Patterns of primes in the Sato-Tate conjecture
Abstract Fix a non-CM elliptic curve $$E/\mathbb {Q}$$E/Q, and let $$a_E(p) = p + 1 - \#E(\mathbb {F}_p)$$aE(p)=p+1-#E(Fp) denote the trace of Frobenius at p. The Sato-Tate conjecture gives the limiting distribution $$\mu _{ST}$$μST of $$a_E(p)/(2\sqrt{p})$$aE(p)/(2p) within $$[-1, 1]$$[-1,1]. We es...
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Springer International Publishing,
2021-09-20T17:17:13Z.
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LEADER | 01474 am a22001813u 4500 | ||
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001 | 131475 | ||
042 | |a dc | ||
100 | 1 | 0 | |a Gillman, Nate |e author |
700 | 1 | 0 | |a Kural, Michael |e author |
700 | 1 | 0 | |a Pascadi, Alexandru |e author |
700 | 1 | 0 | |a Peng, Junyao |e author |
700 | 1 | 0 | |a Sah, Ashwin |e author |
245 | 0 | 0 | |a Patterns of primes in the Sato-Tate conjecture |
260 | |b Springer International Publishing, |c 2021-09-20T17:17:13Z. | ||
856 | |z Get fulltext |u https://hdl.handle.net/1721.1/131475 | ||
520 | |a Abstract Fix a non-CM elliptic curve $$E/\mathbb {Q}$$E/Q, and let $$a_E(p) = p + 1 - \#E(\mathbb {F}_p)$$aE(p)=p+1-#E(Fp) denote the trace of Frobenius at p. The Sato-Tate conjecture gives the limiting distribution $$\mu _{ST}$$μST of $$a_E(p)/(2\sqrt{p})$$aE(p)/(2p) within $$[-1, 1]$$[-1,1]. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval $$I\subseteq [-1, 1]$$I⊆[-1,1], let $$p_{I,n}$$pI,n denote the nth prime such that $$a_E(p)/(2\sqrt{p})\in I$$aE(p)/(2p)∈I. We show $$\liminf _{n\rightarrow \infty }(p_{I,n+m}-p_{I,n}) < \infty $$lim infn→∞(pI,n+m-pI,n)<∞ for all $$m\ge 1$$m≥1 for "most" intervals, and in particular, for all I with $$\mu _{ST}(I)\ge 0.36$$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. | ||
546 | |a en | ||
655 | 7 | |a Article |