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debanjana
debanjana
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Let $K$ be a number field and $E/K$$E$ be an elliptic curve defined over $K$$\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.

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?

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.

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debanjana
debanjana
  • 1.3k
  • 7
  • 16

When is this localization map injective, if at all?

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?