Let $K$ be a number field and $E/K$ be an elliptic curve defined over $K$. 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?
When is this localization map injective, if at all?
debanjana
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