When reading the books or papers on padic Hodge theory, non trivial example of padic representation seems to be only the example of Tate curves. To be sure, I had read the very readable introduction by L.Berger and there are some calculations for elliptic curves by ($\varphi$, Fil)modules. But, are there more concrete examples (coming from given equations) which one can calculate by hand(s)?
Not sure what you mean by "coming from given equations". If you're looking for concrete examples, you can start with abelian representations of $G_K = Gal(K^{alg}/K)$ where $K$ is a finite extension of $Q_p$. Choose a LubinTate character $\chi_K : G_K \to O_K^\times$ and see what kinds of representations you can get by looking at $\eta \circ \chi_K$ where $\eta : O_K^\times \to O_L^\times$ is a character, with $L/K$ a finite extension, and you map $O_L^\times \to GL_d(Z_p)$ using a basis of $O_L$. What HodgeTateSen weights can you get this way? Then what extensions can there be between such representations? Even the abelian representations can teach you quite a bit.

$\begingroup$ Thank you very much. To be sure, they are just abelian representations but I can study lots. What kinds of references do you recommend for the LubinTate characters? $\endgroup$ – J.S.R. Feb 2 '16 at 11:14
Any $p$adic modular form has a $p$adic representation and any classical modular form has padic reprsentation such that it is potentially semistable at $p$.
Generally, any proper smooth scheme $X$ over $\mathbb{Q}$ yields a $p$adic representation given by the action of $G_\mathbb{Q}$ on the $p$adic etale cohomology $H^{i}(X\times \overline{\mathbb{Q}}, \mathbb{Z}_p) \otimes \mathbb{Q}_p$, and it is always potentially semistable at $p$.


$\begingroup$ Example of what ?? there is a lot of examples of $(\varphi,\Gamma)$module but what do you mean by $(\varphi, Fil)$? what is $Fil$? $\endgroup$ – Adel BETINA Feb 1 '16 at 8:38