= Solution
For an integral <Minimal Weierstrass equation>, let $E_0(\mathbb Q_p)$ be the subgroup of points whose reduction is a smooth point of the reduced cubic. Its reduction is a homomorphism to the <smooth-locus group of a singular Weierstrass cubic>, or to the whole reduced <elliptic curve> in the good-reduction case. Define the <formal kernel of a minimal Weierstrass equation> by
$$
\boxed{E_1(\mathbb Q_p)=\ker\bigl(E_0(\mathbb Q_p)\longrightarrow\widetilde E_{\mathrm{ns}}(\mathbb F_p)\bigr).}
$$
It consists of the identity and the points reducing to the identity at infinity. For a nonidentity affine point this is equivalent to $v_p(x)=-2r$ and $v_p(y)=-3r$ for some positive integer $r$.
The <local parameter> $t=-x/y$, with $t(O)=0$, gives the group isomorphism
$$
\boxed{E_1(\mathbb Q_p)\cong\widehat E(p\mathbb Z_p),}
$$
where $\widehat E$ is the <formal group of an elliptic curve>. To see the local coordinates, put $w=-1/y$ in the <Weierstrass equation of an elliptic curve>. The resulting relation is $w=t^3+a_1tw+a_2t^2w+a_3w^2+a_4tw^2+a_6w^3$, and its unit linear coefficient gives a unique integral series $w(t)=t^3+O(t^4)$. Thus $x=t/w$ and $y=-1/w$ have valuations $-2v_p(t)$ and $-3v_p(t)$. The <chord-and-tangent group law> expressed in this parameter is precisely the integral <formal group law> $\widehat E$. This description includes bad reduction; only the smooth identity neighborhood is involved.
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