An example that a $p$ adic Galois representation is crystalline but not $B_e$ admissible

Question

$B_e=B_{\text{cris}}^{\phi=1}$, 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!

Answer 1

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.)