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David Loeffler
David Loeffler
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Let $K$ be a finite extension of $\mathbb{Q}$. Let $T$ be a finite dimensional $G_{K}$ module over $\mathbb{Z}_p$. Does the Bloch Kato-Kato Selmer group $H^{1}_{f}(\mathbb{Q},Ind^{G_{\mathbb{Q}}}_{G_K}T) = H^{1}_{f}(K,T) $$H^{1}_{f}$ satisfy $H^1_f(\mathbb{Q},Ind^{G_{\mathbb{Q}}}_{G_K}T) = H^{1}_{f}(K,T) $?

If not, then can you tell me what kind of condition on $K/\mathbb{Q}$ as well as the condition at the prime $p$ do I need for that equality to hold? Thank you.  (I was trying to use $Ind(Res V\otimes W)\simeq V\otimes Ind W$ for each prime but I got stuck.)

Let $K$ be a finite of $\mathbb{Q}$. Let $T$ be a finite dimensional $G_{K}$ module over $\mathbb{Z}_p$. Does the Bloch Kato Selmer group $H^{1}_{f}(\mathbb{Q},Ind^{G_{\mathbb{Q}}}_{G_K}T) = H^{1}_{f}(K,T) $?

If not then can you tell me what kind of condition on $K/\mathbb{Q}$ as well as the condition at the prime $p$ do I need for that equality to hold? Thank you.(I was trying to use $Ind(Res V\otimes W)\simeq V\otimes Ind W$ for each prime but I got stuck.)

Let $K$ be a finite extension of $\mathbb{Q}$. Let $T$ be a finite dimensional $G_{K}$ module over $\mathbb{Z}_p$. Does the Bloch-Kato Selmer group $H^{1}_{f}$ satisfy $H^1_f(\mathbb{Q},Ind^{G_{\mathbb{Q}}}_{G_K}T) = H^{1}_{f}(K,T) $?

If not, then can you tell me what kind of condition on $K/\mathbb{Q}$ as well as the condition at the prime $p$ do I need for that equality to hold? Thank you.  (I was trying to use $Ind(Res V\otimes W)\simeq V\otimes Ind W$ for each prime but I got stuck.)

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bambootuan1012
bambootuan1012
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Shapiro lemma for Bloch Kato Selmer group

Let $K$ be a finite of $\mathbb{Q}$. Let $T$ be a finite dimensional $G_{K}$ module over $\mathbb{Z}_p$. Does the Bloch Kato Selmer group $H^{1}_{f}(\mathbb{Q},Ind^{G_{\mathbb{Q}}}_{G_K}T) = H^{1}_{f}(K,T) $?

If not then can you tell me what kind of condition on $K/\mathbb{Q}$ as well as the condition at the prime $p$ do I need for that equality to hold? Thank you.(I was trying to use $Ind(Res V\otimes W)\simeq V\otimes Ind W$ for each prime but I got stuck.)