Patterns of primes in the Sato–Tate conjecture

@article{Gillman2019PatternsOP,
title={Patterns of primes in the Sato–Tate conjecture},
author={Nate Gillman and Michael Kural and Alexandru Pascadi and Junyao Peng and Ashwin Sah},
journal={Research in Number Theory},
year={2019}
}
• Published 18 July 2019
• Mathematics
• Research in Number Theory
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 $\liminf_{n\to\infty}(p_{I,n+m}-p_{I,n}) < \infty… 1 Citations • Mathematics Journal of Number Theory • 2020 References SHOWING 1-10 OF 56 REFERENCES • Mathematics • 2013 Let$f(z)=\sum_{n=1}^\infty a(n)q^n\in S^{\text{new}}_ k (\Gamma_0(N))$be a newform with squarefree level$N$that does not have complex multiplication. For a prime$p$, define$\theta_p\in[0,\pi]$Let$H(z)$be a newform of weight$k \geq 4$without complex multiplication on$\Gamma_{0}(N)$with normalized$L$-function$L(H,s) = \prod_{p} (1 - \alpha_{p} p^{-s})^{-1} (1 - \beta_{p}
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