Denote by $G\_p$ a choice of an absolute Galois group of $Q\_p$, the field of $p$-adic numbers. Consider a continuous representations of $G\_p$ on a $3$-dimensional $Q\_p$ vector space that is a successive extension of the trivial character, the cyclotomic character ($\chi\_p$) and the square of the cyclotomic character, i.e. there are exact sequences of $G\_p$-modules :


$$0 \to Q\_p(2) \to V \to W \to 0 $$ and $$0 \to U \to V \to Q\_p \to 0$$


where $W$ is an extension $0 \to Q\_p(1) \to W \to Q\_p \to 0$ and $U$ is an extension $0 \to Q\_p(2) \to U \to Q\_p(1) \to 0$. Here $Q\_p(n)$ denote a 1 dimensional $Q\_p$ vector space endowed with an action of $G\_p$ via the $n$-th power of the cyclotomic character.


Assume $U$ and $W$ to be crystalline, that is the $Q\_p$-vector spaces $D\_{crys}(U) = (B\_{crys} \otimes\_{Q\_p} U)^{G\_p}$ and $D\_{crys}(W) = (B\_{crys} \otimes\_{Q\_p} W)^{G\_p}$ are both of dimension 2 ($B\_{crys}$ is one of the ring of $p$-adic periods introduced by Fontaine).


In this case, is $V$ crystalline ?


It is the case if and only if the cocycle $[V] \in H^1(G\_p, U)$ determined by (the extension determined by) $V$ is in the kernel of the map $H^1(G\_p, U) \to H^1(G\_p, B\_{crys}\otimes\_{Q\_p}U)$, which is usually denoted by $H^1\_{crys}(G\_p, U)$.


One can show that in our case $H^{1}\_{crys}(G\_p,U)$ is 2 dimensional (there is a formula involving the dimension of $U$, the dimension of its invariants under $G\_p$ and the Hodge-Tate weights). Now $H^{1}\_{crys}(G\_p,Q\_p(2))$ and $H^{1}\_{crys}(G\_p,Q\_p)$ are both 1 dimensional over $Q\_p$ so we have an exact sequence of $Q\_p$ vector spaces $$0 \to H^{1}\_{crys}(G\_p,Q\_p(2)) \to H^{1}\_{crys}(G\_p,U) \to H^{1}\_{crys}(G\_p,Q\_p) \to 0.$$


From there I don't know if there is more one can say...



extensions of crystalline representations
One has a natural exact sequence: $$0 \rightarrow H^1(G\_p, \mathbb{Q}\_p(2)) \rightarrow H^1(G\_p,U) \xrightarrow{s} H^1(G\_p,\mathbb{Q}\_p(1)) \rightarrow 0,$$ where $ H^1(G\_p, \mathbb{Q}\_p(2)) \cong H^1\_{crys}(G\_p,\mathbb{Q}\_p(2))$, $\dim H^1(G\_p,U)=\dim H^1\_{crys}(G\_p,U)+1$, $\dim H^1(G\_p, \mathbb{Q}\_p(1))=\dim H^1\_{crys}(G\_p, \mathbb{Q}\_p(1))+1$. So $s^{-1}\big(H^1\_{crys}(G\_p, \mathbb{Q}\_p(1))\big)=H^1\_{crys}(G\_p,U)$. In other words, if $U$ is crystalline, then $V$ is crystalline $\Leftrightarrow$ $W$ is crystalline.


