TITLE: Is there infinitely many prime $p$ such that the normalized trace of Frobenius $\frac{a_p(E)}{2\sqrt{p}}$ is arbitrarily small (but not zero)?
QUESTION [5 upvotes]: I just encountered the following problem and failed to find proper reference, could anyone kindly help to give me some illustration?
For an elliptic curve $E$ without complex multiplication (just like in the Sato-Tate conjecture), let $a_p(E)$ be the trace of Frobenius,  given an arbitrary small  positive number $\epsilon$, is there infinitely many prime numbers $p$ such that $|\frac{a_p(E)}{2\sqrt{p}}|<\epsilon$?
With the proof of Sato-Tate conjecture, the answer should certainly be yes, but what I am wondering is, without the result of Sato-Tate conjecture, do we still have that conclusion? If it is, could you kindly give me some reference? I already knew the supersingular case by Elkies theorem.
With many thanks.

REPLY [4 votes]: The answer is that the proof of such theorems do indeed require the full force of Sato-Tate. For example, if you take a general elliptic curve E defined over a number field $F$ which is not CM, then your problem is still open.