When is this localization map injective, if at all?

Question

Let $K$ be a number field and $E$ be an elliptic curve defined over $\mathbb{Q}$. Consider the localization map $$ E(K)\otimes \mathbb{Q}_p/ \mathbb{Z}_p \rightarrow \bigoplus_{v|p} E(K_v)\otimes \mathbb{Q}_p/ \mathbb{Z}_p. $$ Can we say when is the above map injective. I am wondering if there is some properties of K that can guarantee injectivity.

Answer 1

Often it is, but not always. For instance if $K=\mathbb{Q}$ then the map is injective if and only if the rank of $E(\mathbb{Q})$ is at most $1$. This is because the $p$-adic elliptic logarithm of a non-torsion point $P\in E(\mathbb{Q})$ is non-zero.

The corank of the target is equal to $[K:\mathbb{Q}]$ while the rank of the Mordell-Weil group is often much smaller. In some sense we could expect that the map is as injective as it can be. By composing the map with the logarithm map from $E(K_v)\to K_v$, the question becomes the analogue to the Leopoldt conjecture for units in number fields; it will have to do with the linear independence of the $p$-adic elliptic logarithm maps. So a first guess would be to think that your map is injective when the rank of $E(K)$ is smaller than $[K:\mathbb{Q}]$; but that is not true.

In fact, there is one important difference to the case of units in that the Galois structure of $E(K)$ is not as simple. Suppose the curve $E$ is defined over a subfield $k$ of $K$. For simplicity suppose $k=\mathbb{Q}$. Assume that $K/\mathbb{Q}$ is Galois with finite group $G$. Then the target space $\prod_{v\mid p} E(K_v)\otimes \mathbb{Q}_p$ is isomorphic to a free rank $1$ module over $\mathbb{Q}_p[G]$ by the logarithm map. Now consider the Mordell-Weil group $E(K)\otimes \mathbb{Q}_p$ as a $\mathbb{Q}_p[G]$-module. This could be of $\mathbb{Q}_p$-dimension smaller than $[K:\mathbb{Q}]$, yet your map is not injective because a particular representation $\rho$ appears more often in the Mordell-Weil group than in $\mathbb{Q}_p[G]$, i.e., more often than $\dim(\rho)$. In this situation it would be possible to conjecture the following: Your map is injective if and only if $\dim_{\mathbb{Q}_p}( E(K)\otimes \mathbb{Q}_p)^\rho \leq \dim_{\mathbb{Q}_p}(\rho)$ for all irreducible $\mathbb{Q}_p[G]$-modules $\rho$.

I guess one could make a general conjecture as to when it is injective depending on the field of definition of $E$. Though maybe there are other restrictions. Ultimately an answer to this question will come from transcendence of values of $p$-adic ellitpic logarithms.