O}(\psi))=F/\mathcal{O}$. But
$K_{\infty,w}|K_{w}$ is infinitely ramified, hence $[K_{\infty,w}\cap
K_{w}^{\operatorname{nr}}:K_{w}]<\infty$ and over any finite extension
$L/K_{w}$, $A[p^{\infty}](L)$ is finite by Mattuck’s theorem.∎

######  Lemma 1.3.5.

Let $w\mid p$ in $K$. Then $\text{{H}}^{1}(K_{w},M_{f})$ has $\Lambda$-corank
$2$.

###### Proof.

Let $X\colonequals\text{{H}}^{1}(K_{w},M_{f})^{\vee}$. Consider the exact
sequences

(1.11) $0\to V_{f}/T_{f}\to M_{f}\xrightarrow{\text{$\cdot T$ }}M_{f}\to 0$

and

(1.12) $0\to\overline{\rho}_{f}\to V_{f}/T_{f}\xrightarrow{\text{$\cdot\pi$
}}V_{f}/T_{f}\to 0.$

We aim to study the $\mathcal{O}$-rank of $X/T$ and mimic our argument in
Proposition 1.1.3 (ii). We will show that $X[T]=0$ and the $\mathcal{O}$-rank
is $2$, then the $\Lambda$-rank of $X$ is $2$ by the topological Nakayama
lemma as in Lemma 1.1.2.

We would like to study the following short exact sequence coming from (1.11)

(1.13)
$0\to\text{{H}}^{0}(K_{w},M_{f})/T\to\text{{H}}^{1}(K_{w},V_{f}/T_{f})\to
X^{\vee}[T]\to 0,$

where the first term is finite by Proposition 1.3.4 (i).

By (1.12), there is an exact sequence

(1.14)
$0\to\text{{H}}^{1}(K_{w},V_{f}/T_{f})/\mathfrak{p}\to\text{{H}}^{2}(K_{w},\overline{\rho}_{f})\to\text{{H}}^{2}(K_{w},V_{f}/T_{f})[\mathfrak{p}]\to
0.$

We claim that the last term in the above sequence is $0$. Indeed,
$\text{{H}}^{2}(K_{w},V_{f}/T_{f})=0$ because it’s dual to
$\text{{H}}^{0}(K_{w},T_{f})$ which must vanish because
$\text{{H}}^{0}(K_{w},V_{f}/T_{f})$ is finite by Proposition 1.3.4 (ii). Note
that there is a sequence

$0\to\text{{H}}^{0}(K_{w},\mathbf{F}(\omega))=0\to\text{{H}}^{0}(K_{w},\overline{\rho}_{f})\to\text{{H}}^{0}(K_{w},\mathbf{F}(\mathbf{1}))=\mathbf{F},$

so $\text{{H}}^{0}(K_{w},\overline{\rho}_{f})$ has $\mathbf{F}$-dimension $0$
or $1$. From local duality and self-duality of $\overline{\rho}_{f}$,
$\text{{H}}^{2}(K_{w},\overline{\rho}_{f})=\text{{H}}^{0}(K_{w},\overline{\rho}_{f})^{\vee}$
has the same dimension. Now we consider two cases.

Case I:
${\operatorname{dim}}_{\mathbf{F}}(\text{{H}}^{2}(K_{w},\overline{\rho}_{f}))={\operatorname{dim}}_{\mathbf{F}}(\text{{H}}^{0}(K_{w},\overline{\rho}_{f}))=0$.

In this case, from the local Euler characteristic formula [NSW08, Theorem
7.3.1], $\text{{H}}^{1}(K_{w},\overline{\rho}_{f})$ has $\mathbf{F}$-dimension
$2$. By (1.12), there is an identification

(1.15)
$\text{{H}}^{1}(K_{w},V_{f}/T_{f})[\mathfrak{p}]\simeq\text{{H}}^{1}(K_{w},\overline{\rho}_{f})/(\text{{H}}^{0}(K_{w},V_{f}/T_{f})/\mathfrak{p}).$

Since in this case $\text{{H}}^{0}(K_{w},V_{f}/T_{f})=0$, from (1.15) we see
that $\text{{H}}^{1}(K_{w},V_{f}/T_{f})[\mathfrak{p}]$ has
$\mathbf{F}$-dimension $2$. Therefore
$\text{{H}}^{1}(K_{w},V_{f}/T_{f})/\mathfrak{p}=0$, so
$\text{{H}}^{1}(K_{w},V_{f}/T_{f})$ is $\mathcal{O}$-cofree of rank $2$. Hence
$X/T$ is also $\mathcal{O}$-free of rank $2$.

Case II:
${\operatorname{dim}}_{\mathbf{F}}(\text{{H}}^{2}(K_{w},\overline{\rho}_{f}))={\operatorname{dim}}_{\mathbf{F}}(\text{{H}}^{0}(K_{w},\overline{\rho}_{f}))=1$.

From (1.14), we see that $\text{{H}}^{1}(K_{w},V_{f}/T_{f})/\mathfrak{p}$ also
has $\mathbf{F}$-dimension $1$. Again by the local Euler characteristic
formula, $\text{{H}}^{1}(K_{w},\overline{\rho}_{f})$ has
$\mathbf{F}$-dimension $4$. Since in this case
$\text{{H}}^{0}(K_{w},V_{f}/T_{f})$ is non-trivial, we see from Proposition
1.3.4 (ii) that $\text{{H}}^{0}(K_{w},V_{f}/T_{f})/\mathfrak{p}=\mathbf{F}$.
So by (1.15), $\text{{H}}^{1}(K_{w},V_{f}/T_{f})[\mathfrak{p}]$ has
$\mathbf{F}$-dimension $3$. Hence
${\operatorname{rk}}_{\mathcal{O}}(X/T)={\operatorname{corank}}_{\mathcal{O}}(X^{\vee}[T])={\operatorname{corank}}_{\mathcal{O}}\text{{H}}^{1}(K_{w},V_{f}/T_{f})=2$.

Now we show that $X[T]=0$. Indeed, the same arguments in Proposition 1.1.3
Claim 0 shows $\text{{H}}^{2}(K_{w},M_{f})=0$. We therefore have an
identification
$X^{\vee}/T=\text{{H}}^{1}(K_{w},M_{f})/T\cong\text{{H}}^{2}(K_{w},V_{f}/T_{f})=0$.
Now by Lemma 1.1.2 ((i) for Case I and (ii) for Case II), $X$ has
$\Lambda$-rank $2$. ∎

Similarly as in Proposition 1.1.3, we define the restriction map for $M_{f}$
by

${\operatorname{res}}_{M_{f}}\colon\text{{H}}^{1}(K_{w},M_{f})\to\text{{H}}^{1}(I_{w},M_{f})^{G_{w}/I_{w}}.$

Define ${\operatorname{res}}_{M_{f}[\mathfrak{p}]}$ by replacing $M_{f}$ with
$M_{f}[\mathfrak{p}]$ in the above definition.

######  Lemma 1.3.6.

$\ker({\operatorname{res}}_{M_{f}})$ is finite and cyclic. Moreover, when
$\text{{H}}^{0}(K_{w},M_{f})$ is non-trivial,
$\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})=\mathbf{F}$ and when
$\text{{H}}^{0}(K_{w},M_{f})=0$,
$\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})=0$.

###### Proof.

Similar to (1.2), there are two exact sequences for $M_{f}$ and
$M_{f}[\mathfrak{p}]$ respectively:

(1.16)
$0\to\ker({\operatorname{res}}_{M_{f}})^{\vee}\to(M_{f}^{\vee})_{I_{w}}\to(M_{f}^{\vee})_{I_{w}}\to
M_{f}^{\vee}/T\to 0.$

and

(1.17)
$0\to\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})^{\vee}\to(M_{f}[\mathfrak{p}]^{\vee})_{I_{w}}\to(M_{f}[\mathfrak{p}]^{\vee})_{I_{w}}\to
M_{f}[\mathfrak{p}]^{\vee}/T\to 0.$

Note that the last term in (1.16) is nothing but the dual of
$\text{{H}}^{0}(K_{w},M_{f})$, which is finite by Proposition 1.3.4 (iii).
Therefore
$\text{{H}}^{0}(K_{w},M_{f})^{\vee}=\text{{H}}^{0}(K_{w},M_{f})=\mathcal{O}/\mathfrak{p}^{n}$
for some $n\geq 0$.

Note that $(M_{f}^{\vee})_{I_{w}}$ is $\Lambda$-torsion because $I_{w}$ is
non-trivial by Lemma 1.0.2(iii), so $\ker({\operatorname{res}}_{M_{f}})$ is
also finite because characteristic ideals are multiplicative in exact
sequences of torsion $\Lambda$-modules. Moreover, if
$\text{{H}}^{0}(K_{w},M_{f})=0$, by Noetherian property of $\Lambda$ the
$\Lambda$-module surjection

$(M_{f}^{\vee})_{I_{w}}\to(M_{f}^{\vee})_{I_{w}}$

must be an isomorphism so $\ker({\operatorname{res}}_{M_{f}})=0$. Before
proving it’s always cyclic, we first prove the second half of this
proposition.

First assume $\text{{H}}^{0}(K_{w},M_{f})$ is non-trivial and hence cyclic,
then
$M_{f}[\mathfrak{p}]^{\vee}/T=\text{{H}}^{0}(K_{w},M_{f}[\mathfrak{p}])^{\vee}=(\text{{H}}^{0}(K_{w},M_{f})[\mathfrak{p}])^{\vee}=\mathbf{F}$.
Since
$(M_{f}[\mathfrak{p}]^{\vee})_{I_{w}}=(\text{{H}}^{0}(I_{w},M_{f}[\mathfrak{p}]))^{\vee}=(\text{{H}}^{0}(I_{w,\infty},\overline{\rho}_{f}))^{\vee}$
is finite (the second equality follows from Shapiro’s lemma), it follows from
sequence (1.17) that
$\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})=\mathbf{F}$.

Now if $\text{{H}}^{0}(K_{w},M_{f})=0$, then $M_{f}[\mathfrak{p}]^{\vee}/T=0$
so again from sequence (1.17) we see that
$\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})=0$.

Finally, we show
$\ker({\operatorname{res}}_{M_{f}})[\mathfrak{p}]\subseteq\mathbf{F}$, from
which it follows immediate that $\ker({\operatorname{res}}_{M_{f}})$ is cyclic
because we have showed it’s finite.

Consider the diagram

${0}$${\ker({\operatorname{res}})}$${\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})}$${\ker({\operatorname{res}}_{M_{f}})[\mathfrak{p}]}$${0}$${\text{{H}}^{0}(K_{w},M_{f})/\mathfrak{p}}$${\text{{H}}^{1}(K_{w},M_{f}[\mathfrak{p}])}$${\text{{H}}^{1}(K_{w},M_{f})[\mathfrak{p}]}$${0}$${0}$${\text{{H}}^{0}(I_{w},M_{f})/\mathfrak{p}}$${\text{{H}}^{1}(I_{w},M_{f}[\mathfrak{p}])}$${\text{{H}}^{1}(I_{w},M_{f})[\mathfrak{p}]}$${0}$${{\operatorname{coker}}({\operatorname{res}})}$$\scriptstyle{{\operatorname{res}}}$

When $\text{{H}}^{0}(K_{w},M_{f})=0$, we have seen that
$\ker({\operatorname{res}}_{M_{f}})=0$.

Now assume $\text{{H}}^{0}(K_{w},M_{f})$ is non-trivial so
$\text{{H}}^{0}(K_{w},M_{f})/\mathfrak{p}=\mathbf{F}$. In this case also
$\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})=\mathbf{F}$. Since again
$\text{{H}}^{0}(I_{w},M_{f})[\mathfrak{p}]=\text{{H}}^{0}(I_{w},M_{f}[\mathfrak{p}])=\text{{H}}^{0}(I_{w,\infty},\overline{\rho}_{f})$,
we have an exact sequence

$0\to\text{{H}}^{0}(I_{w,\infty},\mathbf{F}(\omega))=0\to\text{{H}}^{0}(I_{w},M_{f}[\mathfrak{p}])\to\text{{H}}^{0}(I_{w,\infty},\mathbf{F}(\mathbf{1}))=\mathbf{F}.$

Here $\text{{H}}^{0}(I_{w,\infty},\mathbf{F}(\omega))=0$ because $\omega$ is
ramified and $I_{w}/I_{w,\infty}$ is a pro-$p$ group while the trivializing
extension of $\omega$ has order divining $p-1$. Hence
$\text{{H}}^{0}(I_{w},M_{f})[\mathfrak{p}]\subseteq\mathbf{F}$. It follows
that $\text{{H}}^{0}(I_{w},M_{f})\subseteq F/\mathcal{O}$ and
$\text{{H}}^{0}(I_{w},M_{f})/\mathfrak{p}\subseteq\mathbf{F}$.

Now if $\text{{H}}^{0}(I_{w},M_{f})/\mathfrak{p}=0$, from the above diagram we
see that ${\operatorname{coker}}({\operatorname{res}})=0$ and the snake lemma
yields a surjection

$\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})=\mathbf{F}\to\ker({\operatorname{res}}_{M_{f}})[\mathfrak{p}],$

so $\ker({\operatorname{res}}_{M_{f}})\subseteq\mathbf{F}$.

If $\text{{H}}^{0}(I_{w},M_{f})/\mathfrak{p}=\mathbf{F}$, then
$\ker({\operatorname{res}})={\operatorname{coker}}({\operatorname{res}})\subseteq\mathbf{F}$.
If both are $0$,
$\ker({\operatorname{res}}_{M_{f}})[\mathfrak{p}]=\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})=\mathbf{F}$.
If both are $\mathbf{F}$, then the snake lemma yields an injection

$\ker({\operatorname{res}}_{M_{f}})[\mathfrak{p}]\to{\operatorname{coker}}({\operatorname{res}})=\mathbf{F},$

so again
$\ker({\operatorname{res}}_{M_{f}})[\mathfrak{p}]\subseteq\mathbf{F}$. ∎

###  1.4. Selmer groups of $\rho_{f}$

Similarly as for that of $M_{\vartheta}$, we define an _unramified Selmer
group_ for $f$ by

$\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}}(K,M_{f})=\ker\Big{\\{}\text{{H}}^{1}(K^{\Sigma}/K,M_{f})\xrightarrow{{\operatorname{res}}_{{\operatorname{nr}},f}}\prod_{w\in\Sigma,w\nmid
p}\text{{H}}^{1}(I_{w},M_{f})^{G_{w}/I_{w}}\times\text{{H}}^{1}(I_{\overline{v}},M_{f})^{G_{\overline{v}}/I_{\overline{v}}}\Big{\\}},$

and an _$S$ -imprimitive unramified Selmer group_, where
$S=\Sigma-\\{v,\overline{v},\infty\\}$, by

$\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}^{S}}(K,M_{f})=\ker\Big{\\{}\text{{H}}^{1}(K^{\Sigma}/K,M_{f})\xrightarrow{{\operatorname{res}}^{S}_{{\operatorname{nr}},f}}\text{{H}}^{1}(I_{\overline{v}},M_{f})^{G_{\overline{v}}/I_{\overline{v}}}\Big{\\}}.$

The residual Selmer groups
$\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}}(K,M_{f}[\mathfrak{p}])$
and
$\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}^{S}}(K,M_{f}[\mathfrak{p}])$
are defined in the same manner.

One can similarly define the Greenberg Selmer groups for $f$ by imposing
triviality at $\overline{v}$.

By Ribet’s Lemma [Bel09], from the decomposition
$\overline{\rho}_{f}^{\operatorname{ss}}\cong\mathbf{F}(\varphi)\oplus\mathbf{F}(\psi)$
one has an exact sequence

$0\to\mathbf{F}(\varphi)\to\overline{\rho}_{f}\to\mathbf{F}(\psi)\to 0,$

which further induces and exact sequence of
${\operatorname{Gal}}(K^{\Sigma}/K)$-modules

$0\to M_{\varphi}[\mathfrak{p}]\to M_{f}[\mathfrak{p}]\to
M_{\psi}[\mathfrak{p}]\to 0,$

where $\varphi\cdot\psi=\omega$. The cases left out in [CGLS22] are when one
of $\varphi|_{G_{p}},\psi|_{G_{p}}$ is trivial. Hence we assume this is the
case.

Note that by Perrin-Riou’s formula in [KO20, Proposition 2.9], the order of
$\varphi$ and $\psi$ does not matter: Conjecture 1.2 (1) in _op. cit._ is true
(the Selmer group is $\Lambda$-cotorsion), so Conjecture 1.2 (2) (the
anticyclotomic Main Conjecture for our modular forms)111see also Remark 2.2.4.
is independent of the lattice $T_{f}$. Therefore we are free to assume that
$\varphi|_{G_{p}}=\omega$ and $\psi|_{G_{p}}=\mathbf{1}$. We then write the
global character $\varphi$ (resp. $\psi$) as $\tilde{\omega}$ (resp.
$\tilde{\mathbf{1}}$). We remark here that the BSD Conjecture is also
independent of the lattice (see for example [Mil06, I.7]).

From now on, by essentially choosing a lattice $T_{f}$, we fix the following
ordering of the characters appearing in the above sequence

(1.18) $0\to M_{\tilde{\omega}}[\mathfrak{p}]\to M_{f}[\mathfrak{p}]\to
M_{\tilde{\mathbf{1}}}[\mathfrak{p}]\to 0.$

We remark here that the second character $\tilde{\mathbf{1}}$ may be non-
trivial or trivial on $G_{K}$. Therefore,
$\text{{H}}^{0}(K^{\Sigma}/K,M_{\tilde{\mathbf{1}}}[\mathfrak{p}])$ may or may
not be $0$. Since the semisimplification
$\overline{\rho}_{f}^{\operatorname{ss}}$ is independent from the choice of a
lattice $T_{f}$, we must study both cases. However, from [BP19, Theorem 5.2],
we can always choose a non-split extension of
$\overline{\rho}_{f}^{\operatorname{ss}}$ so that $\mathbf{F}(\mathbf{1})$
(the global trivial character) is never a subrepresentation, so we could
always assume that $\text{{H}}^{0}(K^{\Sigma}/K,M_{f})=0$ (equivalently,
$\text{{H}}^{0}(K,\overline{\rho}_{f})=0$), i.e., there is no global torsion
for $f$. We will do so in the rest of this paper. We also include some partial
results without assuming global non-torsionness in the appendix for interested
readers.

Let $\mathfrak{X}_{f}$ and $\mathfrak{X}_{f}^{S}$ denote Pontryagin dual of
the primitive and $S$-imprimitive unramified Selmer groups of $f$. This
section is devoted to proving the following relation between the
$\lambda$-invariants of the imprimitive Selmer groups under the assumption
that there is no global torsion:

######  Theorem 1.4.1.

If $\varphi|_{G_{p}}=\omega$ (so $\psi|_{G_{p}}=\mathbf{1}$) and
$\varphi|_{G_{K}}\neq\omega$ (so $\psi|_{G_{K}}\neq\mathbf{1}$). Then

$\lambda(\mathfrak{X}^{S}_{f})=\lambda(\mathfrak{X}^{S}_{\varphi})+\lambda(\mathfrak{X}^{S}_{\psi}).$

If $\varphi|_{G_{K}}=\omega$ and $\psi|_{G_{K}}=\mathbf{1}$,

$\lambda(\mathfrak{X}^{S}_{f})+1=\lambda(\mathfrak{X}^{S}_{\varphi})+\lambda(\mathfrak{X}^{S}_{\psi}).$

Furthermore, $\mathfrak{X}^{S}_{f}$ and $\mathfrak{X}_{f}$ are both
$\Lambda$-torsion with $\mu$-invariants $0$.

###### Proof of the theorem:.

First notice that the sequence (1.18) yields a commutative diagram

${\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\tilde{\omega}}[\mathfrak{p}])}$${\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{f}[\mathfrak{p}])}$${\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{\tilde{\mathbf{1}}}[\mathfrak{p}])}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{\tilde{\omega}}[\mathfrak{p}])}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{f}[\mathfrak{p}])}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{\tilde{\mathbf{1}}}[\mathfrak{p}])}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\omega}}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{\tilde{\omega}}[\mathfrak{p}]})}}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{f}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})}}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\mathbf{1}}}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{\tilde{\mathbf{1}}}[\mathfrak{p}]})}}$

with exact rows. It then follows from Proposition 1.2.5 that
$\text{{H}}^{1}_{\mathcal{F}_{\operatorname{nr}}^{S}}(K,M_{f}[\mathfrak{p}])$
is finite.

We have the following 2 diagrams for $f$ with exact rows and columns similar
to those of the characters in Section 1.2 and Lemma 1.2.4:

(Gr,$f$)
${0}$${\ker(r_{\overline{v}}^{{\operatorname{Gr}},0})}$${\text{{H}}^{1}_{\mathcal{F}_{\operatorname{Gr}}^{S}}(K,M_{f}[\mathfrak{p}])}$${\text{{H}}^{1}_{\mathcal{F}_{\operatorname{Gr}}^{S}}(K,M_{f})[\mathfrak{p}]}$${0}$${\text{{H}}^{0}(K^{\Sigma}/K,M_{f})/\mathfrak{p}}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{f}[\mathfrak{p}])}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{f})[\mathfrak{p}]}$${0}$${0}$${\text{{H}}^{0}(K_{\overline{v}},M_{f})/\mathfrak{p}}$${\text{{H}}^{1}(K_{\overline{v}},M_{f}[\mathfrak{p}])}$${\text{{H}}^{1}(K_{\overline{v}},M_{f})[\mathfrak{p}]}$${0}$$\scriptstyle{r_{\overline{v}}^{{\operatorname{Gr}},0}}$$\scriptstyle{r_{\overline{v}}^{{\operatorname{Gr}},1}}$

(nr,$f$)
${0}$${\ker(r_{\overline{v}}^{{\operatorname{nr}},0})}$${\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}^{S}}(K,M_{f}[\mathfrak{p}])}$${\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}^{S}}(K,M_{f})[\mathfrak{p}]}$${0}$${\text{{H}}^{0}(K^{\Sigma}/K,M_{f})/\mathfrak{p}}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{f}[\mathfrak{p}])}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{f})[\mathfrak{p}]}$${0}$${0}$${\ker(f)}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{f}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})}}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{f})}{\ker({\operatorname{res}}_{M_{f}})}[\mathfrak{p}]}$$\scriptstyle{r_{\overline{v}}^{{\operatorname{nr}},0}}$$\scriptstyle{r_{\overline{v}}^{{\operatorname{nr}},1}}$$\scriptstyle{f}$

Since $\text{{H}}^{0}(K^{\Sigma}/K,M_{f})=0$ and as we shall see,
$\ker(f)\subseteq\mathbf{F}$,
$\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}^{S}}(K,M_{f})[\mathfrak{p}]$
must also be finite. Then from the structure theorem it follows that
$\mathfrak{X}^{S}_{f}$ (and hence $\mathfrak{X}_{f}$) are $\Lambda$-torsion
with $\mu$-invariants $0$. This proves the last part of the theorem.

######  Remark 1.4.2.

As in Remark 1.2.3, the global-to-local map defining the Selmer groups (both
the Greenberg’s one and the unramified one, both primitive and imprimitive)
for $f$ is also surjective in our case, due to [PW11, Proposition A.2].

(1) is already proved.

(2) is a proved above (both primitive and imprimitive).

To see (3), since $\rho_{f}$ is self-dual, the cohomology group we consider is
identified with $\text{{H}}^{0}(K_{\infty},V_{f}/T_{f})$. By the comments
before [CGLS22, Lemma 3.3.3], we have
$\rho_{f}(G_{K_{\infty}})=\rho_{f}(G_{\mathbf{Q}})$, so we need to show that
$\text{{H}}^{0}(\mathbf{Q},V_{f}/T_{f})$ is finite. Assume this is infinite,
then $V_{f}/T_{f}\cong(F/\mathcal{O})^{2}$ has at least a
$G_{\mathbf{Q}}$-stable copy of $F/\mathcal{O}$. Since we could recover
$T_{f}=\varprojlim V_{f}/T_{f}[p^{n}]$ from $V_{f}/T_{f}$, this implies
$T_{f}$ has a $G_{\mathbf{Q}}$-stable copy of $\mathcal{O}$. However, this
contradicts the irreducibility of $T_{f}$ as a $G_{\mathbf{Q}}$ module, a
result of Ribet in [Rib77].

(4) holds by Lemma 1.3.5 (for Greenberg’s Selmer group) and Lemma 1.3.6 (for
the unramified Selmer group).

To finish the proof of the theorem, it remains to compare the
$\lambda$-invariants. Recall that by Lemma 1.3.6,
$\ker({\operatorname{res}}_{M_{f}})$ is finite. We now have $3$ different
cases to consider:

  1. I

$\text{{H}}^{0}(K_{w},M_{f})$ is non-trivial and
$\ker({\operatorname{res}}_{M_{f}})=0$

  2. II

Both $\text{{H}}^{0}(K_{w},M_{f})$ and $\ker({\operatorname{res}}_{M_{f}})$
are non-trivial

  3. III

$\text{{H}}^{0}(K_{w},M_{f})=0$ (so also
$\ker({\operatorname{res}}_{M_{f}})=\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})=0$)

For simplicity, we first assume $\ker({\operatorname{res}}_{M_{f}})=0$. In
this case, the map $f$ is surjective.

Case I:

Note that $\text{{H}}^{0}(K_{\overline{v}},M_{f})$ is cyclic and non-trivial,
so by Proposition 1.3.4 (iii) one has
$\text{{H}}^{0}(K_{\overline{v}},M_{f})/\mathfrak{p}=\mathbf{F}$. Therefore
$\ker(f)=0$. Also $\text{{H}}^{0}(K^{\Sigma}/K,M_{f})/\mathfrak{p}=0$ by
Proposition 1.3.3.

To apply snake lemma, we add a fourth row to the unramified diagrams and
abbreviate them as follows (here $?=\tilde{\mathbf{1}},f$ or
$\tilde{\omega}$):

(nr,$?$)
${0}$${0}$${{\operatorname{Sel}}(M_{?}[\mathfrak{p}])}$${{\operatorname{Sel}}(M_{?})[\mathfrak{p}]}$${0}$${\text{{H}}^{0}(K^{\Sigma}/K,M_{?})/\mathfrak{p}=0}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{?}[\mathfrak{p}])}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{?})[\mathfrak{p}]}$${0}$${0}$${\ker(f_{?})=0}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{?}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{?}[\mathfrak{p}]})}}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{?})}{\ker({\operatorname{res}}_{M_{?}})}[\mathfrak{p}]}$${0}$${0}$${{\operatorname{coker}}_{?[\mathfrak{p}]}}$${{\operatorname{coker}}_{?}[\mathfrak{p}]}$${0}$$\scriptstyle{r_{?}^{0}}$$\scriptstyle{r_{?[\mathfrak{p}]}}$$\scriptstyle{r_{?}\text{[$\scriptstyle\mathfrak{p}$]}}$$\scriptstyle{f_{?}}$

Here dashed lines are used because in the case
$\ker({\operatorname{res}}_{M_{f}})\neq 0$ they may not be exact. We also
remark that the cokernels appearing in the above diagrams are finite since all
groups have finite $\mathcal{O}$-rank.

Note that by the structure theorem of finitely generated $\Lambda$-modules
(since $\mu({\operatorname{Sel}}(M_{?}))=0$) the $\lambda$-invariants of
${\operatorname{Sel}}(M_{?})$ can be computed as

$\lambda(\mathfrak{X}^{S}_{?})={\operatorname{dim}}_{\mathbf{F}}{\operatorname{Sel}}(M_{?})[\mathfrak{p}]-{\operatorname{dim}}_{\mathbf{F}}{\operatorname{Sel}}(M_{?})/\mathfrak{p},$

whereas
${\operatorname{coker}}_{?}[\mathfrak{p}]={\operatorname{im}}({\operatorname{Sel}}(M_{?})[\mathfrak{p}]\to{\operatorname{Sel}}(M_{?})/\mathfrak{p})$
by Remark 1.2.3 and Remark 1.4.2, so in particular

$\displaystyle{\operatorname{dim}}_{\mathbf{F}}{\operatorname{Sel}}(M_{?})/\mathfrak{p}$
$\displaystyle=$
$\displaystyle{\operatorname{dim}}_{\mathbf{F}}\ker({\operatorname{Sel}}(M_{?})/\mathfrak{p}\to\text{{H}}^{1}(K^{\Sigma}/K,M_{?})/\mathfrak{p})+{\operatorname{dim}}_{\mathbf{F}}{\operatorname{im}}\big{(}{\operatorname{Sel}}(M_{?})/\mathfrak{p}\to\text{{H}}^{1}(K^{\Sigma}/K,M_{?})/\mathfrak{p}\big{)}$
$\displaystyle=$
$\displaystyle{\operatorname{dim}}_{\mathbf{F}}{\operatorname{coker}}_{?}[\mathfrak{p}]+{\operatorname{dim}}_{\mathbf{F}}\ker\Big{(}\text{{H}}^{1}(K^{\Sigma}/K,M_{?})/\mathfrak{p}\to\frac{\text{{H}}^{1}(K_{\overline{v}},M_{?})}{\ker({\operatorname{res}}_{M_{?}})}/\mathfrak{p}\Big{)}.$

Denoting the last kernel by $K_{?}$, we get

$\lambda(\mathfrak{X}^{S}_{?})={\operatorname{dim}}_{\mathbf{F}}{\operatorname{Sel}}(M_{?})[\mathfrak{p}]-{\operatorname{dim}}_{\mathbf{F}}{\operatorname{coker}}_{?}[\mathfrak{p}]-{\operatorname{dim}}_{\mathbf{F}}K_{?}.$

Now by Corollary 1.2.6, the last two terms are $0$ when
$?=\tilde{\mathbf{1}}$. Hence, if we use $d(\cdot)$ to represent the
$\mathbf{F}$-dimension of an $\mathbf{F}$-vector space, we get the following

######  Lemma 1.4.3.

We have

  1. (i)

$d({\operatorname{Sel}}(M_{\tilde{\mathbf{1}}}[\mathfrak{p}]))=\lambda(\mathfrak{X}^{S}_{\tilde{\mathbf{1}}}),$

  2. (ii)

$d({\operatorname{Sel}}(M_{f}[\mathfrak{p}]))=\lambda(\mathfrak{X}^{S}_{f})+d({\operatorname{coker}}_{f[\mathfrak{p}]})+d(K_{f}),$

  3. (iii)

$d({\operatorname{Sel}}(M_{\tilde{\omega}}[\mathfrak{p}]))=\lambda(\mathfrak{X}^{S}_{\tilde{\omega}})+d({\operatorname{coker}}_{\tilde{\omega}[\mathfrak{p}]})+d(K_{\tilde{\omega}}).$

The next step is to relate the left hand sides of the above equations. Unlike
in [CGLS22], they do not simply satisfy an additive relation. Instead, we have

######  Lemma 1.4.4.

If $\tilde{\mathbf{1}}\neq\mathbf{1}$ (i.e., the local trivial character is
not globally trivial), we have

$d({\operatorname{Sel}}(M_{f}[\mathfrak{p}]))=d({\operatorname{Sel}}(M_{\tilde{\omega}}[\mathfrak{p}]))+d(K)+d({\operatorname{coker}}_{f[\mathfrak{p}]})-d({\operatorname{coker}}_{\tilde{\omega}[\mathfrak{p}]})-d(C)+1,$

where $K$ (resp. $C$) is the kernel (resp. cokernel) of the map

${\operatorname{coker}}(\text{{H}}^{1}(K^{\Sigma}/K,M_{\tilde{\omega}}[\mathfrak{p}])\to\text{{H}}^{1}(K^{\Sigma}/K,M_{f}[\mathfrak{p}]))\to{\operatorname{coker}}\Big{(}\frac{\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\omega}}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{\tilde{\omega}}[\mathfrak{p}]})}\to\frac{\text{{H}}^{1}(K_{\overline{v}},M_{f}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})}\Big{)}.$

If $\tilde{\mathbf{1}}=\mathbf{1}$, we have

$d({\operatorname{Sel}}(M_{f}[\mathfrak{p}]))=d({\operatorname{Sel}}(M_{\tilde{\omega}}[\mathfrak{p}]))+d(K)+d({\operatorname{coker}}_{f[\mathfrak{p}]})-d({\operatorname{coker}}_{\tilde{\omega}[\mathfrak{p}]})-d(C).$

###### Proof.

This is just generalized snake lemma:

${0(}$${\mathbf{F})}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{\tilde{\omega}}[\mathfrak{p}])}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{f}[\mathfrak{p}])}$${{\operatorname{coker}}(g)}$${0}$${0}$${\mathbf{F}}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\omega}}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{\tilde{\omega}}[\mathfrak{p}]})}}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{f}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})}}$${{\operatorname{coker}}(g_{\overline{v}})}$${0}$${{\operatorname{coker}}_{\tilde{\omega}[\mathfrak{p}]}}$${{\operatorname{coker}}_{f[\mathfrak{p}]}}$${C}$${0}$$\scriptstyle{g}$$\scriptstyle{g_{\overline{v}}}$

Here the map $g$ is injective if $\tilde{\mathbf{1}}\neq\mathbf{1}$ and it has
kernel $\mathbf{F}$ if $\tilde{\mathbf{1}}=\mathbf{1}$.222This is the only
place where global trivialness of the character makes a difference on the
algebraic side. Similar difference also appears if we choose the other
ordering of the characters, since it determines whether $f$ has global
torsion, as the (local) trivial character is a subrepresentation in that case.
This difference corresponds to a different Iwasawa Main Conjecture for the
(global) trivial character. See Theorem 2.2.3.

To see why the top two rows are exact on the left, first notice that
$\text{{H}}^{0}(K^{\Sigma}/K,M_{?}[\mathfrak{p}])=0$ for all
$?=\tilde{\omega},f$ and $\tilde{\mathbf{1}}$ if $\tilde{\mathbf{1}}$ is
globally non-trivial (it’s $\mathbf{F}$ otherwise). This gives exactness of
the first row. Now
$\text{{H}}^{0}(K_{\overline{v}},M_{\tilde{\mathbf{1}}}[\mathfrak{p}])=\mathbf{F}$
and
$\text{{H}}^{0}(K_{\overline{v}},M_{f}[\mathfrak{p}])=\text{{H}}^{0}(K_{\overline{v}},M_{f})[\mathfrak{p}]=\mathbf{F}$
because we are in Case I.

Thus the map

$\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\omega}}[\mathfrak{p}])\to\text{{H}}^{1}(K_{\overline{v}},M_{f}[\mathfrak{p}])$

is injective. Adding the local conditions
$\ker({\operatorname{res}}_{M_{\tilde{\omega}}[\mathfrak{p}]})=0,\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})=\mathbf{F}$,
we get an exact sequence

$0\to\mathbf{F}\to\frac{\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\omega}}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{\tilde{\omega}}[\mathfrak{p}]})}\xrightarrow{g_{\overline{v}}}\frac{\text{{H}}^{1}(K_{\overline{v}},M_{f}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})}.\qed$

For later use, we remark that we also get from the above argument that

${\operatorname{coker}}(g_{\overline{v}})\cong{\operatorname{coker}}(\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\omega}}[\mathfrak{p}])\to\text{{H}}^{1}(K_{\overline{v}},M_{f}[\mathfrak{p}])).$

To finish the proof of Theorem 1.4.1, we now relate $K_{f}$ to
$K_{\tilde{\omega}}$. The argument in Corollary 1.2.6 (or more precisely,
[Gre89, Proposition 3, Proposition 4] together with Proposition 1.1.3 and
Lemma 1.3.5) implies the vanishing of $\text{{H}}^{2}(K^{\Sigma}/K,M_{?})$. We
can then identify $K_{f}$ (resp. $K_{\tilde{\omega}}$) with $K^{(2)}_{f}$
(resp. $K^{(2)}_{\tilde{\omega}}$) in the following two diagrams (noting that
$\ker({\operatorname{res}}_{M_{f}})=\ker({\operatorname{res}}_{M_{\tilde{\omega}}})=0$)

${0}$${K_{f}}$${K^{(2)}_{f}}$${0}$${0}$${0}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{f})/\mathfrak{p}}$${\text{{H}}^{2}(K^{\Sigma}/K,M_{f}[\mathfrak{p}])}$${\text{{H}}^{2}(K^{\Sigma}/K,M_{f})[\mathfrak{p}]=0}$${0}$${0}$${\text{{H}}^{1}(K_{\overline{v}},M_{f})/\mathfrak{p}}$${\text{{H}}^{2}(K_{\overline{v}},M_{f}[\mathfrak{p}])}$${\text{{H}}^{2}(K_{\overline{v}},M_{f})[\mathfrak{p}]}$${0}$$\scriptstyle{\cong}$

${0}$${K_{\tilde{\omega}}}$${K^{(2)}_{\tilde{\omega}}}$${0}$${0}$${0}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{\tilde{\omega}})/\mathfrak{p}}$${\text{{H}}^{2}(K^{\Sigma}/K,M_{\tilde{\omega}}[\mathfrak{p}])}$${\text{{H}}^{2}(K^{\Sigma}/K,M_{\tilde{\omega}})[\mathfrak{p}]=0}$${0}$${0}$${\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\omega}})/\mathfrak{p}}$${\text{{H}}^{2}(K_{\overline{v}},M_{\tilde{\omega}}[\mathfrak{p}])}$${\text{{H}}^{2}(K_{\overline{v}},M_{\tilde{\omega}})[\mathfrak{p}]}$${0}$$\scriptstyle{\cong}$

It therefore suffices to relate $K^{(2)}_{f}$ to $K^{(2)}_{\tilde{\omega}}$.

Corollary 1.2.6 also gives the vanishing of
$\text{{H}}^{2}(K^{\Sigma}/K,M_{\tilde{\mathbf{1}}}[\mathfrak{p}])$. By local
duality and Shapiro’s lemma, we get
$\text{{H}}^{2}(K_{\overline{v}},M_{\tilde{\mathbf{1}}}[\mathfrak{p}])=\text{{H}}^{0}(K_{\overline{v},\infty},\mathbf{F}(\omega))=0$
as well. We then obtain the following diagram

($K^{(2)}_{f},K^{(2)}_{\tilde{\omega}}$) ${0}$${K^{(2)}_{\tilde{\omega}\to
f}}$${K^{(2)}_{\tilde{\omega}}}$${K^{(2)}_{f}}$${0}$${\ker(h)}$${\text{{H}}^{2}(K^{\Sigma}/K,M_{\tilde{\omega}}[\mathfrak{p}])}$${\text{{H}}^{2}(K^{\Sigma}/K,M_{f}[\mathfrak{p}])}$${\text{{H}}^{2}(K^{\Sigma}/K,M_{\tilde{\mathbf{1}}}[\mathfrak{p}])=0}$${0}$${\ker(h_{\overline{v}})}$${\text{{H}}^{2}(K_{\overline{v}},M_{\tilde{\omega}}[\mathfrak{p}])}$${\text{{H}}^{2}(K_{\overline{v}},M_{f}[\mathfrak{p}])}$${\text{{H}}^{2}(K_{\overline{v}},M_{\tilde{\mathbf{1}}}[\mathfrak{p}])=0}$${0}$$\scriptstyle{h}$$\scriptstyle{h_{\overline{v}}}$

from which we get

(1.19)
$d(K^{(2)}_{\tilde{\omega}})=d(K^{(2)}_{f})+d(K^{(2)}_{\tilde{\omega}\to f}).$

The middle left vertical map is surjective because its domain and codomain are
quotients of those of
$\text{{H}}^{1}(K^{\Sigma}/K,M_{\tilde{\mathbf{1}}}[\mathfrak{p}])\to\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\mathbf{1}}}[\mathfrak{p}])$
which is surjective.

Next, consider the diagram

(Sel,$\tilde{\mathbf{1}}$)
${0}$${K}$${{\operatorname{Sel}}(M_{\tilde{\mathbf{1}}}[\mathfrak{p}])}$${\ker(j)}$${0}$${{\operatorname{coker}}(g)}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{\tilde{\mathbf{1}}}[\mathfrak{p}])}$${\ker(h)}$${0}$${0}$${{\operatorname{coker}}(g_{\overline{v}})}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\mathbf{1}}}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{\tilde{\mathbf{1}}}[\mathfrak{p}]})}}$${{\operatorname{coker}}(i)}$${0}$${C}$${0}$${0}$${0}$$\scriptstyle{j}$$\scriptstyle{i}$

where $g,g_{\overline{v}}$ are the maps in Lemma 1.4.4. Note that by Corollary
1.2.6, the middle vertical map is surjective, so is the right vertical map. We
thus conclude that

(1.20)
$d({\operatorname{Sel}}(M_{\tilde{\mathbf{1}}[\mathfrak{p}]}))=d(K)+d(\ker(j))-d(C).$

The above diagram together with the diagram

${0}$${0}$${\ker({\operatorname{res}}_{M_{\tilde{\mathbf{1}}}})=\mathbf{F}}$${\ker(k)}$${0}$${{\operatorname{coker}}(\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\omega}}[\mathfrak{p}])\to\text{{H}}^{1}(K_{\overline{v}},M_{f}[\mathfrak{p}]))}$${\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\mathbf{1}}}[\mathfrak{p}])}$${\ker(h_{\overline{v}})}$${0}$${0}$${{\operatorname{coker}}(g_{\overline{v}})}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\mathbf{1}}}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{\tilde{\mathbf{1}}}[\mathfrak{p}]})}}$${{\operatorname{coker}}(i)}$${0}$${0}$${0}$${0}$${0}$$\scriptstyle{k}$$\scriptstyle{i}$

then yields the following diagram

${0}$${K^{(2)}_{\tilde{\omega}\to f}}$${K^{(2)}_{\tilde{\omega}\to
f}}$${0}$${0}$${\ker(j)}$${\ker(h)}$${{\operatorname{coker}}(i)}$${0}$${0}$${\ker(k)=\mathbf{F}}$${\ker(h_{\overline{v}})}$${{\operatorname{coker}}(i)}$${0}$${0}$${0}$${0}$${0}$$\scriptstyle{=}$$\scriptstyle{=}$

from which we see

(1.21) $d(\ker(j))=d(K^{(2)}_{\tilde{\omega}\to f})+1.$

Combining equations (1.19), (1.20), (1.21) with Lemma 1.4.3 and Lemma 1.4.4,
we get

$\lambda(\mathfrak{X}^{S}_{f})=\lambda(\mathfrak{X}^{S}_{\varphi})+\lambda(\mathfrak{X}^{S}_{\psi})$

if $\varphi=\tilde{\omega}\neq\omega$ (equivalently, if
$\psi=\tilde{\mathbf{1}}\neq\mathbf{1}$); and

$\lambda(\mathfrak{X}^{S}_{f})+1=\lambda(\mathfrak{X}^{S}_{\varphi})+\lambda(\mathfrak{X}^{S}_{\psi})$

if $\varphi=\tilde{\omega}=\omega$ (equivalently, if
$\psi=\tilde{\mathbf{1}}=\mathbf{1}).$

Case II: $\ker({\operatorname{res}}_{M_{f}})=\mathcal{O}/\pi^{n}$ for some
$n\geq 1$

In this case, it is necessary that
$\ker({\operatorname{res}}_{M_{f}})[\mathfrak{p}]=\mathbf{F}=\ker({\operatorname{res}}_{M_{f}}[\mathfrak{p}])$.
Since $\ker({\operatorname{res}}_{M_{f}})$ is no longer divisible, the natural
injective map

$\frac{\text{{H}}^{1}(K_{\overline{v}},M_{f})[\mathfrak{p}]}{\ker({\operatorname{res}}_{M_{f}})[\mathfrak{p}]}\to\frac{\text{{H}}^{1}(K_{\overline{v}},M_{f})}{\ker({\operatorname{res}}_{M_{f}})}[\mathfrak{p}]$

has cokernel
$\ker(\ker({\operatorname{res}}_{M_{f}})/\mathfrak{p}\to\text{{H}}^{1}(K_{\overline{v}},M_{f})/\mathfrak{p})$
which might be $0$ or $\mathbf{F}$. Denote this cokernel by $C_{f}$.

The map

$\frac{\text{{H}}^{1}(K_{\overline{v}},M_{f}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})}\to\frac{\text{{H}}^{1}(K_{\overline{v}},M_{f})}{\ker({\operatorname{res}}_{M_{f}})}[\mathfrak{p}]$

factors through the above map and has kernel $\mathbf{F}$. The diagram
(nr,$?$) now becomes

(nr,$?=f$)
${0}$${0}$${{\operatorname{Sel}}(M_{f}[\mathfrak{p}])}$${{\operatorname{Sel}}(M_{f})[\mathfrak{p}]}$${0}$${0}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{f}[\mathfrak{p}])}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{f})[\mathfrak{p}]}$${0}$${0}$${\ker(f_{f})=\mathbf{F}}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{f}[\mathfrak{p}])}{\ker({\operatorname{res}}_{M_{f}[\mathfrak{p}]})}}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{f})}{\ker({\operatorname{res}}_{M_{f}})}[\mathfrak{p}]}$${C_{f}}$${0}$${\mathbf{F}}$${{\operatorname{coker}}_{f[\mathfrak{p}]}}$${{\operatorname{coker}}_{f}[\mathfrak{p}]}$${C_{f}}$${0}$$\scriptstyle{r_{f}^{0}}$$\scriptstyle{r_{f[\mathfrak{p}]}}$$\scriptstyle{r_{f}\text{[$\scriptstyle\mathfrak{p}$]}}$$\scriptstyle{f_{f}}$

Therefore as in the Case I we still get

$\lambda(\mathfrak{X}^{S}_{f})={\operatorname{dim}}_{\mathbf{F}}{\operatorname{Sel}}(M_{f})[\mathfrak{p}]-{\operatorname{dim}}_{\mathbf{F}}{\operatorname{coker}}_{f}[\mathfrak{p}]-{\operatorname{dim}}_{\mathbf{F}}K_{f},$

and hence

(1.22)
$d({\operatorname{Sel}}(M_{f}[\mathfrak{p}]))=\lambda(\mathfrak{X}^{S}_{f})+d(C_{f})+d({\operatorname{coker}}_{f[\mathfrak{p}]})+d(K_{f})-1$

by the generalized snake lemma.

Finally, we compare $K_{f}$ to $K_{\tilde{\omega}}$ in this case. Recall that
in Case I we showed that

$d(K_{\tilde{\omega}})=d(\ker(\text{{H}}^{1}(K^{\Sigma}/K,M_{f})/\mathfrak{p}\to\text{{H}}^{1}(K_{\overline{v}},M_{f})/\mathfrak{p}))+d(K^{(2)}_{\tilde{\omega}\to
f})$

(here the first kernel coincides with $K_{f}$ in that case), it suffices to
compare
$K^{I}_{f}\colonequals\ker(\text{{H}}^{1}(K,M_{f})/\mathfrak{p}\to\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{f}})/\mathfrak{p})$
to $K_{f}$. This is done by considering the following diagram

${0}$${0}$${K^{I}_{f}}$${K_{f}}$${0}$${0}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{f})/\mathfrak{p}}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{f})\mathfrak{p}}$${0}$${0}$${C_{f}}$${\ker({\operatorname{res}}_{M_{f}})/\mathfrak{p}=\mathbf{F}}$${\text{{H}}^{1}(K_{\overline{v}},M_{f})/\mathfrak{p}}$${\frac{\text{{H}}^{1}(K_{\overline{v}},M_{f})}{\ker({\operatorname{res}}_{M_{f}})}/\mathfrak{p}}$${0}$${0}$${C_{f}}$${\mathbf{F}}$${0}$${0}$${0}$$\scriptstyle{=}$

If $C_{f}=\mathbf{F}$, $K_{f}=K^{I}_{f}$ and (1.22) becomes exactly as in Case
I, so the result follows by the same computation. If $C_{f}=0$,
$K_{f}=K^{I}_{f}+\mathbf{F}$ and (1.22) becomes

$d({\operatorname{Sel}}(M_{f}[\mathfrak{p}]))=\lambda(\mathfrak{X}^{S}_{f})+d({\operatorname{coker}}_{f[\mathfrak{p}]})+d(K_{f})-1,$

and $d(K_{\tilde{\omega}})=d(K_{f})+d(K^{(2)}_{\tilde{\omega}\to f})-1$. The
same result follows from this observation.

Case III: $\text{{H}}^{0}(K_{w},M_{f})=0$

In this case, it is necessary that
$\ker({\operatorname{res}}_{M_{f}})[\mathfrak{p}]=\mathbf{F}=\ker({\operatorname{res}}_{M_{f}}[\mathfrak{p}])$
by Lemma 1.3.6. We only study what can be different from Case I.

The diagram (nr,$?=f$) now becomes

${0}$${0}$${{\operatorname{Sel}}(M_{f}[\mathfrak{p}])}$${{\operatorname{Sel}}(M_{f})[\mathfrak{p}]}$${0}$${\text{{H}}^{0}(K^{\Sigma}/K,M_{f})/\mathfrak{p}=0}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{f}[\mathfrak{p}])}$${\text{{H}}^{1}(K^{\Sigma}/K,M_{f})[\mathfrak{p}]}$${0}$${0}$${\ker(f_{f})=\text{{H}}^{0}(K_{\overline{v}},M_{f})/\mathfrak{p}=0}$${\text{{H}}^{1}(K_{\overline{v}},M_{f}[\mathfrak{p}])}$${\text{{H}}^{1}(K_{\overline{v}},M_{f})[\mathfrak{p}]}$${0}$${0}$${{\operatorname{coker}}_{f[\mathfrak{p}]}}$${{\operatorname{coker}}_{f}[\mathfrak{p}]}$${0}$$\scriptstyle{r_{f}^{0}}$$\scriptstyle{r_{f[\mathfrak{p}]}}$$\scriptstyle{r_{f}\text{[$\scriptstyle\mathfrak{p}$]}}$$\scriptstyle{f_{f}}$

which agrees with Case I.

$\text{{H}}^{0}(K_{\overline{v}},M_{f})=0$ also implies
$\text{{H}}^{0}(K_{\overline{v}},M_{f}[\mathfrak{p}])=0$, so we get an exact
sequence

$0\to\text{{H}}^{0}(K_{\overline{v}},M_{\tilde{\mathbf{1}}}[\mathfrak{p}])=\mathbf{F}\to\text{{H}}^{1}(K_{\overline{v}},M_{\tilde{\omega}}[\mathfrak{p}])\to\text{{H}}^{1}(K_{\overline{v}},M_{f}[\mathfrak{p}]).$

This is exactly the final ingredient in the proof of Lemma 1.4.4. Now exactly
the same arguments as in Case I give the desired relation. ∎

###  1.5. Comparison I: Algebraic Iwasawa invariants

We’re now ready to state the main result of the algebraic side.

Let

$\mathcal{P}_{w}(f)\colonequals P_{w}(\ell^{-1}\gamma_{w})\in\Lambda$

with $V_{f}=T_{f}\otimes\mathbf{Q}_{\ell}$ and
$P_{w}=\det(1-{\operatorname{Frob}}_{w}X\mid V_{f,I_{w}})$.

Recall that we have fixed the ordering of the characters appearing in the
following decomposition:

$0\to\mathbf{F}(\varphi(=\tilde{\omega}))\to\overline{\rho}_{f}\to\mathbf{F}(\psi(=\tilde{\mathbf{1}}))\to
0.$

######  Theorem 1.5.1.

Assume that $\varphi|_{G_{p}}=\omega$ and $\varphi|_{G_{K}}\neq\omega$.

Then the module $\mathfrak{X}_{f}$ is $\Lambda$-torsion with
$\mu(\mathfrak{X}_{f})=0$ and

$\lambda(\mathfrak{X}_{f})=\lambda(\mathfrak{X}_{\varphi})+\lambda(\mathfrak{X}_{\psi})+\sum_{w\in
S}\big{\\{}\lambda(\mathcal{P}_{w}(\varphi))+\lambda(\mathcal{P}_{w}(\psi))-\lambda(\mathcal{P}_{w}(f))\big{\\}}.$

In the case where $\varphi|_{G_{K}}=\omega$, the same results hold except that
the relation between $\Lambda$-invariants now becomes

$\lambda(\mathfrak{X}_{f})+1=\lambda(\mathfrak{X}_{\varphi})+\lambda(\mathfrak{X}_{\psi})+\sum_{w\in
S}\big{\\{}\lambda(\mathcal{P}_{w}(\varphi))+\lambda(\mathcal{P}_{w}(\psi))-\lambda(\mathcal{P}_{w}(f))\big{\\}}.$

###### Proof.

With our Theorem 1.4.1 and Remark 1.4.2, the proof is the same as that of
[CGLS22, Theorem 1.5.1]. ∎

##  2\. Analytic side

Let $f\in S_{k}(\Gamma_{0}(N))$ with $k=2r$ be a newform, $p\nmid 2N$ and $K$
an imaginary quadratic field satisfying the Heegner hypothesis. As in the
algebraic side, we assume $p=v\overline{v}$ is split in $K$. We further assume
that the discriminant $D_{K}$ of $K$ is odd and $D_{K}\neq-3$. Recall that
$\Lambda=\mathcal{O}\llbracket\Gamma\rrbracket$ denotes the anticyclotomic
Iwasawa algebra, and set
$\Lambda^{\operatorname{nr}}=\Lambda\hat{\otimes}_{\mathbf{Z}_{p}}\mathbf{Z}_{p}^{{\operatorname{nr}}}$,
for $\mathbf{Z}_{p}^{{\operatorname{nr}}}$ the completion of the ring of
integers of the maximal unramified extension of $\mathbf{Q}_{p}$. This is
where the BDP $p$-adic $L$-function lives.

In this section, we continue to study the case
$\overline{\rho}_{f}^{\mathrm{ss}}=\mathbf{F}(\varphi)\oplus\mathbf{F}(\psi)$.
Note that the cases when $\varphi|_{G_{p}}=\mathbf{1}$ (resp. $\omega$) but
$\varphi|_{G_{K}}\neq\mathbf{1}$ (resp. $\omega$) are already studied in
[CGLS22]. We therefore only state the theorems we need and explain how to
extend the results to the case where $\varphi|_{G_{K}}=\mathbf{1}$.

###  2.1. $p$-adic $L$-functions

####  2.1.1. The Bertolini–Darmon–Prasanna $p$-adic $L$-functions.

Thanks to the Heegner hypothesis, we can fix an integral ideal
$\mathfrak{N}\subset\mathcal{O}_{K}$ with

$\mathcal{O}_{K}/\mathfrak{N}\simeq\mathbf{Z}/N.$

######  Proposition 2.1.1 ($p$-adic interpolation property).

There exists an element $\mathcal{L}_{f}\in\Lambda^{{\operatorname{nr}}}$
characterized by the following interpolation property: If
$\hat{\xi}\in\mathfrak{X}_{p^{\infty}}$ is the $p$-adic avatar of a Hecke
character $\xi$ of infinity type $(n,-n)$ with $n\geq 0$ and $p$-power
conductor, then

$\mathscr{L}_{f}(\hat{\xi})=\frac{\Omega_{p}^{4n}}{\Omega_{K}^{4n}}\cdot\frac{4\Gamma(n+\frac{k}{2})\Gamma(n-\frac{k}{2}+1)\xi^{-1}(\mathfrak{N}^{-1})}{(2\pi)^{2n+1}(\sqrt{D_{K}})^{2n-1}}\cdot(1-a_{p}(f)p^{-r}\xi_{\overline{\mathfrak{p}}}(p)+\xi_{\overline{\mathfrak{p}}}(p^{2})p^{-1})^{2}\cdot
L\big{(}f/K,\xi,\tfrac{k}{2}\big{)}.$

Here and in the following $p$-adic units may be ignored in the expressions.

###### Proof.

This is proved in [CGLS22, Theorem 2.1.1], using results from [CH18]. We
remark here that the assumption $p\nmid(2k-1)!$ can be directly removed from
op. cit. so for arbitrary weights we would be able to study all odd primes.

More precisely, [CH18, Prop. 3.8] gives

$\mathscr{L}_{f}(\hat{\xi})=\Omega_{p}^{2k+2n}\cdot
L^{\operatorname{alg}}(\tfrac{1}{2},\pi_{K}\otimes\psi\xi)\cdot
e_{\mathfrak{p}}(f,\psi\xi)\cdot\xi(\mathfrak{N}^{-1})\cdot
2^{\\#A(\psi)+3}c_{0}\varepsilon(f)\cdot u_{K}^{2}\sqrt{D_{K}},$

where

$L^{\operatorname{alg}}(\tfrac{1}{2},\pi_{K}\otimes\xi)=\frac{\Gamma(n+r)\Gamma(n-r+1)}{2(2\pi)^{2n+1}(\sqrt{D_{K}})^{2n}}\frac{L(\frac{1}{2},\pi_{K}\otimes\xi)}{\Omega_{K}^{4n}},$

since $\operatorname{Im}(\vartheta)=\frac{\sqrt{D_{K}}}{2}$. Note that we can
take $c=c_{0}=1$ for our application to the weight $2$ BSD formula (see for
example [CGLS22, Theorem 5.1.3] where the summation is over
$\sigma\in{\operatorname{Gal}}(H/K)$), so in particular $p\nmid c_{0}$ and

$e_{\mathfrak{p}}(f,\xi)=(1-a_{p}(f)p^{-r}\xi_{\overline{\mathfrak{p}}}(p)+\xi_{\overline{\mathfrak{p}}}(p^{2})p^{-1})^{2}.$

Also notice that $u_{K}=1$ if $D_{K}<-4$, $A(\psi)=1$ since $(D_{K},N)=1$. The
global root number $\varepsilon(f)=\pm 1$ is ignored since it’s a unit. Now as
in [CGLS22, Theorem 2.1.1], put $\eta^{-1}\xi$ in place of $\xi$ for a Hecke
character $\eta$ of infinity type $(\frac{k}{2},-\frac{k}{2})$, and notice
that $\eta(\mathfrak{N})$ is also a unit since $p\nmid N$, we get the desired
interpolation property. ∎

######  Remark 2.1.2.

We mention that the period $\Omega_{K}$ is denoted by $\Omega_{\infty}$ in
[CGLS22]. But as is pointed out in [CGS23, Remark 2.3.2], the
$\Omega_{\infty}$ defined in the next theorem (coming from Katz, de Shalit and
Hida–Tilouine) is different from the $\Omega_{K}$ in this proposition (coming
from [BDP13]) and they satisfy the relation $\Omega_{\infty}=2\pi
i\cdot\Omega_{K}$.

####  2.1.2. Katz $p$-adic $L$-functions.

######  Theorem 2.1.3.

There exists an element $\mathcal{L}_{f}\in\Lambda^{\operatorname{nr}}$
characterized by the following interpolation property: For every character
$\xi$ of $\Gamma$ crystalline at both $v$ and $\bar{v}$ and corresponding to a
Hecke character of $K$ of infinity type $(n,-n)$ with $n\in\mathbf{Z}_{>0}$
and $n\equiv 0$ (mod $p-1$), we have

$\displaystyle\mathcal{L}_{\vartheta}(\xi)=\frac{\Omega_{p}^{2n}}{\Omega_{\infty}^{2n}}\cdot
4\Gamma(n+\tfrac{k}{2})\cdot\frac{(2\pi
i)^{n-\frac{k}{2}}}{\sqrt{D_{K}}^{n-\frac{k}{2}}}\cdot(1-\vartheta^{-1}(p)\xi^{-1}(v))\cdot(1-\vartheta(p)\xi(\bar{v})p^{-1})$
$\displaystyle\times\prod_{\ell\mid C}(1-\vartheta(\ell)\xi(w)\ell^{-1})\cdot
L(\vartheta_{K}\xi\mathbf{N}_{K}^{\frac{k}{2}},0).$

###### Proof.

See [Kri16, Theorem 27]. Similarly as in [CGLS22, Theorem 2.1.2], we take
$\chi=\vartheta_{K}\xi\mathbf{N}_{K}^{\frac{k}{2}},k_{1}=n+\frac{k}{2},k_{2}=-n+\frac{k}{2}$
(note that the infinity type of the Hecke characters in these $2$ papers
differ by a sign. Also the weight in [Kri16] is $k$ while the weight in
[CGLS22] is $2k$). ∎

###  2.2. Comparison II: Analytic Iwasawa invariants

######  Theorem 2.2.1.

$\mathcal{L}_{f}\equiv(\mathscr{E}^{\iota}_{\varphi,\psi})^{2}\cdot(\mathcal{L}_{\varphi})^{2}\pmod{p\Lambda^{{\operatorname{nr}}}}.$

###### Proof.

See [CGLS22, Theorem 2.2.1]. Note that when $\varphi|_{G_{K}}\neq\mathbf{1}$,
we always have full Eisenstein descent by [Kri16]. If
$\varphi|_{G_{K}}=\mathbf{1}$, we would only get partial Eisenstein descent in
some cases. However, the proof in [CGLS22] essentially only requires partial
Eisenstein descent because only the constant term of $G$ (a certain Eisenstein
series in loc. cit.) can be different and the $p$-depletion of $G$ does not
see the constant term.

We remark here that the formula in [Kri16, Theorem 37] now becomes

$\displaystyle\mathcal{L}_{G}(\xi)=\frac{\Omega_{p}^{2n}}{\Omega_{K}^{2n}}\cdot\frac{\Gamma(n+\frac{k}{2})\varphi^{-1}(\sqrt{-D_{K}})\xi(\bar{\mathfrak{t}})t}{\mathfrak{g}(\varphi)}\cdot\frac{1}{(2\pi
i)^{n+\frac{k}{2}}\sqrt{D_{K}}^{n-\frac{k}{2}}}$
$\displaystyle\times\Xi_{\xi^{-1}\mathbf{N}_{K}^{-\frac{k}{2}}}(\varphi,\psi\omega^{-1},N_{+},N_{-},N_{0})\cdot
L(\varphi_{K}\xi\mathbf{N}_{K}^{\frac{k}{2}},0),$

where we take $\chi^{-1}=\xi\mathbf{N}_{K}^{\frac{k}{2}},\ \psi_{1}=\varphi,\
\psi_{2}=\varphi^{-1}=\psi\omega^{-1}$, and $j=n-\frac{k}{2}$. Here we’re
using $\mathbf{N}_{K}(\bar{\mathbf{t}})=1$ is a unit since $p\nmid c$ because
$t\mid N$.

From Remark 2.1.2, the above formula becomes

$\displaystyle\mathcal{L}_{G}(\xi)=\frac{\Omega_{p}^{2n}}{\Omega_{\infty}^{2n}}\cdot\frac{\Gamma(n+\frac{k}{2})\varphi^{-1}(\sqrt{-D_{K}})\xi(\bar{\mathfrak{t}})t}{\mathfrak{g}(\varphi)}\cdot\frac{(2\pi
i)^{n-\frac{k}{2}}}{\sqrt{D_{K}}^{n-\frac{k}{2}}}$
$\displaystyle\times\Xi_{\xi^{-1}\mathbf{N}_{K}^{-\frac{k}{2}}}(\varphi,\psi\omega^{-1},N_{+},N_{-},N_{0})\cdot
L(\varphi_{K}\xi\mathbf{N}_{K}^{\frac{k}{2}},0),$

which agrees with the formula in [CGLS22, eq. (2.7)]. ∎

Combining the above inputs, as in [CGLS22], we obtain the following:

######  Theorem 2.2.2.

Assume that
$\overline{\rho}_{f}^{\mathrm{ss}}=\mathbf{F}(\varphi)\oplus\mathbf{F}(\psi)$
as $G_{\mathbf{Q}}$-modules, with the characters $\varphi$, $\psi$ labeled so
that $p\nmid cond(\varphi)$. Then $\mu(\mathcal{L}_{f})=0$ and

$\lambda(\mathcal{L}_{f})=\lambda(\mathcal{L}_{\varphi})+\lambda(\mathcal{L}_{\psi})+\sum_{w\in
S}\\{\lambda(\mathcal{P}_{w}(\varphi))+\lambda(\mathcal{P}_{w}(\psi))-\lambda(\mathcal{P}_{w}(f))\\}.$

######  Theorem 2.2.3.

Assume that
$\overline{\rho}_{f}^{ss}=\mathbf{F}(\psi)\oplus\mathbf{F}(\varphi)$. Then
$\mu(\mathcal{L}_{f})=\mu(\mathfrak{X}_{f})=0$ and

$\lambda(\mathcal{L}_{f})=\lambda(\mathfrak{X}_{f}).$

###### Proof.

The proof relies on the one-variable Iwasawa Main Conjectures for the
characters which is already explained in Theorem 1.2.2. We will add a little
more detail. When $\varphi|_{G_{K}}\notin\\{\mathbf{1},\omega\\}$, the proofs
are identical to those in [CGLS22]. Note also that in those cases the results
on the algebraic side (see Theorem 1.5.1) are also compatible. We explain how
to modify the arguments to the case where $\psi|_{G_{K}}=\mathbf{1}$.

It is essentially proved in [Rub91] that for any character $\vartheta$ of
$G_{K}$, we have

${\operatorname{Char}}(\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}}(K,M_{\vartheta}))={\operatorname{Char}}(U_{\infty}/\bar{\mathscr{C}_{\infty}})^{\vartheta},$

where $U_{\infty}$ and $\mathscr{C}_{\infty}$ are the local units and elliptic
units defined in op. cit. As is mentioned in the remarks after Theorem 4.1
there, the further identification

${\operatorname{Char}}(U_{\infty}/\bar{\mathscr{C}_{\infty}})^{\vartheta}={\operatorname{Char}}(\mathcal{L}_{\vartheta})$

is done in [CW78]. Notice that they need to assume (i) $p>3$; (ii) $p$ is not
anomalous for their elliptic curve $E$; and (iii) $i\neq 0$ where $i$ is the
power of $\chi$ acting on the isotypic components
$(U_{\infty}/\bar{\mathscr{C}_{\infty}})^{(i)}$, with $\chi$ giving the
canonical action of ${\operatorname{Gal}}(K(E[\pi])/K)$ on $E[\pi]$. Notice
that our mod $\mathfrak{p}$ character $\vartheta$ is the reduction of $\chi$,
where the trivial character $\mathbf{1}$ corresponds to $i=0$ and $\omega$
corresponds to $i=1$.

The above assumptions are not needed in a more general theorem [dS87,
III.1.10] of de Shalit, but the result is different when
$\vartheta|_{G_{K}}=\mathbf{1}$. We remark here that the $i=0$ case can also
be studied directly using the idea of [Yag82, Lemma 28] where
$(i_{1},i_{2})=(0,0)$, which also yields

${\operatorname{Char}}(U_{\infty}/\bar{\mathscr{C}_{\infty}})^{\mathbf{1}}={\operatorname{Char}}(\mathcal{L}_{\mathbf{1}}\cdot
T),$

showing that
$\lambda(\mathfrak{X}_{\mathbf{1}})=\lambda(\mathcal{L}_{\mathbf{1}})+1$.
Together with Theorem 1.5.1, this yields the equation
$\lambda(\mathfrak{X}_{f})=\lambda(\mathcal{L}_{f})$, as expected. ∎

######  Remark 2.2.4.

The above arguments imply that the standard Iwasawa Main Conjectures for the
trivial character is different from others, namely the algebraic
$\lambda$-invariant is bigger than the analytic $\lambda$-invariant by $1$.
Actually, it is always predicted that some global torsion terms should appear
(see, for example, [Gre94]). However, in our setting, even if $f$ has global
torsion, it must be finite (from Proposition 1.3.4 (iii)) and will not affect
the $\lambda$-invariants, so our use of Perrin-Riou’s formula in section 1.4
is justified.

##  3\. Proofs of the Iwasawa Main Conjectures

We will directly adopt the notations and the arguments from [CGLS22] to prove
the main theorems of this paper, some of which are further generalized in
[CGS23]. More precisely, we will prove two Iwasawa Main Conjectures at a time.
In the Kolyvagin system argument, we need to replace their assumption
$E(K)[p]=0$ by $\text{{H}}^{0}(K,\overline{\rho}_{f})=0$ so that their
arguments apply to arbitrary weight modular forms. By the arguments in the
beginning of section 1.4, we are allowed to make this assumption. We will list
the intermediate results we need and comment on what should and should not be
changed.

Let $f\in S_{k}(\Gamma_{0}(N))^{\operatorname{new}}$ be a newform and let
$\mathbf{Z}[f]$ be the coefficient ring of $f$ with field of fractions
$\mathbf{Q}(f)$. Let $F$ be a finite extension of the completion of
$\mathbf{Q}(f)$ at a chosen prime above $p\nmid 2N$ with ring of integers
$\mathcal{O}$. Let $\pi\in\mathcal{O}$ be a uniformizer and
$\mathbf{F}\colonequals\mathcal{O}/\pi$ be the residue field of $\mathcal{O}$.
Throughout this section, assume that

(h.1) $\text{{H}}^{0}(K,\overline{\rho}_{f})=0.$

As before, $\Gamma={\operatorname{Gal}}(K_{\infty}/K)$ denote the Galois group
of the anticyclotomic $\mathbf{Z}_{p}$-extension of $K$. We let
$\alpha\colon\Gamma\to R^{\times}$ be a character with values in the ring of
integers $R$ of a finite extension $\Phi/\mathbf{Q}_{p}$. Let

$r\colonequals{\operatorname{rk}}_{\mathcal{O}}R.$

Let $\rho_{f}\colon G_{\mathbf{Q}}\to{\operatorname{GL}}_{2}(F)$ be the unique
semisimple $p$-adic Galois representation unramified outside $pN$ and such
that $\rho_{f}({\operatorname{Frob}}_{\ell})$ has characteristic polynomial

$T^{2}-a_{\ell}\ell^{1-\frac{k}{2}}T+\ell$

for all $\ell\nmid pN$.

Let $T$ be an Galois stable lattice and consider the $G_{K}$-modules

$T_{\alpha}\colonequals T\otimes_{\mathcal{O}}R(\alpha),\ \
V_{\alpha}\colonequals T\otimes_{\mathcal{O}}\Phi,\ \ A_{\alpha}\colonequals