Harmonic analysis and counting field extensions

The Hermite–Minkowski theorem states that there are only finitely many finite extensions of Q, up to isomorphism, with bounded discriminant. We are interested in counting them. Focusing on Galois abelian extensions, we present an approach to this question using harmonic analysis. While the approach extends naturally to the context of finite group schemes and more general height functions, we restrict in this talk to finite abelian groups and the discriminant.

The talk is based on a joint work with Takehiko Yasuda.