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(Comparing to class group cases  : we have an isomorphism $Cl(K)\rightarrow \prod \left(K^\times \backslash K_p^\times /O_p^\times \right)$ for a number field $K$.

Similarly, for an elliptic curve $E/\mathbb{Q}$, I expect the injectivity of $Sel(E/\mathbb{Q})\rightarrow \prod \left( (E/E_{tor})(\mathbb{Q}_p)\right) \left(\hookrightarrow \prod H^1(\mathbb{Q}_p,E_{tor})\right)$, but I cannot prove or disprove it.)

I'm sorry for unclear question. I use Sel$(E/\mathbb{Q})$ as the direct limit of $n$-Selmer groups.

My question comes from comparing two sequences  :  $0\rightarrow E(\mathbb{Q})\otimes \mathbb{Q}/\mathbb{Z} \rightarrow Sel(E/\mathbb{Q})\rightarrow Ш(E/\mathbb{Q}) \rightarrow 0 $ and $0\rightarrow \underset{p}\prod E(\mathbb{Q}_p)\otimes \mathbb{Q}/\mathbb{Z} \rightarrow \underset{p}\prod H^1(\mathbb{Q}_p,E_{tor})\rightarrow \underset{p}\prod H^1(\mathbb{Q}_p,E) \rightarrow ... $

Let $a,b$ be the restriction maps from $E(\mathbb{Q})\otimes \mathbb{Q}/\mathbb{Z},Sel(E/\mathbb{Q})$ respectively. There is an exact sequence $0\rightarrow \ker a\rightarrow \ker b\rightarrow Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~ a\rightarrow \mathrm{coker}~b$.

$Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~a~~$ is an analogue of $r~:~Cl(K)\rightarrow K^\times \backslash \prod \left(K_p^\times /O_p^\times \right)$. The reciprocity map $r$ is known to be bijective and the proof does not assume the finiteness of class groups. Is there similar argument for $Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~a~~$?

(Comparing to class group cases: we have an isomorphism $Cl(K)\rightarrow \prod \left(K^\times \backslash K_p^\times /O_p^\times \right)$ for a number field $K$.

Similarly, for an elliptic curve $E/\mathbb{Q}$, I expect the injectivity of $Sel(E/\mathbb{Q})\rightarrow \prod \left( (E/E_{tor})(\mathbb{Q}_p)\right) \left(\hookrightarrow \prod H^1(\mathbb{Q}_p,E_{tor})\right)$, but I cannot prove or disprove it.)

I'm sorry for unclear question. I use Sel$(E/\mathbb{Q})$ as the direct limit of $n$-Selmer groups.

My question comes from comparing two sequences: 

$0\rightarrow E(\mathbb{Q})\otimes \mathbb{Q}/\mathbb{Z} \rightarrow Sel(E/\mathbb{Q})\rightarrow Ш(E/\mathbb{Q}) \rightarrow 0 $ and $0\rightarrow \underset{p}\prod E(\mathbb{Q}_p)\otimes \mathbb{Q}/\mathbb{Z} \rightarrow \underset{p}\prod H^1(\mathbb{Q}_p,E_{tor})\rightarrow \underset{p}\prod H^1(\mathbb{Q}_p,E) \rightarrow ... $

Let $a,b$ be the restriction maps from $E(\mathbb{Q})\otimes \mathbb{Q}/\mathbb{Z},Sel(E/\mathbb{Q})$ respectively. There is an exact sequence $0\rightarrow \ker a\rightarrow \ker b\rightarrow Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~ a\rightarrow \mathrm{coker}~b$.

$Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~a~~$ is an analogue of $r~:~Cl(K)\rightarrow K^\times \backslash \prod \left(K_p^\times /O_p^\times \right)$. The reciprocity map $r$ is known to be bijective and the proof does not assume the finiteness of class groups. Is there similar argument for $Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~a~~$?

(Comparing to class group cases : we have an isomorphism $Cl(K)\rightarrow \prod \left(K^\times \backslash K_p^\times /O_p^\times \right)$ for a number field $K$.

Similarly, for an elliptic curve $E/\mathbb{Q}$, I expect the injectivity of $Sel(E/\mathbb{Q})\rightarrow \prod \left( (E/E_{tor})(\mathbb{Q}_p)\right) \left(\hookrightarrow \prod H^1(\mathbb{Q}_p,E_{tor})\right)$, but I cannot prove or disprove it.)

I'm sorry for unclear question. I use Sel$(E/\mathbb{Q})$ as the direct limit of $n$-Selmer groups.

My question comes from comparing two sequences : $0\rightarrow E(\mathbb{Q})\otimes \mathbb{Q}/\mathbb{Z} \rightarrow Sel(E/\mathbb{Q})\rightarrow Ш(E/\mathbb{Q}) \rightarrow 0 $ and $0\rightarrow \underset{p}\prod E(\mathbb{Q}_p)\otimes \mathbb{Q}/\mathbb{Z} \rightarrow \underset{p}\prod H^1(\mathbb{Q}_p,E_{tor})\rightarrow \underset{p}\prod H^1(\mathbb{Q}_p,E) \rightarrow ... $

Let $a,b$ be the restriction maps from $E(\mathbb{Q})\otimes \mathbb{Q}/\mathbb{Z},Sel(E/\mathbb{Q})$ respectively. There is an exact sequence $0\rightarrow \ker a\rightarrow \ker b\rightarrow Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~ a\rightarrow \mathrm{coker}~b$.

$Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~a~~$ is an analogue of $r~:~Cl(K)\rightarrow K^\times \backslash \mathbb{A}_K^{\times} /\prod O_p^\times$. The reciprocity map $r$ is known to be bijective and the proof does not assume the finiteness of class groups. Is there similar argument for $Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~a~~$?

(Comparing to class group cases : we have an isomorphism $Cl(K)\rightarrow \prod \left(K^\times \backslash K_p^\times /O_p^\times \right)$ for a number field $K$.

Similarly, for an elliptic curve $E/\mathbb{Q}$, I expect the injectivity of $Sel(E/\mathbb{Q})\rightarrow \prod \left( (E/E_{tor})(\mathbb{Q}_p)\right) \left(\hookrightarrow \prod H^1(\mathbb{Q}_p,E_{tor})\right)$, but I cannot prove or disprove it.)

I'm sorry for unclear question. I use Sel$(E/\mathbb{Q})$ as the direct limit of $n$-Selmer groups.

My question comes from comparing two sequences : $0\rightarrow E(\mathbb{Q})\otimes \mathbb{Q}/\mathbb{Z} \rightarrow Sel(E/\mathbb{Q})\rightarrow Ш(E/\mathbb{Q}) \rightarrow 0 $ and $0\rightarrow \underset{p}\prod E(\mathbb{Q}_p)\otimes \mathbb{Q}/\mathbb{Z} \rightarrow \underset{p}\prod H^1(\mathbb{Q}_p,E_{tor})\rightarrow \underset{p}\prod H^1(\mathbb{Q}_p,E) \rightarrow ... $

Let $a,b$ be the restriction maps from $E(\mathbb{Q})\otimes \mathbb{Q}/\mathbb{Z},Sel(E/\mathbb{Q})$ respectively. There is an exact sequence $0\rightarrow \ker a\rightarrow \ker b\rightarrow Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~ a\rightarrow \mathrm{coker}~b$.

$Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~a~~$ is an analogue of $r~:~Cl(K)\rightarrow K^\times \backslash \prod \left(K_p^\times /O_p^\times \right)$. The reciprocity map $r$ is known to be bijective and the proof does not assume the finiteness of class groups. Is there similar argument for $Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~a~~$?

(Comparing to class group cases : we have an isomorphism $Cl(K)\rightarrow \prod \left(K^\times \backslash K_p^\times /O_p^\times \right)$ for a number field $K$.

Similarly, for an elliptic curve $E/\mathbb{Q}$, I expect the injectivity of $Sel(E/\mathbb{Q})\rightarrow \prod \left( (E/E_{tor})(\mathbb{Q}_p)\right) \left(\hookrightarrow \prod H^1(\mathbb{Q}_p,E_{tor})\right)$, but I cannot prove or disprove it.)

I'm sorry for unclear question. I use Sel$(E/\mathbb{Q})$ as the direct limit of $n$-Selmer groups.

My question comes from comparing two sequences : $0\rightarrow E(\mathbb{Q})\otimes \mathbb{Q}/\mathbb{Z} \rightarrow Sel(E/\mathbb{Q})\rightarrow Ш(E/\mathbb{Q}) \rightarrow 0 $ and $0\rightarrow \underset{p}\prod E(\mathbb{Q}_p)\otimes \mathbb{Q}/\mathbb{Z} \rightarrow \underset{p}\prod H^1(\mathbb{Q}_p,E_{tor})\rightarrow \underset{p}\prod H^1(\mathbb{Q}_p,E) \rightarrow ... $

Let $a,b$ be the restriction maps from $E(\mathbb{Q})\otimes \mathbb{Q}/\mathbb{Z},Sel(E/\mathbb{Q})$ respectively. There is an exact sequence $0\rightarrow \ker a\rightarrow \ker b\rightarrow Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~ a\rightarrow \mathrm{coker}~b$.

$Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~a~~$ is an analogue of $r~:~Cl(K)\rightarrow \prod \left(K^\times \backslash K_p^\times /O_p^\times \right)$. The reciprocity map $r$ is known to be bijective and the proof does not assume the finiteness of class groups. Is there similar argument for $Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~a~~$?

(Comparing to class group cases : we have an isomorphism $Cl(K)\rightarrow \prod \left(K^\times \backslash K_p^\times /O_p^\times \right)$ for a number field $K$.

Similarly, for an elliptic curve $E/\mathbb{Q}$, I expect the injectivity of $Sel(E/\mathbb{Q})\rightarrow \prod \left( (E/E_{tor})(\mathbb{Q}_p)\right) \left(\hookrightarrow \prod H^1(\mathbb{Q}_p,E_{tor})\right)$, but I cannot prove or disprove it.)

I'm sorry for unclear question. I use Sel$(E/\mathbb{Q})$ as the direct limit of $n$-Selmer groups.

My question comes from comparing two sequences : $0\rightarrow E(\mathbb{Q})\otimes \mathbb{Q}/\mathbb{Z} \rightarrow Sel(E/\mathbb{Q})\rightarrow Ш(E/\mathbb{Q}) \rightarrow 0 $ and $0\rightarrow \underset{p}\prod E(\mathbb{Q}_p)\otimes \mathbb{Q}/\mathbb{Z} \rightarrow \underset{p}\prod H^1(\mathbb{Q}_p,E_{tor})\rightarrow \underset{p}\prod H^1(\mathbb{Q}_p,E) \rightarrow ... $

Let $a,b$ be the restriction maps from $E(\mathbb{Q})\otimes \mathbb{Q}/\mathbb{Z},Sel(E/\mathbb{Q})$ respectively. There is an exact sequence $0\rightarrow \ker a\rightarrow \ker b\rightarrow Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~ a\rightarrow \mathrm{coker}~b$.

$Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~a~~$ is an analogue of $r~:~Cl(K)\rightarrow K^\times \backslash \mathbb{A}_K^{\times} /\prod O_p^\times$. The reciprocity map $r$ is known to be bijective and the proof does not assume the finiteness of class groups. Is there similar argument for $Ш(E/\mathbb{Q})\rightarrow \mathrm{coker}~a~~$?