A variant of the Bombieri-Vinogradov theorem in short intervals and some questions of Serre

We generalise the classical Bombieri-Vinogradov theorem for short intervals to a non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are twisted by a splitting condition in a Galois extension of number fields. Using this result in conjunction with the recent work of Maynard, we prove that rational primes with a given splitting condition in a Galois extension L/ exhibit bounded gaps in short intervals. We explore several arithmetic applications related to questions of Serre regarding the non-vanishing Fourier coefficients of cuspidal modular forms. One such application is that for a given modular L-function L(s, f), the fundamental discriminants d for which the d-quadratic twist of L(s, f) has a non-vanishing central critical value exhibit bounded gaps in short intervals.