text:
\section{Introduction}\label{sec: Introduction}

Let $E$ be an elliptic curve defined over $\Q$ and let $f\in S_2(N_f)$ be the newform attached to $E$. Let
\begin{equation*}
g\in S_1(N_g,\chi)_L, \qquad h\in S_1(N_h,\bar\chi)_L
\end{equation*}
be two cuspforms of weight one, inverse nebentype characters and with Fourier coefficients contained in a number field $L$. Let $\rho_g$ and $\rho_h$ be the Artin representations attached to $g$ and $h$. The tensor product $\rho_g\otimes\rho_h$ is a self-dual Artin representation of dimension $4$ of the form $$\rho:=\rho_g\otimes\rho_h:\Gal(H/\Q)\hookrightarrow\Aut(V_g\otimes V_h)\cong\GL_4(L),$$  where $H/\Q$ is a finite extension.

In this setting, the complex $L$-function $L(E\otimes\rho,s)$ attached to the (Tate module $V_p(E)$ of the) elliptic curve $E$ twisted by the Artin representation $\rho$ coincides with the Garrett--Rankin $L$-function $L(f\otimes g\otimes h,s)$ attached to the triple $(f,g,h)$ of modular forms.
By multiplying this $L$-function by an appropriate archimedean factor $L_\infty(f\otimes g\otimes h, s)$ one obtains an entire function $\Lambda(f\otimes g\otimes h,s)$ which satisfies a functional equation of the form
\begin{equation}\label{eq: functional equation triple product (2,1,1)}
\Lambda(f\otimes g\otimes h, s)=\epsilon\cdot\Lambda(f\otimes g\otimes h, 2-s),
\end{equation}
where $\epsilon\in\{ \pm1 \}$.
Moreover, $L_\infty(f\otimes g\otimes h,s)$ does not have zeros nor poles at $s=1$.

Denote $N_g$ and  $N_h$ the level of $g$ and $h$ respectively. The sign can be written as a product of local factors $\epsilon=\prod_{v}\epsilon_v$ where $v$ runs over the places of $\Q$, and $\epsilon_v=+1$ if $v$ is a finite prime which does not divide $\operatorname{lcm}(N_f,N_g,N_h)$ or if $v=\infty$.
We will work under the following assumption
\begin{assumption}\label{ass: local signs}
$	\epsilon_v=+1  \quad \text{ for all } v.$
\end{assumption}
Assumption \ref{ass: local signs} holds most of the time: this is the case for instance if the greatest common divisor of the levels of $f,g$ and $h$ is $1$.

Fix an odd prime number $p$ such that
\begin{equation*}
p\nmid N_fN_gN_h,
\end{equation*}
and denote by $\alpha_g,\beta_g$ the eigenvalues for the action of the Frobenius element at $p$ acting on $V_g$. We use the analogous notation for $h$, and we assume
\begin{equation*}
\alpha_g\neq\beta_g, \text{ and } \ \alpha_h\neq\beta_h.
\end{equation*}
Fix once and for all completions $H_p, L_p$ of the number fields $H, L$ at primes above $p$.

Choose an ordinary $p$-stabilisation of $g$, namely $g_\alpha(z):=g(z)-\beta_gg(pz)$ and define analogously $h_\alpha$. Let ${\bf f},{\bf g},{\bf h}$ be Hida families passing through the unique ordinary $p$-stabilisation of $f$ and  $g_\alpha$ and $h_\alpha$  respectively, and consider the Garret--Hida $p$-adic $L$-function $$\L_p^g({\bf f}, {\bf g},{\bf h}) $$ of \cite{DR1} associated to the specific choice of test vectors $(\breve{{\bf f}},\breve{{\bf g}},\breve{{\bf h}})$ of \cite[Chap. 3]{Hsieh}. This $p$-adic $L$-function interpolates the square-roots of the central values of the classical $L$-function $L(\breve{f}_k\otimes \breve{g}_\ell\otimes \breve{h}_m,s)$ attached to the specializations of the Hida families at classical points of weights $k,\ell,m$  with $k,\ell,m\geq 2$ and $\ell\geq k+m$.
Notice that the point $(2,1,1)$, which corresponds to our triple of modular forms $(f,g,h)$, lies outside the region of classical interpolation for $\L_p^g({\bf f}, {\bf g},{\bf h})$.
We are interested in studying the value
\begin{equation*}
\L_p^g({\bf f}, {\bf g},{\bf h})(2,1,1)
\end{equation*}
under the following assumption:
\begin{assumption}\label{ass: rank 00}
	$L(E\otimes\rho,1)\neq0 \text{ and } \operatorname{Sel}_p(E\otimes\rho)=0.$
\end{assumption}

Here $\operatorname{Sel}_p(E\otimes\rho)$ denotes the Bloch--Kato Selmer group attached to the representation $$V:=V_p(E)\otimes V_g\otimes V_h.$$

Under Assumption \ref{ass: local signs}, the sign $\epsilon$ of the functional equation \eqref{eq: functional equation triple product (2,1,1)}  is $+1$, and thus the order of vanishing of $L(E\otimes \rho,s)$ at $s=1$ is even. One hence expects that $L(E\otimes\rho,1)$ is generically nonzero. If this $L$-value is nonzero, by \cite{DR2} we know that the $\rho$-isotypical component $E(H)^{\rho}:=\hom_{G_\Q}(V_g\otimes V_h,E(H)\otimes L)$ of the Mordell-Weil group $E(H)$ is trivial. By the Shafarefich--Tate conjecture one also expects the Selmer group $\operatorname{Sel}_p(E\otimes\rho)$ to be trivial, although this conjecture is widely open.
It is also worth noting that the value $\L_p^g({\bf f}, {\bf g},{\bf h})(2,1,1)$ in the setting in which the complex $L$-function $L(E\otimes\rho,s)$ vanishes at $s=1$ has been analyzed in \cite{DLR}, where the authors give a conjectural formula for this $p$-adic value as a $2\times 2$-regulator of $p$-adic logarithms of global points.

Under our running assumption \ref{ass: rank 00} one can not expect a similar formula for the above $p$-adic $L$-value, as no global points are naturally present in this scenario.
The {\em main result of this paper} consists in an explicit formula for the value $\L_p^g({\bf f}, {\bf g},{\bf h})(2,1,1)$  which involves the algebraic part of the classical $L$-value $L(E\otimes\rho,1)$ and the {\em logarithm} of a canonical non-crystalline class along a certain {\em crystalline direction}.

In \S\ref{sec: Selmer groups} we recall the basic definitions on Selmer groups and we give a precise description of the relaxed $p$-Selmer group $\operatorname{Sel}_{(p)}(E\otimes\rho)$ under Assumption \ref{ass: rank 00}. More precisely,  the projection to the singular quotient gives an isomorphism
\begin{equation}\label{eq: iso for relaxed selmer}
\partial_p:\operatorname{Sel}_{(p)}(E\otimes\rho)\overset{\cong}{\longrightarrow}\coh^1_s(\Q_p,V).
\end{equation}
Let $V_g^\alpha, V_g^\beta$, with basis $v_g^\alpha, v_g^\beta$ respectively, be the eigenspaces of $V_g$ for the action of $\Frob_p$ with eigenvalues $\alpha_g, \beta_g$, and use the analogous notation for $V_h$. The $G_{\Q_p}$-representation $V$ decomposes as a direct sum as
\begin{equation*}
V= V^{\alpha\alpha}\oplus V^{\alpha\beta}\oplus V^{\beta\alpha}\oplus V^{\beta\beta},
\end{equation*}
where $V^{\alpha\alpha}:=V_pE\otimes V_g^\alpha\otimes V_h^\alpha$ and similarly for the other pieces.
It induces the decomposition
\begin{equation}\label{eq: decomp H1s}
\coh^1_s(\Q_p,V)= \coh^1_s(\Q_p,V^{\alpha\alpha})\oplus \coh^1_s(\Q_p,V^{\alpha\beta})\oplus \coh^1_s(\Q_p,V^{\beta\alpha})\oplus \coh^1_s(\Q_p,V^{\beta\beta}),
\end{equation}
and the Bloch--Kato dual exponential gives isomorphisms
\begin{equation*}
\exp^*_{\alpha\alpha}:\coh^1_s(\Q_p,V^{\alpha\alpha})\overset{\cong}{\longrightarrow}L_p
\end{equation*}
and similarly for the other pieces of the decomposition \eqref{eq: decomp H1s}.
Combining it with \eqref{eq: iso for relaxed selmer}, we get a basis
\begin{equation*}
\xi^{\alpha\alpha}, 	\xi^{\alpha\beta}, 	\xi^{\beta\alpha}, 	\xi^{\beta\beta}
\end{equation*}
for $\operatorname{Sel}_{(p)}(E\otimes\rho)$ characterised by the fact that
\begin{equation*}
\partial_p\xi^{\alpha\alpha}\in\coh_s^1(\Q_p,V^{\alpha\alpha}) \text{ and } \exp^*_{\alpha\alpha}\partial_p\xi^{\alpha\alpha}=1,
\end{equation*}
and similarly for $	\xi^{\alpha\beta}, 	\xi^{\beta\alpha}, 	\xi^{\beta\beta}$.

The $G_{\Q_p}$-cohomology of $V$ and its submodule of crystalline classes $\coh^1_f(\Q_p,V)\subseteq\coh^1(\Q_p,V)$ also have decompositions analogous to \eqref{eq: decomp H1s}. Moreover, if
\begin{equation*}
\pi_{\alpha\beta}:\coh^1(\Q_p,V)\longrightarrow\coh^1(\Q_p,V^{\alpha\beta})
\end{equation*}
denotes the projection, then $\pi_{\alpha\beta}\xi^{\beta\beta}$ lies in $\coh^1_f(\Q_p,V^{\alpha\beta})$.
Finally, we can write
\begin{equation*}
\pi_{\alpha\beta}\xi^{\beta\beta}=R_{\beta\alpha}\otimes v_g^\alpha\otimes v_h^\beta\in(E(H_p)\otimes V_{g}^\alpha\otimes V_h^\beta)^{G_{\Q_p}}\cong\coh^1_f(\Q_p,V^{\alpha\beta})
\end{equation*}
where $R_{\beta\alpha}\in E(H_p)$ is a local point on which $\Frob_p$ acts as multiplication by $\beta_g\alpha_h$.

We can finally state the main result of the paper.

\begin{theorem*}[cf Theorem~\ref{thm: main}]
	Under Assumptions \ref{ass: p doesnt divide the levels} and \ref{ass: rank 00},
	\begin{equation}\label{eq: main thm intro}
	\L_p^g({\bf f}, {\bf g},{\bf h})(2,1,1) = \dfrac{A \cdot{\mathcal{E}}  }{\pi\langle f, f\rangle} \times \dfrac{\log_p(R_{\beta\alpha})}{\L_{g_\alpha}}\times\sqrt{L(E\otimes\rho,1)},
	\end{equation}
	where $A\in\Q^\times$ is an explicit number, ${\mathcal{E}}\in L_p$ is a product of Euler factors, $\langle f, f\rangle$ denotes the Petersson norm of $f$, $\L_{g_\alpha}\in H_p$ is an element on which $\Frob_p$ acts as multiplication by $\frac{\beta_g}{\alpha_g}$ and which only depends on $g_\alpha$, and $\log_p:E(H_p)\rightarrow H_p$ denotes the $p$-adic logarithm.

\end{theorem*}

We refer to Theorem \ref{thm: main} for a more precise statement of the result and of the objects appearing in \eqref{eq: main thm intro}. In particular, the element $\L_{g_\alpha}$ is expected to be related to a so-called Gross--Stark unit attached to $g_\alpha$, as conjectured in \cite[Conjecture 2.1]{DR2b}.

Under the additional assumption that $g$ is not the theta series of a Hecke character of a real quadratic field in which $p$ splits, the value $\L_p^g({\bf f}, {\bf g},{\bf h})(2,1,1)$ can be recast in a more explicit way in terms of \textit{$p$-adic iterated integrals}, as explained in the introduction of \cite{DLR}. The numerical computations we offer in \S \ref{sec: Numerical computations} are obtained by calculating such integrals, where a key input are Lauder's algorithms \cite{Lauder} for the computation of overconvergent projections.

As an application of the main result, in Section \ref{sec: p split in K} we explore the situation where $g$ and $h$ are theta series of the same imaginary quadratic field in which $p$ splits. The following theorem is stated as Theorem \ref{thm: CM case} in the text.

\begin{theorem*}[cf Theorem~\ref{thm: CM case}]
Let $K$ be an imaginary quadratic field in which $p$ is split, and let $\psi_g$ (resp. $\psi_h$) be a finite order Hecke character of $K$ of conductor ${\mathfrak{c}}_g$ (resp. of conductor ${\mathfrak{c}}_h)$.  Denote by $g$ and $h$ the theta series attached to $\psi_g$ and $\psi_h$, respectively. Suppose that $\gcd(N_f,{\mathfrak{c}}_g,{\mathfrak{c}}_h)=1$ and that the Nebentype characters of $g$ and $h$ are inverses to each other. If $L(E,\rho_g\otimes\rho_h,1)\neq 0$ then $\L_p^g({\bf f},{\bf g},{\bf h})(2,1,1)=0$.
\end{theorem*}

\section{The Selmer group of $f\otimes g\otimes h$}\label{sec: Selmer groups}
We begin this section by collecting some standard facts on Selmer groups of $p$-adic Galois representations that we will use. Then we introduce the Galois representation attached to the triple of modular forms $f$, $g$, and $h$ of weights $2$, $1$, $1$, and we study the corresponding Selmer groups. In particular, the structure of the relaxed Selmer group will be key in proving the main theorem of Section \ref{sec: main thm}.

\subsection{Selmer groups}
Let $V$ be a $\Q_p[G_\Q]$-module and let $B_\mathrm{cris}$ be Fontaine's $p$-adic crystalline period ring.  For each prime number $\ell$, denote
\begin{equation}\label{eq: selmer structure}
	\coh_f^1(\Q_\ell,V):=\begin{cases} \coh^1_\mathrm{ur}(\Q_\ell,V):=\coh^1(\Q_\ell^\mathrm{ur}/\Q_\ell,V^{I_\ell}) & \ell\neq p\\ \ker\Big(\coh^1(\Q_p,V)\rightarrow\coh^1(\Q_p,V\otimes_{\Q_p} B_\mathrm{cris})   \Big) & \ell=p,	\end{cases}
\end{equation}
and
\begin{equation*}
	\coh^1_s(\Q_\ell,V):=\coh^1(\Q_\ell,V)/\coh^1_f(\Q_\ell,V).
\end{equation*}
The Bloch--Kato Selmer group of $V$ is $$\operatorname{Sel}_p(\Q,V):=\{x\in\coh^1(\Q,V)\mid \operatorname{res}_\ell(x)\in\coh_f^1(\Q_\ell,V) \text{ for all }\ell \},$$
where $\operatorname{res}_\ell\colon H^1(\Q,V)\rightarrow H^1(\Q_\ell,V)$ denotes the restriction map in Galois cohomology.

For each prime $\ell$, we denote by $\partial_\ell$ the composition
\begin{equation*}
	\partial_\ell: \coh^1(\Q,V)\overset{\operatorname{res}_\ell}{\longrightarrow}\coh^1(\Q_\ell,V)\longrightarrow \coh^1_s(\Q_\ell,V),
\end{equation*}
where the second map is the natural quotient map.

The relaxed Selmer group is defined as $$\operatorname{Sel}_{(p)}(\Q,V):=\{ x\in\coh^1(\Q,V)\mid \operatorname{res}_\ell(x)\in\coh_f^1(\Q_\ell,V) \ \text{ for all }\   \ell\neq p \}\supseteq\operatorname{Sel}_p(\Q,V).$$

Let $V^*:=\Hom_{\Q_p}(V,\Q_p(1))$ be the
Kummer dual of $V$. One can define a Selmer group $\operatorname{Sel}_{p,*}(\Q,V^*)$ for $V^*$ which is dual to (\ref{eq: selmer structure}) with respect to the local Tate pairings
\begin{align}\label{eq: tate pairing}
\langle \ , \ \rangle_\ell:\coh^1(\Q_\ell,V)\times\coh^1(\Q_\ell,V^*)\longrightarrow\Q_p.
\end{align}
For each $\ell$, define $\coh_{f,*}^1(\Q_\ell,V^*)$ to be the orthogonal complement of $\coh_{f}^1(\Q_\ell,V)$ with respect to \eqref{eq: tate pairing}; the Selmer group attached to $V^*$ is then
\begin{align*}
\operatorname{Sel}_{p,*}(\Q,V^*):=\{x\in\coh^1(\Q,V^*)\mid \operatorname{res}_\ell(x)\in\coh_{f,*}^1(\Q_\ell,V^*) \ \text{ for all } \ell  \}.
\end{align*}
Finally, the strict Selmer group of $V^*$ is the subspace of $\operatorname{Sel}_{p,*}(\Q,V^*)$ defined as
\begin{align*}
\operatorname{Sel}_{[p],*}(\Q,V^*):=\{ x\in \coh^1(\Q,V^*)\mid \operatorname{res}_\ell(x)\in \coh_{f,*}^1(\Q_\ell,V^*)  \text{ for all } \ell \text{ and } \operatorname{res}_p(x)=0 \}.
\end{align*}

By Poitou--Tate duality (see, for example, \cite[Theorem 2.3.4]{MR04}) there is an exact sequence
\begin{equation}\label{eq: exact sequence of Selmer groups}
0\rightarrow \operatorname{Sel}_p(\Q,V)\rightarrow\operatorname{Sel}_{(p)}(\Q,V)\rightarrow\coh_s^1(\Q_p,V)\rightarrow\operatorname{Sel}_{p,*}(\Q,V^*)^\lor\rightarrow\operatorname{Sel}_{[p],*}(\Q,V^*)^\lor,
\end{equation}
where ${}^\lor$ stands for the $\Q_p$-dual.

\subsection{Representations attached to modular forms} In this section we review the main features of the representations, both $p$-adic and $\Lambda$-adic, attached to modular forms in the lines of \cite[\S2]{DR2b}, which the reader can consult for more details.

Let $f\in S_2(N_f)$ be a weight two normalized eigenform of level $N_f$, trivial nebentype character and rational Fourier coefficients $a_n(f)$. Denote by $E$ the elliptic curve over $\Q$ of conductor $N_f$ associated to $f$ by the Eichler--Shimura construction.

Let also
\begin{equation*}
	g\in S_1(N_g,\chi) \ \text{ and } \ h\in S_1(N_h,\bar{\chi})
\end{equation*}
be two normalized newforms of weight one, levels $N_g$ and $N_h$, and nebentype characters $\chi$ and $ \bar{\chi}$ respectively. Denote by $K_g$ and $K_h$ their fields of Fourier coefficients, and put $L:=K_g\cdot K_h$ the compositum of these fields.

From now on, we fix a rational prime $p$, and we assume the following hypothesis.
 \begin{assumption}\label{ass: p doesnt divide the levels}
 The prime $p$ does not divide $ N_f N_g N_h.$
 \end{assumption}
Since we will be interested in putting $f$ in a Hida family, we assume moreover that $f$ is ordinary at $p$; that is to say, that $p\nmid a_p(f)$.

We denote the $2$-dimensional $p$-adic representations attached to $f$ by $V_f$. Since $f$ corresponds to the curve $E$, the representation $V_f$ is given by the rational Tate module  $V_p(E)=T_p(E)\otimes\Q_p$. Denote by $\alpha_f,\beta_f$ the roots of the Hecke polynomial $X^2-a_p(f)X + p$. Since $f$ is ordinary at $p$, one of these roots, say $\alpha_f$, is a $p$-adic unit; also, the restriction of $V_f$ to a decomposition group $G_{\Q_p}\subset G_\Q$ admits a filtration of $\Q_p[G_{\Q_p}]$-modules
$$0\rightarrow V^+_f\longrightarrow V_f\longrightarrow V^-_f\rightarrow 0$$
with the following properties:
\begin{enumerate}
\item $\dim_{\Q_p}V^+_f= \dim_{\Q_p}V^-_f=1$;
\item the group $G_{\Q_p}$ acts on the quotient $V_f^-$ via $\psi_f$, where  $\psi_f:G_{\Q_p}\rightarrow\Z_p^\times$ is the unramified character that maps an arithmetic Frobenius $\Frob_p$ to $\alpha_f$.
\item the group $G_{\Q_p}$ acts on  $V_f^+$ via the character $\chi_{\cycl}\psi_f^{-1}$ (here $\chi_{\mathrm{cycl}}$ is the $p$-adic cyclotomic character).
\end{enumerate}

There are Artin representations associated to $g$ and $h$. Without loss of generality we can assume that they are defined over $L$, and that they factor through the same finite extension $H$ of $\Q$. That is to say, they are of the form
\begin{align*}
  \rho_g\colon \Gal(H/\Q)\longrightarrow \Aut(V_g^{\comp})\cong \GL_2(L),\ \ \   \rho_h\colon \Gal(H/\Q)\longrightarrow \Aut(V_h^{\comp})\cong \GL_2(L)
\end{align*}
for certain $2$-dimensional $L$-vector spaces $V_g^{\comp}$ and $V_h^{\comp}$.

Fix once and for all a prime ${\mathfrak{p}}$ of $H$ and a prime ${\mathfrak{P}}$ of $L$ above $p$. Denote the corresponding completions by $H_p:=H_{\mathfrak{p}}$ and $L_p:=L_{\mathfrak{P}}$. There are also $p$-adic Galois representations associated to $g$ and $h$, that we will denote by $V_g$ and $V_h$. There are non-canonical isomorphisms
\begin{align}\label{eq: def of Vg and Vh}
 j_g:V^{\comp}_g\otimes_L L_p\overset{\cong}{\longrightarrow} V_g \ \text{ and } \  j_h: V^{\comp}_h\otimes_L L_p\overset{\cong}{\longrightarrow} V_h.
\end{align}

Since $ p\nmid N_g N_h$ the representations $V_g$ and $V_h$ are unramified at $p$. We assume from now on that $\Frob_p$ acts on $V_g$ and $V_h$ with distinct eigenvalues. Let $\alpha_g$, $ \beta_g$ be the eigenvalues for the action of $\Frob_p$ on $V_g$ and let  $V_g^\alpha, V_g^\beta$ be the corresponding eigenspaces. We will use the analogous notations $\alpha_h$, $\beta_h$, $V_h^\alpha$, and $V_h^\beta$ for $h$.

Denote by $g_\alpha$ the $p$-stabilisation of $g$ such that $U_p(g_\alpha)=\alpha_gg_\alpha$. The theory of Hida families ensures the existence of a Hida family ${\bf g}$ passing through $g_\alpha$. This can be regarded as a power series ${\bf g}\in \Lambda_{\bf g}[[q]]$, where $\Lambda_{\bf g}$ is a finite flat extension of the Iwasawa algebra $\Lambda:=\Z_p[[T]]$, with the property that, if we denote by $y_g\colon \Lambda_{\bf g}\rightarrow L_p$ the weight corresponding to $g$, then $y_g({\bf g})=g_\alpha$. There is a locally free $\Lambda_{\bf g}$-module $V_{{\bf g}}$ and a $\Lambda$-adic representation $\rho_{\bf g}:G_\Q\rightarrow\GL(V_{\bf g})\cong \GL_2(\Lambda_{\bf g})$ that interpolates the $p$-adic representations associated to the specializations of ${\bf g}$.  As a $G_{\Q_p}$-representation, $V_{\bf g}$ is equipped with a filtration of $\Lambda_{\bf g}[G_{\Q_p}]$-modules
\begin{equation}\label{eq: filtration of lambda-adic rep}
0\rightarrow V_{\bf g}^+\longrightarrow V_{\bf g}\longrightarrow V_{\bf g}^-\rightarrow0,
\end{equation}
where $V_{\bf g}^+$ and $V_{\bf g}^-$ are locally free of rank one and the action of $G_{\Q_p}$ on $V_{\bf g}^-$ is unramified, with $\Frob_p$ acting as multiplication by the $p$-th Fourier coefficient of ${\bf g}$. There is a perfect Galois equivariant pairing
\begin{align}\label{eq: lambda-adic pairing}
\langle \ , \ \rangle:V_{\bf g}^-\times V_{\bf g}^+\longrightarrow\Lambda_{\bf g}(\det(\rho_{\bf g})).
\end{align}
For a crystalline $\Q_p[G_{\Q_p}]$-module $W$, denote $D(W):=(W\otimes B_\mathrm{cris})^{G_{\Q_p}}$. Recall that, if $W$ is unramified, then
\begin{equation*}
 D(W)\cong(W\otimes\hat{\Q}_p^{\mathrm{ur}})^{G_{\Q_p}},
\end{equation*}
where $\hat{\Q}_p^{\mathrm{ur}}$ is the $p$-adic completion of the maximal unramified extension of $\Q_p$.  Denote by $\omega_{\bf g}\in D(V_{\bf g}^-)$ the canonical period associated to ${\bf g}$ constructed by  Ohta \cite{Ohta}.

By specializing via $y_g$, we obtain the $L_p$-vector space $y_g(V_{\bf g}):=V_{\bf g}\otimes_{\Lambda_{\bf g},y_g}L_p$, which can be identified with $V_g$. Using the functoriality of $D$ and the identification $y_g(V_{\bf g}^+)=V_g^\beta, \ y_g(V_{\bf g}^-)=V_g^\alpha$ we obtain a pairing
\begin{equation}\label{eq: pairing Vg+}
\langle \ , \ \rangle:D(V_g^\alpha)\times D(V_g^\beta)\longrightarrow D(L_p(\chi))=(H_p\otimes L_p(\chi))^{G_{\Q_p}}.
\end{equation}

Define
\begin{align*}
\omega_g:=y_g(\omega_{\bf g})\in D(V_g^\alpha)
\end{align*}
and let
\begin{align*}
\eta_g\in D(V_g^\beta)
\end{align*}
be the element characterized by the equality
\begin{equation}\label{eq: pairing between etas and omegas}
\langle\omega_g,\eta_g\rangle= \mathfrak{g}(\chi)\otimes 1\in D(L_p(\chi)),
\end{equation}
where $\mathfrak{g}(\chi)$ denotes the Gauss sum of $\chi$ viewed as an element of $H_p$.
We define similarly $\omega_h\in D(V_h^\alpha)$ and  $ \eta_h\in D(V_h^\beta)$.

Using the isomorphisms \eqref{eq: def of Vg and Vh} we can define an $L$ structure on $V_g$ by  $V_g^L:=j_g(V^{\comp}_g)$. Let $v_g^{\alpha}$ (resp. $v_g^{\beta}$) be an $L$-basis of $V_g^L\cap V_g^{\alpha}$ (resp. of $V_g^{\beta}$). Define
\begin{align*}
\Omega_g\in H_p^{1/\alpha_g}, \ \Theta_g\in H_p^{1/\beta_g}
\end{align*}
to be the elements such that
 \begin{equation}\label{eq: definition of Omega and Theta}
 	\Omega_g\otimes v_g^{\alpha}=\omega_g\in D(V_g^\alpha), \ \ \Theta_g\otimes v_g^\beta=\eta_g\in D(V_g^\beta).
 \end{equation}

  Let
 \begin{align*}
   V:=V_{f}\otimes V_{gh}
 \end{align*}
be the $p$-adic representation given by the tensor product $V_f\otimes V_g\otimes V_h$. Since the product of the nebentype characters of $f$, $g$, and $h$ is trivial we have that  $V^*\cong V$. We next study the structure of several Selmer groups associated to $V$.

\subsection{Selmer groups of $V$}

Put $V_{gh}^{\comp}:=V_g^{\comp}\otimes_L V_h^{\comp}$ and denote by $\rho$ the representation afforded by this space:
\begin{align*}
  \rho\colon \Gal(H/\Q)\longrightarrow \Aut(V_{gh}^{\comp}).
\end{align*}
Put $E(H)_L:=E(H)\otimes_\Z L$ and denote by $E(H)^{\rho}$ the $\rho$-isotypical component of the Mordell--Weil group:
\begin{equation*}
E(H)^{\rho}:=\hom_{\Gal(H/\Q)}(V_{gh}^{\comp},E(H)_L).
\end{equation*}

\begin{lemma}\label{lem: first lemma on isomorphisms cohomology triple product}
	There are isomorphisms
	\begin{align}
	\coh^1(\Q,V)\cong(\coh^1(H,V_f)\otimes V_{gh})^{\Gal(H/\Q)}\cong\hom_{\Gal(H/\Q)}(V_{gh},\coh^1(H,V_f));\label{eq: iso cohomology triple product1}\\
	\coh^1(\Q_p,V)\cong(\coh^1(H_ p,V_f)\otimes V_{gh})^{\Gal(H_ p/\Q_p)}\cong\hom_{\Gal(H_ p/\Q_p)}(V_{gh},\coh^1(H_ p,V_f))\label{eq: iso cohomology triple product2}.
	\end{align}
\end{lemma}
\begin{proof}
We prove only \eqref{eq: iso cohomology triple product2}, and \eqref{eq: iso cohomology triple product1} is proven similarly. By the inflation-restriction  exact sequence we have the exact sequence
	\begin{equation*}
0\rightarrow \coh^1(\Gal(H_ p/\Q_p), V^{G_{H_ p}})\rightarrow \coh^1(\Q_p,V)\rightarrow \coh^1(H_ p, V)^{\Gal(H_ p/\Q_p)}\rightarrow\coh^2(\Gal(H_ p/\Q_p), V^{G_{H_ p}}).
	\end{equation*}
Since $\coh^1(\Gal(H_ p/\Q_p), V^{G_{H_ p}})=\coh^2(\Gal(H_ p/\Q_p), V^{G_{H_ p}})=0$, the restriction to $G_{H_ p}$ gives an isomorphism
\begin{equation*}
	\coh^1(\Q_p,V)\longrightarrow \coh^1(H_ p, V)^{\Gal(H_ p/\Q_p)}.
\end{equation*}
Composing it with the identifications
\begin{equation*}
	\coh^1(H_ p, V)^{\Gal(H_ p/\Q_p)}=\coh^1(H_ p, V_f\otimes V_{gh})^{\Gal(H_ p/\Q_p)}=(\coh^1(H_ p, V_f)\otimes V_{gh})^{\Gal(H_ p/\Q_p)},
\end{equation*}
we get the first isomorphism of \eqref{eq: iso cohomology triple product2}.
Finally, the second isomorphism follows from the relation between $\hom$ and tensor and from the selfduality $V_{gh}^\lor\cong V_{gh}$.
\end{proof}

Let $E(H)_L^{V_{gh}}:=\hom_{\Gal(H/\Q)}(V_{gh},E(H)_L)$. The Kummer  homomorphism
\begin{align*}
	 E(H)_L\longrightarrow\coh^1(H,V_f)
\end{align*}
induces a homomorphism
\begin{align*}
	\delta: E(H)_L^{V_{gh}}\longrightarrow\hom_{\Gal(H/\Q)}(V_{gh},\coh^1(H,V_f))\cong\coh^1(\Q,V),
\end{align*}
which using \eqref{eq: iso cohomology triple product1} can be seen as an isomorphism
\begin{align*}
	\delta: E(H)_L^{V_{gh}}\longrightarrow \coh^1(\Q,V).
\end{align*}

For $\triangle, \heartsuit\in\{ \alpha,\beta \}$, denote $$V_{gh}^{\triangle\heartsuit}:=V_g^{\triangle} \otimes V_h^\heartsuit \ \text{ and } \ V^{\triangle\heartsuit}:=V_f\otimes V_g^{\triangle} \otimes V_h^\heartsuit.$$
Specializing \eqref{eq: filtration of lambda-adic rep} via $y_g$ we obtain
\begin{align*}
  0\longrightarrow V_g^\beta \longrightarrow V_g\longrightarrow V_g^\alpha\longrightarrow 0.
\end{align*}
For $\triangle\neq\heartsuit$ the pairing \eqref{eq: lambda-adic pairing} and its analog for $h$ induce perfect pairings
 \begin{align}
 	\langle \ , \ \rangle : V_{gh}^{\triangle\triangle} \times V_{gh}^{\heartsuit\heartsuit}\longrightarrow L_p, \qquad 	\langle \ , \ \rangle : V_g^{\triangle\heartsuit} \times V_g^{\heartsuit\triangle}\longrightarrow L_p.\label{eq: pairing and eigenspaces}
 \end{align}
 The identifications
 \begin{align}\label{eq: identifications}
V_{gh}^{\triangle \triangle }\cong \Hom_{L_p[G_{\Q_p}]}(V_{gh}^{\heartsuit\heartsuit},L_p) \ \   \text{ and } \ \ V_{gh}^{\triangle \heartsuit }\cong \Hom_{L_p[G_{\Q_p}]}(V_{gh}^{\heartsuit\triangle},L_p),
 \end{align}
  together with \eqref{eq: iso cohomology triple product2} give the following isomorphisms:

 	\begin{align}
 		\coh^1(\Q_p,V^{\triangle\triangle})\cong(\coh^1(H_p,V_f)\otimes V_{gh}^{\triangle\triangle})^{\Gal(H_p/\Q_p)}\cong\hom_{\Gal(H_p/\Q_p)}(V_{gh}^{\heartsuit\heartsuit},\coh^1(H_p,V_f));\\
 		\coh^1(\Q_p,V^{\triangle\heartsuit})\cong(\coh^1(H_p,V_f)\otimes V_{gh}^{\triangle\heartsuit})^{\Gal(H_p/\Q_p)}\cong\hom_{\Gal(H_p/\Q_p)}(V_{gh}^{\heartsuit\triangle},\coh^1(H_p,V_f)).
 	\end{align}

It follows from \cite[Lemma 4.1]{DR3} that the submodule $H^1_f(\Q_p,V_f\otimes V_{gh}^{\triangle\heartsuit})$ and the singular quotient $H^1_s(\Q_p,V_f\otimes V_{gh}^{\triangle\heartsuit})$ can be written in terms of the filtration of $V_f$ as follows:
\begin{align}\label{eq: H1s}
  \coh^1_s(\Q_p,V_f\otimes V_{gh}^{\triangle\heartsuit})=\coh^1(\Q_p,V_f^-\otimes V_{gh}^{\triangle\heartsuit})\cong(V_{gh}^{\heartsuit\triangle}\otimes\coh^1_s(H_p,V_f))^{\Gal(H_p/\Q_p)};
\end{align}
\begin{align}\label{eq: H1f}
  \coh^1_f(\Q_p,V_f\otimes V_{gh}^{\triangle\heartsuit}) = \ker(\coh^1(\Q_p,V_f\otimes V_{gh}^{\triangle\heartsuit})\rightarrow\coh^1(I_p,V_f^-\otimes V_{gh}^{\triangle\heartsuit})) = \coh^1(\Q_p,V_f^+\otimes V_{gh}^{\triangle\heartsuit}).
\end{align}

\begin{lemma}\label{lem: kummer maps eigenspaces}
 For $\triangle,\heartsuit\in\{ \alpha,\beta \}$, $\triangle\neq\heartsuit$, there are isomorphisms
	\begin{align*}
& \delta_p: E(H_p)_{L_p}^{V_{gh}}\longrightarrow\coh^1_f(\Q_p,V);\\
& \delta_p^{\triangle\heartsuit}: E(H_p)_{L_p}^{V_{gh}^{\triangle\heartsuit}}\longrightarrow\coh^1_f(\Q_p,V^{\heartsuit\triangle});\\
& \delta_p^{\triangle\triangle}: E(H_p)_{L_p}^{V_{gh}^{\triangle\triangle}}\longrightarrow\coh^1_f(\Q_p,V^{\heartsuit\heartsuit}).
	\end{align*}

\end{lemma}
\begin{proof}
We prove the existence of the isomorphism $\delta^{\triangle\heartsuit}$, the others are similar. By Kummer theory, there is an injective morphism
	\begin{equation*}
E(H_p)_{L_p}\longrightarrow \coh^1(H_p,V_f),
	\end{equation*}
	which is an isomorphism on its image $\coh^1_f(H_p,V_f)\cong\coh^1(H_p,V_f^+)$.
	It induces an isomorphism
	\begin{equation*}
\delta_p^{\triangle\heartsuit}: E(H_p)_{L_p}^{V_{gh}^{\triangle\heartsuit}}\longrightarrow \hom_{\Gal(H_p/\Q_p)}(V_{gh}^{\triangle\heartsuit},\coh^1(H_p,V_f^+)).
\end{equation*}
Using the isomorphisms \eqref{eq: identifications} we obtain
\begin{equation*}
\hom_{\Gal(H_p/\Q_p)}(V_{gh}^{\triangle\heartsuit},\coh^1(H_p,V_f^+))\overset{\cong}{\longrightarrow}(\coh^1(H_p,V_f^+)\otimes V_{gh}^{\heartsuit\triangle})^{\Gal(H_p/\Q_p)}.
\end{equation*}
Arguing as in the proof of Lemma \ref{lem: first lemma on isomorphisms cohomology triple product}, we get the isomorphisms
\begin{align*}
	(\coh^1(H_p,V_f^+)\otimes V_{gh}^{\heartsuit\triangle})^{\Gal(H_p/\Q_p)}& \cong 	\coh^1(H_p,V_f^+\otimes V_{gh}^{\heartsuit\triangle})^{\Gal(H_p/\Q_p)}\\
	& \cong \coh^1(\Q_p,V_f^+\otimes V_{gh}^{\heartsuit\triangle})\cong\coh^1_f(\Q_p,V^{\heartsuit\triangle}).
\end{align*}
\end{proof}

From now on we will make the following assumption on the Selmer group of $V$.
\begin{assumption}\label{ass: rank 0}
	$\operatorname{Sel}_p(\Q,V)=0$.
\end{assumption}
Under this assumption one can identify the relaxed Selmer group with the singular quotient.
\begin{lemma}
 Under Assumptions \ref{ass: rank 0} and \ref{ass: p doesnt divide the levels} the natural map
	\begin{equation*}
		\partial_p: \operatorname{Sel}_{(p)}(\Q,V){\longrightarrow} \coh^1_s(\Q_p,V)
	\end{equation*}
is an isomorphism. In particular, there is an isomorphism
\begin{align}\label{eq: relaxed selmer group as singular quotient}
  \operatorname{Sel}_{(p)}(\Q,V)\cong \coh^1_s(\Q_p,V^{\alpha\alpha})\oplus \coh^1_s(\Q_p,V^{\alpha\beta})\oplus \coh^1_s(\Q_p,V^{\beta\alpha})\oplus \coh^1_s(\Q_p,V^{\beta\beta})
\end{align}
\end{lemma}
\begin{proof}
	Since the representation $V$ is self-dual there is an isomorphism $\operatorname{Sel}_{p,*}(V^*)\cong\operatorname{Sel}_{p}(V),$ (see, for example, \cite{BK90} and \cite[Theorem 2.1]{Belaichenotes}). Then the lemma follows immediately from the exact sequence (\ref{eq: exact sequence of Selmer groups}).
\end{proof}
In the next subsection we will describe the spaces in the right hand side of \eqref{eq: relaxed selmer group as singular quotient} in terms of dual exponential maps.

\subsection{Bloch--Kato logarithms and exponentials} The  $G_{\Q_p}$-representations of the form \mbox{$V_f^+\otimes V_{gh}^{\triangle\heartsuit}$} are one dimensional and, therefore, given by characters. Indeed, $G_{\Q_p}$ acts on $V_f^{+}$ as $\chi_{\mathrm{cycl}}\psi_f^{-1}$,  and it acts as $\psi_g$ (resp. $\psi_g^{-1}$) on $V_g^\alpha$ (resp. $V_g^{\beta}$) and as $\psi_h$ (resp. $\psi_h^{-1}$) on $V_h^\alpha$ (resp. $V_h^{\beta}$). Therefore we have that
 \begin{align*}
		V_f^+\otimes V_{gh}^{\alpha\alpha}=L_p(\chi_{\cycl}\psi_f^{-1}\psi_g\psi_h), \ \qquad  \ V_f^+\otimes V_{gh}^{\alpha\beta}=L_p(\chi_{\cycl}\psi_f^{-1}\psi_g\psi_h^{-1}), \\  V_f^+\otimes V_{gh}^{\beta\alpha}=L_p(\chi_{\cycl}\psi_f^{-1}\psi_g^{-1}\psi_h),\ \qquad \ V_f^+\otimes V_{gh}^{\beta\beta}=L_p(\chi_{\cycl}\psi_f^{-1}\psi_g^{-1}\psi_h^{-1}).
	\end{align*}
In particular $V_f^+\otimes V_{gh}^{\triangle\heartsuit}$ is of the form $L_p(\psi\chi_{\cycl})$ for some nontrivial unramified character $\psi$. By \eqref{eq: H1f} we have that $\coh^1_f(\Q_p,V^{\triangle\heartsuit})\cong \coh^1(\Q_p,V_f^+\otimes V_{gh}^{\triangle\heartsuit})$, and the Bloch--Kato logarithm gives an isomorphism (cf. \cite[Example 1.6 (a)]{DR3}):
\begin{equation}\label{eq: BK}
	\log_{\triangle\heartsuit}:\coh^1_f(\Q_p,V^{\triangle\heartsuit})\longrightarrow D(V_f^+\otimes V_{gh}^{\triangle\heartsuit})=D(L_p(\psi\chi_{\cycl})).
      \end{equation}
       For $(\triangle,\heartsuit)=(\alpha,\alpha)$, the pairings \ref{eq: pairing Vg+} and the analogous pairings for $f$ and $h$ give rise to a pairing
\begin{equation}
	\langle \ , \ \rangle:V_f^+\otimes V_g^\alpha\otimes V_h^\alpha \times V_f^-(-1)\otimes V_g^\beta\otimes V_h^\beta \longrightarrow L_p
\end{equation}
which induces
\begin{equation}\label{eq: pairing D(V)}
\langle \ , \ \rangle:D(V_f^+\otimes V_g^\alpha\otimes V_h^\alpha) \times D(V_f^-(-1)\otimes V_g^\beta\otimes V_h^\beta )\longrightarrow D(L_p)= L_p.
\end{equation}
Denote by $\tilde\omega_f$ the differential form on $X_0(N_f)$ corresponding to $f$. It can be naturally viewed as an element of the de Rham cohomology group $H^1_{\mathrm{dR}}(X_0(N_f)/\Q_p)$. The comparison isomorphisms of $p$-adic Hodge theory provide a natural map
\begin{align*}
 H^1_{\mathrm{dR}}(X_0(N_f)/\Q_p)(1)\longrightarrow D(V_f^-)
\end{align*}
and therefore $\tilde\omega_f$ gives rise to an element $\omega_f\in D(V_f^-(-1))$. In \eqref{eq: pairing D(V)}, pairing with the class $\omega_f\otimes\eta_g\otimes\eta_h$ gives then an isomorphism
\begin{equation}\label{eq: pair with class}
	\langle\cdot, \omega_f\otimes\eta_g\otimes\eta_h\rangle:D(L_p(\psi\chi_{\cycl}))=D(V_f^+\otimes V_g^\alpha\otimes V_h^\alpha)\longrightarrow  L_p.
\end{equation}
There are similar pairings and isomorphisms for the remaining pairs $(\triangle, \heartsuit)$.
We still denote
\begin{equation}\label{eq: block-kato logarithms on V}
\log_{\triangle\heartsuit}:\coh^1_f(\Q_p,V_f\otimes V_{gh}^{\triangle\heartsuit})\longrightarrow  L_p
\end{equation}
the map obtained by composing \eqref{eq: BK} with \eqref{eq: pair with class}.

\begin{remark}\label{rem: logaritmos} The logarithm maps of \eqref{eq: block-kato logarithms on V} are related to the usual $p$-adic logarithm on $E$ as follows. The differential $\omega_f$ gives rise to an invariant differential on $E$, and we denote by
  \begin{align*}
\log_{f,p}\colon E(H_p)\longrightarrow H_p
  \end{align*}
the corresponding formal group logarithm on $E$.  The map $\log_{\alpha\beta}$ coincides with the inverse of the isomorphism of Lemma \ref{lem: kummer maps eigenspaces}
\begin{equation*}
E(H_p)_{L_p}^{V_{gh}^{\beta\alpha}}\cong( E(H_p)^{\beta_g\alpha_h}\otimes V_{gh}^{\alpha\beta})^{G_{\Q_p}}
\end{equation*}
composed with the maps
\begin{align*}
  \begin{array}{ccccc}
	(E(H_p)^{\beta_g\alpha_h}\otimes V_{gh}^{\alpha\beta})^{G_{\Q_p}} & \longrightarrow & (H_p^{\beta_g\alpha_h}\otimes V^{\alpha\beta}_{gh})^{G_{\Q_p}}= D(V^{\alpha\beta}_{gh}) & \longrightarrow & L_p\\
x\otimes v_g^\alpha v_h^\beta & \longmapsto &\log_{f,p}(x)\otimes v_g^\alpha v_h^\beta & & \\
 & & y & \longmapsto & \langle y, \eta_g\omega_h\rangle.
 \end{array}
\end{align*}
Analogous equalities hold for the other maps $\log_{\triangle\heartsuit}$.
\end{remark}

A similar discussion can be applied to the representations of the form $V_f^{-}\otimes V_{gh}^{\triangle\heartsuit}$. In this case we have the following isomorphisms of $1$-dimensional representations:
 \begin{align*}
		V_f^-\otimes V_{gh}^{\alpha\alpha}=L_p(\psi_f\psi_g\psi_h), \ \qquad \ V_f^-\otimes V_{gh}^{\alpha\beta}=L_p(\psi_f\psi_g\psi_h^{-1}\bar\chi), \\ V_f^-\otimes V_{gh}^{\beta\alpha}=L_p(\psi_f\psi_g^{-1}\psi_h\chi),\ \qquad \ V_f^-\otimes V_{gh}^{\beta\beta}=L_p(\psi_f\psi_g^{-1}\psi_h^{-1}).
	\end{align*}
Therefore,  $V_f^-\otimes V_{gh}^{\triangle\heartsuit}$ is isomorphic to a representation of the form $ L_p(\psi)$ for some unramified and nontrivial character  $\psi$. By \eqref{eq: H1s}  there is an identification
\begin{align*}
 \coh^1_s(\Q_p,V^{\triangle\heartsuit})=\coh^1(\Q_p,L_p(\psi)),
\end{align*}
and by \cite[Example 1.8 (b)]{DR2}  the dual exponential gives isomorphisms
\begin{equation}\label{eq: dual exponentials for V}
\exp^*_{\triangle\heartsuit}:\coh^1_s(\Q_p,V^{\triangle\heartsuit})\longrightarrow D(L_p(\psi))\cong  L_p,
\end{equation}
where the last isomorphism is induced by pairing with the appropriate  class of $D(L_p(\psi^{-1}))=D(V_f^+(-1)\otimes V_g^{\heartsuit}\otimes V_h^\triangle)$ similarly as in \eqref{eq: pair with class}.
Arguing as in Remark \ref{rem: logaritmos}, let
$$\exp_{f,p}^*:\coh^1_s(H_p,V_f)\longrightarrow H_p$$ denote the dual exponential on $\coh^1_s(H_p,V_f)$.  Then $\exp^*_{\beta\beta}$ can be identified with the composition
\begin{align}\label{eq: dualexp with pairing}
  \begin{array}{ccccc}
(\coh^1_s(H_p,V_f)^{\alpha_g\alpha_h}\otimes V_{gh}^{\beta\beta})^{G_{\Q_p}} & {\longrightarrow} & (H_p^{\alpha_g\alpha_h}\otimes V^{\beta\beta}_{gh})^{G_{\Q_p}}= D(V^{\beta\beta}_{gh}) & \longrightarrow & L_p\\
x\otimes v_g^\beta v_h^\beta & \longmapsto & \exp_{f,p}^*(x)\otimes v_g^\beta v_h^\beta & & \\
& &  y &\longmapsto & \langle y, \omega_g\omega_h\rangle,
  \end{array}
\end{align}
after taking into account the identification
\begin{align*}
      \coh^1_s(\Q_p,V^{\beta\beta})\cong(\coh^1_s(H_p,V_f)^{\alpha_g\alpha_h}\otimes V_{gh}^{\beta\beta})^{G_{\Q_p}}.
\end{align*}
Analogous formulas hold for the dual exponentials $\exp_{\triangle\heartsuit}$ on the remaining components.

To sum up the discussion of this subsection, we conclude that the relaxed Selmer group of $V$ admits a basis adapted to decomposition \eqref{eq: relaxed selmer group as singular quotient} with respect to the dual exponential maps.
\begin{proposition}
  Under Assumptions \ref{ass: rank 0} and \ref{ass: p doesnt divide the levels}, $\operatorname{Sel}_{(p)}(V)$ has a basis
  \begin{align}\label{eq: basis xi}
    \{ \xi^{\alpha\alpha}, \xi^{\alpha\beta}, \xi^{\beta\alpha}, \xi^{\beta\beta} \}
  \end{align}
  characterized by the fact that there exist elements $\Psi_{\beta\beta}\in\coh^1_s(H_p,V_f)^{\beta_g\beta_h}, \ \Psi_{\beta\alpha}\in\coh^1_s(H_p,V_f)^{\beta_g\alpha_h}, \ \Psi_{\alpha\beta}\in\coh^1_s(H_p,V_f)^{\alpha_g\beta_h}, \ \Psi_{\alpha\alpha}\in\coh^1_s(H_p,V_f)^{\alpha_g\alpha_h}$ such that  $$\partial_p{\xi}^{\alpha\alpha}=(\Psi_{\beta\beta}\otimes v_g^\alpha v_h^\alpha,0,0,0), \ \partial_p{\xi}^{\alpha\beta}=(0,\Psi_{\beta\alpha}\otimes v_g^\alpha v_h^\beta,0,0)$$
	$$\partial_p{\xi}^{\beta\alpha}=(0,0,\Psi_{\alpha\beta}\otimes v_g^\beta v_h^\alpha,0), \ \partial_p{\xi}^{\beta\beta}=(0,0,0,\Psi_{\alpha\alpha}\otimes v_g^\beta v_h^\beta)$$
	and $$\exp_{f,p}^*(\Psi_{\beta\beta})=\exp_{f,p}^*(\Psi_{\beta\alpha})=\exp_{f,p}^*(\Psi_{\alpha\beta})=\exp_{f,p}^*(\Psi_{\alpha\alpha})=1.$$
\end{proposition}
\begin{remark}
  Notice that the basis \eqref{eq: basis xi}  depends on the choice of the $L$-basis $v_g^\alpha, v_g^\beta$ of $V_g$ and the $L$-basis $v_h^\alpha, v_h^\beta$ of $V_h$. Then each element of the basis $\{ \xi^{\alpha\alpha}, \xi^{\alpha\beta}, \xi^{\beta\alpha}, \xi^{\beta\beta}   \}$ depends on this choice up to multiplication by an element of $L^\times$.
\end{remark}

\section{Special value formula for the triple product $p$-adic $\L$-function in rank $0$}\label{sec: main thm}
We continue with the notation and assumptions of the previous section. In particular, $V:=V_{f}\otimes V_g\otimes V_h$ is the tensor product of the $p$-adic representations attached to the newforms $$f\in S_2(N_f)_\Q, \  \ g\in M_1(N_g,\chi)_L, \ \ h\in M_1(N_h,\bar{\chi})_L,$$
and we assume from now on that $\gcd(N_f, N_g, N_h)$ is square free. Recall that $V_g^{\comp}$ (resp. $V_h^{\comp}$) stands for the Artin representation attached to $g$ (resp. $h$) and $\rho$ denotes the tensor product representation
\begin{align*}
  \rho: \Gal(H/\Q)\longrightarrow\GL(V^{\comp}_{g}\otimes V_h^{\comp})\cong\GL_4(L).
\end{align*}
The complex $L$-function $$L(E, \rho,s)=L(f\otimes g\otimes h,s)$$ has entire continuation and satisfies a functional equation relating the value at $s$ with the value at $2-s$.
Let $\epsilon$ be the  sign of this functional equation and denote $N:=\mathrm{lcm}(N_f,N_g,N_h)$. Then $\epsilon$ is the product of local signs
\begin{align*}
  \epsilon=\prod_{v}\epsilon_v,
\end{align*}
where $v$ runs over the places of $\Q$ dividing $N$ or $\infty$. In this setting, $\epsilon_\infty=+1$. Assume also that $\epsilon_v=+1$ for all $v\mid N$. In particular, the global sign is $\epsilon = 1$ and the order of vanishing of $L(E,\rho,s)$ at the central point $s=1$ is even.

Recall that $p$ stands for a prime that does not divide $N$, and that ${\bf g}\in \Lambda_{\bf g}[[q]]$ (resp. ${\bf h}\in\Lambda_{\bf h}[[q]]$) is a Hida family passing through the $p$-stabilization $g_\alpha$ (resp. $h_\alpha$) such that $U_pg_\alpha=\alpha_g g_\alpha$ (resp. $U_ph_\alpha=\alpha_h h_\alpha$). Similarly, denote by ${\bf f}\in \Lambda_{\bf f}[[q]]$ a Hida family passing through the $p$-stabilization $f_\alpha$ of $f$.

Denote by $\L_p^g({\bf f},{\bf g},{\bf h})$ the triple product $p$-adic $\L$-function defined in \cite{DR2}, attached to the choice of $\Lambda$-adic test vector $(\breve{{\bf f}},\breve{{\bf g}},\breve{{\bf h}})$ of \cite[Chap. 3]{Hsieh}. The values $\L_p^g({\bf f},{\bf g},{\bf h})(k,\ell,m)$ of this $p$-adic $\L$-function at triples of integers $(k,\ell,m)$ with $\ell\geq k+m$ interpolate the square root of the algebraic part of
\begin{align}\label{eq: interpolation}
  L(\breve{{\bf f}}_k\otimes \breve{{\bf g}}_\ell\otimes \breve{{\bf h}}_m,\frac{k+\ell+m-2}{2}),
\end{align}
where $\breve{{\bf f}}_k,\breve{{\bf g}}_\ell, \breve{{\bf h}}_m$ denote the specializations of $\breve{{\bf f}}, \breve{{\bf g}}, \breve{{\bf h}}$ at weights $k,\ell,m$.

There is an analogous triple product $p$-adic $\L$-function $\L_p^f({\bf f},{\bf g},{\bf h})$ that interpolates \eqref{eq: interpolation} but for the values $(k,\ell,m)$ with $k\geq \ell + m$. In particular, $\L_p^f({\bf f},{\bf g},{\bf h})(2,1,1)$ is directly related to $L(E,\rho,1)$.

The article \cite{DLR} studies the value $\L_p^g({\bf f},{\bf g},{\bf h})(2,1,1)$ when $L(E,\rho,1)\neq 0$. In particular, the Elliptic Stark Conjecture predicts that when $E(H)^\rho$ is $2$-dimensional $\L_p^g({\bf f},{\bf g},{\bf h})(2,1,1)$ is related to the $p$-adic logarithms of global elements in $E(H)^\rho$.

In the present note, our running Assumption \ref{ass: rank 0} is that $\operatorname{Sel}_p(\Q,V)=0$. This implies that $E(H)^\rho=0$ and, conjecturally, it also implies that $L(E,\rho,1)\neq 0$. The main result of this section is an explicit formula for $\L_p^g({\bf f},{\bf g},{\bf h})(2,1,1)$ in this case, and this can be seen as completing the study of $\L_p^g({\bf f},{\bf g},{\bf h})(2,1,1)$ initiated in \cite{DLR}. The main tool that we will use are the generalized Kato classes
\begin{align*}
  \kappa:=\kappa(f,g_\alpha,h_\alpha)\in\operatorname{Sel}_{(p)}(\Q,V)
\end{align*}
introduced in \cite[\S3]{DR2}. Next, we briefly recall the relation between $\kappa$ and the  $p$-adic $L$-values $\L^f_p({\bf f},{\bf g},{\bf h})(2,1,1)$ and $\L^g_p({\bf f},{\bf g},{\bf h})(2,1,1)$. To lighten the notation, from now on we put $\L^f_p:=\L^f_p({\bf f},{\bf g},{\bf h})(2,1,1)$ and $\L_p^g:=\L^g_p({\bf f},{\bf g},{\bf h})(2,1,1)$. Let also
\begin{align*}
  \pi_{\alpha\beta}:\coh^1(\Q_p,V)\longrightarrow \coh^1(\Q_p,V^{\alpha\beta})
\end{align*}
be the projection map induced by the natural map $V\rightarrow V^{\alpha\beta}$.

\begin{proposition}[Darmon--Rotger]\label{prop: relation with Lpf}
  \begin{enumerate}
  \item The element $\partial_p\kappa$ lies in the image of the natural map
  \begin{align*}
   \coh^1_s(\Q_p,V^{\beta\beta})\longrightarrow \coh_s^1(\Q_p,V) .
  \end{align*}
In particular, $\partial_p\kappa$ can be viewed as an element of  $\coh^1_s(\Q_p,V^{\beta\beta})$. Moreover,
  	\begin{equation}\label{eq: relation with Lpf}
	\exp_{\beta\beta}^*(\partial_p\kappa)=  \dfrac{2(1-p\alpha_f\alpha_g^{-1}\alpha_h^{-1})}{\alpha_g\alpha_h(1-\alpha_f^{-1}\alpha_g\alpha_h)(1-\chi^{-1}(p)\alpha_f^{-1}\alpha_g\alpha_h^{-1})}\times \L_p^f.
      \end{equation}

      \item The element $\pi_{\alpha\beta}\operatorname{res}_p\kappa\in\coh^1(\Q_p,V^{\alpha\beta})$ belongs to $\coh_f^1(\Q_p, V^{\alpha\beta})$, and
	\begin{equation}\label{eq: relation with Lpg}
\log_{\alpha\beta}(\pi_{\alpha\beta}\operatorname{res}_p\kappa)=\L_p^g \times 2(1-\chi(p)p^{-1}\alpha_fa_p(g)^{-1}a_p(h))^{-1}.
\end{equation}

  \end{enumerate}
  \end{proposition}
  \begin{proof} The fact that $\partial_p\kappa$ is the image of an element in $\coh^1_s(\Q_p,V^{\beta\beta})$ is \cite[Proposition 2.8]{DR2}. The equality \eqref{eq: relation with Lpf} follows from Proposition 5.2 and  Theorem 5.3 of \cite{DR2}.
  By part $(1)$ of the proposition  $\pi^s_{\alpha\beta}\partial_p\kappa=0$ in the singular quotient $\coh_s^1(\Q_p, V^{\alpha\beta})$. This means that $\pi_{\alpha\beta}\operatorname{res}_p\kappa$ belongs to $\coh^1_f(\Q_p,V^{\alpha\beta})$. Equality \eqref{eq: relation with Lpg} follows from Proposition 5.1, Theorem 5.3 of \cite{DR2}.
\end{proof}

Using the $\kappa$ recalled above and the basis \eqref{eq: basis xi} of $\operatorname{Sel}_{(p)}(V)$, we can give a precise formula for $\L_p^g$ in the rank $0$ setting. Define $R_{\beta\alpha}\in E(H_p)^{\beta\alpha}$ by the equality
\begin{equation}\label{eq: def of Ralphabeta}
	\pi_{\alpha\beta}\operatorname{res}_p{\xi}^{\beta\beta}={R}_{\beta\alpha}\otimes v_g^{\alpha}v_h^{\beta}\in\coh_f^1(\Q_p,V^{\alpha\beta})=(E(H_p)^{\beta\alpha}\otimes V_{gh}^{\alpha\beta})^{G_{\Q_p}}.
\end{equation}

\begin{theorem}\label{thm: main}
The class $\kappa$ is a multiple of $\xi^{\beta\beta}$; more precisely,
\begin{align*}
  \kappa=\dfrac{\Theta_g\Theta_h2(1-p\alpha_f\alpha_g^{-1}\alpha_h^{-1})\L_p^f }{\alpha_g\alpha_h(1-\alpha_f^{-1}\alpha_g\alpha_h)(1-\chi^{-1}(p)\alpha_f^{-1}a_p(g)a_p(h)^{-1})}\cdot{\xi}^{\beta\beta}.
\end{align*}
Moreover, if we define the quantities
\begin{align*}
  \L_{g_\alpha}:=\dfrac{\Omega_g}{\Theta_g}, \ \ \ {\mathcal{E}}:=\dfrac{(1-\chi(p)p^{-1}\alpha_g^{-1}\alpha_h)(1-p\alpha_f\alpha_g^{-1}\alpha_h^{-1})}{\alpha_g\alpha_h(1-\alpha_f^{-1}\alpha_g\alpha_h)(1-\chi^{-1}(p)\alpha_f^{-1}\alpha_g\alpha_h^{-1})}
\end{align*}
then we have that
\begin{align*}
\L^g_p={{\mathcal{E}} }\times\dfrac{\log_p({R}_{\beta\alpha})}{\L_{g_\alpha}}\times{\L_p^f} \mod L^\times.
\end{align*}
\end{theorem}
\begin{proof}
	By Proposition \ref{prop: relation with Lpf}, $\kappa$ is an element of $\operatorname{Sel}_{(p)}(\Q,V)$ such that
	\begin{equation}\label{eq: exp kappa}
		\exp^*(\partial_p\kappa)=(0,0,0,\dfrac{2(1-p\alpha_f\alpha_g^{-1}\alpha_h^{-1})}{\alpha_g\alpha_h(1-\alpha_f^{-1}\alpha_g\alpha_h)(1-\chi^{-1}(p)\alpha_f^{-1}\alpha_g\alpha_h^{-1})}\times \L_p^f).
	\end{equation}
	Then $\kappa$ is a multiple of the element $\xi^{\beta\beta}$; indeed
	$$\kappa=\dfrac{\exp^*_{\beta\beta}(\partial_p\kappa)}{\exp^*_{\beta\beta}(\partial_p{\xi}^{\beta\beta})}{\xi}^{\beta\beta}.$$
Observe that \eqref{eq: exp kappa} gives us the expression for the numerator. We now compute the denominator.
	\begin{align*}
		\exp^*_{\beta\beta}(\partial_p{\xi}^{\beta\beta})& = \langle  \exp_{f,p}^* (\Psi_{\alpha\alpha})\otimes v_g^\beta v_h^\beta,\omega_g\omega_h \rangle\\
		 &=\dfrac{\exp_{f,p}^* (\Psi_{\alpha\alpha})}{\Theta_g\Theta_h}\\
		 & = \dfrac{1}{\Theta_g\Theta_h}.
	\end{align*}
	Here we used the fact that $\eta_g\eta_h=\Theta_g\Theta_h v_g^\beta v_h^\beta$.
	So we get
	\begin{align*}
	\kappa & =\dfrac{\exp^*_{\beta\beta}(\partial_p\kappa)}{\exp^*_{\beta\beta}(\partial_p{\xi}^{\beta\beta})}\cdot{\xi}^{\beta\beta}\\
	& = \dfrac{2(1-p\alpha_f\alpha_g^{-1}\alpha_h^{-1})\Theta_g\Theta_h}{\alpha_g\alpha_h(1-\alpha_f^{-1}\alpha_g\alpha_h)(1-\chi^{-1}(p)\alpha_f^{-1}\alpha_g\alpha_h^{-1})}\times \L_p^f\cdot{\xi}^{\beta\beta}
	\end{align*}
	By \eqref{eq: relation with Lpg},
	\begin{align*}
		\L_p^g & =\frac{1}{2}(1-\chi(p)p^{-1}\alpha_g^{-1}\alpha_h)\log_{\alpha\beta}(\pi_{\alpha\beta}\operatorname{res}_p\kappa)\\
		& = \dfrac{\Theta_g\Theta_h(1-\chi(p)p^{-1}\alpha_g^{-1}\alpha_h)(1-p\alpha_f\alpha_g^{-1}\alpha_h^{-1})}{\alpha_g\alpha_h(1-\alpha_f^{-1}\alpha_g\alpha_h)(1-\chi^{-1}(p)\alpha_f^{-1}\alpha_g\alpha_h^{-1})}\times \L_p^f\log_{\alpha\beta}(\pi_{\alpha\beta}\operatorname{res}_p{\xi}^{\beta\beta})\\
		& ={\mathcal{E}} \Theta_g\Theta_h \L_p^f \log_{\alpha\beta}(\pi_{\alpha\beta}\operatorname{res}_p{\xi}^{\beta\beta})\\
		& = {\mathcal{E}} \Theta_g\Theta_h \L_p^f\langle \log_p({R}_{\beta\alpha})\otimes v_g^\alpha v_h^\beta,\eta_g\omega_h  \rangle\\
		& = {{\mathcal{E}}\L_p^f }\dfrac{  \Theta_g\Theta_h }{\Omega_g\Theta_h}\log_p({R}_{\beta\alpha})\\
		& = {\mathcal{E}}\L_p^f\dfrac{\Theta_g}{\Omega_g}\log_p({R}_{\beta\alpha})\\
		& = \dfrac{{\mathcal{E}}\L_p^f }{\L_{g_\alpha}}\log_p({R}_{\beta\alpha})
	\end{align*}
	since $\omega_g\eta_h=\Omega_g\Theta_h\otimes v_g^\alpha v_h^\beta$.
\end{proof}

We end this section by noting that $\L_{g_\alpha}$ is often expected to be related to the Gross--Stark Unit $u_{g_\alpha}$ attached to the modular form $g_\alpha$ as defined in \cite[\S1]{DLR}. More precisely, under the additional assumption that $g$ is not the theta series of a Hecke character of a real quadratic field in which $p$ splits, \cite[Conjecture 2.1]{DR2b} predicts that
\begin{align}\label{eq: conj unit}
  \L_{g_\alpha}\stackrel{?}{=}\log_p(u_{g_\alpha}) \ \mod L^\times.
\end{align}
Thus we obtain the following consequence of Theorem \ref{thm: main}, under the aforementioned hypothesis:
\begin{corollary}
	Assuming the equality \eqref{eq: conj unit}, if $\operatorname{Sel}_p(\Q,V)=0$ then
	$$\L^g_p=\mathcal{E}\times\dfrac{\log_p(R_{\beta\alpha})}{\log_p(u_{g_\alpha})}\times \L_p^f.$$
\end{corollary}

\section{The case of theta series of an imaginary quadratic field $K$ where $p$ splits }\label{sec: p split in K}
In this section we will consider a particular case where $g$ and $h$ are theta series of the same imaginary quadratic field in which $p$ splits. We will see that in this setting the representation $V$ decomposes in a way that forces $\L_p^g$ to vanish when the complex $L$-function does not vanish at the central critical point; that is, the special value of the $p$-adic $\L$-function vanishes in analytic rank $0$.

Let $K$ be an imaginary quadratic field of discriminant $D_K$. Let $\psi_g,\psi_h\colon \A_K^\times\rightarrow \C^\times$ be two finite order Hecke characters of $K$ of conductors ${\mathfrak{c}}_g,{\mathfrak{c}}_h$ and central characters $\varepsilon, \bar{\varepsilon}$ respectively. Here $\varepsilon \colon \A_\Q^\times\rightarrow \C^\times$  is a finite order character of  and $\bar{\varepsilon}$ denotes is complex conjugate. Let $g$ and $h$ be the theta series attached to $\psi_g$ and $\psi_h$. They are modular forms of weight one, and their levels and nebentype characters are given by
\begin{align*}
N_g:=D_K\cdot\norm_K({\mathfrak{c}}_g), \  \ N_h:=D_K\cdot\norm_K({\mathfrak{c}}_h), \ \ \chi:=\chi_K\cdot\varepsilon, \ \ \bar{\chi}=\chi_K\cdot\bar{\varepsilon},
\end{align*}
where $\norm_K$ stands for the norm on ideals of $K$ and we regard $\varepsilon$ and $\bar{\varepsilon}$ as Dirichlet characters via class field theory. That is to say,
\begin{align*}
g\in M_1(N_g,\chi), \ \ \text{and} \ \  h\in M_1(N_h,\bar{\chi}).
\end{align*}
Let $f\in S_2(N_f)$ be a newform with rational coefficients and let $E$ be the associated elliptic curve over $\Q$. We will particularize some of the results of the previous sections to this choice of forms $f$, $g$, and $h$, so we will use the same notations as before. In particular, $\rho$ stands for the Artin representation afforded by $V_g\otimes V_h$ and $p$ is a prime that does not divide $N_f\cdot N_g\cdot N_h$. In this section, we will make the following additional assumptions:
\begin{enumerate}
\item $\gcd(N_f,{\mathfrak{c}}_g{\mathfrak{c}}_h)=1;$
\item $p$ splits in $K$.
\end{enumerate}

A finite order Hecke character $\psi$ of $K$ can be regarded, via class field theory, as a Galois character $\psi\colon G_K\rightarrow\A_K^\times$. Let $\sigma_0$ be any element in  $G_\Q\setminus G_K$. We denote by $\psi'$ the character defined by $\psi'(\sigma):=\psi(\sigma_0\sigma\sigma_0^{-1})$ (this does not depend on the particular choice of $\sigma_0$). Also, $\psi$ gives rise to a $1$-dimensional representation of $G_K$, and we let  $V_\psi=\Ind_K^\Q(\psi)$ denote the induced representation; it is a $2$-dimensional representation of $G_\Q$. Observe that, with this notation, $V_g=V_{\psi_g}$ and $V_h=V_{\psi_h}$.

There is a well-known decomposition of $V_g\otimes V_h$ as the direct sum of two representations:
\begin{equation}\label{eq: factorisation of reps attached to gh}
	V_g\otimes V_h=V_{\psi_1}\oplus V_{\psi_2},
\end{equation}
where the characters $\psi_1$ and $\psi_2$ are
\begin{align*}
  \psi_1:=\psi_g\psi_h,  \ \ \text{and} \ \ \psi_2:=\psi_g\psi'_h.
\end{align*}
This induces a decomposition of the representation $V=V_f\otimes V_g\otimes V_h$ as a direct sum of two representations:
\begin{equation}\label{eq: factorisation of reps}
V=V_1\oplus V_2,
\end{equation}
where
\begin{align*}
V_1:=V_f\otimes V_{\psi_1}, \ \ \text{and} \ \  V_2:=V_f\otimes V_{\psi_2}.
\end{align*}
This induces a factorization of complex $L$-functions
\begin{align*}
  L(E,\rho,s)=L(E,\psi_1,s)\cdot L(E,\psi_2,s).
\end{align*}
Under our assumption that $\gcd(N_f,{\mathfrak{c}}_g{\mathfrak{c}}_h)=1$ the local signs of $L(E,\psi_1,s)$ and $L(E,\psi_2,s)$ are equal, so that the local signs of $L(E,\rho,s)$ are all equal to $+1$ and therefore the assumption on local signs of Section \ref{sec: main thm} is satisfied.

\begin{theorem}\label{thm: CM case} In the setting of this section, if   $L(E,\rho,1)\neq 0$ then $\L_p^g=0.$
\end{theorem}
\begin{proof}
If $L(E,\rho,1)\neq 0$ then $L(E,\psi_i,1)\neq 0$ for $i=1,2$. Note that $\psi_1$ and $\psi_2$ are ring  class characters of the imaginary quadratic field $K$. Then, by results of Gross--Zagier and Kolyvagin
\begin{equation}\label{eq: algebraic rank (0,0)}
	\operatorname{Sel}_p(\Q,V_i)=0 \ \ \text{for } i=1,2.
\end{equation}
The decomposition \eqref{eq: factorisation of reps} induces a decomposition of the Selmer groups
\begin{equation}\label{eq: decomposition of Selmer}
	\operatorname{Sel}_p(\Q,V)=\operatorname{Sel}_p(\Q,V_1)\oplus\operatorname{Sel}_p(\Q,V_2),
\end{equation}
 and analogously for the relaxed and the strict Selmer groups of $V$. In particular, $\operatorname{Sel}_p(\Q,V)=0.$

Since $p$ splits in $K$ we can write $p{\mathcal O}_K=\mathfrak p}\def\P{\mathbb P\bar{\mathfrak p}\def\P{\mathbb P}$, and from our assumption that $p\nmid N_f\cdot N_g\cdot N_h$ we see that $p\nmid {\mathfrak{c}}_g{\mathfrak{c}}_h$. Without loss of generality we can suppose that
$$\psi_g({\mathfrak{p}})=\alpha_g, \  \ \psi_g(\bar{\mathfrak p}\def\P{\mathbb P})=\beta_g, \ \ \psi_h({\mathfrak{p}})=\alpha_h, \  \ \psi_h(\bar{\mathfrak p}\def\P{\mathbb P})=\beta_h,$$ so that
$$V_1=V^{\alpha\alpha}\oplus V^{\beta\beta} \ \ \text{and} \ \ V_2=V^{\alpha\beta}\oplus V^{\beta\alpha}.$$

By (\ref{eq: algebraic rank (0,0)}), the same computations as in \S\ref{eq: selmer structure} show that there are isomorphisms
\begin{align*}
  \operatorname{Sel}_{(p)}(\Q,V_1)\overset{\partial_p}{\longrightarrow}\coh^1_s(\Q_p,V_1)\overset{(\pi^s_{\alpha\alpha},\pi^s_{\beta\beta})}{\longrightarrow}\coh^1_s(\Q_p,V_1^{\alpha\alpha})\oplus\coh^1_s(\Q_p,V_1^{\beta\beta}),
\end{align*}
where $\pi_{\alpha\alpha}^s$ denotes the natural map in the singular quotient induced by the projection $V\rightarrow V^{\alpha\alpha}$, and analogously for $\pi_{\beta\beta}^s$.
Similarly, there are dual exponential maps
\begin{align*}
  \exp^*_{\alpha\alpha}:\coh^1_s(\Q_p,V_1^{\alpha\alpha})=\coh^1(\Q_p,V_f^-\otimes V_{gh}^{\alpha\alpha}){\longrightarrow} L_p
\end{align*}
and
\begin{align*}
\exp^*_{\beta\beta}:\coh^1_s(\Q,V_1^{\beta\beta})=\coh^1(\Q_p,V_f^-\otimes V_{gh}^{\beta\beta}){\longrightarrow} L_p
\end{align*}
which are in fact isomorphisms.

Then $\operatorname{Sel}_{(p)}(\Q,V_1)$ has dimension $2$ over $\Q_p$ with the canonical basis
\begin{align*}
  \zeta^{\alpha\alpha}, \ \zeta^{\beta\beta},
\end{align*}
where $\zeta^{\alpha\alpha}$ is characterized (up to scalars in $L^\times$) by the fact that
$$\exp^*_{\alpha\alpha}(\pi_{\alpha\alpha}\partial_p(\zeta^{\alpha\alpha}))=1, \ \ \text{and} \ \ \exp^*_{\beta\beta}(\pi_{\beta\beta}\partial_p(\zeta^{\alpha\alpha}))=0.$$
Similarly,
$$\exp^*_{\alpha\alpha}(\pi_{\alpha\alpha}\partial_p(\zeta^{\beta\beta}))=0, \ \ \text{and} \ \ \exp^*_{\beta\beta}(\pi_{\beta\beta}\partial_p(\zeta^{\beta\beta}))=1.$$
Analogously, $\operatorname{Sel}_{(p)}(\Q,V_2)$ has dimension $2$ with basis $\zeta^{\alpha\beta}, \zeta^{\beta\alpha}$.

By Theorem \ref{thm: main}, the value $\L_p^g$ is a multiple of
$\log_{\alpha\beta}(\operatorname{res}_p\xi^{\beta\beta})$. On the other hand,  using the decomposition $$\operatorname{Sel}_{(p)}(\Q,V)=\operatorname{Sel}_{(p)}(\Q,V_1)\oplus\operatorname{Sel}_{(p)}(\Q,V_2),$$ the element $\xi^{\beta\beta}\in\operatorname{Sel}_{(p)}(\Q,V)$  corresponds to a multiple of $(0,\zeta^{\beta\beta})$, and this implies that $$\pi_{\beta\alpha}\operatorname{res}_p\xi^{\beta\beta}=0.$$
\end{proof}

\section{Numerical computations}\label{sec: Numerical computations}

In this section we present some numerical examples. They have been computed with a Sage (\cite{Sage}) implementation of Lauder's algorithms (\cite{Lauder}), adapted to work in the current setting. The code is available at \url{github.com/mmasdeu/ellipticstarkconjecture}. The data for the weight-one modular forms can be found in Alan Lauder's website.\footnote{See \url{http://people.maths.ox.ac.uk/lauder/weight1/}.}

\subsection{Dihedral case}
\begin{enumerate}[$(a)$]
\item We computed $\L_p^g({\bf f},{\bf g},{\bf g})(2,1,1)$ with ${\bf f}$ the Hida family passing through the modular form $f_E$ of weight $2$ attached to an elliptic curve $E/\Q$ of conductor $N_f$ and ${\bf g}$ attached to the weight-one modular form  $g=\theta(1_K)$ for some imaginary quadratic field $K$. The modular form $g$ belongs then to $M_1(N_g,\chi_K)_\Q$. For each of the entries in the table we give the Cremona label for the elliptic curve $E_f$, its conductor $N_f$, the field $K$, the level $N_g$ of $g$, the level $N$ such that $p^\alpha N=\operatorname{lcm}(N_f,N_g)$ with $\alpha\geq 0$ and  $p\nmid N$. In all of these cases, we obtained $\L_p^g({\bf f},{\bf g},{\bf g})=0$ up to the working precision of $p^{10}$.   Due to computational restrictions, only in the ramified case we have been able to compute examples where $p$ divides the conductor of the elliptic curve.

  Note that all the elliptic curves arising in Table 1 below have rank $0$ over $K$, and thus the zeros obtained in this table are accounted for by Theorem~\ref{thm: CM case}.

\begin{table}[h!]
  \label{table:split}
  \begin{tabular}{lllllc}
  \toprule
$E_f$& $K$ & $N_g$ & $p$ & $N$ & $\L_p^g({\bf f},{\bf g},{\bf g})$\\% &  Atkin--Li\\
  \midrule
11a & $\Q(\sqrt{-5})$&$20$ & $7$ & $220$ & 0\\% & ?\\
11a  & $\Q(\sqrt{-11})$&$11$ & $5$ & $11$& 0\\% & Yes\\
19a & $\Q(\sqrt{-19})$&$19$ & $5$ & $19$& 0\\% & Yes\\
19a  & $\Q(\sqrt{-19})$&$19$ & $7$ & $19$& 0\\%  & Yes\\
39a  & $\Q(\sqrt{-39})$&$39$ & $5$ & $39$& 0\\% & Yes\\
51a  & $\Q(\sqrt{-51})$&$51$ & $5$ & $51$& 0\\% & Yes\\
55a & $\Q(\sqrt{-55})$&$55$ & $7$ & $55$& 0\\% & Yes\\
187a  & $\Q(\sqrt{-187})$&$187$ & $7$ & $187$& 0\\% & Yes\\
\bottomrule\\
  \end{tabular}
    \caption{Cases with $p$ split in $K$.}
  \end{table}

In the next two tables we see instances of zeros which we expect are explained by the sign of the action of the level $N$ Atkin-Lehner operator although we have not verified this in detail.

\begin{table}[h!]
\begin{tabular}{lllllc}
\toprule
$E_f$ & $K$ & $N_g$ & $p$ & $N$ & $\L_p^g({\bf f},{\bf g},{\bf g})$\\% &  Atkin--Li\\
  \midrule
11a  & $\Q(\sqrt{-3})$&$3$ & $5$ & $33$& 0\\% & ?\\
11a & $\Q(\sqrt{-11})$&$11$ & $7$ & $11$& 0\\%  & Yes\\
15a  & $\Q(\sqrt{-15})$&$15$ & $7$ & $15$& 0\\%  & Yes\\
39a  & $\Q(\sqrt{-39})$&$39$ & $7$ & $39$& 0\\%  & Yes\\
51a  & $\Q(\sqrt{-51})$&$51$ & $7$ & $51$& 0\\%  & Yes\\
67a  & $\Q(\sqrt{-67})$&$67$ & $5$ & $67$& 0\\%  & Yes\\
67a  & $\Q(\sqrt{-67})$&$67$ & $7$ & $67$& 0\\%  & Yes\\
187a  & $\Q(\sqrt{-187})$&$187$ & $5$ & $187$& 0\\%  & Yes\\

\bottomrule\\
\end{tabular}
\caption{Cases with $p$ inert in $K$.}
\end{table}

\begin{table}[h!]
  \begin{tabular}{lllllc}
    \toprule
$E_f$ & $K$ & $N_g$ & $p$ & $N$ & $\L_p^g({\bf f},{\bf g},{\bf g})$\\% & Atkin--Li\\
  \midrule
15a  & $\Q(\sqrt{-15})$&$15$ & $5$ & $3$& 0\\%  & ?\\
35a  & $\Q(\sqrt{-35})$&$35$ & $5$ & $7$& 0\\%  & ?\\
35a  & $\Q(\sqrt{-35})$&$35$ & $7$ & $5$& 0\\%  & Yes\\
55a  & $\Q(\sqrt{-55})$&$55$ & $5$ & $11$& 0\\%  & Yes\\

\bottomrule\\
  \end{tabular}

  \caption{Cases with $p$ ramified in $K$.}
\end{table}

In what follows we illustrate with examples the fact that the quantity $\L_p^g({\bf f},{\bf g},{\bf g})$ is not always zero.

\item In this example we fix $f$ to be attached to the elliptic curve $E_f\colon y^2=x^3+x^2-15x+18$, of conductor $N_f=120$. The weight-one form $g$ we consider has level $N_g=120$ also, and has $q$-expansion
  \begin{align*}
    g(q)&=    q + iq^{2} + iq^{3} - q^{4} -iq^{5} - q^{6} -iq^{8} - q^{9} + q^{10} -iq^{12} + q^{15} + q^{16} -iq^{18}\\
    &+ iq^{20} + q^{24} - q^{25} -iq^{27} + iq^{30} - 2q^{31} + iq^{32} + O(q^{34}),
  \end{align*}
  where $i^2=-1$. It is the theta series attached to the Dirichlet character $\epsilon$ modulo $120$ defined by
  \[
    \epsilon(97)=-1, \quad \epsilon(31)=1,\quad \epsilon(41)=-1,\quad\epsilon(61)=-1.
  \]
  The field cut out by $\epsilon$ is $K=\Q(\sqrt{-6})$, and we take $p=5$ which is split in both $L=\Q(\sqrt{-1})$ and $K$. Note that $p$ divides $N_f$ and $N_g$. We compute to precision $10$ the quantity
  \[
\L_5^g({\bf f},{\bf g},{\bf g})(2,1,1)=   4 \cdot 5 + 3 \cdot 5^{2} + 4 \cdot 5^{3} + 3 \cdot 5^{5} + 4 \cdot 5^{6} + 3 \cdot 5^{7} + 5^{8} + 2 \cdot 5^{9} + O(5^{10}) .
    \]

    With the same setting, we take $p=13$ (now $p$ is split in $L$ but inert in $K$). We obtain
    \[
\L_{13}^g({\bf f},{\bf g},{\bf g})(2,1,1) = 7 + 3 \cdot 13 + 10 \cdot 13^{2} + 13^{4} + 11 \cdot 13^{5} + 13^{6} + 6 \cdot 13^{7} + 4 \cdot 13^{8} + 5 \cdot 13^{9} + O(13^{10})
      \]

    \item Let $E_f$ be the elliptic curve $y^2+y = x^3+x^2+42x -131$ with label \texttt{175c1}. It has conductor $N_f=175$ and rank $0$. Let $g=h$ be the theta series of the character $\epsilon_1$ of $K=\Q(\alpha)$ with $\alpha$ satisfying $\alpha^2-\alpha+2=0$, of discriminant $D_K=-7$ and conductor $5{\mathcal O}_K$ (which is inert, of norm $25$), satisfying
      \[
        \epsilon_1(127) = -1,\quad \epsilon_1(101)=-1.
      \]
The modular form $g$ has $q$-expansion
      \[
g(q) = q + iq^{2} - iq^{7} + iq^{8} - q^{9} - q^{11} + q^{14} - q^{16} - iq^{18} - iq^{22} - iq^{23} + q^{29} + O(q^{30}),
\]
where again $i^2=1$. For $p=13$ (which is inert in $K$ and split in $L$), we obtain
\[
\L_{13}^g({\bf f},{\bf g},{\bf g})(2,1,1)= 1 + 3 \cdot 13 + 2 \cdot 13^{2} + 13^{3} + 12 \cdot 13^{4} + 9 \cdot 13^{5} + 3 \cdot 13^{8} + 5 \cdot 13^{9} + O(13^{10}).
\]

	\item Finally, consider the elliptic curve $E_f$ of conductor $175$ from the previous example, and for $g=h$ consider the theta series of another character $\epsilon_2$ of $K=\Q(\alpha)$, $\alpha^2-\alpha+2=0$,  of discriminant $D_K=-7$ and conductor $5{\mathcal O}_K$ (inert, of norm $25$), now taking the values
          \[
            \epsilon_2(127) = 1,\quad \epsilon_2(101) = -1.
      \]
This yields a modular form $g$ with $q$-expansion
      \[
        g(q) = q + q^{2} - q^{7} - q^{8} + q^{9} - q^{11} - q^{14} - q^{16} + q^{18} - q^{22} + q^{23} - q^{29} + O(q^{30}).
      \]
     We numerically obtain for $p=13$ that
      \[
\L_{13}^g({\bf f},{\bf g},{\bf g})(2,1,1) = 0.
\]
Again, we do not have a way to prove that $\L_{13}^g({\bf f},{\bf g},{\bf g})(2,1,1)$ is actually zero.

\end{enumerate}

\subsection{Exotic image case}
In the non-CM setting, we have been able to compute the following example. Consider $E_f \colon y^2=x^3-17x-27$, which has conductor $N_f=124$. Let $g$ be the modular form of level $N_g=124$ and projective image $A_4$, defined as the theta series of the character $\epsilon$ of conductor $124$ having values
\[
  \epsilon(65) = \alpha^2-1,\quad \epsilon(63)=-1,
\]
where $\alpha$ satisfies $\alpha^4-\alpha^2+1=0$. The modular form $g$ has $q$-expansion
\begin{align*}
  g(q)&= q - \alpha^{3}q^{2} + \left(-\alpha^{3} + \alpha\right)q^{3} - q^{4} + \left(\alpha^{2} - 1\right)q^{5} - \alpha^{2}q^{6} + \left(\alpha^{3} - \alpha\right)q^{7} + \alpha^{3}q^{8} + \alpha q^{10} - \alpha q^{11}\\
      & + \left(\alpha^{3} - \alpha\right)q^{12}+ \left(-\alpha^{2} + 1\right)q^{13} + \alpha^{2}q^{14} + \alpha^{3}q^{15} + q^{16} - \alpha^{2}q^{17} + \left(-\alpha^{3} + \alpha\right)q^{19} + \left(-\alpha^{2} + 1\right)q^{20}\\
  &+ \left(\alpha^{2} - 1\right)q^{21} + \left(\alpha^{2} - 1\right)q^{22} + \alpha^{2}q^{24} - \alpha q^{26} + \alpha^{3}q^{27} + O(q^{28}).
\end{align*}
We let $h=g^*$ its complex conjugate, and compute with $p=13$, obtaining
\[
  \L_{13}^g({\bf f},{\bf g},{\bf h})(2,1,1)= 1 + 5 \cdot 13 + 5 \cdot 13^{2} + 4 \cdot 13^{3} + 6 \cdot 13^{4} + 6 \cdot 13^{5} + 6 \cdot 13^{6} + 13^{7} + 3 \cdot 13^{8} + 9 \cdot 13^{9} + 9 \cdot 13^{10} + O(13^{11}).
\]

\bibliographystyle{alpha}

redpajama_set_name: RedPajamaArXiv


Sauter, M., Liesefeld, H. R., & Müller, H. J. (2019). Learning to suppress salient distractors in the target dimension: Region-based inhibition is persistent and transfers to distractors in a nontarget dimension. Journal of Experimental Psychology: Learning, Memory, and Cognition. Advance online publication.


text:
Kentucky Wildcats Morning Quickies: Mizzou in the Rear-View Edition
The Kentucky loss to the Missouri Tigers on Saturday was a big disappointment, especially after a huge rivalry win at Tennessee. What went wrong?
By Glenn Logan Oct 27, 2020, 9:49am EDT
Share All sharing options for: Kentucky Wildcats Morning Quickies: Mizzou in the Rear-View Edition
Jay Biggerstaff-USA TODAY Sports
Good morning, ladies and gentlemen of the Big Blue Nation, and welcome to the Tuesday Morning Quickies.
As per usual on Tuesdays after a game, I'll offer a few brief observations on Kentucky's loss to the Missouri Tigers on Saturday:
This was one of those "so Kentucky" losses — every year we have a couple of head-scratching losses, usually attributable to a poor mental attitude going into the game. Things like overconfidence, distractions, and, as in the instant case, a lack of energy are the usual contributors. The sideline reporter for the game at Mizzou noted that Kentucky's sideline lacked energy. I think that was true for the whole team, given what I saw.
This team doesn't have an offensive identity yet. Eddie Gran is trying desperately to get this offense to balance the run and pass, but unfortunately, I don't think his personnel will support it. The primary culprit seems to be the receiving corps — they are no better, on balance, than the Lynn Bowden-less corps from most of last season, and Terry Wilson cannot run the routes or hold on to well-thrown passes for them.
Continuing in that vein, Josh Ali is a good receiver, but not good enough to be the featured receiver in an SEC offense. He should be either first among equals or in a supporting role next to a guy like Bowden. Unfortunately, our WR development appears to have fallen behind, possibly as a symptom of the run-heavy last couple of years.
The constant threat of Joey Gatewood taking his spot is having a negative effect on Wilson. He is trying too hard, and it showed last Saturday when a bunch of his passes went high. As a former high school QB, it says to me that Wilson is pressing. In Wilson's defense, he is refusing to throw into tight windows and is doing a good job being patient. Unfortunately, his receivers are letting him down and making him look worse than he has been.
Wilson's decision-making and ballhandling on the read option is not as good as last season or two seasons ago, and too many times, we see confusion in the backfield. That shouldn't be happening with a veteran QB. He also seems to be notably avoiding contact more than he should be considering he has a more-than-credible backup.
The defense was hung out to dry by the offense, particularly in the second half. Having said that, tackling was a too-frequent problem for a defense that usually does very well fundamentally. Also, the front seven struggled to get off blocks on Saturday, and in their defense, Mizzou could've been flagged for holds on half their plays. But you must overcome that kind of officiating on the road, and Kentucky didn't.
When Matt Duffy has a below-average punting day to go along with everything else, you know you didn't come to play.
Gotta give props to Mizzou — they executed very sharply on offense, stuck with their game plan, and demoralized Kentucky by being tougher than expected on defense. I thought their coaching staff took a bunch of dumb risks that gave Kentucky opportunities to win the game, but Kentucky was unable to take advantage.
I have read a lot of angry tweets and opinions, particularly about Eddie Gran, and I agree that some of the criticism is well-deserved. Gran is trying to force things that just aren't there. This team is much better being a running, ball-control offense, and trying to force balance with an underachieving receiving corps is not working. This offensive line blocks the run way better, and both the line and receivers need the threat of a running game for play-action to work its magic. Also, what happened to the slants we ran so effectively against the Tennessee Volunteers? I can't believe Missouri was able to take that away consistently.
I won't wade too much into the running back debate — I think we need to do runs by committee this year and don't agree that one of the three needs to be featured. I may be wrong about that, though.
On to the Georgia Bulldogs.
Tweet of the Morning
4-star @BryceHopkins has committed to Kentucky! The Chicago-area power forward (formerly committed to Louisville) becomes the second pledge for Coach Calipari in 2021. #BBN pic.twitter.com/ZbjVrkrvkp
— Ballislife.com (@Ballislife) October 27, 2020
This is a wonderful get, the kind of tough, likely multi-year guy UK always needs to have on the roster.
In case you missed it: Mark Stoops Monday show podcast.
Georgia vs. Kentucky series history.
Georgia must "out-fundamental" Kentucky, according to Kirby Smart.
Pair of Mizzou players win SEC honors with their performance against Kentucky. UK fans aren't giving Missouri enough credit — they earned this win with sharp execution on both sides of the ball.
4-star improving power forward Bryce Hopkins commits to Kentucky, and Ben Roberts explains what makes this recruiting victory so good.
Heh. Hopkins de-committed from Louisville.
Calipari has to take extra glee in stealing Bryce Hopkins after the Chris Mack controversy of a few weeks ago
— Matt Jones (@KySportsRadio) October 26, 2020
The old Chinese curse, "May you live in interesting times," continues to haunt us all: Calipari would shut down UK basketball for two weeks on a positive COVID-19 test.
Looking forward to buying UK basketball tickets? Don't be.
John Calipari pre-season press conference transcript.
Keion Brooks Jr. is nursing a leg injury, likely out a couple more weeks.
Alex Goff goes low for men's golf with a 64 at the Legends Collegiate in Franklin, Tennessee to bull his way into contention.
UK hoops star Rhyne Howard disappointed with being left off the WBCA All-America team.
SEC fines Lane Kiffin for griping about the officiating. The SEC admits he was right, but you know the drill.
Analyzing each CFB Playoff contender after Week 5.
Auburn picked 7th in the SEC by CBS sports. "Rebuilding," you know.
Disney early-season bubble events canceled.
LeBron James might sit out the first part of next season reportedly due to advancing age and a desire to pursue Kareem Abdul-Jabbar's points record. Sounds a bit selfish to me.
NASA telescope discovers evidence of water on the moon. Can the discovery of Swiss cheese just below the surface be far behind?
Question asked: What comes after the International Space Station? Babylon Five, maybe? Or Deep Space Nine?
But I was assured there would be no math: A new formula to conquer mathemaphobia. No, it's not the Babylon Bee or the Onion, I promise...

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Deskop entry which starts .sh script, closing terminal I created the following Desktop Entry:
[Desktop Entry]
Type=Application
Terminal=true
Name=<name>
Exec=<pathTOscript/myScript.sh>

myscript.sh is just one command to start a service.
The service starts the programm itself (CNTLM) and another script (own state).
sudo /etc/init.d/cntlm <ownstate>

Now my problem:
When I type the above command into a terminal: CNTLM stops, the script is executed, CNTLM starts, everything works fine.
However, when I click the shortcut, the terminal closes before CNTLM starts again.
Changing my .desktop to this solved the problem:
Found it in this post.
[Desktop Entry]
Type=Application
Terminal=true
Name=<name>
Exec=gnome-terminal -e "bash -c 'sudo /etc/init.d/cntlm   <ownstate>;$SHELL'"

The important here was "$SHELL".

redpajama_set_name: RedPajamaStackExchange


Found on a boot sale in Weaverham last weekend. unusual to have any matches..
Lovely striking pad on the base.
A wonderful present from our lovely friend Angelica in Texas.
Some information about these banks on the website above ..between 1946 up until 1968 these were used to save your six pence pieces to pay for your radio licence.
Found in the antique fair in Llandudno at the weekend.


text:
package q

import (
	"fmt"
	"sync"
)

// A Tag is an arbitrary key:value pair associate with a resource.
type Tag struct {
	Key   string
	Value string
}

// String represents a tag as a string.
func (t Tag) String() string {
	return fmt.Sprintf("%s:%s", t.Key, t.Value)
}

// Tags are a threadsafe set of tags. A single key may have multiple values.
type Tags struct {
	t map[Tag]bool
	m *sync.RWMutex
}

func (t *Tags) init() {
	t.t = make(map[Tag]bool)
	t.m = &sync.RWMutex{}
}

// Get all tags.
func (t *Tags) Get() []Tag {
	if t.t == nil {
		return nil
	}
	t.m.RLock()
	defer t.m.RUnlock()
	l := make([]Tag, 0, len(t.t))
	for tag := range t.t {
		l = append(l, tag)
	}
	return l
}

// Contains indicates whether the given key value pair exists in a set of tags.
func (t *Tags) Contains(k, v string) bool {
	return t.ContainsTag(Tag{k, v})
}

// ContainsTag indicates whether the given tag exists in a set of tags.
func (t *Tags) ContainsTag(tag Tag) bool {
	if t.t == nil {
		return false
	}
	t.m.RLock()
	defer t.m.RUnlock()
	return t.t[tag]
}

// Add the supplied key value pair to a set of Tags.
func (t *Tags) Add(k, v string) {
	t.AddTag(Tag{k, v})
}

// AddTag adds the supplied Tag to a set of Tags.
func (t *Tags) AddTag(tag Tag) {
	if t.t == nil {
		t.init()
	}
	t.m.Lock()
	defer t.m.Unlock()
	t.t[tag] = true
}

// AddMap adds each key value pair in a map to a set of Tags.
func (t *Tags) AddMap(m map[string]string) {
	if t.t == nil {
		t.init()
	}
	t.m.Lock()
	defer t.m.Unlock()
	for k, v := range m {
		t.t[Tag{Key: k, Value: v}] = true
	}
}

// Remove the supplied key value pair from a set of Tags.
func (t *Tags) Remove(k, v string) {
	t.RemoveTag(Tag{k, v})
}

// RemoveTag removes the supplied tag from a set of Tags.
func (t *Tags) RemoveTag(tag Tag) {
	if t.t == nil {
		return
	}
	t.m.Lock()
	defer t.m.Unlock()
	delete(t.t, tag)
}

redpajama_set_name: RedPajamaGithub


The bike was bought in August 2018 and has been well cared for so it still looks pretty new. It had a rack over the back wheel. Both wheels had anti-theft locks installed as well as the seat. It has a u-lock hanger on the main frame as well as a front-light on the handle and a back-light on the rear side.
I am a student at IIT. I locked my bike on the bike rack in between the life sciences and engineering building at 1:50 pm, the location is a very well transitted spot by members of the IIT community, it is about a 100 feet away from State Street. At around 6-6:30 pm as I was about to go home my bike was not on the rack, I checked for any evidence but there was none, both the lock and the bike were missing. A report was filled with IIT's public safety and with CPD.


So it was somewhat understandable when Senator Suzanne Chun Oakland expressed some reservations about my livestreaming Tuesday nightʻs Homeless Working Group meeting. But a big mahalo to her for allowing me to continue. I sat down and was as unobtrusive as possible.
I was impressed with the community involvement and creativity. I was also shocked that the count of homeless does not take into account 100,000 "hidden" homeless who find ad hoc accommodation with friends or family but have no home or real place to stay. I also found that Hawaii is officially 50,000 housing units short at moment.


Rusape is located in the northeast Zimbabwe, about 120 miles (200 kilometers) from Harare. The community claims both an ancient and modern Jewish heritage. The Jews of Rusape believe to be the descendants of one of the lost tribes of Jacob. These people are believed to be offspring from the Bantu people who came from Northern Africa.
Almost 2,500 years ago, after the destruction of the Temple, a group of Jews left Judea and settled in Yemen. When the economic situation in Yemen began to fail, the Jews left and moved to Africa; with one group settling in Ethiopia and the other in Tanzania. After several years, many Jews left Ethiopia and moved further south into what today is Zimbabwe. They became known as the Ba-Lemba. Today, there exists a Lemba Cultural Association attempting to bring all the various Lemba communities in South Africa, Mozambique, Ethiopia, and Zimbabwe together.
Within the past few years, extensive research has been done on the Lemba communities. These tests verify that many male "Black Jews" have the same genetic structure as the Cohen priests, providing evidence of the relation to Jews of this ancient heritage story.
The Jewish community of Rusape can date itself back to 1903, when an African American Baptist deacon named William Saunders Crowdy met Albert Christian. Although, Crowdy was deemed a deacon of the Church, in the 1880s he had a dream where God instructed him to lead the blacks to Judaism. Crowdy passed his Judaic teachings and traditions onto Christian; afterwards, Christian settled in Southern Africa where he preached about Judaism.
Today, Congregation Betel in Rusape consists of thousands of members, approximately 4,000 people. Rabbi Ambrose "Cohen" Mukawaza leads the community in services and studies. In 1938, the Congregation Betel synagogue was built about seven kilometers (4.34 miles) outside the Rusape city limits. Every Saturday, the building is packed with more than a hundred Jews practicing "prophetic" Judaism. Originating in Virginia a little over a century ago, the practice is associated with African American Beth El Congregations. The main belief is that while Jesus was not the messiah, he was a prophet, as was William Crowdy who delivered the word of God. This congregation believes that while the teachings of Jesus should be respected, Jesus was an active member of the Jewish community of Israel. Besides this difference in Western Judaism, the Rusape community observes all Jewish holidays and prayer services are conducted in Hebrew.
One of the Rusape congregation's high holidays is the Convocation of the Feast of Tevet. This holiday remembers the destruction of the Temple and the migration of the Jews from Yemen into Africa. For eight days the community remains together, with prayer services beginning at 4 a.m. Each day the congregants pray to God in a different position: from standing on the first day to lying on the floor on the eighth day.
1. Belief in only one God, and that he created all things.
2. One day a week is considered holy: On this day the Rusape Jews give thanks and praises to God.
3. Strict observance of Shabbat.
4. Adherence to the Ten Commandments.
5. A bris is performed according to local African tradition, at the age of ten.
6. When praying, a choir melds Shona (the local Zimbabwe dialect), Hebrew, and English words into African melodies to create a distinctive service. Nevertheless, the services contain almost all the traditional Jewish prayers found in Western siddurim.
7. They do not eat pork or any other animal prohibited by the Old Testament; nor do they mix milk and meat. All animals are slaughtered by a professional "kosher" butcher and bled thoroughly.
8. The calendar follows the path of the moon.
9. Stars of David are engraved on tombstones.
The Zimbabwe Orthodox Jewish community does not recognize the Rusape Jews as Jews. Although, the Rusape Jews may have good intentions, according to the Orthodox community they cannot claim to be Jewish until internationally established as Jews.


You need look no further than to the traditions of shamans and witch doctors of native tribes to see that herbs and plants are used to invoke the trance state where they commune with the (demonic) spirits. It's been going on for ages, and still goes on today. That's how the 'white man' learned to smoke tobacco–from Native American rituals. Native peoples also brew or ferment intoxicating beverages. Just because 'civilized people' are the ones now doing it, doesn't mean it is something different.
Smoking weed, using mushrooms (or whatever), smoking tobacco . . . these practices are rooted in witchcraft.


Colorado Secretary of State Wayne Williams recently attended a meeting arranged by the Small Business Administration. He is with Dan Nordberg, the Regional VIII director of the Small Business Administration, and Frances Padilla, director of the Colorado U.S. SBA district office, Becky Fuller, and Raul Acosta.
Secretary of State Wayne Williams and Mike Hardin, the director of business and licensing for the SOS, last week attended Colorado Regional Regulatory Reform Roundtables organized by the U.S. Small Business Administration in Colorado Springs and Fort Collins, respectively.
The SBA's office of advocacy organized round tables across the country to hear from local small businesses from various industries, including transportation, real estate, manufacturing and more. The round tables aim to "discover burdensome regulations that might be hindering the business environment," SBA Regional Administrator Dan Nordberg said.
The SBA advocacy office will take the information to Congress to try to amend statutory regulations or try to address the regulations within the preview of agencies.
In early June, MyBizColorado was launched by the Secretary of State's office in partnership with the Governor. Since then, the Denver Business Journal reported Denver leading the nation in small-business job growth and ranked 3rd in wage growth. Colorado was also the first state in the nation to offer all business certification services online.
"I was particularly happy with the positive comments about our new innovations like free online certificates of good standing and our new MyBiz.Colorado.Gov portal," Williams said.
Mark Hardin, left, the director of Business and Licensing for the Colorado Secretary of State's office, attended the SBA Office of Advocacy event in Fort Collins last week.
The SBA has applauded the vibrant and growing Colorado small business community, but concerns about tourism and trucking industry regulations remain for businesses, said Nordberg, who attended both Colorado round tables.
As a result, smaller trucking companies are struggling to make deadlines.
Beyond licensure concerns, small businesses also reported challenges in transportation law restrictions and the transition to an electronic reporting system, according to Hardin and Nordberg.
Other small business concerns included the "cost of compliance and the impact on subcontracting, rising commercial property expenses, and timing of IRS reporting," Hardin said.
Hardin enjoyed the opportunity to hear from businesses, inding it informative and helpful to gain insight into the challenges they face.


Q: What is an MWT?
A: An MWT is the Maintenance of Wakefulness Test. This test measures a person's ability to stay awake and alert during different times throughout the day and under various circumstances. Some places of employment may require employees to pass this test, such as airlines for pilots and trucking companies for drivers. Also, it is required that a person who has sleep apnea pass this test in order to obtain a CDL driver's license.


text:
Home / Director's Message
Dr. Alex Wilner
Director, IPIS
Associate Professor of International Affairs, NPSIA
www.alexwilner.ca
Dr. Wilner is a leading scholar of contemporary deterrence theory and practice. His research – which explores the nexus between deterrence theory and emerging security considerations, domains, and environments – has shaped the fourth, and now fifth, generation of deterrence scholarship. Among his over two dozen journal publications, his articles on the subject of deterring terrorism and cyber deterrence have been published in top-ranked IR journals, including International Security, Journal of Strategic Studies, Security Studies, and Studies in Conflict & Terrorism. His books include Deterrence by Denial: Theory and Practice (eds., Cambria Press, 2021), Deterring Rational Fanatics (University of Pennsylvania Press, 2015), and Deterring Terrorism: Theory and Practice (eds., Stanford University Press, 2012).
Since joining NPSIA in 2015, his broader scholarship has been awarded over $1M (CAD) in external research funding: he was awarded a Government of Canada SSHRC Insight Development Grant (2016-2017), a prestigious SSHRC Insight Grant (2020-2025), and a Government of Ontario Early Researcher Award (2021-2026) to study state and non-state cyber deterrence; a SSHRC Partnership Engage Grant (2022-2023) to explore foresight in public service; two major IDEaS grants (2018-2021) and several MINDS grant (2019, 2020) from the Department of National Defence to explore Artificial Intelligence (AI) and deterrence; several smaller research grants from the Canadian Network on Terrorism, Security, and Society (TSAS); and a major Mitacs grant (2020-2022) to explore emerging technology and Canadian defence policy and strategy.
Prof Wilner teaches classes on terrorism and violent radicalization, intelligence in international affairs, strategic foresight in international affairs, and a capstone course on Canadian security policy. Past capstone partners have included FINTRAC, Public Safety Canada, Global Affairs Canada, and Policy Horizons Canada. Besides his academic teaching, since 2017 he has trained over 1000 Canadian public servants in strategic foresight, having provided over 40 multi-day training workshops to several government departments, including (among others) the Department of National Defence/Canadian Armed Forces, Global Affairs Canada, Natural Resources Canada, Environment and Climate Change Canada, Agriculture and Agri-Food Canada, Department of Justice, and Standards Council of Canada.
Message from Past IPIS Directors
A. O. Abd El Halim and Director Dane Rowlands speak on behalf of the MIPIS program.


text:
About Luxury Collection
The Luxury Collection® is a selection of hotels and resorts offering unique, authentic experiences that evoke lasting, treasured memories.
For the global explorer, the Luxury Collection offers a gateway to the world's most exciting and desirable destinations. Each hotel and resort is a distinct and cherished expression of its location; a portal to the destination's indigenous charms and treasures. Magnificent décor, spectacular settings, impeccable service and the latest modern conveniences combine to provide a uniquely enriching experience.
75 of the World's finest hotels & resorts
Today, the Luxury Collection is a glittering ensemble of more than 75 of the world's finest hotels and resorts in more than 30 countries. All are noteworthy for their history, architecture, art, furnishings and amenities, highlighting the rich culture of each property's fabulous surroundings. The Luxury Collection seamlessly integrates local customs and location into the guest experience, while fulfilling every desire with extraordinary service. The Hotel Danieli welcomes guests to a masterfully restored 14th century palace bejeweled with Murano glass chandeliers, ornately carved marble columns, and soaring ceilings clad with golden leaves. The Hotel Imperial in Vienna, built in 1863 as a residence for the Prince of Württemberg, features pristine marble, hand-carved statues, and spectacular crystal chandeliers. Guests can retreat to the Convento do Espinheiro in the historic city of Evora, Portugal, a former convent built in the 15th century, now a lavish hotel known for its highly acclaimed butler service.
The Luxury Collection traces its heritage to Compagnia Italiana Grandi Alberghi (CIGA), founded in 1906, owner of some of Italy's most renowned luxury hotels, including the Hotel Danieli in Venice. In 1985, Prince Karim Aga Khan acquired a controlling interest in CIGA and began to expand the company's luxury hotel portfolio beyond Italy, most notably with purchases in Spain and Austria. CIGA later expanded in geographic focus to include legendary properties in Sardinia, including Hotel Cala di Volpe, Hotel Pitrizza and Hotel Romazzino. In 1994, Starwood Hotels & Resorts Worldwide, Inc. acquired CIGA and created the Luxury Collection designed to be the world's most renowned assemblage of hotels and resorts.
Today the collection continues to grow. The newest additions include The Astor Hotel in Tianjin, Tambo del Inka Resort & Spa in Valle Sagrado, Turnberry Resort in Scotland and The Liberty Hotel in Boston. Learn more about The Luxury Collection by browsing our hotels and resorts by destination or pursuit.
Luxury Collection presents The Perfect Day with One Kings Lane
Perfect Day Washington DC Video
We invite you to view our Luxury Collection video
2014 April: Introducing Luxury Collection Spas
2014 January: Featuring Unforgettable Journeys


The Ebb Tide Apartments are the perfect vantage point to watch the receding or outgoing tide. The apartments are situated in Victoria Road opposite the ocean with the most spectacular views of the sunsets that Camps Bay has to offer. The interior design will impress lovers of design and amazing spaces. Modern and minimalistic, featured in many lifestyle magazines and used for catalogue and fashion shoots.
When staying at Ebb Tide, you might decide not to explore the delights of Cape Town, a fantastic view, a fully equipped kitchen, wireless internet etc. The apartments are serviced Mondays to Saturdays and we can arrange a private chef, driver, personal trainer or tours for you.
Security is a key aspect at Ebb Tide and we have staff working 24 hours a day. Access to the building is strictly controlled and the security cameras are monitored by the staff. Our friendly staff can assist with carrying your luggage, cleaning your pool etc. Camps Bay Village offers a host of restaurants, coffee shops and bars and is only 750 meters from Ebb Tide.
It is a flat walk and the same distance to Camps Bay beach. The small hidden beach of Bali Bay is 100 meters away and the quant Bakoven beach is 250 meters away. At Ebb Tide we offer one free basement parking per apartment.
- Rim-flow plunge pool, +/- 4.5m x 3m´s and 1.5m deep.
- Lift access into the apartment.
- Housekeeping Monday – Saturday.
- Under-floor heating in the bathrooms.
- Front patio with a view of the ocean, +/- 2.5m x 8.5m´s.
- 2 Basement parking bays.
- Situated opposite the sea, 100 meters to Bali Beach, 300 meters to Bakoven beach and 780 meters to Camps Bay beach and the Village.
- Size: +/- 200 square meters.
- UPS solar power system.
- 2 Loungers and outdoor table and 4 chairs.
- Nuovo Multi Room Soundsystem.
- Unfortunately no BBQ´s allowed.
When staying at Ebb Tide, you might decide not to explore the delights of Cape Town as you have your own private pool, a fantastic view, a fully equipped kitchen, wireless internet etc. The apartments are serviced Mondays to Saturdays and we can arrange a private chef, driver, personal trainer or tours for you.
Security is a key aspect at Ebb Tide and we have staff working 24 hours a day. Access to the building is strictly controlled and the security cameras are monitored by the staff. Henry and Austin can assist with carrying your luggage, cleaning your pool etc. Camps Bay Village offers a host of restaurants, coffee shops and bars and is only 750 meters from Ebb Tide.


The candidate should have UAE experience at least minimum of 5 years on the same field.
Any nationality may apply and if you don t meet the qualification please don t apply.
Looking for a qualified Civil Project Engineer with 5-10 years of UAE experience in a civil general contracting company. Kindly send your CV at prabhakar.azad.turner@gmail.com and put the position title as your subject.
Our client, a reputable and established semi Government Company based in Abu Dhabi is currently looking to hire Maintenance Supervisor . Responsibilities include maintaining systems and equipment by completing preventive maintenance schedules; restoring, repairing, rebuilding, or replacing faulty or inoperative components and parts; supervising staff. Bachelor s degree / Diploma in Engineering with 5+ years of experience working in Facilities or Hospitality industry is must. CANDIDATES AVAILABLE IN UAE ONLY CAN APPLY.


Back to home page Return to top. For users that wish to play games or work with multimedia applications, it is especially important to have a video board with a high performance graphics processor and enough memory. Category The graphics card, also known as the video card, is the complete graphical subsystem of the PC. Learn more – opens in a new window or tab. The item you've selected was not added to your cart.
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text:
Albon emerges as favourite for Toro Rosso drive
17 October 2018 PlanetF1.com F1 0
Date published: October 17 2018
With Pascal Wehrlein signing up for Formula E, Alexander Albon has emerged as the new favourite to join Toro Rosso for next season.
Unlike in past seasons, Red Bull do not have an abundance of talent waiting in the wings for a Formula 1 drive.
This has resulted in Toro Rosso looking at other options.
Enter Albon.
Although he has signed to race for Nissan e.dams in the Formula E series, he is believed to be Toro Rosso's number one target to partner Daniil Kvyat next season.
Present on the first day of Formula E testing at Valencia on Tuesday, he did not climb into the car with reports claiming he was instead locked in negotiations with Red Bull advisor Helmut Marko.
According to E-racing365, he is thought to have an 'F1 exit clause' in his contract that permits him to break his Nissan contract if F1 comes calling.
Nissan would not be drawn on the rumours.
"Alex is a talented young driver with a lot of potential and is sought after," said a Nissan spokesperson.
"We have ongoing discussions and are not able to make further comments at this time. We're focused on testing the car and getting the data we need to prepare for the first race."
Albon signed a multi-year deal with Nissan earlier this year when he was initially overlooked by the Red Bull family for a F1 drive.
And so, with the Mahindra / Wehrlein news, STR's options shrink again. After years of shunning him are they really going to break Albon out of his FE contract? Fascinating.
— Will Buxton (@wbuxtonofficial) October 16, 2018
He was backed by Red Bull in karting and Formula Renault before being dropped at the end of the 2012 season.
The strange relationship of USA and F1 explained
Charles Leclerc Interview | Beyond The Grid | Official F1 Podcast


text:
New Ferrari Formula 1 Grand Prix Transporter Set From Brumm
A new release from Brumm is always eagerly awaited and it has just has announced a superb new Ferrari 312B transporter set from the 1970 Italian Grand Prix. ...
New Ferrari Formula 1 Grand Prix transporter set from Brumm Images
A new release from Brumm is always eagerly awaited and it has just has announced a superb new Ferrari 312B transporter set from the 1970 Italian Grand Prix.
The set comes complete with an OM 160 Rolfo transporter, the Ferrari 312Bs of Clay Regazzoni, Jacky Ickx and Ignazio Giunti, a Clay Regazzoni standing figure, two mechanics and three sets of wheels and tyres. This is the 4th in Brumm's Transporter Set series and previous issues have proved to be very popular with collectors, and we're sure that this is going to be no different. The set is limited to just 500 pieces worldwide, so don't miss out. Go to www.diecastlegends.com to order yours today.
Remember, to get the latest collecting news delivered direct to your inbox every two weeks, sign up for our e-newsletter.