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.