Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 24 / 4

Work near the identity $O$ of a generalized Weierstrass equation of an elliptic curve. On the projective chart where $Y\ne 0$, use the local parameter $t=-x/y$ and the auxiliary coordinate $w=-1/y$. Substitution givesThe coefficient of $w$ in the linearized equation at $(0,0)$ is one. Consequently, successive coefficient comparison determines a unique formal power series $w(t)$, without dividing by any coefficient of the equation. Its initial terms areRecover the affine coordinates as Laurent series $x=t/w(t)$ and $y=-1/w(t)$; in particular $x=t^{-2}+O(t^{-1})$ and $y=-t^{-3}+O(t^{-2})$.

The elliptic curve group law is a regular morphism near $(O,O)$. Applying the local parameter to the sum and passing to completed local rings producesOne can construct the series without any nonunit division. PutThe divided difference is an integral formal power series, including its tangent line specialization at $T_1=T_2$. The line $w=\lambda t+\nu$ meets the cubic in the two chosen points and a third point $R$. Substitution in the equation shows that its cubic and quadratic coefficients in $t$ are $-A,-B$, whereSince $\lambda$ has total degree at least two and $\nu$ at least three, $A$ is a unit. The sum of the roots therefore givesThe last expression reflects the third intersection according to the chord-and-tangent group law. Both denominators have constant term one. This proves integrality of the universal construction and also provides an explicit coefficient algorithm. Expanding it, the resulting formal group of an elliptic curve beginsThe integral coefficient recursion and the unit-denominator addition calculation work universally over $\mathbb{Z}[a_1,a_2,a_3,a_4,a_6]$; thus these coefficients specialize in every characteristic of a field. This is the initial coefficients of the elliptic formal group at infinity construction, rather than a definition requiring division by the elliptic-curve discriminant.

The identity, commutativity and associativity of the elliptic curve group law give the defining identities of a one-dimensional commutative formal group law:Reflection in the generalized Weierstrass equation of an elliptic curve gives its formal inverse explicitly:Successive addition gives the multiplication series $[n{]}_F(T)=nT+O(T^2)$.

In characteristic zero, an especially useful description comes from the invariant differential on an elliptic curve:Integrating with zero constant term constructs the formal logarithmTranslation invariance of the invariant differential on an elliptic curve implies $L(F(X,Y))=L(X)+L(Y)$. Thus the formal logarithm converts the formal group law into the formal additive group in characteristic zero. Expanding this identity to degree three also recovers the displayed coefficients of $F$; in positive characteristic the integral addition construction remains valid although the logarithmic denominators may not exist.

Choose an integral Minimal Weierstrass equation over ${\mathbb{Z}}_p$. Let $E_0({\mathbb{Q}}_p)$ be the points with nonsingular reduction, and let $E_1({\mathbb{Q}}_p)$ be those reducing to $O$. This distinction matters at bad reduction of an elliptic curve: the whole singular reduced cubic is not a group. The smooth locus is a group under the induced elliptic curve group law, and Hensel's lemma lifts each of its rational points, givingThe formal kernel of a minimal Weierstrass equation identifies $E_1({\mathbb{Q}}_p)$ with $p{\mathbb{Z}}_p$ equipped with $F$. Indeed, the integral series $w(t)$ converges there, and $x=t/w(t)$, $y=-1/w(t)$ give the unique point for each nonzero parameter. If $v_p(t)=r>0$, then $v_p(x)=-2r$ and $v_p(y)=-3r$; the parameter zero gives $O$. Conversely a point reducing to $O$ has these local coordinates of positive valuation, so is recovered by the series.

Define $E_m$ for $m\ge 1$ by $t\in p^m{\mathbb{Z}}_p$. The linear part of the formal group law is addition, and the higher terms have degree at least two. Hence the filtration of elliptic-curve points over a local field hasAt good reduction, $E_0=E$ and $E/E_1\cong \tilde{E}({\mathbb{F}}_p)$. At bad reduction, there is also the finite quotient $E/E_0$: $E({\mathbb{Q}}_p)$ is a compact group because the curve is projective, and $E_0$ is an open subgroup, so its index of a subgroup is finite. Together these quotients and the formal filtration describe the group, including the extra components at bad primes.

The formal logarithm makes the deepest part explicit. Write $L(T)=T+{\sum }_{n\ge 2}{c}_n{T}^n$. Since its derivative is the integral invariant differential of a formal group law, $n{c}_n\in {\mathbb{Z}}_p$. ConsequentlyThe inequality $v_p(n)\le (n-1)/(p-1)$ shows that for an integer $m>1/(p-1)$ every nonlinear term on $p^m{\mathbb{Z}}_p$ has strictly higher valuation than the linear term. After scaling, $L$ is the identity plus a strictly contracting series; solving $L(t)=u$ by successive correction gives its inverse formal group exponential. Thus the deep logarithm subgroup of a formal group givesOne may take $m=1$ for odd $p$ and $m=2$ for $p=2$. In particular, $E({\mathbb{Q}}_p)$ contains an open subgroup isomorphic to the additive p-adic integers, and the remaining information is in finite quotients and their group extensions.

Assume good reduction, and write $N_p=\#\tilde{E}({\mathbb{F}}_p)$. The surjectivity of good reduction over a local field and the formal group law determine the torsion as follows. If $p\nmid n$, multiplication by $n$ is an automorphism of $E_1$, because its multiplication series has an integral compositional inverse with unit linear coefficient. Therefore reduction gives an isomorphismTo construct the inverse, lift a reduced point to $P$, solve $[n]Q=[n]P$ uniquely in $E_1$, and use $P-Q$. This proves the prime-to-p torsion lifts uniquely at good reduction, including uniqueness, rather than merely comparing orders.

For residue-characteristic torsion, use a deep logarithm subgroup of a formal group. Take $m=1$ for odd $p$, or $m=2$ at two. The formal logarithm identifies $E_m$ with $p^m{\mathbb{Z}}_p$, so $E_m$ has no torsion. The finite quotient grouptherefore receives the torsion subgroup injectively. There is a complete test for which cosets actually contain torsion. For a coset $g$ of order $n$, choose a lift $P$. Then $[n]P\in E_m$. Its coset has a torsion representative exactly whenWhen this condition holds, let $Q\in E_m$ be the unique point with $L(Q)=L([n]P)/n$. Then $T=P-Q$ is killed by $n$ and represents $g$. Conversely, any torsion representative has order precisely $n$, since an element killed in the quotient would belong to the torsion-free group $E_m$. It must therefore arise from this construction. Changing $P$ by $R\in E_m$ changes $L([n]P)$ by $nL(R)$, so the test is independent of the chosen lift. This proves the torsion test using a deep formal logarithm subgroup and gives the full torsion by a finite list of cosets and a logarithmic divisibility test.

For odd $p$, one has $E_m=E_1$, so all torsion points of an elliptic curve inject into $\tilde{E}({\mathbb{F}}_p)$. Nevertheless a reduced $p$-primary point need not lift to a torsion point: if its order is $p^r$, the test is $L([p^r]P)\in p^{r+1}{\mathbb{Z}}_p$. For $p=2$, the subgroup $E_2$ is torsion-free but $E_1/E_2$ has order two. Thus $E_1$ can have a two-torsion point, and one must use $G=E/E_2$ of order $2N_2$ rather than assert that all torsion injects under reduction. The prime-two torsion bound for good reduction follows: the reduction kernel contributes at most a subgroup of order two. These distinctions are essential to a correct determination of local torsion subgroups.