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Let $F$ be a formal group law over the valuation ring of a finite extension $K/{\mathbb{Q}}_p$, with $v_K(\pi )=1$ and $e=v_K(p)$. If $m>e/(p-1)$, its formal logarithm is an isomorphism $F({\pi }^m{\mathcal{O}}_K)\to ({\pi }^m{\mathcal{O}}_K,+)$. Indeed, its scaled series is $U+\pi R(U)$ with $R$ integral and convergent on ${\mathcal{O}}_K$; solving by contraction produces the formal group exponential. The subgroup has index $q^{m-1}$ in $F(\pi {\mathcal{O}}_K)$, where $q$ is the residue-field order.

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.

Here a formal group means a one-dimensional commutative formal group law. Over a ring $R$, it is a formal power series $F(X,Y)\in R[[X,Y]]$ satisfyingIn particular, $F(X,Y)=X+Y+$ terms of total degree at least $2$. A formal inverse $\iota (T)=-T+\cdots$ with $F(T,\iota (T))=0$ is obtained recursively. An isomorphism of formal group laws from $F$ to $G$ is a series $h(T)\in TR[[T]]$ with invertible linear coefficient andIts compositional inverse exists over $R$ by coefficient recursion.

For $R={\mathcal{O}}_K$, all these series converge on $\pi {\mathcal{O}}_K$; the operations preserve that set and make it the group $F(\pi {\mathcal{O}}_K)$. Use the uniformizer $\pi$ and normalize the valuation by $v_K(\pi )=1$, and write $e=v_K(p)$.

We explicitly construct its formal logarithm. DefineThen putThis is a formal series over $K$ with linear coefficient $1$. Differentiating associativity in its last variable at $Z=0$ givesIt follows that the $Y$-derivative of ${\log }_F(F(X,Y))$ is $1/f(Y)$, the same as that of ${\log }_F(Y)$. Evaluating at $Y=0$ therefore provesThe differential $dT/f(T)$ is the invariant differential of a formal group law.

Choose an integer $m>e/(p-1)$. In degree $n\ge 2$, the coefficient of ${\log }_F(T)$ has valuation at least $-e{v}_p(n)$. Thus the scaled serieshasHere $v_p(n)\le (n-1)/(p-1)$, and the coefficient valuations tend to infinity. Consequently $H(U)=U+\pi R(U)$ with $R$ a convergent integral power series on ${\mathcal{O}}_K$. Such a series is $1$-Lipschitz. For any $a\in {\mathcal{O}}_K$, the equation $H(U)=a$ is equivalent to $U=a-\pi R(U)$, a strict contraction, to which the contraction mapping theorem applies on the complete ring ${\mathcal{O}}_K$. It has exactly one solution.

The deep logarithm subgroup of a formal group is obtained as follows. The formal group exponential is the formal compositional inverse ${\exp }_F(T)$ of ${\log }_F(T)$, obtained recursively; equivalently it solves ${\exp }_F^{\prime }(T)=f({\exp }_F(T))$, with ${\exp }_F(0)=0$. The preceding contraction, applied to the scaled series, constructs its convergent inverse on ${\pi }^m{\mathcal{O}}_K$ and agrees with that formal inverse. This supplies the required convergence sketch for ${\exp }_F$ as well as a bijectivity proof for the logarithm.

The integral formal law and inverse preserve ${\pi }^m{\mathcal{O}}_K$, so this is a subgroup. The logarithm identity and bijectivity giveReduction of the formal law modulo ${\pi }^m$ has kernel $F({\pi }^m{\mathcal{O}}_K)$. Its underlying image set is $\pi {\mathcal{O}}_K/{\pi }^m{\mathcal{O}}_K$, which has size $q^{m-1}$, where $q=\#({\mathcal{O}}_K/\pi )$. Thus this subgroup has finite index, explicitly $q^{m-1}$.

A one-dimensional commutative formal group law over ${\mathcal{O}}_K$ is a power series $F(X,Y)\in {\mathcal{O}}_K[[X,Y]]$ satisfyingFor $r\ge 1$, the ideal ${\pi }^r{\mathcal{O}}_K$ becomes a group, denoted $F({\pi }^r{\mathcal{O}}_K)$, under $x{+}_{F}y=F(x,y)$; convergence follows because both inputs lie in the maximal ideal.

Over the characteristic-zero field $K$, there is a unique formal logarithmsatisfying ${\log }_F(F(X,Y))={\log }_F(X)+{\log }_F(Y)$. Its coefficients have bounded denominator growth, so for sufficiently large $r$ both ${\log }_F$ and its inverse formal group exponential converge on ${\pi }^r{\mathcal{O}}_K$ and preserve that ideal. They givewhere the final isomorphism is multiplication by ${\pi }^{-r}$.

Let $E/{\mathbb{Q}}_p$ have good reduction, and choose an integer $m>1/(p-1)$. Its filtration of elliptic-curve points over a local field has a torsion-free group $E_m$ identified by the formal logarithm with $p^m{\mathbb{Z}}_p$. The quotient group $G=E({\mathbb{Q}}_p)/E_m$ is finite. For $g\in G$ of order of a group element $N$, choose a lift $P$. There is a torsion point of an elliptic curve in this coset exactly when the displayed condition holds. Indeed, it is equivalent to the existence of $Q\in E_m$ with $[N]Q=[N]P$. Then $P-Q$ is killed by $N$; conversely every torsion representative has order $N$, since $E_m$ is torsion-free. The logarithmic division and the formal group exponential give the unique such $Q$. Changing the lift changes the logarithm by an element of $N{p}^m{\mathbb{Z}}_p$, so the criterion is well-defined. This finite-coset test includes residue-characteristic torsion; only the prime-to-p torsion lifts uniquely at good reduction are automatic.