# Nonvanishing of generalised Kato classes and
Iwasawa main conjectures

Francesc Castella  Department of Mathematics, University of California Santa
Barbara, CA 93106, USA<EMAIL_ADDRESS>
###### Abstract.

A construction due to Darmon–Rotger gives rise to generalised Kato classes
$\kappa_{p}(E)$ in the $p$-adic Selmer group ${\rm Sel}(\mathbf{Q},V_{p}E)$ of
elliptic curves $E/\mathbf{Q}$ of positive even analytic rank, where $p>3$ is
any prime of good ordinary reduction for $E$. In [DR16], they conjectured that
$\kappa_{p}(E)\neq 0$ precisely when ${\rm Sel}(\mathbf{Q},V_{p}E)$ is two-
dimensional. The first cases of this conjecture were obtained by the author
with M.-L. Hsieh [CH22]. In this note we give a new proof of the implication

$\kappa_{p}(E)\neq 0\quad\Longrightarrow\quad{\rm dim}_{\mathbf{Q}_{p}}{\rm
Sel}(\mathbf{Q},V_{p}E)=2$

established in _op. cit._ , and show that the converse implication holds if
_and only if_ the restriction map ${\rm loc}_{p}:{\rm
Sel}(\mathbf{Q},V_{p}E)\rightarrow
E(\mathbf{Q}_{p})\hat{\otimes}\mathbf{Q}_{p}$ is nonzero. The present approach
is an adaptation to the non-CM case of the method introduced by the author
[Cas22] in the case of CM elliptic curves.

###### 2020 Mathematics Subject Classification:

Primary 11G05; Secondary 11G40

##  1\. Introduction

After the grounbreaking works of Gross–Zagier and Kolyvagin in the 1980s, the
construction of non-torsion rational points (or more generally, Selmer
classes) on elliptic curves of $E/\mathbf{Q}$ with ${\rm ord}_{s=1}L(E,s)\geq
2$, akin to the construction of Heegner points in the cases of analytic rank
$0$ or $1$, is widely regarded as one of the central open problems in number
theory.

In a recent series of spectacular works [DR14, DR17, DR22, BSV20, BSV22b]
(culminating in the collective volume [BDR+22] with applications to the theory
of Stark–Heegner points), Darmon–Rotger and Bertolini–Seveso–Venerucci
revisited the construction of diagonal cycle classes due to Gross–Kudla [GK92]
and Gross–Schoen [GS95], obtaining in particular an interpolation of these
classes in $p$-adic families. Directly from their geometric construction, one
obtains classes attached to triples $(f,g,h)$ of cuspidal eigenforms of
“balanced” weights $(k,l,m)$ (meaning that none of $k$, $l$, or $m$ is larger
than the sum of the other two), but after $p$-adic interpolation one also gets
classes for weights beyond this range, such as the “unbalanced” weight
$(2,1,1)$.

Of special relevance to the Birch–Swinnerton-Dyer conjecture and its
equivariant refinements is the case where $f$ is the newform of weight $2$
associated to an elliptic curve $E/\mathbf{Q}$, and $g$ and $h$ are the weight
$1$ cuspidal eigenforms associated (as a consequence of the proof of Serre’s
conjecture [KW09]) to degree $2$ odd irreducible Artin representations
$\varrho_{1}$ and $\varrho_{2}$, respectively. In this case, one of the main
results of [DR17] and [BSV22b] relates the resulting _generalised Kato
classes_

$\kappa_{p}(f,g,h)\in{\mathrm{H}}^{1}(\mathbf{Q},V_{p}E\otimes\varrho),\quad\varrho:=\varrho_{1}\otimes\varrho_{2},$

obtained from a specialisation in weights $(2,1,1)$ of a $p$-adic family of
diagonal cycles classes, to the value at $s=1$ of the twisted Hasse–Weil
$L$-funcion $L(E\otimes\varrho,s)$. Together with global duality, this
relation leads to a proof of the implication

$L(E\otimes\varrho,1)\neq 0\quad\Longrightarrow\quad{\rm
Hom}_{G_{\mathbf{Q}}}(V_{\varrho}^{\vee},E(H)\otimes L)=0,$

where $L$ is any number field over which $\varrho$ is defined (assumed to
admit an embedding into $\mathbf{Q}_{p}$ for simplicity), $V_{\varrho}^{\vee}$
is the linear dual of an $L$-vector space affording $\varrho$, and $H$ is the
fixed field of $\mathop{\rm ker}\nolimits(\varrho)$. More precisely, by virtue
of the explicit reciprocity law obtained in _loc. cit._ , the nonvanishing of
$L(E\otimes\varrho,1)$ implies that the class $\kappa_{p}(f,g,h)$ is _non-
crystalline_ at $p$, from where the bound on the $\varrho$-isotypical
component of $E(H)$ follows by global duality.

Interestingly, the same explicit reciprocity law shows that the generalised
Kato classes $\kappa_{p}(f,g,h)$ are Selmer whenever $L(E\otimes\varrho,1)=0$.
Moreover, the representations $\varrho$ for which the construction applies
necessarily have real traces, and in many cases the sign in the functional of
the self-dual $L$-function $L(E\otimes\varrho,s)$ is $+1$. Thus the results of
[DR17] and [BSV22b] provide a construction of (possibly zero _a priori_!)
Selmer classes in situations where ${\rm ord}_{s=1}L(E\otimes\varrho,s)\geq
2$.

In [DR16], Darmon–Rotger carried out a systematic study of their construction
in relation with the Birch–Swinnerton-Dyer conjecture and the elliptic Stark
conjecture of [DLR15]. (More recently, a similar study was carried out by
Bertolini–Seveso–Venerucci [BSV22a], which in particular allows one to
interpret and refine some of the rationality conjectures in [DR16] in terms of
a $p$-adic Birch–Swinnerton-Dyer conjecture for $p$-adic Garrett–Rankin
$L$-functions.) In particular, when $L(E\otimes\varrho,1)=0$ with sign $+1$
they conjectured the equivalence

(1.1) $\kappa_{p}(f,g,h)\neq 0\quad\overset{?}{\Longleftrightarrow}\quad{\rm
dim}_{\mathbf{Q}_{p}}{\rm Sel}(\mathbf{Q},V_{p}E\otimes\varrho)=2.$

This leads to the expectation, when combined with the equivariant
Birch–Swinnerton-Dyer conjecture, that $\kappa_{p}(f,g,h)$ is a nonzero class
in ${\rm Sel}(\mathbf{Q},V_{p}E\otimes V_{\varrho})$ if and only if ${\rm
ord}_{s=1}L(E\otimes\varrho,s)=2$.

Of special interest to the Birch–Swinnerton-Dyer conjecture is the case in
which $\varrho$ contains the trivial representation. This occurs precisely
when $\varrho_{2}\simeq\varrho_{1}^{\vee}$, so that

$V_{\varrho}\simeq L\oplus{\rm ad}^{0}(V_{\varrho_{1}}),$

where ${\rm ad}^{0}(V_{\varrho_{1}})$ is the $3$-dimensional representation of
$G_{\mathbf{Q}}$ acting on the trace zero endomorphisms of $V_{\varrho_{1}}$.
By the Artin formalism, we then have

$L(E\otimes\varrho,s)=L(E,s)\cdot L(E\otimes{\rm ad}^{0}(\varrho_{1})).$

In particular, if $L(E\otimes{\rm ad}^{0}(\varrho_{1}))\neq 0$, denoting by
$\kappa_{p}(E)$ the image of $\kappa_{p}(f,g,h)$ under the resulting natural
projection
${\mathrm{H}}^{1}(\mathbf{Q},V_{p}E\otimes\varrho)\rightarrow{\mathrm{H}}^{1}(\mathbf{Q},V_{p}E)$,
by the Bloch–Kato conjecture [BK90] (which in this case predicts that the
dimensions of the Selmer groups for $V_{p}E$ and $V_{p}E\otimes\varrho$ are
the same) the equivalence (1.1) amounts to the conjectural equivalence

(1.2) $\kappa_{p}(E)\neq 0\quad\overset{?}{\Longleftrightarrow}\quad{\rm
dim}_{\mathbf{Q}_{p}}{\rm Sel}(\mathbf{Q},V_{p}E)=2.$

In [CH22], the author and M.-L. Hsieh proved the first cases of conjecture
(1.2), in situations where $\varrho_{1}$ is induced from a finite order
character of an imaginary quadratic field in which $p$ splits. The key
ingredient in the proof was a leading coefficient formula for the $p$-adic
$L$-function introduced in [BD96] in terms of Howard’s derived $p$-adic
heights. [How04].

The main result of this note (see Theorem 5.2.3) includes a new proof of
[CH22, Thm. A]. The approach is an adaptation of the method introduced by the
author in [Cas22], where the case of CM elliptic curves $E/\mathbf{Q}$ is
treated, and clarifies the role played by the restriction map at $p$

${\rm loc}_{p}:{\rm Sel}(\mathbf{Q},V_{p}E)\rightarrow
E(\mathbf{Q}_{p})\hat{\otimes}\mathbf{Q}_{p}.$

Indeed, in _loc. cit._ it was shown that (under mild hypotheses) the
nonvanishing of $\kappa_{p}(E)$ implies that ${\rm Sel}(\mathbf{Q},V_{p}E)$ is
$2$-dimensional, while for the proof of the converse implication it was
necessary to assume that

(${\rm Loc}_{p}$) ${\rm Sel}(\mathbf{Q},V_{p}E)\neq{\rm ker}({\rm loc}_{p}).$

A novel insight of the approach in this paper is that, assuming ${\rm
dim}_{\mathbf{Q}_{p}}{\rm Sel}(\mathbf{Q},V_{p}E)=2$, condition (${\rm
Loc}_{p}$) is in fact _necessary_ for the nonvanishing of $\kappa_{p}(E)$.

Similarly as in [Cas22], our approach is based on a study of the relation
between two different formulations of the Iwasawa–Greenberg main conjecture
for triple products studied in [ACR23]:

  1. (IMC-1)

one in terms of Hsieh’s triple product $p$-adic $L$-functions [Hsi21];

  2. (IMC-2)

another without reference to $p$-adic $L$-functions, phrased in terms of the
$p$-adic family of diagonal cycles used in the construction of
$\kappa_{p}(f,g,h)$ and $\kappa_{p}(E)$.

In the case where $\varrho_{1}=\varrho_{2}^{\vee}$ is dual to $\varrho_{2}$,
and is induced from a finite order Hecke character of an imaginary quadratic
field $K$, conjecture (IMC-1) can be related to the Iwasawa main conjecture
for $E/K$ in the anticyclotomic setting, and using global duality and the
explicit reciprocity laws of [DR22] and [BSV22b], one can easily show that (in
general) conjectures (IMC-1) and (IMC-2) are equivalent. Thus, from the
results on the anticyclotomic main conjecture stemming from the works of
Bertolini–Darmon [BD05] and Skinner–Urban [SU14], we deduce under mild
hypotheses a proof of conjecture (IMC-2), from where the proof of our main
result follow easily from a variant of Mazur’s control theorem and global
duality.

We conclude this introduction by noting that even though throughout the paper
$p$ is assumed to be a prime of _good_ ordinary reduction, it seems possible
to extend our results to the multiplicative case; it would then be interesting
to compare the resulting $\kappa_{p}(E)$ with the $p$-adic limits of Heegner
points studied in [DF23, FG23] (see also [Cas18] for a related construction).

###  1.1. Acknowledgements

I heartily thank Matteo Longo, Marco Adamo Seveso, Rodolfo Venerucci, and
Stefano Vigni for the opportunity to contribute this note to the proceedings
in celebration of Massimo Bertolini’s 60th birthday. The chief insight
exploited in this paper is that the nonvanishing of $\kappa_{p}(E)$ can be
related to an Iwasawa main conjecture in the spirit of the Heegner point main
conjecture studied by Massimo about 30 years ago [Ber95]. Some of Massimo’s
results since then also play a key role in this paper, and it is a pleasure to
dedicate it to him as an attestation of his lasting and growing impact on the
subject. During the preparation of this note, the author was partially
supported by the NSF grants DMS-1946136 and DMS-2101458.

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

Fix a prime $p>3$. Let $K$ be an imaginary quadratic field of discriminant
$D_{K}$ and assume that

$\textrm{$(p)=\mathfrak{p}\overline{\mathfrak{p}}$ splits in $K$},$

with $\mathfrak{p}$ the prime of $K$ above $p$ induced by a fixed embedding
$\iota_{p}:\overline{\mathbf{Q}}\hookrightarrow\overline{\mathbf{Q}}_{p}$. In
this section we recall from [Hsi21] the construction of the triple product
$p$-adic $L$-function for Hida families, and its relation with the
anticyclotomic $p$-adic $L$-functions of Bertolini–Darmon [BD96] in the case
where two of the Hida families are CM Hida families attached to dual ray class
characters of $K$.

###  2.1. Triple product $p$-adic $L$-function

Let $\mathbb{I}$ be a normal domain finite flat over

$\Lambda:=\mathscr{O}\llbracket{1+p\mathbf{Z}_{p}}\rrbracket,$

where $\mathscr{O}$ is the ring of integers of a finite extension of
$\mathbf{Q}_{p}$. For a positive integer $N$ prime $p$ and a Dirichlet
character
$\chi:(\mathbf{Z}/Np\mathbf{Z})^{\times}\rightarrow\mathscr{O}^{\times}$, we
denote by $S^{o}(N,\chi,\mathbb{I})\subset\mathbb{I}\llbracket{q}\rrbracket$
the space of ordinary $\mathbb{I}$-adic cusp forms of tame level $N$ and
branch character $\chi$ as defined in [Hsi21, §3.1].

Denote by $\mathfrak{X}_{\mathbb{I}}^{+}\subset{\rm
Spec}\,\mathbb{I}(\overline{\mathbf{Q}}_{p})$ the set of _arithmetic points_
of $\mathbb{I}$, consisting of the ring homomorphisms
$Q:\mathbb{I}\rightarrow\overline{\mathbf{Q}}_{p}$ such that
$Q|_{1+p\mathbf{Z}_{p}}$ is given by $z\mapsto z^{k_{Q}}\epsilon_{Q}(z)$ for
some $k_{Q}\in\mathbf{Z}_{\geq 2}$ called the _weight of $Q$_ and
$\epsilon_{Q}(z)\in\mu_{p^{\infty}}$. As in [Hsi21, §3.1], we say that
$\boldsymbol{f}=\sum_{n=1}^{\infty}a_{n}({\boldsymbol{f}})q^{n}\in
S^{o}(N,\chi,\mathbb{I})$ is a _primitive Hida family_ if for every
$Q\in\mathfrak{X}_{\mathbb{I}}^{+}$ the specialization $\boldsymbol{f}_{Q}$
gives the $q$-expansion of an ordinary $p$-stabilised newform of weight
$k_{Q}$ and tame conductor $N$. Attached to such ${\boldsymbol{f}}$ we let
$\mathfrak{X}_{\mathbb{I}}^{\rm cls}\subset\mathfrak{X}_{\mathbb{I}}^{+}$ be
the set of ring homomorphisms $Q$ as above with $k_{Q}\in\mathbf{Z}$ such that
${\boldsymbol{f}}_{Q}$ is the $q$-expansion of a classical modular form.

For ${\boldsymbol{f}}$ a primitive Hida family of tame level $N$, we let

$\rho_{{\boldsymbol{f}}}:G_{\mathbf{Q}}\rightarrow{\rm GL}_{2}({\rm
Frac}\,\mathbb{I})$

denote the associated Galois representation, where ${\rm Frac}\,\mathbb{I}$ is
the field of fractions of $\mathbb{I}$. It will be convenient for us to take
$\rho_{{\boldsymbol{f}}}$ to be the _dual_ of that in [Hsi21, §3.2]; in
particular,
$\det\,\rho_{\boldsymbol{f}}=\chi_{\mathbb{I}}\cdot\varepsilon_{\rm cyc}$ in
the notations of _loc. cit._ , where $\varepsilon_{\rm cyc}$ is the $p$-adic
cyclotomic character. We assume that the associated the residual
representation $\bar{\rho}_{\boldsymbol{f}}$ is absolutely irreducible, and
also denote by

$\rho_{{\boldsymbol{f}}}:G_{\mathbf{Q}}\rightarrow{\rm
Aut}_{\mathbb{I}}(V_{\boldsymbol{f}})\simeq{\rm GL}_{2}(\mathbb{I})$

the realisation of $\rho_{\boldsymbol{f}}$ on a module
$V_{\boldsymbol{f}}\simeq\mathbb{I}^{\oplus 2}$. By [Wil88, Thm. 2.2.2],
restricted to $G_{\mathbf{Q}_{p}}$ the Galois representation
$V_{\boldsymbol{f}}$ fits into a short exact sequence

$0\rightarrow V_{\boldsymbol{f}}^{+}\rightarrow V_{\boldsymbol{f}}\rightarrow
V_{\boldsymbol{f}}^{-}\rightarrow 0,$

where $V_{\boldsymbol{f}}^{-}$ is free of rank $1$ over $\mathbb{I}$, with the
$G_{\mathbf{Q}_{p}}$-action given by the unramified character sending an
arithmetic Frobenius ${\rm Fr}_{p}$ to $a_{p}({\boldsymbol{f}})$.

Associated with ${\boldsymbol{f}}$ there is a $\mathbb{I}$-algebra
homomorphism

$\lambda_{{\boldsymbol{f}}}:\mathbb{T}(N,\mathbb{I})\rightarrow\mathbb{I}$

where $\mathbb{T}(N,\mathbb{I})$ is the Hecke algebra acting on
$\oplus_{\chi}S^{o}(N,\chi,\mathbb{I})$, where $\chi$ runs over the characters
of $(\mathbf{Z}/pN\mathbf{Z})^{\times}$. Let $\mathbb{T}_{\mathfrak{m}}$ be
the local component of $\mathbb{T}(N,\mathbb{I})$ through which
$\lambda_{{\boldsymbol{f}}}$ factors, and following [Hid88] define the
_congruence ideal_ $C({\boldsymbol{f}})$ of ${\boldsymbol{f}}$ by

$C({\boldsymbol{f}}):=\lambda_{{\boldsymbol{f}}}({\rm
Ann}_{\mathbb{T}_{\mathfrak{m}}}({\rm
ker}\,\lambda_{\boldsymbol{f}}))\subset\mathbb{I}.$

When, in addition to absolutely irreducible, $\bar{\rho}_{{\boldsymbol{f}}}$
is also $p$-distinguished (i.e., the semi-simplification of
$\bar{\rho}_{{\boldsymbol{f}}}|_{G_{\mathbf{Q}_{p}}}$ is non-scalar), it
follows from the results of [Wil95] and [Hid88] that $C({\boldsymbol{f}})$ is
generated by a nonzero element $\eta_{{\boldsymbol{f}}}\in\mathbb{I}$.

####  2.1.1. Triple products of Hida families

Let

$({\boldsymbol{f}},{\boldsymbol{g}},{\boldsymbol{h}})\in
S^{o}(N_{f},\chi_{f},\mathbb{I}_{f})\times
S^{o}(N_{g},\chi_{g},\mathbb{I}_{g})\times
S^{o}(N_{h},\chi_{h},\mathbb{I}_{h})$

be a triple of primitive Hida families with

(2.1) $\chi_{f}\chi_{g}\chi_{h}=\omega^{2a}$ for some $a\in\mathbf{Z}$,

where $\omega$ is the Teichmüller character. Put

$\mathcal{R}=\mathbb{I}_{f}\hat{\otimes}_{\mathscr{O}}\mathbb{I}_{g}\hat{\otimes}_{\mathscr{O}}\mathbb{I}_{h},$

which is a finite extension of the three-variable Iwasawa algebra
$\Lambda\hat{\otimes}_{\mathscr{O}}\Lambda\hat{\otimes}_{\mathscr{O}}\Lambda$,
and let

$\mathfrak{X}_{\mathcal{R}}^{\boldsymbol{f}}:=\left\\{(Q_{1},Q_{2},Q_{3})\in\mathfrak{X}_{\mathbb{I}_{\varphi}}^{+}\times\mathfrak{X}_{\mathbb{I}_{g}}^{\rm
cls}\times\mathfrak{X}_{\mathbb{I}_{h}}^{\rm cls}\;:\;\textrm{$k_{Q_{1}}\geq
k_{Q_{2}}+k_{Q_{3}}$ and $k_{Q_{1}}\equiv k_{Q_{2}}+k_{Q_{3}}\;({\rm
mod}\;2)$}\right\\}$

be the weight space for $\mathcal{R}$ in the so-called _${\boldsymbol{f}}$
-unbalanced range_.

Let
$\mathbf{V}=V_{\boldsymbol{f}}\hat{\otimes}_{\mathscr{O}}V_{{\boldsymbol{g}}}\hat{\otimes}_{\mathscr{O}}V_{{\boldsymbol{h}}}$
be the triple tensor product Galois representation attached to
$({\boldsymbol{f}},{\boldsymbol{g}},{\boldsymbol{h}})$, and writing
$\det\mathbf{V}=\mathcal{X}^{2}\varepsilon_{\rm cyc}$ (as is possible by
(2.1)) define

(2.2) ${\mathbf{V}^{\dagger}}:=\mathbf{V}\otimes\mathcal{X}^{-1},$

which is a self-dual twist of $\mathbf{V}$. Define the rank four
$G_{\mathbf{Q}_{p}}$-invariant subspace
$\mathscr{F}_{p}^{\boldsymbol{f}}({\mathbf{V}^{\dagger}})\subset{\mathbf{V}^{\dagger}}$
by

(2.3)
$\mathscr{F}_{p}^{\boldsymbol{f}}({\mathbf{V}^{\dagger}}):=V_{{\boldsymbol{f}}}^{+}\hat{\otimes}_{\mathcal{O}}V_{\boldsymbol{g}}\hat{\otimes}_{\mathcal{O}}V_{{\boldsymbol{h}}}\otimes\mathcal{X}^{-1}.$

For every
$\underline{Q}=(Q_{1},Q_{2},Q_{3})\in\mathfrak{X}_{\mathcal{R}}^{\boldsymbol{f}}$
we denote by
$\mathscr{F}_{p}^{\boldsymbol{f}}({\mathbf{V}_{\underline{Q}}^{\dagger}})\subset{\mathbf{V}_{\underline{Q}}^{\dagger}}$
the corresponding specialisations. Finally, for every rational prime $\ell$
denote by $\varepsilon_{\ell}({\mathbf{V}_{\underline{Q}}^{\dagger}})$ the
epsilon factor attached to the restriction of
${\mathbf{V}_{\underline{Q}}^{\dagger}}$ to $G_{\mathbf{Q}_{\ell}}$ as in
[Tat79, p. 21], and assume that

(2.4) for some $\underline{Q}\in\mathfrak{X}_{\mathcal{R}}^{\boldsymbol{f}}$,
we have $\varepsilon_{\ell}({\mathbf{V}_{\underline{Q}}^{\dagger}})=+1$ for
all prime factors $\ell$ of $N_{f}N_{g}N_{h}$.

As explained in [Hsi21, §1.2], condition (2.4) is independent of
$\underline{Q}$, and it implies that the sign in the functional equation for
the triple product $L$-function

$L({\mathbf{V}_{\underline{Q}}^{\dagger}},s)$

(with center at $s=0$) is $+1$ for all
$\underline{Q}\in\mathfrak{X}_{\mathcal{R}}^{\boldsymbol{f}}$.

######  Theorem 2.1.1.

Let $({\boldsymbol{f}},{\boldsymbol{g}},{\boldsymbol{h}})$ be a triple of
primitive Hida families as above satisfying conditions (2.1) and (2.4). Assume
in addition that:

  * •

$\gcd(N_{f},N_{g},N_{h})$ is square-free,

  * •

the residual representation $\bar{\rho}_{{\boldsymbol{f}}}$ is absolutely
irreducible and $p$-distinguished,

and fix a generator $\eta_{{\boldsymbol{f}}}$ of the congruence ideal of
${\boldsymbol{f}}$. Then there exists a unique element

$\mathscr{L}_{p}^{{\boldsymbol{f}}}({\boldsymbol{f}},{\boldsymbol{g}},{\boldsymbol{h}})\in\mathcal{R}$

such that for all
$\underline{Q}=(Q_{0},Q_{1},Q_{2})\in\mathfrak{X}_{\mathcal{R}}^{{\boldsymbol{f}}}$
of weight $(k_{0},k_{1},k_{2})$ with $\epsilon_{Q_{0}}=1$ we have

$(\mathscr{L}_{p}^{\boldsymbol{f}}({\boldsymbol{f}},{\boldsymbol{g}},{\boldsymbol{h}})(\underline{Q}))^{2}=\Gamma_{{\mathbf{V}_{\underline{Q}}^{\dagger}}}(0)\cdot\frac{L({\mathbf{V}_{\underline{Q}}^{\dagger}},0)}{(\sqrt{-1})^{2k_{0}}\cdot\Omega_{{\boldsymbol{f}}_{Q_{0}}}^{2}}\cdot\mathcal{E}_{p}(\mathscr{F}_{p}^{{\boldsymbol{f}}}({\mathbf{V}_{\underline{Q}}^{\dagger}}))\cdot\prod_{\ell\in\Sigma_{\rm
exc}}(1+\ell^{-1})^{2},$

where:

  * •

$\Gamma_{{\mathbf{V}_{\underline{Q}}^{\dagger}}}(0)=\Gamma_{\mathbf{C}}(c_{\underline{Q}})\Gamma_{\mathbf{C}}(c_{\underline{Q}}+2-k_{1}-k_{2})\Gamma_{\mathbf{C}}(c_{\underline{Q}}+1-k_{1})\Gamma_{\mathbf{C}}(c_{\underline{Q}}+1-k_{2})$,
with

$c_{\underline{Q}}=(k_{0}+k_{1}+k_{2}-2)/2$

and $\Gamma_{\mathbf{C}}(s)=2(2\pi)^{-s}\Gamma(s)$;

  * •

$\Omega_{{\boldsymbol{f}}_{Q_{0}}}$ is the canonical period

$\Omega_{{\boldsymbol{f}}_{Q_{0}}}:=(-2\sqrt{-1})^{k_{0}+1}\cdot\frac{\|{\boldsymbol{f}}_{Q_{0}}^{\circ}\|_{\Gamma_{0}(N_{f})}^{2}}{\iota_{p}(\eta_{{\boldsymbol{f}}_{Q_{0}}})}\cdot\Bigl{(}1-\frac{\chi_{f}^{\prime}(p)p^{k_{0}-1}}{\alpha_{Q_{0}}^{2}}\Bigr{)}\Bigl{(}1-\frac{\chi_{f}^{\prime}(p)p^{k_{0}-2}}{\alpha_{Q_{0}}^{2}}\Bigr{)},$

with ${\boldsymbol{f}}_{Q_{0}}^{\circ}\in S_{k_{0}}(N_{f})$ the newform of
conductor $N_{f}$ associated with ${\boldsymbol{f}}_{Q_{0}}$,
$\chi_{f}^{\prime}$ the prime-to-$p$ part of $\chi_{f}$, and $\alpha_{Q_{0}}$
the specialization of $a_{p}({\boldsymbol{f}})\in\mathbb{I}_{f}^{\times}$ at
$Q_{0}$;

  * •

$\mathcal{E}_{p}(\mathscr{F}_{p}^{{\boldsymbol{f}}}({\mathbf{V}_{\underline{Q}}^{\dagger}}))$
is the modified $p$-Euler factor

$\mathcal{E}_{p}(\mathscr{F}_{p}^{{\boldsymbol{f}}}({\mathbf{V}_{\underline{Q}}^{\dagger}})):=\frac{L_{p}(\mathscr{F}_{p}^{{\boldsymbol{f}}}({\mathbf{V}_{\underline{Q}}^{\dagger}}),0)}{\varepsilon_{p}(\mathscr{F}_{p}^{{\boldsymbol{f}}}({\mathbf{V}_{\underline{Q}}^{\dagger}}))\cdot
L_{p}({\mathbf{V}_{\underline{Q}}^{\dagger}}/\mathscr{F}_{p}^{{\boldsymbol{f}}}({\mathbf{V}_{\underline{Q}}^{\dagger}}),0)}\cdot\frac{1}{L_{p}({\mathbf{V}_{\underline{Q}}^{\dagger}},0)},$

and $\Sigma_{\rm exc}$ is an explicitly defined subset of the prime factors of
$N_{f}N_{g}N_{h}$, [Hsi21, p. 416].

###### Proof.

This is [Hsi21, Thm. A], which also includes an interpolations formula in the
cases where $\epsilon_{Q_{0}}$ is not necessarily trivial. ∎

###  2.2. CM Hida families

Let $K_{\infty}$ be the unique $\mathbf{Z}_{p}^{2}$-extension of $K$, and let
$K_{\mathfrak{p}^{\infty}}$ be the maximal subfield of $K_{\infty}$ unramified
outside $\mathfrak{p}$. Put

$\Gamma_{\infty}:={\rm
Gal}(K_{\infty}/K)\simeq\mathbf{Z}_{p}^{2},\quad\quad\Gamma_{\mathfrak{p}^{\infty}}:={\rm
Gal}(K_{\mathfrak{p}^{\infty}}/K)\simeq\mathbf{Z}_{p}.$

For every ideal $\mathfrak{C}\subset\mathcal{O}_{K}$ we let $K(\mathfrak{C})$
be the ray class field of $K$ of conductor $\mathfrak{C}$ (so in particular
$K_{\mathfrak{p}^{\infty}}$ is the maximal $\mathbf{Z}_{p}$-extension of $K$
inside $K(\mathfrak{p}^{\infty})$). Let ${\rm Art}_{\mathfrak{p}}$ be the
restriction of the global Artin map to $K_{\mathfrak{p}}^{\times}$, with
geometric normalisation. Identifying $\mathbf{Z}_{p}^{\times}$ and
$\mathcal{O}_{K_{\mathfrak{p}}}^{\times}$ via
$\iota_{p}:\overline{\mathbf{Q}}\hookrightarrow\overline{\mathbf{Q}}_{p}$, the
map ${\rm Art}_{\mathfrak{p}}$ induces an embedding
$1+p\mathbf{Z}_{p}\rightarrow\Gamma_{\mathfrak{p}^{\infty}}$. Write
$I_{\mathfrak{p}}^{\rm w}={\rm
Art}_{\mathfrak{p}}(1+p\mathbf{Z}_{p})|_{K_{\mathfrak{p}^{\infty}}}$ and let
$b\geq 0$ be such that $[\Gamma_{\mathfrak{p}^{\infty}}:I_{\mathfrak{p}}^{\rm
w}]=p^{b}$. (Note that $b=0$ if the class number of $K$ is coprime to $p$.)

Fix a topological generator
$\gamma_{\mathfrak{p}}\in\Gamma_{\mathfrak{p}^{\infty}}$ with
$\gamma_{\mathfrak{p}}^{p^{b}}={\rm
Art}_{\mathfrak{p}}(1+p)|_{K_{\mathfrak{p}^{\infty}}}$, and for each variable
$S$ let
$\Psi_{S}:\Gamma_{\infty}\rightarrow\mathscr{O}\llbracket{S}\rrbracket^{\times}$
be the universal character given by

$\Psi_{S}(\sigma)=(1+S)^{l(\sigma)},$

where $l(\sigma)\in\mathbf{Z}_{p}$ is such that
$\sigma|_{K_{\mathfrak{p}^{\infty}}}=\gamma_{\mathfrak{p}}^{l(\sigma)}$. Let
$\mathbf{v}\in\mathscr{O}$ be such that $\mathbf{v}^{p^{b}}=1+p$ (after
enlarging $\mathscr{O}$ if necessary). For any finite order character
$\psi:G_{K}\rightarrow\mathscr{O}^{\times}$ of conductor $\mathfrak{C}$ put

$\boldsymbol{\theta}_{\psi}(S)(q)=\sum_{(\mathfrak{a},\mathfrak{p}\mathfrak{C})=1}\psi(\sigma_{\mathfrak{a}})\Psi^{-1}_{\mathbf{v}^{-1}(1+S)-1}(\sigma_{\mathfrak{a}})q^{\mathbf{N}(\mathfrak{a})}\in\mathscr{O}\llbracket{S}\rrbracket\llbracket{q}\rrbracket,$

where $\sigma_{\mathfrak{a}}\in{\rm
Gal}(K(\mathfrak{C}\mathfrak{p}^{\infty})/K)$ is the Artin symbol of
$\mathfrak{a}$. Then $\boldsymbol{\theta}_{\psi}(S)$ is a primitive Hida
family defined over $\mathscr{O}\llbracket{S}\rrbracket$ of tame level
$D_{K}\mathbf{N}(\mathfrak{C})$ and tame character
$(\psi\circ\mathscr{V})\eta_{K/\mathbf{Q}}\omega^{-1}$, where

$\mathscr{V}:G_{\mathbf{Q}}^{\rm ab}\rightarrow G_{K}^{\rm ab}$

is the transfer map and $\eta_{K/\mathbf{Q}}$ is the quadratic character
associated to $K/\mathbf{Q}$.

###  2.3. Anticyclotomic $p$-adic $L$-functions

Let $f\in S_{2}(\Gamma_{0}(pN_{f}))$, with $p\nmid N_{f}$, be a $p$-ordinary
$p$-stabilised newform of tame level $N_{f}$ defined over $\mathscr{O}$.
Assume that $f$ is the ordinary $p$-stabilisation of the newform $f^{\circ}\in
S_{2}(\Gamma_{0}(N_{f}))$, and let $\alpha_{p}\in\mathscr{O}^{\times}$ be the
$U_{p}$-eigenvalue of $f$. Write

(2.5) $N_{f}=N^{+}N^{-}$

with $N^{+}$ (resp. $N^{-}$) divisible only by primes which are split (resp.
inert) in $K$, and fix an ideal $\mathfrak{N}^{+}\subset\mathcal{O}_{K}$ with
$\mathcal{O}_{K}/\mathfrak{N}^{+}\simeq\mathbf{Z}/N^{+}\mathbf{Z}$.

Let $\Gamma^{-}$ be the Galois group of the anticyclotomic
$\mathbf{Z}_{p}$-extension $K_{\infty}^{-}/K$. By definition, the map
$\sigma\mapsto l(\sigma^{1-\mathbf{c}}|_{K_{\mathfrak{p}^{\infty}}})$ factor
through $\Gamma^{-}$, and we let $\gamma^{-}$ be the topological generator of
$\Gamma^{-}$ mapping to $1$ under the resulting isomorphism
$\Gamma^{-}\simeq\mathbf{Z}_{p}$. We then identity
$\mathscr{O}\llbracket{\Gamma^{-}}\rrbracket$ with the power series ring
$\mathscr{O}\llbracket{T}\rrbracket$ via $\gamma^{-}\mapsto 1+T$.

######  Theorem 2.3.1.

Let $\chi$ be a ring class character of $K$ of conductor $c\mathcal{O}_{K}$
with values in $\mathscr{O}$, and assume that:

  * (i)

$(pN_{f},cD_{K})=1$,

  * (ii)

$N^{-}$ is the squarefree product of an odd number of primes,

  * (iii)

if $q\mid N^{-}$ is a prime with $q\equiv 1\pmod{p}$, then $\bar{\rho}_{f}$ is
ramified at $q$.

Then there exists a unique
$\Theta_{f/K,\chi}(T)\in\mathscr{O}\llbracket{T}\rrbracket$ such that for
every $p$-power root of unity $\zeta$:

$\Theta_{f/K,\chi}(\zeta-1)^{2}=\frac{p^{n}}{\alpha_{p}^{2n}}\cdot\mathcal{E}_{p}(f,\chi,\zeta)^{2}\cdot\frac{L(f^{\circ}/K\otimes\chi\epsilon_{\zeta},1)}{(2\pi)^{2}\cdot\Omega_{f^{\circ},N^{-}}}\cdot
u_{K}^{2}\sqrt{D_{K}}\chi\epsilon_{\zeta}(\sigma_{\mathfrak{N}^{+}})\cdot\varepsilon_{p},$

where:

  * •

$n\geq 0$ is such that $\zeta$ has exact order $p^{n}$,

  * •

$\epsilon_{\zeta}:\Gamma_{\infty}^{-}\rightarrow\mu_{p^{\infty}}$ be the
character defined by $\epsilon_{\zeta}(\gamma^{-})=\zeta$,

  * •

$\mathcal{E}_{p}(f,\chi,\zeta)=\begin{cases}(1-\alpha_{p}^{-1}\chi(\mathfrak{p}))(1-\alpha_{p}\chi(\overline{\mathfrak{p}}))&\textrm{if
$n=0$,}\\\\[1.99997pt] 1&\textrm{if $n>0$,}\end{cases}$

  * •

$\Omega_{f^{\circ},N^{-}}=4\|f^{\circ}\|_{\Gamma_{0}(N_{f})}^{2}\cdot\eta_{f,N^{-}}^{-1}$
is the _Gross period_ of $f^{\circ}$ (see [Hsi21, p. 524]),

  * •

$u_{K}=|\mathcal{O}_{K}^{\times}|/2$,

  * •

$\varepsilon_{p}\in\\{\pm 1\\}$ is the local root number of $f^{\circ}$ at
$p$.

###### Proof.

See [BD96] for the original construction, and [CH18, Thm. A] for the stated
interpolation property. ∎

When $\chi$ is the trivial character, we write $\Theta_{f/K,\chi}(T)$ simply
as $\Theta_{f/K}(T)$.

###  2.4. Factorisation of triple product $p$-adic $L$-functions

Let $f\in S_{2}(pN_{f})$ be a $p$-stabilised newform as in the preceding
section. By Hida theory, $f$ is the specialisation of a unique primitive Hida
family ${\boldsymbol{f}}\in S^{o}(N_{f},\mathbb{I})$ at an arithmetic point
$Q_{0}\in\mathfrak{X}_{\mathbb{I}}^{+}$ of weight $2$. Let $\ell\nmid pN_{f}$
be a prime split in $K$, and let $\chi$ be a ring class character of $K$ of
conductor $\ell^{m}\mathcal{O}_{K}$ for some $m>0$. Denoting by $\mathbf{c}$
the non-trivial automorphism of $K/\mathbf{Q}$, write
$\chi=\psi/\psi^{\mathbf{c}}$ with $\psi$ a ray class character modulo
$\ell^{m}\mathcal{O}_{K}$ with
$\psi^{\mathbf{c}}(\sigma)=\psi(\mathbf{c}\sigma\mathbf{c}^{-1})$. Let

(2.6)
${\boldsymbol{g}}=\boldsymbol{\theta}_{\psi}(S_{1})\in\mathscr{O}\llbracket{S_{1}}\rrbracket\llbracket{q}\rrbracket,\quad{\boldsymbol{g}}^{*}=\boldsymbol{\theta}_{\psi^{-1}}(S_{2})\in\mathscr{O}\llbracket{S_{2}}\rrbracket\llbracket{q}\rrbracket$

be the primitive CM Hida families (of tame level $C=D_{K}\ell^{2m}$) attached
to $\psi$ and $\psi^{-1}$, respectively.

The triple $({\boldsymbol{f}},{\boldsymbol{g}},{\boldsymbol{g}}^{*})$
satisfies conditions (2.1) and (2.4) and the associated triple product
$p$-adic $L$-function
$\mathscr{L}_{p}^{{\boldsymbol{f}}}({\boldsymbol{f}},{\boldsymbol{g}},{\boldsymbol{g}}^{*})$
of Theorem 2.1.1 is an element in
$\mathcal{R}=\mathbb{I}\hat{\otimes}_{\mathscr{O}}\mathscr{O}\llbracket{S_{1}}\rrbracket\hat{\otimes}_{\mathscr{O}}\mathscr{O}\llbracket{S_{2}}\rrbracket\simeq\mathbb{I}\llbracket{S_{1},S_{2}}\rrbracket$.
Put $S=S_{1}$. In the following, we let

(2.7)
$\mathscr{L}_{p}^{\boldsymbol{f}}(f,\underline{\boldsymbol{gg}}^{*})\in\mathscr{O}\llbracket{S}\rrbracket$

be the image of
$\mathscr{L}_{p}^{\boldsymbol{f}}({\boldsymbol{f}},{\boldsymbol{g}},{\boldsymbol{g}}^{*})$
under the map
$\mathbb{I}\llbracket{S_{1},S_{2}}\rrbracket\rightarrow\mathscr{O}\llbracket{S_{1},S_{2}}\rrbracket$
given by the specialisation $Q_{0}:\mathbb{I}\rightarrow\mathscr{O}$ composed
with the quotient
$\mathscr{O}\llbracket{S_{1},S_{2}}\rrbracket\rightarrow\mathscr{O}\llbracket{S_{1},S_{2}}\rrbracket/(S_{1}-S_{2})$.

Let $K(\chi)$ be the field obtained by adjoining to $K$ the values of $\chi$,
and put $K(\chi,\alpha_{p})=K(\chi)(\alpha_{p})$.

######  Proposition 2.4.1.

Assume that:

  * (i)

$N^{-}$ is the squarefree product of an odd number of primes,

  * (ii)

if $q\mid N^{-}$ is a prime with $q\equiv 1\pmod{p}$, then $\bar{\rho}_{f}$ is
ramified at $q$.

Set $T=\mathbf{v}^{-1}(1+S)-1$. Then

$\mathscr{L}_{p}^{\boldsymbol{f}}(f,\underline{\boldsymbol{gg}}^{*})(S)=\pm\mathbf{w}^{-1}\cdot\Theta_{f/K}(T)\cdot
C_{f,\chi}\cdot\sqrt{L^{\rm
alg}(f/K\otimes\chi,1)}\cdot\frac{\eta_{f^{\circ}}}{\eta_{f^{\circ},N^{-}}},$

where $\mathbf{w}$ is a unit in $\mathscr{O}\llbracket{T}\rrbracket$,
$C_{f,\chi}\in K(\chi,\alpha_{p})^{\times}$, and

$L^{\rm alg}(f/K\otimes\chi,1):=\frac{L(f/K\otimes\chi,1)}{4\pi^{2}\lVert
f^{\circ}\rVert_{\Gamma_{0}(N_{f})}^{2}}\in K(\chi).$

###### Proof.

This is immediate from [Hsi21, Prop. 8.1] and the interpolation property of
$\Theta_{f/K,\chi}(0)$. ∎

##  3\. Selmer group decompositions

In this section we introduce two different Selmer groups associated to triple
products of modular forms, and relate them to anticyclotomic Selmer groups
attached to a single modular form.

###  3.1. Selmer groups for triple products of modular form

Let $({\boldsymbol{f}},{\boldsymbol{g}},{\boldsymbol{h}})$ be a triple of
primitive Hida families as in $\S\ref{subsubsec:triple-hida}$ satisfying
(2.1), and let ${\mathbf{V}^{\dagger}}=\mathbf{V}\otimes\mathcal{X}^{-1}$ be
the self-dual twist of the associated big Galois representation.

######  Definition 3.1.1.

Put

$\mathscr{F}^{\rm
bal}_{p}({\mathbf{V}^{\dagger}})=\mathscr{F}_{p}^{2}({\mathbf{V}^{\dagger}}):=\bigl{(}V_{{\boldsymbol{f}}}^{+}\otimes
V_{{\boldsymbol{g}}}^{+}\otimes
V_{{\boldsymbol{h}}}+V_{{\boldsymbol{f}}}^{+}\otimes
V_{{\boldsymbol{g}}}\otimes
V_{{\boldsymbol{h}}}^{+}+V_{{\boldsymbol{f}}}\otimes
V_{{\boldsymbol{g}}}^{+}\otimes
V_{{\boldsymbol{h}}}^{+}\bigr{)}\otimes\mathcal{X}^{-1},$

and define the _balanced local condition_ ${\mathrm{H}}^{1}_{\rm
bal}(\mathbf{Q}_{p},{\mathbf{V}^{\dagger}})$ by

${\mathrm{H}}^{1}_{\rm bal}(\mathbf{Q}_{p},{\mathbf{V}^{\dagger}}):={\rm
im}\bigl{\\{}{\mathrm{H}}^{1}(\mathbf{Q}_{p},\mathscr{F}_{p}^{\rm
bal}({\mathbf{V}^{\dagger}}))\rightarrow{\mathrm{H}}^{1}(\mathbf{Q}_{p},{\mathbf{V}^{\dagger}})\bigr{\\}}.$

Similarly, put

$\mathscr{F}_{p}^{{\boldsymbol{f}}}({\mathbf{V}^{\dagger}}):=\bigl{(}V_{{\boldsymbol{f}}}^{+}\otimes
V_{{\boldsymbol{g}}}\otimes
V_{{\boldsymbol{h}}}\bigr{)}\otimes\mathcal{X}^{-1},$

and define the _${\boldsymbol{f}}$ -unbalanced local condition_
${\mathrm{H}}^{1}_{{\boldsymbol{f}}}(\mathbf{Q}_{p},{\mathbf{V}^{\dagger}})$
by

${\mathrm{H}}^{1}_{{\boldsymbol{f}}}(\mathbf{Q}_{p},{\mathbf{V}^{\dagger}}):={\rm
im}\bigl{\\{}{\mathrm{H}}^{1}(\mathbf{Q}_{p},\mathscr{F}_{p}^{{\boldsymbol{f}}}({\mathbf{V}^{\dagger}}))\rightarrow{\mathrm{H}}^{1}(\mathbf{Q}_{p},{\mathbf{V}^{\dagger}})\bigr{\\}}.$

It is easy to see that the maps appearing in these definitions are injective,
and in the following we shall use this to identify
${\mathrm{H}}^{1}_{?}(\mathbf{Q}_{p},{\mathbf{V}^{\dagger}})$ with
${\mathrm{H}}^{1}(\mathbf{Q}_{p},\mathscr{F}_{p}^{?}({\mathbf{V}^{\dagger}}))$
for $?\in\\{{\rm bal},{\boldsymbol{f}}\\}$. Fix a finite set of primes $S$
containing $\infty$ and the primes dividing $N_{f}N_{g}N_{g}$, and let
$G_{\mathbf{Q},S}$ be the Galois group of the maximal extension of
$\mathbf{Q}$ unramified outside $S$.

######  Definition 3.1.2.

Let $?\in\\{{\rm bal},{\boldsymbol{f}}\\}$, and define the Selmer group ${\rm
Sel}^{?}(\mathbf{Q},{\mathbf{V}^{\dagger}})$ by

${\rm Sel}^{?}(\mathbf{Q},{\mathbf{V}^{\dagger}}):=\mathop{\rm
ker}\nolimits\biggl{\\{}{\mathrm{H}}^{1}(G_{\mathbf{Q},S},{\mathbf{V}^{\dagger}})\rightarrow\frac{{\mathrm{H}}^{1}(\mathbf{Q}_{p},{\mathbf{V}^{\dagger}})}{{\mathrm{H}}^{1}_{?}(\mathbf{Q}_{p},{\mathbf{V}^{\dagger}})}\biggr{\\}}$

We call ${\rm Sel}^{\rm bal}(\mathbf{Q},{\mathbf{V}^{\dagger}})$ (resp. ${\rm
Sel}^{{\boldsymbol{f}}}(\mathbf{Q},{\mathbf{V}^{\dagger}})$) the _balanced_
(resp. _${\boldsymbol{f}}$ -unbalanced_) Selmer group.

Let ${\mathbf{A}^{\dagger}}={\rm
Hom}_{\mathbf{Z}_{p}}({\mathbf{V}^{\dagger}},\mu_{p^{\infty}})$ be the
arithmetic dual of ${\mathbf{V}^{\dagger}}$, and for $?\in\\{{\rm
bal},{\boldsymbol{f}}\\}$ define
${\mathrm{H}}^{1}_{?}(\mathbf{Q}_{p},{\mathbf{A}^{\dagger}})\subset{\mathrm{H}}^{1}(\mathbf{Q}_{p},{\mathbf{A}^{\dagger}})$
to be the orthogonal complement of
${\mathrm{H}}^{1}_{?}(\mathbf{Q}_{p},{\mathbf{V}^{\dagger}})$ under the local
Tate duality

${\mathrm{H}}^{1}(\mathbf{Q}_{p},{\mathbf{V}^{\dagger}})\times{\mathrm{H}}^{1}(\mathbf{Q}_{p},{\mathbf{A}^{\dagger}})\rightarrow\mathbf{Q}_{p}/\mathbf{Z}_{p}.$

Similarly as above, we then define the balanced and
${\boldsymbol{f}}$-unbalanced Selmer groups with coefficients in
${\mathbf{A}^{\dagger}}$ by

${\rm Sel}^{?}(\mathbf{Q},{\mathbf{A}^{\dagger}}):=\mathop{\rm
ker}\nolimits\biggl{\\{}{\mathrm{H}}^{1}(G_{\mathbf{Q},S},{\mathbf{A}^{\dagger}})\rightarrow\frac{{\mathrm{H}}^{1}(\mathbf{Q}_{p},{\mathbf{A}^{\dagger}})}{{\mathrm{H}}_{?}^{1}(\mathbf{Q}_{p},{\mathbf{A}^{\dagger}})}\times\prod_{\ell\in
S\smallsetminus\\{p\\}}{\mathrm{H}}^{1}(\mathbf{Q}_{\ell},{\mathbf{A}^{\dagger}})\biggr{\\}},$

and let $X^{?}(\mathbf{Q},{\mathbf{A}^{\dagger}})={\rm
Hom}_{\mathbf{Z}_{p}}({\rm
Sel}^{?}(\mathbf{Q},{\mathbf{A}^{\dagger}}),\mathbf{Q}_{p}/\mathbf{Z}_{p})$
denote the Pontryagin dual of ${\rm
Sel}^{?}(\mathbf{Q},{\mathbf{A}^{\dagger}})$.

###  3.2. Anticyclotomic Selmer groups for modular forms

Let $f\in S_{2}(\Gamma_{0}(pN_{f}))$ be an ordinary $p$-stabilised newform and
$K/\mathbf{Q}$ an imaginary quadratic field as in $\S\ref{subsec:anticyc-Lp}$.

Let $V_{f}$ be the $p$-adic Galois representation associated to $f$. We adopt
the convention that if $f$ corresponds to the isogeny class of an elliptic
curve $E/\mathbf{Q}$, then $V_{f}\simeq V_{p}E$ (rather than its dual). By
$p$-ordinarity, restricted to $G_{\mathbf{Q}_{p}}$ the representation $V_{f}$
fits into a short exact sequence

$0\rightarrow V_{f}^{+}\rightarrow V_{f}\rightarrow V_{f}^{-}\rightarrow 0$

with $V_{f}^{\pm}$ both $1$-dimensional, and with the
$G_{\mathbf{Q}_{p}}$-action on $V_{f}^{-}$ given by the unramified character
sending arithmetic Frobenius to $\alpha_{p}\in\mathscr{O}^{\times}$, the
$U_{p}$-eigenvalue of $f$.

Below we fix $\Sigma$ to be any finite set of places of $K$ containing
$\infty$ and the prime dividing $pN_{f}$, and for any field extension $L/K$
let $G_{L,\Sigma}$ the Galois group of the maximal extension of $L$ unramified
outside $\Sigma$.

######  Definition 3.2.1.

Let $L$ be a finite field extension of $K$, and let
$\mathscr{F}=\\{\mathscr{F}_{v}(V_{f})\\}_{v\mid p}$ be a collection a
$G_{K_{v}}$-stable subpaces $\mathscr{F}_{v}(V_{f})\subset V_{f}$ for $v\mid
p$. We define the _Greenberg Selmer group_ ${\rm Sel}_{\mathscr{F}}(L,V_{f})$
by

${\rm Sel}_{\mathscr{F}}(L,V_{f}):={\rm
ker}\biggl{\\{}{\mathrm{H}}^{1}(G_{L,\Sigma},V_{f})\rightarrow\prod_{w}\frac{{\mathrm{H}}^{1}(L_{w},V_{f})}{{\mathrm{H}}^{1}_{\mathscr{F}}(L_{w},V_{f})}\biggr{\\}},$

where $w$ runs over the finite primes of $L$ lying above a prime $v\in\Sigma$,
and

${\mathrm{H}}^{1}_{\mathscr{F}}(L_{w},V_{f}):=\begin{cases}\mathop{\rm
ker}\nolimits\\{{\mathrm{H}}^{1}(L_{w},V_{f})\rightarrow{\mathrm{H}}^{1}(L_{w}^{\rm
ur},V_{f})\\}&\textrm{if $w\nmid p$,}\\\\[1.99997pt] {\rm
im}\\{{\mathrm{H}}^{1}(L_{w},\mathscr{F}_{v}(V_{f}))\rightarrow{\mathrm{H}}^{1}(L_{w},V_{f})\\}&\textrm{if
$w\mid v\mid p$}.\end{cases}$

We shall be particularly interested in the following choices of $\mathscr{F}$:

  * •

The _relaxed-strict_ Selmer group ${\rm Sel}_{\emptyset,0}(L,V_{f})$ obtained
by taking

$\mathscr{F}_{v}(V_{f})=\begin{cases}V_{f}&\textrm{if
$v=\mathfrak{p}$,}\\\\[1.99997pt] 0&\textrm{if
$v=\overline{\mathfrak{p}}$.}\end{cases}$

  * •

The _ordinary_ Selmer group ${\rm Sel}(L,V_{f})$ obtained by taking
$\mathscr{F}_{v}(V_{f})=V_{f}^{+}$ for all $v\mid p$.

For a $G_{K}$-stable lattice $T_{f}\subset V_{f}$, we let
${\mathrm{H}}^{1}_{\mathscr{F}}(L_{w},T_{f})$ be the inverse image of
${\mathrm{H}}^{1}_{\mathscr{F}}(L_{w},V_{f})$ under the natural map
${\mathrm{H}}^{1}(L_{w},T_{f})\rightarrow{\mathrm{H}}^{1}(L_{w},V_{f})$, and
define ${\rm Sel}_{\mathscr{F}}(L,T_{f})$ by the same recipe as above. Then,
for $A_{f}:={\rm Hom}_{\mathbf{Z}_{p}}(T_{f},\mu_{p^{\infty}})$, we define the
_dual Selmer group_ ${\rm Sel}_{\mathscr{F}^{*}}(L,A_{f})$ by

${\rm Sel}_{\mathscr{F}^{*}}(L,A_{f}):=\mathop{\rm
ker}\nolimits\biggl{\\{}{\mathrm{H}}^{1}(G_{L,\Sigma},A_{f})\rightarrow\prod_{w}\frac{{\mathrm{H}}^{1}(L_{w},A_{f})}{{\mathrm{H}}^{1}_{\mathscr{F}^{*}}(L_{w},A_{f})}\biggr{\\}}$

where ${\mathrm{H}}^{1}_{\mathscr{F}^{*}}(L_{w},A_{f})$ is the orthogonal
complement of ${\mathrm{H}}^{1}_{\mathscr{F}}(L_{w},T_{f})$ under local Tate
duality

${\mathrm{H}}^{1}(L_{w},T_{f})\times{\mathrm{H}}^{1}(L_{w},A_{f})\rightarrow\mathbf{Q}_{p}/\mathbf{Z}_{p}.$

Write $K_{m}^{-}$ for the subextension of the the anticyclotomic
$\mathbf{Z}_{p}$-extension $K_{\infty}^{-}$ with $[K_{m}^{-}:K]=p^{m}$, and
put

${\rm Sel}_{\mathscr{F}}(K_{\infty}^{-},T_{f}):=\varprojlim_{m}{\rm
Sel}_{\mathscr{F}}(K_{m}^{-},T_{f}),\quad{\rm
Sel}_{\mathscr{F}}(K_{\infty}^{-},A_{f}):=\varinjlim_{m}{\rm
Sel}_{\mathscr{F}}(K_{m}^{-},A_{f}),$

where the limits are with respect to the corestriction and restriction maps,
respectively. Let
$\Psi:G_{K}\rightarrow\mathscr{O}\llbracket{\Gamma^{-}}\rrbracket^{\times}$ be
the character arising from the projection $G_{K}\rightarrow\Gamma^{-}$.
Writing $T_{f}\otimes\Psi^{-1}$ (resp. $A_{f}\otimes\Psi$) for the module
$T_{f}\otimes_{\mathscr{O}}\mathscr{O}\llbracket{\Gamma^{-}}\rrbracket$ (resp.
$A_{f}\otimes_{\mathscr{O}}\mathscr{O}\llbracket{\Gamma^{-}}\rrbracket$) with
$G_{K}$-action on the second factor given by $\Psi^{-1}$ (resp. $\Psi$), we
then have natural $\mathscr{O}\llbracket{\Gamma^{-}}\rrbracket$-module pseudo-
isomorphisms

(3.1) ${\rm Sel}_{\mathscr{F}}(K_{\infty}^{-},T_{f})\sim{\rm
Sel}_{\mathscr{F}}(K,T_{f}\otimes\Psi^{-1}),\quad{\rm
Sel}_{\mathscr{F}}(K_{\infty}^{-},A_{f})\sim{\rm
Sel}_{\mathscr{F}}(K,A_{f}\otimes\Psi),$

where the Selmer groups on the right-hand side are defined is the same way as
in Definition 3.2.1, with
$\mathscr{F}_{v}(T_{f}\otimes\Psi^{-1}):=(\mathscr{F}_{v}(V_{f})\cap
T_{f})\otimes\Psi^{-1}$ and $\mathscr{F}_{v}(A_{f}\otimes\Psi):={\rm
Hom}_{\mathbf{Z}_{p}}(T_{f}/(\mathscr{F}_{v}(V_{f})\cap
T_{f}),\mu_{p^{\infty}})\otimes\Psi$.

###  3.3. Decomposition of triple product Selmer groups

Suppose now that ${\boldsymbol{f}}$ is the Hida family passing through the
$p$-stabilised newform $f\in S_{2}(\Gamma_{0}(pN_{f}))$, and
$({\boldsymbol{g}},{\boldsymbol{g}}^{*})=(\boldsymbol{\theta}_{\psi}(S_{1}),\boldsymbol{\theta}_{\psi^{-1}}(S_{2}))$
are CM Hida families as in (2.6).

For any primitive Hida family $\boldsymbol{\phi}$, we now let
$V_{\boldsymbol{\phi}}$ be the realization of $\rho_{\boldsymbol{\phi}}$
arising in the $p$-adic étale cohomology of the $p$-tower of modular curves as
in Ohta’s works [Oht99, Oht00], following the conventions in [KLZ17, §7.2]
(except that a “Hida family” for us is a “branch” in their sense).

In the case of ${\boldsymbol{g}}$ (and similarly ${\boldsymbol{g}}^{*}$), when
$\chi:=\psi/\psi^{\mathbf{c}}$ satisfies

(3.2) $\bar{\chi}|_{G_{K_{\mathfrak{p}}}}\neq 1,$

where $G_{K_{\mathfrak{p}}}\subset G_{K}$ is a decomposition group at
$\mathfrak{p}$, it follows from a slight extension of the isomorphism in
[LLZ15, Cor. 5.2.5] (see [BL18, $\S$3.2.3]) that

$V_{{\boldsymbol{g}}}\simeq{\rm Ind}_{K}^{\mathbf{Q}}(\psi^{-1}\Psi_{T_{1}}),$

where $T_{1}=\mathbf{v}^{-1}(1+S_{1})-1$. Suppose
$Q_{0}:\mathbb{I}\rightarrow\mathscr{O}$ is such that $f$ is the
specialisation of ${\boldsymbol{f}}$ at $Q_{0}$, and write
${\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}}$ for the resulting
specialisation of ${\mathbf{V}^{\dagger}}$. Similarly setting
$T_{2}=\mathbf{v}^{-1}(1+S_{2})-1$ and noting that $(\det
V_{\boldsymbol{g}})(\det
V_{{\boldsymbol{g}}^{*}})=\Psi_{T_{1}}\Psi_{T_{2}}\circ\mathscr{V}$, we then
have

(3.3) $\displaystyle{\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}}$
$\displaystyle\simeq T_{f}\otimes{\rm
Ind}_{K}^{\mathbf{Q}}(\psi^{-1}\Psi_{T_{1}})\otimes{\rm
Ind}_{K}^{\mathbf{Q}}(\psi\Psi_{T_{2}})\otimes(\Psi_{T_{1}}^{1/2}\Psi_{T_{2}}^{1/2}\circ\mathscr{V})^{-1}$
$\displaystyle\simeq\bigl{(}T_{f}\otimes{\rm
Ind}_{K}^{\mathbf{Q}}\Psi_{W_{1}}^{1-\mathbf{c}}\bigr{)}\oplus\bigl{(}T_{f}\otimes{\rm
Ind}_{K}^{\mathbf{Q}}\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}}\bigr{)},$

where $T_{f}$ is the specialisation of $V_{{\boldsymbol{f}}}$ at $Q_{0}$ and

$W_{1}=\mathbf{v}^{-1}(1+S_{1})^{1/2}(1+S_{2})^{1/2}-1,\quad
W_{2}=(1+S_{1})^{1/2}(1+S_{2})^{-1/2}-1,$

In particular, together with Shapiro’s lemma this gives

(3.4)
${\mathrm{H}}^{1}(\mathbf{Q},{\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}})\simeq{\mathrm{H}}^{1}(K,T_{f}\otimes\Psi_{W_{1}}^{1-\mathbf{c}})\oplus{\mathrm{H}}^{1}(K,T_{f}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}})$

######  Proposition 3.3.1.

Suppose $\psi$ is a ray class character of $K$ such that
$\chi=\psi/\psi^{\mathbf{c}}$ satisfies (3.2). Then under the isomorphism
(3.4), the balanced Selmer group decomposes as

$\displaystyle{\rm Sel}^{\rm
bal}(\mathbf{Q},{\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}})$
$\displaystyle\simeq{\rm
Sel}_{\emptyset,0}(K,T_{f}\otimes\Psi_{W_{1}}^{1-\mathbf{c}})\oplus{\rm
Sel}(K,T_{f}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}}),$

and the ${\boldsymbol{f}}$-unbalanced Selmer group decomposes as

$\displaystyle{\rm
Sel}^{{\boldsymbol{f}}}(\mathbf{Q},{\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}})$
$\displaystyle\simeq{\rm
Sel}(K,T_{f}\otimes\Psi_{W_{1}}^{1-\mathbf{c}})\oplus{\rm
Sel}(K,T_{f}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}}).$

###### Proof.

This is shown in [CD23, Prop. 5.3.1]; we recall the details for the
convenience of the reader. From (3.3), we see that

${\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}}|_{G_{\mathbf{Q}_{p}}}\simeq\bigl{(}T_{f}\otimes\Psi_{W_{1}}^{1-\mathbf{c}}\bigr{)}\oplus(T_{f}\otimes\Psi_{W_{1}}^{\mathbf{c}-1}\bigr{)}\oplus\bigl{(}T_{f}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}}\bigr{)}\oplus\bigl{(}T_{f}\otimes\chi\Psi_{W_{2}}^{\mathbf{c}-1}\bigr{)}$

noting that $\chi^{\mathbf{c}}=\chi^{-1}$, and we find that the balanced local
condition is given by

(3.5) $\mathscr{F}^{\rm
bal}_{p}({\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}})\simeq\bigl{(}T_{f}\otimes\Psi_{W_{1}}^{1-\mathbf{c}}\bigr{)}\oplus\bigl{(}T_{f}^{+}\otimes\chi^{-1}\Psi_{W_{1}}^{1-\mathbf{c}}\bigr{)}\oplus\bigl{(}T_{f}^{+}\otimes\chi\Psi_{W_{2}}^{\mathbf{c}-1}\bigr{)},$

where $T_{f}^{+}\subset T_{f}$ is the specialisation of
$V_{\boldsymbol{f}}^{+}$ at $Q_{0}$. Put
$\widetilde{\mathbb{V}}_{f\boldsymbol{gg}^{*}}^{\dagger}=(T_{f}\otimes\Psi_{W_{1}}^{1-\mathbf{c}})\oplus(T_{f}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}})$,
and note that

(3.6)
${\mathrm{H}}^{1}(\mathbf{Q},{\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}})\simeq{\mathrm{H}}^{1}(K,\widetilde{\mathbb{V}}_{f\boldsymbol{gg}^{*}}^{\dagger})$

by Shapiro’s lemma. For
$\mathfrak{q}\in\\{\mathfrak{p},\overline{\mathfrak{p}}\\}$, letting
$\mathscr{F}_{\mathfrak{q}}^{\rm
bal}(\widetilde{\mathbb{V}}_{f\boldsymbol{gg}^{*}}^{\dagger})$ be the
submodule of $\widetilde{\mathbb{V}}_{f\boldsymbol{gg}^{*}}^{\dagger}$
corresponding to $\mathscr{F}^{\rm
bal}_{p}({\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}})$, from (3.5) we have

(3.7) $\begin{cases}\mathscr{F}^{\rm
bal}_{\mathfrak{p}}(\widetilde{\mathbb{V}}_{f\boldsymbol{gg}^{*}}^{\dagger})\simeq\bigl{(}T_{f}\otimes\Psi_{W_{1}}^{1-\mathbf{c}}\bigr{)}\oplus\bigl{(}T_{f}^{+}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}}\bigr{)},\\\\[1.99997pt]
\mathscr{F}^{\rm
bal}_{\overline{\mathfrak{p}}}(\widetilde{\mathbb{V}}_{f\boldsymbol{gg}^{*}}^{\dagger})\simeq
T_{f}^{+}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}},\end{cases}$

using again that $\chi^{\mathbf{c}}=\chi^{-1}$ for the second isomorphism. As
a result, under the isomorphism (3.6) we have

$\displaystyle{\rm Sel}^{\rm
bal}(\mathbf{Q},{\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}})$
$\displaystyle\simeq$ $\displaystyle\quad\mathop{\rm
ker}\nolimits\biggl{\\{}{\mathrm{H}}^{1}(G_{K,\Sigma},$
$\displaystyle\widetilde{\mathbb{V}}_{f\boldsymbol{gg}^{*}}^{\dagger})\rightarrow\frac{{\mathrm{H}}^{1}(K_{\mathfrak{p}},\widetilde{\mathbb{V}}_{f\boldsymbol{gg}^{*}}^{\dagger})}{{\mathrm{H}}^{1}(K_{\mathfrak{p}},(T_{f}\otimes\Psi_{W_{1}}^{1-\mathbf{c}})\oplus(T_{f}^{+}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}}))}\times\frac{{\mathrm{H}}^{1}(K_{\overline{\mathfrak{p}}},\widetilde{\mathbb{V}}_{f\boldsymbol{gg}^{*}}^{\dagger})}{{\mathrm{H}}^{1}(K_{\overline{\mathfrak{p}}},T_{f}^{+}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}})}\bigr{\\}},$

yielding the result in this case. Similarly, we find that the
${\boldsymbol{f}}$-balanced local condition is given by

$\begin{cases}\mathscr{F}^{{\boldsymbol{f}}}_{\mathfrak{p}}(\widetilde{\mathbb{V}}_{f\boldsymbol{gg}^{*}}^{\dagger})\simeq\bigl{(}T_{f}^{+}\otimes\Psi_{W_{1}}^{1-\mathbf{c}}\bigr{)}\oplus\bigl{(}T_{f}^{+}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}}\bigr{)},\\\\[1.99997pt]
\mathscr{F}^{{\boldsymbol{f}}}_{\overline{\mathfrak{p}}}(\widetilde{\mathbb{V}}_{f\boldsymbol{gg}^{*}}^{\dagger})\simeq\bigl{(}T_{f}^{+}\otimes\Psi_{W_{1}}^{1-\mathbf{c}}\bigr{)}\oplus\bigl{(}T_{f}^{+}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}}\bigr{)},\end{cases}$

and as above we arrive at the claimed description of ${\rm
Sel}^{\boldsymbol{f}}(\mathbf{Q},{\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}})$.
∎

As a consequence, we also obtain decompositions for corresponding Selmer
groups with coefficients in
${\mathbb{A}_{f\boldsymbol{gg}^{*}}^{\dagger}}={\rm
Hom}_{\mathbf{Z}_{p}}({\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}},\mu_{p^{\infty}})$,
mirroring in the case of ${\rm
Sel}^{{\boldsymbol{f}}}(\mathbf{Q},{\mathbb{A}_{f\boldsymbol{gg}^{*}}^{\dagger}})$
the factorisation of $p$-adic $L$-functions in Proposition 2.4.1. Put
$A_{f}={\rm Hom}_{\mathbf{Z}_{p}}(T_{f},\mu_{p^{\infty}})$.

######  Corollary 3.3.2.

Under the asumption in Proposition 3.3.1, we have the decompositions

$\displaystyle{\rm Sel}^{\rm
bal}(\mathbf{Q},{\mathbb{A}_{f\boldsymbol{gg}^{*}}^{\dagger}})$
$\displaystyle\simeq{\rm
Sel}_{0,\emptyset}(K,A_{f}\otimes\Psi_{W_{1}}^{\mathbf{c}-1})\oplus{\rm
Sel}(K,A_{f}\otimes\chi\Psi_{W_{2}}^{\mathbf{c}-1}),$ $\displaystyle{\rm
Sel}^{{\boldsymbol{f}}}(\mathbf{Q},{\mathbb{A}_{f\boldsymbol{gg}^{*}}^{\dagger}})$
$\displaystyle\simeq{\rm
Sel}(K,A_{f}\otimes\Psi_{W_{1}}^{\mathbf{c}-1})\oplus{\rm
Sel}(K,A_{f}\otimes\chi\Psi_{W_{2}}^{\mathbf{c}-1}).$

###### Proof.

This is immediate from Proposition 3.3.1 and local Tate duality. ∎

##  4\. Iwasawa main conjectures

Keeping the setting from $\S\ref{subsec:dec-S}$, and put
$\Lambda_{2}=\mathscr{O}\llbracket{S_{1},S_{2}}\rrbracket$ and

${\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}}={\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}}\hat{\otimes}_{\Lambda_{2}}\Lambda_{2}/(S_{1}-S_{2}),$

where $\Lambda_{2}=\mathscr{O}\llbracket{S_{1},S_{2}}\rrbracket$. In this
section we recall the diagonal cycle main conjecture formulated in [ACR23]
specialised to our setting, and explain its relation with the anticyclotomic
Iwasawa main conjecture for $f$.

###  4.1. Diagonal cycle main conjecture

Let
$\kappa({\boldsymbol{f}},{\boldsymbol{g}},{\boldsymbol{g}}^{*})\in{\mathrm{H}}^{1}(\mathbf{Q},{\mathbf{V}^{\dagger}})$
be the _big diagonal class_ constructed in [BSV22b, §8.1], where
${\boldsymbol{f}}$ is the primitive Hida family passing through $f$, and
denote by

(4.1)
$\kappa(f,\underline{\boldsymbol{gg}}^{*})\in{\mathrm{H}}^{1}(\mathbf{Q},{\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})$

its specialisation. More precisely, from _loc. cit._ one directly obtains a
class as above with coefficient in a representation
${\mathbf{V}^{\dagger}}(N)$ isomorphic (non-canonically) to finitely many
copies of ${\mathbf{V}^{\dagger}}$, where $N={\rm
lcm}(N_{f},N_{g},N_{g^{*}})={\rm lcm}(N_{f},D_{K}\ell)$; to obtain (4.1) we
apply the projection
${\mathbf{V}^{\dagger}}(N)\rightarrow{\mathbf{V}^{\dagger}}$ corresponding to
the choice of level-$N$ test vectors for
$({\boldsymbol{f}},{\boldsymbol{g}},{\boldsymbol{g}}^{*})$ furnished by
[Hsi21, Thm. A].

From the decompositions in Proposition 3.3.1 we obtain

(4.2) $\displaystyle{\rm Sel}^{\rm
bal}(\mathbf{Q},{\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})$
$\displaystyle\simeq{\rm
Sel}_{\emptyset,0}(K,T_{f}\otimes\Psi_{T}^{1-\mathbf{c}})\oplus{\rm
Sel}(K,T_{f}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}})_{/W_{2}},$
$\displaystyle{\rm
Sel}^{{\boldsymbol{f}}}(\mathbf{Q},{\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})$
$\displaystyle\simeq{\rm Sel}(K,T_{f}\otimes\Psi_{T}^{1-\mathbf{c}})\oplus{\rm
Sel}(K,T_{f}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}})_{/W_{2}},$

where $T=T_{1}=\mathbf{v}^{-1}(1+S_{1})-1$ and the subscript $/W_{2}$ denotes
the cokernel of multiplication by $W_{2}$. Put

$\mathscr{F}_{p}^{3}({\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}}):=T_{f}^{+}\hat{\otimes}_{\mathscr{O}}V_{{\boldsymbol{f}}}^{+}\hat{\otimes}_{\mathscr{O}}V_{{\boldsymbol{g}}^{*}}^{+}\otimes\mathcal{X}^{-1},$

and denote by
$\mathscr{F}_{p}^{3}({\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})$
the resulting submodule of
${\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}}$. Similarly
defining $\mathscr{F}_{p}^{\rm
bal}({\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})$ from
$\mathscr{F}_{p}^{\rm
bal}({\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}})=\mathscr{F}_{p}^{2}({\mathbb{V}_{f\boldsymbol{gg}^{*}}^{\dagger}})$,
in terms of the description in (3.7) we find that

(4.3) $\begin{cases}\mathscr{F}^{\rm
bal}_{\mathfrak{p}}(\widetilde{\mathbb{V}}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger})/\mathscr{F}^{3}_{\mathfrak{p}}(\widetilde{\mathbb{V}}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger})\simeq\bigl{(}T_{f}^{-}\otimes\Psi_{T}^{1-\mathbf{c}}\bigr{)}\oplus\bigl{(}T_{f}^{+}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}}\bigr{)}_{/W_{2}},\\\\[1.99997pt]
\mathscr{F}^{\rm
bal}_{\overline{\mathfrak{p}}}(\widetilde{\mathbb{V}}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger})/\mathscr{F}^{3}_{\overline{\mathfrak{p}}}(\widetilde{\mathbb{V}}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger})\simeq(T_{f}^{+}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}})_{/W_{2}},\end{cases}$

where $T_{f}^{-}:=T_{f}/T_{f}^{+}$.

It follows from [BSV22b, Cor. 8.2] that
$\kappa(f,\underline{\boldsymbol{gg}}^{*})$ lies in the balanced Selmer group
${\rm Sel}^{\rm
bal}(\mathbf{Q},{\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})$.
Therefore, viewing this class in
${\mathrm{H}}^{1}(K,\widetilde{\mathbb{V}}^{\dagger}_{f\underline{\boldsymbol{gg}}^{*}})$
via the isomorphism (3.6), we can consider the image of ${\rm
loc}_{\mathfrak{p}}(\kappa(f,\underline{\boldsymbol{gg}}^{*}))$ under the
natural map

$p_{f}^{-}:{\mathrm{H}}^{1}\bigl{(}K_{\mathfrak{p}},\mathscr{F}^{\rm
bal}_{\mathfrak{p}}(\widetilde{\mathbb{V}}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger})\bigr{)}\rightarrow{\mathrm{H}}^{1}\bigl{(}K_{\mathfrak{p}},\mathscr{F}^{\rm
bal}_{\mathfrak{p}}(\widetilde{\mathbb{V}}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger})/\mathscr{F}^{3}_{\mathfrak{p}}(\widetilde{\mathbb{V}}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger})\bigr{)}\rightarrow{\mathrm{H}}^{1}(K_{\mathfrak{p}},T_{f}^{-}\otimes\Psi_{T}^{1-\mathbf{c}})$

arising from the projection onto the first factor in (4.3).

Put

$\Lambda=\mathscr{O}\llbracket{T}\rrbracket.$

Let $u:=1+p$, and for any $\Lambda$-module $M$ and integer $k$, denote by
$M_{k}$ the specialisation of $M$ at $T=u^{k-2}-1$. Letting $\Phi$ be the
field of fractions of $\mathscr{O}$, it is then easy to see that the
Bloch–Kato logarithm and dual exponential maps define isomorphisms

(4.4) $\displaystyle{\rm
log}_{\mathfrak{p}}:{\mathrm{H}}^{1}(K_{\mathfrak{p}},T_{f}^{-}\otimes\Psi_{T}^{1-\mathbf{c}})_{k}\otimes\mathbf{Q}_{p}$
$\displaystyle\rightarrow\Phi,\quad k\geq 3,$ $\displaystyle{\rm
exp}_{\mathfrak{p}}^{*}:{\mathrm{H}}^{1}(K_{\mathfrak{p}},T_{f}^{-}\otimes\Psi_{T}^{1-\mathbf{c}})_{k}\otimes\mathbf{Q}_{p}$
$\displaystyle\rightarrow\Phi,\quad k=2.$

The following fundamental result due to Bertolini–Seveso–Venerucci and
Darmon–Rotger relates $p_{f}^{-}({\rm
loc}_{\mathfrak{p}}(\kappa(f,\underline{\boldsymbol{gg}}^{*})))$ to the
restricted triple product $p$-adic $L$-function
$\mathscr{L}_{p}^{{\boldsymbol{f}}}(f,\underline{\boldsymbol{gg}}^{*})$ in
(2.7).

######  Theorem 4.1.1 (Explicit reciprocity law).

There is an injective $\Lambda$-module homomorphism

${\rm
Log}^{{\boldsymbol{f}}}:{\mathrm{H}}^{1}(K_{\mathfrak{p}},T_{f}^{-}\otimes\Psi_{T}^{1-\mathbf{c}})\rightarrow\Lambda$

with pseudo-null cokernel satisfying for any
$\mathfrak{Z}\in{\mathrm{H}}^{1}(K_{\mathfrak{p}},T_{f}^{-}\otimes\Psi_{T}^{1-\mathbf{c}})$
the interpolation property

${\rm Log}^{{\boldsymbol{f}}}(\mathfrak{Z})_{k}=c_{k}\times\begin{cases}{\rm
log}_{\mathfrak{p}}(\mathfrak{Z}_{k})&\textrm{if $k\geq 3$,}\\\\[1.99997pt]
{\rm exp}_{\mathfrak{p}}(\mathfrak{Z}_{k})&\textrm{if $k=2$,}\end{cases}$

where $c_{k}$ is an explicit nonzero constant, and such that

${\rm Log}^{{\boldsymbol{f}}}\bigl{(}p_{f}^{-}({\rm
loc}_{\mathfrak{p}}(\kappa(f,\underline{\boldsymbol{gg}}^{*})))\bigr{)}=\mathscr{L}_{p}^{\boldsymbol{f}}(f,\underline{\boldsymbol{gg}}^{*}).$

###### Proof.

The map ${\rm Log}^{\boldsymbol{f}}$ is obtained by specialising the three-
variable big regulator map of [BSV22b, §7.1], and the explicit reciprocity law
is a consequence of the results of [BSV22b, Thm. A] and [DR22, Thm. 10]. The
details are explained in [ACR23, Prop. 7.3] and [CD23, Thm. 5.1.1]. ∎

The next result shows the equivalence between a “diagonal cycle main
conjecture” in the spirit of [PR87, Conj. B] and the Iwasawa–Greenberg main
conjecture for
$\mathscr{L}_{p}^{{\boldsymbol{f}}}(f,\underline{\boldsymbol{gg}}^{*})$.

######  Proposition 4.1.2.

The following statements (1)-(2) are equivalent:

  1. (1)

$\kappa(f,\underline{\boldsymbol{gg}}^{*})$ is not $\Lambda$-torsion,

${\rm rank}_{\Lambda}\bigl{(}{\rm Sel}^{\rm
bal}(\mathbf{Q},{\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})\bigr{)}={\rm
rank}_{\Lambda}\bigl{(}X^{\rm
bal}(\mathbf{Q},{\mathbb{A}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})\bigr{)}=1,$

and the following equality holds in $\Lambda\otimes\mathbf{Q}_{p}$:

${\rm char}_{\Lambda}\bigl{(}X^{\rm
bal}(\mathbf{Q},{\mathbb{A}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})_{\rm
tors}\bigr{)}={\rm char}_{\Lambda}\bigl{(}{\rm Sel}^{\rm
bal}(\mathbf{Q},{\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})/(\kappa(f,\underline{\boldsymbol{gg}}^{*}))\bigr{)}^{2}.$

where the subscript ${\rm tors}$ denotes the $\Lambda$-torsion submodule.

  2. (2)

$\mathscr{L}_{p}^{\boldsymbol{f}}(f,\underline{\boldsymbol{gg}}^{*})$ is
nonzero, the modules ${\rm
Sel}^{{\boldsymbol{f}}}(\mathbf{Q},{\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})$
and
$X^{{\boldsymbol{f}}}(\mathbf{Q},{\mathbb{A}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})$
are both $\Lambda$-torsion, and

${\rm
char}_{\Lambda}\bigl{(}X^{{\boldsymbol{f}}}(\mathbf{Q},{\mathbb{A}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})\bigr{)}=\bigl{(}\mathscr{L}_{p}^{\boldsymbol{f}}(f,\underline{\boldsymbol{gg}}^{*})^{2}\bigr{)}$

in $\Lambda\otimes\mathbf{Q}_{p}$.

###### Proof.

This follows from an argument by now standard using Theorem 4.1.1 and global
duality as in [ACR23, Thm. 7.15]. ∎

###  4.2. Anticyclotomic main conjecture for modular forms

For our later use, here we record some known results on the anticyclotomic
Iwasawa main conjecture for modular forms.

Let $\mathbb{F}$ be the residue field of $\mathscr{O}$, and denote by

$\bar{\rho}_{f}:G_{\mathbf{Q}}\rightarrow{\rm GL}_{2}(\mathbb{F})$

the residual representation associated to $f$. Factor $N_{f}=N^{+}N^{-}$ as in
(2.5). Following [KPW17], we say that Assumption (A) holds if:

  * •

$\bar{\rho}_{f}$ is absolutely irreducible,

  * •

$\bar{\rho}_{f}$ is ramified at every prime $q\mid N^{-}$ with
$q\equiv\pm{1}\pmod{p}$,

  * •

either $p>5$ or the image of $\bar{\rho}_{f}$ contains a conjugate of ${\rm
GL}_{2}(\mathbb{F}_{p})$.

Note that this is the same as Assumption (A) from [KPW17], except that the
latter also includes the non-anomalous condition
$a_{p}(f)\not\equiv{1}\pmod{p}$; that the next result holds without this
condition follows from recent advances on Ihara’s lemma, [MS21].

######  Theorem 4.2.1.

Suppose that the ordinary $p$-stabilised newform $f\in
S_{2}(\Gamma_{0}(pN_{f}))$ is $p$-old, and that $N^{-}$ is the squarefree
product of an odd number of primes. Under Assumption (A), the module ${\rm
Sel}(K,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})$ is $\Lambda$-cotorsion, and

${\rm char}_{\Lambda}\bigl{(}{\rm
Sel}(K,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})^{\vee}\bigr{)}=\bigl{(}\Theta_{f/K}(T)^{2}\bigr{)}$

in $\Lambda\otimes\mathbf{Q}_{p}$.

###### Proof.

This follows from the combination of the main result of Bertolini–Darmon
[BD05], as later refined by several authors (see [PW11, CH15, KPW17]), showing
the $\Lambda$-cotornionness ${\rm Sel}(K,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})$
and the integral divisibility

(4.5) ${\rm char}_{\Lambda}\bigl{(}{\rm
Sel}(K,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})^{\vee}\bigr{)}\supset\bigl{(}\Theta_{f/K}(T)^{2}\bigr{)},$

and the converse divisibility that follows from the two-variable main
conjecture for $f/K$ proved in the work of Skinner–Urban and Wan [SU14, Wan15]
(after inverting $p$ in the latter case) restricted to the anticyclotomic
line. ∎

######  Remark 4.2.2.

By using the Euler system divisibility from [CD23, Thm. 5.6.1] (using diagonal
cycles rather than congruence arguments and Heegner points), the ramification
hypothesis on $\bar{\rho}_{f}$ in Theorem 4.2.1 can be replaced by the “big
image” hypothesis in [_op. cit._ , §3.3.2].

##  5\. Main result

In this section we state and prove the main result of this note toward the
nonvanishing conjectures of [DR16] in the setting of rank two elliptic curves.

###  5.1. Generalised Kato classes

Let $E/\mathbf{Q}$ be an elliptic curve of conductor $N_{f}$, and let $p>3$ a
prime of good ordinary reduction for $E$. Let $f\in S_{2}(\Gamma_{0}(pN_{f}))$
be the ordinary $p$-stabilisation of the newform associated to $E$, and let
$({\boldsymbol{g}},{\boldsymbol{g}}^{*})=(\boldsymbol{\theta}_{\psi}(S_{1}),\boldsymbol{\theta}_{\psi^{-1}}(S_{2}))$
be a dual pair of primitive CM Hida families as in (2.6). When
$\chi=\psi/\psi^{\mathbf{c}}$ satisfies (3.2), we have

(5.1)
${\mathrm{H}}^{1}(\mathbf{Q},{\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})\simeq{\mathrm{H}}^{1}(K,T_{f}\otimes\Psi_{T}^{1-\mathbf{c}})\oplus{\mathrm{H}}^{1}(K,T_{f}\otimes\chi^{-1}\Psi_{W_{2}}^{1-\mathbf{c}})_{/W_{2}}$

by specialising (3.4). The following is a special case of the construction of
generalised Kato classes by Darmon–Rotger [DR16] in the _adjoint case_.

######  Definition 5.1.1.

Let $\kappa_{p}(E)$ be the image of
$\kappa(f,\underline{\boldsymbol{gg}}^{*})$ under the composition

${\mathrm{H}}^{1}(\mathbf{Q},{\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})\rightarrow{\mathrm{H}}^{1}(K,T_{f}\otimes\Psi_{T}^{1-\mathbf{c}})\rightarrow{\mathrm{H}}^{1}(K,T_{f})\simeq{\mathrm{H}}^{1}(K,T_{p}E),$

where the first arrow is the projection onto the first direct summand in
(5.1), and the second is induced by the multiplication by $T$ on
$T_{f}\otimes\Psi_{T}^{1-\mathbf{c}}$.

By (4.2), the inclusion $\kappa(f,\underline{\boldsymbol{gg}}^{*})\in{\rm
Sel}^{\rm
bal}(\mathbf{Q},{\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})$
from [BSV22b, Cor. 8.2] implies that

(5.2) $\kappa_{p}(E)\in{\rm Sel}_{\emptyset,0}(K,T_{p}E).$

In the following we shall view $\kappa_{p}(E)$ as a class with coefficients in
$V_{p}E$.

######  Proposition 5.1.2.

The following implication holds

$L(f/K,1)=0\quad\Longrightarrow\quad\kappa_{p}(E)\in{\rm Sel}(K,V_{p}E).$

###### Proof.

By (5.2), the inclusion $\kappa_{p}(E)\in{\rm Sel}(K,V_{p}E)$ holds if and
only if ${\rm res}_{\mathfrak{p}}(\kappa_{p}(E))$ vanishes under the natural
map
${\mathrm{H}}^{1}(K_{\mathfrak{p}},V_{f})\rightarrow{\mathrm{H}}^{1}(K_{\mathfrak{p}},V_{f}^{-})$.
Since from the combination of Theorem 4.1.1, Proposition 2.4.1, and Theorem
2.3.1 yield the equivalence

${\rm exp}_{\mathfrak{p}}({\rm
res}_{\mathfrak{p}}(\kappa_{p}(E)))=0\quad\Longleftrightarrow\quad
L(f/K,1)\cdot L(f/K\otimes\chi,1)=0,$

and the Bloch–Kato dual in (4.4) is an isomorphism, the result follows. ∎

Denote by $E^{K}$ the twist of $E$ by the quadratic character corresponding to
$K/\mathbf{Q}$.

######  Corollary 5.1.3.

Suppose $L(E,1)=0$ and $L(E^{K},1)\neq 0$. Then $\kappa_{p}(E)\in{\rm
Sel}(\mathbf{Q},V_{p}E)$.

###### Proof.

Since Kato’s results [Kat04] show that ${\rm Sel}(\mathbf{Q},V_{p}E^{K})=0$
when $L(E^{K},1)\neq 0$, this follows from Proposition 5.1.2 and the
isomorphism ${\rm Sel}(K,V_{p}E)\simeq{\rm Sel}(\mathbf{Q},V_{p}E)\oplus{\rm
Sel}(\mathbf{Q},V_{p}E^{K})$. ∎

######  Remark 5.1.4.

The proof of Proposition 5.1.2 show that when $L(f/K,1)=0$, the class
$\kappa_{p}(E)$ lies in fact in the kernel of the restriction map on ${\rm
Sel}(K,V_{p}E)$ at the primes above $p$. In particular, in the setting of
Corollary 5.1.3, $\kappa_{p}(E)$ lands in the _strict_ Selmer group

${\rm Sel}_{0}(\mathbf{Q},V_{p}E)={\rm ker}\left\\{{\rm
Sel}(\mathbf{Q},V_{p}E)\xrightarrow{{\rm
loc}_{p}}E(\mathbf{Q}_{p})\hat{\otimes}\mathbf{Q}_{p}\right\\}.$

###  5.2. Nonvanishing of $\kappa_{p}(E)$ in rank two

Denote by

$\rho_{f}:G_{\mathbf{Q}}\rightarrow{\rm Aut}_{\mathbf{F}_{p}}(E[p])\simeq{\rm
GL}_{2}(\mathbf{F}_{p})$

the mod $p$ Galois representation associated to $E$, and assume that:

  * (h1)

$\bar{\rho}_{f}$ is absolutely irreducible,

  * (h2)

there exists a prime $q\|N_{f}$, and if $q\equiv\pm{1}\pmod{p}$ then
$\bar{\rho}_{f}$ is ramified at $q$,

  * (h3)

either $p>5$ of the image of $\bar{\rho}_{f}$ contains a conjugate of ${\rm
GL}_{2}(\mathbb{F}_{p})$.

We shall consider generalised Kato classes as in §5.1 attached to the
following.

######  Choice 5.2.1.

Let $q$ be a prime as in (h2) above, and let $\ell\nmid 6pN_{f}$ be a prime.
Choose an imaginary quadratic field $K$ and a ring class character $\chi$ of
$K$ of conductor dividing $\ell^{\infty}\mathcal{O}_{K}$ such that:

  1. (1)

$q$ is inert in $K$,

  2. (2)

every prime factor of $N_{f}/q$ splits in $K$,

  3. (3)

$\ell$ splits in $K$,

  4. (4)

$\ell$ is ordinary for $E$,

  5. (5)

$L(E^{K},1)\neq 0$ and $L(E/K,\chi,1)\neq 0$,

  6. (6)

$\chi|_{G_{K_{\mathfrak{p}}}}\neq 1$.

######  Remark 5.2.2.

The existence of (infinitely many) $K$ satisfying (1)–(3) and such that
$L(E^{K},1)\neq 0$ follows from [BFH90]; for any such $K$, the results of
[Vat03] (see also [CH18, Thm. D]) ensure that $L(E/K,\chi,1)\neq 0$ for all
but finitely many $\chi$ of $\ell$-power conductor.

Fix $(K,\chi)$ as in Choice 5.2.1. Write $\chi=\psi/\psi^{\mathbf{c}}$ with
$\psi$ a ray class character modulo $\ell^{m}\mathcal{O}_{K}$ for some $m>0$,
let
$({\boldsymbol{g}},{\boldsymbol{g}}^{*})=(\boldsymbol{\theta}_{\psi}(S_{1}),\boldsymbol{\theta}_{\psi^{-1}}(S_{2}))$
be the associated primitive CM Hida families as in (2.6), and let
$\kappa_{p}(E)\in{\mathrm{H}}^{1}(K,V_{p}E)$ the corresponding generalised
Kato class.

By Corollary 5.1.3, when $L(E,1)=0$ we have $\kappa_{p}(E)\in{\rm
Sel}(\mathbf{Q},V_{p}E)$. Moreover, by our choice of $K$ and $\chi$, when
further $E/\mathbf{Q}$ has sign $+1$ (so in particular ${\rm
ord}_{s=1}L(E,s)\geq 2$), the nonvanishing conjecture of Darmon–Rotger [DR16,
Conj. 3.2] (as specialised in [_op. cit._ , §4.5.3] to the setting of rank two
elliptic curves) predicts the equivalence

$\kappa_{p}(E)\neq 0\quad\overset{?}{\Longleftrightarrow}\quad{\rm
dim}_{\mathbf{Q}_{p}}{\rm Sel}(\mathbf{Q},V_{p}E)=2.$

The following is the main result of this note in the direction of this
conjecture.

######  Theorem 5.2.3.

Suppose that $L(E,s)$ vanishes to positive even order at $s=1$. Then

$\kappa_{p}(E)\neq 0\quad\Longrightarrow\quad{\rm dim}_{\mathbf{Q}_{p}}{\rm
Sel}(\mathbf{Q},V_{p}E)=2.$

Conversely, if ${\rm dim}_{\mathbf{Q}_{p}}{\rm Sel}(\mathbf{Q},V_{p}E)=2$ then
$\kappa_{p}(E)\neq 0$ if and only if the restriction map

${\rm loc}_{p}:{\rm Sel}(\mathbf{Q},V_{p}E)\rightarrow
E(\mathbf{Q}_{p})\hat{\otimes}\mathbf{Q}_{p}$

is nonzero. Moreover, in that case $\kappa_{p}(E)$ spans the strict Selmer
group ${\rm Sel}_{0}(\mathbf{Q},V_{p}E)={\rm ker}({\rm loc}_{p})$.

######  Remark 5.2.4.

Theorem 5.2.3 recovers [CH22, Thm. A] under slightly weaker hypotheses on
$\bar{\rho}_{f}$, and moreover, shows that the condition ${\rm
Sel}(\mathbf{Q},V_{p}E)\neq{\rm ker}({\rm loc}_{p})$ is _necessary_ for the
nonvanishing of $\kappa_{p}(E)$ when ${\rm Sel}(\mathbf{Q},V_{p}E)$ is
$2$-dimensional.

######  Remark 5.2.5.

When ${\rm Sel}(\mathbf{Q},V_{p}E)$ is $2$-dimensional and different from
${\rm ker}({\rm loc}_{p})$, it follows from Theorem 5.2.3 that

$\kappa_{p}(E)=\lambda\cdot\bigl{(}{\rm log}_{p}(Q)P-{\rm
log}_{p}(P)Q\bigr{)},$

for some $\lambda\in\mathbf{Q}_{p}^{\times}$, where $(P,Q)$ is any basis for
${\rm Sel}(\mathbf{Q},V_{p}E)$ and ${\rm log}_{p}$ is the composition of ${\rm
loc}_{p}$ with the formal group logarithm on
$E(\mathbf{Q}_{p})\hat{\otimes}\mathbf{Q}_{p}$. This is consistent with the
refined conjecture in [DR16, Conj. 3.12] specialised to the setting of [_op.
cit._ ,§4.5.3], which further predicts a rationality statement for $\lambda$
(see also [BSV22a, Conj. 3.4, Rem. 3.5.2]).

The rest of this section is devoted to the proof of Theorem 5.2.3.

###  5.3. Some Galois cohomology

Put $K_{p}=K\otimes_{\mathbf{Q}}\mathbf{Q}_{p}\simeq K_{\mathfrak{p}}\oplus
K_{\overline{\mathfrak{p}}}$. Let

(5.3) ${\rm loc}_{p}={\rm loc}_{\mathfrak{p}}\oplus{\rm
loc}_{\overline{\mathfrak{p}}}:{\mathrm{H}}^{1}(G_{K,\Sigma},V_{p}E)\rightarrow{\mathrm{H}}^{1}(K_{p},V_{p}E)$

be the restriction map, and denote by $X_{f}$ (resp. $X_{\mathfrak{p}}$) the
image of ${\rm Sel}(K,V_{p}E)$ (resp. ${\rm Sep}_{\emptyset,0}(K,V_{p}E)$)
under ${\rm loc}_{p}$.

######  Lemma 5.3.1.

We have ${\rm dim}_{\mathbf{Q}_{p}}{\rm Sel}_{\emptyset,0}(K,V_{p}E)={\rm
dim}_{\mathbf{Q}_{p}}{\rm Sel}(K,V_{p}E)+\delta$, where

$\delta=\begin{cases}-1&\textrm{if $X_{f}\neq 0$,}\\\\[1.99997pt] 0&\textrm{if
$X_{f}=X_{\mathfrak{p}}=0$,}\\\\[1.99997pt] 1&\textrm{if $X_{f}=0$ and
$X_{\mathfrak{p}}\neq 0$.}\end{cases}$

###### Proof.

Since $X_{f}$ is invariant under the action of complex conjugation, if $X_{f}$
is nonzero then the restriction maps ${\rm
Sel}(K,V_{p}E)\rightarrow{\mathrm{H}}^{1}(K_{\mathfrak{q}},V_{p}E)$ for
$\mathfrak{q}\in\\{\mathfrak{p},\overline{\mathfrak{p}}\\}$ are both nonzero,
and so in this case the result follows from [CHK+23, Lem. 2.2].

Now suppose $X_{f}=0$, so that ${\rm Sel}(K,V_{p}E)$ is the same as the strict
Selmer group ${\rm Sel}_{\rm str}(K,V_{p}E):={\rm ker}({\rm loc}_{p})$, and
consider the exact sequence

(5.4) $0\rightarrow{\rm Sel}_{\rm str}(K,V_{p}E)\rightarrow{\rm
Sel}_{\emptyset,0}(K,V_{p}E)\xrightarrow{{\rm
loc}_{\mathfrak{p}}}{\mathrm{H}}^{1}(K_{\mathfrak{p}},V_{p}E).$

If $X_{\mathfrak{p}}=0$, then (5.4) yields the result, so it remains to
consider the case $X_{\mathfrak{p}}\neq 0$. If ${\rm
dim}_{\mathbf{Q}_{p}}X_{\mathfrak{p}}=2$, then the image of (5.3) contains
$X_{\mathfrak{p}}\oplus X_{\overline{\mathfrak{p}}}$, where
$X_{\overline{\mathfrak{p}}}$ is the image of $X_{\mathfrak{p}}$ under complex
conjugation (equivalently, the image of ${\rm Sel}_{0,\emptyset}(K,V_{p}E)$),
but this contradicts [Ski20, Lem. 2.3.1]. Therefore if $X_{\mathfrak{p}}\neq
0$ then it is one-dimensional, and the result in this case follows again from
(5.4). ∎

######  Remark 5.3.2.

Standard conjectures predict that $X_{f}\neq 0$ unless ${\rm Sel}(K,V_{p}E)=0$
(indeed, the vanishing of $X_{f}$ when ${\rm Sel}(K,V_{p}E)\neq 0$ would imply
that $\Sha(E/K)[p^{\infty}]$ is infinite).

Recall that we set $\Lambda=\mathcal{O}\llbracket{T}\rrbracket$, and for any
$\Lambda$-module $M$ denote by $M_{/T}=M/TM$ the cokernel of multiplication by
$T$. The following is a variant of Mazur’s control theorem [Maz72].

######  Proposition 5.3.3.

Multiplication by $T$ induces natural maps

$\displaystyle r^{*}:{\rm Sel}_{0,\emptyset}(K,A_{f})$
$\displaystyle\rightarrow{\rm
Sel}_{0,\emptyset}(K,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})[T],$ $\displaystyle
r:{\rm Sel}_{\emptyset,0}(K,T_{f}\otimes\Psi_{T}^{1-\mathbf{c}})_{/T}$
$\displaystyle\rightarrow{\rm Sel}_{\emptyset,0}(K,T_{f})$

with finite kernel and cokernel.

###### Proof.

Letting ${\rm Sel}_{\rm rel}(K,A_{f})$ and ${\rm Sel}_{\rm
rel}(L,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})$ be the Selmer groups defined in
the same manner as ${\rm Sel}_{0,\emptyset}(K,A_{f})$ and ${\rm
Sel}_{\emptyset,0}(L,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})$, respectively, but
with the (propagated in the case of $A_{f}$) relaxed local condition at the
primes above $p$, the map $r^{*}$ fits into the commutative diagram with exact
rows

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Sel}_{0,\emptyset}(K,A_{f})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{*}}$$\textstyle{{\rm
Sel}_{\rm
rel}(K,A_{f})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s^{*}}$$\textstyle{{\mathrm{H}}^{1}(K_{\mathfrak{p}},A_{f})\times\frac{{\mathrm{H}}^{1}(K_{\overline{\mathfrak{p}}},A_{f})}{{\mathrm{H}}^{1}(K_{\overline{\mathfrak{p}}},A_{f})_{\rm
div}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{*}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Sel}_{0,\emptyset}(K,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})[T]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Sel}_{\rm
rel}(K,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})[T]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{1}(K_{\mathfrak{p}},A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})[T]\times\\{0\\},}$

where ${\mathrm{H}}^{1}(K_{\overline{\mathfrak{p}}},A_{f})_{\rm div}$ denotes
the maximal divisible submodule of
${\mathrm{H}}^{1}(K_{\overline{\mathfrak{p}}},A_{f})$. The map $t^{*}$ arises
from the cohomology long exact sequence associated to

$0\rightarrow A_{f}\rightarrow
A_{f}\otimes\Psi_{T}^{\mathbf{c}-1}\xrightarrow{\times
T}A_{f}\otimes\Psi_{T}^{\mathbf{c}-1}\rightarrow 0,$

and therefore is surjective with kernel
${\mathrm{H}}^{0}(K_{\mathfrak{p}},A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})_{/T}$.
Since [KO20, Lem. 2.7] implies that
$\\#{\mathrm{H}}^{0}(K_{\mathfrak{p}},A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})<\infty$,
by the Snake Lemma to prove the stated property for $r^{*}$ it suffices to
show that $s^{*}$ has finite kernel and cokernel.

The latter map fits into the commutative diagram with exact rows

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Sel}_{\rm
rel}(K,A_{f})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s^{*}}$$\textstyle{{\mathrm{H}}^{1}(G_{K,\Sigma},A_{f})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{*}}$$\textstyle{\bigoplus_{w\in\Sigma,w\nmid
p}\frac{{\mathrm{H}}^{1}(K_{w},A_{f})}{{\mathrm{H}}^{1}_{f}(K_{w},A_{f})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v^{*}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Sel}_{\rm
rel}(K,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})[T]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{1}(G_{K,\Sigma},A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})[T]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{w\in\Sigma,w\nmid
p}{\mathrm{H}}^{1}(K_{w},A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})[T],}$

where ${\mathrm{H}}^{1}_{f}(K_{w},A_{f})$ is the natural image of $\mathop{\rm
ker}\nolimits\\{{\mathrm{H}}^{1}(K_{w},V_{f})\rightarrow{\mathrm{H}}^{1}(K_{w}^{\rm
ur},V_{f})\\}$ (see Definition 3.2.1). The finiteness of
${\mathrm{H}}^{0}(K_{\mathfrak{p}},A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})$
implies that of
${\mathrm{H}}^{0}(G_{K,\Sigma},A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})$, and so
the map $u^{*}$ has finite kernel. Since $u^{*}$ is clearly surjective and by
[Gre99, Lem. 3.3] (see also [JSW17, §3.3.6]) the kernel of the map $v^{*}$ is
finite, by the Snake Lemma it follows that $s^{*}$ has the desired properties.
This shows the result for $r^{*}$ and the case of $r$ is shown similarly. ∎

###  5.4. Proof of the main result

###### Proof of Theorem 5.2.3.

The decomposition in Corollary 3.3.2 gives

(5.5)
$X^{{\boldsymbol{f}}}(\mathbf{Q},{\mathbb{A}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})\simeq{\rm
Sel}(K,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})^{\vee}\oplus\bigl{(}{\rm
Sel}(K,A_{f}\otimes\chi\Psi_{T}^{\mathbf{c}-1})[T]\bigr{)}^{\vee}.$

The same argument as in the proof of Proposition 5.3.3 shows that the natural
restriction map

${\rm Sel}(K,A_{f}\otimes\chi)\rightarrow{\rm
Sel}(K,A_{f}\otimes\chi\Psi_{T}^{\mathbf{c}-1})[T]$

has finite kernel and cokernel, and therefore from [BD05, Cor. 4] (a
consequence of the divisibility in the anticyclotomic main conjecture proved
in _op. cit._ with $p=\ell$), as extended in [PW11, CH15, KPW17]), it follows
that

$L(E/K,\chi,1)\neq 0\quad\Longrightarrow\quad\\#{\rm
Sel}(K,A_{f}\otimes\chi\Psi_{T}^{\mathbf{c}-1})[T]<\infty.$

Together with Theorem 4.2.1 and Proposition 2.4.1, in light of (5.5) this
shows that
$X^{{\boldsymbol{f}}}(\mathbf{Q},{\mathbb{A}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})$
is $\Lambda$-torsion, with

${\rm
char}_{\Lambda}\bigl{(}X^{{\boldsymbol{f}}}(\mathbf{Q},{\mathbb{A}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})\bigr{)}=\bigl{(}\mathscr{L}_{p}^{\boldsymbol{f}}(f,\underline{\boldsymbol{gg}}^{*})^{2}\bigr{)}$

in $\Lambda\otimes\mathbf{Q}_{p}$. By Proposition 4.1.2, it follows that
$\kappa(f,\underline{\boldsymbol{gg}}^{*})$ is not $\Lambda$-torsion, that
both ${\rm Sel}^{\rm
bal}(\mathbf{Q},{\mathbb{V}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})$
and $X^{\rm
bal}(\mathbf{Q},{\mathbb{A}_{f\underline{\boldsymbol{gg}}^{*}}^{\dagger}})$
have $\Lambda$-rank one, and that

${\rm char}_{\Lambda}\bigl{(}X^{\rm
bal}(\mathbf{Q},{\mathbb{A}^{\dagger}})_{\rm tors}\bigr{)}={\rm
char}_{\Lambda}\bigl{(}{\rm Sel}^{\rm
bal}(\mathbf{Q},{\mathbb{V}^{\dagger}})/(\kappa(f,{\boldsymbol{g}}{\boldsymbol{g}}^{*}))\bigr{)}^{2}$

in $\Lambda\otimes\mathbf{Q}_{p}$. Denote by
$\kappa_{1}(f,\underline{\boldsymbol{gg}}^{*})\in{\rm
Sel}_{\emptyset,0}(K,T_{f}\otimes\Psi_{T}^{1-\mathbf{c}})$ the projection of
$\kappa(f,\underline{\boldsymbol{gg}}^{*})$ onto the first direct summand in
(4.2). By Proposition 3.3.1 and Corollary 3.3.2, the above shows that both
${\rm Sel}_{\emptyset,0}(K,T_{f}\otimes\Psi_{T}^{1-\mathbf{c}})$ and
$X_{0,\emptyset}(K,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})$ have $\Lambda$-rank
one, with

${\rm
char}_{\Lambda}\bigl{(}X_{0,\emptyset}(K,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})_{\rm
tors}\bigr{)}={\rm char}_{\Lambda}\bigl{(}{\rm
Sel}_{\emptyset,0}(K,T_{f}\otimes\Psi_{T}^{1-\mathbf{c}})/(\kappa_{1}(f,\underline{\boldsymbol{gg}}^{*}))\bigr{)}^{2}$

in $\Lambda\otimes\mathbf{Q}_{p}$. By the control theorem of Proposition
5.3.3, this implies that

(5.6) ${\rm corank}_{\mathbf{Z}_{p}}{\rm Sel}_{0,\emptyset}(K,A_{f})={\rm
rank}_{\mathbf{Z}_{p}}\bigl{(}X_{0,\emptyset}(K,A_{f}\otimes\Psi_{T}^{\mathbf{c}-1})_{/T}\bigr{)}=1+2\,{\rm
rank}_{\mathbf{Z}_{p}}(\mathfrak{Z}_{/T}),$

where $\mathfrak{Z}={\rm
Sel}_{\emptyset,0}(K,T_{f}\otimes\Psi_{T}^{1-\mathbf{c}})/(\kappa_{1}(f,\underline{\boldsymbol{gg}}^{*}))$.
Thus we conclude that

${\rm corank}_{\mathbf{Z}_{p}}{\rm
Sel}_{0,\emptyset}(K,A_{f})=1\quad\Longleftrightarrow\quad(\kappa_{1}(f,\underline{\boldsymbol{gg}}^{*})\;{\rm
mod}\;T)\neq 0,$

where $(\kappa_{1}(f,\underline{\boldsymbol{gg}}^{*})\;{\rm mod}\;T)$ is the
image of $\kappa_{1}(f,\underline{\boldsymbol{gg}}^{*})$ in ${\rm
Sel}_{\emptyset,0}(K,T_{f}\otimes\Psi_{T}^{1-\mathbf{c}})_{/T}$. Since the
natural map

${\rm
Sel}_{\emptyset,0}(K,T_{f}\otimes\Psi_{T}^{1-\mathbf{c}})_{/T}\rightarrow{\rm
Sel}_{\emptyset,0}(K,T_{f})$

has finite kernel by Proposition 5.3.3, and it sends
$(\kappa_{1}(f,\underline{\boldsymbol{gg}}^{*})\;{\rm mod}\;T)$ to
$\kappa_{p}(E)$ by construction, we arrive at

(5.7) $\kappa_{p}(E)\neq 0\quad\Longleftrightarrow\quad{\rm
rank}_{\mathbf{Z}_{p}}{\rm Sel}_{\emptyset,0}(K,T_{f})=1,$

using the action of complex conjugation to reverse the roles of $\mathfrak{p}$
and $\overline{\mathfrak{p}}$. Now, assuming $\kappa_{p}(E)\neq 0$, by Theorem
4.2.1 (resp. the $p$-parity conjecture [Nek01]) the case $\delta=0$ (resp.
$\delta=1$) is excluded in Lemma 5.3.1, and so ${\rm dim}_{\mathbf{Q}_{p}}{\rm
Sel}(K,V_{p}E)=2$; since the nonvanishing of $L(E^{K},1)$ implies that ${\rm
Sel}(\mathbf{Q},V_{p}E)={\rm Sel}(K,V_{p}E)$ by Kato’s work, the first
implication in Theorem 5.2.3 follows.

Conversely, if ${\rm dim}_{\mathbf{Q}_{p}}{\rm Sel}(\mathbf{Q},V_{p}E)=2$ (and
so ${\rm dim}_{\mathbf{Q}_{p}}{\rm Sel}(K,V_{p}E)=2$), from (5.6) we see that
the case $\delta=0$ is excluded in Lemma 5.3.1 and the case $\delta=-1$ holds
(i.e., ${\rm dim}_{\mathbf{Q}_{p}}{\rm Sel}_{\emptyset,0}(K,V_{p}E)=1$) if and
only if ${\rm Sel}(\mathbf{Q},V_{p}E)\neq{\rm ker}({\rm loc}_{p})$. By (5.7),
this concludes the proof. ∎

######  Remark 5.4.1.

Even though the approach in this paper recovers a strenghtened form of [CH22,
Thm. A] under slightly weaker hypotheses, a noteworthy advantage of the
approach in _op. cit._ is that — by relating the derived $p$-adic heights
against $\kappa_{p}(E)$ to the leading coefficient of $\Theta_{f/K}(T)$ (see
[CH22, Thm. 5.3]) — it provides a method to numerically verity the
nonvanishing of $\kappa_{p}(E)$, and to relate its rationality properties
predicted by [DR16, Conj. 3.12] to (the leading coefficient part of) the
anticyclotomic $p$-adic Birch–Swinnerton-Dyer conjectures formulated in
[BD96]. (See also [BSV22a] for a more general setting.)

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