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If $f$ is a classical eigenform of weight $\geq 2$ and ordinary at distinct, odd primes $p$ and $q$ which do not divide the level is it true that the restriction (as a $q$-adic representation) $\rho_{f,q}|G_q$ splits (i.e.is diagonal w.r.t some basis) iff $\rho_{f,p}|G_p$, considered as a $p$-adic representation, splits?

One expects this to be true from the application of the "splitting implies CM" conjecture followed by its converse (which is well-known to be true) but is there a direct way of seeing this result...via compatible systems maybe? Thanks.

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    $\begingroup$ Your proposed equivalence obviously amounts to one implication, and this is expected by the reason that you say, and in fact, by an application of Cebotarev (I believe) equivalent to the "splitting implies CM" conjecture. So the answer to your question would be: it is expected that to be so (i.e. true), but apparently no one knows yet how to prove it in general. $\endgroup$
    – monodromy
    Commented Feb 15, 2012 at 4:03
  • $\begingroup$ Thanks monodromy. Could you possibly expand on the equivalence using Cebotarev (a reference perhaps?) $\endgroup$
    – unramified
    Commented Feb 15, 2012 at 6:29
  • $\begingroup$ Oh wow, I haven't heard of this "split implies CM" conjecture. Is there some reference to this? Maybe a name? $\endgroup$
    – Dror Speiser
    Commented Feb 15, 2012 at 9:04
  • $\begingroup$ @Dror: I've seen this conjecture in papers of Pollack and Stevens on p-adic modular symbols; I don't know if that's where it first originated. $\endgroup$
    – David Loeffler
    Commented Feb 15, 2012 at 10:41
  • $\begingroup$ The conjecture is attributed to Ralph Greenberg although Eknath Ghate and Nike Vatsal have done the most work on it. One can find 3 or 4 papers on work related to this problem on Ghate's page. Emerton has quite a famous preprint "A p-adic variational Hodge conjecture and modular forms with complex multiplication" where he shows how this conjecture follows from the conjecture in the title of the paper. $\endgroup$
    – unramified
    Commented Feb 15, 2012 at 11:12

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