Patterns of primes in joint Sato–Tate distributions

Abstract

For $j=1,2$, let $f_j\left(z\right)={\sum }_{n=1}^{\infty }{a}_j\left(n\right)e^{2\pi inz}$ be a holomorphic, non-CM cuspidal newform of even weight $k_j\ge 2$ with trivial nebentypus. For each prime p, let ${\theta }_j\left(p\right)\in [0,\pi ]$ be the angle such that $a_j\left(p\right)=2p^{(k-1)/2}\cos {\theta }_j\left(p\right)$. The now-proven Sato–Tate conjecture states that the angles $\left({\theta }_j\right(p\left)\right)$ equidistribute with respect to the measure $d{\mu }_{ST}=\frac{2}{\pi }{\sin }^2\theta d\theta$. We show that, if $f_1$ is not a character twist of $f_2$, then for subintervals $I_1,I_2\subseteq [0,\pi ]$, there exist infinitely many bounded gaps between the primes p such that ${\theta }_1\left(p\right)\in I_1$ and ${\theta }_2\left(p\right)\in I_2$. We also prove a common generalization of the bounded gaps with the Green–Tao theorem.