We prove an analogue of the classical Bombieri-Vinogradov estimate for all subsets of the primes whose distribution is determined by Hecke Grossencharaktere. 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 Fp-rational points on certain abelian varieties. In particular, for any fixed 0 < < 1 2 , there exist infinitely many bounded gaps between primes of the form p = a2 + b2 such that jaj < p p. Also, we prove the existence of infinitely many bounded gaps between the primes p 1 (mod 10) for which jp + 1#C(Fp)j < p p, where C=Q is the hyperelliptic curve y2 = x5 + 1.
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