Examples of p-adic representations

Question

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 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)?

Answer 1

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.

Answer 2

Any $p$-adic modular form has a $p$-adic representation and any classical modular form has p-adic reprsentation such that it is potentially semi-stable 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 semi-stable at $p$.