Seven Small Simple Groups Not Previously Known to Be Galois Over <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">Q</mi></semantics></math></inline-formula>

Abstract

In this note we realize seven small simple groups as Galois groups over Q. The technique that we employ is the determination of the images of Galois representations attached to modular and automorphic forms, relying in two cases on recent results of Scholze on the existence of Galois representations attached to non-selfdual automorphic forms.

Keywords

• Galois representations
• modular forms
• inverse Galois theory
• automorphic forms
• finite simple groups