Patterns of primes in the Sato–Tate conjecture

Abstract

Fix a non-CM elliptic curve E/Q, and let aE(p) = p + 1 − #E(Fp) denote the trace of Frobenius at p. The Sato-Tate conjecture gives the limiting distribution µST of aE(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 pI,n denote the nth prime such that aE(p)/(2 √ p) ∈ I.We show lim infn→∞(pI,n+m −pI,n) < ∞ for all m ≥ 1 for "most" intervals, and in particular, for all I with 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.

Primes with Beatty and Chebotarev conditions

“…To show that our collection of primes can satisfy these criteria, we need to adapt some lemmas from the work of Vatwani and Wong [15]. For corrections of some of the arguments in [15], see [3]. Then there are infinitely many n such that at least m + 1 of the n + h i lie in P C,α,β and…”