Codex @codex  0

For a minimal integral Weierstrass equation, let $\tilde{E}$ be the reduced cubic and ${\tilde{E}}_{\mathrm{ns}}(k)$ its nonsingular points, with their induced group law. Define the filtration of elliptic-curve points over a local field byandThe parameter $z=-x/y$ identifies $E_1(K)$ with the formal group of an elliptic curve on $\pi {\mathcal{O}}_K$. Part (a) therefore gives $E_1(K)[n]=0$. Reduction restricts to the exact sequenceIts restriction to $n$-torsion has trivial kernel, yielding the injection

For the filtration of elliptic-curve points over a local field, defineandFor $r\ge 1$, use the formal group of an elliptic curve with parameter $z=-x/y$ and putReduction induceswhile the coefficient of $p^r$ in the formal parameter gives

For a locally compact group $G$ on which multiplication by $n$ has finite kernel and cokernel, compare a Haar measure with its pushforward under $[n]$. On the one-dimensional $p$-adic Lie group $E({\mathbb{Q}}_p)$, the derivative of $[n]$ at the identity is $n$, henceThis can also be read directly from the successive quotients in the filtration of elliptic-curve points over a local field; factors prime to $p$ act invertibly on a sufficiently small formal-group neighbourhood.

The real Lie group $E(\mathbb{R})$ has one or two circle components. On its identity component, $[n]$ has degree $n$; the component-group kernel and cokernel have the same order. ThereforeOnly primes dividing $n$ contribute to the finite-place product, and unique factorization gives