It is a classical result of Ribet that if an eigenform has CM the its residual projective image is "small" (cyclic or dihedral.) Is the converse true, i.e, if f is a form whose associated residual Gal representation has "small" projective image then f has CM? Thanks.
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No. For instance there are plenty of modular forms that are not CM, but are congruent mod p to CM forms or to Eisenstein series, and thus whose residual Galois representations have small image.
answered Feb 16, 2012 at 14:07
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$\begingroup$ Ah, but in a positive density of such p's? $\endgroup$ Commented Feb 16, 2012 at 18:13
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$\begingroup$ @Dror: No. If f does not have CM, there are only finitely many p for which the residual representation is small in the above sense. $\endgroup$ Commented Feb 16, 2012 at 21:02
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$\begingroup$ Thanks all. Hi Kevin, does this finite set of primes consist of those that divide the level of the form? If not, what do we know about this set? $\endgroup$ Commented Feb 17, 2012 at 14:06
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$\begingroup$ I learned this from pages 4-5 of an article of Ribet: math.berkeley.edu/~ribet/Articles/rankin.pdf. His method seems somewhat qualitative, but the remark at the top of page 6 suggests that one might be able to get more precise control on these primes, at least when $p>k+1$ and does not divide the level. $\endgroup$ Commented Feb 17, 2012 at 23:51