Bounded gaps between primes in multidimensional Hecke equidistribution problems

We prove an analogue of the classical Bombieri–Vinogradov estimate for all subsets of the primes whose distribution is determined by Hecke Grössencharaktere. Using this estimate and Maynard’s new sieve techniques, we prove the existence of infinitely many bounded gaps between primes in all such subsets of the primes. We present applications to the study of primes represented by norm forms of number fields and the number of $\mathbb{F}_p$-rational points on certain abelian varieties. In particular, for any fixed $0 \lt \epsilon \lt \frac{1}{2}$, there exist infinitely many bounded gaps between primes of the form $p = a^2 + b^2$ such that $\lvert a \rvert \lt \epsilon \sqrt{p}$. Also, we prove the existence of infinitely many bounded gaps between the primes $p \equiv 1 (\mathrm{mod} \: 10)$ for which $\lvert p + 1 - \# \mathcal{C} (\mathbb{F}_p) \rvert \lt \epsilon \sqrt{p}$, where $\mathcal{C} / \mathrm{Q}$ is the hyperelliptic curve $y^2 = x^5 + 1$.

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Received 14 April 2016

Accepted 20 July 2018

Published 25 October 2019