Ordering of number fields and distribution of class groups

TL;DR

This paper investigates how different orderings of number fields influence the distribution of class groups and related counting problems, establishing invariants and analyzing specific cases like abelian and cubic fields.

Contribution

It introduces an invariant of number fields with parameters to study how various orderings affect counting problems and class group distributions.

Findings

Counting abelian fields yields a main term with parameters.

Infinite moments occur for some orderings but not others in cubic field counting.

Different field orderings impact the distribution and counting results nontrivially.

Abstract

When p divides the ordering of Galois group, the distribution of the Sylow p-subgroup of Cl(K) is closely related to the problem of counting fields with certain specifications. Moreover, different orderings of number fields affect the answers of such questions in a nontrivial way. So, in this paper, we set up an invariant of number fields with parameters, and consider field counting problems with specifications while the parameters change as a variable. The case of abelian extensions shows that the result of counting abelian fields has a main term with parameters. The estimate of counting cubic fields with a parameter shows that infinite moment is true for some ordering but not very likely for the others.