# Patterns of primes in the Sato-Tate conjecture

## TL;DR

This paper proves bounded gaps between primes where the normalized Frobenius traces of a non-CM elliptic curve fall within certain intervals, extending classical prime gap results to the Sato-Tate distribution context.

## Contribution

It establishes bounded gaps for primes in the Sato-Tate distribution and generalizes the Green-Tao theorem to these primes, supported by a Bombieri-Vinogradov type theorem.

## Key findings

- Bounded gaps for primes in the Sato-Tate distribution for most intervals.
- Extension of Green-Tao theorem to Sato-Tate primes.
- Development of a Bombieri-Vinogradov type theorem for these primes.

## 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 $\liminf_{n\to\infty}(p_{I,n+m}-p_{I,n}) < \infty$ 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.