Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 21 / 3

An elliptic curve over $\mathbb{Q}$ has good reduction at $p$ if it admits an integral Weierstrass equation of an elliptic curve over ${\mathbb{Z}}_p$ whose reduced projective cubic is a smooth algebraic curve over ${\mathbb{F}}_p$. Equivalently, its Minimal Weierstrass equation has a unit elliptic-curve discriminant, or its smooth proper model is an elliptic curve over the valuation ring. The existence of an appropriate integral model matters: a nonminimal equation can have discriminant divisible by $p$ even when the curve has good reduction. Equivalently,

The supplied integral equation has elliptic-curve discriminant $\Delta =-11$. It therefore has good reduction at every $p\ne 11$. At eleven its discriminant has valuation one. Under a change of Weierstrass equation of an elliptic curve, discriminant valuations differ by a multiple of twelve. A good integral model would have valuation zero, which is impossible. This is the discriminant valuation obstruction to good reduction; it also shows the displayed model is minimal at eleven. Hence

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