# Galois representation associated with p-ordinary modular form.

Let V be the two dimensional $$\mathbb{Q}_p$$-vector space associated to $$\rho_f$$(where $$f$$ is a p-ordinary form). Then we can choose a $$G_{\mathbb{Q}}$$-invariant lattice T $$\subset$$ V. Set A= V/T.

Question 1: Why the action of $$G_\mathbb{Q}$$ on A is discrete?
Question 2: Why A has a filtration as a $$G_p$$(decomposition group at p) module?

For any $$v\in V$$, the orbit map $$g\mapsto gv:G_{\mathbf{Q}}\to V$$ is continuous; if we follow this with the continuous surjection $$V\to V/T$$, we infer that the orbit map $$g\mapsto g(v+T):G_{\mathbf{Q}}\to V/T$$ is continuous as well. Because $$T$$ is an open subgroup of $$V$$, $$V/T$$ is discrete. Therefore the inverse image of the singleton $$\{v+T\}\subseteq V/T$$ under the aforementioned orbit map is an open neighborhood of the identity in $$G_{\mathbf{Q}}$$. The Krull topology on $$G_{\mathbf{Q}}$$ is determined by the condition that the subgroups of the form $$G_K$$, where $$K/\mathbf{Q}$$ is a finite extension, form basis of open neighborhoods of the identity. So there is an open subgroup $$G_K$$ of $$G_{\mathbf{Q}}$$ such that $$g(v+T)=v+T$$ for every $$g\in G_K$$, i.e., that $$v+T\in (V/T)^{G_K}$$. This is usually what is meant by the phrase "the action of $$G_\mathbf{Q}$$ is discrete" (it amounts to the continuity of the action of $$G_{\mathbf{Q}}$$ on $$V/T$$).
The existence of an "ordinary filtration" on $$V$$ (which gives rise to a similar filtration on $$V/T$$) is (as far as I know) due to Mazur and Wiles ("On $$p$$-adic analytic families of Galois representations"). A more general but also more explicit statement can be found in Theorem 2.1.4 of Wiles's "On ordinary $$\lambda$$-adic representations associated to modular forms."