Assume Elliptic curves over local field, E/k, has good ordinary reduction and $$E:y^2+axy+by=x^3+cx^2+dx+e$$. In this case, we know that with the coordinate system $z$ attached to the formal Lie group $\hat{E}$, $x,y$ can be written like, $$x(z)=z^{-2}+\ldots, y(z)=z^{-3}+\ldots$$. I wanna extend this to abelian variety.
Here, my conjecture is as follows; $X/k$ be an good ordinary abelian variety over local field k with mixed characteristics. (Residue characteristic $p$) Let $H\subset X$ be an ample divisor of X, and in this case, $3H$ is vary ample divisor on $X$. (As mentioned in Milne’s Abelian variety) We take the coordinate of $X$ , $x_1,\ldots,x_n$, by take a basis of $$(H^0(X,\mathcal{O}_X(3H))-\{0\})/k^\times$$. I think , by Bertini's theorem, we can assume that $H$ is irreducible closed subscheme of X passing through the origin of X. (We are considering especially the case, $dim X\ge 2$) There also exists an exact sequence for ordinary abelian variety (proved in Tate's "p-divisible group"), $$0 \rightarrow X(p)\rightarrow X(k)\rightarrow \tilde{X}(\kappa(k)) \rightarrow 0$$, $X(p)$ be a formal Lie group attached to the power of $p$ torsion of X, and $\kappa(k)$ be the residue field of $k$. In this sense, we can expand the $x_1,\ldots,x_n$ by a coordinate system of $A(p)$, $t_1,\ldots,t_g$ where $g=dim X$.
I feel, the generalization of the Elliptic case happens. Suppose, $x_1, \ldots, x_s$ be the basis of $$(H^0(X,\mathcal{O}_X(2H))-\{0\})/k^\times$$ , and assume the origin of $X$ is in some affine neighborhood of codimension $1$ point. Then, $$x_i(t_1,\ldots,t_g)=t_1^{a^i_1}\ldots t_g^{a^i_g}+\ldots$$ , $\sum_k a^i_k=-2$ (if $i\leq s)$ and $\sum_k a^i_k=-3$ (if $i >s)$.
Is this true? Or any other generalization exists?