Unramified extensions of quadratic number fields with Galois group $SL_2(7)$

TL;DR

This paper constructs an infinite family of quadratic number fields with unramified Galois extensions having Galois group $SL_2(7)$, addressing a complex inverse Galois problem for a non-involution-generated perfect group.

Contribution

It presents the first known infinite family of quadratic fields with unramified extensions having Galois group $SL_2(7)$, a significant advance in inverse Galois theory for such groups.

Findings

Infinite family of quadratic fields with unramified $SL_2(7)$ extensions

First such realization for a perfect group not generated by involutions

Addresses delicate local-global problems in inverse Galois theory

Abstract

We provide an infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group $SL_2(7)$. To my knowledge, this is the first instance of infinitely many such realizations for a perfect group which is not generated by involutions, a property which makes it difficult to approach for the problem in question and leads to somewhat delicate local-global problems in inverse Galois theory.