Let $K$ be a finite extension of $\mathbb{Q_p}$. The group $\ker H^1(G_K, \mathbb{Q}_p) \rightarrow H^1(G_K, B_{crys})$ is one-dimensional, which tells us that among all extensions of Galois modules
$$
0 \rightarrow \mathbb{Q}_p \rightarrow E \rightarrow \mathbb{Q}_p \rightarrow 0,
$$
which are classified by $H^1(G_K, \mathbb{Q}_p) \cong Hom_{cts}(K^*, \mathbb{Q}_p)$, there is a one-dimensional subspace corresponding to crystalline representations.
My question is basically: What homomorphism $K^* \rightarrow \mathbb{Q}_p$ does this crystalline cocycle correspond to?
Actually, I know the answer. It was explained to me that a theorem of Sen asserts that a crystalline representation with only $0$ as Hodge-Tate weight must have finite image of inertia, which in this case must be trivial, i.e. unramified. So the homomorphism in question is (a scalar multiple of) $x \mapsto val_p(x)$.
I would like to come to a more elementary and concrete understanding of this puzzle. The fact that this is a coboundary in $H^1(G_K, B_{crys})$ means that I can find $b \in B_{crys}$ such that inertia acts trivially on $b$ and Frobenius translates $b$ by $1$. Is it possible to describe this period explicitly in terms of the construction of $B_{crys}$?
As someone who has never really understood the definition of $B_{crys}$, it would be even more satisfying to me to have an explicit description in terms of elements which I believe should belong in $B_{crys}$ if my intuitive definition of the latter is merely “the ring of periods for cohomology of varieties with good reduction''. In other words, I would love to see this period $b$ expressed in terms of periods coming from cohomology of familiar varieties, such as the period for the cyclotomic character.
Best Answer
The representation $E$ in this case is not only crystalline, it is in fact unramified. This means we don't need much of the complicated machinery of $p$-adic Hodge theory to get a handle on the periods of $E$.
Whereas for general crystalline representations we need to use $\mathbf{B}_\text{cris}$ to find periods, for potentially unramified representations, we can work with $\mathbb{C}_p$ (potentially unramified representations are $\mathbb{C}_p$-admissible). In this case, we don't even need $\mathbb{C}_p$, as the representation is unramified (not just potentially so), and it suffices to work with $(K_0^\text{nr})^\vee$ instead (where $K_0$ is the maximal unramified intermediate extension $K / K_0 / \mathbb{Q}_p$).
To explicitly see that $E$ is $(K_0^\text{nr})^\vee$-admissible, we can start by taking $a$ to be a solution to the Artin–Schreier equation $x^q-x-1=0$ in $\mathcal{O}_K/\mathfrak{m} \cong \mathbb{F}_q$. We then have $\mathrm{Frob}([a]) = [a^q] = [a+1]$, where $[-]$ indicates Teichmüller lifts. This is nearly what we want, save for the fact that the Teichmüller lift is not additive. So you have to remedy that by hand by using the Witt addition polynomials; the upshot is that you'll obtain some element $b \in \mathrm{W}(\overline{\mathbb{F}_q})$ with $\varphi(b) = b+1$ as desired. It is necessary to go all the way up to $\mathrm{W}(\overline{\mathbb{F}_q})$; at any finite level $\mathrm{W}(\mathbb{F}_{q^n})$ there will always be the above issue of non-additivity. Indeed, only potentially trivial representations are detected at finite levels, and we need to pass to the completion $(K_0^\text{nr})^\vee$ to allow a Frobenius of infinite order.
At any rate, this means then that $\{1,b\}$ is a $(K_0^\text{nr})^\vee$-basis of $E \otimes_{\mathbb{Q}_p} (K_0^\text{nr})^\vee$. Using the inclusion $(\mathcal{O}_{K_0^\text{nr}})^\vee \subset \mathbf{A}_\text{cris}$ we can consider $b$ to be an element of $\mathbf{B}_\text{cris}$. It is a crystalline period of $E$; together with $1 \in \mathbf{B}_\text{cris}$ it provides a $\textbf{B}_\text{cris}$-basis of $E \otimes_{\mathbb{Q}_p} \mathbf{B}_\text{cris}$.
As for the motivic question, I think no such variety is expected to exist. We are in the crystalline situation, so we'd like to be able to find $E$ inside the cohomology of some (smooth, separated, finite type) scheme over $\mathcal{O}_K/\mathfrak{m}$. By analogy with the $\ell$-adic and complex situations, I believe such extensions as $E$ shouldn't occur there, because of weight filtration considerations (e.g. there are no non-split extensions of mixed Hodge structures of $\mathbb{Q}$ by $\mathbb{Q}$).