level $N_{g},N_{h}$ passing through the ordinary $p$-stabilizations of $g,h$,
this follows by specializing the three-variable $p$-adic $L$-function in
[Hsi21, Thm. A] attached to $({{\mathbf{f}}},{{\mathbf{g}}},{{\mathbf{h}}})$
and the congruence ideal generator $\xi$. ∎

######  Definition 4.2.

For the test vectors
$(\underline{\breve{{\mathbf{f}}}},\underline{\breve{g}},\underline{\breve{h}})$
of level $N$ provided by Theorem 4.1, we set

$L_{p}({\mathbf{f}},g,h):=\mathscr{L}_{p}^{\xi}(\underline{\breve{{\mathbf{f}}}},\underline{\breve{g}},\underline{\breve{h}})^{2},$

where $\xi$ is any fixed generator of the congruence ideal of ${\mathbf{f}}$.

Note that $L_{p}({\mathbf{f}},g,h)$ depends on the choice of $\xi$, but the
principal ideal in $\Lambda_{{\mathbf{f}}}$ it generates is of course
independent of that choice.

###  4.2. Reciprocity law for diagonal cycles

Let $({\mathbf{f}},g,h)$ be as in $\S\ref{subsec:Lp}$, and without loss of
generality assume that $l\geq m$ (reordering $g$ and $h$ if necessary). Let
${\mathbf{g}}$ and ${\mathbf{h}}$ be the primitive Hida families passing
through $g$ and $h$, respectively; in other words, ${\mathbf{g}}_{l}$ and
${\mathbf{h}}_{m}$ are the ordinary $p$-stabilization of $g$ and $h$,
respectively. Define a character
$\kappa_{{\mathbf{g}}{\mathbf{h}}}:\mathbb{Z}_{p}^{\times}\rightarrow\Lambda_{{\mathbf{g}}}^{\times}\otimes\Lambda_{{\mathbf{h}}}^{\times}$
by

$\kappa_{{\mathbf{g}}{\mathbf{h}}}(u)=\omega(u)^{l+m-4}\langle
u\rangle^{\bm{l}+{\mathbf{m}}-4}$

and choose a square-root of this character defined by
$\kappa_{{\mathbf{g}}{\mathbf{h}}}^{1/2}(u)=\omega(u)^{(l+m-4)/2}\langle
u\rangle^{(\bm{l}+{\mathbf{m}}-4)/2}$.

Recall that

(4.1)
$\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}:=\mathbb{V}_{{\mathbf{f}}}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{g}}}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{h}}}(\Xi_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}),\quad\;\textrm{where\;
$\Xi_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}=\epsilon_{\rm
cyc}^{-1}\kappa_{{\mathbf{f}}}^{-1/2}\kappa_{{\mathbf{g}}{\mathbf{h}}}^{-1/2}$},$

is the self-dual twist of the tensor product of the big Galois representations
associated to ${\mathbf{f}},{\mathbf{g}}$, and ${\mathbf{h}}$. Assume that
these three Galois representations are all irreducible and $p$-distinguished,
and let $\bm{\phi}\in\\{{\mathbf{f}},{\mathbf{g}},{\mathbf{h}}\\}$. As a
$G_{\mathbb{Q}_{p}}$-representation, $\mathbb{V}_{\bm{\phi}}$ admits a
filtration

(4.2)
$0\rightarrow\mathbb{V}_{\bm{\phi}}^{+}\rightarrow\mathbb{V}_{\bm{\phi}}\rightarrow\mathbb{V}_{\bm{\phi}}^{-}\rightarrow
0$

with each $\mathbb{V}_{\bm{\phi}}^{\pm}$ free of rank one over
$\Lambda_{\bm{\phi}}$, and with the $G_{\mathbb{Q}_{p}}$-action on
$\mathbb{V}_{\bm{\phi}}^{-}$ given by the unramified character sending ${\rm
Fr}_{p}\mapsto a_{p}(\bm{\phi})$. This induces an obvious three-step
filtration

$0\subset\mathscr{F}^{3}\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{{\dagger}}\subset\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}\subset\mathscr{F}^{1}\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}\subset\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}$

by $G_{\mathbb{Q}_{p}}$-stable
$\Lambda_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}$-submodules of ranks 1, 4, and
7, respectively, given by

(4.3)
$\begin{split}\mathscr{F}^{1}\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}&=(\mathbb{V}_{{\mathbf{f}}}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{g}}}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{h}}}^{+}+\mathbb{V}_{{\mathbf{f}}}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{g}}}^{+}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{h}}}+\mathbb{V}_{{\mathbf{f}}}^{+}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{g}}}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{h}}})(\Xi_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}),\\\
\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}&=(\mathbb{V}_{{\mathbf{f}}}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{g}}}^{+}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{h}}}^{+}+\mathbb{V}_{{\mathbf{f}}}^{+}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{g}}}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{h}}}^{+}+\mathbb{V}_{{\mathbf{f}}}^{+}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{g}}}^{+}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{h}}})(\Xi_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}),\\\
\mathscr{F}^{3}\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{{\dagger}}&=\mathbb{V}_{{\mathbf{f}}}^{+}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{g}}}^{+}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{h}}}^{+}(\Xi_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}).\end{split}$

The middle term
$\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}$
will play a special role in the following, and we note that

(4.4)
$\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}/\mathscr{F}^{3}\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{{\dagger}}\cong\mathbb{V}_{\mathbf{f}}^{{\mathbf{g}}{\mathbf{h}}}\oplus\mathbb{V}_{\mathbf{g}}^{{\mathbf{f}}{\mathbf{h}}}\oplus\mathbb{V}_{\mathbf{h}}^{{\mathbf{f}}{\mathbf{g}}},$

where
$\mathbb{V}_{\mathbf{f}}^{{\mathbf{g}}{\mathbf{h}}}:=(\mathbb{V}_{{\mathbf{f}}}^{-}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{g}}}^{+}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{h}}}^{+})(\Xi_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}})$
and similarly for the other two summands. We similarly denote
($\mathscr{F}^{i}\mathbb{V}_{{\mathbf{f}}gh}^{\dagger},\mathbb{V}_{{\mathbf{f}}}^{gh}$,
etc.) the induced subquotients on the specializations of
$\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}$; and denote
$\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}(N),\mathscr{F}^{i}\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}(N)$,
etc. the natural extension of all these definitions constructed in level $N$.

Let $\kappa_{{\mathbf{f}}gh}\in
H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}}(N))$ be the class
$\kappa_{n,{\mathbf{f}}gh}^{(3)}$ defined in (3.10) with $n=1$. By [BSV21,
Cor. 8.2], the image of $\kappa_{{\mathbf{f}}gh}$ under the restriction map at
$p$ is contained in

$H^{1}_{\operatorname{bal}}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}(N)):={\rm
im}\bigl{(}H^{1}(\mathbb{Q}_{p},\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}(N))\rightarrow
H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}(N))\bigr{)}.$

It is easily seen that this map is an injection, so we may and will view ${\rm
res}_{p}(\kappa_{{\mathbf{f}}gh})$ as a class in
$H^{1}(\mathbb{Q}_{p},\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}(N))$.
Given a triple $(\breve{{\mathbf{f}}},\breve{g},\breve{h})$ of level-$N$ test
vectors, let

${\rm
pr}_{(\breve{{\mathbf{f}}},\breve{g},\breve{h})}:\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}(N)\longrightarrow\mathbb{V}_{{\mathbf{f}}}^{gh}$

be the map induced by the projection onto the first direct summand in (4.4)
composed with the projection
$\mathbb{V}_{{\mathbf{f}}}^{gh}(N)\rightarrow\mathbb{V}_{{\mathbf{f}}}^{gh}$
corresponding to $(\breve{{\mathbf{f}}},\breve{g},\breve{h})$. The
“reciprocity law” from [BSV21, DR21] recalled in Theorem 4.4 below relates the
image of ${\rm res}_{p}(\kappa_{{\mathbf{f}}gh})$ under the natural projection

${\rm
pr}_{(\breve{{\mathbf{f}}},\breve{g},\breve{h})*}:H^{1}_{\operatorname{bal}}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}(N))\longrightarrow
H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{gh})$

to the triple product $p$-adic $L$-function of $\S\ref{subsec:Lp}$.

Throughout the following, assume that $\bar{\rho}_{{\mathbf{f}}}$ is
absolutely irreducible and $p$-distinguished, and let
$\xi\in\Lambda_{{\mathbf{f}}}$ be a generator of the congruence ideal of
${\mathbf{f}}$.

######  Proposition 4.3.

For every choice of test vectors
$(\breve{{\mathbf{f}}},\breve{{\mathbf{g}}},\breve{{\mathbf{h}}})$ for
$({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$, there is an injective
$\Lambda_{{\mathbf{f}}}$-module homomorphism with pseudo-null cokernel

$\mathfrak{Log}^{\xi}=\mathfrak{Log}^{\xi}(\breve{{\mathbf{f}}},\breve{g},\breve{h}):H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{gh})\longrightarrow\Lambda_{{\mathbf{f}}}$

characterized by the following interpolation property: for all
$\mathfrak{Z}\in H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{gh})$ and all
classical $k\in U$,

$\displaystyle\frac{\mathfrak{Log}^{\xi}(\mathfrak{Z})_{k}}{\xi_{k}}$
$\displaystyle=(p-1)\alpha_{k}\biggl{(}1-\frac{\beta_{k}\alpha_{g}\alpha_{h}}{p^{c}}\biggr{)}\biggl{(}1-\frac{\alpha_{k}\beta_{g}\beta_{h}}{p^{c}}\biggr{)}^{-1}$
$\displaystyle\quad\times\begin{cases}\frac{(-1)^{c-k}}{(c-k)!}\cdot\left\langle{\rm
log}_{p}(\mathfrak{Z}_{k}),\eta_{\breve{{\mathbf{f}}}_{k}}\otimes\omega_{\breve{{\mathbf{g}}}_{l}}\otimes\omega_{\breve{{\mathbf{h}}}_{m}}\right\rangle_{\rm
dR},&\textrm{if $l-m<k<l+m$,}\\\\[5.0pt] (k-c-1)!\cdot\left\langle{\rm
exp}_{p}^{*}(\mathfrak{Z}_{k}),\eta_{\breve{{\mathbf{f}}}_{k}}\otimes\omega_{\breve{{\mathbf{g}}}_{l}}\otimes\omega_{\breve{{\mathbf{h}}}_{m}}\right\rangle_{\rm
dR},&\textrm{if $k\geq l+m$,}\end{cases}$

where $c=(k+l+m-2)/2$.

###### Proof.

The construction of $\mathfrak{Log}^{\xi}$ will follow by specializing the
three-variable $p$-adic regulator constructed in [BSV21, §7.1] (building on a
generalization of the construction in [LZ14] given by Kings–Loeffler–Zerbes
[KLZ17]).

Let
$\vartheta_{gh}:\Lambda_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}\rightarrow\Lambda_{{\mathbf{f}}}$
be the map given by reduction modulo $(\bm{l}-l,\bm{m}-m)$. This induces
isomorphisms

$\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}\otimes_{\Lambda_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}}\Lambda_{\mathbf{f}}\cong\mathbb{V}_{{\mathbf{f}}gh}^{\dagger},\quad\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}\otimes_{\Lambda_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}}\Lambda_{\mathbf{f}}\cong\mathbb{V}_{{\mathbf{f}}}^{gh},$

and a natural map

$\vartheta_{gh*}:H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}})\otimes_{\Lambda_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}}\Lambda_{\mathbf{f}}\longrightarrow
H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{gh}).$

This map is clearly injective, and its surjectivity can be shown easily by an
application of local Tate duality and the Ramanujan–Petersson conjecture
(_cf._ proof of [BSV21, (154)]). Letting

$\mathscr{L}_{{\mathbf{f}}}^{\xi}:H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}})\longrightarrow\Lambda_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}$

be the $p$-adic regulator $\mathscr{L}_{{\mathbf{f}}}$ defined in [BSV21,
Prop. 7.3] (where $\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}$ is
denoted $V({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})_{f}$) attached to the test
vectors $(\breve{{\mathbf{f}}},\breve{{\mathbf{g}}},\breve{{\mathbf{h}}})$ and
multiplied by $\xi$, the map defined by the composition

$\mathfrak{Log}^{\xi}:H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{gh})\xrightarrow{\vartheta_{gh*}^{-1}}H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}})\otimes_{\Lambda_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}}\Lambda_{\mathbf{f}}\xrightarrow{\mathscr{L}^{\xi}_{{\mathbf{f}}}\otimes{\rm
id}}\Lambda_{\mathbf{f}}$

satisfies the interpolation properties in the statement of the proposition.

It remains to see that $\mathfrak{Log}^{\xi}$ is injective with pseudo-null
cokernel. By definition, we have

$\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}=\mathbb{U}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}(\epsilon_{\rm
cyc}\kappa_{{\mathbf{f}}}^{-1/2}\kappa_{{\mathbf{g}}{\mathbf{h}}}^{1/2}),$

where $\mathbb{U}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}$ is an unramified
$G_{\mathbb{Q}_{p}}$-module on which an arithmetic Frobenius ${\rm Fr}_{p}$
acts as multiplication by
$\chi_{f}^{-1}(p)a_{p}({\mathbf{f}})a_{p}({\mathbf{g}})^{-1}a_{p}({\mathbf{h}})^{-1}$,
and $\mathscr{L}_{{\mathbf{f}}}$ is obtained by specializing the four-variable
$p$-adic regulator map in [KLZ17, Thm. 8.2.3] for the module
$\mathbb{U}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}$, paired against the
differential
$\eta_{\breve{{\mathbf{f}}}}\otimes\omega_{\breve{{\mathbf{g}}}}\otimes\omega_{\breve{{\mathbf{h}}}}$.
In light of [KLZ17, Rmk. 8.2.4], the fact that $\mathfrak{Log}^{\xi}$ has the
above properties can therefore be deduced from the vanishing of
$H^{0}(\mathbb{Q}_{p},\mathbb{U}_{{\mathbf{f}}}^{gh})$, where
$\mathbb{U}_{{\mathbf{f}}}^{gh}$ is the image of
$\mathbb{U}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}$ under $\vartheta_{gh}$.
∎

######  Theorem 4.4 (Reciprocity law).

For every triple of test vectors $(\breve{{\mathbf{f}}},\breve{g},\breve{h})$
we have

$\mathfrak{Log}^{\xi}({\rm
pr}_{(\breve{{\mathbf{f}}},\breve{g},\breve{h})*}({\rm
res}_{p}(\kappa_{{\mathbf{f}}gh})))=\mathscr{L}_{p}^{\xi}(\breve{{\mathbf{f}}},\breve{g},\breve{h}),$

where
$\mathfrak{Log}^{\xi}=\mathfrak{Log}^{\xi}(\breve{{\mathbf{f}}},\breve{g},\breve{h})$.

###### Proof.

This is the specialization of the three-variable reciprocity law of Theorem A
in [BSV21] to $({\mathbf{f}},g,h)$ (see also [DR21, Thm. 10]). ∎

######  Definition 4.5.

For the level-$N$ test vectors
$(\underline{\breve{{\mathbf{f}}}},\underline{\breve{g}},\underline{\breve{h}})$
in Theorem 4.1, we let

$\kappa({\mathbf{f}},g,h)\in
H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$

be the image of $\kappa_{{\mathbf{f}}gh}$ under the corresponding projection
$H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}(N))\rightarrow
H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$.

Letting
$p_{{\mathbf{f}}}:\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}/\mathscr{F}^{3}\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}}\rightarrow\mathbb{V}_{\mathbf{f}}^{gh}$
be the projection onto the first direct summand in (4.4), Theorem 4.4 thus
reads

$\mathfrak{Log}^{\xi}(p_{{\mathbf{f}}*}({\rm
res}_{p}(\kappa({\mathbf{f}},g,h)))=\mathscr{L}_{p}^{\xi}(\breve{{\mathbf{f}}},\breve{g},\breve{h}).$

###  4.3. Selmer groups and formulation of the main conjectures

Let $({\mathbf{f}},g,h)$ be as in the preceding section, and let
$\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}$ be the specialization of the “big”
self-dual representation
$\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}$ in (4.1), so
that

$\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}\cong\mathbb{V}_{{\mathbf{f}}}(\kappa_{{\mathbf{f}}}^{-1/2})\hat{\otimes}_{\mathcal{O}}T_{g}\otimes_{\mathcal{O}}T_{h}(1-(l+m)/2),$

where $T_{g}$ and $T_{h}$ are $G_{\mathbb{Q}}$-stable $\mathcal{O}$-lattices
in the Galois representations associated to $g$ and $h$.

Recall the $G_{\mathbb{Q}_{p}}$-stable rank four
$\Lambda_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}$-submodule
$\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}\subset\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{{\dagger}}$
in $(\ref{eq:+})$, and set

$\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{f}:=\mathbb{V}_{{\mathbf{f}}}^{+}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{g}}}\hat{\otimes}_{\mathcal{O}}\mathbb{V}_{{\mathbf{h}}}(\Xi_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}).$

As before, we let $\mathbb{V}_{{\mathbf{f}}gh}^{+}$ and
$\mathbb{V}_{{\mathbf{f}}gh}^{f}$ denote the corresponding specializations.

Fix a finite set $\Sigma$ of places of $\mathbb{Q}$ containing $\infty$ and
the primes dividing $Np$, and let $\mathbb{Q}^{\Sigma}$ be the maximal
extension of $\mathbb{Q}$ unramified outside $\Sigma$.

######  Definition 4.6.

For $\mathcal{L}\in\\{{\rm bal},\mathcal{F}\\}$ define the Selmer group
$\mathrm{Sel}_{\mathcal{L}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$ by

$\mathrm{Sel}_{\mathcal{L}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})=\ker\bigg{(}H^{1}(\mathbb{Q}^{\Sigma}/\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})\longrightarrow\prod_{v\in\Sigma\smallsetminus\\{p,\infty\\}}H^{1}(\mathbb{Q}_{v}^{\rm
nr},\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})\times\frac{H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})}{H^{1}_{\mathcal{L}}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})}\bigg{)},$

where $\mathbb{Q}_{v}^{\rm nr}$ denotes the maximal unramified extension of
$\mathbb{Q}_{v}$, and

$H_{\mathcal{L}}^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})=\begin{cases}\ker\bigl{(}H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})\longrightarrow
H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}}/\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})\bigr{)}&\textrm{if
$\mathcal{L}={\rm bal}$},\\\\[5.0pt]
\ker\bigl{(}H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})\longrightarrow
H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}}/\mathbb{V}_{{\mathbf{f}}gh}^{f})\bigr{)}&\textrm{if
$\mathcal{L}=\mathcal{F}$}.\end{cases}$

We call $\mathrm{Sel}_{\rm bal}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$
(resp. $\mathrm{Sel}_{\mathcal{F}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$)
the _balanced_ (resp. _$f$ -unbalanced_) Selmer group.

######  Remark 4.7.

The pairs

$\bigl{(}\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}gh}^{\dagger},\\{k\in\mathbb{Z}_{\geq
2}\;\colon\;l-m<k<l+m\\}\bigr{)},\quad\bigl{(}\mathbb{V}_{{\mathbf{f}}gh}^{f},\\{k\in\mathbb{Z}\;\colon\;k\geq
l+m\\}\bigr{)}$

satisfy the _Panchishkin condition_ in [Gre94]. Thus ${\rm
Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$ and ${\rm
Sel}_{\mathcal{F}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$ may be viewed as
instances of Greenberg’s Selmer groups attached to different ranges of
critical specializations of $\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}$.

Let

$\mathbb{A}_{{\mathbf{f}}gh}^{\dagger}={\rm
Hom}_{\mathbb{Z}_{p}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger},\mu_{p^{\infty}}).$

Then for $\mathcal{L}\in\\{\operatorname{bal},\mathcal{F}\\}$, we define the
Selmer groups ${\rm Sel}_{\mathcal{L}}(\mathbb{A}_{{\mathbf{f}}gh}^{\dagger})$
as above, taking
$H^{1}_{\mathcal{L}}(\mathbb{Q}_{p},\mathbb{A}_{{\mathbf{f}}gh}^{\dagger})$ to
be the orthogonal complement of
$H^{1}_{\mathcal{L}}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$
under the local Tate duality

$H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})\times
H^{1}(\mathbb{Q}_{p},\mathbb{A}_{{\mathbf{f}}gh}^{\dagger})\longrightarrow\Lambda_{\mathbf{f}},$

and set

$X_{\mathcal{L}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}}):=\operatorname{Hom}_{\operatorname{cts}}(\mathrm{Sel}_{\mathcal{L}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}}),\mathbb{Q}_{p}/\mathbb{Z}_{p}).$

In light of Remark 4.7, the next conjecture may be viewed as an instance of
the Iwasawa–Greenberg main conjectures [Gre94]. In the two formulations below,
we shall assume hypotheses (a)–(c) in Theorem 4.1, so the $p$-adic
$L$-function $L_{p}({\mathbf{f}},g,h)$ in (4.2) is constructed.

######  Conjecture 4.8 (IMC “with $p$-adic $L$-functions”).

The modules
$\mathrm{Sel}_{\mathcal{F}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$ and
$X_{\mathcal{F}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$ are both
$\Lambda_{{\mathbf{f}}}$-torsion, and

$\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}(X_{\mathcal{F}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}}))=(L_{p}({\mathbf{f}},g,h))$

in $\Lambda_{{\mathbf{f}}}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$.

######  Remark 4.9.

An integral formulation of the equality of ideals in Conjecture 4.8 would
involve certain Tamagawa factors, accounting for the fact that by definition
the classes in the Selmer group
$X_{\mathcal{F}}(\mathbb{A}_{{\mathbf{f}}gh}^{\dagger})$ are trivial at the
places $v\in\Sigma\smallsetminus\\{p,\infty\\}$, rather than just unramified.

Under the local root number hypothesis (b) in Theorem 4.1, for all $k\geq 2$
with $l-m<k<l-m$ the sign in the functional equation for
$L(\mathbb{V}_{{\mathbf{f}}_{k}gh}^{\dagger},s)$ is $-1$, and so the central
value $L(\mathbb{V}_{{\mathbf{f}}_{k}gh}^{\dagger},0)$ vanishes. Therefore, in
the spirit of Perrin-Riou’s main conjecture [PR87, Conj. B] in the setting of
Heegner points, a natural formulation of the Iwasawa main conjecture for ${\rm
Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$ takes the
following form.

Note that it follows from [BSV21, Cor. 8.2] that $\kappa({\mathbf{f}},g,h)$
lands in
$\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$.

######  Conjecture 4.10 (IMC “without $p$-adic $L$-functions”).

Suppose
$\kappa({\mathbf{f}},g,h)\in\mathrm{Sel}_{\operatorname{bal}}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$
is not $\Lambda_{{\mathbf{f}}}$-torsion. Then the modules
$\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$
and $X_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$ have
both rank one, and

$\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}(X_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})_{\operatorname{tors}})=\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}\left(\frac{\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})}{\Lambda_{{\mathbf{f}}}\cdot\kappa({\mathbf{f}},g,h)}\right)^{2}$

in $\Lambda_{{\mathbf{f}}}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$, where the
subscript ${\rm tors}$ denotes the $\Lambda_{{\mathbf{f}}}$-torsion submodule.

######  Remark 4.11.

Working under different hypotheses on the local signs ensuring that
$L(\mathbb{V}_{{\mathbf{f}}_{k}gh}^{\dagger},s)$ has sign $+1$ (rather than
$-1$) for weights $k\geq 2$ with $l-m<k<l-m$, the Iwasawa main conjecture
would relate the characteristic ideal of
$X_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$ to the
balanced triple product $p$-adic $L$-function constructed in [Hsi21, Thm. B]
(see also [GS20]), rather than diagonal classes. In this setting, the
$f$-unbalanced Selmer group
$\mathrm{Sel}_{\mathcal{F}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$ should
have $\Lambda_{{\mathbf{f}}}$-rank one, but the expected non-torsion Selmer
class seems to not have been constructed yet.

###  4.4. Equivalence of the formulations

In this section we show that the two formulations of the Iwasawa main
conjecture in the previous section are essentially equivalent, focusing on the
case where ${\mathbf{f}}$ is a CM Hida family111This case will suffice for our
applications in this paper, and makes some of the arguments simpler, but we
expect the equivalence to hold in general. as in $\S\ref{subsec:Iw}$. Similar
equivalences between IMC “with” and “without” $p$-adic $L$-functions appear in
[Yag82], [Kat04, §17], [KLZ17, §11] and, in a setting more closely related to
ours, [Wan20] and [Cas17, Appendix].

The following intermediate Selmer groups will allow us to bridge between ${\rm
Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$ and ${\rm
Sel}_{\mathcal{F}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$ in the comparison.
Set

$\mathbb{V}_{{\mathbf{f}}gh}^{f\cap+}=\mathbb{V}_{{\mathbf{f}}gh}^{f}\cap\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}gh}^{\dagger},\quad\mathbb{V}_{{\mathbf{f}}gh}^{f\cup+}=\mathbb{V}_{{\mathbf{f}}gh}^{f}+\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}gh}^{\dagger},$

which are $G_{\mathbb{Q}_{p}}$-stable $\Lambda_{\mathbf{f}}$-submodules of
$\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}$ of ranks $3$ and $5$, respectively.
Define $\mathrm{Sel}_{\mathcal{L}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$
for $\mathcal{L}\in\\{\mathcal{F}\cap+,\mathcal{F}\cup+\\}$ by the same recipe
as in Definition 4.6, with

$H_{\mathcal{L}}^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})=\begin{cases}\ker\bigl{(}H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})\rightarrow
H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}}/\mathbb{V}_{{\mathbf{f}}gh}^{f\cap+})\bigr{)}&\textrm{if
$\mathcal{L}=\mathcal{F}\cap+$},\\\\[5.0pt]
\ker\bigl{(}H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})\rightarrow
H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}}/\mathbb{V}_{{\mathbf{f}}gh}^{f\cup+})\bigr{)}&\textrm{if
$\mathcal{L}=\mathcal{F}\cup+$}.\end{cases}$

As in the preceding section, we also define the corresponding
$X_{\mathcal{F}\cap+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$ and
$X_{\mathcal{F}\cup+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$.

Throughout this section, we keep the setting from $\S\ref{subsec:Iw}$. In
particular, ${\mathbf{f}}\in\Lambda_{{\mathbf{f}}}[[q]]$ is the CM Hida family
in $(\ref{eq:CM-F})$ associated with the Hecke character $\psi$ of conductor
dividing $\mathfrak{f}$. In addition, we assume conditions (b) and (c) from
Theorem 4.1, so the $p$-adic $L$-function
$L_{p}({\mathbf{f}},g,h)\in\Lambda_{{\mathbf{f}}}$ is defined, and let
$\kappa_{{\mathbf{f}}gh}\in
H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$ be as in the
preceding section.

For every height one prime $\mathfrak{Q}$ of $\Lambda_{{\mathbf{f}}}$ away
from $p$, let $S_{\mathfrak{Q}}$ be the integral closure of
$\Lambda_{{\mathbf{f}}}/\mathfrak{Q}$, and let $\Phi_{\mathfrak{Q}}$ be the
field fractions of $S_{\mathfrak{Q}}$. Let
$\mathbb{V}_{{\mathbf{f}}gh,\mathfrak{Q}}^{\dagger}$ be the extension of
scalars of
$\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}}/\mathfrak{Q}\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}$
to $S_{\mathfrak{Q}}$, and let
$\mathbb{A}^{\dagger}_{{\mathbf{f}}gh,\mathfrak{Q}}=\operatorname{Hom}(\mathbb{V}^{\dagger}_{{\mathbf{f}}gh,\mathfrak{Q}},\mu_{p^{\infty}})$.
Following [MR04], define

(4.5) $H^{1}_{\underline{\rm
bal}}(\mathbb{Q}_{v},\mathbb{V}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}}):=\begin{cases}\ker\bigl{(}H^{1}(\mathbb{Q}_{v},\mathbb{V}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}})\rightarrow
H^{1}(\mathbb{Q}_{v}^{\rm
nr},\mathbb{V}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}}\otimes\Phi_{\mathfrak{Q}})\bigr{)},&\textrm{if
$v\nmid p$,}\\\\[5.0pt]
\ker\bigl{(}H^{1}(\mathbb{Q}_{v},\mathbb{V}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}})\rightarrow
H^{1}(\mathbb{Q}_{v},(\mathbb{V}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}}/\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}gh,\mathfrak{Q}}^{\dagger})\otimes\Phi_{\mathfrak{Q}})\bigr{)},&\textrm{if
$v\mid p$,}\end{cases}$

and let
$H^{1}_{\underline{\operatorname{bal}}}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}})$
be the associated Selmer group. Taking
$H^{1}_{\underline{\operatorname{bal}}}(\mathbb{Q}_{v},\mathbb{A}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}})$
to be the orthogonal complement of
$H^{1}_{\underline{\operatorname{bal}}}(\mathbb{Q}_{v},\mathbb{V}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}})$
under local Tate duality, we define the Selmer group
$H^{1}_{\underline{\operatorname{bal}}}(\mathbb{Q},\mathbb{A}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}})$
similarly.

Define
$\mathbb{V}_{g,h}^{\psi}=\Lambda_{\mathcal{O}}(\bm{\kappa}_{\text{ac}}^{-1})\hat{\otimes}_{\mathcal{O}}T_{g,h}^{\psi}(\kappa_{\operatorname{ac}}^{r_{1}/2})$
and let
$\mathbb{A}_{g,h}^{\psi}=\operatorname{Hom}((\mathbb{V}_{g,h}^{\psi})^{c},\mu_{p^{\infty}})$,
where $(\mathbb{V}_{g,h}^{\psi})^{c}$ denotes $\mathbb{V}_{g,h}^{\psi}$ with
the $G_{K}$-action twisted by complex conjugation. Note that
$\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}}=\text{Ind}_{K}^{\mathbb{Q}}\mathbb{V}_{g,h}^{\psi}$,
so we can define Selmer conditions for this module using Shapiro’s lemma and
for $\mathbb{A}_{g,h}^{\psi}$ by duality. Define
$\mathbb{A}_{g,h,\mathfrak{Q}}^{\psi}=\operatorname{Hom}((\mathbb{V}_{g,h,\mathfrak{Q}}^{\psi})^{c},\mu_{p^{\infty}})$.
We have natural maps

(4.6)
$\mathbb{V}_{g,h}^{\psi}/\mathfrak{Q}\mathbb{V}_{g,h}^{\psi}\rightarrow\mathbb{V}_{g,h,\mathfrak{Q}}^{\psi},\quad\mathbb{A}_{g,h,\mathfrak{Q}}^{\psi}\rightarrow\mathbb{A}_{g,h}^{\psi}[\mathfrak{Q}]$

preserving both the $G_{K}$ and the $\Lambda$-modules structure in the same
way as in [How04, p. 1461]. Note that in the quotient
$\mathbb{V}_{g,h}^{\psi}/\mathfrak{Q}\mathbb{V}_{g,h}^{\psi}$ and in the
submodule $\mathbb{A}_{g,h}^{\psi}[\mathfrak{Q}]$ we can define Selmer
conditions by propagating the balanced conditions for
$\mathbb{V}_{g,h}^{\psi}$ and $\mathbb{A}_{g,h}^{\psi}$, respectively, and we
denote these conditions in the same way.

######  Lemma 4.12.

For every height one prime $\mathfrak{Q}\subset\Lambda_{{\mathbf{f}}}$ as
above and every place $v$ of $K$, the maps $(\ref{eq:spQ})$ induce natural
maps

$\displaystyle
H^{1}_{\operatorname{bal}}(K_{v},\mathbb{V}_{gh}^{\psi}/\mathfrak{Q}\mathbb{V}_{gh}^{\psi})$
$\displaystyle\longrightarrow
H^{1}_{\underline{\operatorname{bal}}}(K_{v},\mathbb{V}_{gh,\mathfrak{Q}}^{\psi}),$
$\displaystyle
H^{1}_{\underline{\operatorname{bal}}}(K_{v},\mathbb{A}_{gh,\mathfrak{Q}}^{\psi})$
$\displaystyle\longrightarrow
H^{1}_{\operatorname{bal}}(K_{v},\mathbb{A}_{gh}^{\psi}[\mathfrak{Q}])$

with finite kernel and cokernel, of order bounded by constants depending only
on $[S_{\mathfrak{Q}}:\Lambda_{{\mathbf{f}}}/\mathfrak{Q}]$.

###### Proof.

For the primes $v\nmid p$, the same argument as in the proof of [MR04, Lem.
5.3.13] applies, so it remains to consider the case $v\mid p$. Put

$\displaystyle\mathcal{F}_{\mathfrak{p}}^{+}(T_{g,h}^{\psi})$
$\displaystyle=(T_{g}^{+}\otimes T_{h}+T_{g}\otimes
T_{h}^{+})(\psi_{\mathfrak{P}}^{-1})(-1-r),$
$\displaystyle\mathcal{F}_{\overline{\mathfrak{p}}}^{+}(T_{g,h}^{\psi})$
$\displaystyle=(T_{g}^{+}\otimes T_{h}^{+})(\psi_{\mathfrak{P}}^{-1})(-1-r).$

Under the isomorphism
$H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})\cong
H^{1}(K,\Lambda_{\mathcal{O}}(\bm{\kappa}_{\text{ac}}^{-1})\hat{\otimes}_{\mathcal{O}}T_{g,h}^{\psi}(\kappa_{\operatorname{ac}}^{r_{1}/2}))$
coming from (3.11) and Shapiro’s lemma, the balanced local condition
$H^{1}_{\rm bal}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$
corresponds to

$H^{1}(K_{\mathfrak{p}},\Lambda_{\mathcal{O}}(\bm{\kappa}_{\text{ac}}^{-1})\hat{\otimes}_{\mathcal{O}}\mathcal{F}_{\mathfrak{p}}^{+}(T_{g,h}^{\psi})(\kappa_{\operatorname{ac}}^{r_{1}/2}))\oplus
H^{1}(K_{\overline{\mathfrak{p}}},\Lambda_{\mathcal{O}}(\bm{\kappa}_{\text{ac}}^{-1})\hat{\otimes}_{\mathcal{O}}\mathcal{F}_{\overline{\mathfrak{p}}}^{+}(T_{g,h}^{\psi})(\kappa_{\operatorname{ac}}^{r_{1}/2})).$

Let $A_{g}^{-}=T_{g}^{-}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$, and define
$A_{g}^{+}$, $A_{h}^{-}$ and $A_{h}^{+}$ similarly. Arguing as in the proof of
[How04, Lem. 2.2.7], we reduce to showing that the groups

$H^{0}(K_{\infty,\mathfrak{p}},(A_{g}^{-}\otimes
A_{h}^{-})(\psi_{\mathfrak{P}}^{-1}\kappa_{\operatorname{ac}}^{r_{1}/2})(-1-r)),\quad
H^{0}(K_{\infty,\overline{\mathfrak{p}}},(A_{g}^{+}\otimes
A_{h}^{-})(\psi_{\mathfrak{P}}^{-1}\kappa_{\operatorname{ac}}^{r_{1}/2})(-1-r))$

are both finite, which follows from the fact that
$\alpha_{g}\alpha_{h}\psi(\mathfrak{p})/p^{k-1}\neq 1$ and
$\beta_{g}\alpha_{h}\psi(\overline{\mathfrak{p}})\neq 1$, and this is a
consequence of the Ramanujan–Petersson conjecture since we are assuming that
$p\nmid N$. Note that the other pieces in the quotient decomposition can be
treated similarly. This yields the required bounds on the kernel and cokernel
of the first map in the statement of the lemma, and the result for the second
map follows as well by local duality. ∎

Let $\Sigma_{\Lambda}$ be the set of height one primes of
$\Lambda_{{\mathbf{f}}}$ away from $p$ for which either
$H^{2}(\mathbb{Q}^{\Sigma}/\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})[\mathfrak{Q}]$
is infinite or
$H^{2}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})[\mathfrak{Q}]$ is
infinite. Since
$H^{2}(\mathbb{Q}^{\Sigma}/\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$
and $H^{2}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$ are both
finitely generated $\Lambda$-modules, the set $\Sigma_{\Lambda}$ is finite.

######  Proposition 4.13.

For every height one prime $\mathfrak{Q}\not\in\Sigma_{\Lambda}$, the maps
(4.6) induce natural maps

$\displaystyle\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})/\mathfrak{Q}\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$
$\displaystyle\longrightarrow\mathrm{Sel}_{\underline{\operatorname{bal}}}(\mathbb{V}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}}),$
$\displaystyle\mathrm{Sel}_{\underline{\operatorname{bal}}}(\mathbb{A}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}})$
$\displaystyle\longrightarrow\mathrm{Sel}_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})[\mathfrak{Q}]$

with finite kernel and cokernel bounded by a constant depending only on
$[S_{\mathfrak{Q}}:\Lambda_{{\mathbf{f}}}/\mathfrak{Q}]$.

###### Proof.

This follows from Lemma 4.12 as in the proof of [MR04, Prop. 5.3.14] (see also
[How04, Lem. 2.2.8]). ∎

For every height one prime $\mathfrak{Q}\subset\Lambda_{{\mathbf{f}}}$ as
above, let $\mathfrak{m}_{\mathfrak{Q}}=(\pi_{\mathfrak{Q}})$ be the maximal
ideal of $S_{\mathfrak{Q}}$.

######  Lemma 4.14.

Assume that there is a perfect pairing $T_{g,h}^{\psi}\times
T_{g,h}^{\psi}\rightarrow\mathcal{O}(1)$ such that $\langle
x^{\sigma},y^{c\sigma c}\rangle=\langle x,y\rangle^{\sigma}$ for all $x,y\in
T_{g,h}^{\psi}$ and for all $\sigma\in G_{K}$, where $c$ stands for complex
conjugation. The following hold:

  1. (1)

${\rm
rank}_{\Lambda_{{\mathbf{f}}}}\,\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})={\rm
rank}_{\Lambda_{{\mathbf{f}}}}\,X_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$.

  2. (2)

${\rm
rank}_{\Lambda_{{\mathbf{f}}}}\,\mathrm{Sel}_{\mathcal{F}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})={\rm
rank}_{\Lambda_{{\mathbf{f}}}}\,X_{\mathcal{F}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$.

  3. (3)

${\rm
rank}_{\Lambda_{{\mathbf{f}}}}\,X_{\mathcal{F}\cup+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})=1+{\rm
rank}_{\Lambda_{{\mathbf{f}}}}\,X_{\mathcal{F}\cap+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$,
and

$\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}(X_{\mathcal{F}\cup+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})_{\operatorname{tors}})=\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}(X_{\mathcal{F}\cap+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})_{\rm
tors}),$

in $\Lambda_{{\mathbf{f}}}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$.

###### Proof.

For part (1), it suffices to show that for all height one primes
$\mathfrak{Q}\subset\Lambda_{{\mathbf{f}}}$ with
$\mathfrak{Q}\not\in\Sigma_{\Lambda}$, the modules
$\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})/\mathfrak{Q}\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$
and
$\mathrm{Sel}_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{\dagger})[\mathfrak{Q}]$
have the same rank over $\Lambda_{{\mathbf{f}}}/\mathfrak{Q}$. Since
$\mathrm{Sel}_{\underline{\operatorname{bal}}}(\mathbb{V}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}})$
is the $\pi_{\mathfrak{Q}}$-adic Tate module of
$\mathrm{Sel}_{\underline{\operatorname{bal}}}(\mathbb{A}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}})$
(indeed, this is a consequence of [How04, Lem. 1.3.3] since the duality
assumption ensures that
$\mathbb{A}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}}\cong\mathbb{V}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$),
the result thus follows from Proposition 4.13.

For part (2), under the isomorphism
$H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})\cong
H^{1}(K,\Lambda_{\mathcal{O}}(\bm{\kappa}_{\text{ac}}^{-1})\hat{\otimes}_{\mathcal{O}}T_{g,h}^{\psi}(\kappa_{\operatorname{ac}}^{r_{1}/2}))$
the $f$-unbalanced local condition
$H^{1}_{\mathcal{F}}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$
corresponds to

$H^{1}(K_{\mathfrak{p}},\Lambda_{\mathcal{O}}(\bm{\kappa}_{\text{ac}}^{-1})\hat{\otimes}_{\mathcal{O}}T_{g,h}^{\psi})\oplus\\{0\\}$

and hence an analogue of Lemma 4.12 for the $f$-unbalanced Selmer groups
follows from the finiteness of
$H^{0}(K_{\infty,\overline{\mathfrak{p}}},A_{g}\otimes
A_{h}(\psi_{\mathfrak{P}}^{-1}\kappa_{\operatorname{ac}}^{r_{1}/2})(-1-r))$.
By the same reason as above, this yields the equality of ranks in part (2).

Finally, for the proof of part (3) we can argue similarly as in [AH06, Thm.
1.2.2]. Keeping with the above notations, let
$\mathrm{Sel}_{\underline{\mathcal{F}\cup+}}(\mathbb{A}_{{\mathbf{f}}gh,\mathfrak{Q}}^{\dagger})$
and
$\mathrm{Sel}_{\underline{\mathcal{F}\cap+}}(\mathbb{A}_{{\mathbf{f}}gh,\mathfrak{Q}}^{\dagger})$
be the Selmer groups defined by the obvious analogues of (4.5), so from
another application of the argument in Lemma 4.12 we obtain natural maps

$\displaystyle\mathrm{Sel}_{\underline{\mathcal{F}\cup+}}(\mathbb{A}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}})$
$\displaystyle\longrightarrow\mathrm{Sel}_{\mathcal{F}\cup+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})[\mathfrak{Q}],$
$\displaystyle\mathrm{Sel}_{\underline{\mathcal{F}\cap+}}(\mathbb{A}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}})$
$\displaystyle\longrightarrow\mathrm{Sel}_{\mathcal{F}\cap+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})[\mathfrak{Q}]$

with finite kernel and cokernel bounded by a constant depending only on
$[S_{\mathfrak{Q}}:\Lambda_{{\mathbf{f}}}/\mathfrak{Q}]$. Since the local
condition $\mathcal{F}\cap+$ is the orthogonal complement of
$\mathcal{F}\cup+$ under the local Tate pairing at $p$ induced by the self-
duality of $\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}$, from [MR04, Thm. 4.1.13]
we obtain

(4.7)
$\mathrm{Sel}_{\underline{\mathcal{F}\cup+}}(\mathbb{A}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}})[\pi_{\mathfrak{Q}}^{i}]\cong(\Phi_{\mathfrak{Q}}/S_{\mathfrak{Q}})^{r}[\pi_{\mathfrak{Q}}^{i}]\oplus\mathrm{Sel}_{\underline{\mathcal{F}\cap+}}(\mathbb{A}_{{\mathbf{f}}gh,\mathfrak{Q}}^{{\dagger}})[\pi_{\mathfrak{Q}}^{i}]$

for all $i$, where $r$ is given (by the Greenberg–Wiles formula in [MR04,
Prop. 2.3.5]) by

${\rm
corank}_{S_{\mathfrak{Q}}}H^{1}(\mathbb{Q}_{p},\mathbb{A}_{{\mathbf{f}}gh,\mathfrak{Q}}^{f\cup+})-{\rm
corank}_{S_{\mathfrak{Q}}}H^{0}(\mathbb{R},\mathbb{A}_{{\mathbf{f}}gh,\mathfrak{Q}}^{\dagger}),$

so $r=5-4=1$. The proof of part (3) now follows from (4.7) as in [AH06, Lem.
1.2.6]. ∎

######  Remark 4.15.

The existence of the pairing in the previous lemma is not too restrictive. In
particular, this holds automatically if $g$ and $h$ are non-Eisenstein.

We are now ready to establish that both formulations of the Iwasawa main
conjecture are equivalent.

######  Theorem 4.16.

Keep the assumptions of the previous lemma and suppose
$\kappa({\mathbf{f}},g,h)$ is not $\Lambda_{{\mathbf{f}}}$-torsion. Then the
following are equivalent:

  1. (1)

${\rm
rank}_{\Lambda_{{\mathbf{f}}}}\,\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})={\rm
rank}_{\Lambda_{{\mathbf{f}}}}\,X_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{\dagger})=1$;

  2. (2)

${\rm
rank}_{\Lambda_{{\mathbf{f}}}}\,\mathrm{Sel}_{\mathcal{F}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})={\rm
rank}_{\Lambda_{{\mathbf{f}}}}\,X_{\mathcal{F}}(\mathbb{A}_{{\mathbf{f}}gh}^{\dagger})=0$,

and in that case we have
$\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})=\mathrm{Sel}_{\mathcal{F}\cup+}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$,
and

$\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}(X_{\mathcal{F}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}}))\cdot\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}\left(\frac{\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})}{\Lambda_{{\mathbf{f}}}\cdot\kappa({\mathbf{f}},g,h)}\right)^{2}=\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}(X_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})_{\operatorname{tors}})\cdot(L_{p}({{\mathbf{f}}},{g},{h}))$

in $\Lambda_{{\mathbf{f}}}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$. In
particular, Conjecture 4.8 and Conjecture 4.10 are equivalent.

###### Proof.

The Poitou–Tate global duality gives rise to the exact sequence

(4.8)
$\begin{split}0\longrightarrow\mathrm{Sel}_{\mathcal{F}\cap+}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})\longrightarrow\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})&\overset{{{\rm
res}_{p}}}{\longrightarrow}H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{gh})\\\
&\longrightarrow
X_{\mathcal{F}\cup+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})\longrightarrow
X_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})\longrightarrow
0.\end{split}$

Assume that
$\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$
and $X_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$ have
both $\Lambda_{{\mathbf{f}}}$-rank one. Since
$H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{gh})$ has
$\Lambda_{{\mathbf{f}}}$-rank one, from (4.8) and Theorem 4.4 we see that
$\mathrm{Sel}_{\mathcal{F}\cap+}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$ is
$\Lambda_{\mathbf{f}}$-torsion and
$X_{\mathcal{F}\cup+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$ has
$\Lambda_{\mathbf{f}}$-rank one. By Lemma 4.14(3), it follows that
$X_{\mathcal{F}\cap+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$ is
$\Lambda_{{\mathbf{f}}}$-torsion, and from the exact sequence

(4.9)
$\begin{split}0\longrightarrow\mathrm{Sel}_{\mathcal{F}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})\longrightarrow\mathrm{Sel}_{\mathcal{F}\cup+}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})&\overset{{{\rm
res}_{p}}}{\longrightarrow}H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{gh})\\\
&\longrightarrow
X_{\mathcal{F}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})\longrightarrow
X_{\mathcal{F}\cap+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})\longrightarrow
0\end{split}$

we get that $X_{\mathcal{F}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$ and
$\mathrm{Sel}_{\mathcal{F}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$ are both
$\Lambda_{{\mathbf{f}}}$-torsion by Theorem 4.4 and Lemma 4.14(1). This proves
the implication ${\rm(1)}\Rightarrow{\rm(2)}$ in the statement of the theorem,
and the converse is shown similarly. Moreover, from (4.9) we see that ${\rm
rank}_{\Lambda_{\mathbf{f}}}\mathrm{Sel}_{\mathcal{F}\cup+}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})=1$,
and hence the quotient
$\mathrm{Sel}_{\mathcal{F}\cup+}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})/\mathrm{Sel}_{\rm
bal}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$ is a torsion
$\Lambda_{{\mathbf{f}}}$-module injecting into
$H^{1}(\mathbb{Q}_{p},\mathbb{V}^{f\cup+}_{{\mathbf{f}}gh}/\mathscr{F}^{2}\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$;
since this is $\Lambda_{\mathbf{f}}$-torsion free, it follows that

(4.10)
$\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})=\mathrm{Sel}_{\mathcal{F}\cup+}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger}).$

Now suppose that either (1) or (2) in the statement of theorem holds. Since
$\bar{\rho}_{{\mathbf{f}}}$ is absolutely irreducible by our hypotheses, the
module
$H^{1}(\mathbb{Q}^{\Sigma}/\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$
is $\Lambda_{{\mathbf{f}}}$-torsion free by [PR00, §1.3.3]. Being
$\Lambda_{{\mathbf{f}}}$-torsion, it follows that the module
$\mathrm{Sel}_{\mathcal{F}\cap+}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$
vanishes, and therefore from (4.8) we deduce the exact sequence

(4.11)
$0\longrightarrow\frac{\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})}{\Lambda_{{\mathbf{f}}}\cdot\kappa({\mathbf{f}},g,h)}\longrightarrow\frac{H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{gh})}{\Lambda_{{\mathbf{f}}}\cdot
p_{{\mathbf{f}}*}({\rm res}_{p}(\kappa({\mathbf{f}},g,h)))}\longrightarrow{\rm
coker}({\rm res}_{p})\longrightarrow 0.$

Together with Theorem 4.4 it follows that

(4.12)
$\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}\left(\frac{\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})}{\Lambda_{{\mathbf{f}}}\cdot\kappa({\mathbf{f}},g,h)}\right)\cdot\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}({\rm
coker}({\rm
res}_{p}))=(\mathscr{L}_{p}^{\xi}(\underline{\breve{{\mathbf{f}}}},\underline{\breve{g}},\underline{\breve{h}})).$

On the other hand, in light of (4.10), from (4.8) and (4.9) we deduce exact
sequences

$\displaystyle 0\longrightarrow{\rm coker}({\rm res}_{p})$
$\displaystyle\longrightarrow
X_{\mathcal{F}\cup+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})\longrightarrow
X_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})\longrightarrow
0,$ $\displaystyle 0\longrightarrow{\rm coker}({\rm res}_{p})$
$\displaystyle\longrightarrow
X_{\mathcal{F}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})\longrightarrow
X_{\mathcal{F}\cap+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})\longrightarrow
0.$

Taking characteristic ideals, these imply

(4.13)
$\displaystyle\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}(X_{\mathcal{F}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$
$\displaystyle=\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}(X_{\mathcal{F}\cap+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}}))\cdot\operatorname{Char}_{\Lambda_{\mathbf{f}}}({\rm
coker}({\rm res}_{p}))$
$\displaystyle=\operatorname{Char}_{\Lambda_{\mathbf{f}}}(X_{\mathcal{F}\cup+}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})_{\rm
tors})\cdot\operatorname{Char}_{\Lambda_{\mathbf{f}}}({\rm coker}({\rm
res}_{p}))$
$\displaystyle=\operatorname{Char}_{\Lambda_{\mathbf{f}}}(X_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})_{\rm
tors})\cdot\operatorname{Char}_{\Lambda_{\mathbf{f}}}({\rm coker}({\rm
res}_{p}))^{2},$

using Lemma 4.14(iv) for the second equality. Multiplying (4.13) by the square
of a generator of the characteristic ideal of
$\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})/\Lambda_{{\mathbf{f}}}\cdot\kappa({\mathbf{f}},g,h)$
and using (4.2) and (4.12), the result follows. ∎

##  5\. Anticyclotomic Euler systems

###  5.1. The general theory

In this subsection we summarize some of the results of [JNS].

Let $K$ be an imaginary quadratic field and let $p\geq 5$ be a rational prime.
If $\mathfrak{n}$ is an integral prime ideal of $K$, we denote by
$K(\mathfrak{n})^{\circ}$ the ray class field of conductor $\mathfrak{n}$; as
in the previous sections, we write $K(\mathfrak{n})$ for the maximal
$p$-subextension in $K(\mathfrak{n})^{\circ}$. For any positive integer $n$,
we denote by $K[n]$ the maximal $p$-subextension in the ring class field of
conductor $n$. We denote by $K_{\infty}^{\circ}=K[p^{\infty}]$ the maximal
anticyclotomic extension of $K$ unramified outside $p$ and by $K_{\infty}$ the
anticyclotomic $\mathbb{Z}_{p}$-extension of $K$.

Let $E$ be a finite extension of $\mathbb{Q}_{p}$ with ring of integers
$\mathcal{O}$ and maximal ideal $\mathfrak{m}$. Let $T$ be a free
$\mathcal{O}$-module of finite rank endowed with a $G_{K}$-action $\varrho$
unramified outside a finite number of places, and let
$V=T\otimes_{\mathcal{O}}E$. Assume that there exists a symmetric
$\mathcal{O}$-bilinear pairing

$\langle\,,\,\rangle\,:\,T\times T\longrightarrow\mathcal{O}(1)$

such that $\langle x^{\sigma},y^{c\sigma c}\rangle=\langle
x,y\rangle^{\sigma}$ for all $x,y\in T$ and for all $\sigma\in G_{K}$, where
$c$ stands for complex conjugation. Thus $V^{c}\simeq V^{\vee}(1)$, where
$V^{c}$ denotes the representation $V$ with the $G_{K}$-action twisted by $c$,
and, if the above pairing is perfect, we also have $T^{c}\simeq T^{\vee}(1)$.
We also define the $G_{K}$-module $A=V/T$.

If $L$ is a finite extension of $K$ and $v$ is a finite place of $L$, we write
$\overline{v}=v^{c}$. Then, the pairing above induces a local pairing

$H^{1}(L_{v},V)\times H^{1}(L_{\overline{v}},V)\longrightarrow E,$

and similarly replacing $V$ by $T$ and $E$ by $\mathcal{O}$. The pair of
compatible maps $G_{L_{v}}\rightarrow G_{L_{\overline{v}}}$ and $V\rightarrow
V^{c}$ defined by $\sigma\mapsto c\sigma c$ and $w\mapsto w$, respectively,
induces an isomorphism $H^{1}(L_{\overline{v}},V)\cong H^{1}(L_{v},V^{c})\cong
H^{1}(L_{v},V^{\vee}(1))$ whereby the previous local pairing is just the
natural cup-product pairing.

For the different results we will discuss, consider two different kinds of
“big image” hypotheses, (HW) for the weaker ones, and (HS) for the stronger
ones.

######  Hypothesis (HW).

  1. (1)

$V$ is absolutely irreducible as a $G_{K}$-representation.

  2. (2)

There exists an element
$\sigma_{0}\in\mathrm{Gal\,}(\bar{K}/K(1)^{\circ}K(\mu_{p^{\infty}}))$ such
that the $E$-dimension of $V/(\sigma_{0}-1)V$ is one.

######  Hypothesis (HS).

  1. (1’)

The residual representation $\bar{T}=T/\mathfrak{m}T$ is absolutely
irreducible.

  2. (2’)

There exists an element
$\sigma_{0}\in\mathrm{Gal\,}(\bar{K}/K(p^{\infty})^{\circ})$ such that
$T/(\sigma_{0}-1)T\simeq\mathcal{O}$ is a free $\mathcal{O}$-module of rank
one.

  3. (3’)

There exists an element $\tau_{0}\in G_{K}$ such that $\tau_{0}-1$ acts on $V$
as multiplication by an element in $\mathcal{O}^{\times}$.

  4. (4’)

The above pairing $T\times T\longrightarrow\mathcal{O}(1)$ is perfect.

For each prime $\mathfrak{p}$ of $K$ above $p$, choose a
$G_{K_{\mathfrak{p}}}$-stable $\mathcal{O}$-submodule
$\mathcal{F}_{\mathfrak{p}}^{+}(T)$ of $T$, and let
$\mathcal{F}_{\mathfrak{p}}^{-}(T)=T/\mathcal{F}_{\mathfrak{p}}^{+}(T)$. We
also define
$\mathcal{F}_{\mathfrak{p}}^{+}(V)=\mathcal{F}_{\mathfrak{p}}^{+}(T)\otimes_{\mathcal{O}}E\subseteq
V$ and
$\mathcal{F}_{\mathfrak{p}}^{-}(V)=V/\mathcal{F}_{\mathfrak{p}}^{+}(V)$. Let
$L$ be a finite extension of $K$. For each place $v$ of $L$, we define a local
condition

$H^{1}_{\operatorname{Gr}}(L_{v},V)=\begin{cases}\ker\left(H^{1}(L_{v},V)\rightarrow
H^{1}(L_{v}^{\text{nr}},V)\right)&\text{if }v\nmid p\\\\[5.0pt]
\ker\left(H^{1}(L_{v},V)\rightarrow
H^{1}(L_{v},\mathcal{F}_{\mathfrak{p}}^{-}(V))\right)&\text{if
}v\mid\mathfrak{p}\text{ for some }\mathfrak{p}\mid p.\end{cases}$

We define the Greenberg Selmer group

$\mathrm{Sel}_{\operatorname{Gr}}(L,V)=\ker\Big{(}H^{1}(L,V)\rightarrow\prod_{v}H^{1}(L_{v},V)/H^{1}_{\operatorname{Gr}}(L_{v},V)\Big{)},$

where the product is over all finite places of $L$.

We also define local conditions for $T$ and $A$ by propagation of the local
conditions for $V$, i.e., for each place $v$ of $L$, we define

  * •

$H^{1}_{\operatorname{Gr}}(L_{v},T)$ as the preimage of
$H^{1}_{\operatorname{Gr}}(L_{v},V)$ by the map $H^{1}(L_{v},T)\rightarrow
H^{1}(L_{v},V)$, and

  * •

$H^{1}_{\operatorname{Gr}}(L_{v},A)$ as the image of
$H^{1}_{\operatorname{Gr}}(L_{v},V)$ by the map $H^{1}(L_{v},V)\rightarrow
H^{1}(L_{v},A)$,

and use these to define the Selmer groups
$\mathrm{Sel}_{\operatorname{Gr}}(L,T)$ and
$\mathrm{Sel}_{\operatorname{Gr}}(L,A)$ as above. Finally, for each positive
integer $n$, we also put

$\displaystyle\mathrm{Sel}_{\operatorname{Gr}}(K[np^{\infty}],T)$
$\displaystyle=\varprojlim_{r}\mathrm{Sel}_{\operatorname{Gr}}(K[np^{r}],T)\quad{\rm
and}\quad\mathrm{Sel}_{\operatorname{Gr}}(K[np^{\infty}],A)=\varinjlim_{r}\mathrm{Sel}_{\operatorname{Gr}}(K[np^{r}],A),$

where the limits are with respect to the corestriction and restriction maps,
respectively.

Let $\mathcal{N}$ be an ideal of $K$ divisible by $p$ and all the primes at
which $T$ is ramified, and let $\mathcal{S}$ be the set of all squarefree
products of primes of $\mathbb{Q}$ which split in $K$ and are coprime to
$\mathcal{N}$.

######  Definition 5.1.

A “split” anticyclotomic Euler system for
$(T,\\{\mathcal{F}_{\mathfrak{p}}^{+}(T)\\}_{\mathfrak{p}\mid p},\mathcal{N})$
is a collection of classes

$\bm{\kappa}=\left\\{\kappa_{n}\in\mathrm{Sel}_{\operatorname{Gr}}(K[n],T)\;\colon\;n\in\mathcal{S}\right\\}$

such that, whenever $q$ is a rational prime and $n,nq\in\mathcal{S}$,

(5.1)
$\operatorname{cor}_{K[nq]/K[n]}(\kappa_{nq})=P_{\mathfrak{q}}(\mathrm{Fr}_{\mathfrak{q}}^{-1})\,\kappa_{n},$

where $\mathfrak{q}$ is any of the primes of $K$ above $q$ and
$P_{\mathfrak{q}}(X)=\det(1-\mathrm{Fr}_{\mathfrak{q}}^{-1}X|T^{\vee}(1))$.

Similarly, a “split” $\Lambda$-adic anticyclotomic Euler system for
$(T,\\{\mathcal{F}_{\mathfrak{p}}^{+}(T)\\}_{\mathfrak{p}\mid p},\mathcal{N})$
is a collection of classes

$\bm{\kappa}_{\infty}=\left\\{\kappa_{n,\infty}\in\mathrm{Sel}_{\operatorname{Gr}}(K[np^{\infty}],T)\;\colon\;n\in\mathcal{S}\right\\}$

satisfying the previous norm relations. In this case, the classes

$\kappa_{n}={\rm
pr}_{K[n]}(\kappa_{n,\infty})\in\mathrm{Sel}_{\operatorname{Gr}}(K[n],T)$

form an anticyclotomic Euler system in the previous sense, and we say that the
Euler system $\bm{\kappa}=\\{\kappa_{n}\\}_{n}$ extends along the
anticyclotomic $\mathbb{Z}_{p}$-extension.

A ($\Lambda$-adic) anticyclotomic Euler system for
$(T,\\{\mathcal{F}_{\mathfrak{p}}^{+}(T)\\}_{\mathfrak{p}\mid p})$ is just a
($\Lambda$-adic) anticylotomic Euler system for
$(T,\\{\mathcal{F}_{\mathfrak{p}}^{+}(T)\\}_{\mathfrak{p}\mid p},\mathcal{N})$
for some $\mathcal{N}$ as above. We shall usually drop
$\\{\mathcal{F}_{\mathfrak{p}}^{+}(T)\\}_{\mathfrak{p}\mid p}$ if there is no
risk of confusion.

If $\bm{\kappa}$ is an anticyclotomic Euler system for $T$, we define

$\kappa_{0}:=\operatorname{cor}_{K[1]/K}(\kappa_{1})\in\mathrm{Sel}_{\operatorname{Gr}}(K,T).$

If it extends along the anticyclotomic $\mathbb{Z}_{p}$-extension, we
similarly define

$\kappa_{\infty}:=\operatorname{cor}_{K[1]/K}(\kappa_{1,\infty})\in\mathrm{Sel}_{\operatorname{Gr}}(K_{\infty},T),$

where $\bm{\kappa}_{\infty}=\\{\kappa_{n,\infty}\\}$ is the $\Lambda$-adic
anticyclotomic Euler system extending $\bm{\kappa}$.

When we have an Euler system as above, we will be interested in ensuring that
the following orthogonality hypothesis holds.

######  Hypothesis (HO).

For all $n\in\mathcal{S}$ and for all places $v$ of $K[n]$ above $p$, the
local conditions $H^{1}_{\operatorname{Gr}}(K[n]_{v},V)$ and
$H^{1}_{\operatorname{Gr}}(K[n]_{\overline{v}},V)$ are orthogonal complements
under the local pairing

$H^{1}(K[n]_{v},V)\times H^{1}(K[n]_{\overline{v}},V)\longrightarrow E.$

######  Remark 5.2.

A condition like the one in (HO) holds automatically for all places away from
$p$, by [Rub00, Prop. 1.4.2]. Observe also that if (HO) holds, then for all
$n\in\mathcal{S}$ and for all places $v$ of $K[n]$ the local conditions
$H^{1}_{\operatorname{Gr}}(K[n]_{v},T)$ and
$H^{1}_{\operatorname{Gr}}(K[n]_{\overline{v}},T)$ are also orthogonal
complements under the local pairing

$H^{1}(K[n]_{v},T)\times
H^{1}(K[n]_{\overline{v}},T)\longrightarrow\mathcal{O},$

as follows easily from the definitions using [Rub00, Prop. B.2.4] and the
commutative diagram

$\textstyle{H^{1}(K[n]_{v},T)\times
H^{1}(K[n]_{\overline{v}},T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(K[n]_{v},V)\times
H^{1}(K[n]_{\overline{v}},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E.}$

We assume in the rest of this subsection that hypothesis (HO) holds for our
choice of local conditions at $p$.

######  Theorem 5.3 ([JNS]).

Assume that $p$ splits in $K$ and that Hypothesis (HW) is satisfied, and let
$\bm{\kappa}=\\{\kappa_{n}\\}_{n}$ be an anticyclotomic Euler system for $T$
which extends along the anticyclotomic $\mathbb{Z}_{p}$-extension. If
$\kappa_{0}\neq 0$, then the Selmer group
$\mathrm{Sel}_{\operatorname{Gr}}(K,T)$ has $\mathcal{O}$-rank one.

######  Remark 5.4.

One can replace the assumptions that $p$ splits in $K$ and the Euler system
extends along the anticyclotomic $\mathbb{Z}_{p}$-extension by the assumption
that there exists an element $\gamma\in G_{K}$ fixing the extension
$K(1)^{\circ}(\mu_{p^{\infty}},(\mathcal{O}_{K}^{\times})^{1/p^{\infty}})$ and
such that $\gamma-1$ acts invertibly on $V$.

Under the stronger Hypothesis (HS), granted the non-triviality of a
$\Lambda$-adic anticyclotomic Euler system, the results of [JNS] yield a
divisibility towards a corresponding Iwasawa main conjecture.

######  Theorem 5.5 ([JNS]).

Assume that $p$ splits in $K$ and that Hypothesis (HS) is satisfied, and let
$\bm{\kappa}$ be a $\Lambda$-adic anticyclotomic Euler system for $T$.

  1. (a)

If $\kappa_{0}\neq 0$, then $\mathrm{Sel}_{\operatorname{Gr}}(K,A)$ has
$\mathcal{O}$-corank one, $\mathrm{Sel}_{\operatorname{Gr}}(K,T)$ has
$\mathcal{O}$-rank one, and

${\rm length}_{\mathcal{O}}(\mathrm{Sel}_{\operatorname{Gr}}(K,A)_{/{\rm
div}})\leq 2\;{\rm
length}_{\mathcal{O}}\left(\frac{\mathrm{Sel}_{\operatorname{Gr}}(K,T)}{\mathcal{O}\cdot\kappa_{0}}\right),$

where $(-)_{/{\rm div}}$ denotes the quotient of $(-)$ by its maximal
divisible submodule.

  2. (b)

If $\kappa_{\infty}$ is not $\Lambda_{\operatorname{ac}}$-torsion, then
$X_{\operatorname{Gr}}(K_{\infty},A)$ and
$\mathrm{Sel}_{\operatorname{Gr}}(K_{\infty},T)$ have both
$\Lambda_{\operatorname{ac}}$-rank one, and

$\operatorname{Char}_{\Lambda_{\operatorname{ac}}}(X_{\operatorname{Gr}}(K_{\infty},A)_{\operatorname{tors}})\supset\operatorname{Char}_{\Lambda_{\operatorname{ac}}}\left(\frac{\mathrm{Sel}_{\operatorname{Gr}}(K_{\infty},T)}{\Lambda_{\operatorname{ac}}\cdot\kappa_{\infty}}\right)^{2}.$

###  5.2. Verifying the hypotheses

We now give conditions under which the previous results can be applied to our
setting. To that end, we will rely on the results of [Loe17].

As before, $K/\mathbb{Q}$ is an imaginary quadratic field of discriminant
$-D_{K}$, $(g,h)$ is a pair of $p$-ordinary newforms of weights $(l,m)$ of the
same parity, levels $(N_{g},N_{h})$ and characters $(\chi_{g},\chi_{h})$, and
$\psi$ is a Grössencharacter of $K$ of infinity type $(1-k,0)$ for some
positive even integer $k$ and of conductor dividing $\mathfrak{f}$. We denote
by $\chi$ the unique Dirichlet character modulo
$N_{K/\mathbb{Q}}(\mathfrak{f})$ such that $\psi((n))=n^{k-1}\chi(n)$ for
integers $n$ coprime to $N_{K/\mathbb{Q}}(\mathfrak{f})$, and we assume that
$\chi\varepsilon_{K}\chi_{g}\chi_{h}=1$.

We now make the further assumptions that

  * •

neither $g$ nor $h$ are of CM type,

  * •

$g$ is not Galois-conjugate to a twist of $h$.

As in [Loe17, §3.1], we define the open subgroups $H_{g}$ and $H_{h}$ of
$G_{\mathbb{Q}}$, the quaternion algebras $B_{g}$ and $B_{h}$, and the
algebraic groups $G_{g}$ and $G_{h}$, and put

$B=B_{g}\times B_{h},\quad G=G_{g}\times_{\mathbb{G}_{m}}G_{h}.$

We define $H$ to be the intersection of $H_{g}$, $H_{h}$ and
$G_{K(\mathfrak{f})^{\circ}}$. Note that in _loc. cit,_ $H$ is defined to be
the intersection of $H_{g}$ and $H_{h}$, so our $H$ might be a finite index
subgroup of his $H$. In any case, we have an adelic representation

$\tilde{\rho}_{g,h}:G_{\mathbb{Q}}\longrightarrow G(\hat{\mathbb{Q}})$

and representations

$\tilde{\rho}_{g,h,p}:G_{\mathbb{Q}}\longrightarrow G(\mathbb{Q}_{p})$

for every rational prime $p$, and, by [Loe17, Thm. 3.2.2],
$\tilde{\rho}_{g,h,p}(H)=G(\mathbb{Z}_{p})$ for all but finitely many $p$.

######  Remark 5.6.

Note that the representations studied in [Loe17] are the dual to the ones
studied in this paper, but as pointed out in [Loe17, Rmk. 2.1.2], this
difference is unimportant when considering the image.

Let $L$ be a finite extension of $K$ containing the Fourier coefficients of
$g$ and $h$ and the image of $\psi$. Let $\mathfrak{P}$ be a prime of $L$
above some rational prime $p$, and let $E=L_{\mathfrak{P}}$.

######  Definition 5.7.

We say that the prime $\mathfrak{P}$ is good if the following conditions hold:

  * •

$p\geq 7$;

  * •

$p$ is unramified in $B$;

  * •

$p$ is coprime to $\mathfrak{f}$, $N_{g}$ and $N_{h}$;

  * •

$\tilde{\rho}_{g,h,p}(H)=G(\mathbb{Z}_{p})$;

  * •

$E=\mathbb{Q}_{p}$.

######  Remark 5.8.

Observe that all but the last condition exclude only finitely many primes. The
last condition could be somewhat relaxed in some cases, and will be used
largely for simplicity. Note also that the above set of conditions holds for a
set of primes of positive density.

######  Lemma 5.9.

Assume that there is at least one prime which divides $D_{K}$ but not $N_{g}$
and one prime which divides $D_{K}$ but not $N_{h}$. Then, if $\mathfrak{P}$
is a good prime,

$(\rho_{g,\mathfrak{P}}\times\rho_{h,\mathfrak{P}})(H\cap
G_{K_{\infty}^{\circ}(\mu_{p^{\infty}})})=\mathrm{SL}_{2}(\mathbb{Z}_{p})\times\mathrm{SL}_{2}(\mathbb{Z}_{p}).$

###### Proof.

Let $\mathbb{Q}(\rho_{g})$ and $\mathbb{Q}(\rho_{h})$ be the Galois extensions
of $\mathbb{Q}$ cut out by the representations $\rho_{g}$ and $\rho_{h}$
attached to $g$ and $h$, respectively. These extensions are unramified outside
$pN_{g}$ and $pN_{h}$, respectively. Therefore, the condition on $D_{K}$
implies that $K\cap\mathbb{Q}(\rho_{g})=\mathbb{Q}$ and
$K\cap\mathbb{Q}(\rho_{h})=\mathbb{Q}$. Moreover, since any Galois extension
of $\mathbb{Q}$ contained in $K_{\infty}$ must itself contain $K$, we also
have $K_{\infty}\cap\mathbb{Q}(\rho_{g})=\mathbb{Q}$ and
$K_{\infty}\cap\mathbb{Q}(\rho_{h})=\mathbb{Q}$.

The conditions on $\mathfrak{P}$ imply that

$(\rho_{g,\mathfrak{P}}\times\rho_{h,\mathfrak{P}})(H\cap
G_{\mathbb{Q}(\mu_{p^{\infty}})})=\mathrm{SL}_{2}(\mathbb{Z}_{p})\times\mathrm{SL}_{2}(\mathbb{Z}_{p}),$

and, from the remarks in the previous paragraph, it follows that

$(\rho_{g,\mathfrak{P}}\times\rho_{h,\mathfrak{P}})(H\cap
G_{K_{\infty}(\mu_{p^{\infty}})})=\mathrm{SL}_{2}(\mathbb{Z}_{p})\times\mathrm{SL}_{2}(\mathbb{Z}_{p}).$

Finally, since the index of $H\cap G_{K_{\infty}^{\circ}(\mu_{p^{\infty}})}$
inside $H\cap G_{K_{\infty}(\mu_{p^{\infty}})}$ divides $p-1$ and there are no
such subgroups in
$\mathrm{SL}_{2}(\mathbb{Z}_{p})\times\mathrm{SL}_{2}(\mathbb{Z}_{p})$, the
lemma follows. ∎

Now we are able to give conditions under which the results of [JNS] can be
applied to our setting, i.e., to the representation $T_{g,h}^{\psi}$ defined
above.

######  Proposition 5.10.

Assume that there is at least one prime which divides $D_{K}$ but not $N_{g}$
and one prime which divides $D_{K}$ but not $N_{h}$. Let $\mathfrak{P}$ be a
good prime. Suppose that there exists
$u\in\mathrm{Gal\,}(K(\mathfrak{f})^{\circ}K_{\infty}^{\circ}(\mu_{p^{\infty}})/K_{\infty}^{\circ}(\mu_{p^{\infty}}))$
such that $\psi_{\mathfrak{P}}(u)\neq\psi_{\mathfrak{P}}^{c}(u)$ modulo $p$.
Then, hypotheses (HS) hold for $T_{g,h}^{\psi}$.

###### Proof.

Since $\psi_{\mathfrak{P}}$ is trivial when restricted to $H\cap
G_{K_{\infty}^{\circ}(\mu_{p^{\infty}})}$, condition (1’) follows easily from
the previous lemma.

To prove condition (2’), we closely follow the proof of [Loe17, Prop. 4.2.1].
Lift $u$ to an element
$u^{\prime}\in\mathrm{Gal\,}(K(\mathfrak{f})^{\circ}K_{\infty}^{\circ}(\mu_{p^{\infty}})(\mu_{N_{g}N_{h}})/K_{\infty}^{\circ}(\mu_{p^{\infty}}))$
and write $\chi_{g}(u^{\prime})$ and $\chi_{h}(u^{\prime})$ for the images of
this element by $\chi_{g}$ and $\chi_{h}$ via the natural identifications
$\mathrm{Gal\,}(\mathbb{Q}(\mu_{N_{g}})/\mathbb{Q})\cong(\mathbb{Z}/N_{g}\mathbb{Z})^{\times}$
and
$\mathrm{Gal\,}(\mathbb{Q}(\mu_{N_{h}})/\mathbb{Q})\cong(\mathbb{Z}/N_{h}\mathbb{Z})^{\times}$.
Then, using [Loe17, Cor. 2.2.3] and the previous lemma, the image of
$G_{K_{\infty}^{\circ}(\mu_{p^{\infty}})}$ under
$\rho_{g,\mathfrak{P}}\times\rho_{h,\mathfrak{P}}$ contains all the elements
of the form

$\left(\begin{pmatrix}x&0\\\
0&x^{-1}\chi_{g}(u^{\prime})\end{pmatrix},\begin{pmatrix}y&0\\\
0&y^{-1}\chi_{h}(u^{\prime})\end{pmatrix}\right),\quad
x,y\in\mathbb{Z}_{p}^{\times}.$

Now choose $x\in\mathbb{Z}_{p}^{\times}$ such that
$x^{-2}\chi_{g}(u^{\prime})\neq 1\pmod{p}$ and
$x^{2}\chi_{h}(u^{\prime})\psi_{\mathfrak{P}}(u)^{-2}\neq 1\pmod{p}$, which is
possible since $p\geq 7$, and let $y=x^{-1}\psi_{\mathfrak{P}}(u)$. Choose
$\sigma_{0}\in G_{K_{\infty}^{\circ}(\mu_{p^{\infty}})}$ whose image under
$\rho_{g,\mathfrak{P}}\times\rho_{h,\mathfrak{P}}$ is given by the element
above, with the choices of $x$ and $y$ which we have just specified. Then, the
eigenvalues of $\sigma_{0}$ acting on $T_{g,h}^{\psi}$ are 1,
$x^{-2}\chi_{g}(u^{\prime})$,
$x^{2}\chi_{h}(u^{\prime})\psi_{\mathfrak{P}}(u)^{-2}$ and
$\psi_{\mathfrak{P}}^{c}(u)\psi_{\mathfrak{P}}(u)^{-1}$, which proves
condition (2’).

To check condition (3’), we can argue as in [KLZ17, Rmk. 11.1.3]. By the
previous lemma, we can find an element $\tau_{0}\in H\cap
G_{K_{\infty}^{\circ}(\mu_{p^{\infty}})}$ such that

$(\rho_{g,\mathfrak{P}}\times\rho_{h,\mathfrak{P}})(\tau_{0})=\left(\begin{pmatrix}-1&0\\\
0&-1\end{pmatrix},\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\right),$

so $\tau_{0}$ acts on $T_{g,h}^{\psi}$ as multiplication by $-1$.

Finally, condition (4’) follows from the assumption that $g$ and $h$ are non-
Eisenstein. ∎

######  Remark 5.11.

If we are just interested in ensuring that hypotheses (HW) hold for
$T_{g,h}^{\psi}$, we can relax some of the assumptions above. For example, we
do not need to require $g$ and $h$ to be non-Eisenstein, and we can require
that there exist
$u\in\mathrm{Gal\,}(K(\mathfrak{f})^{\circ}K_{\infty}^{\circ}(\mu_{p^{\infty}})/K_{\infty}^{\circ}(\mu_{p^{\infty}}))$
such that $\psi_{\mathfrak{P}}(u)\neq\psi_{\mathfrak{P}}^{c}(u)$, without
requiring this inequality to hold modulo $p$.

##  6\. Proof of the main results

Let the setting be as introduced in $\S\ref{subsec:statements}$. In
particular, $g\in S_{l}(N_{g},\chi_{g})$ and $h\in S_{m}(N_{h},\chi_{h})$ are
newforms of weights $l\geq m\geq 2$ of the same parity, $K/\mathbb{Q}$ is an
imaginary quadratic field of discriminant $-D<0$, and $\psi$ is a
Grössencharacter for $K$ of infinity type $(1-k,0)$ for some even integer
$k\geq 2$.

Consider the $G_{K}$-representation

$V_{g,h}^{\psi}=V_{g}\otimes_{E}V_{h}(\psi_{\mathfrak{P}}^{-1})(1-c),$

where $c=(k+l+m-2)/2$.

######  Lemma 6.1.

The Bloch–Kato Selmer group of $V_{g,h}^{\psi}$ is given by

$\mathrm{Sel}(K,V_{g,h}^{\psi})\cong\begin{cases}\mathrm{Sel}_{\operatorname{bal}}(K,V_{g,h}^{\psi})&\textrm{if
$l-m<k<l+m$,}\\\\[1.99997pt]
\mathrm{Sel}_{\mathcal{F}}(K,V_{g,h}^{\psi})&\textrm{if $k\geq
l+m$.}\end{cases}$

###### Proof.

Note that by Shapiro’s lemma $H^{1}(K,V_{g,h}^{\psi})\cong
H^{1}(\mathbb{Q},V_{fgh})$, where $f=\theta_{\psi}$ is the theta series of
$\psi$ and $V_{fgh}$ is the specialization of the big Galois representation
$\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}^{\dagger}$ in (4.1) to
weights $(k,l,m)$. One immediately checks that the Hodge–Tate weights of the
$G_{\mathbb{Q}_{p}}$-subrepresentation $\mathscr{F}^{2}V_{fgh}\subset V_{fgh}$
(resp. $V_{fgh}^{f}\subset V_{fgh}$) are all $<0$ (with the $p$-adic
cyclotomic character $\epsilon_{\rm cyc}$ having Hodge–Tate weight $-1$) if
and only if $l-m<k<l+m$ (resp. $k\geq l+m$). The result follows. ∎

Consider the following set of hypotheses, which (unless otherwise noted) will
be in effect in the results that follow:

######  Hypotheses 6.2.

  * (h1)

$g$ and $h$ are ordinary at $p$,

  * (h2)

$p$ splits in $K$,

  * (h3)

$p$ does not divide the class number of $K$,

  * (h4)

$\psi_{\mathfrak{P}}$ is non-Eisenstein and $p$-distinguished,

  * (h5)

neither $g$ nor $h$ are of CM type,

  * (h6)

$g$ is not Galois-conjugate to a twist of $h$.

  * (h7)

${\rm prime}(D)\cap{\rm prime}^{c}(N_{g})\neq\emptyset$ and ${\rm
prime}(D)\cap{\rm prime}^{c}(N_{h})\neq\emptyset$,

  * (h8)

$\mathfrak{P}$ is a good prime in the sense of Definition 5.7.

Here, for any nonzero $m\in\mathbb{Z}$, ${\rm prime}(m)$ denotes the set of
primes that divide $m$, and ${\rm prime}^{c}(m)$ its complement.

######  Remark 6.3.

Note that the need for hypotheses (h1)–(h4) arises from our construction of
the anticyclotomic Euler system in Theorem 3.5, while (h5)–(h8) are used to
verify the hypotheses in the results from the Euler system machinery developed
in [JNS], as done in §5.2.

###  6.1. Proof of Theorem A

Let ${\kappa_{\psi,g,h,1,\infty}}\in H^{1}_{\rm
Iw}(K[p^{\infty}],T_{g,h}^{\psi})$ be the Iwasawa cohomology class of
conductor $n=1$ from Theorem 3.5, and set

(6.1) ${\kappa_{\psi,g,h}}={\rm cor}_{K[1]/K}({\kappa_{\psi,g,h,1}})\in
H^{1}(K,T_{g,h}^{\psi}),$

where ${\kappa_{\psi,g,h,1}}={\rm pr}_{K[1]}({\kappa_{\psi,g,h,1,\infty}})$.

If $l-m<k<l+m$, the next result recovers Theorem A in the Introduction. Note
however, that the result does not require these inequalities to hold.

######  Theorem 6.4.

Assume hypotheses (h1)–(h8). Then the following implication holds:

${\kappa_{\psi,g,h}}\neq 0\quad\Longrightarrow\quad{\rm
dim}_{E}\,\mathrm{Sel}_{\operatorname{bal}}(K,V^{\psi}_{g,h})=1.$

In particular, if $l-m<k<l+m$ and ${\kappa_{\psi,g,h}}\neq 0$ then the
Bloch–Kato Selmer group $\mathrm{Sel}(K,V^{\psi}_{g,h})$ is one-dimensional.

###### Proof.

By Proposition 3.6, the classes ${\kappa_{\psi,g,h,n}}:={\rm
pr}_{K[n]}({\kappa_{\psi,g,h,n,\infty}})$ land in
$\mathrm{Sel}_{\operatorname{bal}}(K[n],T_{g,h}^{\psi})$, and by Theorem 3.5
they form an anticyclotomic Euler system for $V_{g,h}^{\psi}$. Therefore, the
result follows from Theorem 5.3 and Proposition 5.10. ∎

######  Remark 6.5.

If $k=2$ and $l=m\geq 2$, working with the classes ${\kappa_{\psi,g,h,n}}$
from Theorem 2.12, rather than those from Theorem 3.5 as above, hypotheses
(h2)-(h3) in Theorem 6.4 can be replaced by the assumption that there exists
an element $\gamma\in G_{K}$ satisfying the conditions in Remark 5.4.

###  6.2. Proof of Theorem B

Recall that $\theta_{\psi}\in S_{k}(N_{\psi},\chi\varepsilon_{K})$ is the
theta series attached to $\psi$, and put
$N=\operatorname{lcm}(N_{\psi},N_{g},N_{h})$.

The next theorem, establishing cases of the Bloch–Kato conjecture for
$V_{g,h}^{\psi}$ in analytic rank zero, recovers Theorem B in the
Introduction.

######  Theorem 6.6.

Assume hypotheses (h1)–(h8), and in addition that:

  * •

$\varepsilon_{\ell}(\theta_{\psi},g,h)=+1$ for all primes $\ell\mid N$,

  * •

${\rm gcd}(N_{\psi},N_{g},N_{h})$ is squarefree.

If $k\geq l+m$ then the following implication holds:

$L(V^{\psi}_{g,h},0)\neq
0\quad\Longrightarrow\quad\mathrm{Sel}(K,V^{\psi}_{g,h})=0.$

###### Proof.

We continue to denote by ${\kappa_{\psi,g,h}}$ the image of the class in (6.1)
under the isomorphism

$H^{1}(K,V_{g,h}^{\psi})\cong H^{1}(\mathbb{Q},V_{fgh})$

coming from Shapiro’s lemma. If $k\geq l+m$, the central value
$L(V^{\psi}_{g,h},0)$ is in the range of interpolation of the triple product
$p$-adic $L$-function of Theorem 4.1, and so by Proposition 4.3 and Theorem
4.4 its non-vanishing implies that the image of ${\kappa_{\psi,g,h}}$ under
the natural map

${\rm
res}_{p}:\mathrm{Sel}_{\operatorname{bal}}(\mathbb{Q},V_{fgh})\longrightarrow
H^{1}(\mathbb{Q}_{p},V_{f}^{gh})$

is nonzero. In particular, ${\kappa_{\psi,g,h}}\neq 0$, and therefore by
Theorem 6.4 the balanced Selmer group
$\mathrm{Sel}_{\operatorname{bal}}(K,V_{g,h}^{\psi})=\mathrm{Sel}_{\operatorname{bal}}(\mathbb{Q},V_{fgh})$
is one-dimensional.

From the exact sequence

$\displaystyle
0\longrightarrow\mathrm{Sel}_{\mathcal{F}\cap+}(\mathbb{Q},V_{fgh})\longrightarrow\mathrm{Sel}_{\operatorname{bal}}(\mathbb{Q},V_{fgh})$
$\displaystyle\overset{{\rm
res}_{p}}{\longrightarrow}H^{1}(\mathbb{Q}_{p},V_{f}^{gh})$
$\displaystyle\longrightarrow\mathrm{Sel}_{\mathcal{F}\cup+}(\mathbb{Q},V_{fgh})^{\vee}\longrightarrow\mathrm{Sel}_{\operatorname{bal}}(\mathbb{Q},V_{fgh})^{\vee}\longrightarrow
0$

coming from global duality (adopting notations similar to those in Theorem
4.16), we thus see that
$\mathrm{Sel}_{\mathcal{F}\cap+}(\mathbb{Q},V_{fgh})=0$ and that
$\mathrm{Sel}_{\mathcal{F}\cup+}(\mathbb{Q},V_{fgh})=\mathrm{Sel}_{\operatorname{bal}}(\mathbb{Q},V_{fgh})$.
Together with the exact sequence

$\mathrm{Sel}_{\mathcal{F}\cup+}(\mathbb{Q},V_{fgh})\xrightarrow{{\rm
res}_{p}}H^{1}(\mathbb{Q}_{p},V_{f}^{gh})\longrightarrow\mathrm{Sel}_{\mathcal{F}}(\mathbb{Q},V_{fgh})^{\vee}\longrightarrow\mathrm{Sel}_{\mathcal{F}\cap+}(\mathbb{Q},V_{fgh})^{\vee}\longrightarrow
0,$

it follows that $\mathrm{Sel}_{\mathcal{F}}(\mathbb{Q},V_{fgh})=0$, and
combined with Lemma 6.1 this concludes the proof. ∎

Refining the proof of Theorem 6.6, we can further bound the size of the
Bloch–Kato Selmer group for the discrete module
$A_{g,h}^{\psi}=V^{\psi}_{g,h}/T^{\psi}_{g,h}$ in terms of $L$-values. For the
statement, let ${\mathbf{f}}$ be the Hida family associated to $\psi$ as in
$\S\ref{subsec:Iw}$, so that ${\mathbf{f}}_{k}$ is the ordinary
$p$-stabilization of $\theta_{\psi}$, and, keeping with the notations in
Theorem 4.1, put
$\alpha_{k}=\psi(\overline{\mathfrak{p}}),\beta_{k}=\psi(\mathfrak{p})$.

######  Theorem 6.7.

Assume hypotheses (h1)–(h8), and in addition that:

  * •

$\varepsilon_{\ell}(\theta_{\psi},g,h)=+1$ for all primes $\ell\mid N$,

  * •

${\rm gcd}(N_{\psi},N_{g},N_{h})$ is squarefree.

If $k\geq l+m$ and $L(V^{\psi}_{g,h},0)\neq 0$ then the $\mathcal{O}$-module
$\mathrm{Sel}_{\mathcal{F}}(K,A^{\psi}_{g,h})$ is finite and

${\rm
length}_{{\mathcal{O}}}(\mathrm{Sel}_{\mathcal{F}}(K,A^{\psi}_{g,h}))\leq
2\,v_{\mathfrak{P}}\left((l+m-4)!\cdot\frac{\mathcal{E}_{1}({\mathbf{f}}_{k})}{\mathcal{E}({\mathbf{f}}_{k},g,h)}\cdot\mathscr{L}_{p}^{\xi}(\underline{\breve{{\mathbf{f}}}},\underline{\breve{g}},\underline{\breve{h}})(k)\right),$

where
$\mathcal{E}_{1}({\mathbf{f}}_{k})=\bigl{(}1-\frac{\beta_{k}}{p\alpha_{k}}\bigr{)}$,
$\mathcal{E}({\mathbf{f}}_{k},g,h)=\bigl{(}1-\frac{\alpha_{k}\alpha_{g}\alpha_{h}}{p^{c}}\bigr{)}\bigl{(}1-\frac{\beta_{k}\beta_{g}\alpha_{h}}{p^{c}}\bigr{)}\bigl{(}1-\frac{\beta_{k}\alpha_{g}\beta_{h}}{p^{c}}\bigr{)}\bigl{(}1-\frac{\beta_{k}\beta_{g}\beta_{h}}{p^{c}}\bigr{)}$,
and $c=(k+l+m-2)/2$.

###### Proof.

As in the proof of Theorem 6.6, if $k\geq l+m$ and $L(V^{\psi}_{g,h},0)\neq 0$
then the class ${\kappa_{\psi,g,h}}$ is nonzero. Since by Theorem 3.5 this is
the bottom class of an anticyclotomic Euler system for $V_{gh}^{\psi}$, from
Theorem 5.3 and Proposition 5.10 we deduce that
$\mathrm{Sel}_{\operatorname{bal}}(K,A^{\psi}_{g,h})$ has $\mathcal{O}$-corank
one, with

${\rm
length}_{\mathcal{O}}(\mathrm{Sel}_{\operatorname{bal}}(K,A^{\psi}_{g,h})_{/{\rm
div}})\leq 2\;{\rm
length}_{\mathcal{O}}\biggl{(}\frac{\mathrm{Sel}_{\operatorname{bal}}(K,T_{g,h}^{\psi})}{\mathcal{O}\cdot{\kappa_{\psi,g,h}}}\biggr{)}.$

The finiteness of $\mathrm{Sel}_{\mathcal{F}}(K,A_{g,h}^{\psi})$ with the
stated bound on its ${\mathcal{O}}$-length now follows from this by the same
argument as in the proof of Theorem 4.16, noting that by Theorem 4.4 and the
calculation in [BSV21, §8.5] (see esp. the equality following [_op. cit._ ,
(185)]) the map

$\xi_{k}\cdot\left\langle{\rm
log}_{p}(-),\eta_{\breve{f}}\otimes\omega_{\breve{g}}\otimes\omega_{\breve{h}}\right\rangle$

where $f=\theta_{\psi}$, gives an ${\mathcal{O}}$-linear isomorphism
$H^{1}(\mathbb{Q}_{p},T_{f}^{gh})\rightarrow{\mathcal{O}}$ taking
${\kappa_{\psi,g,h}}$ to

$(l+m-4)!\cdot\frac{\mathcal{E}_{0}({\mathbf{f}}_{k})\cdot\mathcal{E}_{1}({\mathbf{f}}_{k})}{\mathcal{E}({\mathbf{f}}_{k},g,h)}\cdot\mathscr{L}_{p}^{\xi}(\underline{\breve{{\mathbf{f}}}},\underline{\breve{g}},\underline{\breve{h}})(k),$

where
$\mathcal{E}_{0}({\mathbf{f}}_{k})=\bigl{(}1-\frac{\beta_{k}}{\alpha_{k}}\bigr{)}$
is a $p$-adic unit. ∎

######  Remark 6.8.

By Theorem 4.1, the non-vanishing of $L(V^{\psi}_{g,h},0)$ implies that also
$\mathscr{L}_{p}^{\xi}(\underline{\breve{{\mathbf{f}}}},\underline{\breve{g}},\underline{\breve{h}})(k)\neq
0$. Moreover, combined with the interpolation formula in that theorem, the
upper bound in Theorem 6.7 can be expressed in terms of $L(V^{\psi}_{g,h},0)$,
thus giving a result towards the Tamagawa number conjecture of [BK90].

###  6.3. Proof of Theorem C

As before, let ${\mathbf{f}}$ be the Hida family attached to $\psi$ as in
$\S\ref{subsec:Iw}$. Let ${\kappa_{\psi,g,h,1,\infty}}$ be the $\Lambda$-adic
class of conductor $n=1$ constructed in Theorem 3.5, and set

${\kappa_{\psi,g,h,\infty}}:={\rm
cor}_{K[1]/K}({\kappa_{\psi,g,h,1,\infty}})\in H^{1}_{\rm
Iw}(K_{\infty},T_{g,h}^{\psi}(N)).$

As shown in the proof of Proposition 3.6, under the Shapiro isomorphism

$H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})\cong
H^{1}(K,\Lambda_{\mathcal{O}}(\bm{\kappa}_{\text{ac}}^{-1})\hat{\otimes}_{\mathcal{O}}T^{\psi}_{g,h})\cong
H^{1}_{\rm Iw}(K_{\infty},T^{\psi}_{g,h}),$

the Selmer group $\mathrm{Sel}_{\rm
bal}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}gh}^{\dagger})$ of
$\S\ref{subsec:IMC}$ correspond to the Selmer group $\mathrm{Sel}_{\rm
Gr}(K_{\infty},T^{\psi}_{g,h})$ of $\S\ref{sec:ES}$ attached to the local
conditions in (3.13) at the primes above $p$. Moreover, the class
$\kappa({\mathbf{f}},g,h)$ in Definition 4.5 corresponds to the image of
${\kappa_{\psi,g,h,\infty}}$ under a projection $H^{1}_{\rm
Iw}(K_{\infty},T_{g,h}^{\psi}(N))\rightarrow H^{1}_{\rm
Iw}(K_{\infty},T_{g,h}^{\psi})$.

The next result, establishing one of the divisibilities predicted by the
Iwasawa main conjectures from $\S\ref{subsec:IMC}$, recovers Theorem C in the
Introduction.

######  Theorem 6.9.

Assume hypotheses (h1)–(h8), and in addition that:

  * •

$\varepsilon_{\ell}(\theta_{\psi},g,h)=+1$ for all primes $\ell\mid N$,

  * •

${\rm gcd}(N_{\psi},N_{g},N_{h})$ is squarefree.

If $\kappa({\mathbf{f}},g,h)$ is not $\Lambda_{{\mathbf{f}}}$-torsion, then
the following hold:

  1. (a)

The modules
$\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$
and $X_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})$ have
both $\Lambda_{{\mathbf{f}}}$-rank one, and

$\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}(X_{\operatorname{bal}}(\mathbb{A}_{{\mathbf{f}}gh}^{{\dagger}})_{\operatorname{tors}})\supset\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}\left(\frac{\mathrm{Sel}_{\operatorname{bal}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})}{\Lambda_{{\mathbf{f}}}\cdot\kappa({\mathbf{f}},g,h)}\right)^{2}.$

  2. (b)

The modules
$\mathrm{Sel}_{\mathcal{F}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$ and
$X_{\mathcal{F}}(\mathbb{V}_{{\mathbf{f}}gh}^{{\dagger}})$ are both
$\Lambda_{{\mathbf{f}}}$-torsion, and

$\operatorname{Char}_{\Lambda_{{\mathbf{f}}}}(X_{\mathcal{F}}(\mathbb{A}_{{\mathbf{f}}}^{{\dagger}}))\supset(L_{p}({\mathbf{f}},g,h))$

in $\Lambda_{{\mathbf{f}}}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$.

###### Proof.

The non-triviality assumption on $\kappa({\mathbf{f}},g,h)$ implies that
${\kappa_{\psi,g,h,\infty}}$ is not $\Lambda_{{\mathbf{f}}}$-torsion. Since by
Theorem 3.5 the class ${\kappa_{\psi,g,h,\infty}}$ is the bottom class of a
$\Lambda$-adic Euler system for $V_{g,h}^{\psi}$, part (b) follows from
Theorem 5.5 and Proposition 5.10. By Theorem 4.16, part (b) of the theorem
follows from part (a), so this concludes the proof. ∎

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