= Solution
For a short <Weierstrass equation of an elliptic curve> with coefficients in the <p-adic integers>, a point with nonintegral $x$ has valuations $v_p(x)=-2m$ and $v_p(y)=-3m$ for some <integer> $m\ge1$: the term $x^3$ strictly dominates $Ax+B$ in the equation, so $2v_p(y)=3v_p(x)$. Its formal parameter $T=-x/y$ therefore has valuation $m\ge1$. These points, together with the identity, form the formal neighborhood $E_1(\mathbb Q_p)$ of the identity, with parameter in $p\mathbb Z_p$ and the elliptic <formal group> law over the <p-adic integers>. For a point with $x\in\mathbb Z_p$, the equation gives $y\in\mathbb Z_p$ too.
The formal neighborhood has no nontrivial torsion. Multiplication by an <integer> coprime to $p$ has <series> $[m](T)=mT+O(T^2)$ and preserves $|T|_p$, so any torsion there must have $p$-power order. If $p\ge3$, part (i) gives $|T|_p\ge p^{-1/(p-1)}>p^{-1}$, contradicting $T\in p\mathbb Z_p$.
The printed hypotheses also allow $p=2$. In that case, a nontrivial torsion point in the formal neighborhood would have order a power of two and hence yield a point of order two in that neighborhood. But every nonidentity point of order two on a short <Weierstrass equation of an elliptic curve> has $y=0$ and $x$ a root of the monic cubic over the <p-adic integers> $X^3+AX+B$. Such a root lies in the <p-adic integers>, so it cannot belong to the formal neighborhood, whose nonidentity points have nonintegral coordinates. This excludes the remaining case. This is the <torsion-free formal subgroup for a short Weierstrass equation>. Thus every affine torsion point satisfies
$$
\boxed{|x|_p\le1,\qquad|y|_p\le1.}
$$
With <good reduction>, the reduction <homomorphism> $E(\mathbb Q_p)\to\widetilde E(\mathbb F_p)$ has kernel exactly this torsion-free formal neighborhood. Its restriction to the <torsion subgroup> is therefore injective, proving the required subgroup isomorphism. The short model over the <p-adic integers> is essential to the prime-two argument; no blanket torsion-injectivity assertion for arbitrary models is needed.
Back to article page
