MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
What are the results you are alluding to in the second paragraph? I don't known of any such result, at least published. – Joël Nov 30 '12 at 19:40
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. – François Brunault Nov 30 '12 at 23:28
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. – Soroosh Dec 1 '12 at 3:56

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.