Can someone give a really concrete example of such a sequence? I am looking at several notes related with such things, but haven't seen any well-calculated example. And I'm really confused at this point.


Besides asking for a good example, I am also wondering about the following two things:


1. There is an exact sequence for elliptic curves defined over a local field $K$, $0 \rightarrow \hat E(m) \rightarrow E(K) \rightarrow \tilde E(k) \rightarrow 0$, where $\hat E(m)$ is the formal group associated to $E$ and $\tilde E(k)$ is the reduction. (See Silverman AEC I, page 118), is this sequence related with connected-etale sequence?


2.Take the p-torsion kernel $E[p]$ of $[p]: E \rightarrow E$ for $E$ defined over $K$ a local field.Is $E[p]$ a finite flat group scheme over $R$ the valuation ring? And if so, what is its connected-etale sequence? (maybe I should change $p$ to an $n$, but I'm also curious what will happen if $p$ is the characteristic of $k$?)


Thank you.



Example of connected-etale sequence for group schemes over a Henselian field?
Regarding 1. : If you pass to the $n$-torsion parts of the members of this exact sequence,
you will get the $R$-valued points
of the connected-etale sequence for $E[n]$.


Regarding 2. : If $E$ has good reduction, then $E[n]$ is a finite flat group scheme.
If the residue char. $p$ of $R$ does not divide $n$ then it is even etale.


In general, by the Chinese remainder theorem, when analyzing $E[n]$ we may assume $n$
is a prime power, so suppose now that $n$ is a power $p^r$, where $p$ is the residue char.
of $R$.


If $E$ has supersingular reduction, then $E[p^r]$ is connected.


If $E$ has ordinary reduction, then the connected and etale parts of $E[p^r]$ each
have order $p^r$. The special fibre of the connected part is the kernel of $Frob^r$.


Note that if $E$ has bad reduction, then $E[n]$ can still have a finite flat extension
over $R$ for some finite number of $n$. (E.g. $X\_0(11)$ has bad reduction at 11, but its
5-torsion is still finite flat at 11.)


To compute an example, I suggest considering elliptic curves in char. 2, and computing the 2-torsion. 


Consider the two examples: $y^2 + y = x^3 $ and $y^2 + x y = x^3 + x$. If you compute the 2-division equation it will have degree 4. In the first case, it will be purely inseparable: this case is supersingular. In the second case it will have inseparability degree 2; this case is ordinary. In the first case the 2-torsion group scheme is connected; in the second, it has a connected subgroup scheme of order 2.


