Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 32 / 3

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.

Near $O$, put $t=-x/y$ and $w=-1/y$, or projectively $t=-X/Y$, $w=-Z/Y$. The integral Weierstrass equation of an elliptic curve becomesThe linear coefficient of $w$ at the origin is a unit, so successive coefficient comparison gives a unique series $w(t)\in {\mathbb{Z}}_p[[t]]$ beginning $t^3$. Expressing the addition morphism in the local parameters of its two inputs givesIts identity, inverse, commutativity and associativity identities follow from the corresponding elliptic group identities. This one-dimensional commutative formal group law is the formal group of an elliptic curve $\hat{E}$ attached to the model.

A point reducing to $O$ lies in the chart with $Y$ a unit and has $t,w\in p{\mathbb{Z}}_p$. The implicit equation determines $w$ uniquely from $t$. Conversely every $t\in p{\mathbb{Z}}_p$ gives the convergent solution $w(t)$ and the point $[-t:1:-w(t)]$; $t=0$ gives $O$. Hence the parameter is a bijection from the reduction kernel to $p{\mathbb{Z}}_p$. The series $F$ converges there and describes its addition. ThereforeFor nonzero $t$ with $v_p(t)=r>0$, $w(t)=t^3$ times a unit, so $v_p(x)=-2r$ and $v_p(y)=-3r$. These are the formal kernel of a minimal Weierstrass equation valuations used earlier.

Write $E[q^{\infty }]$ for the points killed by some power of $q$, rather than any notion of all points which are $q$-divisible. On the reduction kernel, formal multiplication isSince $q\ne p$, its linear coefficient is a unit. Recursive inversion gives an integral inverse series, convergent on $p{\mathbb{Z}}_p$; equivalently the Hensel lemma gives a unique solution to $[q^m](t)=s$ for every $s\in p{\mathbb{Z}}_p$. Thus multiplication by $q^m$ is bijective on the kernel, proving that its $q$-primary torsion is zero. The reduction group homomorphism is therefore injective on $E({\mathbb{Q}}_p)[q^{\infty }]$.

For surjectivity on this torsion, take a reduced point $\tilde{P}$ killed by $q^m$. Lift it to $P\in E({\mathbb{Q}}_p)$ using smooth reduction. Then $q^{m}P$ belongs to the kernel, so choose its unique kernel division point $R$ with $q^{m}R=q^{m}P$. The point $P-R$ is killed by $q^m$ and reduces to $\tilde{P}$. This proves the prime-to-p torsion lifts uniquely at good reduction statement:The right side is a subgroup of a finite group. Thus the left side is finite and has exactly the requested order. No claim that the whole formal kernel is torsion-free at $p=2$ is needed.