Patterns of primes in the Sato-Tate conjecture

Abstract

Fix a non-CM elliptic curve $E/\mathbb{Q}$, and let $a_E(p)=p+1-\#E({\mathbb{F}}_p)$ denote the trace of Frobenius at $p$. The Sato-Tate conjecture gives the limiting distribution ${\mu }_{ST}$ of $a_E(p)/(2\sqrt{p})$ within $[-1,1]$. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval $I\subseteq [-1,1]$, let $p_{I,n}$ denote the $n$th prime such that $a_E(p)/(2\sqrt{p})\in I$. We show ${\mathrm{liminf}}_{n\to \infty }(p_{I,n+m}-p_{I,n})<\infty$ for all $m\ge 1$ for "most" intervals, and in particular, for all $I$ with ${\mu }_{ST}(I)\ge 0.36$. 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.

26 pages