Patterns of primes in the Sato–Tate conjecture

Fix a non-CM elliptic curve E / Q , and let a E ( p ) = p + 1 - # E ( F p ) denote the trace of Frobenius at p . The Sato–Tate conjecture gives the limiting distribution μ ST of a E ( p ) / ( 2 p ) within [ - 1 , 1 ] . We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval I ⊆ [ - 1 , 1 ] , let p I , n denote the n th prime such that a E ( p ) / ( 2 p ) ∈ I . We show lim inf n → ∞ ( p I , n + m - p I , n ) < ∞ for all m ≥ 1 for “most” intervals, and in particular, for all I with μ ST ( I ) ≥ 0.36 . Furthermore, we prove a common generalization of our bounded gap result with the Green–Tao theorem. To obtain these results, we demonstrate a Bombieri–Vinogradov type theorem for Sato–Tate primes.