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Patterns of primes in the Sato-Tate conjecture. (English) Zbl 1456.11174

Summary: 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 \(\liminf_{n\rightarrow \infty }(p_{I,n+m}-p_{I,n}) < \infty\) 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.

MSC:

11N05 Distribution of primes
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11G05 Elliptic curves over global fields
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