Let $\rho(f):G_\mathbb{Q} \rightarrow GL_2(K_f)$ be the Galois representation attached to some cuspidal modular form $f$ where $K_f$ is a finite extension of $\mathbb{Q}_p$. Let $V_f$ be the vector space of this representation with Tate module $T_f$. Denote $A_f=V_f/T_f$. Let $S$ be the set of ramified primes for the representation (primes dividing p and the level of the modular form). Let $S$ also contain the infinite primes. Let $\mathbb{Q}_S$ be the maximal extension of $\mathbb{Q}$ unramified outside $S$.
I would like to know if the following two modules are finite:
$H^0(\mathbb{Q}_S/\mathbb{Q}(\mu_{p^\infty}),A_{f})$
${\oplus_{v \in S}H^0(\mathbb{Q}(\mu_{p^\infty})_v,A_{f})}$
Here $\mathbb{Q}(\mu_{p^\infty})_v$ is the completion of $\mathbb{Q}(\mu_{p^\infty})$ at prime $v$.
I won't mind if someone gives me references for the cyclotomic $\mathbb{Z}_p$extension of $\mathbb{Q}$ instead of $\mathbb{Q}(\mu_{p^\infty})$.
I know that these results are true for elliptic curves (with good reduction at $p$) and abelian varieties (results of Imiai and Ribet).
I know that both the statements are true for cuspidal modular forms with good ordinary reduction at $p$.
I know that for $v=p$, (2) is true for cuspidal modular forms with supersingular reduction at $p$ with $a_p(f)=0$.
I don't know the references for the other cases. Is there a unified proof of these "classical" facts for all cuspidal modular forms somewhere in the literature? I will be grateful if someone gives me references of these results for all cuspidal modular forms.
