\section{Introduction}

Fix a prime $p>2$. Let $E_{/\mathbf{Q}}$ be an elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field $K$ in which $p=\mathfrak{p}\overline{\mathfrak{p}}$ splits. Let $L(E,s)$ be the Hasse--Weil $L$-function of $E_{/\mathbf{Q}}$, and denote by $w\in\{\pm{1}\}$ the sign in its functional equation:
\[
L(E,s)=wL(E,2-s).
\]


\subsection{Rubin's $p$-converse theorem}

We begin by recalling the following result by Rubin \cite{rubin-BSD}, giving the first ``$p$-converse'' to the theorem of Gross--Zagier and Kolyvagin.

\begin{thm-intro}[Rubin]\label{thm:rubin}
Assume that $w=-1$. Then there exists a Heegner point $P\in E(\mathbf{Q})$ 
such that
\[
\left.
\begin{array}{r}
{\rm rank}_\mathbf{Z} E(\mathbf{Q})=1\\[0.3em]
\#\Sha(E_{/\mathbf{Q}})[p^\infty]<\infty
\end{array}
\right\}\;\Longrightarrow\;P\notin E(\mathbf{Q})_{\rm tors},
\]
and hence ${\rm ord}_{s=1}L(E,s)=1$.
\end{thm-intro}

The proof of this result is deduced in \cite[Thm.~4]{rubin-BSD} as a consequence of the Iwasawa main conjecture for $K$ \cite{rubin-IMC}, Perrin-Riou's calculation of the Euler characteristic of a Selmer group attached to $E_{/K}$ \cite{PR-BSMF}, and Bertrand's transcendence result \cite{bertrand}; the claim that ${\rm ord}_{s=1}L(E,1)$ when $P\notin E(\mathbf{Q})_{\rm tors}$ is of course a consequence of the Gross--Zagier formula \cite{GZ}.

A strenghtening of Rubin's $p$-converse theorem, deducing the non-triviality of the Heegner point $P$ from the only assumption that the $p^\infty$-Selmer group ${\rm Sel}_{p^\infty}(E_{/\mathbf{Q}})\subset{\mathrm{H}}^1(\mathbf{Q},E[p^\infty])$ has $\mathbf{Z}_p$-corank one, was recently proved by Burungale--Tian \cite{BT-conv} and later in \cite{BCST-Q} by a different approach.


\subsection{Statement of the main result}

In this paper we prove an analogue of Rubin's $p$-converse theorem  in rank $2$, in which the Heegner point $P$ is replaced by a certain Selmer class 
\[
\kappa_p\in{\rm Sel}(\mathbf{Q},V_pE).
\]
Here ${\rm Sel}(\mathbf{Q},V_pE)$ denotes the $p$-adic Selmer group of $E_{/\mathbf{Q}}$ fitting into the exact sequence
\[
0\rightarrow E(\mathbf{Q})\otimes_{\mathbf{Z}}\mathbf{Q}_p\rightarrow{\rm Sel}(\mathbf{Q},V_pE)\rightarrow{\rm Ta}_p\Sha(E_{/\mathbf{Q}})\otimes_{\mathbf{Z}_p}\mathbf{Q}_p\rightarrow 0,
\]
where ${\rm Ta}_p\Sha(E_{/\mathbf{Q}})$ is the $p$-adic Tate module of the Tate--Shafarevich group of $E_{/\mathbf{Q}}$. 

The class $\kappa_p$ may be viewed as a variant of the \emph{generalised Kato classes} introduced by Darmon--Rotger \cite{DR2,DR2.5} (see \S\ref{subsec:GKC} for our definition) 

and a natural extension of \cite[Conj.~3.2]{DR2.5} predicts that\footnote{This conjecture appears to be very deep, but in the original setting of \cite[\S{4.5.3}]{DR2} it has been numerically verified in some cases; see the Appendix to \cite{cas-hsieh-ord}.} 
\[
\kappa_p\neq 0 \quad\overset{?}\Longleftrightarrow\quad{\rm ord}_{s=1}L(E,s)=2.
\]

Our first main result in this paper is the following.


\begin{thmintro}\label{thm:1}
Assume that $w=1$ and $L(E,1)=0$ (so ${\rm ord}_{s=1}L(E,s)\geq 2$). Then the following implication holds:
\[
\left.
\begin{array}{r}
{\rm rank}_{\mathbf{Z}}E(\mathbf{Q})=2\\[0.3em]
\#\Sha(E_{/\mathbf{Q}})[p^\infty]<\infty
\end{array}
\right\}\;\Longrightarrow\;\kappa_p\neq 0;
\]
more precisely, $\kappa_p$ is a nonzero class in the fine Selmer group ${\rm Sel}_0(\mathbf{Q},V_pE)$.
\end{thmintro}

Recall that the \emph{fine Selmer group} ${\rm Sel}_0(\mathbf{Q},V_pE)$ is defined as the kernel of the restriction  map 
\[
{\rm Sel}(\mathbf{Q},V_pE)\rightarrow{\mathrm{H}}^1(\mathbf{Q}_p,V_pE).
\] 
More generally, Theorem~\ref{cor:main} in the body of the paper  establishes the nonvanishing of $\kappa_p$ under slightly weaker hypotheses: The assumptions that $E(\mathbf{Q})$ has rank $2$ and $\#\Sha(E_{/\mathbf{Q}})[p^\infty]<\infty$ can be replaced by the assumptions ${\rm corank}_{\mathbf{Z}_p}{\rm Sel}_{p^\infty}(E_{/\mathbf{Q}})=2$ and $E(\mathbf{Q})$ has a point of infinite order.

\subsection{Application to rank $2$ Selmer basis} 

The construction of $\kappa_p$ depends on certain auxiliary choices, but it follows from our results
that different choices give rise to the same Selmer class up to scaling. 
Hence it is natural to ask for a class in the $2$-dimensional ${\rm Sel}(\mathbf{Q},V_pE)$ complementary to the line spanned by $\kappa_p$.

Before stating our application to this problem, we need to introduce some more notation. Let $\lambda$ be the Hecke character of $K$ attached to $E$, so that
\[
L(E,s)=L(\lambda,s),
\]
and denote by $\lambda^{\mathbf{c}}$ the composition of $K$ with complex conjugation. 
Denote by ${\mathcal W}$ the completion of the ring of integers of the maximal unramified extension of $\mathbf{Q}_p$, and let ${\mathcal L}_{\mathfrak{p}}={\mathcal L}_{\mathfrak{p},\mathfrak{c}}\in{\mathcal W}\dBr{Z(\mathfrak{c})}$ be the Katz $p$-adic $L$-function recalled in Theorem~\ref{thm:katz} below, where $Z(\mathfrak{c})$ is the Galois group of the extension $K(E[p^\infty]/K)$.  Following \cite{rubin-BSD}, for $s\in\mathbf{Z}_p$ define
\[
L_\mathfrak{p}(s)={\mathcal L}_{\mathfrak{p}}(\lambda\langle\lambda\rangle^{s-1}),\quad
L_\mathfrak{p}^*(s)={\mathcal L}_\mathfrak{p}(\lambda^\mathbf{c}\langle\lambda^\mathbf{c}\rangle^{s-1}),
\]
where $\langle-\rangle:\mathbf{Z}_p^\times\rightarrow 1+p\mathbf{Z}_p$ is the projection onto the $1$-units.

\begin{thmintro}\label{thm:2}
Assume that $\#\Sha(E_{/\mathbf{Q}})[p^\infty]<\infty$ and
the following conditions hold:
\[
{\rm ord}_{s=1}L_{\mathfrak{p}}(s)=2,\quad{\rm ord}_{s=1}L_{\mathfrak{p}}^*(s)=1.
\]
Then ${\rm Sel}(\mathbf{Q},V_pE)$ is $2$-dimensional, with
\[
{\rm Sel}(\mathbf{Q},V_pE)=\mathbf{Q}_p\kappa_p\oplus\mathbf{Q}_p x_\mathfrak{p},
\]
where $\kappa_p$ is a generalised Kato class and $x_\mathfrak{p}$ is a derived elliptic unit.
\end{thmintro}

\begin{proof}
The assumption that ${\rm ord}_{s=1}L_\mathfrak{p}(s)=2$ implies, on the one hand, that 
\[
r:={\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)\leq 2
\] 
by \cite[Thm.~IV.22]{PR-BSMF} and \cite[Thm.~4.1]{rubin-IMC}, and on the other hand that $r\geq 2$ by 
the theorem of Coates--Wiles \cite{CW} and the $p$-parity conjecture \cite{guo-parity}. Therefore $r=2$, and by \cite[Prop.~4.4]{rubin-points} the construction of derived elliptic units in  [\emph{op.\,cit.}, \S{6}] yields a class
\[
x_\mathfrak{p}\in{\rm Sel}(K,T_{\mathfrak{p}}E)\simeq{\rm Sel}(\mathbf{Q},T_pE)
\]
denoted $x_\mathfrak{p}^{(2)}$ in \emph{loc.\,cit.}, where $T_\mathfrak{p} E$ is the $\mathfrak{p}$-adic Tate module of $E$.
By \cite[Thm.~9.5(ii)]{rubin-points}, the assumptions $\#\Sha(E_{/\mathbf{Q}})[p^\infty]<\infty$ and ${\rm ord}_{s=1}L_{\mathfrak{p}}^*(s)=1$ imply that $x_\mathfrak{p}\notin{\rm Sel}_0(\mathbf{Q},V_pE)$, so the result follows from Theorem~\ref{thm:1}. 
\end{proof}


\subsection{A rank~2 Kolyvagin theorem}  We can also prove a converse to Theorem~\ref{thm:1}, going in the direction of Kolyvagin's theorem \cite{kol88}. 

\begin{thmintro}\label{thm:3}
Assume that $w=1$, $L(E,1)=0$, and $E(\mathbf{Q})$ has a point of infinite order. Then the following implication holds: 
\[
\kappa_p\neq 0\quad\Longrightarrow\quad{\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2.
\]
\end{thmintro}

The proof of Theorem~\ref{thm:3} builds on a ``degeneration'' to weights $(1,1)$ of the diagonal cycle Euler system for Rankin--Selberg convolutions constructed in a joint work with Alonso and Rivero \cite{ACR}, and the general theory of anticyclotomic Euler systems developed by Jetchev--Nekov{\'a}{\v{r}}--Skinner \cite{JNS}. The key new observation here 
is that such a result in rank~$2$ can be deduced  from a rank~$1$ Kolyvagin theorem for a non-classical Selmer group.

\subsection{Relation to other works} 

In the non-CM case, the results of Theorem~\ref{thm:1} and \ref{thm:3} were obtained in a joint work with M.-L.~Hsieh \cite{cas-hsieh-ord}. The generalised Kato classes $\kappa_p\in{\mathrm{H}}^1(\mathbf{Q},V_pE)$ considered in \emph{op.\,cit.} are a special case of the classes introduced by Darmon--Rotger \cite{DR2.5}; they are attached to the choice of:
\begin{itemize}
\item[(i)] an imaginary quadratic field $M/\mathbf{Q}$ in which $p$ splits such that such that $L(E^M,1)\neq 0$;
\item[(ii)] a ring class character $\chi$ of $M$ such that $L(E_{/M},\chi,1)\neq 0$.
\end{itemize}

For elliptic curve $E_{/\mathbf{Q}}$ with CM by $K$, by \cite{BFH90} condition (i) can be arranged by taking $M\neq K$, but condition (ii) is problematic, as Vatsal's nonvanishing results for ring class character twists  $L(E_{/M},\chi,1)$ \cite{vatsal-special} requires $E$ to have some prime of multiplicative reduction, but CM elliptic curves have none. 

Our proof of Theorem~\ref{thm:1} and \ref{thm:3} is therefore completely different. The new approach in this paper suggests a close link
between the nonvanishing criterion for generalised Kato classes conjectured by Darmon--Rotger \cite[Conj.~3.2]{DR2.5} in the ``rank~(2,0) adjoint CM setting'' and a main conjecture in anticyclotomic Iwasawa theory. Moreover, the approach should generalise to other contexts (e.g. the case considered in \cite{cas-hsieh-ord}, or its analogue for supersingular primes, \cite{cas-hsieh-ss}).

On the other hand, the original approach of \cite{cas-hsieh-ord}, based on a generalisation of Rubin's height formula and the theory of derived $p$-adic heights, also allows one to relate the ``refined elliptic Stark conjecture'' in \cite[Conj.~3.12]{DR2.5} to an anticyclotomic $p$-adic Birch and Swinnerton-Dyer conjecture (see \cite[\S{5.6}]{cas-hsieh-ord}, and more recently \cite{BSV-garrett} for a similar relation for general triple product $p$-adic $L$-functions). 


\subsection{Acknowledgements} 

It is a pleasure to thank Ashay Burungale for stimulating exchanges, especially about the idea to consider a setting along the lines of our Choice~\ref{choice:setting}, and also Ming-Lun Hsieh for his comments on an earlier draft. 
During the preparation of this paper, the author was partially supported by the NSF grants DMS-1946136 and DMS-2101458.



\section{$p$-adic $L$-functions}

In this section we recall the two $p$-adic $L$-functions that will appear in our arguments, one due to Katz \cite{Katz49} attached to Hecke characters of an imaginary quadratic fields, and another due to Hsieh \cite{hsieh-triple} attached to triple products of modular forms in Hida families. 

From now on we fix a prime $p>2$ and an imaginary quadratic field $K$ with ring of integers $\mathcal O_K$ in which
\begin{equation}\label{eq:spl}
\textrm{$(p)=\mathfrak{p}\overline{\mathfrak{p}}$ splits,}\tag{spl}
\end{equation}
with $\mathfrak{p}$ the prime of $K$ above $p$ determined by a fixed embedding $\iota_p:\overline{\mathbf{Q}}\hookrightarrow\overline{\mathbf{Q}}_p$. 


\subsection{Katz $p$-adic $L$-function}

Denote by $D_K<0$ the discriminant of $K$, and fix an integral ideal $\mathfrak{c}\subset\mathcal O_K$ coprime to $p$. Let $\mathcal{W}$ be a finite extension of the Witt ring $W(\overline{\mathbf{F}}_p)$, and denote by $Z(\mathfrak{c})$ the Galois group of the ray class field extension $K(\mathfrak{c}p^\infty)/K$.

We say that a Hecke character $\chi:K^\times\backslash\mathbb{A}_K^\times\rightarrow\mathbf{C}^\times$ has infinity type $(a,b)$ if $\chi_\infty(z)=z^a\overline{z}^b$.

\begin{thm}\label{thm:katz}
There exists an element ${\mathcal L}_{\mathfrak{p},\mathfrak{c}}\in{\mathcal W}\dBr{Z(\mathfrak{c})}$ such that for all Hecke characters $\chi$ of conductor dividing $\mathfrak{c}p^\infty$ and infinity type $(k+j,-j)$ with $k\geq 1$, $j\geq 0$, we have
\begin{equation}\label{eq:Katz}
{\mathcal L}_{\mathfrak{p},\mathfrak{c}}(\chi)=\biggl(\frac{\Omega_p}{\Omega_\infty}\biggr)^{k+2j}\cdot\Gamma(k+j)\cdot\biggl(\frac{2\pi}{\sqrt{D_K}}\biggr)^j\cdot\mathcal{E}_\mathfrak{p}(\chi)
\cdot L^{(p\mathfrak{c})}(\chi,0),\nonumber
\end{equation}
where $\mathcal{E}_\mathfrak{p}(\chi)$ is the modified Euler factor
\[
\mathcal{E}_\mathfrak{p}(\chi)=\frac{L(0,\chi_\mathfrak{p})}{\varepsilon(0,\chi_\mathfrak{p})\cdot L(1,\chi_\mathfrak{p}^{-1})}
\]
and $L^{(p\mathfrak{c})}(\chi,s)$ is the $L$-function $L^{(p\mathfrak{c})}(\chi,0)$ deprived from the Euler factors at the primes $\mathfrak{l}\vert p\mathfrak{c}$. Moreover, we have the functional equation 
\[
{\mathcal L}_{\mathfrak{p},\mathfrak{c}}(\chi)={\mathcal L}_{\mathfrak{p},\overline{\mathfrak{c}}}((\chi^{\mathbf{c}})^{-1}\mathbf{N}^{-1}),
\]
where $\chi^{\mathbf{c}}$ is the composition of $\chi$ with the action of the non-trivial automorphism of $K/\mathbf{Q}$, and the equality is up to a $p$-adic unit.

\end{thm}

\begin{proof}
See \cite{Katz49}, \cite{HT-ENS}; our formulation follows of the interpolation property follows \cite[Prop.~4.19]{hsieh-crelle-mu} most closely. The functional equation is shown in \cite[Thm.~II.6.4]{deshalit}. 
\end{proof}



\subsection{Triple product $p$-adic $L$-function}\label{subsec:triple}

Let ${\mathcal R}$ be a normal domain finite flat over 
\[
\Lambda:=\mathcal O\dBr{1+p\mathbf{Z}_p},
\] 
where $\mathcal O$ is the ring of integers of a finite extension of $\mathbf{Q}_p$. For a positive integer $N$ prime $p$ and a Dirichlet character $\chi:(\mathbf{Z}/Np\mathbf{Z})^\times\rightarrow\mathcal O^\times$, we denote by $S^o(N,\chi,{\mathcal R})\subset{\mathcal R}\dBr{q}$ the space of ordinary ${\mathcal R}$-adic cusp forms of tame level $N$ and branch character $\chi$ as defined in \cite[\S{3.1}]{hsieh-triple}. 

Denote by $\mathfrak{X}_{\mathcal R}^+\subset{\rm Spec}\,{\mathcal R}(\overline{\mathbf{Q}}_p)$ the set of \emph{arithmetic points} of ${\mathcal R}$, consisting of the ring homomorphisms $Q:{\mathcal R}\rightarrow\overline{\mathbf{Q}}_p$ such that $Q\vert_{1+p\mathbf{Z}_p}$ is given by $z\mapsto z^{k_Q}\epsilon_Q(z)$ for some $k_Q\in\mathbf{Z}_{\geq 2}$ called the \emph{weight of $Q$} and $\epsilon_Q(z)\in\mu_{p^\infty}$. As in  \cite[\S{3.1}]{hsieh-triple}, we say that $\boldsymbol{f}=\sum_{n=1}^\infty a_n({\boldsymbol{f}})q^n\in S^o(N,\chi,{\mathcal R})$ is a \emph{primitive Hida family} if for every $Q\in\mathfrak{X}_{\mathcal R}^+$ the specialization $\boldsymbol{f}_Q$ gives the $q$-expansion of an ordinary $p$-stabilised newform of weight of $k_\mathbf{Q}$ and tame conductor $N$. Attached to such ${\boldsymbol{f}}$ we let $\mathfrak{X}_{{\mathcal R}}^{\rm cls}\subset\mathfrak{X}_{{\mathcal R}}^+$ 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, we let 
\[
\rho_{{\boldsymbol{f}}}:G_\mathbf{Q}\rightarrow{\rm Aut}_{{\mathcal R}}(V_{\boldsymbol{f}})\simeq{\rm GL}_2({\mathcal R})
\]
denote the associated Galois representation, which  here we take to be the \emph{dual} of that in \cite[\S{3.2}]{hsieh-triple}; in particular, the determinant of $\rho_{\boldsymbol{f}}$ is $\chi_{\mathcal R}\cdot\varepsilon_{\rm cyc}$ in the notations of \emph{loc.\,cit.}, where $\varepsilon_{\rm cyc}$ is the $p$-adic cyclotomic character. By \cite[Thm.~2.2.2]{wiles88}, 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 the quotient $V_{\boldsymbol{f}}^-$ is free of rank one over ${\mathcal R}$, with the $G_{\mathbf{Q}_p}$-action given by the unramified character sending an arithmetic Frobenius ${\rm Fr}_p$ to $a_p({\boldsymbol{f}})$. 

Let
\[
({\boldsymbol{\varphi}},{\boldsymbol{g}},{\boldsymbol{h}})\in S^o(N_\varphi,\chi_\varphi,{\mathcal R}_\varphi)\times S^o(N_g,\chi_g,{\mathcal R}_g)\times S^o(N_h,\chi_h,{\mathcal R}_h)
\]
be a triple of primitive Hida families with 
\begin{equation}\label{eq:a}
\textrm{$\chi_\varphi\chi_g\chi_h=\omega^{2a}$ for some $a\in\mathbf{Z}$,}
\end{equation} 
where $\omega$ is the Teichm\"uller character. Put 
\[
\mathcal{R}={\mathcal R}_\varphi\hat\otimes_{\mathcal O}{\mathcal R}_g\hat\otimes_{\mathcal O}{\mathcal R}_h,
\] 
which is a finite extension of the three-variable Iwasawa algebra $\Lambda\hat\otimes_{\mathcal O}\Lambda\hat\otimes_{\mathcal O}\Lambda$, 
and let
\[
\mathfrak{X}_{\mathcal{R}}^{\boldsymbol{\varphi}}:=\left\{(Q_0,Q_1,Q_2)\in\mathfrak{X}_{{\mathcal R}_\varphi}^+\times\mathfrak{X}_{{\mathcal R}_g}^{\rm cls}\times\mathfrak{X}_{{\mathcal R}_h}^{\rm cls}\;:\;\textrm{$k_{Q_0}\geq k_{Q_1}+k_{Q_2}$ and $k_{Q_0}\equiv k_{Q_1}+k_{Q_2}\;({\rm mod}\;2)$}\right\}
\]
be the weight space for $\mathcal{R}$ in the so-called \emph{${\boldsymbol{\varphi}}$-unbalanced range}. 

Let $\mathbf{V}=V_{\boldsymbol{\varphi}}\hat\otimes_{\mathcal O}V_{{\boldsymbol{g}}}\hat\otimes_{\mathcal O}V_{{\boldsymbol{h}}}$ be the triple tensor product Galois representation attached to $({\boldsymbol{\varphi}},{\boldsymbol{g}},{\boldsymbol{h}})$, and writing $\det\mathbf{V}=\mathcal{X}^2\varepsilon_{\rm cyc}$ define  
\begin{equation}\label{eq:crit-twist}
{\mathbf{V}^\dagger}:=\mathbf{V}\otimes\mathcal{X}^{-1},
\end{equation}
which is a self-dual twist of $\mathbf{V}$. Define the rank four $G_{\mathbf{Q}_p}$-invariant subspace $\mathscr{F}_p^{\boldsymbol{\varphi}}({\mathbf{V}^\dagger})\subset{\mathbf{V}^\dagger}$ by
\begin{equation}\label{eq:unb-intro}
\mathscr{F}_p^{\boldsymbol{\varphi}}({\mathbf{V}^\dagger}):=V_{{\boldsymbol{\varphi}}}^+\hat\otimes_{\mathcal O}V_{\boldsymbol{g}}\hat\otimes_{\mathcal O}V_{{\boldsymbol{h}}}\otimes\mathcal{X}^{-1}.
\end{equation}
For every $\underline{Q}=(Q_0,Q_1,Q_2)\in\mathfrak{X}_{\mathcal{R}}^{\boldsymbol{\varphi}}$ we denote by $\mathscr{F}_p^{\boldsymbol{\varphi}}({\mathbf{V}_{\underline{Q}}^\dagger})\subset{\mathbf{V}_{\underline{Q}}^\dagger}$  the corresponding specializations. 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 \cite[p.\,21]{tate-background}, and assume that 
\begin{equation}\label{eq:+1}
\textrm{for some $\underline{Q}\in\mathfrak{X}_{\mathcal{R}}^{\boldsymbol{\varphi}}$, we have $\varepsilon_\ell({\mathbf{V}_{\underline{Q}}^\dagger})=+1$ for all prime factors $\ell$ of $N_{\boldsymbol{\varphi}} N_{\boldsymbol{g}} N_{\boldsymbol{h}}$.}
\end{equation}
As explained in \cite[\S{1.2}]{hsieh-triple}, it is known that condition (\ref{eq:+1}) is independent of $\underline{Q}$, and it implies that the sign in the functional equation for the triple product $L$-function 
\[
L({\mathbf{V}_{\underline{Q}}^\dagger},s)
\] 
(relating its values at $s$ and $-s$) is $+1$ for all $\underline{Q}\in\mathfrak{X}_{\mathcal{R}}^{\boldsymbol{\varphi}}$. 

For the next statement, we refer the reader to $\S\ref{subsec:congr}$ for a review of the congruence ideal associated with a primitive Hida family.



\begin{thm}\label{thm:hsieh-triple}
Let $({\boldsymbol{\varphi}},{\boldsymbol{g}},{\boldsymbol{h}})$ be a triple of primitive Hida families as above satisfying conditions (\ref{eq:a}) and (\ref{eq:+1}). Assume in addition that:
\begin{itemize}
\item $\gcd(N_\varphi,N_g,N_h)$ is square-free,
\item the residual representation $\bar{\rho}_{{\boldsymbol{\varphi}}}$ is absolutely irreducible and $p$-distinguished,
\end{itemize}
and fix a generator $\eta_{{\boldsymbol{\varphi}}}$ of the congruence ideal of ${\boldsymbol{\varphi}}$. Then there exists a unique element 
\[
\mathcal{L}_p^{{\boldsymbol{\varphi}}}({\boldsymbol{\varphi}},{\boldsymbol{g}},{\boldsymbol{h}})\in\mathcal{R}
\]
such that for all $\underline{Q}=(Q_0,Q_1,Q_2)\in\mathfrak{X}_{\mathcal{R}}^{{\boldsymbol{\varphi}}}$ of weight $(k_0,k_1,k_2)$ with $\epsilon_{Q_0}=1$ we have
\[
(\mathcal{L}_p^{\boldsymbol{\varphi}}({\boldsymbol{\varphi}},{\boldsymbol{g}},{\boldsymbol{h}})(\underline{Q}))^2=\Gamma_{{\mathbf{V}_{\underline{Q}}^\dagger}}(0)\cdot\frac{L({\mathbf{V}_{\underline{Q}}^\dagger},0)}{(\sqrt{-1})^{2k_{0}}\cdot\Omega_{{\boldsymbol{\varphi}}_{Q_0}}^2}\cdot\mathcal{E}_p(\mathscr{F}_p^{{\boldsymbol{\varphi}}}({\mathbf{V}_{\underline{Q}}^\dagger}))\cdot\prod_{\ell\in\Sigma_{\rm exc}}(1+\ell^{-1})^2,
\]
where:
\begin{itemize}
\item $\Gamma_{{\mathbf{V}_{\underline{Q}}^\dagger}}(0)=\Gamma_{\mathbf{C}}(c_{\underline{Q}})\Gamma_\mathbf{C}(c_{\underline{Q}}+2-k_1-k_2)\Gamma_\mathbf{C}(c_{\underline{Q}}+1-k_1)\Gamma_\mathbf{C}(c_{\underline{Q}}+1-k_2)$, with 
\[
c_{\underline{Q}}=(k_0+k_1+k_2-2)/2
\] 
and $\Gamma_\mathbf{C}(s)=2(2\pi)^{-s}\Gamma(s)$;
\item $\Omega_{{\boldsymbol{\varphi}}_{Q_0}}$ is the canonical period
\[
\Omega_{{\boldsymbol{\varphi}}_{Q_0}}:=(-2\sqrt{-1})^{k_0+1}\cdot\frac{\Vert{\boldsymbol{\varphi}}_{Q_0}^\circ\Vert_{\Gamma_0(N_{\varphi})}^2}{\imath_p(\eta_{{\boldsymbol{\varphi}}_{Q_0}})}\cdot\Bigl(1-\frac{\chi_{\varphi}'(p)p^{k_0-1}}{\alpha_{Q_0}^2}\Bigr)\Bigl(1-\frac{\chi_{\varphi}'(p)p^{k_0-2}}{\alpha_{Q_0}^2}\Bigr),
\]
with ${\boldsymbol{\varphi}}_{Q_0}^\circ\in S_{k_0}(N_{\varphi})$ the newform of conductor $N_\varphi$ associated with ${\boldsymbol{\varphi}}_{Q_0}$, $\chi_\varphi'$ the prime-to-$p$ part of $\chi_\varphi$, and $\alpha_{Q_0}$ the specialization of $a_p({\boldsymbol{\varphi}})\in{\mathcal R}_\varphi^\times$ at $Q_0$;
\item $\mathcal{E}_p(\mathscr{F}_p^{{\boldsymbol{\varphi}}}({\mathbf{V}_{\underline{Q}}^\dagger}))$ is the modified $p$-Euler factor
\[
\mathcal{E}_p(\mathscr{F}_p^{{\boldsymbol{\varphi}}}({\mathbf{V}_{\underline{Q}}^\dagger})):=\frac{L_p(\mathscr{F}_p^{{\boldsymbol{\varphi}}}({\mathbf{V}_{\underline{Q}}^\dagger}),0)}{\varepsilon_p(\mathscr{F}_p^{{\boldsymbol{\varphi}}}({\mathbf{V}_{\underline{Q}}^\dagger}))\cdot L_p({\mathbf{V}_{\underline{Q}}^\dagger}/\mathscr{F}_p^{{\boldsymbol{\varphi}}}({\mathbf{V}_{\underline{Q}}^\dagger}),0)}\cdot\frac{1}{L_p({\mathbf{V}_{\underline{Q}}^\dagger},0)},
\]
\end{itemize}
and $\Sigma_{\rm exc}$ is an explicitly defined subset of the prime factors of $N_\varphi N_g N_h$, \cite[p.~416]{hsieh-triple}.
\end{thm}

\begin{proof}
This is \cite[Thm.~A]{hsieh-triple}. 
\end{proof}

\begin{rem}
For simplicity,  
we have stated the interpolation property of ${\mathcal L}_p^{\boldsymbol{\varphi}}({\boldsymbol{\varphi}},{\boldsymbol{g}},{\boldsymbol{h}})$ restricted to $\underline{Q}$ with $\epsilon_{Q_0}=1$, as this will suffice for our purposes; see \cite[Thm.~A]{hsieh-triple} for the interpolation property for all $\underline{Q}\in\mathfrak{X}_{\mathcal{R}}^{\boldsymbol{\varphi}}$.
\end{rem}

\subsection{Congruence ideal}\label{subsec:congr}

Let ${\boldsymbol{f}}\in S^o(N_f,\chi_f,{\mathcal R})$ be a primitive Hida family defined over ${\mathcal R}$. Associated with ${\boldsymbol{f}}$ there is a ${\mathcal R}$-algebra homomorphism 
\[
\lambda_{{\boldsymbol{f}}}:\mathbb{T}(N_f,{\mathcal R})\rightarrow{\mathcal R}
\] 
where $\mathbb{T}(N_f,{\mathcal R})$ is the Hecke algebra acting on $\oplus_\chi S^o(N_f,\chi,{\mathcal R})$, where $\chi$ runs over the characters of $(\mathbf{Z}/pN_f\mathbf{Z})^\times$. Let $\mathbb{T}_{\mathfrak{m}}$ be the local component of $\mathbb{T}(N_f,{\mathcal R})$ through which $\lambda_{{\boldsymbol{f}}}$ factors, and following \cite{hida-AJM-modules} define the \emph{congruence ideal} $C({\boldsymbol{f}})$ of ${\boldsymbol{f}}$ by
\[
C({\boldsymbol{f}}):=\lambda_{{\boldsymbol{f}}}({\rm Ann}_{\mathbb{T}_\mathfrak{m}}({\rm ker}\,\lambda_{\boldsymbol{f}}))\subset{\mathcal R}.
\] 
When the residual representation $\bar{\rho}_{{\boldsymbol{f}}}$ is absolutely irreducible and $p$-distinguished, it follows from the results of \cite{Fermat-Wiles} and \cite{hida-AJM-modules} that $C({\boldsymbol{f}})$ is generated by a nonzero element $\eta_{{\boldsymbol{f}}}\in{\mathcal R}$.



\section{Factorization of $p$-adic triple product $L$-function}

In this section we relate the triple product $p$-adic $L$-function attached to triples of forms with CM by $K$ to a product of anticyclotomic Katz $p$-adic $L$-functions. 

\subsection{Hida families with CM}

We review the construction of CM Hida families following the exposition in \cite[\S{8.1}]{hsieh-triple}. 
Since it will suffice for our purposes, we assume that  the class number $h_K=\vert{\rm Pic}(\mathcal O_K)\vert$ of $K$ is coprime to $p$. Let $K_\infty$ be the unique $\mathbf{Z}_p^2$-extension of $K$, and denote by $K_{\mathfrak{p}^\infty}$ 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 integral ideal $\mathfrak{c}\subset\mathcal O_K$ we denote by $K(\mathfrak{c})$ the ray class field of $K$ of conductor $\mathfrak{c}$. Thus $K_{\mathfrak{p}^\infty}$ is also the maximal $\mathbf{Z}_p$-extension inside $K(\mathfrak{p}^\infty)$. By our assumption on $h_K$, the restriction of the
Artin map to $K_\mathfrak{p}^\times$ induces an isomorphism $1+p\mathbf{Z}_p\simeq\Gamma_{\mathfrak{p}^\infty}$, where we identified $\mathbf{Z}_p^\times$ and $\mathcal O_{K_\mathfrak{p}}^\times$ by the embedding $\iota_p$. Denote by $\gamma_\mathfrak{p}$ the topological generator of $\Gamma_{\mathfrak{p}^\infty}$ corresponding to $1+p$ under this isomorphism, and for each variable $S$ let $\Psi_S:\Gamma_\infty\rightarrow\mathbf{Z}_p\dBr{S}^\times$ be the universal character
given by
\[
\Psi_S(\sigma)=(1+S)^{l(\sigma)},
\]
where $l(\sigma)\in\mathbf{Z}_p$ is such that $\sigma\vert_{K_{\mathfrak{p}^\infty}}=\gamma_\mathfrak{p}^{l(\sigma)}$. Fix $\mathfrak{c}$ prime to $p$, and for any finite order character $\psi:G_K\rightarrow\mathcal O^\times$ of tame conductor $\mathfrak{c}$ put
\[
\boldsymbol{\theta}_\psi(S)(q)=\sum_{(\mathfrak{a},\mathfrak{p}\mathfrak{c})=1}\psi(\sigma_{\mathfrak{a}})\Psi^{-1}_S(\sigma_\mathfrak{a})q^{\mathbf{N}(\mathfrak{a})}\in\mathcal O\dBr{S}\dBr{q},
\]
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 $\mathcal O\dBr{S}$ of 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 $K/\mathbf{Q}$. 


\subsection{Setting}\label{subsec:setting}

Let $E_{/\mathbf{Q}}$ be an elliptic curve of conductor $N$ with CM by the maximal order $\mathcal O_K$. Assume that the prime $p>2$ is a prime of good \emph{ordinary} reduction for $E$, so in particular (\ref{eq:spl}) holds and $h_K=1$. Let $\lambda$ be the Grossencharacter of $K$ associated to $E$ by the theory of complex multiplication, so we have 
\[
L(E,s)=L(\lambda,s).
\] 

Let $\ell, \ell'$ be \emph{distinct} rational primes split in $K$ with $(\ell \ell',Np)=1$. Take $\Phi$ (resp. $\Xi$) a ring class character of conductor $\ell^m\mathcal O_K$ (resp. $(\ell')^n\mathcal O_K$) for some $m,n>0$, and write 
\[
\Phi=\phi^{1-\mathbf{c}},\quad\quad\Xi=\xi^{1-\mathbf{c}}
\]
with $\phi$ (resp. $\xi$) a ray class character modulo $\ell^m\mathcal O_K$ (resp. $q^n\mathcal O_K$), where $\mathbf{c}$ is the non-trivial automorphism of $K/\mathbf{Q}$ (see \cite[Lemma~5.31]{HMI}). As usual, here we write $\phi^\mathbf{c}$ for the composition of $\phi$ with $\mathbf{c}$, and write $\phi^{1-\mathbf{c}}$ (and similarly $\xi^{1-\mathbf{c}}$) to denote the product $\phi(\phi^{\mathbf{c}})^{-1}$. Consider the primitive CM Hida families
\[
{\boldsymbol{g}}=\boldsymbol{\theta}_{\phi}(S_1),\quad\quad{\boldsymbol{h}}=\boldsymbol{\theta}_{\xi}(S_2)
\]
of level $D_K\ell^{2m}$ and $D_K(\ell')^{2n}$, respectively. On the other hand, put $\psi=(\phi^{\mathbf{c}}\xi^{\mathbf{c}})^{-1}$ and let
\begin{equation}\label{eq:varphi}
\varphi:=\theta(\lambda\psi) 
\end{equation}
be the theta series of weight $2$ associated to $\lambda\psi$. Note that
\[
L(E,s)=L(\overline{\lambda},s)=L(\lambda^{-1},s-1),
\]
since $E$ is defined over $\mathbf{Q}$. The follows immediate consequence of the nonvanishing results due to Greenberg and Rohrlich will be a key ingredient in the proof of our main result.

\begin{thm}\label{thm:nonvanishing}
There exists ring class characters $\Phi$ and $\Xi$ as above such that: 
\begin{enumerate}
\item Both $\Phi$ and $\Xi$ have order prime to $p$,
\item The restrictions $\phi\vert_{G_{\mathbf{Q}_p}}$ and  $\xi\vert_{G_{\mathbf{Q}_p}}$ are both non-trivial,
\item Both $\phi$ and $\xi$ have order at $3$,
\end{enumerate}
and the following nonvanishing condition holds:
\[
L(\lambda^{-1}\Phi^{-1},0)\cdot L(\lambda^{-1}\Xi^{-1},0)\cdot L(\lambda^{-1},\Phi^{-1}\Xi^{-1},0)\neq 0.
\]
\end{thm}

\begin{proof}
Since conditions (1)--(3) exclude only finitely many ring class characters, this follows from the nonvanishing results of \cite{greenberg-critical} and \cite{rohrlich-ac}.
\end{proof}

\begin{choice}\label{choice:setting}
From now on, we fix a pair of ring class characters $\Phi$ and $\Xi$ satisfying the conditions of Theorem~\ref{thm:nonvanishing}, and let $(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})$ be the resulting triple as defined above.
\end{choice}

For the construction of nontrivial Selmer classes in the Selmer group of $E$, we will be interested in the Iwasawa main conjecture and the $p$-adic $L$-function for the triple $(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})$.


\subsection{Congruence ideal of CM Hida families}

Denote by $\Gamma^-$ the maximal $\mathbf{Z}_p$-free quotient of $Z^-(\mathfrak{c}):=Z(\mathfrak{c})/Z(\mathfrak{c})^{1+\mathbf{c}}$, and let 
\[
\pi_-:Z(\mathfrak{c})\rightarrow\Gamma^-
\] 
be the natural projection. For a Hecke character $\eta$ of conductor dividing $\mathfrak{c}p^\infty$, we
denote by ${\mathcal L}_{\mathfrak{p},\eta}^-$ the image of ${\mathcal L}_{\mathfrak{p},\mathfrak{c}}$ under the map ${\mathcal W}\dBr{Z(\mathfrak{c})}\rightarrow{\mathcal W}\dBr{\Gamma^-}$ given by $\gamma\mapsto\eta(\gamma)\pi_-(\gamma)$ for $\gamma\in Z(\mathfrak{c})$. Here, with a slight abuse of notation, we enlarge ${\mathcal W}$ if necessary so that is contains the values of $\eta$, but we continue to denote it by ${\mathcal W}$.

\begin{lem}\label{lem:HT}
Let ${\boldsymbol{\varphi}}$ be the primitive CM Hida passing through the ordinary $p$-stabilization of $\varphi$ in (\ref{eq:varphi}). Then the congruence ideal $C({\boldsymbol{\varphi}})$ is generated by
\[
\frac{h_K}{w_K}\cdot\mathcal{L}_{\mathfrak{p},\lambda^{\mathbf{c}-1}\psi^{\mathbf{c}-1}}^-,
\]
where $h_K=\vert{\rm Pic}(\mathcal O_K)\vert$ and $w_K=\vert\mathcal O_K^\times\vert$.
\end{lem}

\begin{proof}
A generator of $C({\boldsymbol{\varphi}})$ is given by a congruence power series $H({\boldsymbol{\varphi}})$ attached to ${\boldsymbol{\varphi}}$ as in \cite{hida-coates}. By our Choice~\ref{choice:setting} of $\Phi$ and $\Xi$, the $H({\boldsymbol{\varphi}})$ corresponds to a branch character satisfying the hypotheses (1)--(4) in \cite[p.\,466]{hida-coates}, so as noted in p.~469 of \emph{op.\,cit}, the result follows from the proof of the anticyclotomic Iwasawa main conjecture by Hida--Tilouine \cite{HT-ENS,HT-117} and Hida \cite{hida-coates}. 
\end{proof}

\subsection{Proof of the factorization}

Let ${\boldsymbol{\varphi}}\in\mathcal O\dBr{S_0}\dBr{q}$ be the primitive CM Hida family associated to the character $\varphi$ in (\ref{eq:varphi}), and let $Q_0\in\mathfrak{X}_{\mathcal O\dBr{S_0}}^+$ be such that ${\boldsymbol{\varphi}}_{Q_0}$ is the ordinary $p$-stabilization of $\varphi$. Letting
\begin{equation}\label{eq:sp}
\mathcal{R}\simeq\mathcal O\dBr{S_0}\hat{\otimes}_\mathcal O\cO\dBr{S_1,S_2}\rightarrow\mathcal O\dBr{S_1,S_2}
\end{equation}
be the specialization map at $Q_0$, in the followiing we denote by
\[
\mathscr{L}_{p}^\varphi(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})(S_1,S_2)\in\mathcal O\dBr{S_1,S_2}
\] 
the image of the triple product $p$-adic $L$-function $\mathscr{L}_{p}^{{\boldsymbol{\varphi}}}({\boldsymbol{\varphi}},{\boldsymbol{g}},{\boldsymbol{h}})\in\mathcal{R}$ under this map. Upon the choice of a topological generator $\gamma^-\in\Gamma^-$, as usual we shall identify ${\mathcal W}\dBr{\Gamma^-}$ with the power series ring ${\mathcal W}\dBr{W}$ via $\gamma^-\mapsto 1+W$. Denote by $\lambda\mapsto\lambda^\iota$ the involution of ${\mathcal W}\dBr{\Gamma^-}$ given by $\gamma^-\mapsto(\gamma^-)^{-1}$.

\begin{prop}\label{prop:factor-L}
Put
\[
W_1=(1+S_1)^{1/2}(1+S_2)^{1/2}-1,\quad
W_2=(1+S_1)^{1/2}(1+S_2)^{-1/2}-1.
\]
Then we have the factorization
\[
\mathscr{L}_{p}^{\varphi}(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})^2(S_1,S_2)=u\cdot{\mathcal L}_{\mathfrak{p},\lambda^{-1}\psi^{\mathbf{c}-1}}^-(W_1)^\iota\cdot{\mathcal L}_{\mathfrak{p},\lambda^{-1}}^-(W_1)\cdot{\mathcal L}_{\mathfrak{p},\lambda^{-1}\phi^{\mathbf{c}-1}}^-(W_2)^\iota\cdot{\mathcal L}_{\mathfrak{p},\lambda^{-1}\xi^{\mathbf{c}-1}}^-(W_2),
\]
where $u\in{\mathcal W}^\times$ is a $p$-adic unit.
\end{prop}

\begin{proof}
Denote by $\mathbf{V}_{Q_0}^\dagger$ the image of $\mathbf{V}^\dagger$ under the specialization map $(\ref{eq:sp})$. Then from (\ref{eq:crit-twist}) and the definition of $\varphi$ we see that
\[
\mathbf{V}_{Q_0}^\dag=
\Ind_K^\mathbf{Q}(\lambda^{-1}\psi^{-1})\otimes\Ind_K^\mathbf{Q}(\phi^{-1}\Psi_{S_1})\otimes\Ind_K^\mathbf{Q}(\xi^{-1}\Psi_{S_2})\otimes\mathcal{X}^{-1},
\]
where $\mathcal{X}=(\Psi_{S_1}^{1/2}\Psi_{S_2}^{1/2}\circ\mathscr{V}):G_\mathbf{Q}\rightarrow\mathbf{Z}_p\dBr{S_1,S_2}^\times$, and so an immediate computation shows that  
\begin{equation}\label{eq:dec-V}
\begin{aligned}
\mathbf{V}_{Q_0}^\dagger
&\simeq\Ind_K^\mathbf{Q}(\lambda^{-1}\psi^{\mathbf{c}-1}\Psi_{W_1}^{1-\mathbf{c}})\oplus\Ind_K^\mathbf{Q}(\lambda^{-1}\Psi_{W_1}^{\mathbf{c}-1})\\
&\quad\oplus
\Ind_K^\mathbf{Q}(\lambda^{-1}\phi^{\mathbf{c}-1}\Psi_{W_2}^{1-\mathbf{c}})\oplus
\Ind_K^\mathbf{Q}(\lambda^{-1}\xi^{\mathbf{c}-1}\Psi_{W_2}^{\mathbf{c}-1}).
\end{aligned}
\end{equation}

For $i=1,2$, let $\zeta_i$ be a primitive $p^{n_i}$-th root of unity with $n_i>0$, and put $Q_1=\zeta_1\zeta_2-1$, $Q_2=\zeta_1\zeta_2^{-1}-1$, so the specializations ${\boldsymbol{g}}_{Q_1},  {\boldsymbol{h}}_{Q_2}$ are both CM forms of weight $1$. Fix a topological generators $\gamma^-\in\Gamma^-$, and denote by $\epsilon_i:\Gamma^-\rightarrow\mu_{p^\infty}$ the finite order character given by $\epsilon_i(\gamma^-)=\zeta_i$. Letting $\underline{Q}=(Q_0,Q_1,Q_2)$, we thus see that
\begin{align*}
{\mathbf{V}_{\underline{Q}}^\dagger}&=
\Ind_K^\mathbf{Q}(\lambda^{-1}\psi^{\mathbf{c}-1}\epsilon_1^{-1})\oplus\Ind_K^\mathbf{Q}(\lambda^{-1}\epsilon_1)\oplus\Ind_K^\mathbf{Q}(\lambda^{-1}\phi^{\mathbf{c}-1}\epsilon_2^{-1})
\oplus\Ind_K^{\mathbf{Q}}(\lambda^{-1}\xi^{\mathbf{c}-1}\epsilon_2),\\
\mathscr{F}_p^{\varphi}({\mathbf{V}_{\underline{Q}}^\dagger})
&=\lambda_\mathfrak{p}^{-1}\psi_\mathfrak{p}^{\mathbf{c}-1}\epsilon_{1,\mathfrak{p}}^{-1}\oplus\lambda_{\mathfrak{p}}^{-1}\epsilon_{1,\mathfrak{p}}\oplus\lambda_\mathfrak{p}^{-1}\phi^{\mathbf{c}-1}_\mathfrak{p}\epsilon_{2,\mathfrak{p}}^{-1}\oplus\lambda_\mathfrak{p}^{-1}\xi^{\mathbf{c}-1}_\mathfrak{p}\epsilon_{2,\mathfrak{p}}.
\end{align*}
Thus the terms appearing in the interpolation formula of Theorem~\ref{thm:hsieh-triple} become:
\begin{equation}\label{eq:explicit}
\begin{aligned}
\Gamma_{{\mathbf{V}_{\underline{Q}}^\dagger}}(0)\cdot L({\mathbf{V}_{\underline{Q}}^\dagger},0)&=\pi^{-4}\cdot L(\lambda^{-1}\psi^{\mathbf{c}-1}\epsilon_1^{-1},0)\cdot L(\lambda^{-1}\epsilon_1,0)\\
&\quad\times L(\lambda^{-1}\phi^{\mathbf{c}-1}\epsilon_2^{-1},0)\cdot L(\lambda^{-1}\xi^{\mathbf{c}-1}\epsilon_2,0),
\end{aligned}
\end{equation}
\begin{align}
\mathcal{E}_p(\mathscr{F}_p^{\varphi}({\mathbf{V}_{\underline{Q}}^\dagger}))&=
\mathcal{E}_\mathfrak{p}(\lambda^{-1}\psi^{\mathbf{c}-1}\epsilon_1^{-1})\cdot\mathcal{E}_\mathfrak{p}(\lambda^{-1}\epsilon_1)\cdot\mathcal{E}_\mathfrak{p}(\lambda^{-1}\phi^{\mathbf{c}-1}\epsilon_2^{-1})\cdot\mathcal{E}_\mathfrak{p}(\lambda^{-1}\xi^{\mathbf{c}-1}\epsilon_2);\label{eq:Ep}
\\
\Omega_{{\boldsymbol{\varphi}}_{Q_0}}&=(-2\sqrt{-1})^3\cdot\frac{\Vert\varphi\Vert_{\Gamma_0(N_\varphi)}^2}{\imath_p(\eta_{{\boldsymbol{\varphi}}_{Q_0}})}\cdot\biggl(1-\frac{\lambda\psi(\mathfrak{p})}{\lambda\psi(\bar\mathfrak{p})}\biggr)\biggl(1-\frac{\lambda\psi(\mathfrak{p})}{p\lambda\psi(\bar\mathfrak{p})}\biggr);\label{eq:period}
\end{align}
and we note that $\Sigma_{\rm exc}=\emptyset$ in this case. 

On the other hand, from Hida's formula for the adjoint $L$-value \cite[Thm.~7.1]{HT-ENS} and Dirichlet's class number formula we obtain
\[
\Vert\varphi\Vert^2_{\Gamma_0(N_\varphi)}=\frac{D_K^2}{2^4\pi^3}\cdot\frac{2\pi h_K}{w_K\sqrt{D_K}}\cdot L(\lambda^{1-\mathbf{c}}\psi^{1-\mathbf{c}},1).
\]
Noting that $L(\lambda^{1-\mathbf{c}}\psi^{1-\mathbf{c}},0)=L(\lambda^{\mathbf{c}-1}\psi^{\mathbf{c}-1}\mathbf{N}^{-1},0)$ and $\lambda^{\mathbf{c}-1}\psi^{\mathbf{c}-1}\mathbf{N}^{-1}$ has infinity type $(2,0)$,  the interpolation property in Theorem~\ref{thm:katz} can thus be rewritten as
\begin{align*}
{\mathcal L}_{\mathfrak{p},\mathfrak{c}\ell\ell'}(\lambda^{\mathbf{c}-1}\psi^{\mathbf{c}-1}\mathbf{N}^{-1})&
=\biggl(\frac{\Omega_p}{\Omega_\infty}\biggr)^2\cdot\frac{2^3\pi^2}{\sqrt{D_K}^{3}}\cdot\biggl(1-\frac{\lambda\psi(\mathfrak{p})}{\lambda\psi(\bar\mathfrak{p})}\biggr)\biggl(1-\frac{\lambda\psi(\mathfrak{p})}{p\lambda\psi(\bar\mathfrak{p})}\biggr)\cdot\frac{w_K}{h_K}\cdot\Vert\varphi\Vert_{\Gamma_0(N_\varphi)}^2\\
&=-\biggl(\frac{\Omega_p}{\Omega_\infty}\biggr)^2\cdot\frac{\pi^2\sqrt{-1}}{\sqrt{D_K}^3}\cdot\Omega_{{\boldsymbol{\varphi}}_{Q_0}}\cdot\eta_{{\boldsymbol{\varphi}}_{Q_0}}\cdot\frac{w_K}{h_K},
\end{align*}
using (\ref{eq:period}) for the second equality, and where $\mathfrak{c}$ denotes the conductor of $\lambda^{\mathbf{c}-1}$. By the functional equation for Katz's $p$-adic $L$-function (see Theorem~\ref{thm:katz})  
together with Lemma~\ref{lem:HT} this shows that
\begin{equation}\label{eq:Omega-bis}
\frac{1}{\Omega_{{\boldsymbol{\varphi}}_{Q_0}}^2}=\biggl(\frac{\Omega_p}{\Omega_\infty}\biggr)^4\cdot\frac{\pi^4}{D_K^{3}},
\end{equation}
where the equality is up to a $p$-adic unit independent of $Q_1, Q_2$. Thus substituting $(\ref{eq:explicit}), (\ref{eq:Ep}), (\ref{eq:period})$ and $(\ref{eq:Omega-bis})$ into the interpolation formula for $\mathscr{L}_p^\varphi(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})$ we thus arrive at
\begin{align*}
\mathscr{L}_p^\varphi(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})^2(\zeta_1\zeta_2-1,\zeta_1\zeta_2^{-1}-1)&=D_K^{-3}\cdot{\mathcal L}_{\mathfrak{p},\lambda^{-1}\psi^{\mathbf{c}-1}}^-(\zeta_1^{-1}-1)\cdot{\mathcal L}_{\mathfrak{p},\lambda^{-1}}^-(\zeta_1-1)\\
&\quad\times{\mathcal L}_{\mathfrak{p},\lambda^{-1}\phi^{\mathbf{c}-1}}^-(\zeta_2^{-1}-1)\cdot{\mathcal L}_{\mathfrak{p},\lambda^{-1}\xi^{\mathbf{c}-1}}^-(\zeta_2-1),
\end{align*}
for all non-trivial $p$-power roots of unity $\zeta_1, \zeta_2$, and the result follows.
\end{proof}




\section{Selmer group decompositions}

In this section we define different Selmer groups attached to Hecke characters and triple products of modular forms. Then, for the so-called unbalanced Selmer groups attached to the latter, we prove a decomposition mirroring the factorization in Proposition~\ref{prop:factor-S}.

\subsection{Selmer groups for Hecke characters}

Let $\nu$ be a Hecke character of $K$ with values in the ring of integers $\mathcal O$ of a finite extension $\Phi$ of $\mathbf{Q}_p$. Denote by $\mathcal O_{\nu}$ the free $\mathcal O$-module of rank $1$ on which $G_K$ acts via $\nu$, and put 
\[
T_\nu=\mathcal O_\nu,\quad V_\nu=T_\nu\otimes_{\mathcal O}\Phi,
\quad A_\nu=V_\nu/T_\nu=T_\nu\otimes_\mathcal O(\Phi/\mathcal O).
\]
Let $\Sigma$ be any finite set of places of $K$ containing $\infty$ and the primes dividing $p$ or the conductor of $\nu$, and for any finite extension $F/K$ denote by $G_{F,\Sigma}$ the Galois group of the maximal extension of $F$ unramified outside the places above $\Sigma$.

\begin{defn}
Let $F/K$ be a finite extension, and for $v\vert p$ a prime of $F$ above $p$ put
\[
{\mathrm{H}}^1_{\emptyset}(F_v,V_\nu)={\mathrm{H}}^1(F_v,V_\nu),\quad{\mathrm{H}}^1_{0}(F_v,V_\nu)=\{0\}.
\]
For $({\mathcal L}_{\mathfrak{p}},{\mathcal L}_{\overline{\mathfrak{p}}})\in\{\emptyset,0\}^{\oplus 2}$ define the Selmer group ${\rm Sel}_{{\mathcal L}_{\mathfrak{p}},{\mathcal L}_{\overline{\mathfrak{p}}}}(F,V_\nu)$ by
\[
{\rm Sel}_{{\mathcal L}_{\mathfrak{p}},{\mathcal L}_{\overline{\mathfrak{p}}}}(F,V_\nu)=\ker\biggl({\mathrm{H}}^1(G_{F,\Sigma},V_\nu)\rightarrow \prod_{v\vert\mathfrak{p}}\frac{{\mathrm{H}}^1(F_v,V_\nu)}{{\mathrm{H}}^1_{{\mathcal L}_\mathfrak{p}}(F_v,V_\nu)}\times
\prod_{v\vert\overline{\mathfrak{p}}}\frac{{\mathrm{H}}^1(F_v,V_\nu)}{{\mathrm{H}}^1_{{\mathcal L}_{\overline{\mathfrak{p}}}}(F_v,V_\nu)}\times
\prod_{v\in\Sigma, v\nmid p\infty}{\mathrm{H}}^1(F_v,V_\nu)\biggr).
\]
\end{defn}

\begin{rem}
In particular, if $\nu$ has infinity type $(-1,0)$, the Selmer group
\[
{\rm Sel}_{\emptyset,0}(F,V_\nu)=\ker\biggl({\mathrm{H}}^1(G_{F,\Sigma},V_\nu)\rightarrow 
\prod_{v\vert\overline{\mathfrak{p}}}{\mathrm{H}}^1(F_v,V_\nu)\times
\prod_{v\in\Sigma, v\nmid p\infty}{\mathrm{H}}^1(F_v,V_\nu)\biggr)
\]
agrees with the Bloch--Kato Selmer group of $V_\nu$ (see e.g.  \cite[\S{1.1}]{AHsplit} or \cite[\S{1.2}]{Arnold}). Similarly, if $\nu$ has infinity type $(0,-1)$ then ${\rm Sel}_{0,\emptyset}(F,V_\nu)$ agrees with the Bloch--Kato Selmer group of $V_\nu$. 
\end{rem}

For $?\in\{\emptyset,0\}$, the local condition ${\mathrm{H}}^1_?(F_v,T_\nu)$ and ${\mathrm{H}}^1_?(F_v,A_\nu)$ are defined from the above by propagation, and using these we define ${\rm Sel}_{{\mathcal L}_\mathfrak{p},{\mathcal L}_{\overline{\mathfrak{p}}}}(F,T_\nu)$ and ${\rm Sel}_{{\mathcal L}_\mathfrak{p},{\mathcal L}_{\overline{\mathfrak{p}}}}(F,A_\nu)$ by the same recipe as before. Then, letting $K_\infty/K$ denote the anticyclotomic $\mathbf{Z}_p$-extension of $K$, we set
\begin{equation}\label{eq:Iw-Sel}
{\rm Sel}_{{{\mathcal L}_\mathfrak{p},{\mathcal L}_{\overline{\mathfrak{p}}}}}(K_\infty,T_\nu)=\varprojlim_n{\rm Sel}_{{{\mathcal L}_\mathfrak{p},{\mathcal L}_{\overline{\mathfrak{p}}}}}(K_n,T_\nu),\quad
{\rm Sel}_{{{\mathcal L}_\mathfrak{p},{\mathcal L}_{\overline{\mathfrak{p}}}}}(K_\infty,A_\nu)=\varinjlim_n{\rm Sel}_{{{\mathcal L}_\mathfrak{p},{\mathcal L}_{\overline{\mathfrak{p}}}}}(K_n,A_\nu).
\end{equation}

Let $\Lambda=\mathcal O\dBr{\Gamma^-}\simeq\mathcal O\dBr{W}$ be the anticyclotomic Iwasawa algebra, and denote by $\Psi_W:G_K\rightarrow\Lambda^\times$ the universal character given $g\mapsto[g\vert_{K_\infty}]$, where $\gamma\mapsto[\gamma]$ is the inclusion of $\Gamma^-$ intro $\Lambda^-$ as a group-like element. Then we have isomorphisms 
\[
{\mathrm{H}}^1(K,T_\nu\otimes\Psi_W^{-1})\simeq\varprojlim_n{\mathrm{H}}^1(K_n,T_\nu),\quad
{\mathrm{H}}^1(K,A_\nu\otimes\Psi_W)\simeq\varinjlim_n{\mathrm{H}}^1(K_n,A_\nu)
\]
by Shapiro's lemma. In the following we let ${\rm Sel}_{{\mathcal L}_\mathfrak{p},{\mathcal L}_{\overline{\mathfrak{p}}}}(K,T_\nu\otimes\Psi_W^{-1})$ and ${\rm Sel}_{{\mathcal L}_\mathfrak{p},{\mathcal L}_{\overline{\mathfrak{p}}}}(K,A_\nu\otimes\Psi_W)$ be the Selmer groups corresponding to \eqref{eq:Iw-Sel} under these isomorphism, and let
\[
X_{{\mathcal L}_\mathfrak{p},{\mathcal L}_{\overline{\mathfrak{p}}}}(K,A_\nu\otimes\Psi_W)={\rm Hom}_{\mathbf{Z}_p}({\rm Sel}_{{\mathcal L}_\mathfrak{p},{\mathcal L}_{\overline{\mathfrak{p}}}}(K,A_\nu\otimes\Psi_W),\mathbf{Q}_p/\mathbf{Z}_p)
\] 
denote the Pontryagin dual of ${\rm Sel}_{{\mathcal L}_\mathfrak{p},{\mathcal L}_{\overline{\mathfrak{p}}}}(K,A_\nu\otimes\Psi_W)$.


\subsection{Selmer groups for triple products}

Let $(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})$ be a triple as in Choice~\ref{choice:setting}. As in the proof of Proposition~\ref{prop:factor-L}, suppose $\varphi$ arising from the specialization of ${\boldsymbol{\varphi}}$ at an arithmetic point $Q_0$, and denote by ${\mathbf{V}^\dagger}$ the triple product Galois representation associated to $({\boldsymbol{\varphi}},{\boldsymbol{g}},{\boldsymbol{h}})$ as in (\ref{eq:crit-twist}).

\begin{defn}\label{def:local-p}
Put
\[
\mathscr{F}^{\rm bal}_p({\mathbf{V}^\dagger})=\mathscr{F}_p^2({\mathbf{V}^\dagger})
:=\bigl(V_{{\boldsymbol{\varphi}}}^+\otimes V_{{\boldsymbol{g}}}^+\otimes V_{{\boldsymbol{h}}}+V_{{\boldsymbol{\varphi}}}^+\otimes V_{{\boldsymbol{g}}}\otimes V_{{\boldsymbol{h}}}^++V_{{\boldsymbol{\varphi}}}\otimes V_{{\boldsymbol{g}}}^+\otimes V_{{\boldsymbol{h}}}^+\bigr)\otimes\mathcal{X}^{-1}
\]
and define the \emph{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 $\mathscr{F}_p^{{\boldsymbol{\varphi}}}({\mathbf{V}^\dagger}):=\bigl(V_{{\boldsymbol{\varphi}}}^+\otimes V_{{\boldsymbol{g}}}\otimes V_{{\boldsymbol{h}}}\bigr)\otimes\mathcal{X}^{-1}$ and define the \emph{${\boldsymbol{\varphi}}$-unbalanced local condition} ${\mathrm{H}}^1_{{\varphi}}(\mathbf{Q}_p,{\mathbf{V}^\dagger})$ by
\[
{\mathrm{H}}^1_{{\boldsymbol{\varphi}}}(\mathbf{Q}_p,{\mathbf{V}^\dagger}):={\rm im}\bigl({\mathrm{H}}^1(\mathbf{Q}_p,\mathscr{F}_p^{{\boldsymbol{\varphi}}}({\mathbf{V}^\dagger}))\rightarrow{\mathrm{H}}^1(\mathbf{Q}_p,{\mathbf{V}^\dagger})\bigr).
\]
\end{defn}

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 $?\in\{{\rm bal},{\boldsymbol{\varphi}}\}$.

\begin{defn}\label{def:Sel-bu}
Let $?\in\{{\rm bal},{\boldsymbol{\varphi}}\}$, and define the Selmer group ${\rm Sel}^?(\mathbf{Q},{\mathbf{V}^\dagger})$ by
\[
{\rm Sel}^?(\mathbf{Q},{\mathbf{V}^\dagger}):=\ker\biggl\{{\mathrm{H}}^1(\mathbf{Q},{\mathbf{V}^\dagger})\rightarrow\frac{{\mathrm{H}}^1(\mathbf{Q}_p,{\mathbf{V}^\dagger})}{{\mathrm{H}}^1_?(\mathbf{Q}_p,{\mathbf{V}^\dagger})}\times\prod_{v\neq p}{\mathrm{H}}^1(\mathbf{Q}_v^{\rm nr},{\mathbf{V}^\dagger})\biggr\}.
\]
We call ${\rm Sel}^{\rm bal}(\mathbf{Q},{\mathbf{V}^\dagger})$ (resp. ${\rm Sel}^{{\boldsymbol{\varphi}}}(\mathbf{Q},{\mathbf{V}^\dagger})$) the \emph{balanced} (resp. \emph{${\boldsymbol{\varphi}}$-unbalanced}) Selmer group.
\end{defn}

Let ${\mathbf{A}^\dagger}={\rm Hom}_{\mathbf{Z}_p}({\mathbf{V}^\dagger},\mu_{p^\infty})$ and for $?\in\{{\rm bal},{\boldsymbol{\varphi}}\}$ 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 ${\boldsymbol{\varphi}}$-unbalanced Selmer groups with coefficients in ${\mathbf{A}^\dagger}$ by
\[
{\rm Sel}^?(\mathbf{Q},{\mathbf{A}^\dagger}):=\ker\biggl\{{\mathrm{H}}^1(\mathbf{Q},{\mathbf{A}^\dagger})\rightarrow\frac{{\mathrm{H}}^1(\mathbf{Q}_p,{\mathbf{A}^\dagger})}{{\mathrm{H}}_?^1(\mathbf{Q}_p,{\mathbf{A}^\dagger})}\times\prod_{v\neq p}{\mathrm{H}}^1(\mathbf{Q}_v^{\rm nr},{\mathbf{A}^\dagger})\biggr\},
\]
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 ${\rm Sel}^?(\mathbf{Q},{\mathbf{A}^\dagger})$.

\subsection{Proof of the decompositions}\label{subsec:dec-S}

Denoting by $\mathbf{V}^\dag_{Q_0}$ the image of ${\mathbf{V}^\dagger}$ under the specialization map $(\ref{eq:sp})$ at $Q_0$,  we have an isomorphism
\begin{equation}\label{eq:shapiro}
\begin{aligned}
{\mathrm{H}}^1(\mathbf{Q},\mathbf{V}_{Q_0}^\dagger)&\simeq{\mathrm{H}}^1(K,T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\oplus{\mathrm{H}}^1(K,T_{\lambda^{-1}}\otimes\Psi_{W_1}^{\mathbf{c}-1})\\
&\quad\oplus{\mathrm{H}}^1(K,T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{1-\mathbf{c}})\oplus{\mathrm{H}}^1(K,T_{\lambda^{-1}\xi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{\mathbf{c}-1})
\end{aligned}
\end{equation}
arising from $(\ref{eq:dec-V})$ and Shapiro's lemma. 

\begin{prop}\label{prop:factor-S}
Under (\ref{eq:shapiro}), the balanced Selmer group ${\rm Sel}^{\rm bal}(\mathbf{Q},\mathbf{V}_{Q_0}^\dagger)$ decomposes as
\begin{align*}
{\rm Sel}^{\rm bal}(\mathbf{Q},\mathbf{V}_{Q_0}^\dagger)&\simeq{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\oplus{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}}\otimes\Psi_{W_1}^{\mathbf{c}-1})\\
&\quad\oplus{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{1-\mathbf{c}})\oplus{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\xi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{\mathbf{c}-1}),
\end{align*}
and the $\varphi$-unbalanced Selmer group ${\rm Sel}^{{\varphi}}(K,\mathbf{V}_{Q_0}^\dagger)$ decomposes as
\begin{align*}
{\rm Sel}^{{\varphi}}(\mathbf{Q},\mathbf{V}_{Q_0}^\dagger)&\simeq
{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\oplus{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}}\otimes\Psi_{W_1}^{\mathbf{c}-1})\\
&\quad\oplus{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{1-\mathbf{c}})\oplus{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\xi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{\mathbf{c}-1}).
\end{align*}
\end{prop}

The proof will follow easily from the following lemma.

\begin{lem}\label{lem:shapiro}
Under $(\ref{eq:shapiro})$, the Selmer group ${\rm Sel}^{\rm bal}(\mathbf{Q},\mathbf{V}_{Q_0}^\dagger)$ corresponds to the submodule of 
\[
{\mathrm{H}}^1(K,(T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\oplus  (T_{\lambda^{-1}}\otimes\Psi_{W_1}^{\mathbf{c}-1})\oplus (T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{1-\mathbf{c}})\oplus (T_{\lambda^{-1}\xi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{\mathbf{c}-1}))
\] 
consisting of unramified-outside-$p$ classes $x$ with
\[
{\rm res}_v(x)\in\begin{cases}
{\mathrm{H}}^1(K_\mathfrak{p},(T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\oplus(T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{1-\mathbf{c}})\oplus (T_{\lambda^{-1}\xi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{\mathbf{c}-1}))&\textrm{if $v=\mathfrak{p}$,}\\[0.2em]
{\mathrm{H}}^1(K_{\overline{\mathfrak{p}}},T_{\lambda^{-1}}\otimes\Psi_{W_1}^{\mathbf{c}-1})&\textrm{if $v=\overline{\mathfrak{p}}$,}
\end{cases}
\]
and the Selmer group ${\rm Sel}^{{\varphi}}(\mathbf{Q},\mathbf{V}_{Q_0}^\dagger)$ similarly corresponds to the submodule consisting of unramified-outside-$p$ classes $x$ with ${\rm res}_{\overline{\mathfrak{p}}}(x)=0$ (and no condition at $\mathfrak{p}$).
\end{lem}

\begin{proof}
Using (\ref{eq:dec-V}) we see that the balanced local condition is given by
\begin{align*}
\mathscr{F}^{\rm bal}_p(\mathbf{V}_{Q_0}^\dagger)
&=(T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\oplus (T_{\lambda^{-\mathbf{c}}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\oplus (T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{1-\mathbf{c}})\oplus(T_{\lambda^{-1}\xi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{\mathbf{c}-1}),
\end{align*}
from where we obtain
\begin{equation}\label{eq:bal-shapiro}
\begin{aligned}
\mathscr{F}^{\rm bal}_\mathfrak{p}(\mathbf{V}_{Q_0}^\dagger)&=(T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\oplus (T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{1-\mathbf{c}})\oplus (T_{\lambda^{-1}\xi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{\mathbf{c}-1}),\\
\mathscr{F}^{\rm bal}_{\overline{\mathfrak{p}}}(\mathbf{V}_{Q_0}^\dagger)&=T_{\lambda^{-1}}\otimes\Psi_{W_1}^{\mathbf{c}-1},
\end{aligned}
\end{equation}
yielding the stated descriptions of ${\rm Sel}^{\rm bal}(K,\mathbf{V}_{Q_0}^\dagger)$. Similarly,
we see that the $\varphi$-unbalanced local condition is given by
\begin{align*}
\mathscr{F}_p^{{\varphi}}(\mathbf{V}_{Q_0}^\dagger)
&=(T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\oplus (T_{\lambda^{-1}}\otimes\Psi_{W_1}^{\mathbf{c}-1})\oplus(T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{1-\mathbf{c}})\oplus(T_{\lambda^{-1}\xi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{\mathbf{c}-1}),
\end{align*}
and this immediately yields the stated description of ${\rm Sel}^{{\varphi}}(\mathbf{Q},{\mathbf{V}^\dagger})$.
\end{proof}




\begin{proof}[Proof of Proposition~\ref{prop:factor-S}]
Put 
\begin{equation}\label{eq:til-V}
\widetilde{\mathbf{V}}_{Q_0}^\dagger=(T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\oplus(T_{\lambda^{-1}}\otimes\Psi_{W_1}^{\mathbf{c}-1})\oplus(T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{1-\mathbf{c}})\oplus(T_{\lambda^{-1}\xi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{\mathbf{c}-1}),
\end{equation} 
so
we have
\[
{\mathrm{H}}^1(\mathbf{Q},\mathbf{V}_{Q_0}^\dagger)\simeq{\mathrm{H}}^1(K,\widetilde{\mathbf{V}}_{Q_0}^\dagger).
\] 

Denoting by ${\rm Sel}^{\{p\}}(K,\widetilde{\mathbf{V}}_{Q_0}^\dagger)$ the submodule of ${\mathrm{H}}^1(K,\widetilde{\mathbf{V}}_{Q_0}^\dagger)$ consisting of unramified-outside-$p$ classes, it follows from Lemma~\ref{lem:shapiro} that under the above isomorphism the balanced Selmer group ${\rm Sel}^{\rm bal}(\mathbf{Q},\mathbf{V}_{Q_0}^\dagger)$ corresponds to the kernel of the restriction map
\begin{align*}
{\rm Sel}^{\{p\}}(K,\widetilde{\mathbf{V}}_{Q_0}^\dagger)&\rightarrow\frac{{\mathrm{H}}^1(K_\mathfrak{p},\widetilde{\mathbf{V}}_{Q_0}^\dagger)}{{\mathrm{H}}^1(K_\mathfrak{p},(T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\oplus(T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{1-\mathbf{c}})\oplus( T_{\lambda^{-1}\xi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{\mathbf{c}-1}))}\\
&\quad\times\frac{{\mathrm{H}}^1(K_{\overline{\mathfrak{p}}},\widetilde{\mathbf{V}}_{Q_0}^\dagger)}{{\mathrm{H}}^1(K_{\overline{\mathfrak{p}}},T_{\lambda^{-1}}\otimes\Psi_{W_1}^{\mathbf{c}-1})},
\end{align*}
and this is isomorphic to
\begin{align*}
&{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\oplus{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{1-\mathbf{c}})\oplus{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\xi^{\mathbf{c}-1}}\otimes\Psi_{W_2}^{\mathbf{c}-1})\\
&\quad\oplus{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}}\otimes\Psi_{W_1}^{\mathbf{c}-1}).
\end{align*}
This shows the result for ${\rm Sel}^{\rm bal}(K,\mathbf{V}_{Q_0}^\dagger)$, and the
case of ${\rm Sel}^{\varphi}(K,\mathbf{V}_{Q_0}^\dagger)$ follows from Lemma~\ref{lem:shapiro} in the same manner.
\end{proof}

As a consequence we also obtain the following decomposition for the Selmer groups with coefficients in $\mathbf{A}_{Q_0}^\dagger={\rm Hom}_{\mathbf{Z}_p}(\mathbf{V}_{Q_0}^\dagger,\mu_{p^\infty})$, mirroring (in the case of ${\rm Sel}^{{\varphi}}(K,\mathbf{A}_{Q_0}^\dagger)$) the factorization of $p$-adic $L$-functions in Proposition~\ref{prop:factor-L}.
 

\begin{cor}\label{cor:factor-S}
The balanced Selmer group ${\rm Sel}^{\rm bal}(\mathbf{Q},\mathbf{A}_{Q_0}^\dagger)$ decomposes as
\begin{align*}
{\rm Sel}^{\rm bal}(\mathbf{Q},\mathbf{A}_{Q_0}^\dagger)&\simeq{\rm Sel}_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\psi^{1-\mathbf{c}}}\otimes\Psi_{W_1}^{\mathbf{c}-1})\oplus{\rm Sel}_{\emptyset,0}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\\
&\quad\oplus{\rm Sel}_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\phi^{1-\mathbf{c}}}\otimes\Psi_{W_2}^{\mathbf{c}-1})\oplus{\rm Sel}_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\xi^{1-\mathbf{c}}}\otimes\Psi_{W_2}^{1-\mathbf{c}}),
\end{align*}
and the $\varphi$-unbalanced Selmer group ${\rm Sel}^{{\varphi}}(\mathbf{Q},\mathbf{A}_{Q_0}^\dagger)$ decomposes as
\begin{align*}
{\rm Sel}^{{\varphi}}(\mathbf{Q},\mathbf{A}_{Q_0}^\dagger)&\simeq
{\rm Sel}_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\psi^{1-\mathbf{c}}}\otimes\Psi_{W_1}^{\mathbf{c}-1})\oplus{\rm Sel}_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{W_1}^{1-\mathbf{c}})\\
&\quad\oplus{\rm Sel}_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\phi^{1-\mathbf{c}}}\otimes\Psi_{W_2}^{\mathbf{c}-1})\oplus{\rm Sel}_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\xi^{1-\mathbf{c}}}\otimes\Psi_{W_2}^{1-\mathbf{c}}).
\end{align*}
\end{cor}

\begin{proof}
This is immediate from Proposition~\ref{prop:factor-S} and local Tate duality.
\end{proof}

Note that the only difference between ${\rm Sel}^{\rm bal}(\mathbf{Q},\mathbf{A}_{Q_0}^\dagger)$ and ${\rm Sel}^{{\varphi}}(\mathbf{Q},\mathbf{A}_{Q_0}^\dagger)$ as described in Corollary~\ref{cor:factor-S} is their second direct summand: The direct summand ${\rm Sel}_{\emptyset,0}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{W_1}^{1-\mathbf{c}})$ in the former is replaced by ${\rm Sel}_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{W_1}^{1-\mathbf{c}})$ in the latter (i.e., their local conditions at the primes above $p$ are \emph{reversed}). 

Under some hypotheses, the Pontryagin dual of ${\rm Sel}_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{W_1}^{1-\mathbf{c}})$ will be $\Lambda$-torsion (see Theorem~\ref{thm:AH}), while ${\rm Sel}_{\emptyset,0}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{W_1}^{1-\mathbf{c}})$ will have $\Lambda$-corank one.



\section{Iwasawa main conjectures}

In this section we explain a variant of the Iwasawa main conjecture for triple products formulated in \cite{ACR}. 

In the notations of $\S\ref{subsec:setting}$ we now set $S_1=S_2$, and call it $S$ (i.e., the weights of ${\boldsymbol{g}}$ and ${\boldsymbol{h}}$ move in tandem). Then the change of variables of Proposition~\ref{prop:factor-L}  
becomes $(W_1,W_2)=(S,0)$,  and we denote by
\[
\mathscr{L}_p^\varphi(\varphi,{\boldsymbol{g}}{\boldsymbol{h}})\in\Lambda:=\mathcal O\dBr{S}
\]
the resulting restriction of $\mathscr{L}_p^\varphi(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})$. Similarly, we put $\mathbb{V}^\dagger=\mathbf{V}^\dagger\vert_{S_1=S_2}$ and denote by 
\begin{equation}\label{eq:diag}
\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}})\in{\mathrm{H}}^1(\mathbf{Q},\mathbb{V}^\dagger)
\end{equation}
the resulting restriction of the three-variable big diagonal class $\kappa({\boldsymbol{\varphi}},{\boldsymbol{g}},{\boldsymbol{h}})$ constructed in \cite[\S{8.1}]{BSV}.

\begin{rem}
Directly from \cite[\S{8.1}]{BSV} one obtains a class in the cohomology of a representation non-canonically isomorphic to finitely many copies of $\mathbb{V}^\dagger$; to obtain a class as in (\ref{eq:diag}) one further needs to pick a choice of level-$N={\rm lcm}(N_\varphi,N_g,N_h)$ test vectors for the triple $(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})$, which we implictly take to be the one furnished by \cite[Thm.~A]{hsieh-triple}.
\end{rem} 

It follows from \cite[Cor.~8.2]{BSV} that the class $\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}})$ lands in ${\rm Sel}^{\rm bal}(K,\mathbb{V}^\dagger)$. Put 
\[
\mathscr{F}_p^{3}({\mathbf{V}^\dagger})=V_{\boldsymbol{\varphi}}^+\hat\otimes_{\mathcal O} V_{{\boldsymbol{g}}}^+\hat\otimes_{\mathcal O} V_{{\boldsymbol{h}}}^+\otimes\mathcal{X}^{-1}\subset{\mathbf{V}^\dagger}
\]
and denote by $\mathscr{F}_p^{3}({\mathbb{V}^\dagger})\subset{\mathbb{V}^\dagger}$ the corresponding restriction to the line $S_1=S_2$. Then we have
\begin{equation}\label{eq:gr2}
\mathscr{F}^{\rm bal}_p({\mathbb{V}^\dagger})/\mathscr{F}_p^3({\mathbb{V}^\dagger})\simeq\mathbb{V}_{{\boldsymbol{\varphi}}}^{{\boldsymbol{g}}{\boldsymbol{h}}}\oplus\mathbb{V}_{{\boldsymbol{g}}}^{{\boldsymbol{\varphi}}{\boldsymbol{h}}}\oplus\mathbb{V}_{{\boldsymbol{h}}}^{{\boldsymbol{\varphi}}{\boldsymbol{g}}},\nonumber
\end{equation}
where $\mathbb{V}_{{\boldsymbol{\varphi}}}^{{\boldsymbol{g}}{\boldsymbol{h}}}$ is the restriction of $\mathbf{V}_{{\boldsymbol{\varphi}}}^{{\boldsymbol{g}}{\boldsymbol{h}}}=\mathbf{V}_{\varphi}^-\hat\otimes_{\mathcal O} V_{{\boldsymbol{g}}}^+\hat\otimes_{\mathcal O} V_{{\boldsymbol{h}}}^+\otimes\mathcal{X}^{-1}$ restricted to the line $S_1=S_2$,  
and similarly for the other two direct summands. In terms of the description given in the proof of Lemma~\ref{lem:shapiro}, we find that 
\[
\mathscr{F}_\mathfrak{p}^3({\mathbb{V}^\dagger})=T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_S^{1-\mathbf{c}},\quad\mathscr{F}_{\overline{\mathfrak{p}}}^3({\mathbb{V}^\dagger})=\{0\},
\]
and so together with $(\ref{eq:bal-shapiro})$ we obtain 
\begin{equation}\label{eq:gh_f}
{\mathrm{H}}^1(\mathbf{Q}_p,\mathbb{V}_\varphi^{{\boldsymbol{g}}{\boldsymbol{h}}})\simeq{\mathrm{H}}^1(K_{\overline{\mathfrak{p}}},T_{\lambda^{-1}}\otimes\Psi_{S}^{\mathbf{c}-1}),
\end{equation}
and under this isomorphism the inclusion $\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}})\in{\rm Sel}_{\rm bal}(\mathbf{Q},{\mathbb{V}^\dagger})$ implies  that
\[
{\rm res}_{\overline{\mathfrak{p}}}(\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}}))\in{\mathrm{H}}^1(K_{\overline{\mathfrak{p}}},\lambda^{-1}\Psi_{S}^{\mathbf{c}-1}).
\] 

Put $u:=1+p$ and for any $\Lambda$-module $M$ and integer $k$ denote by $M_k$ the specialization of $M$ at $S=u^{k-2}-1$. Then in particular we see that there are isomorphisms
\begin{equation}\label{eq:BK}
\begin{aligned}
{\rm log}_{\overline{\mathfrak{p}}}:{\mathrm{H}}^1(K_{\overline{\mathfrak{p}}},T_{\lambda^{-1}}\otimes\Psi_S^{\mathbf{c}-1})_k&\rightarrow \mathbf{Q}_p,\quad k\geq 3,\\
{\rm exp}_{\overline{\mathfrak{p}}}^*:{\mathrm{H}}^1(K_{\overline{\mathfrak{p}}},T_{\lambda^{-1}}\otimes\Psi_S^{\mathbf{c}-1})_k&\rightarrow \mathbf{Q}_p,\quad k=2,
\end{aligned}
\end{equation}
given by the Bloch--Kato logarithm and dual exponential maps.



\begin{thm}[Explicit reciprocity Law]\label{thm:ERL}
There is an injective $\Lambda$-module homomorphism
\[
{\rm Log}^{{\varphi}}:{\mathrm{H}}^1(K_{\overline{\mathfrak{p}}},T_{\lambda^{-1}}\otimes\Psi_{S}^{\mathbf{c}-1})\rightarrow\Lambda
\]
with pseudo-null cokernel satisfying for any $\mathbf{z}\in{\mathrm{H}}^1(K_{\overline{\mathfrak{p}}},T_{\lambda^{-1}}\otimes\Psi_{S}^{\mathbf{c}-1})$ the interpolation property
\[
{\rm Log}^{{\varphi}}(\mathbf{z})_k=\begin{cases}
c_k\cdot{\rm log}_{\overline{\mathfrak{p}}}(\mathbf{z}_k)&\textrm{if $k\geq 3$,}\\
c_k\cdot{\rm exp}_{\overline{\mathfrak{p}}}(\mathbf{z}_k)&\textrm{if $k=2$,}
\end{cases}
\]
where $c_k$ is an explicit nonzero constant, and such that
\[
{\rm Log}^{{\varphi}}\bigl({\rm res}_{\overline{\mathfrak{p}}}(\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}}))\bigr)=\mathscr{L}_p^\varphi(\varphi,{\boldsymbol{g}}{\boldsymbol{h}}).
\]
\end{thm}

\begin{proof}
In light of (\ref{eq:gh_f}), the construction of ${\rm Log}^{{\varphi}}$ follows from specializing the  three-variable $p$-adic regulator of \cite[\S{7.1}]{BSV} similarly as in the proof of \cite[Prop.~7.3]{ACR}; the reciprocity law then follows from \cite[Thm.~A]{BSV} (see also \cite[Thm.~10]{DR3}).
\end{proof}






As explained  
in more detail in Remark~\ref{rem:Gr} below (see also \cite[\S{7.3}]{ACR}), the following can be seen as the equivalence between two different formulations of the Iwasawa--Greenberg main conjecture \cite{Greenberg55} for ${\mathbb{V}^\dagger}$.

\begin{prop}\label{prop:equiv}
The following statements (1)-(2) are equivalent:
\begin{enumerate}
\item{} $\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}})$ is not $\Lambda$-torsion, the modules ${\rm Sel}^{\rm bal}(\mathbf{Q},{\mathbb{V}^\dagger})$ and $X^{\rm bal}(\mathbf{Q},{\mathbb{A}^\dagger})$ have both $\Lambda$-rank one, and
\[
{\rm char}_{\Lambda}\bigl(X^{\rm bal}(\mathbf{Q},{\mathbb{A}^\dagger})_{\rm tors}\bigr)={\rm char}_{\Lambda}\biggl(\frac{{\rm Sel}^{\rm bal}(\mathbf{Q},{\mathbb{V}^\dagger})}{\Lambda\cdot\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}})}\biggr)^2
\]
in $\Lambda\otimes\mathbf{Q}_p$, where the subscript ${\rm tors}$ denotes the $\Lambda$-torsion submodule.
\item{} $\mathscr{L}_p^\varphi(\varphi,{\boldsymbol{g}}{\boldsymbol{h}})$ is nonzero, the modules ${\rm Sel}^{{\varphi}}(\mathbf{Q},{\mathbb{V}^\dagger})$ and $X^{{\varphi}}(\mathbf{Q},{\mathbb{A}^\dagger})$ are both $\Lambda$-torsion, and 
\[
{\rm char}_\Lambda\bigl(X^{{\varphi}}(\mathbf{Q},{\mathbb{A}^\dagger})\bigr)=\bigl(\mathscr{L}_p^\varphi(\varphi,{\boldsymbol{g}}{\boldsymbol{h}})\bigr)^2
\]
in $\Lambda\otimes\mathbf{Q}_p$.
\end{enumerate}
\end{prop}


\begin{proof}
This follows from Theorem~\ref{thm:ERL} and global duality in the same way as \cite[Thm.~7.15]{ACR}. Since the required arguments are virtually the same, we omit the details.
\end{proof}

\begin{rem}\label{rem:Gr}
For any $k\in\mathbf{Z}$ and $\zeta\in\mu_{p^\infty}$ denote by $\mathfrak{P}_{k,\zeta}$ the height one prime of $\Lambda$ given by  $\mathfrak{P}_{k,\zeta}=(S-\gamma^{k-2}\zeta)$, and put
\[
C^{\rm bal}=\{\mathfrak{P}_{k,\zeta}\colon k\geq 2,\;\zeta\in\mu_{p^\infty}\},\
\quad C^{{\varphi}}=\{\mathfrak{P}_{1,\zeta}\colon\zeta\in\mu_{p^\infty}\}.
\] 

Then the pairs $(\mathscr{F}_p^{\rm bal}({\mathbb{V}^\dagger}),C^{\rm bal})$ and $(\mathscr{F}_p^{{\varphi}}({\mathbb{V}^\dagger}),C^{\varphi})$ both satisfy the \emph{Panchishkin condition for ${\mathbb{V}^\dagger}$} introduced in \cite[p.\,216]{Greenberg55}, and statements (1) and (2) of Proposition~\ref{prop:equiv} can be viewed as instances of the corresponding  Iwasawa main conjectures formulated in \emph{op.\,cit.}.
\end{rem}

The results of Proposition~\ref{prop:factor-L}, Proposition~\ref{prop:factor-S}, and Corollary~\ref{cor:factor-S} reduce the study of the Selmer groups ${\rm Sel}^{\rm bal}(\mathbf{Q},{\mathbb{A}^\dagger})$, ${\rm Sel}^{{\varphi}}(\mathbf{Q},{\mathbb{A}^\dagger})$ and their associated main conjectures to a corresponding study for Selmer groups of twists of $E$. 

The next result due to Agboola--Howard and Arnold (building on work of Greenberg, Yager, and Rubin, among others) will therefore be useful.

\begin{thm}\label{thm:AH}
Let $E_{/\mathbf{Q}}$ be an elliptic curve with CM by the ring of integers of $K$ having root number $w=+1$. Let $\nu$ be a ring class character of $K$ of conductor divisible only by primes $\ell\nmid p$ split in $K$. Then ${\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\nu}\otimes\Psi_{S}^{1-\mathbf{c}})=0$, $X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\nu^{-1}}\otimes\Psi_{S}^{\mathbf{c}-1})$ is $\Lambda$-torsion, and
\[
{\rm char}_\Lambda\bigl(X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\nu^{-1}}\otimes\Psi_S^{\mathbf{c}-1})\bigr)=\bigl({\mathcal L}^-_{\mathfrak{p},\lambda^{-1}\nu}(S)\bigr)
\]
as ideals in $\Lambda_{\mathcal W}[1/p]$.
\end{thm}

\begin{proof}
For $\nu=\mathds{1}$ this is \cite[Thm.~2.4.17]{AHsplit}; the general case follows from \cite[Thm.~2.1]{Arnold}.
\end{proof}







\section{Proof of main result}


In this section we prove a strenghtened version of Theorem~\ref{thm:1} in the Introduction. The definition of the class $\kappa_p$ appearing in the statement is given in Definition~\ref{def:GKC} below.

\subsection{Preliminaries}

The link between one of the Selmer groups for characters appeared in $\S\ref{subsec:dec-S}$ and the Selmer group of $E$ arises from the following lemma, whose proof we learnt from \cite[\S{6}]{agboola-rubin-I}.

\begin{lem}\label{lem:-1}
Assume that $E(\mathbf{Q})$ has a point of infinite order. Then
\[
{\rm rank}_{\mathcal O_{K,\mathfrak{p}}}{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}})={\rm rank}_{\mathcal O_{K,\mathfrak{p}}}{\rm Sel}(K,T_{\lambda^{-1}})-1.
\]
\end{lem} 

\begin{proof}
Put $\mathcal{D}_{\overline{\mathfrak{p}}}=K_{\overline{\mathfrak{p}}}/\mathcal O_{K,\overline{\mathfrak{p}}}$. By global duality we have the exact sequence
\begin{equation}\label{eq:PT-bisi}
0\rightarrow{\rm Sel}(K,T_{\lambda^{-1}})\rightarrow{\rm Sel}_{\rm rel}(K,T_{\lambda^{-1}})\xrightarrow{\alpha}\prod_{v\vert p}\frac{{\mathrm{H}}^1(K_v,T_{\lambda^{-1}})}{E(K_v)\otimes\mathcal O_{K,\mathfrak{p}}}\xrightarrow{\beta^\vee}{\rm Sel}(K,A_{\lambda^{-\mathbf{c}}})^\vee,
\end{equation}
where the last arrow is identified with Pontryagin dual of the restriction map 
\[
\beta:{\rm Sel}(K,A_{\lambda^{-\mathbf{c}}})\rightarrow\prod_{v\vert p}E(K_v)\otimes\mathcal{D}_{\overline{\mathfrak{p}}}
\]
via local duality. By assumption, $E(K)$ has positive $\mathcal O_K$-rank, and therefore the image of $\beta$ has $\mathcal O_{K,\overline{\mathfrak{p}}}$-rank one. By (\ref{eq:PT-bisi}) this implies that the map $\alpha$ has finite image (noting that its target has $\mathcal O_{K,\mathfrak{p}}$-rank one), and so ${\rm rank}_{\mathcal O_{K,\mathfrak{p}}}{\rm Sel}_{\rm rel}(K,T_{\lambda^{-1}})={\rm rank}_{\mathcal O_{K,\mathfrak{p}}}{\rm Sel}(K,T_{\lambda^{-1}})$. In particular, this shows that
\[
{\rm rank}_{\mathcal O_{K,\mathfrak{p}}}{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}})=
{\rm rank}_{\mathcal O_{K,\mathfrak{p}}}\ker\bigl({\rm loc}_\mathfrak{p}:{\rm Sel}(K,T_{\lambda^{-1}})\rightarrow E(K_\mathfrak{p})\otimes\mathcal O_{K,\mathfrak{p}}\bigr).
\]
Since the image of the restriction map ${\rm loc}_\mathfrak{p}$ has $\mathcal O_{K,\mathfrak{p}}$-rank one by our hypotheses, this implies the result.
\end{proof}

For any $\mathcal O\dBr{S}$-module $M$ we denote by $M_S=M/SM$ the denote the cokernel of multiplication by $S$. We shall also need the following  variant of Mazur's control theorem. (Note however that the Selmer groups in the statement have reversed local conditions at the primes above $p$ with respect to the usual Selmer groups.)

\begin{prop}\label{prop:control}
Multiplication by $S$ induces natural maps
\begin{align*}
r^*:{\rm Sel}_{\emptyset,0}(K,A_{\lambda^{-\mathbf{c}}})&\rightarrow{\rm Sel}_{\emptyset,0}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})\vert_{S=0},\\
r:{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}}\otimes\Psi_{S}^{\mathbf{c}-1})_S&\rightarrow{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}})
\end{align*}
which are injective with finite cokernel.
\end{prop}

\begin{proof}
We only explain the case of $r^*$, as the case of $r$ follows in the same manner. The map $r^*$ fits into the commutative diagram with exact rows
\begin{equation}\label{eq:diag-res}
\resizebox{\displaywidth}{!}{
\xymatrix{
0\ar[r]&{\rm Sel}_{\emptyset,0}(K,A_{\lambda^{-\mathbf{c}}})\ar[r]\ar[d]^{r^*}&{\rm Sel}^{\{p\}}(K,A_{\lambda^{-\mathbf{c}}})\ar[r]\ar[d]^{s^*}&{\mathrm{H}}^1(K_\mathfrak{p},A_{\lambda^{-\mathbf{c}}})\ar[d]^{t^*}\\
0\ar[r]&{\rm Sel}_{\emptyset,0}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})\vert_{S=0}\ar[r]&{\rm Sel}^{\{p\}}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})\vert_{S=0}\ar[r]&{\mathrm{H}}^1(K_\mathfrak{p},A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})\vert_{S=0}.\nonumber
}}
\end{equation}
It follows from Lubin--Tate theory that the local field extension $K_\mathfrak{p}(E[\overline{\mathfrak{p}}])/K_\mathfrak{p}$ has degree $p-1$ (see \cite[Ch.~I]{deshalit}), and so ${\mathrm{H}}^0(K_{\infty,\mathfrak{p}},E[\overline{\mathfrak{p}}^\infty])=0$ since $\Gamma^-$ is a pro-$p$ group. By inflation-restriction, it follows that the map $t^*$ is an isomorphism. On the other hand, letting $\Sigma$ be any finite set of places of $K$ containing $\infty$ and the primes dividing $pN$, the map $s^*$ above fits into the commutative diagram with exact rows 
\begin{equation}\label{eq:diag-res}
\resizebox{\displaywidth}{!}{
\xymatrix{
0\ar[r]&{\rm Sel}^{\{p\}}(K,A_{\lambda^{\-\mathbf{c}}})\ar[r]\ar[d]^{s^*}&{\mathrm{H}}^1(K_\Sigma/K,A_{\lambda^{-\mathbf{c}}})\ar[r]\ar[d]^{u^*}&\bigoplus_{w\in\Sigma,w\nmid p}{\mathrm{H}}^1(K_w,A_{\lambda^{-\mathbf{c}}})\ar[d]^{v^*}\\
0\ar[r]&{\rm Sel}^{\{p\}}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})\vert_{S=0}\ar[r]&{\mathrm{H}}^1(K_\Sigma/K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})\vert_{S=0}\ar[r]&\bigoplus_{w\in\Sigma,w\nmid p}{\mathrm{H}}^1(K_w,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})\vert_{S=0},\nonumber
}}
\end{equation}
where $K_\Sigma$ is the Galois group of the maximal extension of $K$ unramified outside $\Sigma$. The vanishing of ${\mathrm{H}}^0(K_{\infty,\mathfrak{p}},E[\overline{\mathfrak{p}}^\infty])$ implies that $u^*$ is an isomorphism, and as shown in the proof of \cite[Lem.~IV.3.5]{deshalit} as consequence of \cite[Cor.~4.4]{mazur-18}, the kernel of the map $v^*$ is finite. 
By the Snake Lemma applied to the previous two commutative diagrams,
the result follows.
\end{proof}

Specialised to $(W_1,W_2)=(S,0)$, the decomposition 
of ${\rm Sel}^{\rm bal}(\mathbf{Q},\mathbf{V}_{Q_0}^\dagger)$ given in Proposition~\ref{prop:factor-S} becomes
\begin{equation}\label{eq:bal-Sel}
\begin{aligned}
{\rm Sel}^{\rm bal}(\mathbf{Q},{\mathbb{V}^\dagger})&\simeq
{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_{S}^{1-\mathbf{c}})\oplus{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}}\otimes\Psi_{S}^{\mathbf{c}-1})\\
&\quad
\oplus{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\otimes\Psi_{S}^{1-\mathbf{c}})_S\oplus{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\xi^{\mathbf{c}-1}}\otimes\Psi_{S}^{\mathbf{c}-1})_S.
\end{aligned}
\end{equation}

\subsection{Generalised Kato class}\label{subsec:GKC}

The following may be viewed as a variant of the generalised Kato classes introduced by Darmon--Rotger \cite{DR2,DR2.5} (however, it is crucial for us to take $\varphi=\theta(\lambda\psi)$ for appropriate $\psi$ as in \S\ref{subsec:setting} rather than the newform $\theta(\lambda)$ associated with $E$).

\begin{defn}[Generalised Kato class]\label{def:GKC}
Let $\kappa_p$ be the image of $\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}})$ under the composition
\[
{\rm Sel}^{\rm bal}(\mathbf{Q},{\mathbb{V}^\dagger})\rightarrow{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}}\otimes\Psi_{S}^{\mathbf{c}-1})\rightarrow{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}}),
\]
where the first arrow is given by the projection onto the second direct summand in (\ref{eq:bal-Sel}), and the second arrow is induced by the multiplication by $S$ on $T_{\lambda^{-1}}\otimes\Psi_S^{\mathbf{c}-1}$. 
\end{defn}

Using ${\mathrm{H}}^1(K,V_{\lambda^{-1}})\simeq{\mathrm{H}}^1(\mathbf{Q},V_pE)$, we shall also view $\kappa_p$ in the latter group.

\begin{lem}\label{lem:ERL}
Assume that $L(E,1)=0$. Then $\kappa_p\in{\rm Sel}(\mathbf{Q},V_pE)$.
\end{lem}

\begin{proof}
Viewing $\kappa_p$ as a class in ${\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}})$ it suffices to show that ${\rm res}_{\overline{\mathfrak{p}}}(\kappa_p)=0$ (and therefore $\kappa_p$ lies in the so-called \emph{fine Selmer group} for $V_pE$). Since $L(E,1)=L(\lambda^{-1},0)$, by Theorem~\ref{thm:ERL}, Proposition~\ref{prop:factor-S}, and Theorem~\ref{thm:katz} we see that
\[
L(E,1)=0\quad\Longrightarrow\quad{\rm exp}_{\overline{\mathfrak{p}}}^*({\rm res}_{\overline{\mathfrak{p}}}(\kappa_p))=0,
\]
where ${\rm exp}_{\overline{\mathfrak{p}}}^*$ is the Bloch--Kato dual exponential map in (\ref{eq:BK}). Since for $k=2$ (i.e., $S=0$) this map is an isomorphism, the result follows. 
\end{proof}




\subsection{Proof of Theorem~\ref{thm:1}}

\begin{thm}\label{thm:main}
Assume that $L(E,s)$ has sign $+1$ and $L(E,1)=0$. Then
\[
{\rm rank}_{\mathcal O_{K,\mathfrak{p}}}{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}})=1\quad\Longrightarrow\quad\kappa_{p}\neq 0,
\] 
where $\kappa_p$ is any generalised Kato class associated with a triple $(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})$ as in Choice~\ref{choice:setting}.
\end{thm}

\begin{proof}
Specialised to $(W_1,W_2)=(S,0)$, the factorization in Proposition~\ref{prop:factor-L} reads as the equality
\begin{equation}\label{eq:factor-L}
\mathscr{L}_p^\varphi(\varphi,{\boldsymbol{g}}{\boldsymbol{h}})^2(S)={\mathcal L}^-_{\mathfrak{p},\lambda^{-1}\psi^{\mathbf{c}-1}}(S)^\iota\cdot{\mathcal L}^-_{\mathfrak{p},\lambda^{-1}}(S)\cdot{\mathcal L}_{\mathfrak{p},\lambda^{-1}\psi^{\mathbf{c}-1}}^-(0)\cdot{\mathcal L}_{\mathfrak{p},\lambda^{-1}\xi^{\mathbf{c}-1}}^-(0)\nonumber
\end{equation}
up to a multiplication by a unit $u\in{\mathcal W}^\times$. On the other hand, the decompostion in Corollary~\ref{cor:factor-S} for the unbalanced Selmer group becomes
\begin{align*}
X^{{\varphi}}(\mathbf{Q},{\mathbb{A}^\dagger})&\simeq
X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\psi^{1-\mathbf{c}}}\otimes\Psi_{S}^{\mathbf{c}-1})\oplus X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})\\
&\quad\oplus X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\phi^{1-\mathbf{c}}}\otimes\Psi_{S}^{\mathbf{c}-1})_S\oplus X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\xi^{1-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})_S.
\end{align*}
Since clearly $X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\psi^{1-\mathbf{c}}}\otimes\Psi_{S}^{\mathbf{c}-1})=X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\psi^{1-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})^\iota$, from Theorem~\ref{thm:AH} we deduce that $X^{{\varphi}}(\mathbf{Q},{\mathbb{A}^\dagger})$ is $\Lambda$-torsion with 
\[
{\rm char}_{\Lambda}\bigl(X^{{\varphi}}(\mathbf{Q},{\mathbb{A}^\dagger})\bigr)=\bigl(\mathscr{L}_p^\varphi(\varphi,{\boldsymbol{g}}{\boldsymbol{h}})\bigr)^2
\]
as ideals in $\Lambda_{\mathcal W}\otimes\mathbf{Q}_p$. By Proposition~\ref{prop:equiv}, it follows that 
\begin{equation}\label{eq:rank-bal}
{\rm rank}_\Lambda\bigl(X^{\rm bal}(\mathbf{Q},{\mathbb{A}^\dagger})\big)={\rm rank}_\Lambda\bigl({\rm Sel}^{\rm bal}(\mathbf{Q},{\mathbb{V}^\dagger})\bigr)=1,
\end{equation}
and
\begin{equation}\label{eq:char-bal}
{\rm char}_{\Lambda}\bigl(X^{\rm bal}(\mathbf{Q},{\mathbb{A}^\dagger})_{\rm tors}\bigr)={\rm char}_{\Lambda}\biggl(\frac{{\rm Sel}^{\rm bal}(\mathbf{Q},{\mathbb{V}^\dagger})}{\Lambda\cdot\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}})}\biggr)^2
\end{equation}
as ideals in $\Lambda_{\mathcal W}\otimes\mathbf{Q}_p$. 

Now, from Theorem~\ref{thm:katz}, Theorem~\ref{thm:AH}, and Mazur's control theorem (for the usual Selmer groups) we have the implications
\begin{align*}
L(\lambda^{-1}\psi^{\mathbf{c}-1},0)\neq 0\quad&\Longrightarrow\quad{\mathcal L}_{\mathfrak{p},\lambda^{-1}\psi^{\mathbf{c}-1}}^-(0)\neq 0\\
&\Longrightarrow\quad\left\vert X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\psi^{1-\mathbf{c}}}\otimes\Psi_{S}^{\mathbf{c}-1})_S\right\vert<\infty.
\end{align*}
Similarly, the nonvanishing of $L(\lambda^{-1}\phi^{\mathbf{c}-1},0)$ and $L(\lambda^{-1}\xi^{\mathbf{c}-1},0)$ implies that
\[
\left\vert X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\phi^{1-\mathbf{c}}}\otimes\Psi_{S}^{\mathbf{c}-1})_S\right\vert<\infty,\quad\left\vert X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\xi^{1-\mathbf{c}}}\otimes\Psi_{S}^{\mathbf{c}-1})_S\right\vert<\infty,
\]
and so from the balanced Selmer group decomposition from Corollary~\ref{cor:factor-S}:  
\begin{align*}
X^{\rm bal}(\mathbf{Q},{\mathbb{A}^\dagger})&\simeq
X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\psi^{1-\mathbf{c}}}\otimes\Psi_{S}^{\mathbf{c}-1})\oplus X_{\emptyset,0}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})\\
&\quad\oplus X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\phi^{1-\mathbf{c}}}\otimes\Psi_{S}^{\mathbf{c}-1})_S\oplus X_{0,\emptyset}(K,A_{\lambda^{-\mathbf{c}}\xi^{1-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})_S,
\end{align*}
we deduce that
\begin{equation}\label{eq:reversed-rank1}
{\rm rank}_\Lambda\bigl(X_{\emptyset,0}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})\bigr)=1.
\end{equation}
Moreover, by Proposition~\ref{prop:control} we have
\begin{equation}\label{eq:coinv}
\begin{aligned}
{\rm rank}_{\mathcal O_{K,\mathfrak{p}}}\bigl(X_{\emptyset,0}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_{S}^{1-\mathbf{c}})_S\bigr)
&={\rm corank}_{\mathcal O_{K,\mathfrak{p}}}\bigl({\rm Sel}_{\emptyset,0}(K,A_{\lambda^{-\mathbf{c}}})\bigr)\\
&={\rm rank}_{\mathcal O_{K,\mathfrak{p}}}\bigl({\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-\mathbf{c}}})\bigr)\\
&={\rm rank}_{\mathcal O_{K,\mathfrak{p}}}\bigl({\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}})\bigr),
\end{aligned}
\end{equation}
using the isomorphism ${\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-\mathbf{c}}})\simeq{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}})$ given by the action of complex conjugation for the last equality.

On the other hand, denote by ${\rm pr}_\lambda(\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}}))$ the projection of $\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}})$ onto the second direct summand in the balanced Selmer groups decomposition from Proposition~\ref{prop:factor-S}:
\begin{align*}
{\rm Sel}^{\rm bal}(\mathbf{Q},{\mathbb{V}^\dagger})&\simeq
{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\otimes\Psi_{S}^{1-\mathbf{c}})\oplus{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}}\otimes\Psi_{S}^{\mathbf{c}-1})\\
&\quad
\oplus{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\otimes\Psi_{S}^{1-\mathbf{c}})_S\oplus{\rm Sel}_{\emptyset,0}(K,T_{\lambda^{-1}\xi^{\mathbf{c}-1}}\otimes\Psi_{S}^{\mathbf{c}-1})_S.
\end{align*}
From the above we see that (\ref{eq:rank-bal}) and (\ref{eq:char-bal}) imply that ${\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}}\otimes\Psi_{S}^{\mathbf{c}-1})$ has $\Lambda$-rank one, and we have the following equality of ideals in $\Lambda_{\mathcal W}\otimes\mathbf{Q}_p$:
\begin{equation}\label{eq:IMC-lambda}
{\rm char}_\Lambda\bigl(X_{\emptyset,0}(K,A_{\lambda^{-\mathbf{c}}}\otimes\Psi_S^{1-\mathbf{c}})_{\rm tors}\bigr)={\rm char}_\Lambda\bigl(\mathfrak{Z}\bigr)^2,\quad\textrm{where $\mathfrak{Z}=\frac{{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}}\otimes\Psi_{S}^{\mathbf{c}-1})}{\Lambda\cdot{\rm pr}_\lambda(\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}}))}$.}
\end{equation}

The combination of (\ref{eq:reversed-rank1}), (\ref{eq:coinv}), and (\ref{eq:IMC-lambda}) shows that
\[
{\rm rank}_{\mathcal O_{K,\mathfrak{p}}}\bigl({\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}})\bigr)=1+2\,{\rm rank}_{\mathcal O_{K,\mathfrak{p}}}\bigl(\mathfrak{Z}_S\bigr).
\]
Thus if ${\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}})$ has $\mathcal O_{K,\mathfrak{p}}$-rank one then $\mathfrak{Z}_S$ is $\mathcal O_{K,\mathfrak{p}}$-torsion, and so the image 
of ${\rm pr}_\lambda(\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}}))$ in 
${\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}}\otimes\Psi_{S}^{\mathbf{c}-1})_S$ is non-torsion. Since the injection
\[
{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}}\otimes\Psi_{S}^{\mathbf{c}-1})_S\rightarrow{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}})
\]
of Proposition~\ref{prop:control} sends 
$({\rm pr}_\lambda(\kappa(\varphi,{\boldsymbol{g}}{\boldsymbol{h}}))\;{\rm mod}\;S)$ into $\kappa_p$, this concludes the proof.
\end{proof}

The following implies in particular Theorem~\ref{thm:1} in the Introduction.

\begin{thm}\label{cor:main}
Assume that $L(E,s)$ has sign $+1$, $L(E,1)=0$, and $E(\mathbf{Q})$ has a point of infinite order. 
Then the following implication holds:
\[
{\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2\quad\Longrightarrow\quad\kappa_p\neq 0,
\] 
where $\kappa_p$ is any generalised Kato class associated with a triple $(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})$ as in Choice~\ref{choice:setting}.
\end{thm}

\begin{proof}
Since ${\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)={\rm rank}_{\mathcal O_{K,\mathfrak{p}}}{\rm Sel}(K,T_{\lambda^{-1}})$, this follows from Theorem~\ref{thm:main} and Lemma~\ref{lem:-1}.
\end{proof}


\begin{rem}
Of course, by the theorem of Coates--Wiles \cite{CW} and the $p$-parity conjecture \cite{guo-parity}, the first two conditions in Theorem~\ref{cor:main} are superfluous.
\end{rem}

\begin{rem}
The construction of $\kappa_p$ depends on the choice of a triple $(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})$ as in Choice~\ref{choice:setting}, but it follows from Lemma~\ref{lem:-1} and the proof of Lemma~\ref{lem:ERL} that under the hypotheses of Theorem~\ref{cor:main} they all differ by a nonzero scalar; in fact, under those hypotheses they all span the  $1$-dimensional subspace ${\rm Sel}_0(\mathbf{Q},V_pE)\simeq{\rm Sel}_{0,\emptyset}(K,V_{\lambda^{-1}})$ inside the $2$-dimensional ${\rm Sel}(\mathbf{Q},V_pE)$. 
\end{rem}


\section{Proof of Theorem~\ref{thm:3}}\label{sec:converse}

We conclude with the proof of the rank~$2$ Kolyvagin theorem stated
in the Introduction. 

\begin{thm}\label{thm:converse}
Assume that $L(E,s)$ has sign $+1$, $L(E,1)=0$, and $E(\mathbf{Q})$ has a point of infinite order. Then the following implication holds:
\[
\kappa_p\neq 0\quad\Longrightarrow\quad
{\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2,
\] 
where $\kappa_p$ is any generalised Kato class associated with a triple $(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})$ as in Choice~\ref{choice:setting}.
\end{thm}

\begin{proof}
By Lemma~\ref{lem:-1}, it suffices to prove the implication 
\begin{equation}\label{eq:conv}
\kappa_p\neq 0\quad\Longrightarrow\quad{\rm rank}_{\mathcal O_{K,\mathfrak{p}}}{\rm Sel}_{0,\emptyset}(K,T_{\lambda^{-1}})=1.
\end{equation}
Let $\widetilde{\mathbf{V}}_{Q_0}^\dagger$ be as in (\ref{eq:til-V}). A straightforward modification of the constructions in \cite[\S{6}]{ACR} building on the big diagonal cycle classes of \cite[\S{8.1}]{BSV} give rise to a collection of cohomology classes
\[
\boldsymbol{\kappa}(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})=\left\{\kappa(\varphi,{\boldsymbol{g}},{\boldsymbol{h}})_n\in{\rm Sel}^{\rm bal}(K[n],\widetilde{\mathbf{V}}_{Q_0}^\dagger)\;:\;n\in\mathcal{S}\right\},
\]
where $n$ runs over the set $\mathcal{S}$ of all squarefree products of primes $q\nmid p\ell\ell'N$ split in $K$ and $K[n]$ denotes the maximal $p$-extension inside the ring class field of $K$ of conductor $n$, satisfying the obvious extension of the norm-compatibility relations of \cite[Prop.~6.3]{ACR}. Projecting these classes to the cohomology of the direct summand $T_{\lambda^{-1}}\otimes\Psi_{W_1}^{\mathbf{c}-1}$ in (\ref{eq:til-V}) and specialising to $W_1=0$ we obtain a collection of cohomology classes
\[
\left\{\widetilde{\kappa}_\lambda(\varphi,g,h)_n\in{\rm Sel}_{0,\emptyset}(K[n],T_{\lambda^{-1}})\;:\;n\in\mathcal{S}\right\}
\]
such that, for all $nq\in\mathcal{S}$ with $q$ a prime, we have 
\[
{\rm cor}_{K[nq]/K[n]}\left(\widetilde{\kappa}_\lambda(\varphi,g,h)_{nq}\right)=Q({\rm Fr}_{\mathfrak{q}}^{-1})\,\widetilde{\kappa}_\lambda(\varphi,g,h)_n,
\]
where ${\rm Fr}_{\mathfrak{q}}$ is the Frobenius element in ${\rm Gal}(K[n]/K)$ for any of the primes above $q$, and
\begin{align*}
Q_\mathfrak{q}({\rm Fr}_{\mathfrak{q}}^{-1})&=\frac{1}{q^{2}}\Bigg\{\chi_g(q)\chi_h(q)q\left(\frac{\lambda\psi(\mathfrak{q})}{q}{\rm Fr}_\mathfrak{q}^{-1}\right)^2-a_q(g)a_q(h)q\left(\frac{\lambda\psi(\mathfrak{q})}{q}{\rm Fr}_\mathfrak{q}^{-1}\right) \\
& +\chi_g(q)^{-1}a_q(g)^2q+\chi_h(q)^{-1}a_q(h)^2-\frac{q^2+1}{q} \\
& -a_q(g)a_q(h)q\left(\frac{\lambda\psi(\overline{\mathfrak{q}})}{q}{\rm Fr}_{{\mathfrak{q}}}\right)+\chi_g(q)\chi_h(q)q\left(\frac{\psi(\overline{\mathfrak{q}})}{q}{\rm Fr}_{\mathfrak{q}}\right)^2\Bigg\}.
\end{align*}
Moreover, by definition the generalised Kato class $\kappa_p$ is related to this construction via
\[
{\rm cor}_{K[1]/K}(\widetilde{\kappa}_\lambda(\varphi,g,h)_1)=\kappa_p. 
\]
Denote by $\mathbb{V}_S^\dagger$ the specialisation of $\widetilde{\mathbf{V}}_{Q_0}^\dagger$ at $(W_1,W_2)=(0,0)$ so by (\ref{eq:til-V}) we have
\[
\mathbb{V}_S^\dagger=T_{\lambda^{-1}\psi^{\mathbf{c}-1}}\oplus T_{\lambda^{-1}}\oplus T_{\lambda^{-1}\phi^{\mathbf{c}-1}}\oplus T_{\lambda^{-1}\xi^{\mathbf{c}-1}}.
\]
Then as in \cite[Thm.~4.6]{ACR} we find the congruence
\[
\widetilde{P}_{\mathfrak{q}}({\rm Fr}_{\mathfrak{q}}^{-1})\chi_g(q)\chi_h(q)\lambda\psi(\overline{\mathfrak{q}})^2{\rm Fr}_{\mathfrak{q}}^2\equiv Q_\mathfrak{q}({\rm Fr}_{\mathfrak{q}}^{-1})\pmod{q-1},
\]
where $\widetilde{P}_{\mathfrak{q}}(X)=\det(1-{\rm Fr}_{\mathfrak{q}}^{-1}X\,\vert\,(\mathbb{V}^\dagger_S)^\vee(1))$. Thus by \cite[Lem.~9.6.1]{Rubin-ES} and 
after restricting to a subset $\mathcal{S'}\subset\mathcal{S}$ consisting of all squarefree products of primes in a positive density set $\mathcal{P}'$ as in \cite[\S{5.1}]{ACR-2}, the classes $\widetilde{\kappa}_\lambda(\varphi,g,h)_n$ can be modified similarly as in Theorem~5.1 of \emph{op.\,cit.} to yield an anticyclotomic Euler system for $T_{\lambda^{-1}}$ with base class $\kappa_p$, i.e., a collection of classes
\[
\left\{\kappa_\lambda(\varphi,g,h)_n\in{\rm Sel}_{0,\emptyset}(K[n],T_{\lambda^{-1}})\;:\;n\in\mathcal{S}'\right\}
\]
satisfying, for all $nq\in\mathcal{S}'$ with $q$ a prime, 
\[
{\rm cor}_{K[nq]/K[n]}\left(\kappa_\lambda(\varphi,g,h)_{nq}\right)=P_\mathfrak{q}({\rm Fr}_{\mathfrak{q}}^{-1})\,\kappa_\lambda(\varphi,g,h)_n,
\]
where $P_{\mathfrak{q}}(X)=\det(1-{\rm Fr}_{\mathfrak{q}}^{-1}X\,\vert\,(T_{\lambda^{-1}})^\vee(1))$ and such that
\[
{\rm cor}_{K[1]/K}(\kappa_\lambda(\varphi,g,h)_1)=\kappa_p. 
\]
Therefore, by Kolyvagin's methods in the form extended in
\cite{JNS} to general anticyclotomic Euler systems (see also \cite[Thm.~8.3]{ACR}), the proof of the implication (\ref{eq:conv}) follows, and this gives the result.
\end{proof}




\bibliographystyle{amsalpha}
