Suppose one has an elliptic curve $E$ over $\mathbb{Q}$ with conductor $N < k^3$ for some (large) positive $k$, with the property that its Fourier coefficients satisfy $$ a_p=0, \; \mbox{ for all } \; p \equiv -1 \mod{4}, \; \; \; k/3 < p < k. $$ Can one prove unconditionally that $E$ has CM? This follows from work of Serre (under GRH) or Elkies (under something like Szpiro's conjecture), since otherwise we'd have a surplus of supersingular primes. It does not appear to be a consequence of, say, Serre's argument without additional hypotheses (though I'd be happy to be wrong on this score).

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    $\begingroup$ Dear Mike: Can one make "effective" (in the spirit of your question) any information coming from the knowledge that Sato-Tate holds in the non-CM case? (Sorry to answer a question with another question...). $\endgroup$ – BCnrd Dec 20 '10 at 8:18
  • $\begingroup$ Dear BCrd : I've pondered that, but I really don't know. My admittedly uneducated guess would be "probably not" with the level of precision I'm after.... $\endgroup$ – Mike Bennett Dec 20 '10 at 9:57

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