$B\_e=B\_{\text{cris}}^{\phi=1}$, so if a $p$-adic Galois representation $V$ is $B\_e$ admissible, then it is crystalline, so I want to know an example that $V$ is crystalline but not $B\_e$ admissible.


Thanks!



An example that a $p$ adic Galois representation is crystalline but not $B_e$ admissible
We have $(V \otimes B\_e)^{G\_K} = D\_{\mathrm{cris}}(V)^{\varphi = 1}$. So any representation which is crystalline, but such that $\mathbf{D}\_{\mathrm{cris}}(V)$ has zero $\varphi$-invariants, is an example of a representation which is $B\_{\mathrm{cris}}$-admissible but not $B\_e$-admissible. 


For instance, taking $K = \mathbf{Q}\_p$ and $V$ any non-zero power of the cyclotomic character) will do the trick -- note that $D\_{\mathrm{cris}}(\mathbf{Q}\_p(n))$ is 1-dimensional with $\varphi$ acting as multiplication by $p^{-n}$.


(Moral: although the ring $B\_e$ is useful in many ways, it is too small for the category of $B\_e$-admissible representations to be interesting.)


