I am studying the paper of R. Greenberg and G. Stevens, $p$-adic $L$-functions and $p$-adic periods of modular forms.
I am currently trying to understand the definition of $\mathcal{L}$-invariant for an abelian variety $A$ over $\mathbb{Q}_p$ with split multiplicative reduction. They define $X$ and $Y$ to be respectively the character groups of the reductions of $B$ and $A$ modulo $p$ and say that these are free abelian groups of rank dim($A$) with trivial Galois action. Then there is what I am not familiar with, they say that the theory of $p$-adic uniformization gives us a bi-multiplicative pairing \begin{equation} j\,\colon\,\, X\,\times\, Y \,\rightarrow\,\mathbb{Q}_p^\times \end{equation} and an exact sequence (plus the analogous one inverting $X$, $Y$ and $A$, $B$) of $G_{\mathbb{Q}_p}$-modules \begin{equation} 0\quad\longrightarrow\quad X \quad \longrightarrow\quad \text{Hom}(Y,\overline{\mathbb{Q}_p}^\times) \quad \longrightarrow\quad A(\overline{\mathbb{Q}_p}) \quad \longrightarrow\quad 0 \end{equation} where the first nonzero arrow is given by the previous pairing. They then proced to say that the pairing $\alpha = \text{ord}_p\circ j$ with values in $\mathbb{Z}$ is nondegenerate.
My first question is about the nature of this pairing, in which sense it comes from the theory of $p$-adic uniformization? What is $j$ in the case of elliptic curves?
Moreover I would also be interested in understanding how the exact sequence comes to existence, how does the second arrow arise?
Thank you in advance for any answer and reference.
