When reading the books or papers on p-adic Hodge theory, non trivial example of p-adic representation seems to be only the example of Tate curves. To be sure, I had read the very readable introduction by [L.Berger](http://perso.ens-lyon.fr/laurent.berger/articles/article05.pdf) 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)? 



Examples of p-adic representations
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 Lubin-Tate 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 Hodge-Tate-Sen 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.


