Statistics of the Genus Number of $S_3 \times C_q$ and $D_4$-fields

TL;DR

This paper studies the distribution and statistical properties of the genus number across various families of number fields, including $S_3 imes C_q$, $D_4$, and pure quartic fields, providing average, higher moments, and conjectures.

Contribution

It establishes the statistics of genus numbers for specific families of number fields and proposes conjectures on genus density based on heuristics.

Findings

Derived the average and higher moments of genus distribution in $S_3 \times C_q$-fields.

Provided precise results on genus numbers for $D_4$-fields and pure quartic fields.

Formulated conjectures predicting zero genus density in certain families.

Abstract

The genus number of a number field is a fundamental invariant which measures the contribution of ramification to its ideal class group. In this paper, we establish the statistics for the genus number for $S_3\times C_q$-fields for $q\neq 3$ a prime number, $D_4$-fields and pure quartic fields. We also obtain precise results on the average and higher moments of the genus distribution within the family of $S_3\times C_q$-fields. Finally, based on heuristics, we formulate a conjecture identifying families for which one should expect the genus density to be zero, i.e., only a density zero subset of fields in the family attains any fixed genus number.