# 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 $\liminf_{n\to\infty}(p_{I,n+m}-p_{I,n}) < \infty… 1 Citations ## References SHOWING 1-10 OF 56 REFERENCES • Mathematics • 2013 Let$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}
• Mathematics
• 2015
Let $K/\mathbb{Q}$ be a number field. Let $\pi$ and $\pi^\prime$ be cuspidal automorphic representations of $\mathrm{GL}_d(\mathbb{A}_K)$ and $\mathrm{GL}_{d^\prime}(\mathbb{A}_K)$, and suppose that
• Mathematics
Compositio Mathematica
• 2014
Abstract As the simplest case of Langlands functoriality, one expects the existence of the symmetric power $S^n(\pi )$, where $\pi$ is an automorphic representation of ${\rm GL}(2,{\mathbb{A}})$ and
• Mathematics
Mathematische Zeitschrift
• 2018
We give a simple proof of a standard zero-free region in the t-aspect for the Rankin–Selberg L-function $$L(s,\pi \times \widetilde{\pi })$$L(s,π×π~) for any unitary cuspidal automorphic
• Mathematics
• 2002
In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on GL2 × GL3 as automorphic forms on GL6, from which we obtain our second case, the
• Mathematics
• 2004
In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on $GL_2\times GL_3$ as automorphic forms on $GL_6$, from which we obtain our second
• Mathematics
• 1999
In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E/Q is an elliptic curve, then E is modular.
• Mathematics
• 1999
where is the reduction of p. These results were subject to hypotheses on the local behavior of p at X, i.e., the restriction of p to a decomposition group at X, and to irreducibility hypotheses on
After the proof of Zhang about the existence of infinitely many bounded gaps between consecutive primes the author showed the existence of a bounded d such that there are arbitrarily long arithmetic