Representations attached to p-adic modular forms

Question

A theorem of Gouvea and Hida (or rather a consequence of it) states that there exist a Galois representation attached to a $p$-adic eigenform $f$ provided the residual representation attached to a classical eigenform $g$ congruent to $f$ is absolutely irreducible. I believe there might be generalizations due to Skinner-Wiles. So my question is: do there always exist a Galois representation attached to an ordinary $p$-adic modular form? What if $g$ has some extra properties like CM but I still want the residual representation to be reducible?

Answer 1

I am not sure this answer will satisfy you totally because I am not sure what you mean exactly by $p$-adic modular forms. But at least, for a $p$-adic modular form $f$ in the sense of Serre, which is an eigenform for almost all the Hecke operators $T_\ell$ (with eigenvalue $a_\ell)$ there always exists a semi-simple Galois representation $r:G_{\mathbb Q} \rightarrow Gl_2( \bar Q_p)$ such that $tr(\rm Frob_\ell)=a_\ell$ for almost all $\ell$. This is quite easy to prove with modern techniques which were not available at the time Serre, Hida, and Gouvêa worked on the subject. Here is how:

Lemma : for every classical normalized form $g$ of weight $k$ and level $\Gamma$ with coefficient in $\mathbb Z_p$ such that $T_\ell g \equiv a_\ell g \pmod{p^n}$ for almost all $\ell$, there exists a unique continuous pseudo-character $T_g : G_{\mathbb Q} \rightarrow \mathbb{Z}/p^n \mathbb{Z}$ such that $T_g(\rm Frob_\ell) = a_\ell$ in $\mathbb Z/p^n \mathbb Z$

Proof: consider the Hecke algebra $A$ generated by the Hecke operators $T_\ell$ acting on $M_k(\Gamma,\mathbb Z_p)$ (the module of modular forms of weight $k$, level $\Gamma$, coefficients in $\mathbb Z_p$). There is pseudo-character $T : G_Q \rightarrow A$ such that $T(Frob_\ell)=T_\ell$. To prove this, first prove it with $A$ replaced by $A \otimes_{\mathbb Z_p} K$ where $K$ is is some sufficiently large extension of $\mathbb Q_p$. Then $A \otimes_{\mathbb Z_p} K = K^d$ where each factor corresponds to an eigenform in $M_k(\Gamma,K)$. Hence to construct $T : G \rightarrow K^d$, one just takes the sum of the trace of the representations attached to these eigenforms. Then $T(\rm Frob_\ell)=T_\ell$ by construction, so $T$ takes values in $A$ on a dense subset of $G_{\mathbb Q}$, so $T$ takes values in $A$ everywhere since $A$ is closed in $A \otimes_{Z_p} K$. To finish the proof of the existence part of the lemma, just compose $T$ with the morphism of ring $A \rightarrow \mathbb Z/p^n \mathbb Z$ which sends $T_\ell$ on $a_\ell$. The uniqueness is clear by Cebotarev.

Now, back to the main claim, for every $n$, $f$ is by definition congruent to a classical form $g$ mod $p^n$, hence by the lemma we get a pseudo-character $T_n: G_{\mathbb Q} \rightarrow \mathbb Z/p^n \mathbb Z$ such that $T_n(\rm Frob_\ell)$ $ = a_\ell \pmod{p^n}$. By uniqueness, we can glue those pseudo-characters into
a pseudo-character $T_\infty: G_{\mathbb Q} \rightarrow \mathbb Z_p \rightarrow \mathbb Q_p$ and by a theorem of Taylor, this pseudo-character is the trace of a representation over $\bar Q_p$.