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

Abstract

For each finite subgroup $G$ of $PG{L}_2(\mathbb{Q})$, 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.

24 pages; added Lemma 3.8. To appear in International Journal of Number Theory