Let $K$ be a local field with residue field $k$ and $E/K$ an elliptic curve. I'm interested for which $K$ and $E$ the group of torsion points on the curve is finite. I can prove that this group is finite for an $n$-dimensional local field $K$ such that its $(n-1)$'th residue field has zero characteristics i.e. a finite extension of $\mathbb Q_p$. The proof goes by induction on $n$.

Basis.

Let $K$ be a one-dimensional local field i.e. $k = \mathbb F_q$,$q=p^r$. We use the standard notation $E_0(K) = \{P\in E(K)|\tilde{P}\in \tilde{E_{ns}}(k)\}$ and $E_1(K) = \{P\in E(K)|\tilde{P} = \tilde{O}\}$. Then we have an exact sequence of abelian groups:

$$ 0\to E_1(K) \to E_0(K) \to \tilde{E_{ns}}(k) \to 0. $$

We know that $E_1(K) \cong \hat{E}(\mathcal M)$ where $\hat{E}$ is the formal group associated with $E$, hence $E_1(K)$ has no non-trivial elements of order $m$. From the exact sequence we see that $E_0(K)[m]$ injects to $\tilde{E_{ns}}(k)$ so all non-trivial torsion elements of $E_0(K)$ go to non trivial torsion elements of $\tilde{E_{ns}}(k)$. But $\#\tilde{E_{ns}}(k)$ is finite so the number of such torsion elements is finite. By Kodaira-Néron theorem the index of $E_0(K)$ in $E(K)$ is finite so we have proved that the torsion of $E(K)$ is finite.

Step.

Suppose we have proved the theorem for all $(n-1)$-dimensional local fields. Let $K$ be an $n$-dimensional local field i.e. $k$ is an $(n-1)$-dimensional local field. Then if the curve has good reduction the same argument reduces the case to $k$. If not $\tilde{E_{ns}}(k)$ is isomorphic either to $k^*$ or $k^+$ which have finite torsion.

Q.E.D.

I have two hypotheses. The first is that this is true for all local field of finite dimension. The second is that the only fields that have this property are subfields of higher local fields. In particular, having the property that the torsion of $E(K)$ is finite depends only on the field but not on the curve. Can you prove these statements or provide a counterexample?

EDIT There are some mistakes in my proof. First of all, there might exist $p$-torsion in $E_1(K)$ where $p$ is the characteristics but it is finite because $E_1(K)$ is a pro-$p$-finite group such that $p^r$-torsion is finite for every $r$. But I don't know how to generalize this to the case of higher-dimensional fields because now $E_1(K)$ is not at all pro-$p$-group.

Local bounds on torsion in abelian varietiesin Canad. J. Math. (its full journal data can be found as item [2] in this note that I have written: arxiv.org/pdf/1603.03789.pdf ) $\endgroup$ – Vesselin Dimitrov Jul 6 '18 at 21:275more comments