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.)



Shapiro lemma for Bloch Kato Selmer group
This is a nice exercise. You start by making some reduction steps. Firstly, the BK Selmer group for $T$ is by definition saturated (i.e. it is the preimage in $H^1(K, T)$ of a $\mathbf{Q}\_p$-subspace of $H^1\_{\mathrm{f}}(K, T \otimes \mathbf{Q}\_p)$) so you can invert $p$ everywhere and ask the question for finite-dimensional $\mathbf{Q}\_p$-linear representations $V$


Secondly, since you know Shapiro's lemma for the global cohomology, it is sufficient to check that the local conditions match, i.e. that for each prime we have
$$H^1\_{\mathrm{f}}(\mathbf{Q}\_\ell, \operatorname{Ind}\_K^{\mathbf{Q}} V) = \bigoplus\_{v \mid \ell} H^1\_{\mathrm{f}}(K\_v, V).$$


The next reduction step is to use [Mackey's theorem](http://math.stanford.edu/~conrad/210BPage/handouts/mackey.pdf) to write $(\operatorname{Ind}\_K^{\mathbf{Q}} V) |\_{\mathbf{G}\_{\mathbf{Q}\_\ell}}$ as a direct sum of pieces indexed by the quotient $G\_{\mathbf{Q}\_{\ell}} \backslash G\_\mathbf{Q} / G\_K$. This double quotient turns out to naturally biject with the set of primes of $K$ above $\ell$, and the summand corresponding to the prime $v$ is $\operatorname{Ind}\_{G\_{K\_v}}^{G\_{\mathbf{Q}\_\ell}}\left( V\_{G\_{K\_v}}\right)$. So now we are down to a *purely local* question: if $M/L$ is a finite extension of $\ell$-adic fields for some $\ell$, we need to check that
$$ H^1\_{\mathrm{f}}(L, Ind\_{G\_{M}}^{G\_{L}} V) = H^1\_{\mathrm{f}}(M, V).$$
This I gather you already know how to do.


