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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.

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    $\begingroup$ Do you want to ask "is there infinitely many prime numbers $p$ such that $0<|\frac{a_p(E)}{2\sqrt{p}}|<\epsilon$", as in the title ? $\endgroup$
    – Watson
    Commented Jun 22, 2018 at 15:14
  • $\begingroup$ @Watson Thanks Watson, certainly yes $\endgroup$
    – YC ZHOU
    Commented Jun 22, 2018 at 15:16
  • $\begingroup$ This would also follow from the Lang–Trotter conjecture. So for example, the answer is "yes" for almost all elliptic curves, by work of David and Pappalardi. $\endgroup$
    – Greg Martin
    Commented Jun 22, 2018 at 17:12
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    $\begingroup$ Knowing that there are infinitely many supersingular primes ($a_p=0$) is enough. If the base field is $\mathbf{Q}$ (which seems implicit in the question), this is a famous result of Elkies. $\endgroup$
    – Denis Chaperon de Lauzières
    Commented Jun 22, 2018 at 20:43
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    $\begingroup$ @DenisChaperondeLauzières The question in the subject asks for non-zero values, and the question already mentions Elkies result. And the comment by Watson and response by the OP indicates he really want $a_p\ne0$. $\endgroup$
    – Joe Silverman
    Commented Jun 23, 2018 at 0:05

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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.

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