# Question on the Sato-Tate conjecture

Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM. For a good prime $p$, define $\theta_{E}(p)$ by $$\cos\theta_{E}(p)=\frac{p+1-N_{p}(E)}{2\sqrt{p}}\quad (0\leq \theta_{E}(p)\leq \pi).$$ I wonder if the function $\theta_{E}$ is injective?

No. If $E_p$ is a supersingular elliptic curve and $p>3$ then trace of Frobenius on $E_p$ is zero, so $\theta_E(p)=\pi/2$.
By a result of Elkies any elliptic curve over $\mathbb{Q}$ has supersingular reduction in infinitely many primes, so for infinitely many $p$ this function takes the same value.
• On the other hand, this is the only answer. If $(p+1-N)/\sqrt{p} = (q+1-M)/\sqrt{q}$ with $p+1-N$ and $q+1-M$ nonzero, then $\sqrt{p/q}$ is rational, which it isn't. So the only solutions are when $p+1-N=q+1-M=0$, the supersingular case. – David E Speyer Oct 10 '16 at 13:26