The Corollary follows by a repeated application of Lemma 4.24 combined with
the weight $\lambda_{k}$-specialization of (4.7). Compare with [Sev12,
Corollary 4.7] and [BD07, Proposition 2.12]. Since our form
$\mathcal{F}_{k_{0}}$ is split–multiplicative (See Remark 4.17), we don’t need
to restrict to even vertices as in [Sev12]. ∎

######  Definition 4.27.

The semidefinite integral attached to $r,s\in\mathbb{P}^{1}(F)$;
$\tau\in\mathcal{H}_{\mathfrak{p}}^{\mathrm{ur}}$ and $P(x,y)\in
V_{k_{0},k_{0}}$ is defined as

$\displaystyle\int_{r}^{s}\int^{\tau}P(x,y)\omega_{\mathcal{F}}$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=|\mathscr{L}_{\tau}|^{k_{0}/2}\int_{\mathscr{Y}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau
y)\right)d\Phi_{\widetilde{L_{\tau}}}\\{{r-s\\}}(x,y)$

where $[\mathscr{L}_{\tau}]=\mathrm{red}_{\mathfrak{p}}(\tau)$.

######  Lemma 4.28.

Let $\alpha(\lambda_{\kappa})\in\mathcal{O}(U)$. For all
$e\in\mathcal{E}(\mathcal{T})$, $\tau\in\mathcal{H}_{\mathfrak{p}}$,
$r,s\in\mathbb{P}^{1}(F)$ and $P(x,y)\in V_{k_{0},k_{0}}$, we have

$\int_{Y_{e}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau
y)\right)d\alpha\Phi_{\widetilde{L_{\tau}}}\\{r-s\\}(x,y)\\\
=\alpha^{\prime}(\lambda_{k_{0}})|\mathscr{L}_{\tau}|^{-k_{0}/2}\int_{U_{e}}P(t)d\Phi_{\mathcal{F}}^{\mathrm{har}}\\{r-s\\}(t)\\\
+\alpha(\lambda_{k_{0}})\int_{Y_{e}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau
y)\right)d\Phi_{\widetilde{L_{\tau}}}\\{r-s\\}(x,y).$

###### Proof.

By Definition 4.14,

$\displaystyle\int_{Y_{e}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau
y)\right)d\alpha(\lambda_{\kappa})\Phi_{\widetilde{L_{\tau}}}\\{r-s\\}(x,y)$
$\displaystyle=\frac{d}{d\lambda_{\kappa}}\left(\alpha(\lambda_{\kappa})\int_{Y_{e}}\widetilde{P}(x,y)\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau
y)\right)^{\lambda_{\kappa}-\lambda_{k_{0}}}d\Phi_{\widetilde{L_{\tau}}}\\{r-s\\}(x,y)\right)_{\lambda_{\kappa}=\lambda_{k_{0}}}$
$\displaystyle=\alpha^{\prime}(\lambda_{k_{0}})\int_{Y_{e}}\widetilde{P}(x,y)d\Phi_{\widetilde{L_{\tau}}}\\{r-s\\}(x,y)$
$\displaystyle+\alpha(\lambda_{k_{0}})\int_{Y_{e}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau
y)\right)d\Phi_{\widetilde{L_{\tau}}}\\{r-s\\}(x,y)$
$\displaystyle=\alpha^{\prime}(\lambda_{k_{0}})|\mathscr{L}_{\tau}|^{-k_{0}/2}\int_{U_{e}}P(t)d\Phi_{\mathcal{F}}^{\mathrm{har}}\\{r-s\\}(t)$
$\displaystyle+\alpha(\lambda_{k_{0}})\int_{Y_{e}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau
y)\right)d\Phi_{\widetilde{L_{\tau}}}\\{r-s\\}(x,y).$

∎

######  Proposition 4.29.

For $r,s\in\mathbb{P}^{1}(F)$ and
$\tau_{1},\tau_{2}\in\mathcal{H}_{\mathfrak{p}}^{\mathrm{ur}}$, we have

$\int_{r}^{s}\int^{\tau_{2}}P(x,y)\omega_{\mathcal{F}}-\int_{r}^{s}\int^{\tau_{1}}P(x,y)\omega_{\mathcal{F}}=\\\
\int_{r}^{s}\int_{\tau_{1}}^{\tau_{2}}P(x,y)\omega^{\mathrm{log}_{p}}_{\mathcal{F}}+2\frac{a_{\mathfrak{p}}^{\prime}(\mathcal{F})}{a_{\mathfrak{p}}(\mathcal{F})}\int_{r}^{s}\int_{\tau_{1}}^{\tau_{2}}P(x,y)\omega^{\mathrm{ord}_{\mathfrak{p}}}_{\mathcal{F}}$

where $a_{\mathfrak{p}}^{\prime}(\mathcal{F})$ is defined as the derivative of
the $U_{\mathfrak{p}}$-eigenvalue of the specialisations of the Coleman family
$\mathbfcal{F}(g)$ at $\lambda_{k_{0}}$.

###### Proof.

We may suppose without loss of generality that
$\mathscr{L}_{\tau_{2}}\subset\mathscr{L}_{\tau_{1}}$ and that
$[\mathscr{L}_{\tau_{1}}:\mathscr{L}_{\tau_{2}}]=\mathrm{N}_{F/\mathbb{Q}}(p)$.
We denote by $e\in\mathcal{E}(\mathcal{T})$ to be the ordered edge between
$\mathscr{L}_{\tau_{1}}$ and $\mathscr{L}_{\tau_{2}}$. By Definition 4.27
above, we have

$\displaystyle\int_{r}^{s}\int^{\tau_{2}}P(x,y)\omega_{\mathcal{F}}$
$\displaystyle-\int_{r}^{s}\int^{\tau_{1}}P(x,y)\omega_{\mathcal{F}}$
$\displaystyle=|\mathscr{L}_{\tau_{2}}|^{k_{0}/2}\int_{\mathscr{Y}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{2}y)\right)d\Phi_{\widetilde{L_{\tau_{2}}}}\\{{r-s\\}}(x,y)$
$\displaystyle-|\mathscr{L}_{\tau_{1}}|^{k_{0}/2}\int_{\mathscr{Y}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{1}y)\right)d\Phi_{\widetilde{L_{\tau_{1}}}}\\{{r-s\\}}(x,y)$
$\displaystyle=|\mathscr{L}_{\tau_{2}}|^{k_{0}/2}\int_{\mathscr{Y}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{2}y)\right)d\Phi_{\widetilde{L_{\tau_{2}}}}\\{{r-s\\}}(x,y)$
$\displaystyle-|\mathscr{L}_{\tau_{2}}|^{k_{0}/2}\int_{\mathscr{Y}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{1}y)\right)d\Phi_{\widetilde{L_{\tau_{2}}}}\\{{r-s\\}}(x,y)$
$\displaystyle+|\mathscr{L}_{\tau_{2}}|^{k_{0}/2}\int_{\mathscr{Y}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{1}y)\right)d\Phi_{\widetilde{L_{\tau_{2}}}}\\{{r-s\\}}(x,y)$
$\displaystyle-|\mathscr{L}_{\tau_{1}}|^{k_{0}/2}\int_{\mathscr{Y}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{1}y)\right)d\Phi_{\widetilde{L_{\tau_{1}}}}\\{{r-s\\}}(x,y)$
$\displaystyle=I_{\mathrm{ord}}+I_{\log}$

where

$\displaystyle I_{\mathrm{ord}}$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=|\mathscr{L}_{\tau_{2}}|^{k_{0}/2}\int_{\mathscr{Y}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{1}y)\right)d\Phi_{\widetilde{L_{\tau_{2}}}}\\{{r-s\\}}(x,y)$
$\displaystyle-|\mathscr{L}_{\tau_{1}}|^{k_{0}/2}\int_{\mathscr{Y}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{1}y)\right)d\Phi_{\widetilde{L_{\tau_{1}}}}\\{{r-s\\}}(x,y).$

and

$\displaystyle I_{\log}$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=|\mathscr{L}_{\tau_{2}}|^{k_{0}/2}\int_{\mathscr{Y}}\widetilde{P}(x,y)\log_{p}\circ\mathrm{N}_{F/\mathbb{Q}}\left(\frac{x-\tau_{2}y}{x-\tau_{1}y}\right)d\Phi_{\widetilde{L_{\tau_{2}}}}\\{{r-s\\}}(x,y).$

By Corollary 4.26 above, we have

$\displaystyle I_{\log}$
$\displaystyle=\int_{\mathbb{P}^{1}_{\mathfrak{p}}}P(t)\log_{p}\circ\mathrm{N}_{F/\mathbb{Q}}\left(\frac{t-\tau_{2}}{t-\tau_{1}}\right)d\Phi_{\mathcal{F}}^{\mathrm{har}}\\{r-s\\}(t)$
$\displaystyle=\int_{r}^{s}\int_{\tau_{1}}^{\tau_{2}}P(x,y)\omega^{\mathrm{log}_{p}}_{\mathcal{F}}.$

We express $I_{\mathrm{ord}}$ as the sum of two contributions $I_{e}$ and
$I_{\overline{e}}$ obtained by evaluating the integrals over disjoint subsets
$Y_{e}$ and $Y_{\overline{e}}$ of $\mathscr{Y}$ respectively. By Lemma 4.24,

$\displaystyle I_{e}$
$\displaystyle=|\mathscr{L}_{\tau_{2}}|^{k_{0}/2}\frac{a_{\mathfrak{p}}(\mathcal{F}_{k_{0}})}{\mathrm{N}_{F/\mathbb{Q}}(p)^{k_{0}}}\int_{{Y_{e}}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{1}y)\right)d\Phi_{\widetilde{L_{\tau_{1}}}}\\{{r-s\\}}(x,y)$
$\displaystyle-|\mathscr{L}_{\tau_{1}}|^{k_{0}/2}\int_{{Y_{e}}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{1}y)\right)d\Phi_{\widetilde{L_{\tau_{1}}}}\\{{r-s\\}}(x,y)$
$\displaystyle=|\mathscr{L}_{\tau_{1}}|^{k_{0}/2}\left(\frac{a_{\mathfrak{p}}(\mathcal{F}_{k_{0}})}{\mathrm{N}_{F/\mathbb{Q}}(p)^{k_{0}/2}}-1\right)\int_{{Y_{e}}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{1}y)\right)d\Phi_{\widetilde{L_{\tau_{1}}}}\\{{r-s\\}}(x,y).$

Applying Lemma 4.28 with
$\alpha(\lambda_{\kappa})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=a_{\mathfrak{p}}(\mathcal{F}_{\lambda_{\kappa}})/\mathrm{N}_{F/\mathbb{Q}}(p)^{k_{0}/2}-1$
upon noting that $\alpha(\lambda_{k_{0}})=0$, we get

$I_{e}=\frac{a_{\mathfrak{p}}^{\prime}(\mathcal{F})}{\mathrm{N}_{F/\mathbb{Q}}(p)^{k_{0}/2}}\sum\limits_{e:\textup{red}_{\mathfrak{p}}(\tau_{1})\rightarrow\textup{red}_{\mathfrak{p}}(\tau_{2})}\int_{U_{e}}P(t)d\Phi_{\mathcal{F}}^{\mathrm{har}}\\{r-s\\}(t)$

Similarly for $\overline{e}$ \- the ordered edge between
$(1/p)\mathscr{L}_{2}$ and $\mathscr{L}_{1}$, noting that
$[(1/p)\mathscr{L}_{2}:\mathscr{L}_{1}]=\mathrm{N}_{F/\mathbb{Q}}(p)$, the
same computation using Lemma 4.24 gives

$\displaystyle I_{\overline{e}}$
$\displaystyle=|\mathscr{L}_{\tau_{2}}|^{k_{0}/2}\frac{a_{\mathfrak{p}}(\mathcal{F}_{k_{0}})}{\mathrm{N}_{F/\mathbb{Q}}(p)^{k_{0}}}\int_{Y_{\overline{e}}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{1}y)\right)d\Phi_{\widetilde{L_{\tau_{1}}}}\\{{r-s\\}}(x,y)$
$\displaystyle-|\mathscr{L}_{\tau_{1}}|^{k_{0}/2}\int_{Y_{\overline{e}}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{1}y)\right)d\Phi_{\widetilde{L_{\tau_{1}}}}\\{{r-s\\}}(x,y)$
$\displaystyle=$
$\displaystyle-|\frac{1}{p}\mathscr{L}_{\tau_{2}}|^{k_{0}/2}\left(\frac{a_{\mathfrak{p}}(\mathcal{F}_{k_{0}})}{\mathrm{N}_{F/\mathbb{Q}}(p)^{k_{0}/2}}-1\right)\int_{Y_{\overline{e}}}\widetilde{P}(x,y)\log_{p}\left(\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{1}y)\right)d\Phi_{\widetilde{(1/p)L_{\tau_{2}}}}\\{{r-s\\}}(x,y)$
$\displaystyle=-\frac{a_{\mathfrak{p}}^{\prime}(\mathcal{F})}{\mathrm{N}_{F/\mathbb{Q}}(p)^{k_{0}/2}}\sum\limits_{e:\textup{red}_{\mathfrak{p}}(\tau_{1})\rightarrow\textup{red}_{\mathfrak{p}}(\tau_{2})}\int_{U_{\overline{e}}}P(t)d\Phi_{\mathcal{F}}^{\mathrm{har}}\\{r-s\\}(t).$

Since $\mathbb{P}^{1}_{\mathfrak{p}}$ is a disjoint union of $U_{e}$ and
$U_{\overline{e}}$, we have

$\int_{U_{\overline{e}}}P(t)d\Phi_{\mathcal{F}}^{\mathrm{har}}\\{r-s\\}(t)+\int_{U_{e}}P(t)d\Phi_{\mathcal{F}}^{\mathrm{har}}\\{r-s\\}(t)=\int_{\mathbb{P}^{1}_{\mathfrak{p}}}P(t)d\Phi_{\mathcal{F}}^{\mathrm{har}}\\{r-s\\}(t)=0$

where the final equality follows from [BSW19, Proposition 5.8(i)]. Hence

$\displaystyle I_{\mathrm{ord}}=I_{e}+I_{\overline{e}}$
$\displaystyle=2\frac{a_{\mathfrak{p}}^{\prime}(\mathcal{F})}{\mathrm{N}_{F/\mathbb{Q}}(p)^{k_{0}/2}}\sum\limits_{e:\textup{red}_{\mathfrak{p}}(\tau_{1})\rightarrow\textup{red}_{\mathfrak{p}}(\tau_{2})}\int_{U_{\overline{e}}}P(t)d\Phi_{\mathcal{F}}^{\mathrm{har}}\\{r-s\\}(t)$
$\displaystyle=2\frac{a_{\mathfrak{p}}^{\prime}(\mathcal{F})}{\mathrm{N}_{F/\mathbb{Q}}(p)^{k_{0}/2}}\int_{r}^{s}\int_{\tau_{1}}^{\tau_{2}}P(x,y)\omega^{\mathrm{ord}_{\mathfrak{p}}}_{\mathcal{F}}$

by Definition 2.10(iii). Since
$a_{\mathfrak{p}}(\mathcal{F})=\mathrm{N}_{F/\mathbb{Q}}(p)^{k_{0}/2}$, we get
that

$\displaystyle\int_{r}^{s}\int^{\tau_{2}}P(x,y)\omega_{\mathcal{F}}$
$\displaystyle-\int_{r}^{s}\int^{\tau_{1}}P(x,y)\omega_{\mathcal{F}}$
$\displaystyle=I_{\log}+I_{\mathrm{ord}}$
$\displaystyle=\int_{r}^{s}\int_{\tau_{1}}^{\tau_{2}}P(x,y)\omega^{\mathrm{log}_{p}}_{\mathcal{F}}+2\frac{a_{\mathfrak{p}}^{\prime}(\mathcal{F})}{a_{\mathfrak{p}}(\mathcal{F})}\int_{r}^{s}\int_{\tau_{1}}^{\tau_{2}}P(x,y)\omega^{\mathrm{ord}_{\mathfrak{p}}}_{\mathcal{F}}.$

∎

######  Proposition 4.30.

The $\mathcal{L}$-invariant attached to the Bianchi cusp form $\mathcal{F}\in
S_{\underline{k_{0}}+2}(U_{0}(\mathcal{N}))^{\mathrm{new}}$ is given by

$\mathcal{L}_{\mathfrak{p}}^{\mathrm{BW}}=-2\frac{a_{\mathfrak{p}}^{\prime}(\mathcal{F})}{a_{\mathfrak{p}}(\mathcal{F})}$

###### Proof.

Recall that the Bianchi cuspform $\mathcal{F}$ is the base–change to $F$ of
the elliptic cuspidal newform $f\in S_{k_{0}+2}(\Gamma_{0}(N))$. By [VW21,
Lemma 4.4], we know that
$\mathcal{L}_{\mathfrak{p}}^{\mathrm{BW}}=2\mathcal{L}_{p}(f)$. Moreover,
since $p$ is inert in $F$, we know that
$a_{\mathfrak{p}}(\mathcal{F})=a_{p}(f)^{2}$ where $a_{p}$ is the
$U_{p}$-eigenvalue of the form $f$. The result follows since it is known that
(See for eg. [Sev12, Theorem 4.11])

$\mathcal{L}_{p}(f)=-2\frac{a_{p}^{\prime}(f)}{a_{p}(f)}$

where $a_{p}^{\prime}(f)$ is defined as the derivative of the
$U_{p}$-eigenvalue of the specialisations of the Coleman family
$\mathbf{f}(q)$ at $\lambda_{k_{0}}$. ∎

######  Corollary 4.31.

The semidefinite integral $\int_{r}^{s}\int^{\tau}P(x,y)\omega_{\mathcal{F}}$
satisfies

$\int_{r}^{s}\int^{\tau_{2}}P(x,y)\omega_{\mathcal{F}}-\int_{r}^{s}\int^{\tau_{1}}P(x,y)\omega_{\mathcal{F}}=\int_{r}^{s}\int^{\tau_{2}}_{\tau_{1}}P(x,y)\omega_{\mathcal{F}}^{\mathrm{log}_{\mathcal{L}_{p}(f)}}$

where $\log_{\mathcal{L}_{p}(f)}$ is the branch of the $p$-adic logarithm such
that $\log_{\mathcal{L}_{p}(f)}(p)=-\mathcal{L}_{p}(f)$.

###### Proof.

By Propositions 4.29 and 4.30 above, we have

$\int_{r}^{s}\int^{\tau_{2}}P(x,y)\omega_{\mathcal{F}}-\int_{r}^{s}\int^{\tau_{1}}P(x,y)\omega_{\mathcal{F}}=\\\
\int_{r}^{s}\int^{\tau_{2}}_{\tau_{1}}P(x,y)\omega_{\mathcal{F}}^{\mathrm{log}_{p}}-\mathcal{L}_{\mathfrak{p}}^{\mathrm{BW}}\int_{r}^{s}\int^{\tau_{2}}_{\tau_{1}}P(x,y)\omega_{\mathcal{F}}^{\mathrm{ord}_{\mathfrak{p}}}.$

By Remark 3.2 above, we have

$\Phi_{\mathcal{F}}^{\log_{p}}-\mathcal{L}_{\mathfrak{p}}^{\mathrm{BW}}\Phi_{\mathcal{F}}^{\mathrm{ord}_{\mathfrak{p}}}=\Phi_{\mathcal{F}}^{\log_{\mathcal{L}_{p}(f)}}$

or in terms of double integrals, we have

(4.16)
$\int_{r}^{s}\int^{\tau_{2}}_{\tau_{1}}P(x,y)\omega_{\mathcal{F}}^{\mathrm{log}_{p}}-\mathcal{L}_{\mathfrak{p}}^{\mathrm{BW}}\int_{r}^{s}\int^{\tau_{2}}_{\tau_{1}}P(x,y)\omega_{\mathcal{F}}^{\mathrm{ord}_{\mathfrak{p}}}=\int_{r}^{s}\int^{\tau_{2}}_{\tau_{1}}P(x,y)\omega_{\mathcal{F}}^{\mathrm{log}_{\mathcal{L}_{p}(f)}}$

∎

######  Theorem 4.32.

Let $\Psi\in\mathrm{Emb}(\mathcal{O},\mathcal{R})$ and let
$r\in\mathbb{P}^{1}(F)$ be an arbitrary base point. Then

$\int_{r}^{\gamma_{\Psi}r}\int^{\tau_{\Psi}}P_{\Psi}^{k_{0}/2}\omega_{\mathcal{F}}=\bigg{(}\sqrt{\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})}\bigg{)}^{k_{0}/2}\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{[\Psi]})(\Phi_{\mathcal{F}}^{\mathrm{har}})$

###### Proof.

By Corollary 4.31,

$\int_{r}^{\gamma_{\Psi}r}\int^{\tau_{\Psi}}P_{\Psi}^{k_{0}/2}\omega_{\mathcal{F}}-\int_{r}^{\gamma_{\Psi}r}\int^{\tau_{\Psi}^{\theta}}P_{\Psi}^{k_{0}/2}\omega_{\mathcal{F}}=\int_{r}^{\gamma_{\Psi}r}\int^{\tau_{\Psi}}_{\tau_{\Psi}^{\theta}}P_{\Psi}^{k_{0}/2}\omega_{\mathcal{F}}^{\mathrm{log}_{\mathcal{L}_{p}(f)}}\\\
=\Phi_{\mathcal{F}}^{\mathrm{log}_{\mathcal{L}_{p}(f)}}\left(\\{\gamma_{\Psi}r-r\\}\otimes\\{\tau_{\Psi}-\tau_{\Psi}^{\theta}\\}\otimes
P_{\Psi}^{k_{0}/2}\right)$

which by Remark 3.2 is equivalent to

(4.17)
$\mathrm{log}\;\Phi_{\mathcal{F}}\left(\\{\gamma_{\Psi}r-r\\}\otimes\\{\tau_{\Psi}-\tau_{\Psi}^{\theta}\\}\otimes
P_{\Psi}^{k_{0}/2}\right)\in\mathbf{MS}_{\Gamma}(L)_{(\mathcal{F})}^{\vee}.$

Then,

(4.18)
$\mathrm{log}\;\Phi^{\mathrm{AJ}}\left(\\{\gamma_{\Psi}r-r\\}\otimes\\{\tau_{\Psi}\\}\otimes
P_{\Psi}^{k_{0}/2}\right)\in\mathbf{MS}_{\Gamma}(L)_{(\mathcal{F})}^{\vee}$

is a lift of (4.17) above as in Remark 3.3. The theorem now follows Theorem
2.22 (See [VW21, Theorem 6.5] for more details) since the $\mathfrak{p}$-adic
Abel–Jacobi image of Stark–Heegner cycles are independent of the choice of the
$\mathfrak{p}$-adic Abel–Jacobi map. ∎

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

In this section, we briefly recall the construction and interpolation
properties of the various $p$-adic $L$-functions that are of relevance to this
article.

###  5.1. Bellaiche-Stevens-Mazur-Kitagawa $p$-adic $L$-functions

Let $\chi$ be a quadratic Dirichlet character of conductor $c(\chi)$
relatively prime to $N$. For $\lambda_{k}\in U$ a classical weight, which we
view as a classical weight on the weight space
$\mathcal{W}_{\mathbb{Q}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{Hom}_{\mathrm{cts}}(\mathbb{Z}_{p}^{\times},L^{\times})$
when the context is clear, let $L(f_{k}^{\\#},\chi,s)$ be the analytic
continuation of the $\chi$-twisted Dirichlet $L$-series $\sum\limits_{n\geq
1}\frac{\chi(n)a_{n}(f_{k}^{\\#})}{n^{s}}$ for $s\in\mathbb{C}$. We denote by
$\Omega_{\infty}(f_{k}^{\\#})^{\pm}\in\mathbb{C}^{\times}$ to be the Shimura
periods that satisfy

$\Omega_{\infty}(f_{k}^{\\#})^{+}\cdot\Omega_{\infty}(f_{k}^{\\#})^{-}=\langle
f_{k}^{\\#},f_{k}^{\\#}\rangle_{Mp^{r(k)}}$

where $r(k_{0})=1$ and zero otherwise, and $\langle,\rangle$ is the usual
Petersson norm. Then the _algebraic part_ of the special $L$-values, for
$1\leq j\leq k+1$, are given by

$L^{\mathrm{alg}}(f_{k}^{\\#},\chi,j)=\frac{(j-1)!\cdot\tau(\chi)\cdot
c(\chi)^{j}\cdot L(f_{k}^{\\#},\chi,j)}{(-2\pi
i)^{j-1}\cdot\Omega_{\infty}(f_{k}^{\\#})^{\pm}}\in\mathbb{Q}(f_{k}^{\\#},\chi)$

where
$\tau(\chi)=\sum_{a\in(\mathbb{Z}/c(\chi)\mathbb{Z})^{*}}\chi(a)\cdot\zeta^{a}_{c(\chi)}$
is the Gauss sum of $\chi$. The sign of the Shimura period is chosen according
to the sign of

$\chi(-1)=\pm(-1)^{j-1}$

######  Theorem 5.1 (Bellaiche, Stevens, Mazur, Kitagawa).

There exists $p$-adic periods $\Lambda^{\pm}(k)\in L$ for $\lambda_{k}\in
U\cap\mathbb{Z}$ (the set of classical weights in the affinoid $U$) such that
for any quadratic Dirichlet character $\chi$, there is a locally analytic
$p$-adic function $L_{p}(\mathbf{f},\chi)$ on $U\times\mathbb{Z}_{p}$ that
satisfies, for $1\leq j\leq k+1$,

(5.1)
$L_{p}(\mathbf{f},\chi)(\lambda_{k},j)=\begin{cases}\Lambda^{\pm}(k_{0})\Big{(}1-\frac{\chi(p)p^{j-1}}{a_{p}(k)}\Big{)}L^{\mathrm{alg}}(f_{k},\chi,j),&\text{if
}k=k_{0}\\\
\Lambda^{\pm}(k)\Big{(}1-\frac{\chi(p)p^{j-1}}{a_{p}(k)}\Big{)}\Big{(}1-\frac{\chi(p)p^{k-j+1}}{a_{p}(k)}\Big{)}L^{\mathrm{alg}}(f_{k}^{\\#},\chi,j),&\text{if
}k\neq k_{0}\end{cases}$

###### Proof.

See [Bel12] for instance. ∎

######  Remark 5.2.

We can normalize $L_{p}(\mathbf{f},\chi)$ such that $\Lambda^{\pm}(k_{0})=1$.

$L_{p}(\mathbf{f},\chi)(\lambda_{\kappa},s)$ is the two-variable $p$-adic
$L$-function attached to the Coleman family $\mathbf{f}(q)$ and the character
$\chi$ constructed by Mazur (unpublished) and Kitagawa ([Kit94]) in the slope
zero case (Hida families) and by Stevens in the finite slope scenario ([PS11]
& [PS13]). We will be primarily interested in the slice of the two variable
$p$-adic $L$-function along the central critical line $s=k/2+1$. Namely, set

$\mathcal{L}_{p}(\mathbf{f},\chi,\lambda_{k})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=L_{p}(\mathbf{f},\chi)(\lambda_{k},k/2+1)$

for all classical weights $\lambda_{k}\in U^{\mathrm{cl}}$. Theorem 5.1 above
implies that

(5.2)
$\mathcal{L}_{p}(\mathbf{f},\chi,\lambda_{k})=\begin{cases}\Big{(}1-\frac{\chi(p)p^{k/2}}{a_{p}(k)}\Big{)}L^{\mathrm{alg}}(f_{k},\chi,k/2+1),&\quad\text{if
}k=k_{0}\\\
\Lambda^{\pm}(k)\Big{(}1-\frac{\chi(p)p^{k/2}}{a_{p}(k)}\Big{)}^{2}L^{\mathrm{alg}}(f_{k}^{\\#},\chi,k/2+1),&\quad\text{if
}k\neq k_{0}\end{cases}$

###  5.2. $p$-adic $L$-functions attached to Bianchi modular forms

We briefly sumarize the construction of $p$-adic $L$-functions attached to
Bianchi modular forms in [Wil17] following the exposition of [BSW19, §3.4]
where the $p$-adic $L$-function is described in terms of analytic functions on
$\mathcal{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}\cong\mathcal{O}_{F_{\mathfrak{p}}}$
(since $p$ is inert in $F$) rather than as locally analytic distributions on
$\mathrm{Cl}_{F}(p^{\infty})$ \- the ray class group of $F$ of conductor
$p^{\infty}$. Let $\mathfrak{g}\subseteq\mathcal{O}_{F}$ be any ideal
relatively prime to $p$. It can be shown that

$\mathrm{Cl}_{F}(\mathfrak{g}p^{\infty})\cong[(\mathcal{O}_{F}/\mathfrak{g})^{\times}\times(\mathcal{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}_{p})^{\times}]/\mathcal{O}_{F}^{\times}$

Let $\mathcal{G}\in S_{\underline{k_{0}}+2}(U_{0}(\mathcal{N}))$ be a small
slope cuspidal Bianchi eigenform and let $\Phi_{\mathcal{G}}$ be the
_overconvergent modular symbol_ of [Wil17] attached to $\mathcal{G}$. Let
$\mu^{\prime}_{a(\textrm{mod }\mathfrak{g})}$ be a distribution on
$\\{{[a]\\}}\times(\mathcal{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}_{p})\subset(\mathcal{O}_{F}/\mathfrak{g})^{\times}\times(\mathcal{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}_{p})$
(which can be seen as a copy of
$(\mathcal{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}_{p})$) defined as

(5.3) $\mu^{\prime}_{a(\textrm{mod
}\mathfrak{g})}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(g\overline{g})^{k_{0}/2}\Bigg{[}\Phi_{\mathcal{G}}\Biggl{\lvert}\begin{pmatrix}1&b\\\
0&g\end{pmatrix}\Bigg{]}\\{0-\infty\\}$

where $b$ is some lift of $a(\textrm{mod }\mathfrak{g})$ and
$\mathfrak{g}\mathcal{O}_{F}=(g)$. Combining the distributions for different
$a\in(\mathcal{O}_{F}/\mathfrak{g})^{\times}$, we get a distribution $\mu_{p}$
on
$(\mathcal{O}_{F}/\mathfrak{g})^{\times}\times(\mathcal{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}_{p})$.
On restricting to units in the second variable and then to restricting to
functions invariant under $\mathcal{O}_{F}^{\times}$, we obtain a distribution
on $\mathrm{Cl}_{F}(\mathfrak{g}p^{\infty})$ afforded by the identification
above.

######  Definition 5.3.

Let $\chi$ be a finite order Hecke character of conductor $\mathfrak{gf}$
where $\mathfrak{g}$ is coprime to $p$ and $\mathfrak{f}\mid p^{\infty}$
(which can be seen as a finite order character of
$\mathrm{Cl}_{F}(\mathfrak{g}p^{\infty})$). The $p$-adic $L$-function
associated to $\mathcal{G}\in S_{\underline{k_{0}}+2}(U_{0}(\mathcal{N}))$ is
defined to be the analytic function on
$(\mathcal{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}_{p})$ given by

$L_{p}(\mathcal{G},\chi,s)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{\mathrm{Cl}(\mathfrak{g}p^{\infty})}\langle\mathbf{z}_{p}\rangle^{s}\chi(\mathbf{z})d\mu_{p}(\mathbf{z}),$

where
$s\in\mathcal{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}\cong\mathcal{O}_{F_{\mathfrak{p}}}$
and $\mathbf{z}_{p}$ is the projection of
$\mathbf{z}\in\mathrm{Cl}(\mathfrak{g}p^{\infty})$ to
$\mathrm{Cl}(p^{\infty})$.

Germane to this article is the two variable base–change Bianchi $p$-adic
$L$-function constructed by Seveso in [Sev12, Section 5.3],

(5.4) $L_{p}(\mathbfcal{F},\chi,\lambda_{\kappa}):U\rightarrow\mathbb{C}_{p}$

where $\chi$ is a finite order Hecke character of $F$. This two variable
$p$-adic $L$-function interpolates the central critical values of the weight
$k$-specializations of the Coleman family $\mathbfcal{F}=\mathbf{f}/F$ which
we recall below :-

######  Theorem 5.4.

Let $\chi$ be a finite order Hecke character of $F$ of conductor
$\mathfrak{c}_{\chi}$. Then, for all classical weights $\lambda_{k}\in U$,

(5.5)
$L_{p}(\mathbfcal{F},\chi,\lambda_{k})=\begin{cases}\left(1-\chi(p)\frac{\mathrm{N}_{F/\mathbb{Q}}(p)^{k/2}}{a_{\mathfrak{p}}(\mathcal{F}_{k})}\right)\cdot
L^{\mathrm{alg}}(\mathcal{F},\chi,k_{0}/2+1)&\text{if }k=k_{0}\\\
C(k)\Big{(}1-\chi(p)\frac{\mathrm{N}_{F/\mathbb{Q}}(p)^{k/2}}{a_{\mathfrak{p}}(\mathcal{F}_{k})}\Big{)}^{2}\cdot
L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#},\chi,k/2+1)&\text{if }k\neq
k_{0}\end{cases}$

where $C(k)\in L^{\times}$ is the $p$-adic period of Theorem 4.20 and

(5.6)
$L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#},\chi,k/2+1)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=u_{F}\frac{D_{F}^{k/2}(k/2)!^{2}\tau(\chi^{-1})(\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{c}_{\chi}))^{k/2}}{(2\pi
i)^{k}\Omega_{k}^{\\#}}L(\mathcal{F}_{k}^{\\#},\chi,k/2+1)\in\overline{\mathbb{Q}}$

for
$u_{F}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left[\mathcal{O}_{F}^{\times}:\mathbb{Z}^{\times}\right]=|\mu(\mathcal{O}_{F})|/2$.

###### Proof.

See [BSW19, Theorem 3.12], [Sev12, Theorem 5.17] and [BD07, Theorem 3.8]. ∎

We then have the following factorisation of the Seveso $p$-adic $L$-function
from above as a product of two Stevens–Mazur–Kitagawa $p$-adic $L$-functions
of Section 5.1.

######  Theorem 5.5.

Let $\epsilon_{F/\mathbb{Q}}$ denote the quadratic Dirichlet character
associated to the imaginary quadratic field $F/\mathbb{Q}$. Then there exists
a $p$-adic analytic function $\eta$ of $\lambda_{\kappa}\in U$ such that

(5.7)
$L_{p}(\mathbfcal{F},\lambda_{\kappa})=L_{p}(\mathbf{f}/F,\lambda_{\kappa})=\eta(\lambda_{\kappa})u_{F}\mathcal{L}_{p}(\mathbf{f},\lambda_{\kappa})\mathcal{L}_{p}(\mathbf{f},\epsilon_{F/\mathbb{Q}},\lambda_{\kappa})$

###### Proof.

We refer the reader to [Sev12, Theorem 5.21] which is based on [BD07,
Corollary 5.3]. ∎

###  5.3. Heegner cycles

We briefly recall the connection between Heegner cycles and the $p$-adic
$L$-functions considered above, primarily following the exposition in [IS03],
[Sev12], [Sev14] and [GSS16]. Recall the factorization $N=pM=pN^{+}N^{-}$. Let
$\mathcal{B}$ (resp. $B$) be the indefinite (resp. definite) quaternion
algebra ramified at the primes dividing $pN^{-}$ (resp. $N^{-}\infty$). Let
$\mathcal{R}^{\prime}=\mathcal{R}_{N^{+},pN^{-}}$ (resp. $R^{\prime}$) be a
fixed Eichler order of level $N^{+}$ in $\mathcal{O}_{\mathcal{B}}$ (resp. of
level $pN^{+}$ in $\mathcal{O}_{B}$) where $\mathcal{O}_{\mathcal{B}}$ and
$\mathcal{O}_{B}$ are maximal orders in $\mathcal{B}$ and $B$ respectively. We
set
$\widehat{B}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=B\otimes\widehat{\mathbb{Z}}$
and
$\widehat{R^{\prime}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=R^{\prime}\otimes\widehat{\mathbb{Z}}$.
Let
$\Sigma=\prod\limits_{\ell}\Sigma_{\ell}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\widehat{R^{\prime}}^{\times}$.
Fix an identification
$\iota_{p}:B\otimes\mathbb{Q}_{p}\cong\mathrm{M}_{2}(\mathbb{Q}_{p})$ and set

$\widetilde{\Gamma}^{\prime}_{\Sigma}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\iota_{p}\left(\mathcal{O}_{B}[1/p]\cap\prod\limits_{\ell\neq
p}\Sigma_{\ell}\right)=\iota_{p}\left(R^{\prime}[1/p]^{\times}\right)$

and by $\Gamma^{\prime}_{\Sigma}=\Gamma^{\prime}$ to be the subgroups of
elements of reduced norm one of $\widetilde{\Gamma}^{\prime}_{\Sigma}$. Denote
by $X\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=X_{N^{+},pN^{-}}$
the Shimura curve attached to $\mathcal{B}$ and by $f^{\mathrm{JL}}\in
M_{k_{0}+2}(X)$ the weight $k_{0}+2$ modular form on the Shimura curve $X$
attached to $f$ via the Jacquet–Langlands correspondence. By the
Cerednik–Drinfeld Theorem of $p$-adic uniformization, we have a rigid analytic
isomorphism

$X_{\Gamma^{\prime}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\Gamma^{\prime}\backslash\mathcal{H}_{p}\cong
X^{\mathrm{an}}$

This identification between the Mumford curve $X_{\Gamma^{\prime}}$ and the
rigid analytification of the Shimura curve $X^{\mathrm{an}}$ is defined over
$F_{\mathfrak{p}}\cong\mathbb{Q}_{p^{2}}$ (See [BC91]). We denote by
$f^{\mathrm{rig}}\in M_{k_{0}+2}(\Gamma^{\prime},F_{\mathfrak{p}})$ the rigid
analytic modular form associated to $f$ via this identification.

#####

Let $\mathcal{M}_{k_{0}}$ be the Chow motive over $\mathbb{Q}$, attached to
the space of weight $k_{0}+2$ modular forms on the Shimura curve $X$ and let
$V(k_{0}/2+1)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=H_{p}(\mathcal{M}_{k_{0},\overline{\mathbb{Q}}},\mathbb{Q}_{p}(k_{0}/2+1))$
be its $p$-adic realization. See [IS03, Appendix 10.1] for the construction.
By [IS03, Lemma 5.8], the $G_{\mathbb{Q}}$-representation $V(k_{0}/2+1)$ maybe
realized as the representation attached to weight $k_{0}+2$ cusp forms that
are new at the primes dividing $pN^{-}$. In particular, $V_{p}(f)(k_{0}/2+1)$
maybe realized as the idempotent component (corresponding to $f$) of the
representation $V(k_{0}/2+1)$. For $H/\mathbb{Q}$ any number field, we have
the global $p$-adic étale Abel–Jacobi map

$\mathrm{cl}_{H}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{cl}_{0,H}^{k_{0}/2+1}:\mathrm{CH}^{k_{0}/2+1}\left(\mathcal{M}_{k_{0}}\otimes
H\right)\rightarrow\mathrm{Sel}_{\mathrm{st}}(H,V(k_{0}/2+1))$

where $\mathcal{M}_{k_{0}}\otimes H$ is the base–change of
$\mathcal{M}_{k_{0}}/\mathbb{Q}$ to $H$ and $\mathrm{CH}^{k_{0}/2+1}$ denotes
the Chow group of co-dimension $k_{0}/2+1$ cycles. We may also consider the
projection $V(k_{0}/2+1)\rightarrow V_{p}(f)(k_{0}/2+1)$ to obtain

$\mathrm{cl}_{f,H}:\mathrm{CH}^{k_{0}/2+1}\left(\mathcal{M}_{k_{0}}\otimes
H\right)\rightarrow\mathrm{Sel}_{\mathrm{st}}(H,V_{p}(f)(k_{0}/2+1))$

Let $\mathfrak{P}$ be a prime in $H$ above $p$ and let $H_{\mathfrak{P}}$ be
its $\mathfrak{P}$-adic completion. Then we have a commutative diagram

(5.8) ${\mathrm{CH}^{k_{0}/2+1}\left(\mathcal{M}_{k_{0}}\otimes
H\right)}$${\mathrm{Sel}_{\mathrm{st}}(H,V_{p}(f)(k_{0}/2+1))}$${\mathrm{CH}^{k_{0}/2+1}\left(\mathcal{M}_{k_{0}}\otimes
H_{\mathfrak{P}}\right)}$${\mathrm{H}^{1}_{\mathrm{st}}(H_{\mathfrak{P}},V_{p}(f)(k_{0}/2+1))}$$\scriptstyle{\mathrm{cl}_{f,H}}$$\scriptstyle{\mathrm{res}_{\mathfrak{P}}}$$\scriptstyle{\mathrm{cl}_{f,H_{\mathfrak{P}}}}$

Let

$\mathbb{D}_{f}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbb{D}_{\mathrm{st}}(V_{p}(f)|_{G_{\mathbb{Q}_{p}}})$

denote the rank-two $(\varphi,N)$-module attached to $V_{p}(f)$. By [IS03,
(49)], we have the following identification

$\displaystyle\mathrm{IS}:\mathrm{H}^{1}_{\mathrm{st}}(H_{\mathfrak{P}},V_{p}(f)(k_{0}/2+1))$
$\displaystyle\overset{\mathrm{log_{BK}}}{\cong}\frac{\mathbb{D}_{f}\otimes
H_{\mathfrak{P}}}{\mathrm{Fil}^{k_{0}/2+1}(\mathbb{D}_{f}\otimes
H_{\mathfrak{P}})}$ $\displaystyle\cong
M_{k_{0}+2}(X,H_{\mathfrak{P}})_{(f^{\mathrm{JL}})}^{\vee}\cong
M_{k_{0}+2}(\Gamma^{\prime},H_{\mathfrak{P}})_{(f^{\mathrm{rig}})}^{\vee}$

where $\mathrm{log}_{\mathrm{BK}}$ is the Bloch–Kato logarithm and the final
identification holds assuming $H_{\mathfrak{P}}\supseteq
F_{\mathfrak{p}}\cong\mathbb{Q}_{p^{2}}$. Here $(-)^{\vee}$ stands for the
$H_{\mathfrak{P}}$-dual and $(f^{?})$ stands for the $f^{?}$-isotypic
component. It would be useful to consider the composition

(5.9)
$\mathrm{log\;cl}_{f,L}:\mathrm{CH}^{k_{0}/2+1}\left(\mathcal{M}_{k_{0}}\otimes
F\right)\rightarrow\mathrm{Sel}_{\mathrm{st}}(F,V_{p}(f)(k_{0}/2+1))\rightarrow\mathrm{Sel}_{\mathrm{st}}(K,V_{p}(f)(k_{0}/2+1))\\\
\xrightarrow{\mathrm{res}_{\mathfrak{p}}}\mathrm{H}^{1}_{\mathrm{st}}(L,V_{p}(f)(k_{0}/2+1))\rightarrow\frac{\mathbb{D}_{f}\otimes
L}{\mathrm{Fil}^{k_{0}/2+1}(\mathbb{D}_{f}\otimes L)}\\\ \rightarrow
M_{k_{0}+2}(X,L)_{(f^{\mathrm{JL}})}^{\vee}\rightarrow
M_{k_{0}+2}(\Gamma^{\prime},L)_{(f^{\mathrm{rig}})}^{\vee}$

where $L/\mathbb{Q}_{p}$ is as before (recall that $L\supseteq
K_{\mathfrak{p}}$) and

$\mathrm{Sel}_{\mathrm{st}}(F,V_{p}(f)(k_{0}/2+1))\rightarrow\mathrm{Sel}_{\mathrm{st}}(K,V_{p}(f)(k_{0}/2+1))$

is the usual restriction map.

#####

The following result is proved in [Sev14] generalising the weight the
$k_{0}=0$ setting of [BD07]. See also [Sev12, Section 5.3.2] for more details.

######  Theorem 5.6.

There exists a global cycle

$\mathcal{Y}\in\mathrm{CH}^{k_{0}/2+1}\left(\mathcal{M}_{k_{0}}\otimes
F\right)$

such that

$\frac{d^{2}}{d\lambda_{\kappa}^{2}}\left[L_{p}(\mathbfcal{F},\lambda_{\kappa})\right]_{\lambda_{\kappa}=\lambda_{k_{0}}}=\frac{d^{2}}{d\lambda_{\kappa}^{2}}\left[L_{p}(\mathbf{f}/F,\lambda_{\kappa})\right]_{\lambda_{\kappa}=\lambda_{k_{0}}}=2\mathrm{log\;cl}_{f,L}(\mathcal{Y})(f^{\mathrm{rig}})^{2}$

###### Proof.

This is [Sev12, Corollary 5.27] which is in turn a special case of [Sev14,
Corollary 9.2]. ∎

######  Remark 5.7.

The global cycle
$\mathcal{Y}\in\mathrm{CH}^{k_{0}/2+1}\left(\mathcal{M}_{k_{0}}\otimes
F\right)$ is the Heegner cycle (associated to the trivial character $\chi=1$)
constructed in [IS03, Section 8] using the theory of Complex Multiplication.

###  5.4. $p$-adic $L$-functions over $K$

Recall that $K/F$ is a relative quadratic extension that satisfies the
Stark–Heegner hypothesis (SH–Hyp)

  * •

$\mathfrak{p}$ is inert in $K$

  * •

All primes $\mathfrak{l}\mid\mathcal{M}$ split in $K$

Let $\Psi\in\mathrm{Emb}(\mathcal{O},\mathcal{R})$ be an optimal embedding of
conductor $\mathcal{C}$ relatively prime to $\mathcal{N}\mathcal{D}_{K/F}$.
Let
$G_{\mathcal{C}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{Gal}(H_{\mathcal{C}}/K)$
be the corresponding Galois group of the ring class field of conductor
$\mathcal{C}$ and $(\tau_{\Psi},P_{\psi},\gamma_{\Psi})$ be the data attached
to the embedding $\Psi$ as in §2.6. Let $\mathscr{L}_{\Psi}$ be an
$\mathcal{O}_{F_{\mathfrak{p}}}$-lattice corresponding to the vertex
$v_{\Psi}$ and let
$\widetilde{L}_{\Psi}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mu(\mathcal{O}_{F})\backslash\mathscr{L}_{\Psi}$.
In particular
$\mathscr{L}_{\Psi}=\mathscr{L}_{\tau_{\Psi}}=\mathscr{L}_{\tau^{\theta}_{\Psi}}$
since
$v_{\Psi}=\mathrm{red}_{\mathfrak{p}}(\tau_{\Psi})=\mathrm{red}_{\mathfrak{p}}(\tau^{\theta}_{\Psi})$.

Following [BD09] and [Sev12], we define a _partial square root $p$-adic
$L$-function_ to such an embedding $\Psi$ as follows

######  Definition 5.8.

Let $r\in\mathbb{P}^{1}(F)$ be any base point.

  * (i)

The partial square root $p$-adic $L$-function attached to
$(\mathbfcal{F}/K,\Psi)$ is defined as

$\mathcal{L}_{p}(\mathbfcal{F}/K,\Psi,\lambda_{\kappa})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=|\mathscr{L}_{\Psi}|^{k_{0}/2}\int_{\widetilde{L}_{\Psi}^{\prime}}\langle
P_{\Psi}(x,y)\rangle^{\frac{\lambda_{\kappa}-\lambda_{k_{0}}}{2}}P_{\Psi}^{k_{0}/2}(x,y)d\Phi_{\widetilde{L}_{\Psi}}\\{r-\gamma_{\Psi}r\\}.$

  * (ii)

The partial square root $p$-adic $L$-function attached to
$(\mathbfcal{F}/K,\psi_{K})$ for
$\psi_{K}:G_{\mathcal{C}}\rightarrow\mathbb{C}^{\times}$ is then defined as

$\mathcal{L}_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{\kappa})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum\limits_{\sigma\in
G_{\mathcal{C}}}\psi_{K}^{-1}(\sigma)\mathcal{L}_{p}(\mathbfcal{F}/K,\sigma\Psi,\lambda_{\kappa})$

and finally

  * (iii)

the _$p$ -adic $L$-function_ attached to $(\mathbfcal{F}/K,\psi_{K})$ is
defined as

$L_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{k})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathcal{L}_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{\kappa})^{2}.$

######  Remark 5.9.

A priori the $p$-adic $L$-functions defined above depend on the
$\mathbb{D}_{U}^{\dagger}$-valued modular symbol $\Phi_{\widetilde{L}_{\Psi}}$
of Proposition 4.22 associated to the lattice $\mathscr{L}_{\Psi}$. It can be
shown that the definition depends only on the class of optimal embeddings
$[\Psi]\in\Gamma\backslash\mathrm{Emb}(\mathcal{O},\mathcal{R})$ (See [GSS16,
Lemma 5.1]). Following [Sev12, Remark 5.6], we choose a lattice
$\mathscr{L}_{\Psi}$ as follows. Since $\Gamma$ acts transitively on the set
of vertices $\mathcal{V}(\mathcal{T})$, let $\gamma\in\Gamma$ be such that
$\gamma v_{\Psi}=v_{*}$. Then $v_{*}=v_{\gamma\Psi\gamma^{-1}}$ and
$\mathscr{L}_{*}=\mathscr{L}_{\gamma\Psi\gamma^{-1}}$ is the lattice
associated to the optimal embedding $\gamma\Psi\gamma^{-1}\in[\Psi]$. We show
later on in Theorem 5.12 that this choice of a lattice associated to a class
of optimal embeddings $[\Psi]$ is the natural one to consider.

#####

The $p$-adic $L$-function $L_{p}(\mathbfcal{F}/K,\psi_{K},-)$ defined above
interpolates $L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#}/K,\psi_{K},k/2+1)$ – the
_algebraic part_ of central critical $L$-values of the newforms
$\mathcal{F}_{k}^{\\#}$,

(5.10)
$L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#}/K,\psi_{K},k/2+1)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\\
\frac{Tu_{K}^{2}}{(\Omega_{k}^{\\#})^{2}}\cdot\frac{\prod_{\nu\in\Sigma_{\infty}^{F}}C^{\prime}_{\nu}(K,\pi_{k},\psi_{K})}{\sqrt{\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})}}\cdot\frac{((k/2)!)^{4}}{(2\pi)^{2k+4}}L(\mathcal{F}_{k}^{\\#}/K,\psi_{K},k/2+1)\in\overline{\mathbb{Q}}.$

where
$u_{K}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left[\mu(\mathcal{O}_{K}):\mu(\mathcal{O}_{F})\right]$
and $T$ and $C^{\prime}_{\nu}(K,\pi_{k},\psi_{K})$ are explicit constants (See
Appendix A). This entails rewriting the adelic toric periods appearing in
Waldspurger’s formula (See [MW09] and [FMP17]) in terms of certain geodesic
cycles.

Let

(5.11)
$\mathbb{L}(\mathcal{F}_{k}^{\\#},\psi_{K})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\Bigg{(}\sum\limits_{\sigma\in
G_{\mathcal{C}}}\psi_{K}^{-1}(\sigma)\phi_{k}^{\\#}\\{\tau-\gamma_{\sigma\Psi}\tau\\}\Big{(}(P_{\sigma\Psi}(x,y))^{k/2}\Big{)}\Bigg{)}^{2}$

where $\phi_{k}^{\\#}$ is the Bianchi modular symbol attached to the newform
$\mathcal{F}_{k}^{\\#}$. Then, the following generalization of Alexandru
Popa’s result ([Pop06, Theorem 6.3.1]) holds.

######  Theorem 5.10.

Let $\psi_{K}:\mathrm{Gal}(H_{K}/K)\rightarrow\mathbb{C}^{\times}$ be an
unramified character. Then

$\mathbb{L}(\mathcal{F}_{k}^{\\#},\psi_{K})=L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#}/K,\psi_{K},k/2+1).$

###### Proof.

The proof follows from Santiago Molina’s _Waldspurger formula in higher
cohomology_ ([Mol22, Theorem 4.6]) and an explicit form of Waldspurger formula
as in [MW09, Theorem 4.2] or [CST14, Theorem 1.8]. Given that such a result
could be of independent interest, we explain this in detail in Appendix A. ∎

######  Theorem 5.11.

The $p$-adic $L$-functions, for all $\Psi$ and $\psi_{K}$ as above, vanish at
$\lambda_{k_{0}}$, i.e.

$\mathcal{L}_{p}(\mathbfcal{F}/K,\Psi,\lambda_{k_{0}})=\mathcal{L}_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{k_{0}})=L_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{k_{0}})=0.$

Further, we also have

$\frac{d}{d\lambda}[L_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{\kappa})]_{\kappa=k_{0}}=0$

###### Proof.

By Definition 5.8 above, we have

$\displaystyle\mathcal{L}_{p}(\mathbfcal{F}/K,\Psi,\lambda_{k_{0}})$
$\displaystyle=|\mathscr{L}_{\Psi}|^{k_{0}/2}\int_{\widetilde{L}_{\Psi}^{\prime}}P_{\Psi}^{k_{0}/2}(x,y)d\Phi_{\widetilde{L}_{\Psi}}\\{\tau-\gamma_{\Psi}\tau\\}$
$\displaystyle=\int_{\mathbb{P}^{1}(F_{\mathfrak{p}})}P_{\Psi}^{k_{0}/2}(t)d\pi_{*}(\Phi_{\widetilde{L}_{\Psi}})\\{\tau-\gamma_{\Psi}\tau\\}$

where the second equality follows from (4.15). By Corollary 4.26, we then have
that

$\mathcal{L}_{p}(\mathbfcal{F}/K,\Psi,\lambda_{k_{0}})=\int_{\mathbb{P}^{1}(F_{\mathfrak{p}})}P_{\Psi}^{k_{0}/2}(t)d\Phi_{\mathcal{F}}^{\mathrm{har}}\\{\tau-\gamma_{\Psi}\tau\\}$

The vanishing now follows from [BSW19, Proposition 5.8(i)]. The defining
properties of the other $p$-adic $L$-functions implies the vanishing
simultaneously at $\lambda_{k_{0}}$. ∎

By Theorem 5.10, we have the following interpolation property of the $p$-adic
$L$-functions introduced in this section

######  Theorem 5.12.

For all classical weights $\lambda_{k}\in U$, $k\neq k_{0}$, we have

$L_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{k})=C(k)^{2}\left(1-\frac{\mathrm{N}_{F/\mathbb{Q}}(p)^{k}}{a_{\mathfrak{p}}(\mathcal{F}_{k})^{2}}\right)^{2}L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#}/K,\psi_{K},k/2+1)$

###### Proof.

By definition,

$\mathcal{L}_{p}(\mathbfcal{F}/K,\Psi,\lambda_{k})=|\mathscr{L}_{\Psi}|^{k_{0}/2}\int_{\widetilde{L}_{\Psi}^{\prime}}P_{\Psi}^{k/2}(x,y)d\Phi_{\widetilde{L}_{\Psi}}\\{\tau-\gamma_{\Psi}\tau\\}$

since $\langle
P_{\Psi}\rangle^{\frac{\lambda_{k}-\lambda_{k_{0}}}{2}}P_{\Psi}^{k_{0}/2}=P_{\Psi}^{k/2}[P_{\Psi}]^{\frac{\lambda_{k}-\lambda_{k_{0}}}{2}}=P_{\Psi}^{k/2}$
(as $[z]^{k}=[z]^{k_{0}}$ for all $\lambda_{k}\in U$ classical). By Remark 5.9
above, we may choose $\mathscr{L}_{\Psi}$ to be the lattice
$\mathscr{L}_{*}=\mathcal{O}_{F_{\mathfrak{p}}}\oplus\mathcal{O}_{F_{\mathfrak{p}}}$
and hence we get

$\mathcal{L}_{p}(\mathbfcal{F}/K,\Psi,\lambda_{k})=\int_{\widetilde{L}_{*}^{\prime}}P_{\Psi}^{k/2}(x,y)d\Phi_{\widetilde{L}_{*}}\\{\tau-\gamma_{\Psi}\tau\\}$

which by Lemma 4.25 implies that

$\mathcal{L}_{p}(\mathbfcal{F}/K,\Psi,\lambda_{k})=C(k)\left(1-\frac{\mathrm{N}_{F/\mathbb{Q}}(p)^{k}}{a_{\mathfrak{p}}(\mathcal{F}_{k})^{2}}\right)\phi_{k}^{\\#}\\{\tau-\gamma_{\Psi}\tau\\}\Big{(}(P_{\Psi}(x,y))^{k/2}\Big{)}.$

Hence, by Definition 5.8

$L_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{k})=C(k)^{2}\left(1-\frac{\mathrm{N}_{F/\mathbb{Q}}(p)^{k}}{a_{\mathfrak{p}}(\mathcal{F}_{k})^{2}}\right)^{2}\mathbb{L}(\mathcal{F}_{k}^{\\#},\psi_{K}).$

The proof follows from Theorem 5.10. ∎

###  5.5. Factorization of $p$-adic $L$-functions I

Let $\epsilon_{K/F}$ be the quadratic idèle class character of $F$ that cuts
out the relative quadratic extension $K/F$ and let $\chi_{1}$ and $\chi_{2}$
be a pair of quadratic Hecke characters such that
$\chi_{1}\cdot\chi_{2}=\epsilon_{K/F}$. Let $\psi_{K}$ be the _genus
character_ of $K$ associated to the pair $\chi_{1},\chi_{2}$, i.e.

$\mathrm{Ind}_{K}^{F}\psi_{K}=\chi_{1}\oplus\chi_{2}.$

Then by the classical Artin formalism, we have the following factorization of
$L$-functions :-

(5.12)
$L(\mathcal{F}_{k}^{\\#}/K,\psi_{K},s)=L(\mathcal{F}_{k}^{\\#},\chi_{1},s)L(\mathcal{F}_{k}^{\\#},\chi_{2},s).$

The goal of this section is to show that a similar factorization ($p$-adic
Artin formalisim) also holds at the level of $p$-adic $L$-functions. Set

(5.13)
$\eta\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{u_{K}^{2}.T.\prod_{\nu\in\Sigma_{\infty}^{F}}C^{\prime}_{\nu}(K,\pi_{k},\psi_{K})}{u_{F}^{2}(2\pi)^{4}\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})}.$

######  Theorem 5.13.

For all classical weights $\lambda_{k}\in U$,

$(D_{K})^{k/2}L_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{k})=\eta.L_{p}(\mathbfcal{F},\chi_{1},\lambda_{k}).L_{p}(\mathbfcal{F},\chi_{2},\lambda_{k})$

###### Proof.

Let $\lambda_{k}\in U(k\neq k_{0})$ be a classical weight. Then by (5.12)
above, we have

$L(\mathcal{F}_{k}^{\\#}/K,\psi_{K},k/2+1)=L(\mathcal{F}_{k}^{\\#},\chi_{1},k/2+1)L(\mathcal{F}_{k}^{\\#},\chi_{2},k/2+1)$

Recall that we have the following identities (See [Mok11, Appendix I] for
instance)

$\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{c}_{\chi_{1}})\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{c}_{\chi_{2}})=\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})$
$\tau(\chi_{1}^{-1})\tau(\chi_{2}^{-1})=\tau(\chi_{1})\tau(\chi_{2})=\sqrt{\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})}$

Further, upon combining (5.6) and (5.10), we get

(5.14)
$\frac{(2\pi)^{4}\sqrt{\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})}}{u_{K}^{2}.T.\prod_{\nu\in\Sigma_{\infty}^{F}}C^{\prime}_{\nu}(K,\pi_{k},\psi_{K})}L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#}/K,k/2+1)\\\
=\frac{1}{u_{F}^{2}\sqrt{\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})}D_{K}^{k/2}}L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#},k/2+1)L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#},\epsilon_{K/F},k/2+1)$

By Theorem 5.4 and Theorem 5.12 above, we have

$(D_{K})^{k/2}.L_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{k})=\eta.L_{p}(\mathbfcal{F},\chi_{1},\lambda_{k}).L_{p}(\mathbfcal{F},\chi_{2},\lambda_{k})$

for all $\lambda_{k}\in U$ classical. The proof for $\lambda_{k_{0}}$ is
simpler as both sides vanish. ∎

Note that since by assumption that $D_{K}$ is relatively prime to $p$, the
function $D_{K}^{k/2}$ extends to an analytic function on $U$,
$D_{K}^{\lambda_{\kappa}/2}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\langle
D_{K}\rangle^{\lambda_{\kappa}/2}$.

######  Corollary 5.14.

For all genus characters $\psi_{K}$ of $K$ associated to a pair of quadratic
Hecke characters $(\chi_{1},\chi_{2})$ of $F$ and for all $\lambda_{\kappa}\in
U$,

(5.15)
$(D_{K})^{\lambda_{\kappa}/2}L_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{\kappa})=\eta.L_{p}(\mathbfcal{F},\chi_{1},\lambda_{\kappa}).L_{p}(\mathbfcal{F},\chi_{2},\lambda_{\kappa}).$

###### Proof.

Since the set of classical points in $U$ is Zariski–dense and the two sides of
(5.15) are continuous functions on $U$, they agree on all points of $U$. ∎

###  5.6. A $p$-adic Gross–Zagier formula

We will now show a $p$-adic Gross–Zagier formula relating the (second
derivative of the) base–change $p$-adic $L$-function
$L_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{\kappa})$ and the $p$-adic
Abel–Jacobi image of Stark–Heegner cycles introduced in §2.6.

######  Theorem 5.15.

$\frac{d}{d\lambda_{\kappa}}[\mathcal{L}_{p}(\mathbfcal{F}/K,\Psi,\lambda_{\kappa})]_{\kappa=k_{0}}=\\\
\frac{1}{2}\left(\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})\right)^{k_{0}/4}\left(\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{[\Psi]})(\Phi_{\mathcal{F}}^{\mathrm{har}})+(-1)^{\frac{k_{0}+2}{2}}\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{[\Psi^{\theta}]})(\Phi_{\mathcal{F}}^{\mathrm{har}})\right)$

###### Proof.

Consider the factorization

$\displaystyle P_{\Psi}(x,y)$
$\displaystyle=A(x-\tau_{\Psi}y)(x-\tau_{\Psi}^{\theta}y)\overline{A}(\overline{x}-\tau_{\Psi}\overline{y})(\overline{x}-\tau_{\Psi}^{\theta}\overline{y})$
$\displaystyle=\mathrm{N}_{F/\mathbb{Q}}(A(x-\tau_{\Psi}y)(x-\tau_{\Psi}^{\theta}y)).$

Here $\theta$ is the non-trivial automorphism of $\mathrm{Gal}(K/F)$ (and the
over line denotes the non-trivial automorphism of
$\mathrm{Gal}(F/\mathbb{Q})$). Then, we can write

$\mathcal{L}_{p}(\mathbfcal{F}/K,\Psi,\lambda_{\kappa})=|\mathscr{L}_{\Psi}|^{k_{0}/2}\langle\mathrm{N}_{F/\mathbb{Q}}(A)\rangle^{\frac{\lambda_{\kappa}-\lambda_{k_{0}}}{2}}\\\
\int_{\widetilde{L}_{\Psi}^{\prime}}\langle\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{\Psi}y)\rangle^{\frac{\lambda_{\kappa}-\lambda_{k_{0}}}{2}}\langle\mathrm{N}_{F/\mathbb{Q}}(x-\tau_{\Psi}^{\theta}y)\rangle^{\frac{\lambda_{\kappa}-\lambda_{k_{0}}}{2}}P_{\Psi}^{k_{0}/2}(x,y)d\Phi_{\widetilde{L}_{\Psi}}\\{r-\gamma_{\Psi}r\\}$

By the product rule for derivatives,

$\frac{d}{d\lambda_{\kappa}}[\mathcal{L}_{p}(\mathbfcal{F}/K,\Psi,\lambda_{\kappa})]_{\kappa=k_{0}}=|\mathscr{L}_{\Psi}|^{k_{0}/2}\langle\mathrm{N}_{F/\mathbb{Q}}(A)\rangle^{\frac{\lambda_{k_{0}}-\lambda_{k_{0}}}{2}}\\\
\frac{d}{d\lambda_{\kappa}}\left[\int_{\widetilde{L}_{\Psi}^{\prime}}\langle\mathrm{N}_{F/\mathbb{Q}}((x-\tau_{\Psi}y)(x-\tau_{\Psi}^{\theta}y))\rangle^{\frac{\lambda_{\kappa}-\lambda_{k_{0}}}{2}}P_{\Psi}^{k_{0}/2}(x,y)d\Phi_{\widetilde{L}_{\Psi}}\\{r-\gamma_{\Psi}r\\}\right]_{\kappa=k_{0}}\\\
+|\mathscr{L}_{\Psi}|^{k_{0}/2}\int_{\widetilde{L}_{\Psi}^{\prime}}P_{\Psi}^{k_{0}/2}(x,y)d\Phi_{\widetilde{L}_{\Psi}}\\{r-\gamma_{\Psi}r\\}\cdot\frac{d}{d\lambda_{\kappa}}\left[\langle\mathrm{N}_{F/\mathbb{Q}}(A)\rangle^{\frac{\lambda_{\kappa}-\lambda_{k_{0}}}{2}}\right]$

By the proof of Theorem 5.11, we have the vanishing,

$|\mathscr{L}_{\Psi}|^{k_{0}/2}\int_{\widetilde{L}_{\Psi}^{\prime}}P_{\Psi}^{k_{0}/2}(x,y)d\Phi_{\widetilde{L}_{\Psi}}\\{r-\gamma_{\Psi}r\\}=0$

Hence, we get

$\frac{d}{d\lambda_{\kappa}}[\mathcal{L}_{p}(\mathbfcal{F}/K,\Psi,\lambda_{\kappa})]_{\kappa=k_{0}}=|\mathscr{L}_{\Psi}|^{k_{0}/2}\\\
\frac{d}{d\lambda_{\kappa}}\left[\int_{\widetilde{L}_{\Psi}^{\prime}}\langle\mathrm{N}_{F/\mathbb{Q}}((x-\tau_{\Psi}y)(x-\tau_{\Psi}^{\theta}y))\rangle^{\frac{\lambda_{\kappa}-\lambda_{k_{0}}}{2}}P_{\Psi}^{k_{0}/2}(x,y)d\Phi_{\widetilde{L}_{\Psi}}\\{r-\gamma_{\Psi}r\\}\right]_{\kappa=k_{0}}$

By Proposition 4.15,

$|\mathscr{L}_{\Psi}|^{k_{0}/2}\frac{d}{d\lambda_{\kappa}}\left[\int_{\widetilde{L}_{\Psi}^{\prime}}\langle\mathrm{N}_{F/\mathbb{Q}}((x-\tau_{\Psi}y)(x-\tau_{\Psi}^{\theta}y))\rangle^{\frac{\lambda_{\kappa}-\lambda_{k_{0}}}{2}}P_{\Psi}^{k_{0}/2}(x,y)d\Phi_{\widetilde{L}_{\Psi}}\\{r-\gamma_{\Psi}r\\}\right]_{\kappa=k_{0}}\\\
=\frac{1}{2}|\mathscr{L}_{\Psi}|^{k_{0}/2}\frac{d}{d_{\lambda_{\kappa}}}\left(\int_{\widetilde{L}_{\Psi}^{\prime}}\langle\mathrm{N}_{F/\mathbb{Q}}((x-\tau_{\Psi}y)\rangle^{\lambda_{\kappa}-\lambda_{k_{0}}}P_{\Psi}^{k_{0}/2}(x,y)d\Phi_{\widetilde{L}_{\Psi}}\\{r-\gamma_{\Psi}r\\}\right)_{\kappa=k_{0}}\\\
+\frac{1}{2}|\mathscr{L}_{\Psi}|^{k_{0}/2}\frac{d}{d_{\lambda_{\kappa}}}\left(\int_{\widetilde{L}_{\Psi}^{\prime}}\langle\mathrm{N}_{F/\mathbb{Q}}((x-\tau^{\theta}_{\Psi}y)\rangle^{\lambda_{\kappa}-\lambda_{k_{0}}}P_{\Psi}^{k_{0}/2}(x,y)d\Phi_{\widetilde{L}_{\Psi}}\\{r-\gamma_{\Psi}r\\}\right)_{\kappa=k_{0}}$

Note now that
$\mathscr{L}_{\Psi}=\mathscr{L}_{\tau_{\Psi}}=\mathscr{L}_{\tau^{\theta}_{\Psi}}$
and recall from Remark 2.18 that
$(\tau_{\Psi^{\theta}},P_{\Psi^{\theta}},\gamma_{\Psi^{\theta}})=(\tau_{\Psi}^{\theta},-P_{\Psi},\gamma_{\Psi}^{-1})$.
By Definition 4.27 on semidefinite integrals, we have

$\frac{d}{d\lambda_{\kappa}}[\mathcal{L}_{p}(\mathbfcal{F}/K,\Psi,\lambda_{\kappa})]_{\kappa=k_{0}}=\\\
\frac{1}{2}\left(\int_{r}^{\gamma_{\Psi}r}\int^{\tau_{\Psi}}P_{\Psi}(x,y)^{k_{0}/2}\omega_{\mathcal{F}}+(-1)^{k_{0}/2}\int_{r}^{\gamma_{\Psi^{\theta}}^{-1}r}\int^{\tau_{\Psi^{\theta}}}P_{\Psi^{\theta}}(x,y)^{k_{0}/2}\omega_{\mathcal{F}}\right)$

Replacing the arbitrary base point $r\in\mathbb{P}^{1}(F)$ by
$\gamma_{\Psi^{\theta}}r$, the second term on the RHS becomes

$\int_{r}^{\gamma_{\Psi^{\theta}}^{-1}r}\int^{\tau_{\Psi^{\theta}}}P_{\Psi^{\theta}}(x,y)^{k_{0}/2}\omega_{\mathcal{F}}=\int^{r}_{\gamma_{\Psi^{\theta}}r}\int^{\tau_{\Psi^{\theta}}}P_{\Psi^{\theta}}(x,y)^{k_{0}/2}\omega_{\mathcal{F}}=-\int_{r}^{\gamma_{\Psi^{\theta}}r}\int^{\tau_{\Psi^{\theta}}}P_{\Psi^{\theta}}(x,y)^{k_{0}/2}\omega_{\mathcal{F}}$

The result now follows from Theorem 4.32 above. ∎

An immediate consequence of Theorem 5.11 and Theorem 5.15 above is

######  Corollary 5.16.

For all unramified characters
$\psi_{K}:\mathrm{Gal}(H_{K}/K)\rightarrow\mathbb{C}^{\times}$,

$\frac{d^{2}}{d\lambda_{\kappa}^{2}}[L_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{\kappa})]_{\kappa=k_{0}}=\\\
\frac{1}{2}\left(\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})\right)^{k_{0}/2}\left(\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\psi_{K}})(\Phi_{\mathcal{F}}^{\mathrm{har}})+(-1)^{\frac{k_{0}+2}{2}}\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\psi_{K}}^{\theta})(\Phi_{\mathcal{F}}^{\mathrm{har}})\right)^{2}$

In particular, for $\psi_{K}=\psi_{\mathrm{triv}}$ \- the trivial character,
we may further simplify the expression of Corollary 5.16 above as

######  Corollary 5.17.

$\frac{d^{2}}{d\lambda_{\kappa}^{2}}[L_{p}(\mathbfcal{F}/K,\psi_{\mathrm{triv}},\lambda_{\kappa})]_{\kappa=k_{0}}=\\\
\frac{1}{2}\left(\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})\right)^{k_{0}/2}\left(1+(-1)^{\frac{k_{0}+2}{2}}\omega_{\mathcal{M}}\right)^{2}\left(\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\mathbbm{1}})(\Phi_{\mathcal{F}}^{\mathrm{har}})\right)^{2}$

###### Proof.

For $\sigma\in\mathrm{Gal}(H_{K}/K)$, let
$\sigma\Psi\in\Gamma/\mathrm{Emb}^{\mathfrak{o}_{\sigma\Psi}}(\mathcal{O},\mathcal{R})$
be a $\Gamma$-conjugacy class of oriented optimal embeddings. Since
$\Phi_{\mathcal{F}}^{\mathrm{har}}$ is an eigensymbol for the Atkin–Lehner
involution $W_{\mathcal{M}}$, we have

$\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{(\sigma\Psi)^{\theta}})(\Phi_{\mathcal{F}}^{\mathrm{har}}\mid
W_{\mathcal{M}})=\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\alpha_{\mathcal{M}}(\sigma\Psi)^{\theta}\alpha_{\mathcal{M}}^{-1}})(\Phi_{\mathcal{F}}^{\mathrm{har}})=\omega_{\mathcal{M}}\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{(\sigma\Psi)^{\theta}})(\Phi_{\mathcal{F}}^{\mathrm{har}}).$

Whilst $(\sigma\Psi)^{\theta}$ doesn’t have the same orientation (at
$\mathcal{M}$) as $\sigma\Psi$, we know that
$\alpha_{\mathcal{M}}(\sigma\Psi)^{\theta}\alpha_{\mathcal{M}}^{-1}\in\Gamma/\mathrm{Emb}^{\mathfrak{o}_{\sigma\Psi}}(\mathcal{O},\mathcal{R})$
(See Remark 2.26). By Proposition 2.25 which exhibits the set of the
$\Gamma$-conjugacy class of oriented optimal embeddings as a
$\mathrm{Gal}(H_{K}/K)$-torsor, we know that there exists
$\delta_{\sigma\Psi}\in\mathrm{Gal}(H_{K}/K)$ such that

$\alpha_{\mathcal{M}}(\sigma\Psi)^{\theta}\alpha_{\mathcal{M}}^{-1}=\delta_{\sigma\Psi}\sigma\Psi.$

Thus we have

$\displaystyle\sum\limits_{\sigma\in\mathrm{Gal}(H_{K}/K)}\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{(\sigma\Psi)^{\theta}})(\Phi_{\mathcal{F}}^{\mathrm{har}})=\omega_{\mathcal{M}}\sum\limits_{\sigma\in\mathrm{Gal}(H_{K}/K)}\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\delta_{\sigma\Psi}\sigma\Psi})(\Phi_{\mathcal{F}}^{\mathrm{har}})$

Yet again using the fact that the $\mathrm{Gal}(H_{K}/K)$-action on
$\Gamma/\mathrm{Emb}^{\mathfrak{o}_{\sigma\Psi}}(\mathcal{O},\mathcal{R})$ is
transitive, we get

$\displaystyle\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\mathbbm{1}}^{\theta})(\Phi_{\mathcal{F}}^{\mathrm{har}})$
$\displaystyle=\sum\limits_{\sigma\in\mathrm{Gal}(H_{K}/K)}\psi_{\mathrm{triv}}^{-1}(\sigma)\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{(\sigma\Psi)^{\theta}})(\Phi_{\mathcal{F}}^{\mathrm{har}})$
$\displaystyle=\omega_{\mathcal{M}}\sum\limits_{\sigma\in\mathrm{Gal}(H_{K}/K)}\psi_{\mathrm{triv}}^{-1}(\sigma)\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\delta_{\sigma\Psi}\sigma\Psi})(\Phi_{\mathcal{F}}^{\mathrm{har}})$
$\displaystyle=\omega_{\mathcal{M}}\sum\limits_{\sigma\in\mathrm{Gal}(H_{K}/K)}\psi_{\mathrm{triv}}^{-1}(\sigma)\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\sigma\Psi})(\Phi_{\mathcal{F}}^{\mathrm{har}})$
$\displaystyle=\omega_{\mathcal{M}}\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\mathbbm{1}})(\Phi_{\mathcal{F}}^{\mathrm{har}})$

and the result follows from Corollary 5.16 above. ∎

######  Corollary 5.18.

$\frac{d^{2}}{d\lambda_{\kappa}^{2}}[L_{p}(\mathbfcal{F}/K,\psi_{\mathrm{triv}},\lambda_{\kappa})]_{\kappa=k_{0}}=\\\
\begin{cases}2\left(\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})\right)^{\frac{k_{0}}{2}}\left(\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\mathbbm{1}})(\Phi_{\mathcal{F}}^{\mathrm{har}})\right)^{2}&\text{if
}\omega_{\mathcal{M}}=(-1)^{\frac{k_{0}+2}{2}}\\\ 0&\text{if
}\omega_{\mathcal{M}}=(-1)^{\frac{k_{0}}{2}}\end{cases}$

##  6\. Proof of the main result

We begin by noting the vanishing of the several $p$-adic $L$-functions
introduced above. For $\mathcal{F}\in
S_{\underline{k_{0}}+2}(U_{0}(\mathcal{N}))^{\mathrm{new}}$, we know that the
sign of the functional equation of the base–change $L$-function
$L(\mathcal{F}/K,s)$ is $-1$ by the (SH–Hyp). In particular the central
critical $L$-value $L(\mathcal{F}/K,k_{0}/2+1)$ vanishes to odd order. The
classical Artin formalism (5.12) above;

$L(\mathcal{F}/K,s)=L(\mathcal{F},s)L(\mathcal{F},\epsilon_{K/F},s)$

along with the Heegner hypothesis (Heeg–Hyp) shows that the sign of the
functional equation of $L(f/F,s)=L(\mathcal{F},s)$ is $-1$, which forces an
odd order of vanishing of the central critical $L$-value
$L(\mathcal{F},k_{0}/2+1)$, whilst that of
$L(f/F,\epsilon_{K/F},s)=L(\mathcal{F},\epsilon_{K/F},s)$ is $+1$ (See [Pal23,
Theorem 1.1]). We shall assume that
$L(\mathcal{F},\epsilon_{K/F},k_{0}/2+1)\neq 0$.

#####

By Remark 4.17, $\omega_{\mathfrak{p}}=1$ and hence by [BSW19, Theorem 9.3],
we know that the $p$-adic $L$-function $L_{p}(\mathcal{F},s)$ has a trivial
zero at $s=k_{0}/2+1$. Similarly, the $p$-adic $L$-function
$L_{p}(\mathcal{F},\epsilon_{K/F},s)$ doesn’t have a trivial zero at
$s=k_{0}/2+1$ since $\epsilon_{K/F}(p)=-1$ (recall that $p$ is inert in $K$ by
(SH–Hyp)). In particular, we may summarize that

$\mathrm{ord}_{s=k_{0}/2+1}L_{p}(\mathcal{F},s)\geq 2$

from which we conclude, by Theorem 5.4 above, that

(6.1)
$\mathrm{ord}_{\lambda_{\kappa}=\lambda_{k_{0}}}L_{p}(\mathbfcal{F},\lambda_{\kappa})\geq
2.$

We know, a priori, that the quantity

(6.2)
$S_{\mathcal{F}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=2\frac{L^{\mathrm{alg}}(\mathcal{F},\epsilon_{K/F},k_{0}/2+1)}{\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})}\in\mathbb{Q}(\mathcal{F})^{\times}.$

We shall work under the following assumption for the rest of the paper which
is consistent with the Birch and Swinnerton–Dyer conjecture (See [Mok11,
Remark 6.6] for instance) :-

######  Assumption 6.1.

We shall assume that the quantity $S_{\mathcal{F}}$ is a square, i.e.

$S_{\mathcal{F}}\in\left(\mathbb{Q}(\mathcal{F})^{\times}\right)^{2}$

We can now compare the $\mathfrak{p}$-adic Abel–Jacobi image of Stark–Heegner
cycles introduced in §2.6 with that of the Heegner cycles that appear in §5.3

######  Theorem 6.2.

Suppose that $\omega_{\mathcal{M}}=(-1)^{\frac{k_{0}+2}{2}}$. Then, under
Assumption 6.1, there exists
$\mathcal{Y}\in\mathrm{CH}^{k_{0}/2+1}\left(\mathcal{M}_{k_{0}}\otimes
F\right)\subset\mathrm{CH}^{k_{0}/2+1}\left(\mathcal{M}_{k_{0}}\otimes
K\right)$ and $s_{\mathcal{F}}\in\mathbb{Q}(\mathcal{F})^{\times}$ such that

$\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\mathbbm{1}})(\Phi_{\mathcal{F}}^{\mathrm{har}})=s_{\mathcal{F}}\cdot\mathrm{log\;cl}_{f,L}(\mathcal{Y})(f^{\mathrm{rig}}).$

###### Proof.

By Corollary 5.14 above, we have

$(D_{K})^{\lambda_{\kappa}/2}L_{p}(\mathbfcal{F}/K,\psi_{K},\lambda_{\kappa})=\eta.L_{p}(\mathbfcal{F},\chi_{1},\lambda_{\kappa}).L_{p}(\mathbfcal{F},\chi_{2},\lambda_{\kappa})$

Further, by Theorem 5.11 and (6.1) (See also Theorem 5.4), we have

(6.3)
$\frac{d^{2}}{d\lambda_{\kappa}^{2}}\left[L_{p}(\mathbfcal{F}/K,\lambda_{\kappa})\right]_{\lambda_{\kappa}=\lambda_{k_{0}}}=\frac{\eta}{D_{K}^{k_{0}/2}}\frac{d^{2}}{d\lambda_{\kappa}^{2}}\left[L_{p}(\mathbfcal{F},\lambda_{\kappa}))\right]_{\lambda_{\kappa}=\lambda_{k_{0}}}L_{p}(\mathbfcal{F},\epsilon_{K/F},\lambda_{k_{0}})$

By (5.5), we know that

$\displaystyle L_{p}(\mathbfcal{F},\epsilon_{K/F},\lambda_{k_{0}})$
$\displaystyle=\left(1-\epsilon_{K/F}(p)\frac{\mathrm{N}_{F/\mathbb{Q}}(p)^{k_{0}/2}}{a_{\mathfrak{p}}(\mathcal{F}_{k_{0}})}\right)\cdot
L^{\mathrm{alg}}(\mathcal{F},\epsilon_{K/F},k_{0}/2+1)$
$\displaystyle=2L^{\mathrm{alg}}(\mathcal{F},\epsilon_{K/F},k_{0}/2+1).$

By Corollary 5.18 & Theorem 5.6 above and Remark A.4 in Appendix A, we know
that

(6.4)
$\left(\mathrm{log}\>\Phi^{\mathrm{AJ}}(D_{\mathbbm{1}})(\Phi_{\mathcal{F}}^{\mathrm{har}})\right)^{2}=\frac{D_{F}^{k_{0}}u_{K}^{2}T(16\pi)^{2}}{D_{K}^{k_{0}}u_{F}^{2}16\pi^{4}}S_{\mathcal{F}}\cdot\left(\mathrm{log\;cl}_{f,L}(\mathcal{Y})(f^{\mathrm{rig}})\right)^{2}$

for $\mathcal{Y}\in\mathrm{CH}^{k_{0}/2+1}\left(\mathcal{M}_{k_{0}}\otimes
F\right)\subset\mathrm{CH}^{k_{0}/2+1}\left(\mathcal{M}_{k_{0}}\otimes
K\right)$. Note that
$T=\left(\phi,\phi\right)/\left(\phi_{\pi},\phi_{\pi}\right)$ is a square
since $\phi$ is a translate of $\phi_{\pi}$ (See Appendix A for more details).
The result now follows, under Assumption 6.1, upon extracting square-roots on
both sides of (6.4). ∎

Recall that since $\mathcal{F}$ is the base–change $f/F$, at the level of
Galois representations, we have

(6.5) $V_{p}(\mathcal{F})=V_{p}(f)|_{G_{F}}$

Let
$\mathbb{D}_{\mathcal{F}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbb{D}_{\mathrm{st}}(V_{p}(\mathcal{F}))$
(resp.
$\mathbb{D}_{f}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbb{D}_{\mathrm{st}}(V_{p}(f))$
be Fontaine’s semistable Dieudonné module attached to the Galois
representation $V_{p}(\mathcal{F})$ (resp. $V_{p}(f)$). By (6.5) above, we
have an identification

$\mathbb{D}_{\mathcal{F}}\cong\mathbb{D}_{f}\otimes_{\mathbb{Q}_{p}}F_{\mathfrak{p}}$

Further by [VW21, Theorem 4.5], we have an isomorphism of
$(\varphi,N)$-modules over $F_{\mathfrak{p}}$ with coefficients in $L$

$\bigoplus\limits_{\sigma}\mathbf{D}^{\sigma}_{\mathcal{F},L}=\mathbf{D}_{\mathcal{F}}\overset{\varphi}{\cong}\mathbb{D}_{\mathcal{F}}=\bigoplus\limits_{\sigma}\mathbb{D}_{\mathcal{F},L}^{\sigma}$

where
$\mathbb{D}_{\mathcal{F},L}^{\sigma}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbb{D}_{\mathcal{F}}\otimes_{F_{\mathfrak{p}}\otimes
L,\sigma}L$, which induces an identification of the tangent spaces

$\frac{\mathbf{D}_{\mathcal{F},L}^{\sigma}}{\mathrm{Fil}^{\frac{k_{0}+2}{2}}(\mathbf{D}_{\mathcal{F},L}^{\sigma})}\cong\frac{\mathbb{D}_{\mathcal{F},L}^{\sigma}}{\mathrm{Fil}^{\frac{k_{0}+2}{2}}(\mathbb{D}_{\mathcal{F},L}^{\sigma})}\cong\frac{\mathbb{D}_{f,L}}{\mathrm{Fil}^{\frac{k_{0}+2}{2}}(\mathbb{D}_{f,L})}$

for each $\sigma:F_{\mathfrak{p}}\hookrightarrow L$. We fix an isomorphism

$\mathbf{MS}_{\Gamma}(L)_{(\mathcal{F})}^{\vee}\overset{\alpha}{\cong}M_{k_{0}+2}(\Gamma^{\prime},L)_{(f^{\mathrm{rig}})}^{\vee}$

defined as follows

(6.6)
$\displaystyle\alpha:\mathbf{MS}_{\Gamma}(L)_{(\mathcal{F})}^{\vee}\overset{(\mathrm{Pr}^{\sigma})^{-1}}{\cong}$
$\displaystyle\frac{\mathbf{D}_{\mathcal{F},L}^{\sigma}}{\mathrm{Fil}^{\frac{k_{0}+2}{2}}(\mathbf{D}^{\sigma}_{\mathcal{F},L})}\overset{\varphi}{\cong}\frac{\mathbb{D}^{\sigma}_{\mathcal{F},L}}{\mathrm{Fil}^{\frac{k_{0}+2}{2}}(\mathbb{D}^{\sigma}_{\mathcal{F},L})}$
$\displaystyle\cong$
$\displaystyle\frac{\mathbb{D}_{f,L}}{\mathrm{Fil}^{\frac{k_{0}+2}{2}}(\mathbb{D}_{f,L})}\overset{\mathrm{exp}_{\mathrm{BK}}}{\cong}\mathrm{H}^{1}_{\mathrm{st}}(L,V_{p}(f)(k_{0}/2+1))$
$\displaystyle\overset{\mathrm{IS}}{\cong}$ $\displaystyle
M_{k_{0}+2}(X,L)_{(f^{\mathrm{JL}})}^{\vee}\cong
M_{k_{0}+2}(\Gamma^{\prime},L)_{(f^{\mathrm{rig}})}^{\vee}$

for either choice of an embedding $\sigma:F_{\mathfrak{p}}\hookrightarrow L$
(See Remark 3.4). In particular, we have a commutative diagram

(6.7)
${(\Delta_{0}\otimes\textup{Div}(\mathcal{H}_{\mathfrak{p}}^{\textup{ur}})\otimes
V_{k_{0},k_{0}})_{\Gamma}}$${\frac{\mathbf{D}^{\sigma}_{\mathcal{F},L}}{\mathrm{Fil}^{\frac{k_{0}+2}{2}}(\mathbf{D}^{\sigma}_{\mathcal{F},L})}}$${\mathrm{H}^{1}_{\mathrm{st}}(L,V_{p}(f)(k_{0}/2+1))}$${(\Delta_{0}\otimes\textup{Div}(\mathcal{H}_{\mathfrak{p}}^{\textup{ur}})\otimes
V_{k_{0},k_{0}})_{\Gamma}}$${\mathbf{MS}_{\Gamma}(L)_{(\mathcal{F})}^{\vee}}$${M_{k_{0}+2}(\Gamma^{\prime},L)_{(f^{\mathrm{rig}})}^{\vee}}$$\scriptstyle{\Phi^{\mathrm{AJ}}}$$\scriptstyle{\mathrm{exp}_{\mathrm{BK}}\circ\varphi}$$\scriptstyle{\mathrm{Pr^{\sigma}}}$$\scriptstyle{\mathrm{IS}}$$\scriptstyle{\mathrm{log}\;\Phi^{\mathrm{AJ}}}$$\scriptstyle{\alpha}$

######  Remark 6.3.

Note that the isomorphism of [VW21, Theorem 4.5],
$\mathbf{D}_{\mathcal{F}}\overset{\varphi}{\cong}\mathbb{D}_{\mathcal{F}}$ is
conditional on [VW21, Conjecture 4.2] in addition to the semistability of the
local Galois representation $V_{p}(\mathcal{F})|_{G_{F_{\mathfrak{p}}}}$.
However, in the base–change scenario both these conditions are satisfied,
making the isomorphism $\varphi$ unconditional. See [VW21, Lemma 4.4] for more
details.

We can now prove our main theorem, Theorem 1.4 from the Introduction, that
sheds evidence towards the global rationality conjectures formulated in [VW21,
Section 6.2] :-

#####

By Theorem 6.2 above, we have

(6.8)
$\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\mathbbm{1}})(\Phi_{\mathcal{F}}^{\mathrm{har}})=\mathrm{log\;cl}_{f,L}(s_{\mathcal{F}}\cdot\mathcal{Y})(f^{\mathrm{rig}}),$

and hence

(6.9)
$\alpha\left(\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\mathbbm{1}})\right)=\mathrm{log\;cl}_{f,L}(s_{\mathcal{F}}\mathcal{Y})=\mathrm{IS}\left(\mathrm{cl}_{f,L}(s_{\mathcal{F}}\mathcal{Y})\right)$

where the last equality follows from (5.9). The commutative diagram (6.7)
implies that

$\alpha\left(\mathrm{log}\>\Phi^{\mathrm{AJ}}(\mathrm{D}_{\mathbbm{1}})\right)=\alpha\left(\mathrm{Pr}^{\sigma}\left(\Phi^{\mathrm{AJ}}(\mathrm{D}_{\mathbbm{1}})\right)\right)=\mathrm{IS}\left(\mathrm{exp}_{\mathrm{BK}}\circ\varphi\left(\Phi^{\mathrm{AJ}}(\mathrm{D}_{\mathbbm{1}})\right)\right).$

In particular,

$\mathrm{IS}\left(\mathrm{exp}_{\mathrm{BK}}\circ\varphi\left(\Phi^{\mathrm{AJ}}(\mathrm{D}_{\mathbbm{1}})\right)\right)=\mathrm{IS}\left(\mathrm{cl}_{f,L}(s_{\mathcal{F}}\mathcal{Y})\right)$

from which we conclude that

$\mathrm{exp}_{\mathrm{BK}}\circ\varphi\left(\Phi^{\mathrm{AJ}}(\mathrm{D}_{\mathbbm{1}})\right)=\mathrm{cl}_{f,L}(s_{\mathcal{F}}\mathcal{Y})\in\mathrm{H}^{1}_{\mathrm{st}}(L,V_{p}(f)(k_{0}/2+1)).$

Let $\mathcal{S}_{K}\in\mathrm{Sel}_{\mathrm{st}}(K,V_{p}(f)(k_{0}/2+1))$
denote the image of the global cycle $s_{\mathcal{F}}\mathcal{Y}$ under

$\mathrm{cl}_{f,K}:\mathrm{CH}^{k_{0}/2+1}\left(\mathcal{M}_{k_{0}}\otimes
K\right)\rightarrow\mathrm{Sel}_{\mathrm{st}}(K,V_{p}(f)(k_{0}/2+1)).$

Then,

$\mathrm{exp}_{\mathrm{BK}}\circ\varphi\left(\Phi^{\mathrm{AJ}}(\mathrm{D}_{\mathbbm{1}})\right)=\mathrm{res}_{\mathfrak{p}}\left(\mathcal{S}_{K}\right)\in\mathrm{H}^{1}_{\mathrm{st}}(L,V_{p}(f)(k_{0}/2+1)).$

Theorem 1.4 now follows since we have an identification of the Bloch–Kato
Selmer groups owing to (6.5) above

$\mathrm{Sel}_{\mathrm{st}}(K,V_{p}(\mathcal{F})(k_{0}/2+1))=\mathrm{Sel}_{\mathrm{st}}(K,V_{p}(f)(k_{0}/2+1)).$

######  Remark 6.4.

Let $\mathbb{T}_{(\mathcal{F})}$ denote the $\mathcal{F}$-isotypic component
of the usual Hecke algebra $\mathbb{T}$ acting on
$S_{\underline{k_{0}}+2}(U_{0}(\mathcal{N}))^{\mathrm{new}}$. Then, via the
isomorphism $\mathbb{T}_{(\mathcal{F})}\cong\mathbb{Q}(\mathcal{F})$, we may
regard $s_{\mathcal{F}}\in\mathbb{Q}(\mathcal{F})^{\times}$ as a Hecke
operator in $\mathbb{T}_{(\mathcal{F})}$ acting on the Chow groups.

##  7\. Concluding Remarks

  * (1)

Conjectures of Calegari–Mazur ([CM09, Conjecture 1.3]) and Barrera
Salazar–Williams ([BSW21, Conjecture 5.13]) predict that the only cuspidal
$p$-adic families of Bianchi eigenforms come from (twisted) base–change or CM
families over $\mathbb{Q}$. Since the ideas explored in this article heavily
rely on Hida/Coleman families of Bianchi eigenforms, the crucial assumption
that the Bianchi eigenform $\mathcal{F}$ is the base–change to $F$ of a
classical cuspidal eigenform $f$ is indispensable. In fact, the reader will
realize that no genuine Stark–Heegner cycles are constructed in this article.
Similar to [BD09] and [Sev12], in scenarios where the theory of Heegner cycles
overlaps with that of their Stark–Heegner counterparts such as base–change, we
show that the Stark–Heegner cycles can be expressed in terms of Heegner
cycles.

  * (2)

It would be evident to the reader that we have restricted ourselves to the
_analytic rank one_ setting in this article. In a forthcoming work with
Lennart Gehrmann, we will consider _Plectic Stark–Heegner cycles_ (See [FG21]
for example) that would account for higher orders of vanishing of the
$L$-series $L(\mathcal{F}/K,s)$ at the central critical point.

  * (3)

It would be interesting to give some computational evidence when the Bianchi
eigenform $\mathcal{F}$ corresponds to the quadratic base–change of an
elliptic curve $E_{/F}$ as in §1.3. We hope to get back to this in the future.

##  Appendix A A Central $L$-value formula

The goal of this appendix is to prove Theorem 5.10. In particular, we have to
show that, for each classical weight $\lambda_{k}\in U$,

$\mathbb{L}(\mathcal{F}_{k}^{\\#},\psi_{K})=L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#}/K,\psi_{K},k/2+1)\in
E_{k}(\psi_{K})$

where

(A.1)
$\mathbb{L}(\mathcal{F}_{k}^{\\#},\psi_{K})=\Bigg{(}\sum\limits_{\sigma\in\mathrm{Gal}(H_{K}/K)}\psi_{K}^{-1}(\sigma)\phi_{k}^{\\#}\\{\tau-\gamma_{\sigma\Psi}\tau\\}\Big{(}(P_{\sigma\Psi}(x,y))^{k/2}\Big{)}\Bigg{)}^{2}.$

###  A.1. Classical formulation

Recall that $\psi_{K}:\mathrm{Gal}(H_{K}/K)\rightarrow\mathbb{C}^{\times}$ is
an unramified character and that

$\phi_{k}^{\\#}\in\textup{H}^{0}(\Gamma_{0}(\mathcal{M}),\Delta(V_{k,k}(E_{k})^{\vee}))$

is the Bianchi modular symbol attached to the normalised newform
$\mathcal{F}_{k}^{\\#}\in
S_{\underline{k}+2}(U_{0}(\mathcal{M}))^{\mathrm{new}}.$ Note that we have a
natural injective map

$\textup{H}^{0}(\Gamma_{0}(\mathcal{M}),\Delta(V_{k,k}(E_{k})^{\vee}))\hookrightarrow\textup{H}^{1}(\Gamma_{0}(\mathcal{M}),V_{k,k}(E_{k})^{\vee})$

and hence we may consider $\phi_{k}^{\\#}$ as a cohomology class in
$\textup{H}^{1}(\Gamma_{0}(\mathcal{M}),V_{k,k}(E_{k})^{\vee})$.

For an optimal embedding $\Psi\in\mathrm{Emb}(\mathcal{O}_{K},\mathcal{R})$,
let $P_{\Psi}(x,y)\in(V_{2,2})^{\Gamma_{\Psi}}$ be as in Section 2.6. By
Dirichlet’s Unit theorem, we have that
$\mathcal{O}_{K}^{\times}/\\{\mathrm{torsion}\\}\cong\mathbb{Z}.$ In
particular, $\textup{H}_{1}(\Gamma_{\Psi},\mathbb{Z})\cong\mathbb{Z}$ where
$\Gamma_{\Psi}$ is the cyclic group generated by $\Psi(u)$ for $u$ a
fundamental unit of $\mathcal{O}_{K}^{\times}$. We fix a generator
$\eta\in\textup{H}_{1}(\Gamma_{\Psi},\mathbb{Z})$. Consider the cap product

$C_{\Psi}:=\eta\cap(P_{\Psi}(x,y))^{k/2}\in\textup{H}_{1}(\Gamma_{\Psi},\mathbb{Z})\times\textup{H}^{0}(\Gamma_{\Psi},V_{k,k})\cong\textup{H}_{1}(\Gamma_{\Psi},V_{k,k})$

Note that $\Gamma_{\Psi}\subseteq\Gamma_{0}(\mathcal{M})$. We set

$C_{[\Psi]}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{corres}^{\Gamma_{0}(\mathcal{M})}_{\Gamma_{\Psi}}[C_{\Psi}]\in\textup{H}_{1}(\Gamma_{0}(\mathcal{M}),V_{k,k}).$

For
$\psi_{K}:\operatorname{\mathrm{Gal}}(H_{K}/K)\rightarrow\mathbb{C}^{\times}$
as above, we define the $\psi_{K}$-twisted cycle

$C^{\psi_{K}}_{[\Psi]}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum\limits_{\sigma\in\mathrm{Cl}(K)}\psi_{K}^{-1}(\sigma)C_{\sigma\cdot[\Psi]}\in\textup{H}_{1}(\Gamma_{0}(\mathcal{M}),V_{k,k}).$

Under the pairing given by cap product,

(A.2)
$\langle\cdot,\cdot\rangle:\textup{H}_{1}(\Gamma_{0}(\mathcal{M}),V_{k,k})\times\textup{H}^{1}(\Gamma_{0}(\mathcal{M}),V_{k,k}(E_{k})^{\vee})\rightarrow
E_{k}(\psi_{K}),$

we have

$\mathbb{L}(\mathcal{F}_{k}^{\\#},\psi_{K})=\langle
C^{\psi_{K}}_{[\Psi]},\phi_{k}^{\\#}\rangle^{2}.$

###  A.2. Waldspurger formulas in higher cohomology

In this section, we relate the cap product $\langle
C^{\psi_{K}}_{[\Psi]},\phi_{k}^{\\#}\rangle$ with the Waldspurger period
integral $\mathcal{P}(\psi_{K},\phi_{k}^{\\#})$ considered by Santiago Molina
in [Mol22, Theorem 4.6]. Since the formulation of [Mol22] is of an adelic
nature, we introduce some notation. Let
$B\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=M_{2}(F)$ be the
split quaternion algebra over $F$. We may view the newform
$\mathcal{F}_{k}^{\\#}$ as an automorphic newform of $B^{\times}$ and let
$\pi_{k}$ be the automorphic representation it generates. Note that $\pi_{k}$
has trivial central character. Let $G$ (resp. $T$) be the algebraic group
associated to $B^{\times}/F^{\times}$ (resp. to $K^{\times}/F^{\times}$). Via
the embedding $\Psi$, we have an inclusion of algebraic groups $T\subseteq G$.

####  A.2.1. Fundamental classes associated to tori

We briefly recall the construction of fundamental classes in homology
following the exposition in [Mol22, Section 2.1]. Note that by Dirichlet’s
Unit theorem, the $\mathbb{Z}$-rank of $T(\mathcal{O}_{F})$ is $1$ and we fix
a generator

$\xi\in\textup{H}_{1}(T(\mathcal{O}_{F}),\mathbb{Z})\cong\mathbb{Z}$

Let us fix the compact subgroup
$U\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=T(\mathcal{O}_{F}\otimes\hat{\mathbb{Z}})\subset
T(\mathbb{A}_{F}^{\infty})$. Note that $T(\mathcal{O}_{F})=T(F)\cap U$.
Further, let us fix a fundamental domain $\widetilde{\mathcal{F}}\subset
T(\mathbb{A}_{F}^{\infty})$ for the action of $T(F)/T(\mathcal{O}_{F})$ on
$T(\mathbb{A}_{F}^{\infty})/U$. Then the set of continuous functions
$C(\widetilde{\mathcal{F}},\mathbb{Z})$ has an action of $T(\mathcal{O}_{F})$
(Note that $\widetilde{\mathcal{F}}$ is $U$-invariant). In particular, the
characteristic function
$\mathbbm{1}_{\widetilde{\mathcal{F}}}\in\textup{H}^{0}(T(\mathcal{O}_{F}),C(\widetilde{\mathcal{F}},\mathbb{Z}))$
is $T(\mathcal{O}_{F})$-invariant.

######  Definition A.1.

The fundamental class associated to the tori $T\subset G$ is defined as the
cap product

$\widetilde{\eta}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\xi\cap\mathbbm{1}_{\widetilde{\mathcal{F}}}\in\textup{H}_{1}(T(\mathcal{O}_{F}),C(\widetilde{\mathcal{F}},\mathbb{Z})).$

Further, by Shapiro’s lemma

(A.3)
$\textup{H}_{1}(T(\mathcal{O}_{F}),C(\widetilde{\mathcal{F}},\mathbb{Z}))\cong\textup{H}_{1}(T(F),C_{c}(T(\mathbb{A}_{F}^{\infty})/U,\mathbb{Z}))$

from which we conclude that the class $\widetilde{\eta}$ is $U$-invariant.
Here $C_{c}(T(\mathbb{A}_{F}^{\infty})/U,\mathbb{Z}))$ is the set of
$\mathbb{Z}$-valued continuous functions on $T(\mathbb{A}_{F}^{\infty})/U$
with compact support.

On the other hand, let $C^{\emptyset}(T(\mathbb{A}_{F}),\mathbb{Z})$ denote
the set of locally constant functions on $T(\mathbb{A}_{F})$ and
$C_{c}^{\emptyset}(T(\mathbb{A}_{F}),\mathbb{Z})$ be the subset of functions
in $C^{\emptyset}(T(\mathbb{A}_{F}),\mathbb{Z})$ that are compactly supported
when restricted to $T(\mathbb{A}_{F}^{\infty})$. By [Mol22, Lemma 2.4], there
is an isomorphism of $T(F)$-modules

$\mathrm{Ind}_{T(\mathcal{O}_{F})}^{T(F)}(C(\widetilde{\mathcal{F}},\mathbb{Z}))\cong
C_{c}^{\emptyset}(T(\mathbb{A}_{F}),\mathbb{Z})$

and hence by Shapiro’s Lemma, we have an isomorphism

$\textup{H}_{1}(T(F),C_{c}^{\emptyset}(T(\mathbb{A}_{F}),\mathbb{Z}))\cong\textup{H}_{1}(T(\mathcal{O}_{F}),C(\widetilde{\mathcal{F}},\mathbb{Z}))$

This way we may regard the fundamental class $\widetilde{\eta}$ as a class in
$\textup{H}_{1}(T(F),C_{c}^{\emptyset}(T(\mathbb{A}_{F}),\mathbb{Z}))$.

####  A.2.2. Cohomology of arithmetic groups and Eichler–Shimura morphism

Let $\mathcal{A}^{\infty}(V_{k,k}^{\vee})$ be the set of functions
$\phi:G(\mathbb{A}_{F}^{\infty})/U_{0}(\mathcal{M})\longrightarrow
V_{k,k}^{\vee}$ with a natural $G(F)$-action defined by

$\gamma.\phi(g)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\phi(\gamma^{-1}g).$

Since the class number of $F$ is one, the double quotient space
$G(F)\backslash G(\mathbb{A}_{F}^{\infty})/U_{0}(\mathcal{M})$ is trivial and
$\Gamma_{0}(\mathcal{M})=G(F)\cap U_{0}(\mathcal{M})$. Shapiro’s lemma then
induces

(A.4)
$\textup{H}^{1}(G(F),\mathcal{A}^{\infty}(V_{k,k}^{\vee}))\cong\textup{H}^{1}(\Gamma_{0}(\mathcal{M}),V_{k,k}^{\vee}).$

In particular, by (A.4), we may consider
$\phi_{k}^{\\#}\in\textup{H}^{1}(G(F),\mathcal{A}^{\infty}(V_{k,k}^{\vee}))$.

Let

$C^{(k,k)}(T(\mathbb{A}_{F}),\mathbb{C})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=C^{0}(T(\mathbb{A}_{F}),\mathbb{C})\otimes
V_{k,k}$

be the space of locally polynomial functions in $T(\mathbb{A}_{F})$ considered
by Molina in [Mol22, Section 4.2] which has a natural $T(F)$-action. Now, via
the Artin reciprocity map we may view the unramified character
$\psi_{K}:\operatorname{\mathrm{Gal}}(H_{K}/K)\rightarrow\mathbb{C}^{\times}$
as a locally constant character
$T(\mathbb{A}_{F})\rightarrow\mathbb{C}^{\times}$ that is $T(F)$-invariant.
Similarly the element $P_{\Psi}(x,y)^{k/2}\in V_{k,k}$ is $T(F)$-invariant. In
particular, we have

$\psi_{K}\otimes
P_{\Psi}(x,y)^{k/2}\in\textup{H}^{0}(T(F),C^{(k,k)}(T(\mathbb{A}_{F}),\mathbb{C})).$

We recall the $T(F)$-equivariant pairing of [Mol22, Section 4.3] (See Remark
4.2 in particular)

$\displaystyle\varphi:C^{(k,k)}(T(\mathbb{A}_{F}),\mathbb{C})\otimes\mathcal{A}^{\infty}(V_{k,k}^{\vee})$
$\displaystyle\longrightarrow C^{\emptyset}(T(\mathbb{A}_{F}),\mathbb{C})$
$\displaystyle\varphi((f\otimes P)\otimes\phi)(z,t)$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=f(z,t)\cdot\phi(t)(P)$

for all $z\in T(F_{\infty})$ and $t\in T(\mathbb{A}_{F}^{\infty})$. The
natural pairing
$\langle\cdot,\cdot\rangle_{T}:C^{\emptyset}(T(\mathbb{A}_{F}),\mathbb{C})\times
C_{c}^{\emptyset}(T(\mathbb{A}_{F}),\mathbb{Z})\rightarrow\mathbb{C}$ given by
the Haar measure on $T(\mathbb{A}_{F})$ induces a cap product

(A.5)
$\langle\cdot,\cdot\rangle:\textup{H}^{1}(T(F),C^{\emptyset}(T(\mathbb{A}_{F}),\mathbb{C}))\times\textup{H}_{1}(T(F),C_{c}^{\emptyset}(T(\mathbb{A}_{F}),\mathbb{Z}))\longrightarrow{\mathbb{C}}$

We set

$\mathcal{P}(\psi_{K},\phi_{k}^{\\#})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varphi((\psi_{K}\otimes
P_{\psi}(x,y))\otimes\phi_{k}^{\\#})\cap\widetilde{\eta}$

under the cap product of (A.5).

######  Proposition A.2.

With notation as above, we have an equality of cap products, i.e.

$\langle
C_{[\Psi]}^{\psi_{K}},\phi_{k}^{\\#}\rangle=\mathcal{P}(\psi_{K},\phi_{k}^{\\#})$

###### Proof.

Note that it suffices to show that the two fundamental homology classes viz.
$\eta$ and $\widetilde{\eta}$ coincide. Recall that
$\eta\in\textup{H}_{1}(\Gamma_{\Psi},\mathbb{Z}).$ Since
$\Gamma_{\Psi}=T(F)\cap U$, where
$U=T(\mathcal{O}_{F}\otimes\hat{\mathbb{Z}})$, we have

$C_{c}^{\emptyset}(T(\mathbb{A}_{F})/U,\mathbb{Z})=\mathrm{Ind}_{\Gamma_{\Psi}}^{T(F)}\mathbb{Z}.$

Again, by Shapiro’s Lemma, we may hence consider the class
$\eta\in\textup{H}_{1}(T(F),C_{c}^{\emptyset}(T(\mathbb{A}_{F})/U,\mathbb{Z}))$.
Now, since
$\widetilde{\eta}\in\textup{H}_{1}(T(F),C_{c}^{\emptyset}(T(\mathbb{A}_{F}),\mathbb{Z}))$
is $U$-invariant (See (A.3) above), we conclude that the two fundamental
homology classes $\eta$ and $\widetilde{\eta}$ coincide upto renormalizing. In
particular, it follows that the two pairings defined via cap product in (A.2)
and (A.5) respectively are the same. ∎

####  A.2.3. A Waldspurger type formula

We can now prove Theorem 5.10 in the main text which relates the cap products
considered above to the central $L$-value of the quadratic base change of
Bianchi newforms.

######  Theorem A.3.

Let $\psi_{K}:\mathrm{Gal}(H_{K}/K)\rightarrow\mathbb{C}^{\times}$ be an
unramified character as above. Then

$\mathbb{L}(\mathcal{F}_{k}^{\\#},\psi_{K})=L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#}/K,\psi_{K},k/2+1)\in
E_{k}(\psi_{K})$

where

(A.6)
$L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#}/K,\psi_{K},k/2+1)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{Tu_{K}^{2}}{(\Omega_{k}^{\\#})^{2}}\times\frac{\prod_{\nu\in\Sigma_{\infty}^{F}}C^{\prime}_{\nu}(K,\pi_{k},\psi_{K})}{\sqrt{\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})}}\cdot\frac{((k/2)!)^{4}}{(2\pi)^{2k+4}}L(\mathcal{F}_{k}^{\\#}/K,\psi_{K},k/2+1)$

where
$u_{K}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left[\mu(\mathcal{O}_{K}):\mu(\mathcal{O}_{F})\right]$
and $T$ and $C^{\prime}_{\nu}(K,\pi_{k},\psi_{K})$ are some explicit
constants.

###### Proof.

The main ingredient in the proof is Molina’s Waldspurger formula in higher
cohomology. More precisely, it is shown in [Mol22, Theorem 4.6] that

$\mathcal{P}(\psi_{K},\phi_{k}^{\\#})^{2}=\frac{u_{K}^{2}}{(\Omega_{k}^{\\#})^{2}}\cdot
L(1/2,\pi_{k}\times\pi_{\psi_{K}})\cdot\prod_{\nu\nmid\infty}\alpha_{\pi_{k,\nu},\psi_{K,\nu}}(\phi_{k}^{\\#})$

where $\pi_{k}$ is the automorphic representation of $G(\mathbb{A}_{F})$ that
the newform $\mathcal{F}_{k}^{\\#}\in S_{\underline{k}+2}(U_{0}(\mathcal{M}))$
generates, $\Omega_{k}^{\\#}\in\mathbb{C}^{\times}$ is the complex period of
(4.9) and $\alpha_{\pi_{k,\nu},\psi_{K,\nu}}(\phi_{k}^{\\#})$ are explicit
local factors (almost all one). Since $\phi_{k}^{\\#}$ is the modular symbol
associated to the newform $\mathcal{F}_{k}^{\\#}$, we may use the explicit
Waldspurger formula of [MW09] given in terms of period integrals of
Gross–Prasad test vectors $\phi\in\pi_{k}$. In particular, by [MW09, Theorem
4.2], we get that

$\displaystyle\mathcal{P}(\psi_{K},\phi_{k}^{\\#})^{2}$
$\displaystyle=\frac{u_{K}^{2}}{(\Omega_{k}^{\\#})^{2}}\times\lvert\int\phi(t)\psi_{K}^{-1}(t)dt\rvert^{2}$
$\displaystyle=\frac{u_{K}^{2}}{(\Omega_{k}^{\\#})^{2}}\times\frac{L^{S_{2}(\pi_{k},K)}(1/2,\pi_{k}\times\pi_{\psi_{K}})\left(\phi,\phi\right)}{\left(\phi_{\pi},\phi_{\pi}\right)}\times\frac{L_{S(\psi_{K})}(1,\epsilon_{K/F})^{2}}{\sqrt{\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})}}$
$\displaystyle\times\frac{L_{S_{1}(\pi_{k},K)}(1,\epsilon_{K/F})}{L_{S_{1}(\pi_{k},K)}(1,1_{F})}\times\prod\limits_{\nu\in
S_{2}(\pi_{k},K)}C^{\prime}(\pi_{k,\nu})\times\prod\limits_{\nu\in\Sigma_{\infty}^{F}}C^{\prime}_{\nu}(K,\pi_{k},\psi_{K}).$

where

  * •

$\phi_{\pi}$ is a new vector for $\pi_{k}$,

  * •

$S_{1}(\pi,K)=\\{\mbox{places of }F\mbox{ where }\pi_{k}\mbox{ ramifies but
}K\mbox{ doesn't.}\\}$

  * •

$S_{2}(\pi,K)=\\{\mbox{places of }F\mbox{ where both }\pi_{k}\mbox{ and
}K\mbox{ ramify.}\\}$

  * •

$S(\psi_{K})=\\{\mbox{places of }F\mbox{ above which }\psi_{K}\mbox{
ramifies.}\\}$

  * •

$C^{\prime}(\pi_{k,\nu})$ and $C^{\prime}_{\nu}(K,\pi_{k},\psi_{K})$ are
certain non-archimedean and archimedean constants defined in Section 4.2.1 and
4.2.2 of [MW09] respectively.

In our setting, note that $S(\psi_{K})$ and $S_{2}(\pi,K)$ are both empty
while $S_{1}(\pi,K)=\\{\mathfrak{l}\mid\mathcal{M}\\}$. In particular, all
places in $S_{1}(\pi,K)$ split in $K$ under (SH-Hyp). Further, the
Gross–Prasad test vector $\phi$ can be chosen to be a translate of the new
vector $\phi_{\pi}$ so that
$\left(\phi,\phi\right)=T\left(\phi_{\pi},\phi_{\pi}\right)$ for some scalar
$T$. We can simplify the above equation as

(A.7)
$\mathbb{L}(\mathcal{F}_{k}^{\\#},\psi_{K})=\frac{Tu_{K}^{2}}{(\Omega_{k}^{\\#})^{2}}\times\frac{\prod_{\nu\in\Sigma_{\infty}^{F}}C^{\prime}_{\nu}(K,\pi_{k},\psi_{K})}{\sqrt{\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})}}\times
L(1/2,\pi_{k}\times\pi_{\psi_{K}}).$

Using the relation

$L(s,\pi_{k}\times\pi_{\psi_{K}})=\left(\frac{\Gamma(s+(k+1)/2)}{(2\pi)^{s+(k+1)/2}}\right)^{4}L(\mathcal{F}_{k}^{\\#}/K,\psi_{K},k/2+1),$

we get that

$\displaystyle\mathbb{L}(\mathcal{F}_{k}^{\\#},\psi_{K})$
$\displaystyle=\frac{Tu_{K}^{2}}{(\Omega_{k}^{\\#})^{2}}\times\frac{\prod_{\nu\in\Sigma_{\infty}^{F}}C^{\prime}_{\nu}(K,\pi_{k},\psi_{K})}{\sqrt{\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})}}\cdot\frac{((k/2)!)^{4}}{(2\pi)^{2k+4}}\times
L(\mathcal{F}_{k}^{\\#}/K,\psi_{K},k/2+1)$
$\displaystyle=L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#}/K,\psi_{K},k/2+1)\in
E_{k}(\psi_{K}).$

∎

######  Remark A.4.

When $\psi_{K,\nu}$ is trivial for each archimedean place $\nu$ of $F$, then
[MW09, Section 4.2.2] calculates $C^{\prime}_{\nu}(K,\pi_{k},\psi_{K})=16\pi$.
In particular, for the trivial character, we have

(A.8)
$L^{\mathrm{alg}}(\mathcal{F}_{k}^{\\#}/K,k/2+1)=\frac{Tu_{K}^{2}}{(\Omega_{k}^{\\#})^{2}}\times\frac{(16\pi)^{2}}{\sqrt{\mathrm{N}_{F/\mathbb{Q}}(\mathcal{D}_{K/F})}}\cdot\frac{((k/2)!)^{4}}{(2\pi)^{2k+4}}\times
L(\mathcal{F}_{k}^{\\#}/K,k/2+1)$

######  Remark A.5.

Note that $P_{\Psi}(x,y)^{k/2}\in V_{k,k}$ that appears in the definition of
the period integral $\mathcal{P}(\psi_{K},\phi_{k}^{\\#})$ corresponds to the
dual element $\mu_{\underline{0}}\in V_{k,k}^{\vee}$, for
$\underline{0}=(0,0)$, in [Mol22] via the isomorphism (cf. [Mol22, Section
3.1])

$V_{k,k}^{\vee}\cong V_{k,k},\quad\quad\mu\mapsto\mu((Xy-Yx)^{k}).$

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