# Representations attached to p-adic modular forms

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?

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 $\bar Q_p$.

• Ahh this is wonderful. Thank you. As far as I can remember from Gouvea's book(quoting it from memory) he needed the condition of absolute irreducibility as he was using Mazur's result on the existence of universal deformation ring. But this completely circumvent that problem. Thanks again for your answer. Sep 28 '12 at 19:33
• Yes, I should add that the argument outlined above is essentially present (in a different context) in Richard Taylor's thesis. I am not exactly sure if it is due to Taylor or Wiles (who invented the pseudo-characters in dimension 2), or Mazur-Wiles, but assuredly it comes from this group of people in the late 80's.
– Joël
Sep 28 '12 at 20:00
• @Filipo. I apologize, I haven't seen your comment earlier. I don't know if Taylor and Wiles definition are equivalent in dim 2 and char 2. What I know is that both notions are not the good one. To summarize, Wiles' notion os the older, and is very peculiar to the situation he was studying. Taylor's notion makes senses in any dimension, and any characteristic, but the main theorem (that over an alg. closed field any character comes from a true representation) is proved only by him in char 0. (Note that this is enough for the argument of my answer to the PO's question)...
– Joël
Dec 11 '12 at 1:43
• Then come's Rouquier (and Nyssen). Rouqier's notion of pseudo-character is essentially the same as Taylor's. Ho offers a nice, self-contained treatment, and states and prove Taylor's theorem for an alg. closed field of any characteristic. Unfortunately, there is an unfixable mistake if the characteristic is less or equal than the dimension. So his theorem is proved only in char greater than dimension. The problem is in the notion of Rouquier's or Taylor's pseudocharacter/pseudorepresentation: that notion focuses on the trace of a representation, while in low characteristic, the full...
– Joël
Dec 11 '12 at 1:47
• characteristic polynomial is needed. The correct notion was developed by Chenevier, under the name "determinants", and it works fine in any characteristic. The article is on his web page.
– Joël
Dec 11 '12 at 1:48