# 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)?

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.

• 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 ? Jun 22 '18 at 15:14
• @Watson Thanks Watson, certainly yes Jun 22 '18 at 15:16
• 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. Jun 22 '18 at 17:12
• 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. Jun 22 '18 at 20:43
• @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 $a_p\ne0$. Jun 23 '18 at 0:05

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.