Patterns of primes in the Sato–Tate conjecture

Abstract

Abstract Fix a non-CM elliptic curve $$E/\mathbb {Q}$$E/Q, and let $$a_E(p) = p + 1 - \#E(\mathbb {F}_p)$$aE(p)=p+1-#E(Fp) denote the trace of Frobenius at p. The Sato–Tate conjecture gives the limiting distribution $$\mu _{ST}$$μST of $$a_E(p)/(2\sqrt{p})$$aE(p)/(2p) within $$[-1, 1]$$[-1,1]. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval $$I\subseteq [-1, 1]$$I⊆[-1,1], let $$p_{I,n}$$pI,n denote the nth prime such that $$a_E(p)/(2\sqrt{p})\in I$$aE(p)/(2p)∈I. We show $$\liminf _{n\rightarrow \infty }(p_{I,n+m}-p_{I,n}) < \infty $$lim infn→∞(pI,n+m-pI,n)<∞ for all $$m\ge 1$$m≥1 for “most” intervals, and in particular, for all I with $$\mu _{ST}(I)\ge 0.36$$μST(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.