# Tamagawa number conjecture for CM modular forms and Rankin–Selberg
convolutions

Francesc Castella  University of California Santa Barbara, South Hall, Santa
Barbara, CA 93106, USA<EMAIL_ADDRESS>In memory of Jan Nekovář

(Date: August 28, 2024)

###### Abstract.

Let $E/F$ be an elliptic curve defined over a number field $F$ with complex
multiplication by the ring of integers of an imaginary quadratic field $K$
such that the torsion points of $E$ generate over $F$ an abelian extension of
$K$. In this paper we prove the $p$-part of the Birch and Swinnerton-Dyer
formula for $E/F$ in analytic rank $1$ for primes $p>3$ split in $K$. This was
previously known for $F=\mathbb{Q}$ by work of Rubin [Rub91] as a consequence
of his proof of the Mazur–Swinnerton-Dyer “main conjecture” for rational CM
elliptic curves, but the problem remained wide open for general $F$.

The approach in this paper, based on a novel application of an idea of
Bertolini–Darmon–Prasanna to consider a carefully chosen decomposable
Rankin–Selberg convolution of two CM modular forms having the Hecke
$L$-function of interest as one of the factors, circumvents the use of
$p$-adic heights and Bertrand’s $p$-adic transcendence results in previous
approaches. It also yields a proof of similar results for CM abelian varieties
$A/K$, and for CM modular forms of higher weight.

This research was partially supported by the NSF grants DMS-2101458 and
DMS-2401321.

###### Contents

  1. 1 Introduction
    1. 1.1 Statement of the main results
    2. 1.2 About the proofs
    3. 1.3 About the hypotheses
    4. 1.4 Acknowledgements
  2. 2 $p$-adic $L$-functions
    1. 2.1 $p$-adic $L$-functions for self-dual pairs
    2. 2.2 Katz $p$-adic $L$-functions
    3. 2.3 Factorization
  3. 3 Selmer groups and main conjectures
    1. 3.1 Selmer groups for self-dual pairs
    2. 3.2 Generalized Heegner cycles
    3. 3.3 Main conjectures for self-dual pairs
    4. 3.4 Selmer groups for Hecke characters
    5. 3.5 Decompositions
    6. 3.6 Equivalence between the main conjectures
  4. 4 Proof of the Iwasawa main conjectures
    1. 4.1 Statement
    2. 4.2 Explicit reciprocity law
    3. 4.3 Main conjectures for characters
    4. 4.4 $p$-parity conjecture
    5. 4.5 Non-triviality of $\mathbf{z}_{g,\chi}$
    6. 4.6 Proof of Theorem 4.1.1 and Heegner cycle main conjecture
  5. 5 Formula for the Bloch–Kato $\text{\wncyr{W}}_{{\rm BK}}(W_{g,\chi}/K)$
    1. 5.1 Strong good pairs
    2. 5.2 Proving Selmer rank $1$
    3. 5.3 Proof of Theorem C
  6. 6 The $p$-part of the Birch Swinnerton-Dyer formula
    1. 6.1 Periods and heights of Heegner points
    2. 6.2 Theorem B implies Theorem A
    3. 6.3 Proof of Theorem B

##  1\. Introduction

###  1.1. Statement of the main results

In this paper we prove the following result towards the Birch and Swinnerton-
Dyer conjecture for elliptic curves with complex multiplication.

######  Theorem A.

Let $E$ be an elliptic curve over a number field $F$ with complex
multiplication by the ring of integers of an imaginary quadratic field $K$
such that $F(E_{\rm tors})/K$ is abelian. Let
$\psi_{E}:F^{\times}\backslash\mathbb{A}_{F}^{\times}\rightarrow\mathbb{C}^{\times}$
be the Hecke character associated to $E/F$. Suppose

${\rm ord}_{s=1}L(\psi_{E},s)=1,$

and let $p\nmid 6h_{K}$ be a prime split in $K$, where $h_{K}:=\\#{\rm
Pic}(\mathcal{O}_{K})$ is the class number of $K$. Assume the conductor
$\mathfrak{f}_{\psi}$ of $\psi$ is prime to $p$ and satisfies
$\mathfrak{d}_{K}\|N_{F/K}(\mathfrak{f}_{\psi})$, where
$\mathfrak{d}_{K}:=(\sqrt{-D_{K}})$. Then

${\rm rank}_{\mathbb{Z}}E(F)={\rm ord}_{s=1}L(E/F,s),$

Moreover, for all primes $\wp\mid p$ in $K$ we have
$\\#\text{\wncyr{W}}(E/F)[\wp^{\infty}]<\infty$, with

${\rm ord}_{\wp}\biggl{(}\frac{L^{*}(E/F,1)}{{\rm
Reg}(E)\cdot\Omega(E)}\biggr{)}={\rm
ord}_{\wp}\biggl{(}\frac{\\#\text{\wncyr{W}}(E/F)[\wp^{\infty}]\cdot{\rm
Tam}(E/F)}{\\#(E(F)_{\rm tors})^{2}}\biggr{)}.$

where $L^{*}(E/F,1)$ is the leading Taylor coefficient of the Hasse–Weil
$L$-function for $E/F$ at $s=1$.

######  Remark 1.1.1.

  1. (1)

Classical results of Deuring [Deu53] (see also [Sil94, Thm. 10.5]) show that

$L(E/F,s)=L(\psi_{E},s)\cdot L(\overline{\psi}_{E},s)$

if $K$ is contained in $F$, where $\overline{\psi}_{E}$ is the complex
conjugate of $\psi_{E}$, while

$L(E/F,s)=L(\psi_{E},s)=L(\overline{\psi}_{E},s)$

otherwise. Hence the first assertion in Theorem A is that ${\rm
rank}_{\mathbb{Z}}E(F)=2$ in the former case, and ${\rm
rank}_{\mathbb{Z}}E(F)=1$ in the latter.

  2. (2)

The relevance of the condition that the extension $F(E_{\rm tors})/K$ be
abelian for the application of Iwasawa theory of $K$ to the arithmetic of
elliptic curves with CM by $K$ was first highlighted in work of Arthaud
[Art78] generalizing Coates–Wiles [CW77] (see also [Rub81, GS81]).

For $F=\mathbb{Q}$ (which forces $K$ to have class number one), Theorem A was
obtained by Rubin [Rub91] as a consequence of his proof of the Iwasawa main
conjecture for $K$, the Gross–Zagier formula [GZ86], and Perrin-Riou’s work
[PR87a, PR87b] on a $p$-adic Gross–Zagier formula and Iwasawa theory. The
requirement that $F=\mathbb{Q}$ is essential to Rubin’s result, as it relies
on a link with the main conjecture of Mazur and Swinnerton-Dyer [Maz72, MSD74]
for rational elliptic curves. As a consequence, prior to Theorem A the case
$F\neq\mathbb{Q}$ remained wide open.

Our proof of Theorem A also gives a new proof of Rubin’s result when
$F=\mathbb{Q}$, circumventing the use of $p$-adic heights and of Bertrand’s
$p$-adic transcendence results [Ber84] (which remain unknown in higher
dimensions). The new method also yields a similar result on the Birch and
Swinnerton-Dyer conjecture for higher-dimensional CM abelian varieties $A/K$
(Theorem B), and for higher weight CM forms (see Theorem C and Remark 1.2.2
below).

######  Theorem B.

Let $A/K$ be an abelian variety with ${\rm End}_{K}(A)\simeq\mathcal{O}_{L}$
for a CM field $L$ with $[L:K]={\rm dim}(A)$, and let
$\lambda:K^{\times}\backslash\mathbb{A}_{K}^{\times}\rightarrow\mathbb{C}^{\times}$
be the associated Hecke character of conductor $\mathfrak{c}$. Suppose

${\rm ord}_{s=1}L(\lambda,s)=1,$

and let $p\nmid 6h_{K}$ be a prime split in $K$. Assume $\mathfrak{c}$ is
prime to $p$ and satisfies $\mathfrak{d}_{K}\|\mathfrak{c}$. Then

${\rm rank}_{\mathbb{Z}}A(K)={\rm ord}_{s=1}L(A/K,s).$

Moreover, for all primes $\mathfrak{P}\mid p$ in $L$ we have
$\\#\text{\wncyr{W}}(A/K)[\mathfrak{P}^{\infty}]<\infty$, with

${\rm ord}_{\mathfrak{P}}\biggl{(}\frac{L^{*}(A/K,1)}{{\rm
Reg}(A)\cdot\Omega(A)}\biggr{)}={\rm
ord}_{\mathfrak{P}}\biggl{(}\frac{\\#\text{\wncyr{W}}(A/K)[\mathfrak{P}^{\infty}]\cdot{\rm
Tam}(A/K)}{\\#(A(K)_{\rm tors})\cdot\\#(A^{\vee}(K)_{\rm tors})}\biggr{)},$

where $A^{\vee}/K$ is the abelian variety dual to $A/K$.

The periods $\Omega(A)$ and $\Omega(E)$ in the above results are as defined in
[FS22, Def. 30], which as explained in [BF24, §1] agree with the periods in
the conjecture of Birch Swinnerton-Dyer for abelian varieties over number
fields.

###  1.2. About the proofs

The starting point in the proof of our main results is an idea introduced by
Bertolini–Darmon–Prasanna [BDP12] in their proof of (a generalization of)
Rubin’s formula [Rub92] expressing the $p$-adic logarithm of Heegner points in
terms of special values of Katz $p$-adic $L$-functions. In Rubin’s original
proof of the formula, the arithmetic of CM elliptic curves $E/\mathbb{Q}$ is
studied using Heegner points for an auxiliary imaginary quadratic field
$K^{\prime}$ satisfying the Heegner hypothesis relative to the conductor of
$E$. In particular, one is forced to take $K^{\prime}\neq K$, as the
$L$-function

$L(E/K,s)=L(E,s)^{2}$

has always sign $+1$. Letting $\lambda$ be the Hecke character of $K$ attached
to $E/\mathbb{Q}$, the ingenious idea of [BDP12] is to write

$\lambda=\psi\chi$

as a product of a suitable Hecke character $\psi$ of infinity type $(-1,0)$
and a ray class character $\chi$, so that $L(E,s)=L(\lambda,s)$ appears as a
factor of the Rankin–Selberg convolution

$L(g/K,\chi,s)=L(\lambda,s)\cdot L(\psi^{\tau}\chi,s)$

for $g=\theta_{\psi}$, where $\psi^{\tau}$ denotes the composition of $\psi$
with the action of the non-trivial automorphism of $K/\mathbb{Q}$, and use
Heegner cycles for the pair $(g,\chi)$ as in the general Gross–Zagier formula
[YZZ13]. Combined with a factorization and special value formula for the
$p$-adic Rankin $L$-series for $(g,\chi)$ obtained in [BDP13], this leads to a
new proof of Rubin’s formula.

Towards the application of this idea to our problem, the first part of the
paper is devoted to the study of the anticyclotomic Iwasawa main conjecture
for “self-dual pairs” $(g,\chi)$ of the form

$(g,\chi)=(\theta_{\psi},\chi)$

with $\psi$ a Hecke a character of $K$ of infinity type $(1-2r,0)$ for any
$r\geq 1$ and $\chi$ a finite order Hecke character with central character
$\varepsilon_{\chi}=\varepsilon_{g}^{-1}$, where $\varepsilon_{g}$ is the
nebentypus of $g$. Expanding on the work of Agboola–Howard [AH06] and Arnold
[Arn07], incorporating key ideas and results from Kato [Kat04] and Johnson-
Leung–Kings [JLK11], we prove the main conjecture for $(g,\chi)$ in this
setting, both

  * •

in terms of the $p$-adic $L$-function $\mathscr{L}_{w}(g,\chi)$ of
Bertolini–Darmon–Prasanna (Theorem 4.1.1),

  * •

in terms of the $\Lambda_{\mathscr{O}}$-adic Heegner classes
$\mathbf{z}_{g,\chi}$ of [CH18] (Corollary 4.6.1).

After inverting $p$ and for $r=1$, the “lower bound” divisibility predicted by
these main conjectures was obtained in earlier work of the author with
Burungale, Skinner, and Tian in [BCST22]; here we further develop the method
to prove the equality predicted by the main conjecture without any ambiguity
by powers of $p$ (as is essential for results such as Theorem A). An
interesting new insight here is the dependence on a certain root number that
seems to not have been observed before (see Theorem 4.5.1 and Remark 4.5.2).

The next step is to deduce from the equality of characteristic ideals in the
Iwasawa main conjectures, a formula for the order of the Tate–Shafarevich
group attached (after Bloch–Kato [BK90]) to the pair $(g,\chi)$. In our rank
$1$ case, the formula we obtain is in terms of the index of a Heegner class
$z_{g,\chi}$ (introduced Theorem 3.2.1) inside the Bloch–Kato Selmer group
${\rm Sel}_{\rm BK}(K,T_{g,\chi})$.

With future arithmetic applications in mind (see Remark 1.2.2), this result
also applies in arbitrary weights $2r\geq 2$. Below we let
$\mathscr{L}_{v}(\lambda\mathbf{N}^{-r})$ denote the anticyclotomic Katz
$p$-adic $L$-function introduced in $\S\ref{subsubsec:katz-ac}$, which has the
trivial character $\mathds{1}$ outside the range of interpolation.

######  Theorem C.

Let $\lambda$ be a Hecke character of infinity type $(1-2r,0)$ for some $r\geq
1$ and conductor $\mathfrak{c}$ prime to $p$ such that
$\mathfrak{d}_{K}\|\mathfrak{c}$. Suppose
$\mathscr{L}_{v}(\lambda\mathbf{N}^{-r})(\mathds{1})\neq 0$ and
$\lambda\mathbf{N}^{-r}$ has root number $-1$. Let $(\psi,\chi)$ be a good
pair for $\lambda$ in the sense of Definition 5.1.1, and let

$z_{g,\chi}\in{\rm Sel}_{\rm BK}(K,T_{g,\chi})$

be the Heegner class associated to the self-dual pair
$(g,\chi)=(\theta_{\psi},\chi)$. Then ${\rm rank}_{\mathscr{O}}\,{\rm
Sel}_{\rm BK}(K,T_{g,\chi})=1$, $z_{g,\chi}$ is non-torsion, and

$\\#\bigl{(}{\rm Sel}_{\rm BK}(K,T_{g,\chi})/\mathscr{O}\cdot
z_{g,\chi}\bigr{)}^{2}=\frac{\\#\text{\wncyr{W}}_{\rm
BK}(W_{g,\chi}/K)}{\\#\mathrm{H}^{0}(K,W_{g,\chi})\cdot\\#\mathrm{H}^{0}(K,W_{g,\chi}^{\tau})}\cdot\prod_{\begin{subarray}{c}w\in\Sigma,v\nmid
p\end{subarray}}c_{w}(W_{g,\chi}/K),$

where $c_{w}(W_{g,\chi}/K)$ is the Tamagawa number of $W_{g,\chi}$ at $w$.

######  Remark 1.2.1.

When $r=1$, the assumption
$\mathscr{L}_{v}(\lambda\mathbf{N}^{-r})(\mathds{1})\neq 0$ and
$\lambda\mathbf{N}^{-r}$ has root number $-1$ can be shown to follow from
${\rm ord}_{s=r}L(\lambda,s)=1$; the same implication is expected to hold for
$r>1$, but this is not known at present (see Remark 5.2.2 for further
details).

Let $\Gamma={\rm Gal}(K_{\infty}/K)$ denote the Galois group of the
anticyclotomic $\mathbb{Z}_{p}$-extension of $K$. The proof of Theorem C is
based on a new calculation of the $\Gamma$-Euler characteristic (in the sense
of [Gre99, §4]) of a certain the Selmer group $\mathcal{X}_{v}(g,\chi)$. In
the non-CM case (and for $r=1$), a related computation was obtained by
Jetchev–Skinner–Wan [JSW17, §4]. As one of the most novel features in our
approach, here we exploit the decomposition

(1.1)
$\mathcal{X}_{v}(g,\chi)\simeq\mathcal{X}_{v}(\lambda\mathbf{N}^{-r})\oplus\mathcal{X}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})$

of Proposition 3.5.1, where $\mathcal{X}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})$
interpolates the Bloch–Kato Selmer group of $\psi^{\tau}\chi\mathbf{N}^{-r}$
over $K_{\infty}/K$, while $\mathcal{X}_{v}(\lambda\mathbf{N}^{-r})$ is seen
to agree with the result of _reversing_ the local conditions at the primes
above $p$ defining the Bloch–Kato Selmer group of $\lambda\mathbf{N}^{-r}$
over the anticyclotomic tower. After computing the $\Gamma$-Euler
characteristic for each of the summands in (1.1), the expressions we obtain
combine quite pleasantly into the formula of Theorem C.

The proof of Theorem A is easily deduced from an application of Theorem B to
(the isogeny factors of) $B={\rm Res}_{F/K}(E)$, so it remains to explain how
to go from Theorem C to Theorem B, for which we rely on the general
Gross–Zagier formula [YZZ13] (in its explicit form in [CST14]). Letting
$B_{g,\chi}/K$ be the Serre tensor attached to the pair $(g,\chi)$, the
deduction relies on the factorization

$L(B_{g,\chi}/K,s)=\prod_{\sigma:L\hookrightarrow\mathbb{C}}L(g^{\sigma}/K,\chi^{\sigma},s)$

and the explicit description of the motivic structure attached to Hecke
characters in the work of Kato [Kat04] and Burungale–Flach [BF24]. The
existence of _strong good pairs_ in the sense introduced in Definition 5.1.1
(whose existence is established in Lemma 5.1.3 building on Finis’ mod $p$ non-
vanishing results [Fin06]) is also used at this point to address a certain
period comparison building on work of Hida–Tilouine [HT93, HT94], as completed
by Hida [Hid06], on the anticyclotomic Iwasawa main conjecture for CM fields.

######  Remark 1.2.2.

Granted a Gross–Zagier type formula for the generalized Heegner cycles of
[BDP13] in the style of S.-W. Zhang [Zha97]111See forthcoming work of
Lilienfeldt–Shnidman for progress in this direction., Theorem C should yield a
proof of the $p$-part of the equivariant Tamagawa number conjecture of
Burns–Flach [BF01] for CM motives in analytic rank $1$ under standard
hypotheses222That is, injectivity of the $p$-adic Abel–Jacobi map and non-
degeneracy of a (Gillet–Soulé) height pairing.. (See [LV23] for results in
this direction in the non-CM case building on [Zha97].) This will be studied
in the forthcoming PhD thesis by Mychelle Parker [Par24], which will also
include a proof, building on the methods of this paper, of a higher weight
analogue of Skinner’s $p$-converse theorem [Ski20] in the CM case.

###  1.3. About the hypotheses

The restriction on the conductor in our results arises from the _Heegner
hypothesis_ present in some of our sources (such as [CH18] and [JLZ21]), and
could be removed with a suitable refinement of the results in [LZZ18]. Our
second additional hypothesis that $p\nmid h_{K}$ would also become superfluous
with a better understanding of the relation between modular degrees and
congruence numbers studied in [ARS12]. Lastly, removing the hypotheses that
$p>3$ and $p$ be split in $K$ would seem to require new ideas. We hope to come
back to this in future work.

###  1.4. Acknowledgements

It is a pleasure to thank Chris Skinner was stimulating exchanges related to
this work; the idea that a proof of the $p$-part of the Birch and Swinnerton-
Dyer formula for CM forms in analytic rank $1$ might be within reach of the
techniques developed in this paper occurred to us during conversations with
him. We would also like to thank Ashay Burungale, Henri Darmon, Giada Grossi,
Ye Tian, and Xin Wan for interesting conversations related to the topics in
this paper. We also thank Matthias Flach, Dick Gross, and Haruzo Hida for
comments on an earlier draft.

We would like to dedicate this work to the memory of the late Jan Nekovář,
whose encouragement was heartfelt and instrumental to us, specially in the
early career stages.

##  2\. $p$-adic $L$-functions

In this section we describe the $p$-adic $L$-functions needed for our
arguments. The key result is the factorization of Proposition 2.3.1. The
discussion in this section parallels [BCST22, §4], but here we need to pay
more attention to integrality properties.

###  2.1. $p$-adic $L$-functions for self-dual pairs

####  2.1.1. Self-dual pairs

Let $g\in S_{2r}(\Gamma_{0}(N_{g}),\varepsilon_{g})$ be a newform of even
weight $k=2r\geq 2$, and let $K/\mathbb{Q}$ be an imaginary quadratic field of
discriminant $-D_{K}<0$ satisfying the _Heegner hypothesis_ :

(Heeg) there exists an ideal $\mathfrak{N}_{g}$ with
$\mathcal{O}_{K}/\mathfrak{N}_{g}\simeq\mathbb{Z}/N_{g}\mathbb{Z}$.

We also fix once and for all embedding
$\imath_{\infty}:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$ and
$\imath_{p}:\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{p}$,
and assume that

(spl) $(p)=v\overline{v}$ splits in $K$,

with $v$ the prime of $K$ above $p$ induced by $\imath_{p}$.

As in [BCST22, §2.1], we say that a Hecke character
$\psi=(\psi_{w})_{w}:K^{\times}\backslash\mathbb{A}_{K}^{\times}\rightarrow\mathbb{C}^{\times}$
has _infinity type_ $(a,b)\in\mathbb{Z}^{2}$ if
$\psi_{\infty}(z)=z^{a}\overline{z}^{b}$ for all
$z\in(K\otimes_{\mathbb{Q}}\mathbb{R})^{\times}\simeq\mathbb{C}^{\times}$
under the identification induced by $\imath_{\infty}$. In particular, the norm
character $\mathbf{N}$ given by
$\mathfrak{a}\mapsto\\#(\mathcal{O}_{K}/\mathfrak{a})$ on ideals of
$\mathcal{O}_{K}$, has infinity type $(-1,-1)$. Then the _central character_
of $\psi$ is the Dirichlet character $\varepsilon_{\psi}$ defined by

$\psi|_{\mathbb{A}^{\times}}=\varepsilon_{\psi}\cdot|\;|_{\mathbb{A}^{\times}}^{-a-b}.$

We say that $\psi$ is _anticyclotomic_ if $\psi|_{\mathbb{A}^{\times}}=1$; in
particular, such $\psi$ has trivial central character and its infinity type is
of the form $(n,-n)$.

######  Definition 2.1.1.

Let $\chi$ be a finite order character of $K$. We say that $(g,\chi)$ is a
_self-dual pair_ if

$\varepsilon_{\chi}=\varepsilon_{g}^{-1}.$

Then the Rankin–Selberg $L$-function $L(g/K,\chi,s)$ is self-dual, with a
functional equation relating its values at $s$ and $2r-s$. The sign
$\epsilon(g,\chi)\in\\{\pm{1}\\}$ in the functional equation is a product of
local signs:

$\epsilon(g,\chi)=\prod_{q}\epsilon_{q}(g,\chi),$

where $q$ runs over all places of $\mathbb{Q}$.

Let $\hat{K}^{\times}$ denote group of finite idèles of $K$. Attached to a
character $\psi$ of $K$ of infinity type $(a,b)$ is its _$p$ -adic avatar_
$\hat{\psi}:K^{\times}\backslash\hat{K}^{\times}\rightarrow\mathbb{C}_{p}^{\times}$
defined by

$\hat{\psi}(x)=\imath_{p}\circ\imath_{\infty}^{-1}(\psi(x))\sigma(x_{p})^{a}\overline{\sigma}(x_{p})^{b}$

for all $x\in\hat{K}^{\times}$, where
$\sigma:K\otimes_{\mathbb{Q}}\mathbb{Q}_{p}\rightarrow\mathbb{C}_{p}$ is the
map induced by $\imath_{p}$, and $\overline{\sigma}:=\sigma\circ{\tau}$ for
the non-trivial automorphism ${\tau}$ of $K/\mathbb{Q}$. Via the
(geometrically normalized) reciprocity map ${\rm
rec}_{K}:K^{\times}\backslash\hat{K}^{\times}\rightarrow G_{K}^{\rm ab}$, we
shall view $\hat{\psi}$ as a Galois character. Since it should not lead to
confusion, in the following we shall still use $\psi$ to denote its $p$-adic
avatar $\hat{\psi}$ (except for $\psi=\mathbf{N}$, whose $p$-adic avatar is
$\varepsilon_{\rm cyc}^{-1}$ for the $p$-adic cyclotomic character
$\varepsilon_{\rm cyc}:G_{K}\rightarrow\mathbb{Z}_{p}^{\times}$).

For $\mathscr{O}$ the ring of integers of a finite extension $\Phi$ of
$\mathbb{Q}_{p}$, we say that the pair $(g,\chi)$ is _defined over
$\mathscr{O}$_ if (under our fixed embeddings $\imath_{\infty},\imath_{p}$)
the Fourier coefficients of $g$ and the values of $\chi$ are all valued in
$\mathscr{O}$.

######  Definition 2.1.2.

For a positive integer $c$ prime to $N_{g}$, we let $\Sigma_{\rm
cc}(c,\mathfrak{N}_{g},\varepsilon_{g})$ denote the set of finite order
characters $\chi$ such that $(g,\chi)$ is a self-dual pair and moreover:

  * (i)

The conductor of $\chi$ is

$\mathfrak{f}_{\chi}=(c)\mathfrak{N}_{\varepsilon_{g}},$

where $\mathfrak{N}_{\varepsilon_{g}}$ is the unique divisor of
$\mathfrak{N}_{g}$ with norm equal to the conductor of $\varepsilon_{g}$;

  * (ii)

$\epsilon_{q}(g,\chi)=+1$ for all finite primes $q$.

In particular, for $\chi\in\Sigma_{\rm
cc}(c,\mathfrak{N}_{g},\varepsilon_{g})$ and $\xi$ any anticyclotomic Hecke
character of $K$ of infinity type $(n,-n)$ and conductor divisible only by
primes that split in $K$, the $L$-series $L(g/K,\chi\xi,s)$ is self-dual with
center at $s=r$ and sign

$\epsilon(g,\chi\xi)=\begin{cases}+1&\textrm{if $|n|\geq r$,}\\\\[1.99997pt]
-1&\textrm{if $|n|<r$.}\end{cases}$

The $p$-adic $L$-functions in this section interpolate the central $L$-values
$L(g/K,\chi\xi,r)$ for $|n|\geq r$.

####  2.1.2. CM periods

Fix an elliptic curve $A_{0}/F$ defined over a number field $F$ with complex
multiplication by $\mathcal{O}_{K}$, and let
$(\Omega_{K},\Omega_{p})\in\mathbb{C}^{\times}\times\mathcal{O}_{\mathbb{C}_{p}}^{\times}$
be the complex and $p$-adic CM periods in [CH18, §2.5] (with $A_{0}$
corresponding to the CM elliptic curve denoted $A$ in _loc. cit._). Put also

$\Omega:=2\pi i\cdot\Omega_{K},$

and note that this recovers (up to $F^{\times}$) the complex CM period
appearing in [dS87, II.4.2]. For any embedding
$\sigma:F\rightarrow\mathbb{C}$, we define
$\Omega_{\sigma}\in\mathbb{C}^{\times}$ by replacing $A_{0}$ by
$A_{0}^{\sigma}$ in the above definition.

####  2.1.3. $p$-adic interpolation

For any abelian extension $K^{\prime}/K$, let
$\Lambda_{\mathscr{O}}(K^{\prime})$ denote the Iwasawa algebra
$\mathscr{O}[\\![{\rm
Gal}(K^{\prime}/K)]\\!]:=\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits\mathscr{O}[{\rm
Gal}(K^{\prime\prime}/K)]$, where $K^{\prime\prime}$ runs over the finite
extensions of $K$ contained in $K^{\prime}$, and the inverse limit is with
respect to the natural projection maps. We also put

$\Lambda_{\mathscr{O}}^{\rm
ur}(K^{\prime}):=\Lambda_{\mathscr{O}}(K^{\prime})\hat{\otimes}_{\mathbb{Z}_{p}}\mathbb{Z}_{p}^{\rm
ur},$

where $\mathbb{Z}_{p}^{\rm ur}$ denotes the completion of the ring of integers
of the maximal unramified extension of $\mathbb{Q}_{p}$.

Let $H_{p^{\infty}}$ be the union of the ring class fields of $K$ of $p$-power
conductor; the maximal torsion-free quotient of $\widetilde{\Gamma}:={\rm
Gal}(H_{p^{\infty}}/K)$ is by definition the Galois group

$\Gamma:={\rm Gal}(K_{\infty}/K)$

of the anticyclotomic $\mathbb{Z}_{p}$-extension of $K$. Since anticyclotomic
Iwasawa algebras will play a prominent role in the paper, we set

$\Lambda_{\mathscr{O}}:=\Lambda_{\mathscr{O}}(K_{\infty}),\quad\quad\Lambda_{\mathscr{O}}^{\rm
ur}:=\Lambda_{\mathscr{O}}^{\rm ur}(K_{\infty})$

for the ease of notation.

######  Theorem 2.1.3.

Let $(g,\chi)\in S_{2r}(\Gamma_{1}(N_{g}))\times\Sigma_{\rm
cc}(c,\mathfrak{N}_{g},\varepsilon_{g})$ be a self-dual pair for some positive
integer $c$ prime to $pN_{g}$ defined over $\mathscr{O}$. There exists a
“square-root” $p$-adic $L$-function

$\mathscr{L}_{v}(g,\chi)\in\Lambda_{\mathscr{O}}^{\rm ur}(H_{p^{\infty}})$

characterized by the property that for every character $\xi$ of
$\widetilde{\Gamma}$ crystalline at both $v$ and $\overline{v}$ corresponding
to a Hecke character of $K$ of infinity type $(n,-n)$ with $n\geq r$ and
$n\equiv 0\pmod{p-1}$, we have

$\displaystyle\mathscr{L}_{v}(g,\chi)^{2}(\xi)$
$\displaystyle=\frac{\Omega_{p}^{4n}}{\Omega^{4n}}\cdot\Gamma(r+n)\Gamma(n+1-r)\cdot\biggl{(}\frac{2\pi}{\sqrt{D_{K}}}\biggr{)}^{2n-1}$
$\displaystyle\quad\times\bigl{(}1-a_{p}(g)\chi\xi(\overline{v})p^{-r}+\varepsilon_{g}(p)\chi\xi(\overline{v})^{2}p^{-1}\bigr{)}^{2}\cdot
L(g/K,\chi\xi,r).$

###### Proof.

This follows from the results of [CH18, §3] as in [BCST22, Thm. 4.1]. (Note
that in the above interpolation property we have replaced the CM period
$\Omega_{K}$ from [CH18] by the above $\Omega$, and have omitted terms
interpolated by a unit in $\Lambda_{\mathscr{O}}^{\rm
ur}(H_{p^{\infty}})^{\times}$.) ∎

######  Remark 2.1.4.

Reversing the roles of $v$ and $\overline{v}$ in Theorem 2.1.3, we obtain an
element $\mathscr{L}_{\overline{v}}(g,\chi)\in\Lambda_{\mathscr{O}}^{\rm
ur}(H_{p^{\infty}})$ interpolating the central critical values
$L(g/K,\chi\xi,r)$ for $\xi$ of infinity type $(-n,n)$ with $n\geq r$ and
$n\equiv 0\pmod{p-1}$. We shall use both $\mathscr{L}_{v}(g,\chi)$ and
$\mathscr{L}_{\overline{v}}(g,\chi)$ in the following.

With a slight abuse of notation, we also denote by $\mathscr{L}_{v}(g,\chi)$
its image under the natural projection $\Lambda_{\mathscr{O}}^{\rm
ur}(H_{p^{\infty}})\rightarrow\Lambda_{\mathscr{O}}^{\rm ur}$, and similarly
for $\mathscr{L}_{\overline{v}}(g,\chi)$.

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

The construction of $p$-adic $L$-functions in the next result is originally
due to Katz [Kat78], but the alternative construction by de Shalit [dS87],
whose exposition we follow, will be most convenient for our purposes.

####  2.2.1. Two-variable $p$-adic $L$-functions

For an integral ideal $\mathfrak{c}$ of $K$, let $K(\mathfrak{c}p^{\infty})$
be the union of the ray class fields of $K$ conductor $\mathfrak{c}p^{n}$ for
$n\geq 0$, and put

$G_{\mathfrak{c}}:={\rm Gal}(K(\mathfrak{c}p^{\infty})/K).$

For a character $\xi$ of $G_{\mathfrak{c}}$, let $\xi^{\tau}$ denote the
composition of $\xi$ with action of the non-trivial automorphism of
$K/\mathbb{Q}$, and put $\xi^{-\tau}:=(\xi^{\tau})^{-1}$.

######  Theorem 2.2.1.

Let $\mathfrak{c}$ be an integral ideal of $K$ prime to $p$. There exists an
element

$\mathscr{L}_{v,\mathfrak{c}}\in\Lambda_{\mathscr{O}}^{\rm
ur}(K(\mathfrak{c}p^{\infty}))$

characterized by the property that for every character $\xi$ of
$G_{\mathfrak{c}}$ crystalline at both $v$ and $\overline{v}$ corresponding to
a Hecke character of infinity type $(a,b)$ with $a>0$ and $b\leq 0$ we have

$\mathscr{L}_{v,\mathfrak{c}}(\xi)=\frac{\Omega_{p}^{a-b}}{\Omega^{a-b}}\cdot\Gamma(a)\cdot\biggl{(}\frac{\sqrt{D_{K}}}{2\pi}\biggr{)}^{b}\cdot(1-\xi^{-1}(v)p^{-1})(1-\xi(\overline{v}))\cdot
L_{\mathfrak{c}}(\xi,0),$

where $L_{\mathfrak{c}}(\xi,s)$ denotes the Hecke $L$-function of $\xi$ with
the Euler factors at the primes $w\mid\mathfrak{c}$ removed. Similarly, there
exists an element
$\mathscr{L}_{\overline{v},\mathfrak{c}}\in\Lambda_{\mathscr{O}}^{\rm
ur}(K(\mathfrak{c}p^{\infty}))$ such that for every character $\xi$ of
$G_{\mathfrak{c}}$ crystalline at both $v$ and $\overline{v}$ corresponding to
a Hecke character of infinity type $(b,a)$ with $a>0$ and $b\leq 0$ we have

$\mathscr{L}_{\overline{v},\mathfrak{c}}(\xi)=\frac{\Omega_{p}^{a-b}}{\Omega^{a-b}}\cdot\Gamma(a)\cdot\biggl{(}\frac{\sqrt{D_{K}}}{2\pi}\biggr{)}^{b}\cdot(1-\xi^{-1}(\overline{v})p^{-1})(1-\xi(v))\cdot
L_{\mathfrak{c}}(\xi,0).$

Moreover, we have the functional equation

$\mathscr{L}_{v,\mathfrak{c}}(\xi)=\mathscr{L}_{v,\overline{\mathfrak{c}}}(\xi^{-\tau}\mathbf{N}^{-1}),$

where the equality is up to a $p$-adic unit, and similarly for
$\mathscr{L}_{\overline{v},\mathfrak{c}}$.

###### Proof.

The construction of the $p$-adic $L$-functions with the stated interpolation
properties is given in [dS87, Thm. II.4.14]: $\mathscr{L}_{v,\mathfrak{c}}$
(resp. $\mathscr{L}_{\overline{v},\mathfrak{c}}$) corresponds the $p$-adic
measure $\mu(\mathfrak{c}\overline{v}^{\infty})$ (resp.
$\mu(\mathfrak{c}v^{\infty})$) on $G_{\mathfrak{c}}$ constructed in _loc.
cit._. The functional equation is proved in [dS87, Thm. II.6.4], which implies
that the interpolation formulae extend from the range $0\leq-b<a$ to the range
$a>0$ and $b\leq 0$ (see [dS87, Cor. II.6.7]). ∎

####  2.2.2. Anticyclotomic projection

We shall be particularly interested in the anticyclotomic projection of Katz’s
$p$-adic $L$-functions twisted by characters that are self-dual in the
following sense.

######  Definition 2.2.2.

We say that a Hecke character $\phi$ of $K$ is _self-dual_ if it satisfies

$\phi^{\tau}=\phi^{-1}\mathbf{N}^{-1}\quad\textrm{and}\quad\varepsilon_{\phi}=\eta_{K},$

where $\eta_{K}$ is the quadratic Dirichlet character corresponding to
$K/\mathbb{Q}$.

In particular, if $\phi$ is self-dual then its $p$-adic avatar is a conjugate
self-dual character of the absolute Galois group $G_{K}={\rm
Gal}(\overline{\mathbb{Q}}/K)$, i.e.

$\phi^{\tau}=\varepsilon_{\rm cyc}\phi^{-1}.$

Note that for self-dual $\phi$, the Hecke $L$-function $L(\phi,s)$ is self-
dual, with a functional equation relating its values at $s$ and $-s$.
Moreover, the infinity type of such $\phi$ is of the form $(1-r,r)$ for some
$r\in\mathbb{Z}$ and its conductor is invariant under complex conjugation.

Let $F/K$ be an abelian extension contained in $K(\mathfrak{c}p^{\infty})$.
For a character $\phi$ of ${\rm Gal}(F/K)$ valued in $\mathscr{O}$ and having
$\mathfrak{c}$ as the prime-to-$p$ part of its conductor, we denote by
$\mathscr{L}_{v}(\phi)$ the image of $\mathscr{L}_{v,\mathfrak{c}}$ under the
composite map

$\Lambda_{\mathscr{O}}^{\rm ur}(K(\mathfrak{c}p^{\infty}))\xrightarrow{{\rm
Tw}_{\phi}}\Lambda_{\mathscr{O}}^{\rm
ur}(K(\mathfrak{c}p^{\infty}))\twoheadrightarrow\Lambda_{\mathscr{O}}^{\rm
ur},$

where ${\rm Tw}_{\phi}$ is the $\mathscr{O}$-linear isomorphism given by
$\gamma\mapsto\phi(\gamma)\gamma$ for $\gamma\in G_{\mathfrak{c}}$ and the
second arrow is the natural projection (noting that
$K(\mathfrak{c}p^{\infty})$ contains $H_{p^{\infty}}$, and hence also
$K_{\infty}$).

######  Remark 2.2.3.

Although not reflected in the notation for simplicity, when we write
$\mathscr{L}_{v}(\psi)$ (resp. $\mathscr{L}_{\overline{v}}(\psi)$) it will be
understood that this is obtained from $\mathscr{L}_{v,\mathfrak{c}}$ (resp.
$\mathscr{L}_{\overline{v},\mathfrak{c}}$) are above taking for $\mathfrak{c}$
the prime-to-$p$ part of the conductor of $\phi$.

###  2.3. Factorization

Recall that if $\psi$ is a Hecke character of infinity type $(1-2r,0)$ and
conductor $\mathfrak{f}_{\psi}$, then the theta series $g=\theta_{\psi}$ is an
eigenform in $S_{2r}(\Gamma_{0}(N_{g}),\varepsilon_{g})$ with
$N_{g}=D_{K}\mathbf{N}(\mathfrak{f}_{\psi})$ and
$\varepsilon_{g}=\varepsilon_{\psi}\eta_{K}$.

######  Proposition 2.3.1.

Let $\psi$ be a Hecke character of $K$ of infinity type $(1-2r,0)$ for some
$r\geq 1$ and conductor a cyclic ideal $\mathfrak{f}_{\psi}$ of norm prime to
$pD_{K}$. Let

$(g,\chi)=(\theta_{\psi},\chi)\in
S_{2r}(\Gamma_{0}(N_{g}),\varepsilon_{g})\times\Sigma_{\rm
cc}(c,\mathfrak{N}_{g},\varepsilon_{g})$

be a self-dual pair for some positive integer $c$ prime to $pN_{g}$ defined
over $\mathscr{O}$. Then for every $w\in\\{v,\overline{v}\\}$ we have

$\mathscr{L}_{w}(g,\chi)^{2}=u\cdot\mathscr{L}_{w}(\psi\chi\mathbf{N}^{-r})\cdot\mathscr{L}_{w}(\psi^{\tau}\chi\mathbf{N}^{-r}),$

where $u$ is a unit in $\Lambda_{\mathscr{O}}^{{\rm ur},\times}$.

###### Proof.

We only explain the proof in the case $w=v$, since the argument for
$w=\overline{v}$ is virtually the same. By our assumption on
$\mathfrak{f}_{\psi}$, the imaginary quadratic field $K$ satisfies (Heeg) with
$\mathfrak{N}_{g}=\mathfrak{d}_{K}\cdot\mathfrak{f}_{\psi}$. Let $\xi$ be a
character as in the statement of Theorem 2.1.3, corresponding to a Hecke
character of $K$ of infinity type $(n,-n)$ with $n\geq r$, and assume that
$\xi$ factors through $\Gamma$. Then the central $L$-value
$L(\theta_{\psi}/K,\chi\xi,r)$ factors as

$L(\theta_{\psi}/K,\chi\xi,r)=L(\psi\chi\xi,r)\cdot
L(\psi^{\tau}\chi\xi,r)=L(\psi\chi\mathbf{N}^{-r}\xi,0)\cdot
L(\psi^{\tau}\chi\mathbf{N}^{-r}\xi,0).$

The Hecke characters $\psi\chi\mathbf{N}^{-r}\xi$ and
$\psi^{\tau}\chi\mathbf{N}^{-r}\xi$ have infinity type $(1-r+n,r-n)$ and
$(n+r,1-n-r)$, and one easily checks (see [BDP12, Lem. 3.16]) that they are
both self-dual, with prime-to-$p$ conductor $(c)\mathfrak{d}_{K}$ and
$(cM)\mathfrak{d}_{K}$, respectively, where
$\mathfrak{d}_{K}:=(\sqrt{-D_{K}})$ and $M:=\mathbf{N}(\mathfrak{f}_{\psi})$;
thus they are in the range of interpolation of
$\mathscr{L}_{v,(c)\mathfrak{d}_{K}}$ and
$\mathscr{L}_{v,(cM)\mathfrak{d}_{K}}$, respectively. From the self-duality of
$\psi\chi\mathbf{N}^{-r}\xi$ and $\psi^{\tau}\chi\mathbf{N}^{-r}\xi$ we
immediately get the relations

$(1-\psi^{-1}\chi^{-1}\xi^{-1}(v)p^{r-1})=(1-\psi\chi\xi(\overline{v})p^{-r}),\quad(1-\psi^{-{\tau}}\chi^{-1}\xi^{-1}(\overline{v})p^{r-1})=(1-\psi^{\tau}\chi\xi(\overline{v})p^{-r}),$

and so from the interpolation formula in Theorem 2.2.1 we obtain

(2.1)
$\displaystyle\mathscr{L}_{v,(c)\mathfrak{d}_{K}}(\psi\chi\mathbf{N}^{-r}\xi)$
$\displaystyle\cdot\mathscr{L}_{v,(c)\mathfrak{d}_{K}M}(\psi^{\tau}\chi\mathbf{N}^{-r}\xi)=\frac{\Omega_{p}^{2n-2r+1}}{\Omega^{2n-2r+1}}\cdot\frac{\Omega_{p}^{2n+2r-1}}{\Omega^{2n+2r-1}}\cdot\Gamma(n-r+1)\cdot\Gamma(n+r)$
$\displaystyle\quad\times\biggl{(}\frac{2\pi}{\sqrt{D_{K}}}\biggr{)}^{n-r}\cdot\biggl{(}\frac{2\pi}{\sqrt{D_{K}}}\biggr{)}^{n+r-1}\cdot\bigl{(}1-\psi\chi\xi(\overline{v})p^{-r}\bigr{)}^{2}\cdot\bigl{(}1-\psi^{\tau}\chi\xi(\overline{v})p^{-r}\bigr{)}^{2}$
$\displaystyle\quad\times L(\psi\chi\mathbf{N}^{-r}\xi,0)\cdot
L(\psi^{\tau}\chi\mathbf{N}^{-r}\xi,0),$

using that the prime-to-$p$ conductors of $\psi\chi\mathbf{N}^{-r}\xi$ and
$\psi^{\tau}\chi\mathbf{N}^{-r}\xi$ are $(c)\mathfrak{d}_{K}$ and
$(cM)\mathfrak{d}_{K}$, respectively, to equivalently write the primitive
Hecke $L$-values. Noting that

$(1-\psi\chi\xi(\bar{v})p^{-r}\bigr{)}\cdot\bigl{(}1-\psi^{\tau}\chi\xi(\bar{v})p^{-r}\bigr{)}=(1-a_{p}(g)\chi\xi(\overline{v})p^{-r}+\varepsilon_{g}(p)\chi\xi(\overline{v})^{2}p^{-1}),$

comparing (2.1) with the interpolation property of Theorem 2.1.3, the result
follows. ∎

##  3\. Selmer groups and main conjectures

Throughout this section, we fix a prime $p>3$ and integer $N_{g}\geq 1$ with
$p\nmid N_{g}$, and let

$(g,\chi)\in S_{2r}(\Gamma_{1}(N_{g}))\times\Sigma_{\rm
cc}(c,\mathfrak{N}_{g},\varepsilon_{g})$

be a self-dual pair defined over $\mathscr{O}$ for some positive integer $c$
prime to $pN_{g}$ and an imaginary quadratic field $K$ satisfying (Heeg) and
(spl).

###  3.1. Selmer groups for self-dual pairs

Let $V_{g}$ denote the $p$-adic Galois representation associated to $g$ by
Deligne, and put

$V_{g,\chi}:=V_{g}(r)|_{G_{K}}\otimes\chi.$

######  Definition 3.1.1.

Let $w\in\\{v,\overline{v}\\}$ be a prime of $K$ above $p$, and let $F/K$ be a
finite extension. For $\eta$ a prime of $F$ above $w$, define the local
condition $\mathrm{H}^{1}_{w}(F_{\eta},V_{g,\chi})$ by

$\mathrm{H}^{1}_{w}(F_{\eta},V_{g,\chi}):=\begin{cases}\mathrm{H}^{1}(F_{\eta},V_{g,\chi})&\textrm{if
$\eta\mid w$,}\\\\[1.99997pt] 0&\textrm{if
$\eta\mid\overline{w}$,}\end{cases}$

and let $\mathrm{H}^{1}_{f}(F_{\eta},V_{g,\chi})$ denote the Bloch–Kato finite
subspace, i.e.

$\mathrm{H}^{1}_{f}(F_{\eta},V_{g,\chi}):=\ker\bigl{\\{}\mathrm{H}^{1}(F_{\eta},V_{g,\chi})\rightarrow\mathrm{H}^{1}(F_{\eta},V_{g,\chi}\otimes\mathbf{B}_{\rm
cris})\bigr{\\}},$

where $\mathbf{B}_{\rm cris}$ is Fontaine’s ring of crystalline periods. For a
finite prime $\eta\nmid p$ of $F$, let

$\mathrm{H}^{1}_{\rm ur}(F_{\eta},V_{g,\chi}):={\rm
ker}\bigl{\\{}\mathrm{H}^{1}(F_{\eta},V_{g,\chi})\rightarrow\mathrm{H}^{1}(I_{\eta},V_{g,\chi})\bigr{\\}}$

be the unramified subspace.

  * •

The _$w$ -Selmer group_ of $V_{g,\chi}$ is

${\rm
Sel}_{w}(F,V_{g,\chi}):=\ker\biggl{\\{}\mathrm{H}^{1}(F,V_{g,\chi})\rightarrow\prod_{\eta\mid
p}\frac{\mathrm{H}^{1}(F_{\eta},V_{g,\chi})}{\mathrm{H}_{w}^{1}(F_{\eta},V_{g,\chi})}\times\prod_{\eta\nmid
p}\frac{\mathrm{H}^{1}(F_{\eta},V_{g,\chi})}{\mathrm{H}^{1}_{\rm
ur}(F_{\eta},V_{g,\chi})}\biggr{\\}}.$

  * •

The _Bloch–Kato Selmer group_ of $V_{g,\chi}$ is

${\rm Sel}_{\rm
BK}(F,V_{g,\chi}):=\ker\biggl{\\{}\mathrm{H}^{1}(F,V_{g,\chi})\rightarrow\prod_{\eta|p}\frac{\mathrm{H}^{1}(F_{\eta},V_{g,\chi})}{\mathrm{H}_{f}^{1}(F_{\eta},V_{g,\chi})}\times\prod_{w\nmid
p}\frac{\mathrm{H}^{1}(F_{\eta},V_{g,\chi})}{\mathrm{H}^{1}_{\rm
ur}(F_{\eta},V_{g,\chi})}\biggr{\\}}.$

(Note that since $p$ is odd, for $\eta\mid\infty$ the groups
$\mathrm{H}^{1}(F_{\eta},V_{g,\chi})$ all vanish.)

Let $T_{g,\chi}\subset V_{g,\chi}$ be a $G_{K}$-stable $\mathscr{O}$-lattice,
and define $W_{g,\chi}$ by the exact sequence

(3.1) $0\rightarrow T_{g,\chi}\rightarrow V_{g,\chi}\rightarrow
W_{g,\chi}\rightarrow 0.$

Then we define the Selmer groups ${\rm Sel}_{w}(F,T_{g,\chi}),{\rm
Sel}_{w}(F,W_{g,\chi}),{\rm Sel}_{\rm BK}(F,T_{g,\chi}),{\rm Sel}_{\rm
BK}(F,W_{g,\chi})$ as above, with the corresponding local conditions obtained
from those of $V_{g,\chi}$ by propagating via (3.1).

Finally, for $\star\in\\{v,\overline{v},{\rm BK}\\}$, we define the
$\Lambda$-adic Selmer groups

$\check{\mathcal{S}}_{\star}(g,\chi):=\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F}{\rm
Sel}_{\star}(F,T_{g,\chi}),\quad\quad\mathcal{S}_{\star}(g,\chi):=\mathop{\mathchoice{\varlim@\displaystyle{\rightarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\rightarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\rightarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\rightarrowfill@\scriptscriptstyle}}}\displaylimits_{F}{\rm
Sel}_{\star}(F,W_{g,\chi}),$

where $F$ runs over the finite extensions of $K$ contained in $K_{\infty}$,
and the limits are with respect to the corestriction and restriction maps,
respectively, and put

$\mathcal{X}_{\star}(g,\chi):={\rm Hom}_{\rm
cts}(\mathcal{S}_{\star}(g,\chi),\mathbb{Q}_{p}/\mathbb{Z}_{p})$

for the Pontryagin dual of $\mathcal{S}_{\star}(g,\chi)$.

###  3.2. Generalized Heegner cycles

We recall the construction of $\Lambda_{\mathscr{O}}$-adic classes
$\mathbf{z}_{g,\chi}$ interpolating the generalized Heegner cycles of
Bertolini–Darmon–Prasanna [BDP13] over the anticyclotomic tower
$K_{\infty}/K$, as well as their link with the $p$-adic $L$-functions
$\mathscr{L}_{w}(g,\chi)$.

A main result of [BDP13] is the following formula for the value of
$\mathscr{L}_{w}(g,\chi)$ at the trivial character $\xi=\mathds{1}$ of
$\Gamma$, which lies outside the range of $p$-adic interpolation in Theorem
2.1.3. For $w\in\\{v,\overline{v}\\}$ a prime of $K$ above $p$, denote by

${\rm
log}_{V_{g,\chi}}:\mathrm{H}^{1}_{f}(K_{w},V_{g,\chi})\rightarrow\frac{\mathbf{D}_{\rm
dR}(V_{g,\chi})}{{\rm Fil}^{0}\mathbf{D}_{\rm dR}(V_{g,\chi})}\simeq{\rm
Fil}^{0}\mathbf{D}_{\rm dR}(V_{g,\chi}^{*}(1))^{\vee}$

the Bloch–Kato logarithm map, and recall the CM elliptic curve $A_{0}/F$ used
for the definition of the CM periods in $\S\ref{subsub:CM-periods}$.

######  Theorem 3.2.1 (Bertolini–Darmon–Prasanna).

There exists a class $z_{g,\chi}\in{\rm Sel}_{\rm BK}(K,V_{g,\chi})$ such that

$\displaystyle\mathscr{L}_{w}(g,\chi)(\mathds{1})$
$\displaystyle=\frac{c^{-r}}{\Gamma(r)}\cdot\bigl{(}1-a_{p}(g)\chi(\overline{w})p^{-r}+\varepsilon_{g}(p)\chi(\overline{w})^{2}p^{-1}\bigr{)}\cdot\left\langle{\rm
log}_{V_{g,\chi}}({\rm
loc}_{\overline{w}}(z_{g,\chi})),\omega_{g}^{\prime}\wedge\omega_{A_{0}}^{r-1}\eta_{A_{0}}^{r-1}\right\rangle,$

where $\omega_{g}^{\prime}\in\mathbf{D}_{\rm dR}(V_{g}^{*})$ is a differential
attached to $g$ as in [KLZ20], and $\omega_{A_{0}}$ and $\eta_{A_{0}}$ are
differentials attached to $A_{0}$ as in [BDP13, §1.4].

###### Proof.

This is [BDP13, Thm. 5.13], as reformulated in [JLZ21, Thm. 7.2.4]. ∎

By interpolating $p$-power conductor variants of the classes $z_{g,\chi}$ into
a $\Lambda_{\mathscr{O}}$-adic class
$\mathbf{z}_{g,\chi}\in\check{\mathcal{S}}_{\rm BK}(g,\chi)$, a generalization
of Theorem 3.2.1 allows one to recover $\mathscr{L}_{v}(g,\chi)$ as the image
of $\mathbf{z}_{g,\chi}$ under a generalized Coleman power series map. This
was first done in a joint work of the author with M.-L. Hsieh [CH18] for
$\chi=\mathds{1}$, and later in [JLZ21] in the level of generality required
for this paper.

To state the result, given a class $\mathbf{z}\in\check{\mathcal{S}}_{\rm
BK}(g,\chi)$, we let $\mathbf{z}(\mathds{1})$ denote the image of $\mathbf{z}$
under the natural projection

$\check{\mathcal{S}}_{\rm BK}(g,\chi)\rightarrow{\rm Sel}_{\rm
BK}(K,V_{g,\chi}).$

Let $\alpha$ be the $p$-adic unit root of
$x^{2}-a_{p}(g)x+\varepsilon_{g}(p)p^{2r-1}$, and let $\varpi\in\mathscr{O}$
be a uniformizer.

######  Theorem 3.2.2.

Let $w\in\\{v,\overline{v}\\}$ be a prime of $K$ above $p$. There exits a
class $\mathbf{z}_{g,\chi}\in\check{\mathcal{S}}_{\rm BK}(g,\chi)$ with

$\mathbf{z}_{g,\chi}(\mathds{1})=\mathcal{E}_{p}(g,\chi)\cdot z_{g,\chi},$

where
$\mathcal{E}_{p}(g,\chi)=\bigl{(}1-\frac{\chi(v)p^{r-1}}{\alpha}\bigr{)}\bigl{(}1-\frac{\chi(\overline{v})p^{r-1}}{\alpha}\bigr{)}$,
and an injective $\Lambda_{\mathscr{O}}^{\rm ur}$-module homomorphism

${\rm
Col}_{\overline{w}}:\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|\overline{w}}\mathrm{H}^{1}_{f}(F_{\eta},T_{g,\chi})\hat{\otimes}_{\mathscr{O}}\Lambda_{\mathscr{O}}^{\rm
ur}\rightarrow\Lambda_{\mathscr{O}}^{\rm ur}$

with finite cokernel for which we have the “explicit reciprocity law”

${\rm Col}_{\overline{w}}\bigl{(}{\rm
loc}_{\overline{w}}(\mathbf{z}_{g,\chi})\bigr{)}=\mathscr{L}_{w}(g,\chi).$

###### Proof.

The construction of $\mathbf{z}_{g,\chi}$ is given in [CH18, §5.2] in the case
$\varepsilon_{g}=1$ and in [JLZ21, Thm. 5.4.1]333Note that this theorem
achieves more, also interpolating $z_{g,\chi}$ along a “weight” variable for
$g$, vastly generalizing the results of [How07, Cas13, Cas20, Ota20] in the
$p$-ordinary case; cf. [Dis22, BL21]. Here we only need the interpolation
result in the anticyclotomic direction. in general. Similarly, the
construction of ${\rm Col}_{\overline{w}}$ and the proof of the explicit
reciprocity law is given in [CH18, §5.3] in the case $\varepsilon_{g}=1$, and
in [JLZ21, §8] in general. ∎

###  3.3. Main conjectures for self-dual pairs

We now state the anticyclotomic Iwasawa main conjectures that will be studied
in this paper.

######  Conjecture 3.3.1 (Iwasawa–Greenberg main conjecture).

Let $w\in\\{v,\overline{v}\\}$ be a prime of $K$ above $p$, and assume that
$\mathscr{L}_{w}(g,\chi)\neq 0$. Then:

  * (i)

${\rm
rank}_{\Lambda_{\mathscr{O}}}\big{(}\check{\mathcal{S}}_{w}(g,\chi)\bigr{)}={\rm
rank}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{w}(g,\chi)\bigr{)}=0$;

  * (ii)

We have

${\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{w}(g,\chi)\bigr{)}=\bigl{(}\mathscr{L}_{w}(g,\chi)^{2}\bigr{)}$

as ideals in $\Lambda_{\mathscr{O}}^{\rm ur}$.

######  Remark 3.3.2.

It follows from Perrin-Riou’s results [PR89] that the characteristic ideal of
$\mathcal{X}_{w}(g,\chi)$ is independent of the choice of $G_{K}$-stable
$\mathscr{O}$-lattice $T_{g,\chi}$ in $V_{g,\chi}$ used in the definition of
$\mathcal{X}_{w}(g,\chi)$ (see [KO20, Prop. 2.9]). Note that this is
consistent with Conjecture 3.3.1(ii), since the periods used in the
construction of $\mathscr{L}_{w}(g,\chi)$ depend only on $K$.

The following is a natural higher weight extension of the Heegner point main
conjecture of [BT20] and [BCST22] (see also [LV19] for an analogous higher
weight extension of the original Heegner point main conjecture by Perrin-Riou
[PR87a]).

######  Conjecture 3.3.3 (Heegner cycle main conjecture).

The following hold:

  * (i)

$\mathbf{z}_{g,\chi}$ is not $\Lambda_{\mathscr{O}}$-torsion;

  * (ii)

${\rm rank}_{\Lambda_{\mathscr{O}}}\bigl{(}\check{\mathcal{S}}_{{\rm
BK}}(g,\chi)\bigr{)}={\rm
rank}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{{\rm
BK}}(g,\chi)\bigr{)}=1$;

  * (iii)

We have

${\rm char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{{\rm BK}}(g,\chi)_{\rm
tors}\bigr{)}={\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\check{\mathcal{S}}_{{\rm
BK}}(g,\chi)/\Lambda_{\mathscr{O}}\cdot\mathbf{z}_{g,\chi}\bigr{)}^{2},$

where the subscript ${\rm tors}$ denotes the maximal
$\Lambda_{\mathscr{O}}$-torsion submodule.

######  Remark 3.3.4.

It follows from Nekovář’s results [Nek06] that there is a
$\Lambda_{\mathscr{O}}$-module pseudo-isomorphism

$\mathcal{X}_{\rm BK}(g,\chi)_{\rm tors}\,\sim\,M\oplus M$

for a torsion $\Lambda_{\mathscr{O}}$-module $M$ with ${\rm
char}_{\Lambda_{\mathscr{O}}}(M)^{\iota}={\rm
char}_{\Lambda_{\mathscr{O}}}(M)$, where $\iota$ denotes the involution on
$\Lambda_{\mathscr{O}}$ given by $\gamma\mapsto\gamma^{-1}$ for
$\gamma\in\Gamma$ (see also [How04, p. 1464]). Thus we see that the equality
of characteristic ideals in Conjecture 3.3.3 can alternatively be written as

${\rm char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{{\rm BK}}(g,\chi)_{\rm
tors}\bigr{)}={\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\check{\mathcal{S}}_{{\rm
BK}}(g,\chi)/\Lambda_{\mathscr{O}}\cdot\mathbf{z}_{g,\chi}\bigr{)}\cdot{\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\check{\mathcal{S}}_{{\rm
BK}}(g,\chi)/\Lambda_{\mathscr{O}}\cdot\mathbf{z}_{g,\chi}\bigr{)}^{\iota}.$

###  3.4. Selmer groups for Hecke characters

In this subsection, the imaginary quadratic field $K$ is only required to
satisfy (spl) for our fixed prime $p>3$.

Let $\xi$ be a Hecke character of $K$ of infinity type
$(a,b)\in\mathbb{Z}^{2}$ with $p$-adic avatar valued in the ring of integers
$\mathscr{O}$ of a finite extension $\Phi/\mathbb{Q}_{p}$. Put

$V_{\xi}=\Phi(\xi),$

where $\Phi(\xi)$ denotes the one-dimensional $\Phi$-vector space on which
$G_{K}$ act via $\xi$.

######  Definition 3.4.1.

Let $w\in\\{v,\overline{v}\\}$ be a prime of $K$ above $p$, and let $F/K$ be a
finite extension. For $\eta$ a prime of $F$ above $p$, put

$\mathrm{H}^{1}_{f}(F_{\eta},V_{\xi}):=\ker\big{\\{}\mathrm{H}^{1}(F_{\eta},V_{\xi})\rightarrow\mathrm{H}^{1}(F_{\eta},V_{\xi}\otimes\mathbf{B}_{\rm
cris})\bigl{\\}}$

and

$\mathrm{H}^{1}_{w}(F_{\eta},V_{\xi}):=\begin{cases}\mathrm{H}^{1}(F_{\eta},V_{\xi})&\textrm{if
$\eta\mid w$,}\\\\[1.99997pt] 0&\textrm{if
$\eta\mid\overline{w}$},\end{cases}$

and for $\eta$ a finite prime of $F$ not dividing $p$, let
$\mathrm{H}^{1}_{\rm ur}(F_{\eta},V_{\xi})={\rm
ker}\\{\mathrm{H}^{1}(F_{\eta},V_{\xi})\rightarrow\mathrm{H}^{1}(I_{\eta},V_{\xi})\\}$
be the unramified subspace.

  * •

The _Bloch–Kato_ Selmer group of $V_{\xi}$ is

${\rm Sel}_{\rm
BK}(F,V_{\xi}):=\ker\biggl{\\{}\mathrm{H}^{1}(F,V_{\xi})\rightarrow\prod_{\eta|p}\frac{\mathrm{H}^{1}(F_{\eta},V_{\xi})}{\mathrm{H}_{f}^{1}(F_{\eta},V_{\xi})}\times\prod_{\eta\nmid
p}\frac{\mathrm{H}^{1}(F_{\eta},V_{\xi})}{\mathrm{H}^{1}_{\rm
ur}(F_{\eta},V_{\xi})}\biggr{\\}}.$

  * •

The _$w$ -Selmer group_ of $V_{\xi}$ is

${\rm
Sel}_{w}(F,V_{\xi}):=\ker\biggl{\\{}\mathrm{H}^{1}(F,V_{\xi})\rightarrow\prod_{\eta|p}\frac{\mathrm{H}^{1}(F_{\eta},V_{\xi})}{\mathrm{H}_{w}^{1}(F_{\eta},V_{\xi})}\times\prod_{\eta\nmid
p}\frac{\mathrm{H}^{1}(F_{\eta},V_{\xi})}{\mathrm{H}^{1}_{\rm
ur}(F_{\eta},V_{\xi})}\biggr{\\}}.$

Let $T_{\xi}$ denote the free rank $1$ $\mathscr{O}$-module on which $G_{K}$
acts via $\xi$, and put $W_{\xi}=V_{\xi}/T_{\xi}$. Similarly as above, for
$\star\in\\{{\rm BK},v,\overline{v}\\}$, we define ${\rm
Sel}_{\star}(F,T_{\xi})$ and ${\rm Sel}_{\star}(F,W_{\xi})$ by propagation via
$0\rightarrow T_{\xi}\rightarrow V_{\xi}\rightarrow W_{\xi}\rightarrow 0$,
define the $\Lambda_{\mathscr{O}}$-adic Selmer groups

$\check{\mathcal{S}}_{\star}(\xi):=\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}{\rm
Sel}_{\star}(F,T_{\xi}),\quad\quad\mathcal{S}_{\star}(\xi):=\mathop{\mathchoice{\varlim@\displaystyle{\rightarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\rightarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\rightarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\rightarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}{\rm Sel}_{\star}(F,W_{\xi}),$

with $F$ running over the finite extensions of $K$ contained in $K_{\infty}$,
and let

$\mathcal{X}_{\star}(\xi):={\rm Hom}_{\rm
cts}(\mathcal{S}_{\star}(\xi),\mathbb{Q}_{p}/\mathbb{Z}_{p})$

be the Pontryagin dual of $\mathcal{S}_{\star}(\xi)$.

######  Lemma 3.4.2.

Suppose $\xi$ has infinity type $(a,b)$. Let $F/K$ be a finite extension and
let $\eta$ be a prime of $F$ above $p$. Then

$\mathrm{H}^{1}_{f}(F_{\eta},V_{\xi})=\begin{cases}\mathrm{H}^{1}(F_{\eta},V_{\xi})&\textrm{if
$\eta\mid v$ and $a>0$, or $\eta\mid\overline{v}$ and $b>0$,}\\\\[1.99997pt]
0&\textrm{else.}\end{cases}$

In particular,

$\mathcal{X}_{\rm BK}(\xi)=\begin{cases}\mathcal{X}_{v}(\xi)&\textrm{if $a>0$
and $b\leq 0$,}\\\\[1.99997pt] \mathcal{X}_{\overline{v}}(\xi)&\textrm{if
$b>0$ and $a\leq 0$.}\end{cases}$

###### Proof.

With the convention that the $p$-adic cyclotomic character $\varepsilon_{\rm
cyc}:G_{\mathbb{Q}}\rightarrow\mathbb{Z}_{p}^{\times}$ has Hodge–Tate weight
$+1$, our convention on infinity types implies that the $p$-adic avatar
$\xi:G_{K}\rightarrow\mathscr{O}$ has Hodge–Tate weight $a$ (resp. $b$) at $v$
(resp. $\overline{v}$). In view of [BK90, Thm. 4.1(ii)], this implies the
result. ∎

###  3.5. Decompositions

When $g$ is the theta series of a Hecke character $\psi$ of $K$, then
$V_{g}\simeq{\rm Ind}_{K}^{\mathbb{Q}}V_{\psi}$ and so

(3.2) $V_{g,\chi}\simeq V_{\psi\chi}(r)\oplus V_{\psi^{\tau}\chi}(r).$

We fix an isomorphism as above, and let $T_{g,\chi}\subset V_{g,\chi}$ be the
$G_{K}$-stable $\mathscr{O}$-lattice with $T_{g,\chi}\simeq
T_{\psi\chi}(r)\oplus T_{\psi^{\tau}\chi}(r)$ under that isomorphism, so that

$W_{g,\chi}:=V_{g,\chi}/T_{g,\chi}\simeq W_{\psi\chi}(r)\oplus
W_{\psi^{\tau}\chi}(r)$

as $G_{K}$-modules.

######  Proposition 3.5.1.

Suppose $g=\theta_{\psi}\in S_{2r}(\Gamma_{1}(N_{g}))$ is the theta series of
a Hecke character $\psi$ of $K$, and $\chi$ is a finite order character with
$\chi\in\Sigma_{\rm cc}(c,\mathfrak{N}_{g},\varepsilon_{g})$. Then there are
$\Lambda_{\mathscr{O}}$-module isomorphisms

$\displaystyle\check{\mathcal{S}}_{v}(g,\chi)$
$\displaystyle\simeq\check{\mathcal{S}}_{v}(\psi\chi\mathbf{N}^{-r})\oplus\check{\mathcal{S}}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r}),$
$\displaystyle\check{\mathcal{S}}_{{\rm BK}}(g,\chi)$
$\displaystyle\simeq\check{\mathcal{S}}_{\overline{v}}(\psi\chi\mathbf{N}^{-r})\oplus\check{\mathcal{S}}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r}),$

and similarly

$\displaystyle\mathcal{X}_{v}(g,\chi)$
$\displaystyle\simeq\mathcal{X}_{v}(\psi\chi\mathbf{N}^{-r})\oplus\mathcal{X}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r}),$
$\displaystyle\mathcal{X}_{{\rm BK}}(g,\chi)$
$\displaystyle\simeq\mathcal{X}_{\overline{v}}(\psi\chi\mathbf{N}^{-r})\oplus\mathcal{X}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r}).$

###### Proof.

The first and third isomorphisms are immediate from (3.2) and the definitions.
On the other hand, note that the character $\psi\chi\varepsilon_{\rm cyc}^{r}$
has Hodge–Tate weight $1-r$ and $r$ above $v$ and $\overline{v}$,
respectively, while $\psi^{\tau}\chi\varepsilon_{\rm cyc}^{r}$ has Hodge–Tate
weight $r$ and $1-r$ above $v$ and $\overline{v}$, respectively. In view of
Lemma 3.4.2, the second and fourth isomorphisms thus follow the decomposition

$\mathrm{H}^{1}_{f}(F_{\eta},V_{g,\chi})=\mathrm{H}^{1}_{f}(F_{\eta},V_{\psi\chi}(r))\oplus\mathrm{H}^{1}_{f}(F_{\eta},V_{\psi^{\tau}\chi}(r))$

induced by (3.2) for any $\eta\mid p$. ∎

######  Remark 3.5.2.

In the situation of Proposition 3.5.1, the local conditions defining the
Selmer groups $\check{\mathcal{S}}_{v}(\psi\chi\mathbf{N}^{-r})$ and
$\mathcal{X}_{v}(\psi\chi\mathbf{N}^{-r})$ at the primes above $p$ are
_reversed_ with respect to those defining the corresponding Bloch–Kato Selmer
groups $\check{\mathcal{S}}_{\rm
BK}(\psi\chi\mathbf{N}^{-r})=\check{\mathcal{S}}_{\overline{v}}(\psi\chi\mathbf{N}^{-r})$
and $\mathcal{X}_{\rm
BK}(\psi\chi\mathbf{N}^{-r})=\mathcal{X}_{\overline{v}}(\psi\chi\mathbf{N}^{-r})$
(see Lemma 3.4.2).

###  3.6. Equivalence between the main conjectures

In the case where $V_{g,\chi}|_{G_{K}}$ is irreducible, one can show that
$\mathscr{L}_{w}(g,\chi)\neq 0$ for both values of $w\in\\{v,\overline{v}\\}$
(see [CH18, Thm. 3.9]), and Conjecture 3.3.3 can be shown to be equivalent to
either version ($v$ or $\overline{v}$) of Conjecture 3.3.1. In the weight $2$
case, the argument for this appears in several places in the literature (see
e.g. [Wan21, §3] or [BCK21, §5]).

In Theorem 4.5.1 below, we shall see that when $g=\theta_{\psi}$ has CM by $K$
(and so $V_{g,\chi}|_{G_{K}}$ is reducible), the vanishing or not of
$\mathscr{L}_{w}(g,\chi)$ _depends_ on the value of the root number

$w(\psi\chi\mathbf{N}^{-r})\in\\{\pm{1}\\}$

giving the sign in the functional equation relating $L(\psi\chi,s)$ and
$L(\psi\chi,2r-s)$. In fact, in this case we shall see that
$\mathscr{L}_{w}(g,\chi)$ vanishes for one value of
$w(\psi\chi\mathbf{N}^{-r})$, and it is nonzero for the other (see Remark
4.5.2). Thus in this section we carefully explain the relation between
Conjecture 3.3.3 and Conjecture 3.3.1 in general.

######  Lemma 3.6.1.

For any prime $w$ of $K$ above $p$, the quotient

$\mathcal{H}_{w}:=\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|w}\frac{\mathrm{H}^{1}(F_{\eta},T_{g,\chi})}{\mathrm{H}^{1}_{f}(F_{\eta},T_{g,\chi})}$

is $\Lambda_{\mathscr{O}}$-torsion-free.

###### Proof.

From Lemma 3.4.2 we immediately see that

$\mathcal{H}_{v}\simeq\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|v}\mathrm{H}^{1}(F_{\eta},T_{\psi\chi}(r)),\quad\quad\mathcal{H}_{\overline{v}}\simeq\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|\overline{v}}\mathrm{H}^{1}(F_{\eta},T_{\psi^{\tau}\chi}(r)).$

The result thus follows from [PR92, Prop. 2.1.6] as in [Arn07, Lem. 2.8].
(Note that [Arn07, Lem. 2.5] also applies in our case, since neither of the
characters $\psi\chi\mathbf{N}^{-r}$ and $\psi^{\tau}\chi\mathbf{N}^{-r}$ has
infinity type of the form $(a,b)$ with $a=-b$.) ∎

######  Proposition 3.6.2.

Suppose $\mathbf{z}_{g,\chi}$ is non-torsion, and the localization map

(3.3) ${\rm loc}_{\overline{v}}:\check{\mathcal{S}}_{\rm
BK}(g,\chi)\rightarrow\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|\overline{v}}\mathrm{H}^{1}_{f}(F_{\eta},T_{g,\chi})$

is nonzero. Then the following are equivalent:

  * $(i^{+})$

${\rm
rank}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{{v}}(g,\chi)\bigr{)}={\rm
rank}_{\Lambda_{\mathscr{O}}}\bigl{(}\check{\mathcal{S}}_{v}(g,\chi)\bigr{)}=0$;

  * $(i^{-})$

${\rm rank}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{\rm
BK}(g,\chi)\bigr{)}={\rm
rank}_{\Lambda_{\mathscr{O}}}\bigl{(}\check{\mathcal{S}}_{\rm
BK}(g,\chi)\bigr{)}=1$,

and in that case, the following are equivalent:

  * $(ii^{+})$

${\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{v}(g,\chi)\bigr{)}\subset\bigl{(}\mathscr{L}_{v}(g,\chi)^{2}\bigr{)}$
in $\Lambda_{\mathscr{O}}^{\rm ur}$;

  * $(ii^{-})$

${\rm char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{\rm BK}(g,\chi)_{\rm
tors}\bigr{)}\subset{\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\check{\mathcal{S}}_{\rm
BK}(g,\chi)/{\Lambda_{\mathscr{O}}}\cdot\mathbf{z}_{g,\chi}\bigr{)}^{2}$ in
${\Lambda_{\mathscr{O}}}$,

and the same holds true for the opposite divisibilities. In particular,
Conjecture 3.3.1 for $w=v$ and Conjecture 3.3.3 are equivalent.

###### Proof.

The argument is similar to that in [BCST22, Thm. 5.2]. For
$\circ,\bullet\in\\{{\rm str},f,{\rm rel}\\}$, let
$\check{\mathcal{S}}_{\circ,\bullet}(g,\chi)$ and
$\mathcal{X}_{\circ,\bullet}(g,\chi)$ denote the Selmer groups defined as in
$\S\ref{subsec:Sel-sd}$ by with the local conditions $\circ$ and $\bullet$ at
the primes above $v$ and $\overline{v}$, respectively. Thus, for instance,
$\mathcal{X}_{{\rm rel},{\rm str}}(g,\chi)$ and $\mathcal{X}_{f,f}(g,\chi)$
are the previously defined $\mathcal{X}_{{v}}(g,\chi)$ and $\mathcal{X}_{{\rm
BK}}(g,\chi)$, respectively.

By global duality, ${\rm coker}({\rm loc}_{\overline{v}})$ is the same444Using
the fact that the representation $V_{g,\chi}$ is conjugate self-dual, so the
Selmer group dual to $\check{\mathcal{S}}_{f,{\rm str}}(g,\chi)={\rm ker}({\rm
loc}_{\overline{v}})$ is identified with $\mathcal{X}_{{\rm rel},f}(g,\chi)$.
as the kernel of the projection $\mathcal{X}_{{\rm
rel},f}(g,\chi)\rightarrow\mathcal{X}_{\rm BK}(g,\chi)$. Similarly, by global
duality the cokernel of the localization map

${\rm loc}_{\overline{v}}^{\prime}:\check{\mathcal{S}}_{{\rm
rel},f}(g,\chi)\rightarrow\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|\overline{v}}\mathrm{H}^{1}_{f}(F_{\eta},T_{g,\chi})$

is identified with the kernel of the projection
$\mathcal{X}_{v}(g,\chi)\rightarrow\mathcal{X}_{{f},{\rm str}}(g,\chi)$. Since
the same argument as in [Cas17, Lem. 2.3] shows that ${\rm
rank}_{\Lambda_{\mathscr{O}}}(\mathcal{X}_{{\rm rel},f}(g,\chi))=1+{\rm
rank}_{\Lambda_{\mathscr{O}}}(\mathcal{X}_{f,{\rm str}}(g,\chi))$, under the
assumption on ${\rm loc}_{\overline{v}}$ in the statement we thus conclude
that

${\rm rank}_{\Lambda_{\mathscr{O}}}(\mathcal{X}_{\rm BK}(g,\chi))=1+{\rm
rank}_{\Lambda_{\mathscr{O}}}(\mathcal{X}_{v}(g,\chi)),$

whence the equivalence $(i^{+})\Longleftrightarrow(i^{-})$.

Now assume that $(i^{+})$ (and therefore also $(i^{-})$) holds. As above, from
global duality we have the short exact sequences

(3.4) $\displaystyle 0\rightarrow{\rm coker}({\rm loc}_{\overline{v}})$
$\displaystyle\rightarrow\mathcal{X}_{{\rm
rel},f}(g,\chi)\rightarrow\mathcal{X}_{\rm BK}(g,\chi)\rightarrow 0,$
$\displaystyle 0\rightarrow{\rm coker}({\rm loc}_{\overline{v}}^{\prime})$
$\displaystyle\rightarrow\mathcal{X}_{v}(g,\chi)\rightarrow\mathcal{X}_{f,{\rm
str}}(g,\chi)\rightarrow 0.$

Since the target of ${\rm loc}_{\overline{v}}^{\prime}$ has
$\Lambda_{\mathscr{O}}$-rank $1$, and ${\rm
rank}_{\Lambda_{\mathscr{O}}}(\check{\mathcal{S}}_{v}(g,\chi))=0$, the
quotient $\check{\mathcal{S}}_{{\rm rel},f}(g,\chi)/\check{\mathcal{S}}_{\rm
BK}(g,\chi)$ is $\Lambda_{\mathscr{O}}$-torsion; since this quotient injects
via ${\rm loc}_{v}$ into $\mathcal{H}_{v}$, which is
$\Lambda_{\mathscr{O}}$-torsion-free by Lemma 3.6.1, we conclude that
$\check{\mathcal{S}}_{{\rm rel},f}(g,\chi)=\check{\mathcal{S}}_{\rm
BK}(g,\chi)$, so in particular ${\rm loc}_{\overline{v}}={\rm
loc}_{\overline{v}}^{\prime}$. From (3.4) we thus obtain

(3.5) $\displaystyle{\rm
char}_{\Lambda_{\mathscr{O}}}(\mathcal{X}_{v}(g,\chi))$ $\displaystyle={\rm
char}_{\Lambda_{\mathscr{O}}}(\mathcal{X}_{f,{\rm str}}(g,\chi))\cdot{\rm
char}_{\Lambda_{\mathscr{O}}}({\rm coker}({\rm loc}_{\overline{v}}))$
$\displaystyle={\rm char}_{\Lambda_{\mathscr{O}}}(\mathcal{X}_{{\rm
rel},f}(g,\chi)_{\rm tors})\cdot{\rm char}_{\Lambda_{\mathscr{O}}}({\rm
coker}({\rm loc}_{\overline{v}}))$ $\displaystyle={\rm
char}_{\Lambda_{\mathscr{O}}}(\mathcal{X}_{\rm BK}(g,\chi)_{\rm
tors})\cdot{\rm char}_{\Lambda_{\mathscr{O}}}({\rm coker}({\rm
loc}_{\overline{v}}))^{2},$

using a variant of [Cas17, Lem. 2.3(3)] for the second equality. On the other
hand, since
$\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F}\mathrm{H}^{1}(F,T_{g,\chi})$
is $\Lambda_{\mathscr{O}}$-torsion–free, the map ${\rm loc}_{\overline{v}}$
defines the short exact sequence

$0\rightarrow\check{\mathcal{S}}_{\rm
BK}(g,\chi)/\Lambda_{\mathscr{O}}\cdot\mathbf{z}_{g,\chi}\rightarrow\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\oplus_{\eta|\overline{v}}\mathrm{H}^{1}_{f}(F_{\eta},T_{g,\chi})/\Lambda_{\mathscr{O}}\cdot{\rm
loc}_{\overline{v}}(\mathbf{z}_{g,\chi})\rightarrow{\rm coker}({\rm
loc}_{\overline{v}})\rightarrow 0,$

which together with Theorem 3.2.2 yields the equality

(3.6) ${\rm char}_{\Lambda_{\mathscr{O}}}\bigl{(}\check{\mathcal{S}}_{\rm
BK}(g,\chi)/\Lambda_{\mathscr{O}}\cdot\mathbf{z}_{g,\chi}\bigr{)}\cdot{\rm
char}_{\Lambda_{\mathscr{O}}}({\rm coker}({\rm
loc}_{\overline{v}}))=\bigl{(}\mathscr{L}_{v}(g,\chi)\bigr{)}$

in $\Lambda_{\mathscr{O}}^{\rm ur}$. Combining (3.5) and (3.6) we deduce

${\rm char}_{\Lambda_{\mathscr{O}}}(\mathcal{X}_{v}(g,\chi))\cdot{\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\check{\mathcal{S}}_{\rm
BK}(g,\chi)/\Lambda_{\mathscr{O}}\cdot\mathbf{z}_{g,\chi}\bigr{)}^{2}={\rm
char}_{\Lambda_{\mathscr{O}}}(\mathcal{X}_{v}(g,\chi)_{\rm
tors})\cdot\bigl{(}\mathscr{L}_{v}(g,\chi)^{2}\bigr{)},$

which readily yields the equivalence $(ii^{+})\Longleftrightarrow(ii^{-})$. ∎

######  Remark 3.6.3.

Theorem 3.2.2 also relates the image of $\mathbf{z}_{g,\chi}$ under the
localization map

${\rm loc}_{v}:\check{\mathcal{S}}_{\rm
BK}(g,\chi)\rightarrow\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|v}\mathrm{H}^{1}_{f}(F_{\eta},T_{g,\chi})$

to the $p$-adic $L$-function $\mathscr{L}_{\overline{v}}(g,\chi)$ of Theorem
2.1.3. The same argument as in Proposition 3.6.2 shows that, assuming the non-
vanishing of both $\mathbf{z}_{g,\chi}$ and ${\rm
loc}_{v}(\check{\mathcal{S}}_{\rm BK}(g,\chi))$, the following are equivalent:

  * (i)

Conjecture 3.3.1 for $w=\overline{v}$;

  * (ii)

Conjecture 3.3.3.

##  4\. Proof of the Iwasawa main conjectures

In this section we prove Conjecture 3.3.1 and Conjecture 3.3.3 for self-dual
pairs $(g,\chi)$ in the case where $g=\theta_{\psi}$ is the theta series of a
Hecke character $\psi$. In the course of the proof we shall establish the non-
triviality of $\mathbf{z}_{g,\chi}$ in this case (which in weights $2r>2$
appears to be new).

###  4.1. Statement

The main result of this section is the following, where $K$ is an imaginary
quadratic field satisfying (spl) for a prime $p>3$.

For a self-dual character $\phi$, we denote by $w(\phi)\in\\{\pm{1}\\}$ the
sign in the functional equation

$L(\phi,s)=w(\phi)L(\phi,-s).$

######  Theorem 4.1.1.

Let $\psi$ be a Hecke character of infinity type $(1-2r,0)$ for some $r\geq 1$
and conductor a cyclic ideal $\mathfrak{f}_{\psi}$ of norm prime to $pD_{K}$.
Set

$g=\theta_{\psi},\quad
N_{g}=D_{K}\cdot\mathbf{N}(\mathfrak{f}_{\psi}),\quad\mathfrak{N}_{g}=\mathfrak{d}_{K}\cdot\mathfrak{f}_{\psi}.$

Let $c$ be a positive integer prime to $pN_{g}$, let $\chi$ be a finite order
character in $\Sigma_{\rm cc}(c,\mathfrak{N}_{g},\varepsilon_{g})$, and
suppose $(g,\chi)$ is defined over $\mathscr{O}$.

  * (i)

If $w(\psi\chi\mathbf{N}^{-r})=-1$, then $\check{\mathcal{S}}_{v}(g,\chi)=0$
and $\mathcal{X}_{v}(g,\chi)$ is $\Lambda_{\mathscr{O}}$-torsion, with

${\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{v}(g,\chi)\bigr{)}={\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathscr{L}_{v}(g,\chi)^{2}\bigr{)}$

in $\Lambda_{\mathscr{O}}^{\rm ur}$.

  * (ii)

If $w(\psi\chi\mathbf{N}^{-r})=+1$, the same results hold with $v$ replaced by
$\overline{v}$.

Hence Conjecture 3.3.1 holds for $(g,\chi)$.

######  Remark 4.1.2.

The above assumption on $\mathfrak{f}_{\psi}$ implies that
$\mathcal{O}_{K}/\mathfrak{N}_{g}\simeq\mathbb{Z}/N_{g}\mathbb{Z}$, i.e. $K$
satisfies (Heeg) relative to $N_{g}$. Moreover, it is easy to see that in the
setting of Theorem 4.1.1, the character $\psi\chi\mathbf{N}^{-r}$ (and hence
also $\psi^{\tau}\chi\mathbf{N}^{-r}$) is self-dual (see [BDP12, Lem. 3.16]).

By Proposition 2.3.1 and Proposition 3.5.1, the proof of Theorem 4.1.1 is
reduced to the study of the relation between the $\Lambda_{\mathscr{O}}$-adic
Selmer groups attached to the self-dual characters $\psi\chi\mathbf{N}^{-r}$
and $\psi^{\tau}\chi\mathbf{N}^{-r}$ and the $p$-adic $L$-functions
$\mathscr{L}_{w}(\psi\chi\mathbf{N}^{-r})$ and
$\mathscr{L}_{w}(\psi^{\tau}\chi\mathbf{N}^{-r})$, respectively. The main
difficulty arises from the fact that, as noted in Remark 3.5.2, the Selmer
group $\mathcal{X}_{v}(\psi\chi\mathbf{N}^{-r})$ is different from the
Bloch–Kato Selmer group over $K_{\infty}/K$, and so a relation between the
characteristic ideal of $\mathcal{X}_{v}(\psi\chi\mathbf{N}^{-r})$ and the
$p$-adic $l$-function $\mathscr{L}_{v}(\psi\chi\mathbf{N}^{-r})$ is not
immediate from the main conjecture in this setting.

The proof of Theorem 4.1.1 is concluded in $\S\ref{subsec:IMC-proof-end}$,
where we also deduce from it a similar result on Conjecture 3.3.3.

###  4.2. Explicit reciprocity law

For an ideal $\mathfrak{c}\subset\mathcal{O}_{K}$ prime to $p$ and a non-
trivial ideal $\mathfrak{a}$ prime to $6\mathfrak{c}p$, let
$c_{\mathfrak{a}}(K(\mathfrak{c}p^{k}))\in\mathrm{H}^{1}(K(\mathfrak{c}p^{k}),\mathbb{Z}_{p}(1))$
be the elliptic unit denoted
$\vartheta_{\mathfrak{a}}(\mathfrak{\mathfrak{c}}p^{k})$ in [AH06, §2.3] and
${}_{\mathfrak{a}}z_{\mathfrak{c}p^{k}}$ in [Kat04, §15.5]; these are norm-
compatible at $k$ varies.

As a piece of notation, for an infinite abelian extension $K^{\prime}/K$ and a
$\mathbb{Z}_{p}$-module $T$ with a continuous linear $G_{K}$-action, put

$\mathrm{H}^{1}_{\rm
Iw}(K^{\prime},T):=\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{K^{\prime\prime}\subset
K^{\prime}}\mathrm{H}^{1}(K^{\prime\prime},T),$

with $K^{\prime\prime}$ running over the finite extensions of $K$ contained in
$K^{\prime}$, and limit with respect to corestriction. For a character
$\phi:{\rm Gal}(K(\mathfrak{c}p^{\infty})/K)\rightarrow\mathscr{O}^{\times}$,
we put $\phi^{*}=\varepsilon_{\rm cyc}\phi^{-1}$ and let
$c_{\mathfrak{a}}(K(\mathfrak{c}p^{\infty}))^{\phi^{-1}}$ denote the image of
$\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{k}c_{\mathfrak{a}}(K(\mathfrak{c}p^{k}))$
under the twisting homomorphism

$\mathrm{H}^{1}_{\rm
Iw}(K(\mathfrak{c}p^{\infty}),\mathbb{Z}_{p}(1))\xrightarrow{\otimes\phi^{-1}}\mathrm{H}^{1}_{\rm
Iw}(K(\mathfrak{c}p^{\infty}),T_{\phi^{*}}).$

For any subextension $L$ of $K(\mathfrak{c}p^{\infty})$ (not necessarily
finite over $K$), we define $c_{\mathfrak{a}}(L)^{\phi^{-1}}$ to be the image
of $c_{\mathfrak{a}}(K(\mathfrak{c}p^{\infty}))^{\phi^{-1}}$ under the
corestriction map $\mathrm{H}^{1}_{\rm
Iw}(K(\mathfrak{c}p^{\infty}),T_{\phi^{*}})\rightarrow\mathrm{H}^{1}_{\rm
Iw}(L,T_{\phi^{*}})$.

The next result is a refinement of Yager’s work [Yag82] (specialized to the
anticyclotomic setting) building on Kato’s explicit reciprocity law [Kat04].

######  Theorem 4.2.1.

Let $w\in\\{v,\overline{v}\\}$ be a prime of $K$ above $p$, and let $\phi$ be
a self-dual Hecke character of $K$ with values in $\mathscr{O}$. Then there is
an injective $\Lambda_{\mathscr{O}}^{\rm ur}$-module homomorphism

${\rm
Col}_{w}:\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|w}\mathrm{H}^{1}(F_{\eta},T_{\phi^{\tau}})\hat{\otimes}_{\mathbb{Z}_{p}}\mathbb{Z}_{p}^{\rm
ur}\rightarrow\Lambda_{\mathscr{O}}^{\rm ur}$

with finite cokernel, and a “twisted elliptic unit”
$\mathbf{c}_{\phi^{\tau}}\in\mathrm{H}^{1}_{\rm
Iw}(K_{\infty},T_{\phi^{\tau}})$, such that

${\rm Col}_{w}\bigl{(}{\rm
loc}_{w}(\mathbf{c}_{\phi^{\tau}})\bigr{)}=\mathscr{L}_{w}(\phi).$

###### Proof.

We first explain the construction of the map ${\rm Col}_{w}$ for $w=v$. For
any field $F\subset\overline{\mathbb{Q}}$ over $K$, let $F_{v}$ denote the
completion of $F$ at the prime above $p$ induced by $\imath_{p}$. Then
$H_{p^{\infty},v}$ contains $K_{\infty,v}$, and it follows by local Class
Field Theory that $H_{p^{\infty},v}$ is obtained by adjoining to $H_{v}$ the
torsion points of a height $1$ Lubin–Tate formal group relative to the
extension $H_{v}/K_{v}$ (see [Shn16, Prop. 39]). Thus, as explained in [CH22,
§3], from the Perrin-Riou big exponential map for $H_{p^{\infty}}/H_{v}$ (see
[PR94, Kob23]) we have a $\mathbb{Z}_{p}[\\![\Gamma_{v}]\\!]$-linear map

${\rm
Col}_{v}:\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\mathrm{H}^{1}(F_{v},T_{\phi^{\tau}})\hat{\otimes}_{\mathbb{Z}_{p}}\mathbb{Z}_{p}^{\rm
ur}\rightarrow\mathscr{O}[\\![\Gamma_{v}]\\!]\hat{\otimes}_{\mathbb{Z}_{p}}\mathbb{Z}_{p}^{\rm
ur}$

interpolating the Bloch–Kato logarithm and dual exponential maps for varying
specializations (see [CH22, Thm. 3.4]) which gives the map ${\rm Col}_{v}$ in
the statement after tensoring with $\mathbb{Z}_{p}[\\![\Gamma]\\!]$ over
$\mathbb{Z}_{p}[\\![\Gamma_{v}]\\!]$. The maps ${\rm Col}_{\overline{v}}$ is
constructed in the same manner, replacing $\imath_{p}$ by
$\imath_{p}\circ{\tau}$, where ${\tau}\in G_{\mathbb{Q}}$ is the complex
conjugation induced by $\imath_{\infty}$; the claim that both maps ${\rm
Col}_{w}$ are injective with finite cokernel follows form the theory of
Coleman power series as in [Kat04, §17.10].

On the other hand, letting $\mathfrak{c}$ be the prime-to-$p$ part of the
conductor of $\phi$, we can find $\mathfrak{a}$ coprime to $6\mathfrak{c}p$
with $\sigma_{\mathfrak{a}}-\phi^{\tau}(\sigma_{\mathfrak{a}})$ invertible in
$\Lambda_{\mathscr{O}}^{\rm ur}$ (cf. [dS87, p. 77]), and setting

$\mathbf{c}_{\phi^{\tau}}:=(\sigma_{\mathfrak{a}}-\phi^{\tau}(\sigma_{\mathfrak{a}}))^{-1}\cdot
c_{\mathfrak{a}}(K_{\infty})^{\phi^{-1}}$

(which is independent of $\mathfrak{a}$ by [Kat04, (15.4.4] and belongs to
$\mathrm{H}^{1}_{\rm Iw}(K_{\infty},T_{\phi^{*}})=\mathrm{H}^{1}_{\rm
Iw}(K_{\infty},T_{\phi^{\tau}})$ by the self-duality of $\phi$), the last
assertion follows directly from the interpolations properties of
$\mathscr{L}_{w}(\phi)$ and ${\rm Col}_{w}$ together with Kato’s explicit
reciprocity law [Kat04, Prop. 15.9]. ∎

######  Remark 4.2.2.

Specializing Yager’s result [Yag82] to the anticyclotomic line as in [AH06,
Prop. 2.3.4] and [Arn07, Prop. 2.6] yields an analogue of Theorem 4.2.1 under
the assumption that $p\nmid[K(\mathfrak{c}):K]$ and
$\phi|_{G_{K_{w}}}\not\equiv 1\pmod{\varpi}$ (i.e. $\phi$ is “non-anomalous”
at $w$). Indeed, following the argument in _loc. cit._ , by local Tate duality
one can show that under these additional hypotheses the corestriction map

$\biggl{(}\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K(\mathfrak{c}p^{\infty})}\prod_{\eta|w}\mathrm{H}^{1}(F_{\eta},T_{\phi^{*}})\biggr{)}\otimes_{\Lambda_{\mathscr{O}}(K(\mathfrak{c}p^{\infty}))}\Lambda_{\mathscr{O}}\rightarrow\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|w}\mathrm{H}^{1}(F_{\eta},T_{\phi^{*}})$

is an isomorphism.

###  4.3. Main conjectures for characters

The $2$-variable main conjecture for $K$ was proved by Rubin [Rub91] under
mild hypotheses on the prime $p$, and by Johnson-Leung–Kings [JLK11] in
general.

The problem of deducing from the $2$-variable main conjecture a proof of the
anticyclotomic main conjecture for self-dual Hecke characters $\phi$ (which is
particularly subtle in the case of root number $w(\phi)=-1$), was first
studied in detail by Agboola–Howard [AH06] in the case of CM elliptic curves
$E/\mathbb{Q}$, and by Arnold [Arn07] for general CM forms.

In this section, we extend their methods to deduce the results we need on the
relation between the Selmer groups $\mathcal{X}_{v}(\phi)$ and the $p$-adic
$L$-functions $\mathscr{L}_{v}(\phi)$.

####  4.3.1. The case of $\lambda=\psi\chi$

######  Theorem 4.3.1.

Let $\lambda$ be a Hecke character of infinity type $(1-2r,0)$ for some $r\geq
1$ with $\varepsilon_{\lambda}=\eta_{K}$, conductor $\mathfrak{c}$ prime to
$p$. Suppose

$w(\lambda\mathbf{N}^{-r})=-1.$

Then $\check{\mathcal{S}}_{v}(\lambda\mathbf{N}^{-r})=0$ and
$\mathcal{X}_{v}(\lambda\mathbf{N}^{-r})$ is $\Lambda_{\mathscr{O}}$-torsion,
with

${\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{v}(\lambda\mathbf{N}^{-r})\bigr{)}=\bigl{(}\mathscr{L}_{v}(\lambda\mathbf{N}^{-r})\bigr{)}$

as ideals in $\Lambda^{\rm ur}$.

######  Remark 4.3.2.

Since $\mathcal{X}_{v}(\lambda\mathbf{N}^{-r})$ is not the same as
$\mathcal{X}_{{\rm BK}}(\lambda\mathbf{N}^{-r})$ (see Remark 3.5.2; in fact,
we shall see that their $\Lambda_{\mathscr{O}}$-ranks are always different),
the result of Theorem 4.3.1 is not covered by the main results of [AH06] and
[Arn07], which only concern the Bloch–Kato Selmer groups.

By anticyclotomic descent, the proof of Theorem 4.3.1 will be obtained from a
twisted variant of the results by Rubin [Rub91] and Johnson-Leung–Kings
[JLK11] on the $2$-variable Iwasawa main conjecture for $K$. The starting
point is the following key consequence of Greenberg’s nonvanishing results
[Gre85].

######  Proposition 4.3.3.

Let $w(\lambda\mathbf{N}^{-r})\in\\{\pm{1}\\}$ be the root number of
$\lambda\mathbf{N}^{-r}$. Then

$\displaystyle\mathscr{L}_{v}(\lambda\mathbf{N}^{-r})\neq 0\quad$
$\displaystyle\Longleftrightarrow\quad w(\lambda\mathbf{N}^{-r})=-1.$

###### Proof.

We adapt the argument in the proof of [Arn07, Prop. 2.3]. Put
$\phi_{0}=\overline{\lambda}/\lambda$, and take $m>0$ (to be possibly enlarged
later) large enough so that $\phi:=\phi_{0}^{m}$ factors through $\Gamma$ and
has trivial conductor. Then for $n\gg 0$, the character
$\lambda\mathbf{N}^{-r}\phi^{n}$ is in the range of interpolation of
$\mathscr{L}_{v,\mathfrak{c}}$ (indeed, its infinity type
$(1-r+mn(2r-1),r+mn(1-2r))$) and so from Theorem 2.2.1 and the functional
equation for Hecke $L$-functions555See e.g. [dS87, p. 37], whose conventions
on infinity type are _opposite_ to ours. we immediately obtain

$\mathscr{L}_{v}(\lambda\mathbf{N}^{-r})(\phi^{n})\doteq
L(\lambda^{2mn-1},c),$

where $c=mn(2r-1)-r+1$ is the center of the functional equation. Applying
Weil’s formula for root numbers as stated in [AH06, Prop. 2.1.6], arguing as
in [Arn07, p. 51] we see that (after possibly replacing $m$ by $2m$)
$w(\lambda^{2mn-1})=-w(\lambda\mathbf{N}^{-r})$, and the result follows from
[Gre85, Thm. 1]. ∎

######  Proposition 4.3.4.

If $w(\lambda\mathbf{N}^{-r})=-1$, then
$\mathcal{X}_{v}(\lambda\mathbf{N}^{-r})$ is $\Lambda_{\mathscr{O}}$-torsion.

###### Proof.

By Theorem 4.2.1 (with $\phi=\lambda\mathbf{N}^{-r}$) and Proposition 4.3.3,
if $w(\lambda\mathbf{N}^{-r})=-1$ then the map

${\rm loc}_{v}:\check{\mathcal{S}}_{\rm
rel}(\lambda^{\tau}\mathbf{N}^{-r})\rightarrow\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|v}\mathrm{H}^{1}(F_{\eta},T_{\lambda^{\tau}}(r))$

is nonzero666Note that the implicit inclusion
$\mathbf{c}_{\lambda^{\tau}\mathbf{N}^{-r}}\in\check{\mathcal{S}}_{\rm
rel}(\lambda^{\tau}\mathbf{N}^{-r})$ follows from [AH06, Lem. 2.4.2].. On the
other hand, by [Arn07, Thm. 2.14] (an application of the Euler system
machinery, [Rub00, Thm. 2.3.3]), the $\Lambda_{\mathscr{O}}$-module
$\mathcal{X}_{\rm str}(\lambda\mathbf{N}^{-r})$ is torsion. By the global
duality exact sequence

$\check{\mathcal{S}}_{\rm
rel}(\lambda^{\tau}\mathbf{N}^{-r})\rightarrow\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|v}\mathrm{H}^{1}(F_{\eta},T_{\lambda^{\tau}}(r))\rightarrow\mathcal{X}_{v}(\lambda\mathbf{N}^{-r})\rightarrow\mathcal{X}_{\rm
str}(\lambda\mathbf{N}^{-r})\rightarrow 0,$

this yields the result. ∎

Together with Proposition 4.3.4, the next result completes the proof of
Theorem 4.3.1.

######  Proposition 4.3.5.

Suppose $w(\lambda\mathbf{N}^{-r})=-1$. Then

${\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{v}(\lambda\mathbf{N}^{-r})\bigr{)}=\bigl{(}\mathscr{L}_{v}(\lambda\mathbf{N}^{-r})\bigr{)}$

as ideals in $\Lambda_{\mathscr{O}}^{\rm ur}$.

###### Proof.

Denote by $\mathcal{C}_{\lambda^{\tau}\mathbf{N}^{-r}}(K_{\infty})$ the
$\Lambda_{\mathscr{O}}$-submodule of $\check{\mathcal{S}}_{\rm
rel}(\lambda^{\tau}\mathbf{N}^{-r})$ generated by
$\mathbf{c}_{\lambda^{{\tau}}\mathbf{N}^{-r}}$. By Theorem 4.2.1 (with
$\phi=\lambda\mathbf{N}^{-r}$) and Proposition 4.3.3,
$\mathcal{C}_{\lambda^{\tau}\mathbf{N}^{-r}}(K_{\infty})$ is not
$\Lambda_{\mathscr{O}}$-torsion, and by Theorem 2.14 in [Arn07] we know that
$\check{\mathcal{S}}_{\rm rel}(\lambda^{\tau}\mathbf{N}^{-r})$ is torsion-free
of $\Lambda_{\mathscr{O}}$-rank $1$, $\mathcal{X}_{\rm
str}(\lambda\mathbf{N}^{-r})$ is $\Lambda_{\mathscr{O}}$-torsion, and we have
the divisibility

${\rm char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{\rm
str}(\lambda\mathbf{N}^{-r})\bigr{)}\supset{\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\check{\mathcal{S}}_{\rm
rel}(\lambda^{\tau}\mathbf{N}^{-r})/\mathcal{C}_{\lambda^{\tau}\mathbf{N}^{-r}}(K_{\infty})\bigr{)}.$

Note that this divisibility is independent of the value of
$w(\lambda\mathbf{N}^{-r})$, and in _loc. cit._ the result is stated for
$T_{\lambda}(r)$ and $W_{\lambda^{\tau}}(r)$ in our notations (corresponding
to $T$ and $W^{*}$ in _loc. cit._), rather than for our
$T_{\lambda^{\tau}}(r)$ and $W_{\lambda}(r)$ as above.

On the other hand, if $w(\lambda\mathbf{N}^{-r})=-1$, in view of Proposition
4.3.4 the arguments in [Arn07, §3]777Using [JLK11, Thm. 5.2], which proves an
extension of Rubin’s result on the $2$-variable Iwasawa main conjecture for
$K$ used in [Arn07, Thm. 3.2] under weaker hypotheses on $p$. leading to the
proof of Proposition 3.8 in _loc. cit._ apply _verbatim_ with $T_{\lambda}(r)$
and $W_{\lambda^{\tau}}(r)$ replaced by $T_{\lambda^{\tau}}(r)$ and
$W_{\lambda}(r)$, hence yielding a proof of the converse divisibility, and so

(4.1) ${\rm char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{\rm
str}(\lambda\mathbf{N}^{-r})\bigr{)}={\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\check{\mathcal{S}}_{\rm
rel}(\lambda^{\tau}\mathbf{N}^{-r})/\mathcal{C}_{\lambda^{\tau}\mathbf{N}^{-r}}(K_{\infty})\bigr{)}.$

Now, from Poitou–Tate duality we have the exact sequence

(4.2)
$0\rightarrow\check{\mathcal{S}}_{\overline{v}}(\lambda^{\tau}\mathbf{N}^{-r})\rightarrow\check{\mathcal{S}}_{\rm
rel}(\lambda^{\tau}\mathbf{N}^{-r})\xrightarrow{{\rm
loc}_{v}}\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|v}\mathrm{H}^{1}(F_{\eta},T_{\lambda^{\tau}}(r))\rightarrow\mathcal{X}_{v}(\lambda\mathbf{N}^{-r})\rightarrow\mathcal{X}_{\rm
str}(\lambda\mathbf{N}^{-r})\rightarrow 0,$

which from the above it implies that
$\check{\mathcal{S}}_{\overline{v}}(\lambda^{\tau}\mathbf{N}^{-r})$ is
$\Lambda_{\mathscr{O}}$-torsion, and so being also
$\Lambda_{\mathscr{O}}$-torsion-free ,it vanishes. Thus from (4.2) we obtain
the exact sequence

$\displaystyle 0\rightarrow\check{\mathcal{S}}_{\rm
rel}(\lambda^{\tau}\mathbf{N}^{-r})/\mathcal{C}_{\lambda^{\tau}\mathbf{N}^{-r}}(K_{\infty})$
$\displaystyle\xrightarrow{{\rm
loc}_{v}}\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|v}\mathrm{H}^{1}(F_{\eta},T_{\lambda^{\tau}}(r))/{\rm
loc}_{v}(\mathcal{C}_{\lambda^{\tau}\mathbf{N}^{-r}}(K_{\infty}))$
$\displaystyle\quad\rightarrow\mathcal{X}_{v}(\lambda\mathbf{N}^{-r})\rightarrow\mathcal{X}_{\rm
str}(\lambda\mathbf{N}^{-r})\rightarrow 0.$

Since by Theorem 4.2.1 the map ${\rm Col}_{v}$ defines a
$\Lambda_{\mathscr{O}}^{\rm ur}$-module pseudo-isomorphism

$\biggl{(}\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{F\subset
K_{\infty}}\prod_{\eta|v}\mathrm{H}^{1}(F_{\eta},T_{\lambda^{\tau}}(r))/{\rm
loc}_{v}(\mathcal{C}_{\lambda^{\tau}\mathbf{N}^{-r}}(K_{\infty}))\biggr{)}\hat{\otimes}_{\mathscr{O}}\mathbb{Z}_{p}^{\rm
ur}\rightarrow\Lambda_{\mathscr{O}}^{\rm
ur}/(\mathscr{L}_{v}(\lambda\mathbf{N}^{-r}))$

together with (4.3.1) this concludes the proof. ∎

####  4.3.2. The case of $\psi^{\tau}\chi$

######  Theorem 4.3.6.

Let $\psi$ be a Hecke character of infinity type $(1-2r,0)$ for some $r\geq 1$
and let $\chi$ be a finite order character such that
$\psi^{\tau}\chi\mathbf{N}^{-r}$ is self-dual with root number

$w(\psi^{\tau}\chi\mathbf{N}^{-r})=+1$

and conductor prime to $p$. Then
$\check{\mathcal{S}}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})=0$ and
$\mathcal{X}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})$ is
$\Lambda_{\mathscr{O}}$-torsion, with

${\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})\bigr{)}=\bigl{(}\mathscr{L}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})\bigr{)}.$

as ideals in $\Lambda_{\mathscr{O}}^{\rm ur}$.

###### Proof.

We being by noting that $L(\psi^{\tau}\chi,s)=L(\psi\chi^{\tau},s)$, and by
our assumption the self-dual character $\psi\chi^{\tau}\mathbf{N}^{-r}$ has
root number $w(\psi\chi^{\tau}\mathbf{N}^{-r})=+1$. Since $\mathcal{X}_{{\rm
BK}}(\psi\chi^{\tau}\mathbf{N}^{-r})=\mathcal{X}_{\overline{v}}(\psi\chi^{\tau}\mathbf{N}^{-r})$
by Lemma 3.4.2 (indeed, the infinity type of $\psi\chi^{\tau}\mathbf{N}^{-r}$
is $(1-r,r)$), by the same argument as in the proof of [AH06, Thm. 2.4.17(1)]
and [Arn07, Thm. 3.9], but replacing the use of [AH06, Prop. 2.3.4] and
[Arn07, Prop. 2.6] by an appeal to Theorem 4.2.1 above, we deduce that
$\check{\mathcal{S}}_{\overline{v}}(\psi^{\tau}\chi\mathbf{N}^{-r})=0$ and
$\mathcal{X}_{\overline{v}}(\psi\chi^{\tau}\mathbf{N}^{-r})$ is
$\Lambda_{\mathscr{O}}$-torsion, with

${\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{\overline{v}}(\psi\chi^{\tau}\mathbf{N}^{-r})\bigr{)}=\bigl{(}\mathscr{L}_{\overline{v}}(\psi\chi^{\tau}\mathbf{N}^{-r})\bigr{)}$

as ideals in $\Lambda_{\mathscr{O}}^{\rm ur}$.

Let $\iota$ denote the involution of $\Lambda_{\mathscr{O}}$ given by
$\gamma\mapsto\gamma^{-1}$ for $\gamma\in\Gamma$. Since by [Arn07, Prop. 4.1],
and as a direct consequence of the interpolation property of Theorem 2.2.1
(see e.g. [BCG+20, Lem. 3.3.2(a)]), we have the equalities

$\displaystyle{\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{\overline{v}}(\psi\chi^{\tau}\mathbf{N}^{-r})\bigr{)}$
$\displaystyle={\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})^{\iota}\bigr{)},$
$\displaystyle\bigl{(}\mathscr{L}_{\overline{v}}(\psi\chi^{\tau}\mathbf{N}^{-r})\bigr{)}$
$\displaystyle=\bigl{(}\mathscr{L}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})^{\iota}\bigr{)},$

the result follows. ∎

###  4.4. $p$-parity conjecture

By Nekovář’s methods, we deduce from some of the preceding results a proof of
the $p$-parity conjecture for CM forms of higher weight (see [Guo93] for
earlier results in this context). This will be an important ingredient in the
proof a higher weight CM $p$-converse theorem (in the style of [Ski20] and
[BCST22]) in forthcoming work [CP24].

######  Corollary 4.4.1.

Let $\lambda$ be a Hecke character of infinity type $(1-2r,0)$ for some $r\geq
1$ with $\varepsilon_{\lambda}=\eta_{K}$ and conductor $\mathfrak{c}$ coprime
to $p$ and such that $\mathfrak{d}_{K}\|\mathfrak{c}$. Then

${\rm ord}_{s=r}\,L(\lambda,s)\equiv{\rm dim}_{\Phi}\,{\rm Sel}_{\rm
BK}(K,V_{\lambda}(r))\pmod{2},$

and hence the $p$-parity conjecture holds for $\lambda$.

###### Proof.

We argue similarly as in the proof of [CH18, Thm. 6.4]. By Nekovář’s general
result [Nek07, Cor. (5.3.2)] (see also [Nek09]), it suffices to show that if
$w(\lambda\mathbf{N}^{-r})=-1$ (resp. $w(\lambda\mathbf{N}^{-r})=+1$), there
exists a finite order character $\phi$ of $\Gamma$ such that
$\dim_{\Phi}\,{\rm Sel}_{\rm BK}(K,V_{\lambda\phi}(r))=1$ (resp. ${\rm
Sel}_{\rm BK}(K,V_{\lambda\phi}(r))=0$).

Suppose first that $w(\lambda\mathbf{N}^{-r})=-1$. In view of Proposition
4.3.3, we can take a finite order character $\phi$ of $\Gamma$ such that
$\mathscr{L}_{v}(\lambda\mathbf{N}^{-r})(\phi)\neq 0$. By Theorem 4.3.1 and a
variant of Mazur’s control theorem for
$\mathcal{X}_{v}(\lambda\phi\mathbf{N}^{-r})$ (see [Cas23, Prop. 6.2.1]), it
follows that ${\rm Sel}_{v}(K,V_{\lambda\phi}(r))=0$, and by Theorem 4.2.1 it
follows that the localization map

(4.3) ${\rm loc}_{v}:{\rm
Sel}_{v}(K,V_{\lambda^{\tau}\phi^{\tau}}(r))\rightarrow\mathrm{H}^{1}(K_{v},V_{\lambda^{\tau}\phi^{\tau}}(r))$

is nonzero888Indeed, letting
$\mathbf{c}_{\lambda^{\tau}\mathbf{N}^{-r}}(\phi^{\tau})$ denote the
specialization of $\mathbf{c}_{\lambda^{\tau}\mathbf{N}^{-r}}$ at
$\phi^{\tau}$, the inclusion
$\mathbf{c}_{\lambda^{\tau}\mathbf{N}^{-r}}(\phi^{\tau})\in{\rm
Sel}_{v}(K,V_{\lambda^{\tau}\phi^{\tau}}(r))$ follows from Theorem 4.2.1, the
interpolation property in Theorem 2.2.1, and the fact that $L(\lambda,r)=0$..
From the Poitou–Tate exact sequence

${\rm
Sel}_{v}(K,V_{\lambda^{\tau}\phi^{\tau}}(r))\rightarrow\mathrm{H}^{1}(K_{v},V_{\lambda^{\tau}\phi^{\tau}}(r))\rightarrow{\rm
Sel}_{\rm rel}(K,V_{\lambda\phi}(r))^{*}\rightarrow{\rm
Sel}_{\overline{v}}(K,V_{\lambda\phi}(r))^{*}\rightarrow 0,$

it follows that

(4.4) ${\rm Sel}_{\rm rel}(K,V_{\lambda\phi}(r))={\rm
Sel}_{\overline{v}}(K,V_{\lambda\phi}(r))={\rm Sel}_{{\rm
BK}}(K,V_{\lambda\phi}(r)),$

using Lemma 3.4.2 for the last equality. Since the nonvanishing of (4.3) gives
${\rm loc}_{\overline{v}}({\rm Sel}_{\overline{v}}(K,V_{\lambda\phi}(r)))\neq
0$, from the tautological exact sequence

${\rm Sel}_{v}(K,V_{\lambda\phi}(r))\rightarrow{\rm Sel}_{\rm
rel}(K,V_{\lambda\phi}(r))\rightarrow\mathrm{H}^{1}(K_{\overline{v}},V_{\lambda\phi}(r))$

and (4.4) we obtain ${\rm dim}_{\Phi}\,{\rm Sel}_{\rm
BK}(K,V_{\lambda\phi}(r))={\rm dim}_{\Phi}\,{\rm
Sel}_{v}(K,V_{\lambda\phi}(r))+1=1$, as desired.

The case $w(\lambda\mathbf{N}^{-r})=+1$ is easier: By [Arn07, Prop. 2.3], we
can take a finite order character $\phi$ of $\Gamma$ such that
$\mathscr{L}_{\overline{v}}(\lambda\mathbf{N}^{-r})(\phi)\neq 0$, and then by
Theorem 3.9 in _op. cit._ it follows that ${\rm
Sel}_{\overline{v}}(K,V_{\lambda\phi}(r))=0$, so by Lemma 3.4.2 the Bloch–Kato
Selmer group ${\rm Sel}_{{\rm BK}}(K,V_{\lambda\phi}(r))$ vanishes as desired.
∎

###  4.5. Non-triviality of $\mathbf{z}_{g,\chi}$

For $g$ of weight $k=2$, the non-triviality of $\mathbf{z}_{g,\chi}$ follows
from the work of Cornut–Vatsal [CV05, CV07]. For $g$ of even weight $k>2$,
assuming that the residual representation attached to $V_{g}|_{G_{K}}$ is
absolutely _irreducible_ , the non-triviality of $\mathbf{z}_{g,\chi}$ follows
from the combination of Theorem 3.9 and Theorem 5.7 in [CH18], and also from
[Bur20, Thm. 4.3]; in both cases, the result is deduced from Hida’s methods
[Hid10, Hid04], showing the non-vanishing of $\mathscr{L}_{v}(g,\chi)$, and a
form of Theorem 3.2.2.

Here we are interested in the case where $g=\theta_{\psi}$ has CM by $K$, so
in particular $V_{g}|_{G_{K}}$ is reducible. In the weight $2$ case, an
alternative proof of Cornut–Vatsal’s nonvanishing result was given in [BD20]
building on the $\Lambda_{\mathscr{O}}$-adic Gross–Zagier formula of [Dis17]
and the nonvanishing result of [Bur15]. Here we prove the non-triviality of
$\mathbf{z}_{g,\chi}$ for $g=\theta_{\psi}$ of arbitrary even weight.

######  Theorem 4.5.1.

Let $\psi$ be a Hecke character of infinity type $(1-2r,0)$ for some $r\geq 1$
and conductor a cyclic ideal $\mathfrak{f}_{\psi}$ of norm prime to $pD_{K}$.
Let

$(g,\chi)=(\theta_{\psi},\chi)\in S_{2r}(\Gamma_{1}(N_{g}))\times\Sigma_{\rm
cc}(c,\mathfrak{N}_{g},\varepsilon_{g})$

be a self-dual pair for some positive integer $c$ prime to $pN_{g}$. Then:

  1. (i)

If $w(\psi\chi\mathbf{N}^{-r})=-1$, then $\mathscr{L}_{v}(g,\chi)\neq 0$ and
${\rm loc}_{\overline{v}}(\check{\mathcal{S}}_{\rm BK}(g,\chi))\neq 0$.

  2. (ii)

If $w(\psi\chi\mathbf{N}^{-r})=+1$, then
$\mathscr{L}_{\overline{v}}(g,\chi)\neq 0$ and ${\rm
loc}_{v}(\check{\mathcal{S}}_{\rm BK}(g,\chi))\neq 0$.

In particular, $\mathbf{z}_{g,\chi}\neq 0$ regardless of the sign of
$w(\psi\chi\mathbf{N}^{-r})$.

###### Proof.

Since $\mathscr{L}_{v}(g,\chi)$ is a unit multiple of
$\mathscr{L}_{v}(\psi\chi\mathbf{N}^{-r})\mathscr{L}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})$
by Proposition 2.3.1, in the case $w(\psi\chi\mathbf{N}^{-r})=-1$ the
nonvanishing of $\mathscr{L}_{v}(g,\chi)$ follows from Proposition 4.3.3 and
Theorem 4.3.6; the nonvanishing of ${\rm
loc}_{\overline{v}}(\check{\mathcal{S}}_{\rm BK}(g,\chi))$ then follows from
Theorem 3.2.2.

In the case $w(\psi\chi\mathbf{N}^{-r})=+1$, the same argument as in
Proposition 2.3.1 shows that $\mathscr{L}_{\overline{v}}(g,\chi)$ is a unit
multiple of
$\mathscr{L}_{\overline{v}}(\psi\chi\mathbf{N}^{-r})\mathscr{L}_{\overline{v}}(\psi^{\tau}\chi\mathbf{N}^{-r})$.
Directly from Theorem 2.2.1 (see [BCG+20, Lem. 3.3.2(a)]), we have the
relations

$\bigr{(}\mathscr{L}_{\overline{v}}(\psi\chi\mathbf{N}^{-r})\bigr{)}=\big{(}\mathscr{L}_{v}(\psi^{\tau}\chi^{\tau}\mathbf{N}^{-r})^{\iota}\bigr{)},\quad\bigr{(}\mathscr{L}_{\overline{v}}(\psi^{\tau}\chi\mathbf{N}^{-r})\bigr{)}=\big{(}\mathscr{L}_{v}(\psi\chi^{\tau}\mathbf{N}^{-r})^{\iota}\bigr{)}.$

Note that the character $\psi^{\tau}\chi^{\tau}\mathbf{N}^{-r}$ lies inside
the range of $p$-adic interpolation for $\mathscr{L}_{v,\mathfrak{c}}$, while
$\psi\chi^{\tau}\mathbf{N}^{-r}$ lies outside this range. As in Theorem 4.3.6
and Proposition 4.3.3, we see that

$\displaystyle w(\psi^{\tau}\chi^{\tau}\mathbf{N}^{-r})=+1\quad$
$\displaystyle\Longrightarrow\quad\mathscr{L}_{v}(\psi^{\tau}\chi^{\tau}\mathbf{N}^{-r})\neq
0,$ $\displaystyle w(\psi\chi^{\tau}\mathbf{N}^{-r})=-1\quad$
$\displaystyle\Longrightarrow\quad\mathscr{L}_{v}(\psi\chi^{\tau}\mathbf{N}^{-r})\neq
0,$

respectively, which together with Theorem 3.2.2 yields the result. ∎

######  Remark 4.5.2.

It is interesting to note that the preceding results show in fact the
equivalences

$\displaystyle{\rm loc}_{\overline{v}}(\check{\mathcal{S}}_{\rm
BK}(g,\chi))\neq 0\quad\Longleftrightarrow\quad\mathscr{L}_{v}(g,\chi)\neq 0$
$\displaystyle\quad\Longleftrightarrow\quad w(\psi\chi\mathbf{N}^{-r})=-1,$
$\displaystyle{\rm loc}_{v}(\check{\mathcal{S}}_{\rm BK}(g,\chi))\neq
0\quad\Longleftrightarrow\quad\mathscr{L}_{\overline{v}}(g,\chi)\neq 0$
$\displaystyle\quad\Longleftrightarrow\quad w(\psi\chi\mathbf{N}^{-r})=+1.$

Indeed, the right pair of equivalences and the backward direction for the left
pair directly follow from the proof of Theorem 4.5.1. For the left pair of
implications $\Longrightarrow$, note that by [AH06, Thm. 2.4.17] (as extended
in [Arn07, Thm. 3.9]) we have the implications

$\displaystyle w(\psi\chi\mathbf{N}^{-r})=+1$
$\displaystyle\quad\Longrightarrow\quad\check{\mathcal{S}}_{\rm
BK}(\psi\chi\mathbf{N}^{-r})=0,$ $\displaystyle\quad
w(\psi\chi\mathbf{N}^{-r})=-1$
$\displaystyle\quad\Longrightarrow\quad\check{\mathcal{S}}_{\rm
BK}(\psi^{\tau}\chi\mathbf{N}^{-r})=0,$

using that $w(\psi^{\tau}\chi\mathbf{N}^{-r})=-w(\psi\chi\mathbf{N}^{-r})$ for
the second implication. Since Proposition 3.5.1 gives the equalities

${\rm loc}_{\overline{v}}(\check{\mathcal{S}}_{\rm
BK}(\psi\chi\mathbf{N}^{-r}))={\rm
loc}_{\overline{v}}(\check{\mathcal{S}}_{\rm BK}(g,\chi)),\quad{\rm
loc}_{v}(\check{\mathcal{S}}_{\rm BK}(\psi^{\tau}\chi\mathbf{N}^{-r}))={\rm
loc}_{v}(\check{\mathcal{S}}_{\rm BK}(g,\chi)),$

this yields the claim.

###  4.6. Proof of Theorem 4.1.1 and Heegner cycle main conjecture

###### Proof of Theorem 4.1.1.

Since $(g,\chi)=(\theta_{\psi},\chi)$ is a self-dual pair, the $L$-function
$L(g/K,\chi,s)$ is self-dual with sign $-1$ and center at $s=r$, and the
decomposition (3.2) gives

$L(g/K,\chi,s)=L(\psi\chi,s)\cdot L(\psi^{\tau}\chi,s).$

The $L$-functions in the right-hand side of this factorization are self-dual
with opposite signs,

$w(\psi\chi\mathbf{N}^{-r})=-w(\psi^{\tau}\chi\mathbf{N}^{-r}).$

Suppose first that $w(\psi\chi\mathbf{N}^{-r})=-1$. By Proposition 3.5.1, the
assertions that $\check{\mathcal{S}}_{v}(g,\chi)$ vanishes and
$\mathcal{X}_{v}(g,\chi)$ is $\Lambda_{\mathscr{O}}$-torsion follow from
Theorem 4.3.1 and Theorem 4.3.6. Together with the factorization in
Proposition 2.3.1, the same theorems give the equalities

(4.5) $\displaystyle{\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{v}(g,\chi)\bigr{)}$
$\displaystyle={\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{v}(\psi\chi\mathbf{N}^{-r})\bigr{)}\cdot{\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})\bigr{)}$
$\displaystyle=\bigl{(}\mathscr{L}_{v}(\psi\chi\mathbf{N}^{-r})\cdot\mathscr{L}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})\bigr{)}$
$\displaystyle=\bigl{(}\mathscr{L}_{v}(g,\chi)^{2}\bigr{)}$

in $\Lambda_{\mathscr{O}}^{\rm ur}$, yielding the result in this case.

In the case $w(\psi\chi\mathbf{N}^{-r})=+1$, we apply the result of Theorem
4.3.1 and Theorem 4.3.6 to $\psi\chi^{\tau}\mathbf{N}^{-r}$ (which has
$w(\psi\chi^{\tau}\mathbf{N}^{-r})=-1$) and
$\psi^{\tau}\chi^{\tau}\mathbf{N}^{-r}$ (which has
$w(\psi^{\tau}\chi^{\tau}\mathbf{N}^{-r})=+1$), respectively, to obtain

$\check{\mathcal{S}}_{v}(\psi\chi^{\tau}\mathbf{N}^{-r})=\check{\mathcal{S}}_{v}(\psi^{\tau}\chi^{\tau}\mathbf{N}^{-r})=0$

and that $\mathcal{X}_{v}(\psi\chi^{\tau}\mathbf{N}^{-r})$ and
$\mathcal{X}_{v}(\psi^{\tau}\chi^{\tau}\mathbf{N}^{-r})$ are both
$\Lambda_{\mathscr{O}}$-torsion, with

${\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{v}(\psi\chi^{\tau}\mathbf{N}^{-r})\bigr{)}=\bigl{(}\mathscr{L}_{v}(\psi\chi^{\tau}\mathbf{N}^{-r})\bigr{)},\quad{\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{v}(\psi^{\tau}\chi^{\tau}\mathbf{N}^{-r})\bigr{)}=\bigl{(}\mathscr{L}_{v}(\psi^{\tau}\chi^{\tau}\mathbf{N}^{-r})\bigr{)}$

as ideals in $\Lambda_{\mathscr{O}}^{\rm ur}$. As in the proof of Theorem
4.3.6, by the action of complex conjugation this gives
$\check{\mathcal{S}}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})=\check{\mathcal{S}}_{v}(\psi\chi\mathbf{N}^{-r})=0$
and that $\mathcal{X}_{\overline{v}}(\psi^{\tau}\chi\mathbf{N}^{-r})$ and
$\mathcal{X}_{\overline{v}}(\psi\chi\mathbf{N}^{-r})$ are both
$\Lambda_{\mathscr{O}}$-torsion, with

${\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{\overline{v}}(\psi^{\tau}\chi\mathbf{N}^{-r})\bigr{)}=\bigl{(}\mathscr{L}_{\overline{v}}(\psi^{\tau}\chi\mathbf{N}^{-r})\bigr{)},\quad{\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{\overline{v}}(\psi\chi\mathbf{N}^{-r})\bigr{)}=\bigl{(}\mathscr{L}_{\overline{v}}(\psi\chi\mathbf{N}^{-r})\bigr{)},$

which by the $\overline{v}$-versions of Proposition 2.3.1 and Proposition
3.5.1 yields the result as above. ∎

By the nonvanishing results established in the course of proving Theorem
4.1.1, we can deduce the following result on Conjecture 3.3.3.

######  Corollary 4.6.1.

Let $(g,\chi)$ be a self-dual pair as in Theorem 4.1.1. Then
$\check{\mathcal{S}}_{{\rm BK}}(g,\chi)$ and $\mathcal{X}_{{\rm BK}}(g,\chi)$
both have $\Lambda_{\mathscr{O}}$-rank one, and

${\rm char}_{\Lambda_{\mathscr{O}}}\bigl{(}\mathcal{X}_{{\rm BK}}(g,\chi)_{\rm
tors}\bigr{)}={\rm
char}_{\Lambda_{\mathscr{O}}}\bigl{(}\check{\mathcal{S}}_{{\rm
BK}}(g,\chi)/\Lambda_{\mathscr{O}}\cdot\mathbf{z}_{g,\chi}\bigr{)}^{2}.$

In other words, Conjecture 3.3.3 holds for $(g,\chi)$.

###### Proof.

By Theorem 4.5.1, $\mathbf{z}_{g,\chi}$ is not
$\Lambda_{\mathscr{O}}$-torsion, and ${\rm
loc}_{\overline{v}}(\check{\mathcal{S}}_{\rm BK}(g,\chi))\neq 0$ (resp. ${\rm
loc}_{v}(\check{\mathcal{S}}_{\rm BK}(g,\chi))\neq 0$) when
$w(\psi\chi\mathbf{N}^{-r})=-1$ (resp. $w(\psi\chi\mathbf{N}^{-r})=+1$). Hence
by Proposition 3.6.2 (see also Remark 3.6.3) the result follows from Theorem
4.1.1. ∎

##  5\. Formula for the Bloch–Kato $\text{\wncyr{W}}_{{\rm BK}}(W_{g,\chi}/K)$

Let $K$ be an imaginary quadratic field satisfying (spl) for a prime $p>3$.
Let $\lambda$ be a Hecke character of $K$ of infinity type $(1-2r,0)$ for some
$r\geq 1$ and central character

(sd) $\varepsilon_{\lambda}=\eta_{K}.$

In particular, $\lambda\mathbf{N}^{-r}$ is self-dual in the sense of
Definition 2.2.2, and the conductor $\mathfrak{c}$ of $\lambda$ is divisible
by $\mathfrak{d}_{K}:=(\sqrt{-D_{K}})$. Throughout this section we assume that

(div)
$p\nmid\mathfrak{c}\quad\textrm{and}\quad\mathfrak{d}_{K}\|\mathfrak{c}.$

By self-duality, $\mathfrak{c}$ is invariant under complex conjugation, and so
by (div) can write

(5.1) $\mathfrak{c}=(c)\mathfrak{d}_{K}$

for a unique positive integer $c$ prime to $pD_{K}$.

###  5.1. Strong good pairs

The following definition is a natural higher weight extension (and
strengthening) of the notion of “good pair” for $\lambda$ introduced in
[BDP12].

######  Definition 5.1.1.

Suppose the self-dual character $\lambda\mathbf{N}^{-r}$ has root number
$w(\lambda\mathbf{N}^{-r})=-1$. We say that a pair of Hecke characters
$(\psi,\chi)$ is a _good pair_ for $\lambda$ if it satisfies

  1. (S1)

$\psi$ has infinity type $(1-2r,0)$ and conductor a cyclic ideal
$\mathfrak{f}_{\psi}$ of norm prime to $pD_{K}$.

  2. (S2)

$\chi$ is a finite order character such that

$(g,\chi)=(\theta_{\psi},\chi)\in S_{2r}(\Gamma_{1}(N_{g}))\times\Sigma_{\rm
cc}(c,\mathfrak{N}_{g},\varepsilon_{g}),$

where $\mathfrak{N}_{g}:=\mathfrak{d}_{K}\cdot\mathfrak{f}_{\psi}$ and
$N_{g}=D_{K}\mathbf{N}(\mathfrak{f}_{\psi})$.

  3. (S3)

$\lambda=\psi\chi$.

  4. (S4)

$L(\psi^{\tau}\chi,r)\neq 0$.

If $r=1$, we say that $(\psi,\chi)$ is a _strong good pair_ for $\lambda$ if
in addition

  1. (S5)

$\psi^{-}:=\psi^{\tau}/\psi$ has order prime to $p$, and
$p\nmid\mathfrak{f}_{\psi^{-}}$.

  2. (S6)

Every prime $\mathfrak{l}\mid\mathfrak{f}_{\psi^{-}}$ splits in $K$.

  3. (S7)

$\psi^{-}|_{G_{v}}\neq\mathds{1}$, where $G_{v}\subset G_{K}$ is a
decomposition group at $v$.

  4. (S8)

$\psi^{-}$ has order $\geq 3$.

  5. (S9)

The value

$L^{\rm
alg}(\psi^{-},0):=\biggl{(}\frac{2\pi}{\sqrt{D_{K}}}\biggr{)}\cdot(1-\psi^{-}(\overline{v}))(1-\psi^{-}(\overline{v})p^{-1})\cdot\frac{L(\psi^{-},0)}{\Omega}$

is a $p$-adic unit.

######  Remark 5.1.2.

Note that by Theorem 2.2.1 the normalized $L$-value $L^{\rm alg}(\psi^{-},0)$
is the same as the $p$-adic $L$-value $\mathscr{L}_{v,\mathfrak{c}}(\psi^{-})$
for $\mathfrak{c}=\mathfrak{f}_{\psi^{-}}$; having infinity type $(1,-1)$,
$\psi^{-}$ lies within its range of $p$-adic interpolation of
$\mathscr{L}_{\mathfrak{c},v}$. In particular, by the functional equation of
Theorem 2.2.1, if $(\psi,\chi)$ is a strong good pair for $\lambda$, then the
$p$-adic $L$-function
$\mathscr{L}_{v}(\psi^{-}\mathbf{N}^{-1})\in\Lambda_{\mathscr{O}}^{\rm ur}$ is
invertible.

For $r=1$, the existence of good pairs for $\lambda$ is shown in [BDP12, Prop.
3.28] building on Greenberg [Gre85] and Rohrlich [Roh84] nonvanishing results.
Adapting their argument, and building on Finis’ mod $p$ non-vanishing results
[Fin06], we can prove the following.

######  Lemma 5.1.3.

Suppose $\lambda\mathbf{N}^{-r}$ has root number $-1$. Then there exist good
pairs for $\lambda$. Moreover, if $r=1$, then there exist strong good pairs
for $\lambda$.

###### Proof.

Take $\psi$ a Hecke character of $K$ of infinity type $(1-2r,0)$ and conductor
a cyclic ideal $\mathfrak{f}_{\psi}$ of norm coprime to $p\mathfrak{c}$, and
put

$\chi:=\psi^{-1}\lambda.$

By (5.1), $\psi$ satisfies (S1), and by construction, the pair $(\psi,\chi)$
satisfies (S3). By [BDP12, Rem. 3.20], it follows that $\chi$ satisfies (S2),
so it remains to verify the much subtler condition (S4).

We begin by noting (as used in the proof of Theorem 4.1.1) the sign in the
functional equation for $L(\psi^{\tau}\chi,s)$ relating its values at $s$ and
$2r-s$ has sign $w(\psi^{\tau}\chi\mathbf{N}^{-r})=+1$. Indeed, by the
inclusion $\chi\in\Sigma_{\rm cc}(c,\mathfrak{N}_{g},\varepsilon_{g})$, the
sign in the functional equation for $L(g/K,\chi,s)=L(\theta_{\psi}/K,\chi,s)$
is $\epsilon(g,\chi)=-1$, and so the claim follow from the factorization

(5.2) $\displaystyle L(\theta_{\psi}/K,\chi,s)$
$\displaystyle=L(\lambda,s)\cdot L(\psi^{\tau}\chi,s)$

and the assumption $w(\lambda\mathbf{N}^{-r})=-1$. Let
$\ell=\lambda\overline{\lambda}$ be a prime split in $K$, with $\ell\neq p$
and prime to the conductors of $\psi$ and $\chi$. Given a pair $(\psi,\chi)$
satisfying conditions (S1)–(S3), for any finite order character $\alpha$ of
conductor dividing $\lambda^{\infty}$, the pair
$(\psi^{\prime},\chi^{\prime}):=(\psi\alpha,\chi\alpha^{-1})$ satisfies the
same conditions (with $\mathfrak{f}_{\psi}$ replaced by
$\mathfrak{f}_{\psi}\lambda^{m}$ for some $m\geq 0$), while (5.2) becomes

$L(\theta_{\psi^{\prime}}/K,\chi^{\prime},s)=L(\lambda,s)\cdot
L(\psi^{\tau}\chi\alpha^{-},s),$

where $\alpha^{-}:=\alpha^{\tau}/\alpha$. Since $\alpha^{-}$ is anticyclotomic
of $\ell$-power conductor, by Theorem 1.1 of [Fin06] all but finitely many of
the values $L(\psi^{\tau}\chi\alpha^{-},r)$ are nonzero (even mod $\varpi$ as
$\alpha$ varies999Reversing the roles of $\ell$ and $p$. Note that
$\psi^{\tau}\chi\alpha^{-}\mathbf{N}^{-r}$ has infinity type $(r,1-r)$, and
our conventions are _opposite_ to those in [Fin06].), this shows the first
assertion.

Moreover, when $r=1$ noting that

$L((\psi^{\prime})^{-},0)=L(\psi^{-}\alpha^{-},0),$

the same [Fin06, Thm. 1.1] ensures we can find $\alpha$ as above so that
further $L^{\rm alg}(\psi^{-}\alpha^{-},0)$ is a $p$-adic unit and conditions
(S5)–(S8) hold for $(\psi^{\prime})^{-}$, whence the result. ∎

######  Remark 5.1.4.

In the proof of Lemma 5.1.3, the use of the main result of [Fin06] could
alternatively be replaced by an appeal to Hsieh’s mod $p$ nonvanishing result
[Hsi12, Thm. A].

###  5.2. Proving Selmer rank $1$

Let $f\in S_{2r}(\Gamma_{0}(N_{f}))$ be an eigenform of even weight $2r\geq 2$
and level $N_{f}$ prime to $p$. Granted the injectivity of a certain $p$-adic
Abel–Jacobi map, the works of Nekovář [Nek92] and S.-W. Zhang [Zha97]
(extending to weights $2r>2$ landmark results of Kolyvagin [Kol88] and
Gross–Zagier [GZ86]) yield a proof of the implication

(5.3) ${\rm ord}_{s=r}L(f,s)=1\quad\Longrightarrow\quad{\rm dim}_{\Phi}\,{\rm
Sel}_{\rm BK}(\mathbb{Q},V_{f}(r))=1.$

Key to the proof of this result are Heegner cycles attached to an auxiliary
imaginary quadratic field $K^{\prime}$ satisfying the Heegner hypothesis
(Heeg) relative to $N_{f}$.

Here we are interested in the case where $f=\theta_{\lambda}$ is the theta
series of a Hecke character $\lambda$ of infinity type $(1-2r,0)$, so
$\varepsilon_{\lambda}=\eta_{K}$. In particular, the conductor $\mathfrak{c}$
of $\lambda$ is divisible by $\mathfrak{d}_{K}$, and so $K$ does _not_ satisfy
the Heegner hypothesis (Heeg) relative to
$N_{f}=D_{K}\mathbf{N}(\mathfrak{c})$.

For later use, here we prove the following analogue of (5.3) for
$f=\theta_{\lambda}$ with CM by $K$ using only arithmetic over $K$.

######  Theorem 5.2.1.

Let $\lambda$ be a Hecke character of $K$ of infinity type $(1-2r,0)$ for some
$r\geq 1$, conductor $\mathfrak{c}$ prime to $p$, and central character
$\varepsilon_{\lambda}=\eta_{K}$. Then

$\mathscr{L}_{v}(\lambda\mathbf{N}^{-r})(0)\neq 0\quad\Longrightarrow\quad{\rm
dim}_{\Phi}\,{\rm Sel}_{\rm BK}(K,V_{\lambda}(r))=1.$

Equivalently, if $\mathscr{L}_{v}(\lambda\mathbf{N}^{-r})(0)$ then ${\rm
Sel}_{\rm BK}(\mathbb{Q},V_{f}(r))$ is $1$-dimensional, where
$f=\theta_{\lambda}$.

######  Remark 5.2.2.

When $r=1$ and $\mathfrak{d}_{K}\|\mathfrak{c}$, the assumption
$\mathscr{L}_{v}(\lambda\mathbf{N}^{-r})(0)\neq 0$ follows from ${\rm
ord}_{s=1}L(\lambda,s)=1$. Indeed, letting $(\psi,\chi)$ be a good pair for
$\lambda$ (as they exist by Lemma 5.1.3; note that
$w(\lambda\mathbf{N}^{-1})=-1$ by the assumption that ${\rm
ord}_{s=1}L(\lambda,s)=1$), putting $g=\theta_{\psi}$ and letting
$z_{g,\chi}\in{\rm Sel}_{\rm BK}(K,T_{g,\chi})$ be the Heegner class of
Theorem 3.2.1, which in this case arises as the image under the Kummer map

(5.4) $B_{g,\chi}(K)\otimes_{\mathcal{O}_{L}}\mathscr{O}\rightarrow{\rm
Sel}_{\rm BK}(K,T_{g,\chi})$

of a Heegner cycle on a certain CM abelian variety $B_{g,\chi}/K$, we have

(by (5.2)) $\displaystyle{\rm ord}_{s=1}L(\lambda,s)=1$
$\displaystyle\quad\Longrightarrow\quad{\rm ord}_{s=1}L(g/K,\chi,s)=1$ (by
[YZZ13]) $\displaystyle\quad\Longrightarrow\quad z_{g,\chi}\neq 0$ (by Theorem
3.2.1) $\displaystyle\quad\Longrightarrow\quad\mathscr{L}_{v}(g,\chi)(0)\neq
0$ (by Proposition 2.3.1)
$\displaystyle\quad\Longrightarrow\quad\mathscr{L}_{v}(\lambda\mathbf{N}^{-1})(0)\neq
0.$

Note that the second implication relies on the injectivity (5.4), and the
third on the non-triviality of the localization map

${\rm loc}_{\overline{v}}:{\rm Sel}_{\rm
BK}(K,T_{g,\chi})\rightarrow\mathrm{H}^{1}_{f}(K_{\overline{v}},T_{g,\chi}),$

both of which are expected—but not known—to continue to hold in weight $2r>2$,
with (5.4) replaced by the $p$-adic étale Abel–Jacobi map on generalized
Kuga–Sato varieties studied in [BDP13].

###### Proof of Theorem 5.2.1.

Let $\mathbf{c}_{\lambda\mathbf{N}^{-r}}\in\mathrm{H}^{1}_{\rm
Iw}(K_{\infty},T_{\lambda}(r))$ be the $\Lambda_{\mathscr{O}}$-adic class (of
twisted elliptic units) of Theorem 4.2.1, and denote by
$\mathbf{c}_{\lambda\mathbf{N}^{-r}}(\mathds{1})$ its image under the
projection

$\mathrm{H}^{1}_{\rm
Iw}(K_{\infty},T_{\lambda}(r))\rightarrow\mathrm{H}^{1}(K,T_{\lambda}(r)).$

As a consequence of Theorem 4.2.1, we have

$\mathscr{L}_{v}(\lambda\mathbf{N}^{-r})(0)\neq 0\quad\Longrightarrow\quad{\rm
loc}_{\overline{v}}(\mathbf{c}_{\lambda\mathbf{N}^{-r}}(\mathds{1}))\neq 0.$

We claim that $\mathbf{c}_{\lambda\mathbf{N}^{-r}}(\mathds{1})$ lands in ${\rm
Sel}_{\rm BK}(K,T_{\lambda}(r))$. Indeed, since $\lambda\mathbf{N}^{-r}$ has
infinity type $(1-r,r)$, by Lemma 3.4.2 the claim amounts to the assertion
that

(5.5) ${\rm loc}_{v}(\mathbf{c}_{\lambda\mathbf{N}^{-r}}(\mathds{1}))=0,$

which by Theorem 4.2.1 for $w=\overline{v}$ amounts to the assertion that
$\mathscr{L}_{v}(\lambda^{\tau}\mathbf{N}^{-r})(0)=0$. Since
$\lambda^{\tau}\mathbf{N}^{-r}$ is within the range of interpolation of
$\mathscr{L}_{v,\mathfrak{c}}$ (as its infinity type is $(r,1-r)$), by Theorem
2.2.1 we have

$\mathscr{L}_{v}(\lambda^{\tau}\mathbf{N}^{-r})(0)=0\quad\Longleftrightarrow\quad
L(\lambda^{\tau},r)=L(\lambda,r)=0,$

whence the claim. In particular, this shows that the map

(5.6) ${\rm loc}_{\overline{v}}:{\rm Sel}_{\rm
BK}(K,T_{\lambda}(r))\rightarrow\mathrm{H}^{1}_{f}(K_{\overline{v}},T_{\lambda}(r))=\mathrm{H}^{1}(K_{\overline{v}},T_{\lambda}(r))$

is nonzero, using Lemma 3.4.2 for the last equality.

It follows from the discussion in $\S{3.3}$ and Theorem 6.4.1 in [Rub00] that
the class $\mathbf{c}_{\lambda\mathbf{N}^{-r}}(\mathds{1})$ extends to an
anticyclotomic Euler system for $T_{\lambda}(r)$; since the above shows that
this Euler system is non-trivial, by [Rub00, Thm. 2.2.3] (see also [_loc.
cit._ , §9.3]) it follows that ${\rm rank}_{\mathscr{O}}\,{\rm Sel}_{\rm
rel}(K,T_{\lambda}(r))=1$. From the global duality exact sequence

${\rm Sel}_{\rm BK}(K,T_{\lambda^{\tau}}(r))\xrightarrow{{\rm
loc}_{v}}\mathrm{H}^{1}(K_{v},T_{\lambda^{{\tau}}}(r))\rightarrow{\rm
Sel}_{\rm rel}(K,W_{\lambda}(r))^{\vee}\rightarrow{\rm Sel}_{\rm
BK}(K,W_{\lambda}(r))^{\vee}\rightarrow 0,$

noting that the non-vanishing of (5.6) implies the non-vanishing of the map
${\rm loc}_{v}$ in this sequence, we conclude that ${\rm
corank}_{\mathscr{O}}\,{\rm Sel}_{\rm BK}(K,W_{\lambda}(r))={\rm
corank}_{\mathscr{O}}\,{\rm Sel}_{\rm rel}(K,W_{\lambda}(r))=1$, whence the
result. ∎

###  5.3. Proof of Theorem C

We begin by noting that by the assumption that $\lambda\mathbf{N}^{-r}$ has
root number $-1$, Lemma 5.1.3 ensures the existence of good pairs for
$\lambda$. We fix once and for all a good pair $(\psi,\chi)$ for $\lambda$,
and put $g=\theta_{\psi}$. Exploiting the decomposition

$\mathcal{X}_{v}(g,\chi)\simeq\mathcal{X}_{v}(\lambda\mathbf{N}^{-r})\oplus\mathcal{X}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})$

from Proposition 3.5.1, the formula for $\\#\text{\wncyr{W}}_{\rm
BK}(W_{g,\chi}/K)$ of Theorem C will be obtained by computing the
$\Gamma$-Euler characteristic for the constituent Selmer groups in this
decomposition and combining the resulting formulae. Note that some of the
intermediate results will be obtained in greater generality than needed for
the proof of Theorem C.

For a finite extension $\Phi$ of $\mathbb{Q}_{p}$ with ring of integers
$\mathscr{O}$, and a $\Phi$-linear $G_{K}$-representation $V$ with a fixed
$G_{K}$-stable $\mathscr{O}$-lattice $T\subset V$ with $W:=V/T$, recall that
the Bloch–Kato Tate–Shafarevic group $\text{\wncyr{W}}_{\rm BK}(W/K)$ is
defined by

$\text{\wncyr{W}}_{\rm BK}(W/K):={\rm Sel}_{\rm BK}(K,W)/{\rm Sel}_{\rm
BK}(K,W)_{\rm div},$

where ${\rm Sel}_{\rm BK}(K,W)_{\rm div}$ denotes the maximal divisible
submodule of ${\rm Sel}_{\rm BK}(K,W)$, and for a finite prime $w\nmid p$ of
$K$, the Tamagawa number of $W$ at $w$ is defined by

$c_{w}(W/K):=\\#\bigl{(}\mathrm{H}^{1}_{\rm
ur}(K_{w},W)/\mathrm{H}^{1}_{f}(K_{w},W)\bigr{)},$

where $\mathrm{H}^{1}_{\rm ur}(K_{w},W):={\rm
ker}\\{\mathrm{H}^{1}(K_{w},W)\rightarrow\mathrm{H}^{1}(I_{w},W)\\}$ is the
unramified submodule, and $\mathrm{H}^{1}_{f}(K_{w},W)$ is the image of the
unramified subspace $\mathrm{H}^{1}_{\rm ur}(K,V)$ under the natural map
$\mathrm{H}^{1}(K_{w},V)\rightarrow\mathrm{H}^{1}(K_{w},W)$ (which is thus
contained in $\mathrm{H}^{1}_{\rm ur}(K_{w},W)$).

####  5.3.1. Formula for $\psi^{\tau}\chi$

######  Proposition 5.3.1.

Let $(\psi,\chi)$ be as in Theorem 4.3.6, and let
$\mathscr{F}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})\in\Lambda_{\mathscr{O}}$ be a
generator of the characteristic ideal of
$\mathcal{X}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})$. If ${\rm Sel}_{\rm
BK}(K,W_{\psi^{\tau}\chi}(r))$ is finite, then
$\mathscr{F}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})(0)\neq 0$ with

$\mathscr{F}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})(0)\;\sim_{p}\;\\#\mathrm{H}^{0}(K_{\overline{v}},W_{\psi^{\tau}\chi}(r))^{2}\cdot\frac{\\#\text{\wncyr{W}}_{{\rm
BK}}(W_{\psi^{\tau}\chi}(r)/K)}{\\#\mathrm{H}^{0}(K,W_{\psi^{\tau}\chi}(r))^{2}}\cdot\prod_{\begin{subarray}{c}w\in\Sigma,w\nmid
p\end{subarray}}c_{w}(W_{\psi^{\tau}\chi}(r)/K).$

###### Proof.

By Lemma 3.4.2, $\mathcal{X}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})$ interpolates
Selmer groups ${\rm Sel}_{\rm BK}(F,W_{\psi^{\tau}\chi}(r))$ as $F$ varies
over the finite extensions of $K$ contained in $K_{\infty}$. Therefore, by the
control theorem of [Arn07, Prop. 4.3] the natural restriction map

${\rm Sel}_{{\rm BK}}(K,W_{\psi^{\tau}\chi}(r))={\rm
Sel}_{v}(K,W_{\psi^{\tau}\chi}(r))\rightarrow\mathcal{S}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})^{\Gamma}$

has finite kernel and cokernel, and so the finiteness of ${\rm Sel}_{{\rm
BK}}(K,W_{\psi^{\tau}\chi}(r))$ gives
$\\#\mathcal{S}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})^{\Gamma}<\infty$. Letting
$\mathscr{F}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})\in\Lambda_{\mathscr{O}}$ be
any generator of ${\rm
char}_{\Lambda_{\mathscr{O}}}(\mathcal{X}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r}))$,
it is then easily checked that the $\Gamma$-coinvariants
$\mathcal{S}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})_{\Gamma}$ are also finite,
that $\mathscr{F}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})(0)\neq 0$, and that

(5.7)
$\mathscr{F}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})(0)\,\sim_{p}\,\frac{\\#\mathcal{S}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})^{\Gamma}}{\\#\mathcal{S}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})_{\Gamma}}$

(see [Gre99, Lem. 4.2] for example). For the proof of the formula in the
Proposition, we shall adapt the methods of [Gre99, §4] and [JSW17, §3].

First we set up some notations. Fix $\Sigma$ a finite set of primes of $K$
containing the archimedean prime, the primes above $p$, the primes dividing
the conductor of $\psi^{\tau}$ or $\chi$ and their complex conjugates, and let
$K^{\Sigma}$ denote the maximal extension of $K$ unramified outside $\Sigma$.
Put

$\mathcal{P}^{\Sigma}(K):=\frac{\mathrm{H}^{1}(K_{v},W_{\psi^{\tau}\chi}(r))}{\mathrm{H}^{1}(K_{v},W_{\psi^{\tau}\chi}(r))_{\rm
div}}\times\mathrm{H}^{1}(K_{\overline{v}},W_{\psi^{\tau}\chi}(r))\times\prod_{w\in\Sigma,w\nmid
p}\mathrm{H}^{1}(K_{w},W_{\psi^{\tau}\chi}(r)).$

Similarly, consider the $\Lambda_{\mathscr{O}}$-module
$\mathbf{W}_{\psi^{\tau}\chi}(r):=W_{\psi^{\tau}\chi}(r)\otimes_{\mathscr{O}}\Lambda_{\mathscr{O}}^{\vee}$
with diagonal $G_{K}$-action, letting $G_{K}$ act on the second factor through
the inverse of the tautological character
$G_{K}\twoheadrightarrow\Gamma\hookrightarrow\Lambda_{\mathscr{O}}^{\times}$,
and put

$\mathcal{P}^{\Sigma}(K_{\infty}):=\\{0\\}\times\mathrm{H}^{1}(K_{\overline{v}},\mathbf{W}_{\psi^{\tau}\chi}(r))\times\prod_{w\in\Sigma,w\nmid
p}\mathrm{H}^{1}(K_{w},\mathbf{W}_{\psi^{\tau}\chi}(r)),$

thinking of $\\{0\\}$ as a subspace of
$\mathrm{H}^{1}(K_{v},\mathbf{W}_{\psi^{\tau}\chi}(r))$. Then from the
definitions and Shapiro’s lemma we have exact sequences

(5.8) $\displaystyle 0$ $\displaystyle\rightarrow{\rm Sel}_{\rm
BK}(K,W_{\psi^{\tau}\chi}(r))\rightarrow\mathrm{H}^{1}(K^{\Sigma}/K,W_{\psi^{\tau}\chi}(r))\rightarrow\mathcal{P}^{\Sigma}(K),$
(5.9) $\displaystyle 0$
$\displaystyle\rightarrow\mathcal{S}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})\rightarrow\mathrm{H}^{1}(K^{\Sigma}/K,\mathbf{W}_{\psi^{\tau}\chi}(r))\rightarrow\mathcal{P}^{\Sigma}(K_{\infty}).$

Denote by $\mathcal{G}^{\Sigma}(K)\subset\mathcal{P}^{\Sigma}(K)$ and
$\mathcal{G}^{\Sigma}(K_{\infty})\subset\mathcal{P}^{\Sigma}(K_{\infty})$ the
image of the right maps in the above exact sequence, whose $\Gamma$-invariants
then fit into the commutative diagram

(5.10)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Sel}_{{\rm
BK}}(K,W_{\psi^{\tau}\chi}(r))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{*}}$$\textstyle{\mathrm{H}^{1}(K^{\Sigma}/K,W_{\psi^{\tau}\chi}(r))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s^{*}}$$\textstyle{\mathcal{G}^{\Sigma}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{*}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{S}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(K^{\Sigma}/K,\mathbf{W}_{\psi^{\tau}\chi}(r))^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}^{\Sigma}(K_{\infty})^{\Gamma}.}$

The surjectivity of $s^{*}$ follows from the fact that $\Gamma$ has
$p$-cohomological dimension $1$, while the kernel of $s^{*}$ is given by

$\mathrm{H}^{1}(\Gamma,\mathrm{H}^{0}(K,\mathbf{W}_{\psi^{\tau}\chi}(r)))=\mathrm{H}^{1}(\Gamma,\mathrm{H}^{0}(K_{\infty},W_{\psi^{\tau}\chi}(r))),$

whose order is the same as $\mathrm{H}^{0}(K,W_{\psi^{\tau}\chi}(r))$ (using
that $\\#\mathrm{H}^{1}(K_{\infty},W_{\psi^{\tau}\chi}(r))<\infty$ by [Arn07,
Lem. 2.5]). Hence from the Snake Lemma applied to (5.10) we immediately get
the relation

(5.11) $\\#\mathcal{S}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})^{\Gamma}=\\#{\rm
Sel}_{\rm BK}(K,W_{\psi^{\tau}\chi}(r))\cdot\frac{\\#{\rm
ker}(t^{*})}{\\#\mathrm{H}^{0}(K,W_{\psi^{\tau}\chi}(r))}.$

To compute the order of ${\rm ker}(t^{*})$, we first the order of the kernel
of the map

$\tau^{*}=(\tau_{w}^{*})_{w\in\Sigma,w\nmid
p}:\mathcal{P}^{\Sigma}(K)\rightarrow\mathcal{P}^{\Sigma}(K_{\infty})^{\Gamma}.$

For $w=v$ we find

(5.12) $\displaystyle{\rm
ker}(\tau_{v}^{*})=\frac{\mathrm{H}^{1}(K_{v},W_{\psi^{\tau}\chi}(r))}{\mathrm{H}^{1}(K_{v},W_{\psi^{\tau}\chi}(r))_{\rm
div}}$ $\displaystyle\simeq\mathrm{H}^{1}(K_{v},T_{\psi\chi^{\tau}}(r))_{\rm
tors}$ $\displaystyle={\rm
ker}\bigl{(}\mathrm{H}^{1}(K_{v},T_{\psi\chi^{\tau}}(r))\rightarrow\mathrm{H}^{1}(K_{v},V_{\psi\chi^{\tau}}(r))\bigr{)}$
$\displaystyle={\rm
coker}\bigl{(}\mathrm{H}^{0}(K_{v},V_{\psi\chi^{\tau}}(r))\rightarrow\mathrm{H}^{0}(K_{v},W_{\psi\chi^{\tau}}(r))\bigr{)}$
$\displaystyle=\mathrm{H}^{0}(K_{v},W_{\psi\chi^{\tau}}(r)),$

using local Tate duality, the cohomology long exact sequence associated to
$0\rightarrow T_{\psi\chi^{\tau}}(r)\rightarrow
V_{\psi\chi^{\tau}}(r)\rightarrow W_{\psi\chi^{\tau}}(r)\rightarrow 0$, and
the vanishing of $\mathrm{H}^{0}(K_{v},V_{\psi\chi^{\tau}}(r))$ (as follows
from [Arn07, Lem. 2.5]). Similarly, we find

(5.13) $\\#{\rm
ker}(\tau_{\overline{v}}^{*})=\\#\mathrm{H}^{1}(\Gamma,\mathrm{H}^{0}(K_{\overline{v}},\mathbf{W}_{\psi^{\tau}\chi}(r)))=\\#\mathrm{H}^{0}(K_{\overline{v}},W_{\psi^{\tau}\chi}(r)),$

using the finiteness of
$\mathrm{H}^{0}(K_{\overline{v}},\mathbf{W}_{\psi^{\tau}\chi}(r))$ (a
consequence of [Arn07, Lem. 2.5]) for the last equality. On the other hand,
consider a prime $w\in\Sigma$ with $w\nmid p$. If $W_{\psi^{\tau}\chi}$ is
unramified at $w$, then ${\rm ker}(\tau_{w}^{*})=0$ by the same argument as in
[JSW17, p. 389]; if $W_{\psi^{\tau}\chi}$ is ramified at $w$, then the group
$\mathrm{H}^{0}(K_{w},\mathbf{W}_{\psi^{\tau}\chi}(r))$ is finite (see [Arn07,
Lem. 3.4]), and as in (5.13) we find $\\#{\rm
ker}(\tau_{w}^{*})=\\#\mathrm{H}^{0}(K_{w},W_{\psi^{\tau}\chi}(r))$. Together
with [BK90, Lem. 5.10] and the discussion in [_op. cit._ , p. 373], this shows

(5.14) $\\#{\rm ker}(\tau_{w}^{*})=c_{w}(W_{\psi^{\tau}\chi}(r)/K)$

for all $w\in\Sigma$ with $w\nmid p$.

Next, the relation between $\\#{\rm ker}(t^{*})$ and $\\#{\rm ker}(\tau^{*})$
can be found similarly as in [Gre99, Lem. 4.7]. Indeed, by [PW11, Prop. A.2]
(extending and generalizing [GV00, Prop. 2.1] to the anticyclotomic setting),
the $\Lambda_{\mathscr{O}}$-torsionness of
$\mathcal{X}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})$ (as shown in Theorem 4.3.6)
implies surjectivity of the right arrow in (5.9), i.e.
$\mathcal{G}^{\Sigma}(K_{\infty})=\mathcal{P}^{\Sigma}(K_{\infty})$. On the
other hand, from Poitou–Tate duality we have the exact sequence

$0\rightarrow{\rm Sel}_{\rm
BK}(K,W_{\psi^{\tau}\chi}(r))\rightarrow\mathrm{H}^{1}(K,W_{\psi^{\tau}\chi}(r))\rightarrow\mathcal{P}^{\Sigma}(K)\rightarrow{\rm
Sel}_{\overline{v}}(K,T_{\psi\chi^{\tau}}(r))^{\vee}\rightarrow\mathrm{H}^{2}(K,W_{\psi^{\tau}\chi}(r))^{\vee}\rightarrow
0,$

using that ${\rm Hom}(W_{\psi^{\tau}\chi}(r),\mu_{p^{\infty}})\simeq
T_{\psi\chi^{\tau}}(r)$ by the self-duality of
$\psi^{\tau}\chi\mathbf{N}^{-r}$. Since $\\#{\rm Sel}_{\rm
BK}(K,W_{\psi^{\tau}\chi}(r))<\infty$ by assumption, the generalization of
Cassels’ theorem in [Gre99, Prop. 4.13] gives101010Taking
$M=W_{\psi^{\tau}\chi}(r)$ in the notations of _loc. cit._. Note also that the
hypothesis in _loc. cit._ that $\mathrm{H}^{0}(K_{w},M)$ is finite for some
finite $w\in\Sigma$ holds in our case taking $w=v$ thanks to [Arn07, Lem.
2.5].

$\mathcal{P}^{\Sigma}(K)/\mathcal{G}^{\Sigma}(K)\simeq\mathrm{H}^{0}(K,W_{\psi\chi^{\tau}}(r))^{\vee},$

and with this isomorphism, the argument in [Gre99, Prop. 4.7] gives

$\\#{\rm ker}(t^{*})=\\#{\rm
ker}(\tau^{*})\cdot\frac{\\#\mathcal{S}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})_{\Gamma}}{\\#\mathrm{H}^{0}(K,W_{\psi\chi^{\tau}}(r))}.$

Substituting (5.12), (5.13), and (5.14) into this, and the resulting formula
for ${\rm ker}(t^{*})$ into (5.11), we obtain

$\\#\mathcal{S}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})^{\Gamma}=\\#\text{\wncyr{W}}_{{\rm
BK}}(K,W_{\psi^{\tau}\chi}(r))\cdot\\#\mathrm{H}^{0}(K_{\overline{v}},W_{\psi^{\tau}\chi}(r))^{2}\cdot\frac{\mathcal{S}_{v}(\psi^{\tau}\chi\mathbf{N}^{-r})_{\Gamma}}{\\#\mathrm{H}^{0}(K,W_{\psi^{\tau}\chi}(r))^{2}}\cdot\prod_{\begin{subarray}{c}w\in\Sigma,w\nmid
p\end{subarray}}c_{w}(W_{\psi^{\tau}\chi}(r)/K),$

using
$\mathrm{H}^{0}(K_{v},W_{\psi\chi^{\tau}}(r))\simeq\mathrm{H}^{0}(K_{\overline{v}},W_{\psi^{\tau}\chi}(r))$
and
$\mathrm{H}^{0}(K,W_{\psi\chi^{\tau}}(r))\simeq\mathrm{H}^{0}(K,W_{\psi^{\tau}\chi}(r))$
by the action of complex conjugation and the equality $\text{\wncyr{W}}_{\rm
BK}(K,W_{\psi^{\tau}\chi}(r))={\rm Sel}_{\rm BK}(K,W_{\psi^{\tau}\chi}(r))$
that follows from the finiteness of the latter. Together with (5.7) this
concludes the proof. ∎

####  5.3.2. Interlude: $p$-part of TNC in rank $0$

Combined with Theorem 4.3.6, the preceding proposition yields a result on the
Tamagawa number conjecture of Bloch–Kato [BK90] that will play an important
role in the proof of Theorem B.

Recall that a Hecke character
$\xi:K\backslash\mathbb{A}_{K}^{\times}\rightarrow\mathbb{C}^{\times}$ of
infinity type $(a,b)\in\mathbb{Z}^{2}$ in the sense of §2.1 can alternatively
be viewed as homomorphism

$\xi:\mathbb{A}_{K}^{\times}/U\rightarrow\mathbb{C}^{\times}$

trivial on an open subgroup $U$ of $\mathbb{A}_{K}^{\times}$ (that can be
described explicitly in terms of the conductor of $\xi$) satisfying
$\xi(\alpha)=\alpha^{-a}(\alpha^{\tau})^{-b}$ for all $\alpha\in K^{\times}$.
Under our fixed embedding $\imath_{\infty}$, $\xi$ then takes values in a
number field $L$, and for every embedding $\sigma\in{\rm Hom}(L,\mathbb{C})$
we obtain the character

$\xi_{\sigma}:\mathbb{A}_{K}^{\times}/U\xrightarrow{\xi}L^{\times}\xrightarrow{\sigma}\mathbb{C}^{\times}$

by composition. Combining their associated Hecke $L$-functions, can consider
the $L_{\mathbb{C}}:=L\otimes_{\mathbb{Q}}\mathbb{C}$-valued $L$-function

$(L(\xi_{\sigma},s))_{\sigma:L\hookrightarrow\mathbb{C}},$

with $L(\xi,s)$ corresponding to
$\sigma:L\subset\overline{\mathbb{Q}}\xrightarrow{\imath_{\infty}}\mathbb{C}$.

######  Theorem 5.3.2.

Let $(\psi,\chi)$ be as in Theorem 4.3.6, with values in a number field $L$,
and suppose that $L(\psi^{\tau}\chi,r)\neq 0$. Then

$\biggl{(}\frac{L((\psi^{\tau}\chi)_{\sigma},r)}{\Omega_{\sigma}}\biggr{)}\in
L^{\times}\subset L_{\mathbb{C}}^{\times},$

and for all primes $\mathfrak{P}$ of $L$ above $p$ we have

${\rm
ord}_{\mathfrak{P}}\biggl{(}\frac{L(\psi^{\tau}\chi,r)}{\Omega}\biggr{)}={\rm
ord}_{\mathfrak{P}}\biggl{(}\frac{\\#\text{\wncyr{W}}_{{\rm
BK}}(W_{\psi^{\tau}\chi}(r)/K)}{\\#\mathrm{H}^{0}(K,W_{\psi^{\tau}\chi}(r))^{2}}\biggr{)}+\sum_{\begin{subarray}{c}w\in\Sigma,w\nmid
p\end{subarray}}{\rm ord}_{\mathfrak{P}}(c_{w}(W_{\psi^{\tau}\chi}(r)/K)).$

In other words, the $p$-part of the Tamagawa number conjecture holds for
$\psi^{\tau}\chi$.

###### Proof.

The first assertion is proved in [GS81] and (more generally) in [Bla86] (cf.
[BDP12, Thm. 2.12]), and the second is immediate from the combination of
Theorem 2.2.1, Theorem 4.3.6, and Proposition 5.3.1. ∎

####  5.3.3. Formula for $\lambda$

######  Proposition 5.3.3.

Let $\lambda$ be as in Theorem 4.3.1, and suppose
$\mathscr{L}_{v}(\lambda\mathbf{N}^{-r})(0)\neq 0$. Then the map

(5.15) ${\rm loc}_{\overline{v}}:{\rm Sel}_{{\rm
BK}}(K,T_{\lambda}(r))\rightarrow\mathrm{H}^{1}_{f}(K_{\overline{v}},T_{\lambda}(r))$

is nonzero, and letting
$\mathscr{F}_{v}(\lambda\mathbf{N}^{-r})\in\Lambda_{\mathscr{O}}$ be any
generator of the characteristic ideal of
$\mathcal{X}_{v}(\lambda\mathbf{N}^{-r})$ we have

$\mathscr{F}_{v}(\lambda\mathbf{N}^{-r})(0)\;\sim_{p}\;\\#\mathrm{H}^{0}(K_{\overline{v}},W_{\lambda}(r))^{2}\cdot\frac{\\#\text{\wncyr{W}}_{{\rm
BK}}(W_{\lambda}(r)/K)}{\\#\mathrm{H}^{0}(K,W_{\lambda}(r))^{2}}\cdot\\#{\rm
coker}({\rm loc}_{\overline{v}/{\rm
tors}})^{2}\cdot\prod_{\begin{subarray}{c}w\in\Sigma,w\nmid
p\end{subarray}}c_{w}(W_{\lambda}(r)/K),$

where ${\rm loc}_{\overline{v}/{\rm tors}}$ is the composition of ${\rm
loc}_{\overline{v}}$ with