Quantitative inverse Galois problem for semicommutative finite group schemes

TL;DR

This paper establishes a lower bound on the number of connected torsors for a specific class of finite group schemes, advancing understanding in the inverse Galois problem for these algebraic structures.

Contribution

It introduces a lower bound estimate for connected torsors of semicommutative finite group schemes, extending inverse Galois theory to new algebraic contexts.

Findings

Lower bound on the number of connected G-torsors of bounded height.

Application to the inverse Galois problem for semicommutative finite group schemes.

Abstract

A semicommutative finite group scheme is a finite group scheme which can be obtained from commutative finite group schemes by iterated performing semidirect products with commutative kernels and taking quotients by normal subgroups. In this article, for an \'etale tame semicommutative finite group scheme $G$, we give a lower bound on the number of connected $G$-torsors of bounded height (such as discriminant).