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$\textbf{First question}$: $B_e=B_{\text{cris}}^{\phi=1}$, 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.

$\textbf{Second question}$: I know any elliptic curve over a $p$-adic field with $v_p(j)\leq 0$ is semi-stable and not crystalline, and we can construct an extension of $\mathbb{Q}_p(1)$ by $\mathbb{Q}_p$ to show it is Hodge-Tate but not de Rham. So I want to know an example that $V$ is de Rham but not semi-stable. By Fontaine's famous theorem $B$, de Rham is equivalent to potentially semi-stable. So I only need a concrete example that $V$ is potentially semi-stable but not semi-stable.

Thanks!

$\textbf{First question}$: $B_e=B_{\text{cris}}^{\phi=1}$, 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.

$\textbf{Second question}$: I know any elliptic curve over a $p$-adic field with $v_p(j)\leq 0$ is semi-stable and not crystalline, and we can construct an extension of $\mathbb{Q}_p(1)$ by $\mathbb{Q}_p$ to show it is Hodge-Tate but not de Rham. So I want to know an example that $V$ is de Rham but not semi-stable. By Fontaine's famous theorem $B$, de Rham is equivalent to potentially semi-stable. So I only need a concrete example that $V$ is potentially semi-stable but not semi-stable.

Thanks!

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

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user141691
user141691

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

$\textbf{First question}$: $B_e=B_{\text{cris}}^{\phi=1}$, 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.

$\textbf{Second question}$: I know any elliptic curve over a $p$-adic field with $v_p(j)\leq 0$ is semi-stable and not crystalline, and we can construct an extension of $\mathbb{Q}_p(1)$ by $\mathbb{Q}_p$ to show it is Hodge-Tate but not de Rham. So I want to know an example that $V$ is de Rham but not semi-stable. By Fontaine's famous theorem $B$, de Rham is equivalent to potentially semi-stable. So I only need a concrete example that $V$ is potentially semi-stable but not semi-stable.

Thanks!