In the paper $p$-adic L-functions and $p$-adic periods of modular forms of Greenberg and Stevens $\S3$ page $420$ they make the following statement:
Let $A$ over $\mathbf{Q}_p$ be an abelian variety with split multiplicative reduction and $B$ be its dual. Let $X,Y$ be the character groups of $B^0_{\mathbf{F}_p}$ and $A^0_{\mathbf{F}_p}$ with trivial $G_{\mathbf{Q}_p}$ action. Then from the theory of $p$-adic uniformization we get a multiplicative pairing
$$ j : X \times Y \to \mathbf{Q}_p^* $$ and exact sequences of $G_{\mathbf{Q}_p}$ modules $$ 0 \to X \xrightarrow{\;j\;} \text{Hom}(Y,\overline{\mathbf{Q}}_p^*) \longrightarrow A(\overline{\mathbf{Q}}_p) \longrightarrow 0 $$ $$ 0 \to Y \xrightarrow{\;j\;} \text{Hom}(X,\overline{\mathbf{Q}}_p^*) \longrightarrow B(\overline{\mathbf{Q}}_p) \longrightarrow 0 $$ Furthermore the pairing $\alpha : ord_p \circ j : X \times Y \to \mathbf{Z}$ is non degenerate.
They give two references for this : McCabe's thesis at harvard "$p$-adic theta functions" which I haven't found on line and Morikawa : "On theta functions and abelian varieties over valuation fields of rank one, part I and part II" from where I haven't been able to extract the mentioned result.
My question is : does anyone know of a good reference for this or can explain to me how to obtain the result from Morikawa's paper ?
