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

TL;DR

This work advances the unramified inverse Galois problem by constructing infinitely many quadratic fields that admit everywhere unramified extensions with Galois group $SL_2(7)$. The approach blends function-field pullbacks from $PGL_2(p)$-extensions with central embedding-problem solvability, leveraging local ramification control via results on specialization and a local-global principle. A concrete, explicit family for $p=7$ is provided, together with a broader, non-explicit existence result for infinitely many primes $p$ with similar phenomena, and an accompanying discussion of the method's limitations and potential extensions. The findings enrich our understanding of realizing non-involution-generated quasisimple groups in the unramified setting over quadratic fields and offer concrete arithmetic objects for further study and computational verification.

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.

Theorems & Definitions (18)

• Conjecture 1
• Theorem 1.1
• Theorem 1.2
• Theorem 3.1
• Theorem 3.2
• Remark 1
• Lemma 3.3
• proof
• Remark 2
• Proposition 3.4