Statistics of genus numbers of cubic fields

Abstract:

The class number is among the most important invariants associated to a number field. Conjecturally, its behavior (at the “good” primes) is governed by the heuristics of Cohen-Lenstra-Martinet. By contrast, the genus number is supported at the “bad” primes, but is easier to understand. It is very natural to ask about the density of genus number one fields among all number fields of a fixed degree and signature. We prove that approximately 96.23% of cubic fields, ordered by discriminant, have genus number one. Finally, we show that a positive proportion of totally real cubic fields with genus number one fail to be norm-Euclidean. Time permitting, I will further discuss norm-Euclidean questions and the case of quintic fields. This is joint work with Amanda Tucker.