Patterns of primes in joint Sato–Tate distributions

Abstract

For j=1,2, let fj(z)=∑n=1∞aj(n)e2πinz be a holomorphic, non-CM cuspidal newform of even weight kj≥2 with trivial nebentypus. For each prime p, let θj(p)∈[0,π] be the angle such that aj(p)=2p(k−1)/2cos⁡θj(p). The now-proven Sato–Tate conjecture states that the angles (θj(p)) equidistribute with respect to the measure dμST=2πsin2⁡θdθ. We show that, if f1 is not a character twist of f2, then for subintervals I1,I2⊆[0,π], there exist infinitely many bounded gaps between the primes p such that θ1(p)∈I1 and θ2(p)∈I2. We also prove a common generalization of the bounded gaps with the Green–Tao theorem.