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."


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