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Let $f$ be a modular form (say of weight 2) and let $S$ be the set of primes $p$, the reductions modulo $p$ of $\rho_{f,p}$ restricted to $G_p$ splits. Equivalently, $S$ is the set of prime $p$ such that $f$ has a companion form modulo $p$. Now if $f$ has CM then $S$ is an infinite set. Is the converse true? That is, does there exist a modular form $f$ without CM such that $S$ is infinite?

I think that there are results that says that if the $p$-adic representation $\rho_{f,p}$ restricted to $G_p$ is split then $f$ has CM, but I don't know how (or if) that can be helpful for this problem.

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  • $\begingroup$ What are the results you are alluding to in the second paragraph? I don't known of any such result, at least published. $\endgroup$
    – Joël
    Commented Nov 30, 2012 at 19:40
  • $\begingroup$ There is a nice discussion regarding the relation between splitting of local Galois representations and CM forms in Ghate's article On the local behaviour of ordinary modular Galois representations. $\endgroup$
    – François Brunault
    Commented Nov 30, 2012 at 23:28
  • $\begingroup$ I came across that claim from Rajender Adibhatla's thesis, who references Ghate's article from Francois comment. Ghate proves this in section 2, but gives references to others who proved parts of it before. $\endgroup$
    – Soroosh
    Commented Dec 1, 2012 at 3:56

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