and

$\bar{\xi}_{\chi}^{\varepsilon}(\ell)=\bar{\xi}_{\ell}^{\varepsilon}(g,\ell)\in{}H^{1}(K,T_{\bar{g}})\otimes_{\mathbf{F}_{p}}\mathbf{F}_{\chi}$

as the image of $\hat{\kappa}_{\chi}^{\varepsilon}(\ell)$ under the map
induced in cohomology by the map $\varpi_{\chi}$:
$T_{g}(\chi)\twoheadrightarrow{}T_{\bar{g}}\otimes_{\mathbf{F}_{p}}\mathbf{F}_{\chi}$,
where $\bar{g}\in{}S_{2}(N^{+},LN^{-};\mathbf{F}_{p})$ is the reduction of $g$
modulo $p$.

######  Lemma 7.2 (_cf. Lemma 4.5 of_ [3]).

  1. (1)

$0\not=\xi_{\chi}^{\varepsilon}(\ell)\in{}\mathfrak{Sel}_{\ell}^{\varepsilon}(K,T_{g}(\chi))$
and $v_{\ell}\big{(}\xi_{\chi}^{\varepsilon}(\ell)\big{)}=0$.

  2. (2)

$\mathrm{ord}_{\chi}\big{(}\partial_{\ell}(\xi_{\chi}^{\varepsilon}(\ell))\big{)}=t_{\chi}^{\varepsilon}(g)-t_{\chi}^{\varepsilon}(g,\ell).$

  3. (3)

$0\not=\bar{\xi}_{\chi}^{\varepsilon}(\ell)\in{}\mathfrak{Sel}_{\ell}^{\varepsilon}(K,T_{\bar{g}})\otimes_{\mathbf{F}_{p}}\mathbf{F}_{\chi}$
and $\partial_{\ell}(\bar{\xi}_{\chi}^{\varepsilon}(\ell))$ is non-zero if and
only if $t_{\chi}^{\varepsilon}(g,\ell)=t_{\chi}^{\varepsilon}(g)$.

###### Proof.

(1) Because the kernel of
$\mathfrak{Sel}_{S}^{\varepsilon}(K,\mathbf{T}_{\check{g}})\stackrel{{\scriptstyle}}{{\rightarrow}}\mathfrak{Sel}_{S}^{\varepsilon}(K,\mathbf{T}_{g})$
is killed by $p^{k}$ and
$\mathrm{ord}_{\chi}(\tilde{\kappa}_{\chi}^{\varepsilon}(\ell))=0$ by Equation
$(\ref{eq:divcla})$, the class $\hat{\kappa}_{\chi}^{\varepsilon}(\ell)$ is
not zero, hence so is its image $\xi_{\chi}^{\varepsilon}(\ell)$ under the map
$\texttt{s}_{\chi}$, which is injective by Proposition 5.9. Let $q$ be a prime
divisor of $S/\ell$. To prove the first statement one has to show that the
residue

$\partial_{q}(\xi_{\chi}^{\varepsilon}(\ell))\in{}H^{1}_{\mathrm{sing}}(K_{q},T_{g,{\mathscr{O}_{\chi}}})\cong{}\mathscr{O}_{\chi}/p^{k}$

of $\xi_{\chi}^{\varepsilon}(\ell)$ at $q$ is zero. Fix isomorphisms
$H^{1}_{\mathrm{sing}}(K_{q},\mathbf{T}_{\check{g},{\mathscr{O}_{\chi}}})\cong{}\Lambda_{\mathscr{O}_{\chi}}/p^{2k}$
and
$H^{1}_{\mathrm{sing}}(K_{q},\mathbf{T}_{g,{\mathscr{O}_{\chi}}})\cong{}\Lambda_{\mathscr{O}_{\chi}}/p^{k}$
such that the map
$H^{1}_{\mathrm{sing}}(K_{q},\mathbf{T}_{\check{g},{\mathscr{O}_{\chi}}})\stackrel{{\scriptstyle}}{{\rightarrow}}H^{1}_{\mathrm{sing}}(K_{q},\mathbf{T}_{g,{\mathscr{O}_{\chi}}})$
becomes identified with the natural projection
$\Lambda_{\mathscr{O}_{\chi}}/p^{2k}\twoheadrightarrow{}\Lambda_{\mathscr{O}_{\chi}}/p^{k}$.
Since $\partial_{q}(\kappa_{\infty}^{\varepsilon}(\ell))$ is zero by
Proposition 6.1 and
$t_{\chi}^{\varepsilon}(g,\ell)<\mathrm{ord}_{\chi}(p^{k})$ by Equation
$(\ref{eq:frkol})$, it follows that
$\partial_{q}(\tilde{\kappa}_{\chi}^{\varepsilon}(\ell))\in{}\mathscr{O}_{\chi}/p^{2k}$
has $\mathscr{O}_{\chi}$-adic valuation at least $\mathrm{ord}_{\chi}(p^{k})$,
hence its projection
$\partial_{q}(\hat{\kappa}_{\chi}^{\varepsilon}(\ell))\in{}\mathscr{O}_{\chi}/p^{k}$
modulo $p^{k}$ vanishes (here and in the following we wrote $\partial_{q}$ for
the scalar extension $\partial_{q}\otimes\mathrm{id}$ to simplify the notation
as before). This gives

$\partial_{q}(\xi_{\chi}^{\varepsilon}(\ell))=\partial_{q}\circ{}\texttt{s}_{\chi}(\hat{\kappa}_{\chi}^{\varepsilon}(\ell))=\partial_{q}(\xi_{\chi}^{\varepsilon}(\ell))=0,$

as was to be shown. The second statement is proved similarly, using that
$v_{\ell}(\kappa_{\infty}^{\varepsilon}(\ell))=0$ by Proposition 6.1.

(2) Equations $(\ref{eq:frkol})$, $(\ref{eq:divcla})$ and $(\ref{eq:divcla2})$
show that $\partial_{\ell}(\tilde{\kappa}_{\chi}^{\varepsilon}(\ell))$ has
$\mathscr{O}_{\chi}$-adic valuation
$t_{\chi}^{\varepsilon}(g)-t_{\chi}^{\varepsilon}(g,\ell)$. Since
$\mathrm{ord}_{\chi}(p^{k})>t_{\chi}^{\varepsilon}(g)$ this is also the
$\mathscr{O}_{\chi}$-adic valuation of
$\partial_{\ell}(\hat{\kappa}_{\chi}^{\varepsilon}(\ell))$, which is equal to
that of $\partial_{\ell}(\xi_{\chi}^{\varepsilon}(\ell))$ (_cf._ the proof of
(1)).

(3) Note that the class $\bar{\xi}_{\chi}^{\varepsilon}(\ell)$ is equal to the
image of $\tilde{\kappa}_{\chi}^{\varepsilon}(\ell)$ under the composition

$\mathfrak{Sel}_{S}^{\varepsilon}(K,\mathbf{T}_{\check{g}})\otimes\mathscr{O}_{\chi}\stackrel{{\scriptstyle}}{{\longrightarrow}}\mathfrak{Sel}_{S}^{\varepsilon}(K,\mathbf{T}_{\bar{g}})\otimes\mathscr{O}_{\chi}\stackrel{{\scriptstyle\texttt{s}_{\chi}}}{{\longrightarrow}}\mathfrak{Sel}_{S}^{\varepsilon}(K,T_{\bar{g}}(\chi))\otimes_{\mathbf{F}_{p}}\mathbf{F}_{\chi}.$

As above this implies that $\bar{\xi}_{\chi}^{\varepsilon}(\ell)$ is not zero,
since $\texttt{s}_{\chi}$ is injective and
$\mathrm{ord}_{\chi}(\tilde{\kappa}_{\chi}^{\varepsilon}(\ell))=0$. Together
with (1) this implies the first statement. Since
$\partial_{\ell}\big{(}\bar{\xi}_{\chi}^{\varepsilon}(\ell)\big{)}\in{}H^{1}_{\mathrm{sing}}(K_{\ell},T_{\bar{g}}(\chi))\otimes_{\mathbf{F}_{p}}\mathbf{F}_{\chi}\cong{}\mathbf{F}_{\chi}$
is the projection of
$\partial_{\ell}(\xi_{\chi}^{\varepsilon}(\ell))\in{}\mathscr{O}_{\chi}/p^{k}$
modulo $\varpi_{\chi}$, the second statement follows from (2). ∎

###  7.2. Proof of Theorem 7.1

The proof of Theorem 7.1 is divided into several steps. Steps 1, 2 and 3
consist in a generalization to the present context of similar results of [3].
The direct generalizations of the techniques in (7.1) only allow to prove the
inequality
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\big{)}\leqslant{}2\mathrm{ord}_{\chi}\left(\mathcal{L}_{g}^{\varepsilon}(\bar{\chi})\right)$,
in Theorem 7.1 which also holds in the exceptional case. We can prove the
opposite inequality in the non-exceptional case with a further inductive
argument on the length of $\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))$,
developed in Steps 4, 5, 6 and 7. The key ingredient is Step 4 (the basis of
the inductive argument, _i.e._ the case when
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\big{)}=0$)
which combines Gross formula and Lemma 4.5 with results of Shinner-Urban
(ordinary case) and Wan (supersingular case); the inductive argument then
follows in Steps 6 and 7 using a a structure theorem for
$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))$, which we prove in Step 5.

####  7.2.1. Step 1

If $\mathcal{L}_{g}^{\varepsilon}(\bar{\chi})$ is a $p$-adic unit then
$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))$ is trivial.

###### Proof.

(Cf. [3, Proposition 4.7].) Assume _ad absurdum_ that there exists a
nontrivial class $x$ in the Selmer group
$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))$. Choose a $2k$-admissible prime
$\ell$ such that
$v_{\ell}(x)\in{}H^{1}_{\mathrm{fin}}(K_{\ell},A_{g}(\chi))\cong{}\mathscr{O}_{\chi}/p^{k}$
is not zero, which exists by Theorem 3.2 of [3]. Since
$\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi}))$ is the dual Selmer group
of $\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))$, Lemma $\ref{basickol}(1)$ and
the reciprocity law of global class field theory yield

$0=\sum_{v}\left<{}\mathrm{res}_{v}(x),\mathrm{res}_{v}(\xi_{\bar{\chi}}^{\varepsilon}(\ell))\right>_{v}=\left<{}\mathrm{res}_{\ell}(x),\mathrm{res}_{\ell}(\xi_{\bar{\chi}}^{\varepsilon}(\ell))\right>_{\ell},$

where the sum is taken over all the primes of $K$ and $\left<{}-,-\right>_{v}$
denotes the local Tate pairing at $v$ induced by the duality
$T_{g}(\chi)\times{}A_{g}(\chi)\stackrel{{\scriptstyle}}{{\rightarrow}}\mathscr{O}_{\chi}/p^{k}(1)$
(cf. [21, Chapter 1]). Since $\partial_{\ell}(x)=0$, $v_{\ell}(x)\not=0$ and
$H^{1}_{\mathrm{fin}}(K_{\ell},T_{g}(\bar{\chi}))$ is the orthogonal
complement of $H^{1}_{\mathrm{fin}}(K_{\ell},A_{g}(\chi))$ under the perfect
pairing $\left<{}-,-\right>_{\ell}$, the previous equation implies that the
residue at $\ell$ of $\xi_{\bar{\chi}}^{\varepsilon}(\ell)$ has positive
$\mathscr{O}_{\chi}$-adic valuation. According to Lemma $\ref{basickol}(2)$
this in turn implies that $\mathcal{L}_{g}^{\varepsilon}(\bar{\chi})$ has
positive $\mathscr{O}_{\chi}$-adic valuation, contradicting the assumption. ∎

####  7.2.2. Step 2.

Assume that $\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))$ is non-trivial. Then
there exist two distinct $2k$-admissible primes $\ell_{1}$ and $\ell_{2}$
satisfying the following properties.

  * $\mathbf{I}_{1}$.

$t_{\bar{\chi}}^{\varepsilon}(g,\ell_{1})=t_{\bar{\chi}}^{\varepsilon}(g,\ell_{2})<t_{\bar{\chi}}^{\varepsilon}(g)$.

  * $\mathbf{I}_{2}.$

If $h\in{}S_{2}(N^{+},L\ell_{1}\ell_{2}N^{-};\mathbf{Z}/p^{k})$ denotes the
$\ell_{1}\ell_{2}$-level raising of $g$, then

$\mathrm{Sel}_{\varepsilon}(K,A_{h}(\chi))=\mathrm{Sel}_{\varepsilon}^{\ell_{1}\ell_{2}}(K,A_{g}(\chi)).$

  * $\mathbf{I}_{3}.$

The $\varpi_{\chi}$-adic valuation of
$\mathcal{L}_{h}^{\varepsilon}(\bar{\chi})\in\mathscr{O}_{\chi}/p^{k}$ is
equal to $t_{\bar{\chi}}^{\varepsilon}(g,\ell_{i})$ (for $i=1,2$):

$t_{\bar{\chi}}^{\varepsilon}(h)=\mathrm{ord}_{\chi}\big{(}\mathcal{L}_{h}^{\varepsilon}(\bar{\chi})\big{)}=t_{\bar{\chi}}^{\varepsilon}(g,\ell_{i})<\infty.$

  * $\mathbf{I}_{4}.$

$\mathrm{ord}_{\chi}\big{(}v_{\ell_{1}}\big{(}\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})\big{)}\big{)}=0$
and
$\mathrm{ord}_{\chi}\big{(}v_{\ell_{2}}\big{(}\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1})\big{)}\big{)}=0$.

###### Proof.

We first prove that there exist infinitely many $2k$-admissible primes $\ell$
such that $t_{\bar{\chi}}(g,\ell)<t_{\bar{\chi}}(g).$ Let
$\mathfrak{m}_{\Lambda_{\mathscr{O}_{\chi}}}$ be as above the maximal ideal of
$\Lambda_{\mathscr{O}_{\chi}}$, so that
$A_{g}(\chi)[\mathfrak{m}_{\Lambda_{\mathscr{O}_{\chi}}}]\cong{}A_{\bar{g}}\otimes_{\mathbf{F}_{p}}\mathbf{F}_{\chi}$
(as $\chi(g)\equiv{}1\pmod{\varpi_{\chi}}$ for every $g\in{}G_{\infty}$). The
control theorem Proposition 5.8 yields

$\mathrm{Sel}_{\varepsilon}(K,A_{\bar{g},{\mathscr{O}_{\chi}}})\cong{}\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))[\mathfrak{m}_{{\Lambda}_{\mathscr{O}_{\chi}}}],$

hence $\mathrm{Sel}_{\varepsilon}(K,A_{\bar{g},{\mathscr{O}_{\chi}}})$ is
nontrivial by Nakayama’s Lemma. Fix a non-zero class

$0\not=x\in{}\mathrm{Sel}_{\varepsilon}(K,A_{\bar{g},{\mathscr{O}_{\chi}}}).$

According to (a slight generalization of) Theorem 3.2 of [3] there exist
infinitely many $2k$-admissible primes $\ell$ such that
$v_{\ell}(x)\in{}H^{1}_{\mathrm{fin}}(K_{\ell},A_{\bar{g},{\mathscr{O}_{\chi}}})$
is non zero. We claim that for every such prime $\ell$ one has

(7.6) $t_{\bar{\chi}}^{\varepsilon}(g,\ell)<t_{\bar{\chi}}^{\varepsilon}(g).$

Recall the class
$\bar{\xi}_{\bar{\chi}}^{\varepsilon}(\ell)\in{}H^{1}(K,T_{\bar{g}})\otimes_{\mathbf{F}_{p}}\mathbf{F}_{\chi}$
constructed in Section 7.1. Lemma $\ref{basickol}(3)$ shows that
$\bar{\xi}_{\bar{\chi}}^{\varepsilon}(\ell)$ belongs to
$\mathfrak{Sel}_{\ell}^{\varepsilon}(K,T_{\bar{g}})\otimes_{\mathbf{F}_{p}}\mathbf{F}_{\chi}$,
hence (as in the proof of Step 1) the reciprocity law of global class field
theory yields

$\left<{}\partial_{\ell}\big{(}\bar{\xi}_{\bar{\chi}}^{\varepsilon}(\ell)\big{)},v_{\ell}(x)\right>_{\ell}=0,$

where $\left<{}-,-\right>_{\ell}$ is the $\mathbf{F}_{\chi}$-linear extension
of the perfect local Tate pairing

$H^{1}_{\mathrm{sing}}(K_{\ell},T_{\bar{g}})\otimes_{\mathbf{F}_{p}}H^{1}_{\mathrm{fin}}(K_{\ell},A_{\bar{g}})\longrightarrow\mathbf{F}_{p}.$

Since $v_{\ell}(x)\not=0$ this gives
$\partial_{\ell}\big{(}\bar{\xi}_{\bar{\chi}}^{\varepsilon}(\ell)\big{)}=0$,
and the claim $(\ref{eq:nmn1})$ follows from another application of Lemma
$\ref{basickol}(3)$.

Fix a $2k$-admissible prime $\ell_{1}$ such that
$t_{\bar{\chi}}^{\varepsilon}(g,\ell_{1})<t_{\bar{\chi}}^{\varepsilon}(g)$,
and such that
$t_{\bar{\chi}}^{\varepsilon}(g,\ell_{1})\leqslant{}t_{\bar{\chi}}^{\varepsilon}(g,\ell)$
for every $2k$-admissible prime $\ell$. Since
$\bar{\xi}^{\varepsilon}_{\bar{\chi}}(\ell_{1})$ is non-zero by Lemma
$\ref{basickol}(3)$, Theorem 3.2 of [3] proves that there exists a
$2k$-admissible prime $\ell_{2}\not=\ell_{1}$ such that
$v_{\ell_{2}}\big{(}\bar{\xi}_{\bar{\chi}}^{\varepsilon}(\ell_{1})\big{)}\in{}H^{1}_{\mathrm{fin}}(K_{\ell_{2}},T_{\bar{g}})\otimes_{\mathbf{F}_{p}}\mathbf{F}_{\chi}\cong{}\mathbf{F}_{\chi}$
is non-zero. By construction (cf. Section 7.1) the latter condition is
equivalent to

$\mathrm{ord}_{\chi}\big{(}v_{\ell_{2}}(\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1}))\big{)}=0.$

The second reciprocity law Theorem 6.3 and the definition of
$\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1})$ show that the identities (where we
write $v_{\ell}$ for $v_{\ell}\otimes{}\mathrm{id}$ for $\ell=\ell_{1}$ and
$\ell=\ell_{2}$ as before)

(7.7)
$\varpi_{\chi}^{t_{\bar{\chi}}^{\varepsilon}(g,\ell_{1})}\cdot{}v_{\ell_{2}}\big{(}\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1})\big{)}=v_{\ell_{2}}\big{(}\kappa_{\bar{\chi}}^{\varepsilon}(\ell_{1})\big{)}\stackrel{{\scriptstyle\text{Th.}\
\ref{secondrec}}}{{=}}\mathcal{L}_{h}^{\varepsilon}(\bar{\chi})\stackrel{{\scriptstyle\text{Th.}\
\ref{secondrec}}}{{=}}v_{\ell_{1}}\big{(}\kappa_{\bar{\chi}}^{\varepsilon}(\ell_{2})\big{)}=\varpi_{\chi}^{t_{\bar{\chi}}^{\varepsilon}(g,\ell_{2})}\cdot{}v_{\ell_{1}}\big{(}\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})\big{)}$

hold in $\mathscr{O}_{\chi}/p^{k}$ up to multiplication by $p$-adic units
(_cf._ the proof of Lemma $\ref{basickol}(1)$ for the first and last
identities). Since
$t_{\bar{\chi}}^{\varepsilon}(g,\ell)<\mathrm{ord}_{\chi}(p^{k})$ for
$\ell=\ell_{1},\ell_{2}$ by Equation $(\ref{eq:frkol})$, and since by
construction
$t_{\bar{\chi}}^{\varepsilon}(g,\ell_{1})\leqslant{}t_{\bar{\chi}}^{\varepsilon}(g,\ell_{2})$,
the previous two equations and Lemma $\ref{basickol}(1)$ show that

(7.8)
$t_{\bar{\chi}}^{\varepsilon}(g,\ell_{1})=t_{\bar{\chi}}^{\varepsilon}(g,\ell_{2})<t_{\bar{\chi}}^{\varepsilon}(g)$

and that the identities

(7.9)
$\left(v_{\ell_{1}}\big{(}\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})\big{)},v_{\ell_{2}}\big{(}\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})\big{)}\right)=(1,0),$
(7.10)
$\left(v_{\ell_{1}}\big{(}\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1})\big{)},v_{\ell_{2}}\big{(}\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1})\big{)}\right)=(0,1)$

hold in $\mathscr{O}_{\chi}/p^{k}\oplus{}\mathscr{O}_{\chi}/p^{k}$ up to
multiplication by $p$-adic units. (Here for $\ell=\ell_{1}$ or $\ell=\ell_{2}$
one fixes an isomorphism
$H^{1}_{\mathrm{fin}}(K_{\ell},T_{g}(\bar{\chi}))\cong{}\mathscr{O}_{\chi}/p^{k}$.)
It follows from the definitions (cf. Section 5.2) that

$\mathrm{Sel}^{\ell_{1}\ell_{2}}_{\varepsilon}(K,A_{g}(\chi))=\mathrm{Sel}^{\ell_{1}\ell_{2}}_{\varepsilon}(K,A_{h}(\chi)),$
$\mathfrak{Sel}_{\ell_{1}\ell_{2}}^{\varepsilon}(K,T_{g}(\bar{\chi}))=\mathfrak{Sel}_{\ell_{1}\ell_{2}}^{\varepsilon}(K,T_{h}(\bar{\chi})),$

and a class
$z\in{}\mathfrak{Sel}^{\varepsilon}_{\ell_{1}\ell_{2}}(K,T_{g}(\bar{\chi}))$
belongs to $\mathfrak{Sel}^{\varepsilon}(K,T_{h}(\bar{\chi}))$ precisely if
$v_{\ell_{1}}(z)$ and $v_{\ell_{2}}(z)$ are both trivial. Poitou–Tate duality
(see Theorem 7.3 of [28] or Chapter 1 of [21]) then yields a short exact
sequence of $\mathscr{O}_{\chi}/p^{k}$-modules

(7.11)
$\mathfrak{Sel}_{\ell_{1}\ell_{2}}^{\varepsilon}(K,A_{g}(\bar{\chi}))\stackrel{{\scriptstyle
v_{\ell_{1}}\oplus{}v_{\ell_{2}}}}{{\longrightarrow}}\mathscr{O}_{\chi}/p^{k}\oplus\mathscr{O}_{\chi}/p^{k}\stackrel{{\scriptstyle\partial_{\ell_{1}}^{\vee}\oplus{}\partial_{\ell_{2}}^{\vee}}}{{\longrightarrow}}\mathrm{Sel}_{\varepsilon}(K,A_{h}(\bar{\chi}))^{\vee}\stackrel{{\scriptstyle}}{{\longrightarrow}}\mathrm{Sel}_{\varepsilon}^{\ell_{1}\ell_{2}}(K,A_{g}(\chi))^{\vee}\stackrel{{\scriptstyle}}{{\longrightarrow}}0,$

where
$(\cdot{})^{\vee}=\mathrm{Hom}_{\mathbf{Z}_{p}}(\cdot{},\mathbf{Q}_{p}/\mathbf{Z}_{p})$
and for $\ell=\ell_{1},\ell_{2}$ one identifies
$H^{1}_{\mathrm{sing}}(K_{\ell},A_{g}(\chi))$ with the Pontrjagin dual of
$H^{1}_{\mathrm{fin}}(K_{\ell},T_{g}(\bar{\chi}))\cong{}\mathscr{O}_{\chi}/p^{k}$
under the local Tate duality. Equation $(\ref{eq:finalStep3})$ shows that the
first map is surjective, hence

$\mathrm{Sel}_{\varepsilon}(K,A_{h}(\chi))=\mathrm{Sel}_{\varepsilon}^{\ell_{1}\ell_{2}}(K,A_{g}(\chi)).$

Together with Equations $(\ref{eq:semifinalStep3})$–$(\ref{eq:finalStep3})$
this concludes the proof. ∎

####  7.2.3. Step 3.

$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\big{)}\leqslant{}2t_{\bar{\chi}}^{\varepsilon}(g)$.

###### Proof.

As in [3] one proceeds by induction on $t_{\bar{\chi}}(g)$. Step 1 shows that
the statement holds if $t_{\bar{\chi}}(g)=0$. Assume then
$t_{\bar{\chi}}(g)>0$. If $\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))=0$ the
statement is trivially verified, hence assume that
$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))$ is non-trivial. According to Step
2 there exists two distinct $2k$-admissible primes $\ell_{1}$ and $\ell_{2}$
satisfying the properties $\mathbf{I}_{1}$–$\mathbf{I}_{3}$. As in loc. cit.
denote by $h\in{}S_{2}(N^{+},L\ell_{1}\ell_{2}L;\mathbf{Z}/p^{k})$ the
$\ell_{1}\ell_{2}$-level raising of $g$.

Let
$\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{1})\in{}\mathfrak{Sel}_{\ell_{1}}^{\varepsilon}(K,T_{g}(\bar{\chi}))$
be a global class such that
$\partial_{\ell_{1}}\big{(}\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{1})\big{)}$
generates the image of the residue map
$\partial_{\ell_{1}}:\mathfrak{Sel}_{\ell_{1}}^{\varepsilon}(K,T_{g}(\bar{\chi}))\stackrel{{\scriptstyle}}{{\rightarrow}}H^{1}_{\mathrm{sing}}(K_{\ell},T_{g}(\bar{\chi}))\cong{}\mathscr{O}_{\chi}/p^{k}$,
viz. $\partial_{\ell_{1}}$ induces an isomorphism

(7.12)
$\partial_{\ell_{1}}:\mathfrak{Sel}_{\ell_{1}}^{\varepsilon}(K,T_{g}(\bar{\chi}))\big{/}\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi}))\cong{}\partial_{\ell_{1}}\big{(}\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{1})\big{)}\cdot{}\mathscr{O}_{\chi}/p^{k}.$

Since $\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1})$ belongs to the Selmer group
$\mathfrak{Sel}_{\ell_{1}}^{\varepsilon}(K,T_{g}(\bar{\chi}))$ by Lemma
$\ref{basickol}(1)$, multiplying $\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{1})$
by a $p$-adic unit if necessary one can assume that there exists an integer
$m_{1}\geqslant{}0$ such that

$\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1})-\varpi_{\chi}^{m_{1}}\cdot{}\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{1})\in{}\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi})).$

Equation $(\ref{eq:step21})$, Lemma $\ref{basickol}(2)$ and property
$\mathbf{I}_{3}$ then yield

(7.13)
$\displaystyle\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathfrak{Sel}_{\ell_{1}}^{\varepsilon}(K,T_{g}(\bar{\chi}))\big{/}\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi}))\big{)}=\mathrm{ord}_{\chi}(p^{k})-t_{\bar{\chi}}^{\varepsilon}(g)+t_{\bar{\chi}}^{\varepsilon}(h)+m_{1}.$

Similarly let
$\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{2})\in{}\mathfrak{Sel}_{\ell_{1}\ell_{2}}^{\varepsilon}(K,T_{g}(\bar{\chi}))$
be a class such that the residue map at $\ell_{2}$ induces an isomorphism

$\partial_{\ell_{2}}:\mathfrak{Sel}_{\ell_{1}\ell_{2}}^{\varepsilon}(K,T_{g}(\bar{\chi}))\big{/}\mathfrak{Sel}^{\varepsilon}_{\ell_{1}}(K,T_{g}(\bar{\chi}))\cong{}\partial_{\ell_{2}}\big{(}\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{2})\big{)}\cdot{}\mathscr{O}_{\chi}/p^{k}.$

Because
$\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})\in{}\mathfrak{Sel}^{\varepsilon}_{\ell_{1}\ell_{2}}(K,T_{g}(\bar{\chi}))$
by Lemma $\ref{basickol}(1)$, one can assume that there exists
$m_{2}\geqslant{}0$ such that

$\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})-\varpi_{\chi}^{m_{2}}\cdot{}\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{2})\in{}\mathfrak{Sel}^{\varepsilon}_{\ell_{1}}(K,T_{g}(\bar{\chi})),$

and apply as above Lemma $\ref{basickol}(2)$ and property $\mathbf{I}_{3}$ to
deduce the equality

(7.14)
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathfrak{Sel}_{\ell_{1}\ell_{2}}^{\varepsilon}(K,T_{g}(\bar{\chi}))\big{/}\mathfrak{Sel}^{\varepsilon}_{\ell_{1}}(K,T_{g}(\bar{\chi}))\big{)}=\mathrm{ord}_{\chi}(p^{k})-t_{\bar{\chi}}^{\varepsilon}(g)+t_{\bar{\chi}}^{\varepsilon}(h)+m_{2}.$

When combined together Equations $(\ref{eq:step22})$ and $(\ref{eq:step23})$
give the equality

(7.15)
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathfrak{Sel}_{\ell_{1}\ell_{2}}^{\varepsilon}(K,T_{g}(\bar{\chi}))\big{/}\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi}))\big{)}=2\cdot{}\mathrm{ord}_{\chi}(p^{k})-2\cdot{}t_{\bar{\chi}}^{\varepsilon}(g)+2\cdot{}t_{\bar{\chi}}^{\varepsilon}(h)+m_{1}+m_{2}.$

By construction $\mathrm{Sel}_{\varepsilon}^{\cdot}(K,A_{g}(\chi))$ is the
dual Selmer group of
$\mathfrak{Sel}_{\cdot}^{\varepsilon}(K,T_{g}(\bar{\chi}))$, hence Poitou–Tate
duality gives a short exact sequence of $\mathscr{O}_{\chi}/p^{k}$-modules
(cf. Equation $(\ref{eq:PTStep3})$ in the proof of Step 2)

$0\stackrel{{\scriptstyle}}{{\longrightarrow}}\frac{\mathfrak{Sel}^{\varepsilon}_{\ell_{1}\ell_{2}}(K,T_{g}(\bar{\chi}))}{\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi}))}\stackrel{{\scriptstyle\partial_{\ell_{1}}\oplus{}\partial_{\ell_{2}}}}{{\longrightarrow}}\mathscr{O}_{\chi}/p^{k}\oplus{}\mathscr{O}_{\chi}/p^{k}\stackrel{{\scriptstyle
v_{\ell_{1}}^{\vee}\oplus{}v_{\ell_{2}}^{\vee}}}{{\longrightarrow}}\left(\frac{\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))}{\mathrm{Sel}_{\varepsilon}^{\ell_{1}\ell_{2}}(K,A_{g}(\chi))}\right)^{\vee}\stackrel{{\scriptstyle}}{{\longrightarrow}}0,$

where for $\ell=\ell_{1},\ell_{2}$ one identifies
$H^{1}_{\mathrm{sing}}(K_{\ell},T_{g}(\bar{\chi}))\cong{}H^{1}_{\mathrm{fin}}(K_{\ell},A_{g}(\chi))^{\vee}$
with $\mathscr{O}_{\chi}/p^{k}$ under a fixed isomorphism. Together with
Equation $(\ref{eq:step24})$ and property $\mathbf{I}_{2}$ this implies

(7.16)
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\big{)}-2\cdot{}t_{\bar{\chi}}^{\varepsilon}(g)=\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{h}(\chi))\big{)}-2\cdot{}t_{\bar{\chi}}^{\varepsilon}(h)-m_{1}-m_{2}.$

Properties $\mathbf{I}_{1}$ and $\mathbf{I}_{3}$ give
$t_{\bar{\chi}}^{\varepsilon}(h)<t_{\bar{\chi}}^{\varepsilon}(g)$, hence

(7.17)
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{h}(\chi))\big{)}-2\cdot{}t_{\bar{\chi}}^{\varepsilon}(h)\leqslant{}0$

by the induction hypothesis. The statement follows from Equations
$(\ref{eq:step45})$ and $(\ref{eq:shshsh})$. ∎

####  7.2.4. Step 4

Assume that $(f,K,p,\varepsilon)$ is not exceptional and that
$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))=0$. Then
$t_{\bar{\chi}}^{\varepsilon}(g)=0$.

###### Proof.

Theorem B of [11] implies that there exists a newform
$\xi=\sum_{n=1}^{\infty}a_{n}(\xi)\cdot{}q^{n}$ in
$S_{2}(\Gamma_{0}(NL))^{\mathrm{new}}$ which is congruent to $f$ modulo $p$.
More precisely, if $\mathbf{Q}(\xi)$ denotes the field generated over
$\mathbf{Q}$ by the Fourier coefficients of $\xi$, then there exists a prime
$\bar{\mathfrak{P}}$ of $\bar{\mathbf{Q}}$ dividing $p$ such that
$a_{l}(\xi)\equiv{}a_{l}(E)\pmod{\bar{\mathfrak{P}}}$ for every rational prime
$l\nmid{}NLp$. (_Loc. cit._ proves the existence of an eigenform
$\xi\in{}S_{2}(\Gamma_{1}(N)\cap{}\Gamma_{0}(L))$ of conductor divisible by
$L$ which is congruent to $f$ modulo $p$. It is not difficult to prove that an
eigenform with these properties has trivial character and conductor $NL$.) Let
$J_{\xi}^{o}/\mathbf{Q}$ be the quotient of
$\mathrm{Pic}^{0}(X_{0}(NL)/\mathbf{Q})$ associated to $\xi$ by the
Eichler–Shimura construction. It is an abelian variety of dimension
$[\mathbf{Q}(\xi):\mathbf{Q}]$ equipped with a morphism of
$\mathbf{Q}$-algebras
$\mathbf{Q}(\xi)\stackrel{{\scriptstyle}}{{\rightarrow}}\mathrm{End}_{\mathbf{Q}}(J_{\xi}^{o})\otimes_{\mathbf{Z}}\mathbf{Q}$.
Let $J_{\xi}/\mathbf{Q}$ be an abelian variety in the isogeny class of
$J_{\xi}^{o}$ which has real multiplication by the ring of integers
$\mathcal{O}$ of $\mathbf{Q}(\xi)$ and set
$\mathfrak{P}=\bar{\mathfrak{P}}\cap{}\mathcal{O}$. Since $E_{p}$ is an
irreducible $\mathbf{F}_{p}[G_{\mathbf{Q}}]$-module by Hypothesis
$\ref{thehy}(1)$, the Eichler–Shimura relations and the Brauer–Nesbitt theorem
imply that there are isomorphisms of
$\mathcal{O}/\mathfrak{P}[G_{\mathbf{Q}}]$-modules

$J_{\xi}[\mathfrak{P}]\cong{}E_{p}\otimes_{\mathbf{F}_{p}}\mathcal{O}/\mathfrak{P}\cong{}A_{\bar{g}}\otimes_{\mathbf{F}_{p}}\mathcal{O}/\mathfrak{P}.$

Identify in what follows $J_{\xi}[\mathfrak{P}]$ and
$A_{\bar{g}}\otimes_{\mathbf{F}_{p}}\mathcal{O}/\mathfrak{P}$ under a fixed
isomorphism, and let

$\mathrm{Sel}_{\mathfrak{P}}(J_{\xi}/K)\subset{}H^{1}(K,A_{\bar{g}})$

be the $\mathfrak{P}$-Selmer group of $J_{\xi}$ over $K$ (cf. [13]). It
follows from the results of [13, Sections 3–5] that

(7.18)
$\mathrm{Sel}_{\mathfrak{P}}(J_{\xi}/K)=\mathrm{Sel}(K,A_{\bar{g}})\otimes_{\mathbf{F}_{p}}\mathcal{O}/\mathfrak{P}$

inside $H^{1}(K,A_{\bar{g}})\otimes_{\mathbf{F}_{p}}\mathcal{O}/\mathfrak{P}$,
where $\mathrm{Sel}(K,A_{\bar{g}})$ is the Selmer group defined by imposing
the finite local condition
$H^{1}_{\mathrm{fin}}(K_{\mathfrak{p}},A_{\bar{g}})\cong{}E(K_{\mathfrak{p}})\otimes\mathbf{F}_{p}$
at every prime $\mathfrak{p}$ of $K$ dividing $p$ (viz.
$\mathrm{Sel}(K,A_{\bar{g}})=\mathrm{Sel}_{\emptyset}(K,A_{\bar{g}})$ with the
notations of Section 5.2, independently of whether $E$ has ordinary or
supersingular reduction at $p$). Note that since $(f,K,p,\varepsilon)$ is not
exceptional, then by definition
$E(K_{\mathfrak{p}})_{\varepsilon}=E(K_{\mathfrak{p}})$, hence

$H^{1}_{\mathrm{fin},\varepsilon}(K_{\mathfrak{p}},A_{\bar{g}})=H^{1}_{\mathrm{fin},\varepsilon}(K_{\mathfrak{p}},\mathbf{A}_{f})[\mathfrak{m}_{\Lambda}]=E(K_{\mathfrak{p}})_{\varepsilon}\otimes\mathbf{F}_{p}=E(K_{\mathfrak{p}})\otimes\mathbf{F}_{p}=H^{1}_{\mathrm{fin}}(K_{\mathfrak{p}},A_{\bar{g}})$

where the first equality follows from Corollary 5.5(2) and the second from
Proposition 5.3. It follows that
$\mathrm{Sel}_{\varepsilon}(K,A_{\bar{g}})=\mathrm{Sel}(K,A_{\bar{g}})$. We
have an isomorphism
$H^{1}(K,A_{\bar{g}})\otimes_{\mathbf{Z}_{p}}\mathscr{O}_{\chi}\cong
H^{1}(K,A_{\bar{g},{\mathscr{O}_{\chi}}})$ and an injection
$\mathrm{Sel}(K,A_{\bar{g}})\otimes_{\mathbf{Z}_{p}}\mathscr{O}_{\chi}\hookrightarrow
H^{1}(K,A_{\bar{g}})\otimes_{\mathbf{Z}_{p}}\mathscr{O}_{\chi}$ by the
flatness of $\mathscr{O}_{\chi}/\mathbf{Z}_{p}$, and therefore
$\mathrm{Sel}(K,A_{\bar{g}})\otimes_{\mathbf{Z}_{p}}\mathscr{O}_{\chi}$
injects into $\mathrm{Sel}(K,A_{\bar{g}}(\chi))$. Since by assumption
$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))$ is trivial, it follows from the
Control Theorem 5.8 and the irreducibility of $\bar{\varrho}_{E,p}$ that the
same is true for $\mathrm{Sel}(K,A_{\bar{g}})$ and Equation (7.18) gives

(7.19) $\mathrm{Sel}_{\mathfrak{P}}(J_{\xi}/K)=0.$

Let $L(\xi/K,1)_{\mathrm{alg}}$ denote the algebraic part of the special value
of the complex $L$-function of $\xi$ over $K$, normalized as in Section 4 of
[1]. Results of Skinner–Urban–Wan ([31] and Theorem B of [32] in the ordinary
case, [34] and [35] in the supersingular case) prove the inequality

$\mathrm{ord}_{\mathfrak{P}}\big{(}L(\xi/K,1)_{\mathrm{alg}}\big{)}\leqslant{}\mathrm{length}_{\mathcal{O}_{\mathfrak{P}}}\big{(}\mathrm{Sel}_{\mathfrak{P}^{\infty}}(J_{\xi}/K)\big{)}+\sum_{q|NL}t_{\xi}(q).$

Because $J_{\xi}[\mathfrak{P}]$ is an irreducible $G_{K}$-module,
$\mathrm{Sel}_{\mathfrak{P}}(J_{\xi}/K)$ is equal to the
$\mathfrak{P}$-torsion submodule of the Selmer group
$\mathrm{Sel}_{\mathfrak{P}^{\infty}}(J_{\xi}/K)$, so that
$\mathrm{Sel}_{\mathfrak{P}^{\infty}}(J_{\xi}/K)$ is trivial by Equation
$(\ref{eq:trivP})$. In addition $t_{\xi}(q)=0$ for every prime $q|N^{+}$ under
our assumptions, and the previous equation yields

$\mathrm{ord}_{\mathfrak{P}}\big{(}L(\xi/K,1)_{\mathrm{alg}}\big{)}\leqslant{}\sum_{q|LN^{-}}t_{\xi}(q).$

On the other hand Gross’ formula (see Theorem 4.2 of [1] for the formulation
in the form required in this paper) gives the identity

$\mathrm{ord}_{\mathfrak{P}}\big{(}L(\xi/K,1)_{\mathrm{alg}}\big{)}=2\cdot{}\mathrm{ord}_{\mathfrak{P}}\big{(}\psi_{\xi}\big{(}P_{K}(L)\big{)}\big{)}+\sum_{q|LN^{-}}t_{\xi}(q).$

It follows combining the two previous formulas that
$\psi_{\xi}\big{(}P_{K}(L)\big{)}$ has trivial $\mathfrak{P}$-adic valuation:

(7.20)
$\psi_{\xi}\big{(}P_{K}(L)\big{)}\in{}\mathcal{O}_{\mathfrak{P}}^{\ast}.$

Since $\xi$ is congruent to $f$ modulo $p$, one has
$\psi_{\xi}\big{(}P_{K}(L)\big{)}\equiv{}\psi_{\bar{g}}\big{(}P_{K}(L)\big{)}\pmod{\mathfrak{P}}$.
In addition, as $(f,K,p,\varepsilon)$ is not exceptional, Lemmas 4.1 and 4.5
show that the equalities
$\psi_{\bar{g}}\big{(}P_{K}(L)\big{)}=\mathcal{L}_{\bar{g}}^{\varepsilon}(\mathbf{1})=\mathcal{L}_{g}^{\varepsilon}(\mathbf{1})\pmod{p}$
hold in $\mathbf{F}_{p}$ up to multiplication by non-zero elements. Equation
$(\ref{eq:Stepmain4})$ then yields
$\mathcal{L}_{g}^{\varepsilon}(\mathbf{1})\in{}\mathbf{Z}_{p}^{\ast}.$ This
implies that the $p$-adic $L$-function $\mathcal{L}_{g}^{\varepsilon}$ is a
unit in $\Lambda/p^{k}$, which in turn gives
$t_{\bar{\chi}}^{\varepsilon}(g)=0$. ∎

####  7.2.5. Step 5

There exist a $\mathscr{O}_{\chi}/p^{k}$-module M and an integer
$s\in\\{0,1\\}$ such that

$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\cong{}(\mathscr{O}_{\chi}/p^{k})^{s}\oplus\mathtt{M}\oplus\mathtt{M}\cong{}\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi})).$

###### Proof.

To prove this result, we need to show that the Selmer groups
$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))$ and
$\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi}))$ are isomorphic and that
the structure result holds for one of them. To prove the structure results
$\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi}))\cong{}(\mathscr{O}_{\chi}/p^{k})^{s}\oplus\mathtt{M}\oplus\mathtt{M}$,
we use Theorem 1.4.2 of [17], for which we need to show that local conditions
defining $\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi}))$ are maximal
isotropic for the local Tate pairing. It turns out that proving this isotropic
condition is sufficient to also show the isomorphism
$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\simeq\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi}))$.

To begin with, fix an isomorphism of $\mathbf{Z}/p^{k}[G_{K}]$-modules
$T_{g}\cong{}E_{p^{k}}$, and let
$(\cdot{},\cdot{}):T_{g}\otimes_{\mathbf{Z}_{p}}T_{g}\stackrel{{\scriptstyle}}{{\rightarrow}}\mu_{p^{k}}$
be the perfect, skew-symmetric Kummer duality corresponding to the Weil
pairing on $E_{p^{k}}$. As
$\mathscr{O}_{\chi}\cong{}\mathrm{Hom}_{\mathbf{Z}_{p}}(\mathscr{O}_{\chi},\mathbf{Z}_{p})$
as $\mathscr{O}_{\chi}$-modules, $(\cdot{},\cdot{})$ induces an isomorphism of
$\mathscr{O}_{\chi}/p^{k}[G_{K}]$-modules

(7.21)
$A_{g}(\chi)=\mathrm{Hom}(T_{g}\otimes_{\mathbf{Z}_{p}}\mathscr{O}_{\chi}(\chi),\mu_{p^{k}})\cong{}\mathrm{Hom}(T_{g},\mu_{p^{k}})\otimes_{\mathbf{Z}_{p}}\mathscr{O}_{\chi}(\bar{\chi})\cong{}T_{g}\otimes_{\mathbf{Z}_{p}}\mathscr{O}_{\chi}(\bar{\chi})=T_{g}(\bar{\chi})$

under which one identifies $A_{g}(\chi)$ with $T_{g}(\bar{\chi})$ as
$\mathscr{O}_{\chi}/p^{k}[G_{K}]$-modules. We show that this isomorphism
induces an isomorphism

(7.22)
$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\cong{}\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi})).$

To check (7.22), it is enough to show that the local conditions defining the
two Selmer groups correspond to each other under the previous identification
of $A_{g}(\chi)$ and $T_{g}(\bar{\chi})$ as
$\mathscr{O}_{\chi}/p^{k}[G_{K}]$-modules. First, recall that given any finite
field extension $F$ of $\mathbf{Q}_{\ell}$, where $\ell$ is a prime, the Weil
pairing identifies the subgroup
$E(F)\otimes_{\mathbf{Z}}(\mathbf{Q}_{p}/\mathbf{Z}_{p})_{p^{k}}$ of
$H^{1}(F,A_{g})$ with the subgroup
$E(F)\otimes_{\mathbf{Z}}\mathbf{Z}/p^{k}\mathbf{Z}$ of $H^{1}(F,T_{g})$, and
these groups are the exact annihilators of each other under the local Tate
pairing. For rational primes $\ell\nmid Np$ and prime ideals $w\mid\ell$, it
is well known that $H^{1}_{\mathrm{fin}}(K_{w},\mathbf{A}_{g})$ is equal to
the image of
$E(K_{\infty,w})\otimes_{\mathbf{Z}}(\mathbf{Q}_{p}/\mathbf{Z}_{p})_{p^{k}}$,
where $K_{\infty,w}$ is the completion of $K_{\infty}$ above a prime dividing
$w$; for primes $\ell\mid N^{-}$, the same follows from the theory of
$\ell$-adic uniformisation, and finally for primes $\ell\mid N^{+}$ the local
cohomology vanishes. By definition
$H^{1}_{\mathrm{fin}}(K_{w},{T}_{g}(\bar{\chi})))$ is the orthogonal
complement of $H^{1}_{\mathrm{fin}}(K_{w},{A}_{g}(\chi))$ under the local Tate
pairing, hence the local conditions in $H^{1}(K_{w},{T}_{g}(\bar{\chi}))$
corresponding to $H^{1}_{\mathrm{fin}}(K_{w},A_{g}(\chi))$ under the
isomorphism $A_{g}(\chi)\cong T_{g}(\bar{\chi})$ coincide with
$H^{1}_{\mathrm{fin}}(K_{w},T_{g}(\bar{\chi}))$. Let now $\ell=p$ and $w\mid
p$ be a prime of $K$. Recall that
$H^{1}_{\mathrm{fin},\varepsilon}(K_{w},\mathbf{A}_{g})$ is a subgroup of
$E(K_{\infty,w})\otimes_{\mathbf{Z}}(\mathbf{Q}_{p}/\mathbf{Z}_{p})_{p^{k}}$
(where again $K_{\infty,w}$ is the completion of $K_{\infty}$ at the unique
prime above $w$). It follows that the isomorphism $A_{g}(\chi)\cong
T_{g}(\bar{\chi})$ takes
$H^{1}_{\mathrm{fin},\varepsilon}(K_{w},A_{g}({\chi}))$ to a subgroup of
$H^{1}(K_{w},T_{g}(\bar{\chi}))$ contained in the orthogonal complement of
$H^{1}_{\mathrm{fin},\varepsilon}(K_{w},A_{g}({\chi}))$; by definition, this
orthogonal complement is
$H^{1}_{\mathrm{fin},\varepsilon}(K_{w},T_{g}(\bar{\chi}))$. However,
$H^{1}_{\mathrm{fin},\varepsilon}(K_{w},A_{g}({\chi}))$ and
$H^{1}_{\mathrm{fin},\varepsilon}(K_{w},T_{g}(\bar{\chi}))$ have the same
cardinality by Proposition 5.4, and therefore the isomorphism
$A_{g}(\chi)\cong T_{g}(\bar{\chi})$ takes
$H^{1}_{\mathrm{fin},\varepsilon}(K_{\mathfrak{p}},A_{g}({\chi}))$ exactly to
$H^{1}_{\mathrm{fin},\varepsilon}(K_{\mathfrak{p}},T_{g}(\bar{\chi}))$. This
shows the isomorphism (7.22), and to conclude Step 5 it therefore enough to
prove that, for any character $\chi$, we have

$\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\chi))\cong{}(\mathscr{O}_{\chi}/p^{k})^{s}\oplus\mathtt{M}\oplus\mathtt{M}.$

This follows from Theorem 1.4.2 of [17] since, as noticed above, the the local
conditions defining $\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\chi))$ are maximal
isotropic for the local Tate pairing. ∎

####  7.2.6. Step 6

Assume that $(f,K,p,\varepsilon)$ is not exceptional and let $\ell_{1}$ and
$\ell_{2}$ be $2k$-admissible primes which satisfy the conditions
$\mathbf{I}_{1}$–$\mathbf{I}_{4}$ (cf. Step 2). Then (with the notations of
_loc. cit._)

(7.23)
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\big{)}-2\cdot{}t_{\bar{\chi}}^{\varepsilon}(g)=\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{h}(\chi))\big{)}-2\cdot{}t_{\bar{\chi}}^{\varepsilon}(h).$

###### Proof.

We first prove that the dimension of
$\mathrm{Sel}_{\varepsilon}(K,A_{\bar{g}})$ over $\mathbf{F}_{p}$ is even:

(7.24)
$\dim_{\mathbf{F}_{p}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{\bar{g}})\big{)}\equiv{}0\pmod{2}.$

Let $x\in{}\mathrm{Sel}_{\varepsilon}(K,A_{\bar{g}})$ be a nonzero class.
Choose an admissible prime $\ell$ such that

$v_{\ell}(x_{1})\in{}H^{1}_{\mathrm{fin}}(K_{\ell_{1}},A_{\bar{g}})\cong{}\mathbf{F}_{p}$

is non-zero, which exists by Theorem 3.2 of [3]. Let
$h\in{}S_{2}(N^{+},\ell{}LN^{-};\mathbf{F}_{p})$ be the $\ell$-level raising
of $\bar{g}$. Note that $\mathrm{Sel}_{\varepsilon}(K,A_{\bar{g}})$ is
identified with $\mathfrak{Sel}^{\varepsilon}(K,T_{\bar{g}})$ under the
isomorphism $T_{\bar{g}}\cong{}A_{\bar{g}}$ induced by the Weil pairing, viz.
$\mathrm{Sel}_{\varepsilon}(K,A_{\bar{g}})$ is equal to its own dual Selmer
group. (This can either be seen as a special case of Step $5$ or, more simply,
follows from the discussion in the proof of Step $4$ under the current
assumptions.) As in the proof of Step 1, Poitou–Tate duality then implies that
$\mathrm{Sel}_{\varepsilon}(K,A_{h})$ is equal to
$\mathrm{Sel}_{\varepsilon}^{\ell}(K,A_{\bar{g}})$, hence

$\dim_{\mathbf{F}_{p}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{h})\big{)}=\dim_{\mathbf{F}_{p}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{\bar{g}})\big{)}-1$

since $v_{\ell}(x)\not=0$. If $\mathrm{Sel}_{\varepsilon}(K,A_{h})\not=0$ we
can apply the same argument after replacing $\bar{g}$ with $h$. In this way
one constructs a squarefree product $T\in{}\mathscr{S}_{1}$ of
$\dim_{\mathbf{F}_{p}}\mathrm{Sel}_{\varepsilon}(K,A_{\bar{g}})$ admissible
primes such that $\mathrm{Sel}_{\varepsilon}(K,A_{h})=0$, where
$h\in{}S_{2}(N^{+},TLN^{-};\mathbf{F}_{p})$ denotes now the $T$-level raising
of $\bar{g}$. As in the proof of Step 4, the theorem of Skinner–Urban–Wan then
implies that $L(\xi/K,1)\not=0$, where $\xi$ is a newform of weight
$\Gamma_{0}(TLN)$ which is congruent to $f$ modulo $p$. As a consequence

$-1=\epsilon_{K}(TLN)=\epsilon_{K}(LN^{-})\cdot{}(-1)^{\dim_{\mathbf{F}_{p}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{\bar{g}})\big{)}},$

and since by assumption $LN^{-}$ has an _odd_ number of prime divisors, this
proves $(\ref{eq:parity})$.

We now show (7.23), which is equivalent to show that the integers $m_{1}$ and
$m_{2}$ in Equation (7.16) are both equal to $0$. Preliminarily, note that if
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\big{)}=2\cdot{}t_{\bar{\chi}}^{\varepsilon}(g)$,
then, since
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{h}(\chi))\big{)}\leqslant
2\cdot{}t_{\bar{\chi}}^{\varepsilon}(h)$ by Step 3, we have $m_{1}+m_{2}=0$
(where $m_{1}$ and $m_{2}$ are defined in Step 3; we also have
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{h}(\chi))\big{)}=2\cdot{}t_{\bar{\chi}}^{\varepsilon}(h)$)
directly from Equation (7.16). Therefore we assume in the following that
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\big{)}<2\cdot{}t_{\bar{\chi}}^{\varepsilon}(g)$.

We first show that $m_{1}=0$. By (7.1),
$t_{\bar{\chi}}^{\varepsilon}(g)<\mathrm{ord}_{\chi}(p^{k})$. Since the
$\mathbf{F}_{p}$-dimension of $\mathrm{Sel}_{\varepsilon}(K,A_{\bar{g}})$ is
even, combining Step 5 and Nakayama Lemma shows that

$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\cong{}\mathtt{M}\oplus\mathtt{M}.$

If follows that
$\mathrm{length}_{\mathscr{O}_{\chi}}(\mathtt{M})<\mathrm{ord}_{\chi}(p^{k})$,
and therefore

$\varpi_{\chi}^{\mathrm{ord}_{\chi}(p^{k})-1}\cdot\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))=0.$

We now consider the class
$\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1})-\varpi_{\chi}^{m_{1}}\cdot{}\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{1})$
in $\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi}))$ appearing in the proof
of Step 3. Since $\varpi^{\mathrm{ord}_{\chi}(p^{k})-1}$ kills
$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))$, and since
$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))$ and
$\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi}))$ are dual to each other,
the same is true for $\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi}))$.
Therefore we obtain the equality

(7.25)
$\varpi_{\chi}^{\mathrm{ord}_{\chi}(p^{k})-1}\cdot\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1})=\varpi_{\chi}^{\mathrm{ord}_{\chi}(p^{k})-1+m_{1}}\cdot\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{1}).$

We now show that the left hand side of this equality is always non-trivial.
First, enlarge $\\{\ell_{1}\\}$ to a freeing set $S$ as in Section 5.5; then
by Proposition 5.11,
$\mathfrak{Sel}_{S}^{\varepsilon}\big{(}K,T_{g}(\chi)\big{)}$ is free over
$\mathscr{O}_{\chi}/p^{k}$ of rank $\delta(S)$, the number of prime divisors
of $S$. By Lemma $\ref{basickol}(3)$, the class
$\bar{\xi}_{\chi}^{\varepsilon}(\ell)$ in
$\mathfrak{Sel}_{S}^{\varepsilon}(K,T_{\bar{g}})\otimes_{\mathbf{F}_{p}}\mathbf{F}_{\chi}$
is not trivial, therefore $\varpi_{\chi}$ does not divide
${\xi}_{\chi}^{\varepsilon}(\ell)$ and it follows that
$\varpi_{\chi}^{\mathrm{ord}_{\chi}(p^{k})-1}\cdot\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1})\neq
0$ from the freeness result
$\mathfrak{Sel}_{S}^{\varepsilon}\big{(}K,T_{g}(\chi)\big{)}\cong(\mathscr{O}_{\chi}/p^{k})^{\delta(S)}$
recalled above. On the other hand, if $m_{1}>0$, then the right hand side of
(7.25) is zero, which is a contradition. Therefore $m_{1}$ must be equal to
$0$.

We now show that $m_{2}=0$ with a similar argument. Consider the class
$\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})-\varpi_{\chi}^{m_{2}}\cdot{}\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{2})$
in $\mathfrak{Sel}^{\varepsilon}_{\ell_{1}}(K,T_{g}(\bar{\chi}))$ appearing in
the proof of Step 3. Since $m_{1}=0$, we know that
$\partial_{\ell_{1}}(\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1}))$ generates the
image of the residue map
$\partial_{\ell_{1}}:\mathfrak{Sel}_{\ell_{1}}^{\varepsilon}(K,T_{g}(\bar{\chi}))\stackrel{{\scriptstyle}}{{\rightarrow}}H^{1}_{\mathrm{sing}}(K_{\ell},T_{g}(\bar{\chi}))$,
and therefore there exists an integer $m_{3}\geqslant 0$ such that the class
$\partial_{\ell_{1}}\left(\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})-\varpi_{\chi}^{m_{2}}\cdot{}\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{2})\right)$
is equal to the class
$\varpi_{\chi}^{m_{3}}\cdot\partial_{\ell_{1}}(\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1}))$,
i.e.

$\partial_{\ell_{1}}\left(\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})-\varpi_{\chi}^{m_{2}}\cdot{}\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{2})-\varpi_{\chi}^{m_{3}}\cdot\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1})\right)=0.$

Therefore by definition the class
$\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})-\varpi_{\chi}^{m_{2}}\cdot{}\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{2})-\varpi_{\chi}^{m_{3}}\cdot\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1})$
belongs to $\mathfrak{Sel}^{\varepsilon}(K,T_{g}(\bar{\chi}))$. Since this
group is annihilated by $\varpi_{\chi}^{\mathrm{ord}_{\chi}(p^{k})-1}$ we
obtain the equality

$\varpi_{\chi}^{\mathrm{ord}_{\chi}(p^{k})-1}\cdot\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})-\varpi_{\chi}^{\mathrm{ord}_{\chi}(p^{k})-1+m_{2}}\cdot{}\zeta_{\bar{\chi}}^{\varepsilon}(\ell_{2})=\varpi_{\chi}^{\mathrm{ord}_{\chi}(p^{k})-1+m_{3}}\cdot\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1}).$

We now suppose _ad absurdum_ that $m_{2}>0$. Then the above equation implies

(7.26)
$\varpi_{\chi}^{\mathrm{ord}_{\chi}(p^{k})-1}\cdot\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})=\varpi_{\chi}^{\mathrm{ord}_{\chi}(p^{k})-1+m_{3}}\cdot\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1}).$

By $\mathbf{I}_{4}$,
$\mathrm{ord}_{\chi}\big{(}v_{\ell_{1}}\big{(}\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})\big{)}\big{)}=0$,
and therefore, again using the freeness argument as above, we see that
$\varpi_{\chi}^{\mathrm{ord}_{\chi}(p^{k})-1}\cdot\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})$
is not trivial. Equation (7.26) then shows that $m_{3}=0$. Therefore, applying
$v_{\ell_{1}}$ to Equation (7.26), we obtain the equality

$\varpi_{\chi}^{\mathrm{ord}_{\chi}(p^{k})-1}\cdot
v_{\ell_{1}}\big{(}\xi_{\bar{\chi}}^{\varepsilon}(\ell_{2})\big{)}=\varpi_{\chi}^{\mathrm{ord}_{\chi}(p^{k})-1}\cdot
v_{\ell_{1}}\big{(}\xi_{\bar{\chi}}^{\varepsilon}(\ell_{1})\big{)}$

By $\mathbf{I}_{4}$, the left hand side of this equality is not trivial, while
by Lemma $\ref{basickol}(1)$, the right hand side is trivial, which is a
contradiction. Therefore, $m_{2}=0$, concluding the proof of Step 6. ∎

####  7.2.7. Step 7

Assume that $(f,K,p,\varepsilon)$ is not exceptional. Then

$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\big{)}=2\cdot{}t_{\bar{\chi}}^{\varepsilon}(g).$

###### Proof.

The proof is by induction on
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\big{)}$.
If
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\big{)}=0$,
then the equality follows from Step 4. When
$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))$ is not trivial, choose a pair of
$2k$-admissible primes $\ell_{1}$ and $\ell_{2}$ satisfying conditions
$\mathbf{I}_{1}$-$\mathbf{I}_{4}$, and let $h$ be the $\ell_{1}\ell_{2}$-level
raising of $g$. Since
$t_{\bar{\chi}}^{\varepsilon}(h)<t_{\bar{\chi}}^{\varepsilon}(g)$, we see from
Step 6 that
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{h}(\chi))\big{)}$
is strictly smaller than
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\big{)}$,
and therefore by the inductive hypothesis
$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{h}(\chi))\big{)}=2t_{\bar{\chi}}^{\varepsilon}(h)$.
A further application of the equality in Step 6 implies then the result. ∎

######  Definition 7.3.

Let $\mathfrak{X}_{p}^{\varepsilon}(f)$ be the Pontryagin dual of
$\mathrm{Sel}_{\varepsilon}(K,\mathbf{A}_{f})$, which is a compact
$\Lambda$-module, and denote $\mathrm{Char}_{p}^{\varepsilon}(f)$ its
characteristic power series.

##  8\. $\varepsilon$-BSD formulas in the indefinite case

Assume that $N^{-}$ is _indefinite_ (i.e. $\epsilon_{K}(N^{-})=+1$).

######  Proposition 8.1.

The compact $\Lambda_{\mathscr{O}}$-module
$\mathfrak{Sel}^{\varepsilon}(K,\mathbf{T}_{f,\mathscr{O}})$ is free of finite
rank.

###### Proof.

By Proposition 5.9 (for $\mathfrak{P}$ equal to the augmentation ideal of
$\Lambda_{\mathscr{O}}$) and Shapiro’s Lemma, the
$\Lambda_{\mathscr{O}}$-quotient of $G_{\infty}$-coinvariants of
$\mathfrak{Sel}^{\varepsilon}(K,\mathbf{T}_{f,\mathscr{O}})$ injects into
$\mathfrak{Sel}^{\varepsilon}(K,T_{f,{\mathscr{O}}})$, and therefore is
${\mathscr{O}}$-free. Thanks to $E_{p}(K)=0$, we have
$\mathbf{T}_{f,\mathscr{O}}^{G_{\infty}}=0$, so the
$\Lambda_{\mathscr{O}}$-module
$\mathfrak{Sel}^{\varepsilon}(K,\mathbf{T}_{f,\mathscr{O}})$ is torsion free
by [25, Lemma 1.3.3], hence its $\Lambda_{\mathscr{O}}$-submodule of
$G_{\infty}$-invariants is trivial. The result follows then from a standard
argument (_e.g._ [23, Proposition 5.3.19(ii)]). ∎

###  8.1. $\Lambda$-adic classes

We first construct global classes, in a way similar to §6.1. Since
$\epsilon_{K}(N^{-})=+1$, $J_{N^{+},N^{-}}$ is the Picard variety of the
Shimura curve $X_{N^{+},N^{-}}$. Let $I_{f}\subset{}\mathbf{T}_{N^{+},N^{-}}$
denote the kernel of $f$. Modularity implies that there is an isomorphism of
$\mathbf{Z}_{p}[G_{\mathbf{Q}}]$-modules

$\pi_{f}:\mathrm{Ta}_{p}(J_{N^{+},N^{-}})/I_{f}\cong{}T_{f},$

which is unique up to multiplication by a $p$-adic unit by Hypothesis
$\ref{thehy}(1)$. For every integer $n\geqslant{}0$ define

$\psi_{f,n}:J_{N^{+},N^{-}}(K_{n})\stackrel{{\scriptstyle}}{{\longrightarrow}}H^{1}(K_{n},\mathrm{Ta}_{p}(J_{N^{+},N^{-}})/I_{f})\cong{}H^{1}(K_{n},T_{f}),$

where the first (resp., second) map is induced by the Kummer map (resp., by
$\pi_{f}$). For every point $x\in{}J_{N^{+},N^{-}}(K_{n})$ the classe
$\psi_{f,n}(x)$ is finite at every prime of $K_{n}$ dividing $p$. Moreover,
since $J_{N^{+},N^{-}}$ has purely toric reduction at every prime divisor of
$N^{-}$, Mumford–Tate theory of $p$-adic uniformisation implies that these
classes are ordinary at every such prime. Therefore, we obtain a map

$\psi_{f,n}:J_{N^{+},LN^{-}}(K_{n})\stackrel{{\scriptstyle}}{{\longrightarrow}}\mathfrak{Sel}(K_{n},T_{f}).$

Recall the compatible sequence of Heegner points $P_{n}=P_{n}(1)$, for
$n\geqslant 0$, introduced in Section 2.5 and define
$\tilde{\kappa}_{n}=\psi_{g,n}(P_{n})$.

####  8.1.1. Ordinary case

Suppose that $E$ has ordinary reduction at $p$. The classes

$\kappa_{n}=\frac{1}{\alpha_{p}(g)^{n}}\Big{(}\tilde{\kappa}_{n-1}-\alpha_{p}(g)\cdot{}\tilde{\kappa}_{n}\big{)}\Big{)}$

belong to $\mathfrak{Sel}(K_{n},T_{f})$ by the previous discussion, and
Equation (2.5) shows that they are norm-compatible. As in §6.1, define

(8.1)
$\kappa_{\infty}=\mathop{\varprojlim}\limits_{n}\kappa_{n}\in\mathop{\varprojlim}\limits_{n}\mathfrak{Sel}(K_{n},T_{f})$

where the inverse limit is computed with respect to the canonical norm maps.

####  8.1.2. Supersingular case

Using the freeness result of Proposition 8.1, by the same argument in §6.1 one
can define classes

$\tilde{\kappa}_{n}^{\varepsilon}\in\mathfrak{Sel}^{\varepsilon}(K_{n},T_{f})/\omega_{n}^{\varepsilon}$

such that
$\tilde{\omega}_{n}^{-\varepsilon}\cdot\tilde{\kappa}_{n}^{\varepsilon}=\tilde{\kappa}_{n}$
if $p$ is split in $K$ or $p$ is inert in $K$ and $\varepsilon=-1$ (the non-
exceptional case), and
$\omega_{n}^{-}\cdot\tilde{\kappa}_{n}^{+}=\tilde{\kappa}_{n}$ if $p$ is inert
in $K$ and $\varepsilon=+1$ (the exceptional case). Define
$\kappa_{n}^{+}=(-1)^{n/2}\bar{\kappa}_{n}^{+}$ if $n$ is even and
$\kappa_{n}^{-}=(-1)^{(n-1)/2}\tilde{\kappa}_{n}^{-}$ if $n$ is odd. A
calculation using Equation (2.5) shows that the classes
$\kappa_{n}^{\varepsilon}$ are compatible with respect to the canonical
projection maps. Define as in §6.1

$\kappa_{\infty}^{\varepsilon}=\mathop{\varprojlim}\limits_{n}\kappa_{n}^{\varepsilon}\in\mathop{\varprojlim}\limits_{n\in\mathbf{N}^{\varepsilon}}\mathfrak{Sel}^{\varepsilon}(K_{n},T_{f})/\omega_{n}^{\varepsilon},$

where $\mathbf{N}^{\varepsilon}$ is the set of positive integers verifying the
condition $(-1)^{n}=\varepsilon$.

###  8.2. Lengths of Selmer groups

Fix a morphism $\chi:\Lambda\rightarrow\mathscr{O}_{\chi}$ of
$\mathbf{Z}_{p}$-algebras, where as above $\mathscr{O}_{\chi}$ is the integral
closure of $\Lambda/\mathfrak{P}_{\chi}$, and
$\mathfrak{P}_{\chi}=\ker(\chi)$. Denote

$\kappa_{\chi}^{\varepsilon}\in\mathfrak{Sel}^{\varepsilon}(K,\mathbf{T}_{f})\otimes\mathscr{O}_{\chi}\overset{\mathrm{def}}{=}\mathfrak{Sel}^{\varepsilon}(K,\mathbf{T}_{f,{\mathscr{O}_{\chi}}})\otimes_{\Lambda_{\mathscr{O}_{\chi}}}\mathscr{O}_{\chi}$

the image of $\kappa_{\infty}^{\varepsilon}$ via the canonical map described
above, where recall that the tensor product
$\otimes_{\Lambda_{\mathscr{O}_{\chi}}}$ is taken with respect to $\chi$. We
assume that $\mathrm{ord}_{\chi}(\kappa_{\chi}^{\varepsilon})$ is finite.
Using that
$\mathfrak{Sel}^{\varepsilon}(K,\mathbf{T}_{f,{\mathscr{O}_{\chi}}})$ is
$\Lambda_{\mathscr{O}_{\chi}}$-free by Proposition 8.1, define the integer

$t_{\chi}^{\varepsilon}(f)=\mathrm{ord}_{\chi}\left(\kappa_{\chi}^{\varepsilon}\right)<\infty.$

For any group $p$-power torsion group $G$, let $G_{/\mathrm{div}}$ denote the
quotient of $G$ by its maximal $p$-divisible subgroup.

######  Theorem 8.2.

Suppose that $t_{\chi}^{\varepsilon}(f)<\infty$. Then the
$\mathscr{O}_{\chi}$-corank of $\mathrm{Sel}_{\varepsilon}(K,A_{f}(\chi))$ is
$1$ and we have

$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\varepsilon}(K,A_{f}(\chi))_{/\mathrm{div}}\right)\leqslant
2\cdot\mathrm{length}_{\mathscr{O}_{\chi}}\left((\mathfrak{Sel}^{\varepsilon}(K,\mathbf{T}_{f})\otimes\mathscr{O}_{\chi})/\mathscr{O}_{\chi}\cdot\kappa_{\bar{\chi}}^{\varepsilon}\right),$

and the equality holds if $(f,K,p,\varepsilon)$ is not exceptional.

###### Proof.

It follows from the freeness of
$\mathfrak{Sel}^{\varepsilon}(K,\mathbf{T}_{f,\mathscr{O}_{\chi}})$ that there
exists $\tilde{\kappa}_{\chi}^{\varepsilon}$ in
$\mathfrak{Sel}^{\varepsilon}(K,\mathbf{T}_{f})\otimes\mathscr{O}_{\chi}$ such
that $\mathrm{ord}_{\chi}(\tilde{\kappa}_{\chi}^{\varepsilon})=0$ and
$\kappa_{\chi}^{\varepsilon}=\varpi_{\chi}^{t_{\chi}^{\varepsilon}(f)}\cdot{}\tilde{\kappa}_{\chi}^{\varepsilon}.$
Define
$\xi_{\chi}^{\varepsilon}\in{}\mathfrak{Sel}^{\varepsilon}(K,{T}_{f}(\chi))$
to be the image of $\tilde{\kappa}_{\chi}^{\varepsilon}$ under the (injective)
specialization map
$\texttt{s}_{\chi}:\mathfrak{Sel}^{\varepsilon}_{S}(K,\mathbf{T}_{f})\otimes\mathscr{O}_{\chi}\hookrightarrow\mathfrak{Sel}^{\varepsilon}(K,{T}_{f}(\chi))$.
We also denote $\kappa_{\chi,k}^{\varepsilon}$ the image of
$\kappa_{\chi}^{\varepsilon}$ in
$\mathfrak{Sel}^{\varepsilon}(K,T_{f,k}(\chi))$, for all integers $k\geqslant
1$, and $\xi^{\varepsilon}_{\chi,k}$ the image of $\xi^{\varepsilon}_{\chi}$
in $H^{1}(K,T_{f,k}(\chi))$. If $k=1$, the element
$\xi^{\varepsilon}_{\chi,1}$ will be denoted $\bar{\xi}^{\varepsilon}_{\chi}$.
As before (_cf._ Step 5 in §7.2.5) we have

$\mathrm{Sel}_{\varepsilon}(K,A_{f}(\chi))\cong{}(\mathscr{K}_{\chi}/\mathscr{O}_{\chi})^{s}\oplus
M_{\chi}\oplus M_{\chi}.$

for some integer $s$ and a finite torsion $\mathscr{O}_{\chi}$-module
$M_{\chi}$. Choose an integer

$k>\max\\{\mathrm{length}_{\mathscr{O}_{\chi}}(M_{\chi}),t_{\bar{\chi}}^{\varepsilon}(f)\\}.$

Using [3, Theorem 3.2], choose an admissible prime $\ell\in\mathscr{S}_{k}$
such that $v_{\ell}(\bar{\xi}_{\bar{\chi}}^{\varepsilon})\neq 0$. Let $g$ be
the $\ell$-level raising of $f$. Then since
$H^{1}_{\mathrm{fin}}(K_{\ell},T_{f,\mathscr{O}_{\chi}})$ is a free
$\mathscr{O}_{\chi}$-module of rank $1$, and
$v_{\ell}(\bar{\xi}_{\bar{\chi}}^{\varepsilon})\neq 0$, using Proposition 5.5,
we have

(8.2)
$\mathrm{ord}_{\chi}\left(v_{\ell}\left(\kappa_{\bar{\chi},k}^{\varepsilon}\right)\right)=\mathrm{ord}_{\chi}\left(v_{\ell}\left(\kappa_{\bar{\chi}}^{\varepsilon}\right)\right).$

_Step 1._ Theorem 7.1 for $g$ shows that

$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\right)\leqslant
2\cdot\mathrm{ord}_{\chi}\left(\mathcal{L}_{g}^{\varepsilon}(\bar{\chi})\right),$

with equality in the non-exceptional case, and Theorem 6.3 shows that

$\mathrm{ord}_{\chi}\left(v_{\ell}\left(\kappa_{\bar{\chi},k}^{\varepsilon}\right)\right)=\mathrm{ord}_{\chi}\left(\mathcal{L}_{g}^{\varepsilon}(\bar{\chi})\right).$

Thus, by (8.2) and the injectivity of the map $v_{\ell}$ (which follows from
$v_{\ell}(\bar{\xi}_{\bar{\chi}}^{\varepsilon})\neq 0$), we have

$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\right)\leqslant
2\cdot\mathrm{ord}_{\chi}\left(v_{\ell}\left(\kappa_{\bar{\chi}}^{\varepsilon}\right)\right)=2\cdot\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathfrak{Sel}^{\varepsilon}(K,\mathbf{T}_{f})\otimes\mathscr{O}_{\chi}/\mathscr{O}_{\chi}\cdot\kappa_{\bar{\chi}}^{\varepsilon}\right)$

with equality in the non-exceptional case.

_Step 2._ Recall the relaxed Selmer group
$\mathrm{Sel}_{\varepsilon}^{(\ell)}(K,A_{f,k}(\chi))\supseteq\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))$,
_i.e._ the set of cohomology classes defined requiring the same conditions as
$\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))$ at primes different from $\ell$,
and no condition at $\ell$. We claim that

(8.3)
$\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))=\mathrm{Sel}_{\varepsilon}^{(\ell)}(K,A_{f,k}(\chi)).$

To prove this, let $x\in\mathrm{Sel}_{\varepsilon}^{(\ell)}(K,A_{f,k}(\chi))$.
We have to show that $x$ is in the kernel of the residue map at $\ell$. By
global class field theory, using the orthogonality of $\mathrm{res}_{v}(x)$
and $\mathrm{res}_{v}(\xi_{\bar{\chi},k}^{\varepsilon})$ outside $\ell$ as in
Step 1 of the proof of Theorem 7.1, one then obtains

$0=\sum_{v}\left<{}\mathrm{res}_{v}(x),\mathrm{res}_{v}\big{(}\xi_{\bar{\chi},k}^{\varepsilon})\big{)}\right>_{v}=\left<{}\mathrm{res}_{\ell}(x),\mathrm{res}_{\ell}\big{(}\xi_{\bar{\chi},k}^{\varepsilon})\big{)}\right>_{\ell}=\left<{}\partial_{\ell}(x),v_{\ell}\big{(}\xi_{\bar{\chi},k}^{\varepsilon})\big{)}\right>_{\ell}.$

Since $v_{\ell}\big{(}\bar{\xi}_{\bar{\chi}}^{\varepsilon})\big{)}\not=0$, and
since $\left<{}-,-\right>_{\ell}$ is a perfect pairing, this implies that
$\partial_{\ell}(x)=0$, as was to be shown.

_Step 3_. We claim that there is an exact sequence:

$0\longrightarrow\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\longrightarrow\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))\overset{v_{\ell}}{\longrightarrow}H^{1}_{\mathrm{fin}}(K_{\ell},A_{f,k}(\chi))\longrightarrow
0.$

To show this, first note that

$\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\subseteq\mathrm{Sel}_{\varepsilon}^{(\ell)}(K,A_{g}(\chi))=\mathrm{Sel}_{\varepsilon}^{(\ell)}(K,A_{f,k}(\chi))=\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))$

where the last equality follows from Step 2; this shows the exactness on the
left. By definition, the kernel of the map
$v_{\ell}:\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))\rightarrow
H^{1}_{\mathrm{fin}}(K_{\ell},A_{f,k}(\chi))$ is
$H^{1}_{\mathrm{ord}}(K_{\ell},A_{f,k}(\chi))$, proving the exactness in the
middle. Finally, $v_{\ell}$ is surjective because, under the isomorphism
$T_{f,k}(\chi)\simeq A_{f,k}(\chi)$, $\xi_{\bar{\chi}}^{\varepsilon}$ is a
class in $\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))$ which satisfies
$v_{\ell}\big{(}\bar{\xi}_{\bar{\chi}}^{\varepsilon})\big{)}\not=0$.

_Step 4._ From Step 3 we obtain the equality

$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\varepsilon}(K,A_{g}(\chi))\right)=\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))\right)-\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathscr{O}_{\chi}/p^{k}\mathscr{O}_{\chi}\right)$

and combinig with Step 1 we get

$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))\right)-\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathscr{O}_{\chi}/p^{k}\mathscr{O}_{\chi}\right)\leqslant
2\cdot\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathfrak{Sel}^{\varepsilon}(K,\mathbf{T}_{f}\otimes\mathscr{O}_{\chi})/\mathscr{O}_{\chi}\cdot\kappa_{\bar{\chi}}^{\varepsilon}\right)$

with equality in the non-exceptional case. Since the left hand side has finite
order, bounded independently of $k$, we see that the
$\mathscr{O}_{\chi}$-corank of $\mathrm{Sel}_{\varepsilon}(K,A_{f}(\chi))$ is
$1$. By the choice of $k$,

$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))\right)-\mathrm{ord}_{\chi}\left(\mathscr{O}_{\chi}/p^{k}\mathscr{O}_{\chi}\right)=\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\varepsilon}(K,A_{f}(\chi))_{/\mathrm{div}}\right),$

concluding the proof. ∎

The next two definitions intervene in the statement of the IAMC.

######  Definition 8.3.

Let $L_{p}^{\varepsilon}(f)$ denote the characteristic power series of
$\mathfrak{Sel}(K,\mathbf{T}_{f})/\Lambda\cdot\kappa_{\infty}^{\varepsilon}$.

Let $\mathfrak{X}_{p}^{\varepsilon}(f)$ be the Pontryagin dual of
$\mathrm{Sel}_{\varepsilon}(K,\mathbf{A}_{f})$. Then the compact
$\Lambda$-module $\mathfrak{X}_{p}^{\varepsilon}(f)$ is pseudo-isomorphic to
$\Lambda\oplus\mathfrak{M}\oplus\mathfrak{M}$ for a torsion $\Lambda$-module
$\mathfrak{M}$, supported only on primes of height $1$; this follows from
Theorem 8.2, the structure results in Step 5 of the proof of Theorem 7.1, and
Proposition 5.8.

######  Definition 8.4.

Let $\mathrm{Char}_{p}^{\varepsilon}(f)$ be the characteristic ideal of the
$\Lambda$-module $\mathfrak{M}$.

##  9\. Proof of Theorems B and C

Fix throughout this section a finite order character
$\chi:G_{\infty}\twoheadrightarrow\mathscr{O}_{\chi}^{\times}$ of conductor
$p^{n}$.

###  9.1. Comparison of Selmer groups

Suppose that $p$ is supersingular. The aim of this section is to compare the
discrete Selmer groups $\mathrm{Sel}^{\varepsilon}(K,A_{f}(\chi))$ and
$\mathrm{Sel}(K,A_{f}(\chi))$ and the compact Selmer groups
$\mathfrak{Sel}^{\varepsilon}(K,T_{f}(\chi))$ and
$\mathfrak{Sel}(K,T_{f}(\chi))$. Let $\mathfrak{p}\mid p$ be a prime and fix
$k\in\mathbf{N}\cup\\{\infty\\}$. Set as before $\Phi=K_{\mathfrak{p}}$,
$\Phi_{n}=K_{n,\mathfrak{p}}$ and $\Phi_{\infty}=K_{\infty,\mathfrak{p}}$. Let
$\mathfrak{P}_{\chi}=(\mathfrak{p}_{\chi})$ be the kernel of the character
$\chi:\Lambda_{\mathscr{O}_{\chi}}\twoheadrightarrow\mathscr{O}_{\chi}$
obtained from $\chi$, where $\mathfrak{p}_{\chi}=\gamma-\chi(\gamma)$. We also
view $\chi$ as a character
$\chi:\mathscr{O}_{\chi}[G_{n}]\twoheadrightarrow\mathscr{O}_{\chi}$, whose
kernel we still denote by $\mathfrak{P}_{\chi}=(\mathfrak{p}_{\chi})$. To
simplify the notation, define

$\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\mathrm{div}}=\mathbf{E}(\Phi_{n})\otimes_{\mathbf{Z}}(\mathscr{K}_{\chi}/\mathscr{O}_{\chi}),$
$\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\pm,\mathrm{div}}=\mathbf{E}(\Phi_{n})_{\pm}\otimes_{\mathbf{Z}}(\mathscr{K}_{\chi}/\mathscr{O}_{\chi}).$

######  Lemma 9.1.

Let $\varepsilon=(-1)^{n}$.

  1. (1)

In the non-exceptional case,
$\frac{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}$
is finite and

$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\frac{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}\right)=[\Phi:\mathbf{Q}_{p}]\cdot\mathrm{ord}_{\chi}(\tilde{\omega}_{n}^{-\varepsilon}).$

  2. (2)

In the exceptional case,

    1. (a)

If $n=0$, so $\chi=\mathbf{1}$ is the trivial character

$\frac{\left(\mathbf{E}_{\mathbf{Z}_{p}}(\Phi)_{\mathrm{div}}\right)[(\gamma-1)]}{\left(\mathbf{E}_{\mathbf{Z}_{p}}(\Phi)_{+,\mathrm{div}}\right)[(\gamma-1)}=\mathbf{E}(\Phi)\otimes_{\mathbf{Z}_{p}}\mathbf{Q}_{p}/\mathbf{Z}_{p}\cong(\mathbf{Q}_{p}/\mathbf{Z}_{p})^{[\Phi:\mathbf{Q}_{p}]},$

    2. (b)

If $n\geqslant 2$,
$\frac{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{+,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}$
is finite and

$\mathrm{length}\left(\frac{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{+,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}\right)=[\Phi:\mathbf{Q}_{p}]\cdot\mathrm{ord}_{\chi}(\omega_{n}^{-}).$

###### Proof.

Suppose first $n=0$. Then $\chi$ is the trivial character,
$\mathscr{O}_{\chi}=\mathbf{Z}_{p}$ and we suppress the index
$\mathscr{O}_{\chi}$ from the notation, thus writing
$\mathbf{E}(\Phi)_{\mathrm{div}}$ for
$\mathbf{E}_{\mathbf{Z}_{p}}(\Phi)_{\mathrm{div}}$ and
$\mathbf{E}(\Phi)_{\pm,\mathrm{div}}$ for
$\mathbf{E}_{\mathbf{Z}_{p}}(\Phi)_{\pm,\mathrm{div}}$. If $p$ is split, then
$\mathbf{E}(\Phi)_{\mathrm{div}}=\mathbf{E}(\Phi)_{+,\mathrm{div}}$, so the
quotient in the statement is trivial; on the other hand,
$\tilde{\omega}_{0}^{-}=1$, and the statement is proved. If $p$ is inert,
$\mathbf{E}(\Phi)_{\mathrm{div}}=\mathbf{E}(\Phi)_{-,\mathrm{div}}$ and
$\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)_{+,\mathrm{div}}=0$, so the quotient in
the statement is
$\mathbf{E}(\Phi)_{\mathrm{div}}[(\gamma-1)]\cong(\mathbf{Q}_{p}/\mathbf{Z}_{p})^{[\Phi:\mathbf{Q}_{p}]}$,
where the last isomorphism follows from Theorem 5.1.

Suppose $n\geqslant 1$. We first observe that we have an exact sequence:

(9.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]\oplus\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{-\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\mathrm{div}}\right)[\mathfrak{P}_{\chi}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $C=0$ if $p$ is inert in $K$ and
$C=\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)_{\mathrm{div}}\right)[\mathfrak{P}_{\chi}]$
if $p$ is split in $K$; in this exact sequence the second arrow is the map
$x\mapsto(x,x)$, and the third arrow is the map $(x,y)\mapsto x-y$. If $p$ is
inert in $K$, it follows from Theorem 5.1 that
$\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\mathrm{div}}$ is the direct sum
of $\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\varepsilon,\mathrm{div}}$ and
$\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{-\varepsilon,\mathrm{div}}$, which
proves (9.1) (also in the exceptional case). In the split case, it follows
again from Theorem 5.1 that

$\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\varepsilon,\mathrm{div}}\cap\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{-\varepsilon,\mathrm{div}}=\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)_{\mathrm{div}},$

so we need to show that the exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)_{\mathrm{div}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\varepsilon,\mathrm{div}}\oplus\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{-\varepsilon,\mathrm{div}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\mathrm{div}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

remains exact after taking $\mathfrak{P}_{\chi}$-torsion, so we need to show
that the map

$\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]\oplus\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{-\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]\longrightarrow\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\mathrm{div}}\right)[\mathfrak{P}_{\chi}]$

is surjective. The cokernel of this map injects into the quotient
$\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)_{\mathrm{div}}/\mathfrak{p}_{\chi}\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)_{\mathrm{div}}$,
and we need to show that this group is trivial. Since
$\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)$ is $\mathscr{O}_{\chi}$-free, it is
enough to show that

(9.2)
$\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)/\mathfrak{p}_{\chi}\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)\right)\otimes_{\mathscr{O}_{\chi}}\mathscr{K}_{\chi}/\mathscr{O}_{\chi}=0.$

Now, $\mathfrak{p}_{\chi}$ acts on the $\mathscr{O}_{\chi}$-free module
$\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)$ as multiplication by $1-\chi(\gamma)$,
and since $\chi(\gamma)$ is a primitive $p^{n}$-root of unity, and $n\geqslant
1$, the quotient
$\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)/\mathfrak{p}_{\chi}\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)$
is finite, and (9.2) follows.

We have therefore a commutative diagram with exact rows

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]\oplus\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{-\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\mathrm{div}}\right)[\mathfrak{P}_{\chi}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $C$ is defined before, and the middle vertical arrow is the map
$x\mapsto(x,0)$. By the snake lemma we obtain an exact sequence

$0\longrightarrow
C\longrightarrow\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{-\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]\longrightarrow\frac{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}\longrightarrow
0.$

The Pontryagin dual of the middle term is
$\Lambda_{\mathscr{O}_{\chi}}/(\omega_{n}^{-\varepsilon},\mathfrak{p}_{\chi})$,
because the Pontryagin dual of
$\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{-\varepsilon,\mathrm{div}}$ is
$\Lambda_{\mathscr{O}_{\chi}}/(\omega_{n}^{-\varepsilon})$; since
$\Lambda_{\mathscr{O}_{\chi}}/\mathfrak{P}_{\chi}\cong\mathscr{O}_{\chi}$, the
length of the middle term is equal to the length of
$\mathscr{O}_{\chi}/\chi(\omega_{n}^{-\varepsilon})$. Similarly, if $p$ is
split, the length of
$C=\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)_{\mathrm{div}}\right)[\mathfrak{P}_{\chi}]$
is equal to the length of $\mathscr{O}_{\chi}/\chi(\gamma-1)$ and therefore
the length of the quotient is equal to the length of
$\mathscr{O}_{\chi}/\chi(\tilde{\omega}_{n}^{-\varepsilon})$, completing the
proof in this case. If $p$ is inert, then $C$ is trivial. If $\varepsilon=-1$
(the non-exceptional case), then $\tilde{\omega}_{n}^{+}=\omega_{n}^{+}$, and
the length of the last term is equal to the length of
$\mathscr{O}_{\chi}/\chi(\omega_{n}^{+})=\mathscr{O}_{\chi}/\chi(\tilde{\omega}_{n}^{+})$,
while if $\varepsilon=+1$ (the exceptional case) then the length is
$\mathscr{O}_{\chi}/\chi(\omega_{n}^{-})$, completing the proof. ∎

######  Proposition 9.2.

In the exceptional case, assume that $n\neq 0$. The discrete Selmer groups
$\mathrm{Sel}_{\varepsilon}(K,A_{f}(\chi))$ and $\mathrm{Sel}(K,A_{f}(\chi))$
have the same $\mathscr{O}_{\chi}$-corank. Moreover,

  1. (1)

If $p$ is split in $K$ or $p$ is inert in $K$ and $\varepsilon=-1$ (the non-
exceptional case),

$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\frac{\mathrm{Sel}(K,A_{f}(\chi))}{\mathrm{Sel}_{\varepsilon}(K,A_{f}(\chi))}\right)=2\cdot\mathrm{ord}_{\chi}(\tilde{\omega}_{n}^{-\varepsilon}).$

  2. (2)

If $p$ is inert in $K$ and $\varepsilon=+1$ (the exceptional case),

$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\frac{\mathrm{Sel}(K,A_{f}(\chi))}{\mathrm{Sel}_{+}(K,A_{f}(\chi))}\right)=2\cdot\mathrm{ord}_{\chi}(\omega_{n}^{-}).$

###### Proof.

We have the Poitou-Tate exact sequence

$\begin{split}0\longrightarrow\mathrm{Sel}_{\varepsilon}(K,A_{f}(\chi))\longrightarrow{\mathrm{Sel}}(K,A_{f}(\chi))\longrightarrow\prod_{\mathfrak{p}\mid
p}\frac{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)_{\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}\longrightarrow\\\
\longrightarrow\left(\mathfrak{Sel}^{\varepsilon}(K,T_{f}(\bar{\chi}))\right)^{\vee}\longrightarrow\left(\mathfrak{Sel}(K,T_{f}(\bar{\chi}))\right)^{\vee}\longrightarrow
0.\end{split}$

Now $\mathfrak{Sel}^{\varepsilon}(K,T_{f}(\bar{\chi}))$ is
$\mathscr{O}_{\chi}$-free, and therefore
$\left(\mathfrak{Sel}^{\varepsilon}(K,T_{f}(\bar{\chi}))\right)^{\vee}$ is
$p$-divisible. If we show that the kernel of the map
$\left(\mathfrak{Sel}^{\varepsilon}(K,T_{f}(\bar{\chi}))\right)^{\vee}\rightarrow\left(\mathfrak{Sel}(K,T_{f}(\bar{\chi}))\right)^{\vee}$
is divisible, then, since the local quotient in the middle of the above exact
sequence is finite by Lemma 9.1, we have an exact sequence

$0\longrightarrow\mathrm{Sel}_{\varepsilon}(K,A_{f}(\chi))\longrightarrow{\mathrm{Sel}}(K,A_{f}(\chi))\longrightarrow\prod_{\mathfrak{p}\mid
p}\frac{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi)_{\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}{\left(\mathbf{E}_{\mathscr{O}_{\chi}}(\Phi_{n})_{\varepsilon,\mathrm{div}}\right)[\mathfrak{P}_{\chi}]}\longrightarrow
0$

and the result follows from Lemma 9.1. So to complete the proof we need to
show that the kernel of the (surjective) map

$\textstyle{\left(\mathfrak{Sel}^{\varepsilon}(K,T_{f}(\bar{\chi}))\right)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\mathfrak{Sel}(K,T_{f}(\bar{\chi}))\right)^{\vee}}$

is divisible. For this, it is enough to show that the cokernel of the
(injective) map

(9.3)
$\textstyle{\mathfrak{Sel}(K,T_{f}(\bar{\chi}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{Sel}^{\varepsilon}(K,T_{f}(\bar{\chi}))}$

is torsion free. Let $x\in\mathfrak{Sel}^{\varepsilon}(K,T_{f}(\bar{\chi}))$
and let $M\geqslant 1$ be such that $\varpi_{\chi}^{M}\cdot
x\in\mathfrak{Sel}(K,T_{f}(\bar{\chi}))$; to conclude that the cokernel of the
map (9.3) is torsion-free, it is then enough to show that
$x\in\mathfrak{Sel}(K,T_{f}(\bar{\chi}))$. Since $\varpi_{\chi}^{M}\cdot
x\in\mathfrak{Sel}(K,T_{f}(\bar{\chi}))$, we have (writing $\langle-,-\rangle$
for the local Tate pairing
$\langle-,-\rangle_{\mathfrak{P}_{\chi},\mathfrak{p}}$ to simplify the
notation) $\langle\mathrm{res}_{\mathfrak{p}}(\varpi_{\chi}^{M}\cdot
x),y\rangle=0$ for all $y\in
H^{1}_{\mathrm{fin}}(K_{\mathfrak{p}},A_{f}(\chi))$, and since
$\langle\mathrm{res}_{\mathfrak{p}}(\varpi_{\chi}^{M}\cdot
x),y\rangle=\langle\mathrm{res}_{\mathfrak{p}}(x),\varpi_{\chi}^{M}\cdot
y\rangle$, we also have
$\langle\mathrm{res}_{\mathfrak{p}}(x),\varpi_{\chi}^{M}\cdot y\rangle=0$ for
all $y\in H^{1}_{\mathrm{fin}}(K_{\mathfrak{p}},A_{f}(\chi))$. Recall that
$H^{1}_{\mathrm{fin}}(K_{\mathfrak{p}},A_{f}(\chi))$ is co-free over
$\mathscr{O}_{\chi}$ by Proposition 5.4, hence
$H^{1}_{\mathrm{fin}}(K_{\mathfrak{p}},A_{f}(\chi))$ is
$\varpi_{\chi}$-divisible. So the function
$y\mapsto\langle\mathrm{res}_{\mathfrak{p}}(x),y\rangle$ is zero on
$H^{1}_{\mathrm{fin}}(K_{\mathfrak{p}},A_{f}(\chi))$ and therefore $x$ belongs
to $\mathfrak{Sel}(K,T_{f}(\bar{\chi}))$, concluding the proof. ∎

###  9.2. Proof of Theorem B

Recall
$L_{p,n}(f)=\mathcal{L}_{f,n}\cdot\left(\mathcal{L}_{f,n}\right)^{\iota}\in\mathbf{Z}_{p}[G_{n}]$.
By Remark 1.3 we may assume $n\neq 0$ in the exceptional case.

_Step 1._ We first show that

$\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}(K,A_{f}(\chi))\big{)}\leqslant\mathrm{ord}_{\chi}\big{(}\chi(L_{p,n}(f))\big{)}$

with equality in the non-exceptional case. Take $g=f_{k}$ for $L=\emptyset$ in
Theorem 7.1. By Theorem 7.1, $\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))$ is
finite, of order bounded independently of $k$, so
$\mathrm{Sel}_{\varepsilon}(K,A_{f}(\chi))$ is finite, and by Proposition 9.2
the Selmer group $\mathrm{Sel}(K,A_{f}(\chi))$ is finite. Let
$t_{\bar{\chi}}^{\varepsilon}(f)=\mathrm{ord}_{\bar{\chi}}\big{(}\bar{\chi}(\mathcal{L}_{f}^{\varepsilon})\big{)}$
and choose

$k>\max\left\\{\mathrm{length}_{\mathscr{O}_{\chi}}\big{(}\mathrm{Sel}_{\varepsilon}(K,A_{f}(\chi))\big{)},t_{\bar{\chi}}^{\varepsilon}(f),\mathrm{ord}_{\chi}(\tilde{\omega}_{n}^{-\varepsilon})\right\\}.$

For such a $k$, we have
$\mathrm{Sel}_{\varepsilon}(K,A_{f}(\chi))\cong\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))$,
and, by Proposition 9.2,

  * •

$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}(K,A_{f}(\chi))\right)=\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))\right)+2\cdot\mathrm{ord}_{\chi}(\tilde{\omega}_{n}^{-\varepsilon})$
in the non-exceptional case;

  * •

$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}(K,A_{f}(\chi))\right)=\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))\right)+2\cdot\mathrm{ord}_{\chi}(\omega_{n}^{-\varepsilon})$
in the exceptional case.

By [2, Proposition 2.6],
$(\mathcal{L}_{f}^{\varepsilon})^{\iota}=\pm\gamma_{\infty}\mathcal{L}_{f}^{\varepsilon}$,
for some $\gamma_{\infty}\in{}G_{\infty}$, and therefore
$t_{\chi}^{\varepsilon}(f)=t_{\bar{\chi}}^{\varepsilon}(f)$. We thus have
$\mathrm{ord}_{\chi}\big{(}\chi(L_{p}^{\varepsilon}(f))\big{)}=2\cdot
t_{\bar{\chi}}^{\varepsilon}(f)$. If $p$ is split in $K$ or $p$ is inert in
$K$ and $\varepsilon=-1$ (non-exceptional case) we have
$\tilde{\omega}_{n}^{-\varepsilon}\cdot\mathcal{L}_{f,n}^{\varepsilon}\equiv\pm\mathcal{L}_{f,n}$
modulo $\omega_{n}$, and therefore

$\begin{split}\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}(K,A_{f}(\chi))\right)&=\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))\right)\\\
&=2\cdot\mathrm{ord}_{\chi}\left(\mathcal{L}_{f,n}^{\varepsilon}(\bar{\chi})\right)+2\cdot\mathrm{ord}_{\chi}(\tilde{\omega}_{n}^{-\varepsilon})\text{
(by Theorem \ref{BSD0})}\\\
&=2\cdot\mathrm{ord}_{\chi}\left(\mathcal{L}_{f,n}(\bar{\chi})\right).\end{split}$

If $p$ is inert in $K$ and $\varepsilon=+1$ (exceptional case), we have
$\omega_{n}^{-}\cdot\mathcal{L}_{f,n}^{+}\equiv\pm\mathcal{L}_{f,n}$ modulo
$\omega_{n}$, and therefore

$\begin{split}\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}(K,A_{f}(\chi))\right)&=\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\varepsilon}(K,A_{f,k}(\chi))\right)+2\cdot\mathrm{ord}_{\chi}(\omega_{n}^{-})\\\
&\leqslant{}2\cdot\mathrm{ord}_{\chi}\left(\mathcal{L}_{f,n}^{+}(\bar{\chi})\right)+2\cdot\mathrm{ord}_{\chi}(\omega_{n}^{-})\text{
(by Theorem \ref{BSD0})}\\\
&\leqslant{}2\cdot\mathrm{ord}_{\chi}\left(\mathcal{L}_{f,n}(\bar{\chi})\right).\end{split}$

_Step 2._ We now use explicit formulas for special values to conclude the
proof of Theorem B. Recall that the Gross formula gives the equality

$L(E/K,\chi,1)=C\cdot\chi(L_{p}(f))$

where the constant $C=\frac{\sqrt{D}p^{n}}{\Omega}$, and $\Omega$ is Gross’
period, defined in Lemma 2.5 of [33]; see Sections 2.4 and 2.5 of [33] for a
complete description of these constants. From Theorem A and Gross’ formula we
see that $\mathrm{Sel}(K,A_{f}(\chi))$ is finite if and only if
$L(E/K,\chi,1)\neq 0$ and

$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\mathrm{}}(K,A_{f}(\chi)\right)\leqslant\mathrm{ord}_{\chi}\left(\frac{L(E/K,\chi,1)}{C}\right)$

with equality in the non-exceptional case.

###  9.3. Proof of Theorem C

As noted in Remark 1.3 we may assume that $n\neq 0$ in the exceptional case.
We define the _regulator_

$\mathrm{Reg}_{\chi}(E/K)=\frac{h_{\mathrm{NT}}(P_{\bar{\chi}})}{2\cdot\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathfrak{Sel}(K,T_{f}(\chi))/\mathscr{O}_{\chi}\cdot\tilde{\kappa}_{\bar{\chi}}\right)}$

and the _Stafarevich-Tate group_

$\sha(K,A_{f}(\chi))=\mathrm{Sel}(K,A_{f}(\chi))_{/\mathrm{div}}.$

_Step 1._ We first show that

$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\mathrm{}}(K,A_{f}(\chi))_{/\mathrm{div}}\right)\leqslant\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathfrak{Sel}(K,T_{f}(\chi))/\mathscr{O}_{\chi}\cdot\tilde{\kappa}_{\bar{\chi}})\right)$

with equality in the non-exceptional case. Combining Theorem 8.2 and
Proposition 9.2, if $p$ is split in $K$ or $p$ is inert in $K$ and
$\varepsilon=-1$ (non-exceptional case) we have

(9.4)
$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\mathrm{}}(K,A_{f}(\chi))_{/\mathrm{div}}\right)=2\cdot\mathrm{length}_{\mathscr{O}_{\chi}}\left(\left(\mathfrak{Sel}^{\varepsilon}(K,\mathbf{T}_{f,\mathscr{O}_{\chi}})\otimes_{\Lambda_{\mathscr{O}_{\chi}}}\Lambda_{\mathscr{O}_{\chi}}/\mathfrak{P}_{\bar{\chi}}\right)/\left(\mathscr{O}_{\chi}\cdot\kappa_{\bar{\chi}}^{\varepsilon}\right)\right)+2\cdot\mathrm{ord}_{\chi}(\tilde{\omega}_{n}^{-\varepsilon}).$

From (9.6), using Proposition 5.9 and Proposition 9.2(2), we obtain

(9.5)
$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\mathrm{}}(K,A_{f}(\chi))_{/\mathrm{div}}\right)=2\cdot\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathfrak{Sel}(K,{T}_{f}(\chi))/\left(\mathscr{O}_{\chi}\cdot\kappa_{\bar{\chi}}^{\varepsilon}\right)\right)+2\cdot\mathrm{ord}_{\chi}(\tilde{\omega}_{n}^{-\varepsilon}).$

Now
$\tilde{\kappa}_{\bar{\chi}}=\chi(\tilde{\omega}_{n}^{-\varepsilon})\cdot\kappa_{\bar{\chi}}^{\varepsilon}$,
and the result follows. If $p$ is inert in $K$ and $\varepsilon=+1$ (the
exceptional case), combining Theorem 8.2 and Proposition 9.2, we have

(9.6)
$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\mathrm{}}(K,A_{f}(\chi))_{/\mathrm{div}}\right)\leqslant
2\cdot\mathrm{length}_{\mathscr{O}_{\chi}}\left(\left(\mathfrak{Sel}^{\varepsilon}(K,\mathbf{T}_{f,\mathscr{O}_{\chi}})\otimes_{\Lambda_{\mathscr{O}_{\chi}}}\Lambda_{\mathscr{O}_{\chi}}/\mathfrak{P}_{\bar{\chi}}\right)/\left(\mathscr{O}_{\chi}\cdot\kappa_{\bar{\chi}}^{+}\right)\right)+2\cdot\mathrm{ord}_{\chi}(\omega_{n}^{-}).$

From (9.6), using Proposition 5.9 and Proposition 9.2(2), we obtain

(9.7)
$\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathrm{Sel}_{\mathrm{}}(K,A_{f}(\chi))_{/\mathrm{div}}\right)\leqslant
2\cdot\mathrm{length}_{\mathscr{O}_{\chi}}\left(\mathfrak{Sel}(K,{T}_{f}(\chi))/\left(\mathscr{O}_{\chi}\cdot\kappa_{\bar{\chi}}^{+}\right)\right)+2\cdot\mathrm{ord}_{\chi}(\omega_{n}^{-}).$

Now
$\tilde{\kappa}_{\bar{\chi}}=\chi(\omega_{n}^{-\varepsilon})\cdot\kappa_{\bar{\chi}}^{\varepsilon}$,
and the result follows.

_Step 2._ We now use Gross-Zagier formulas to conclude the proof of Theorem C.
By the Gross-Zagier formula ([14], [37])

${L^{\prime}(E/K,\chi,1)}=C\cdot h_{\mathrm{NT}}(P_{\bar{\chi}})$

where the non-zero complex constant $C$ is defined by

$C=\begin{cases}\frac{8\pi^{2}(f,f)}{h_{K}u_{K}\sqrt{D_{K}}},\text{ if
}N^{-}=1\\\ \frac{4(\phi^{\sharp},\phi^{\sharp})^{2}}{\sqrt{D_{K}}},\text{ if
}N^{-}\neq 1;\end{cases}$

here
$h_{\mathrm{NT}}:E(K_{n})\otimes_{\mathbf{Z}}\mathscr{O}_{\chi}\rightarrow\mathbf{R}$
is the $\mathscr{O}_{\chi}$-linear extension of the usual Néron-Tate height on
$E(K_{n})$, $(f,f)$ is the Petersson norm of $f$, $\phi^{\sharp}$ is the quasi
newform associated with $f$ as in [37], Theorem 1.2.1, and
$(\phi^{\sharp},\phi^{\sharp})$ is the $L^{2}$-norm of $\phi^{\sharp}$ with
respect to the Haar measure normalised as in _loc. cit._ ; see also [36],
Theorem 1.3.1, and [38], Theorem 8.1 and the discussion that follows. The
result follows combing Step 1 with the definitions of
$\mathrm{Reg}_{\chi}(E/K)$ and $\sha(K,A_{f}(\chi))$ introduced above and the
Gross-Zagier formula.

##  10\. Statements and declarations

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On behalf of all authors, the corresponding author states that there is no
conflict of interest.

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