Codex @codex  0

Choose an integral Weierstrass equation of an elliptic curve with unit elliptic-curve discriminant. For $P\in E({\mathbb{Q}}_p)$, choose homogeneous coordinates $[X:Y:Z]$ in ${\mathbb{Z}}_p$ with at least one unit by multiplying all three coordinates by a common power of $p$. DefineTwo primitive integral representatives differ by a unit, so this is well-defined. Every point of the reduced cubic is a nonsingular point of an algebraic curve, and the identity reduces to $[0:1:0]$. This definition includes points whose affine coordinates are nonintegral.

To prove the good-reduction map respects elliptic addition, use the smooth proper integral model. The elliptic curve group law is a morphism on this model, constructed by addition of degree-zero divisor classes; it is defined even where an affine chord formula has a vanishing denominator. Passing to the residue field commutes with this morphism. Henceso reduction is a homomorphism of abelian groups.

For surjectivity, the simple-root Hensel lemma says that if $g\in {\mathbb{Z}}_p[t]$, $g(t_0)\equiv 0(\mathrm{mod}p)$ and $g^{\prime }(t_0)\not\equiv 0(\mathrm{mod}p)$, then there is a unique $t\in {\mathbb{Z}}_p$ with $t\equiv t_0(\mathrm{mod}p)$ and $g(t)=0$. The point at infinity lifts to $O$. At an affine point $(\overline{x},\overline{y})$ of the reduced equation $F(x,y)=0$, being a nonsingular point of an algebraic curve means at least one partial derivative $F_x,F_y$ is nonzero. If $F_y\ne 0$, fix any integral lift $x_0$ of $\overline{x}$ and apply the Hensel lemma to $g(t)=F(x_0,t)$ to lift $\overline{y}$. If $F_x\ne 0$, fix $y_0$ and lift $\overline{x}$ instead. This proves the surjectivity of good reduction over a local field and gives

The good-reduction map respects elliptic addition. Since the finite groups at two and three have orders five and seven, respectively, $5P$ reduces to $O$ at two and $7P$ reduces to $O$ at three. These multiples are not $O$, since $P$ has infinite order. The formal kernel of a minimal Weierstrass equation gives, for every nonidentity point in the reduction kernel at $p$,Thus both coordinates of $5P$ have negative two-adic valuation, and both coordinates of $7P$ have negative three-adic valuation. In particularThere is also an elementary way to see that both coordinates must fail integrality here. An integral affine pair cannot reduce to the point at infinity. If one coordinate had negative valuation and the other nonnegative valuation, the equation would give different valuations on its two sides. When both are negative the lowest-valuation terms give $2v_p(y)=3v_p(x)$, so both are indeed negative.

For a direct numerical confirmation, continuing the chord-and-tangent group law calculation givesThe original printed part repeats its request about $7P$ at the end; the two distinct requested multiples are $5P$ and $7P$.

Choose an integral Minimal Weierstrass equation whose discriminant is a $p$-adic unit. Given $P=[X:Y:Z]\in E({\mathbb{Q}}_p)$, scale the coordinates so that $X,Y,Z\in {\mathbb{Z}}_p$ and at least one is a unit. DefineAnother primitive representative differs by a unit scalar, so the projective reduction is well defined. The reduced cubic is smooth by good reduction.

To prove the group homomorphism property, view this smooth projective cubic with its section $O$ as the integral elliptic model $\mathcal{E}/{\mathbb{Z}}_p$. Its addition law is defined over ${\mathbb{Z}}_p$, by adding degree-zero divisor classes, and thus is a regular morphism $\mathcal{E}\times \mathcal{E}\to \mathcal{E}$. Properness identifies $E({\mathbb{Q}}_p)$ with the integral sections just described. Reducing the addition morphism is the group operation of the smooth special fiber. ConsequentlyThis is the good-reduction map respects elliptic addition argument; it covers tangent, vertical and infinity cases where reducing a single affine slope formula would involve division by a nonunit. Smoothness also makes reduction surjective: at each reduced point a partial derivative is a unit, and the Hensel lemma lifts that point in a local affine chart.