Galois covers of $\mathbb{P}^1$ and number fields with large class groups

TL;DR

The paper constructs infinitely many Galois extensions of the rationals with specified Galois groups and large class group ranks, providing new records for class group sizes in number fields.

Contribution

It explicitly constructs infinite families of number fields with prescribed Galois groups and large class group ranks, advancing understanding of class group distribution.

Findings

Constructed Galois extensions with prescribed groups

Achieved large class group ranks in these extensions

Established new records for class group sizes

Abstract

For each finite subgroup $G$ of $PGL_2(\mathbb{Q})$, and for each integer $n$ coprime to $6$, we construct explicitly infinitely many Galois extensions of $\mathbb{Q}$ with group $G$ and whose ideal class group has $n$-rank at least $\#G-1$. This gives new $n$-rank records for class groups of number fields.