The Poisson spacing conjecture for unfolded Sato–Tate angles

Let $E$ be an elliptic curve for which the Sato–Tate conjecture holds. If $\theta_p\in[0,\pi]$ are the Sato–Tate angles and $\rho(\theta)$ is their density, define the unfolded variables by

$$\Theta(\theta)=\int_0^{\theta}\rho(\theta')\,\mathrm{d}\theta',$$

and arrange $(\Theta(\theta_p))_{p\text{ prime}}$ as $(\Theta_1\leq\Theta_2\leq\cdots)$. For $k\geq0$, let $p_k(s)$ denote the next-$k$ nearest-neighbor spacing distribution. Spacing distribution of the unfolded angles. The distribution is Poisson:

$$p_k(s):=\lim_{X\to\infty}\mathbb{P}_X\left[\Theta_{i+k+1}-\Theta_i=\frac{s}{\#\{p\leq X\}}\right]=\frac{s^k e^{-s}}{k!}.$$

It is normalized by

$$\int_0^\infty p_k(s)\,\mathrm{d}s=1,\qquad \int_0^\infty s p_k(s)\,\mathrm{d}s=k+1.$$

This is a refinement of the Sato–Tate conjecture concerning the spacing statistics, rather than only the one-point distribution, of elliptic-curve angles. The claim is supported in the source by numerical computations; no proof or resolution is supplied.