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.

equivalentto 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. – monodromy Feb 15 '12 at 4:03