Inverse Galois realizations for simple groups of Lie type over Q

Determine, for each finite simple group of Lie type, whether there exists a Galois extension of the rational numbers Q whose Galois group is isomorphic to that group.

Background

Finite simple groups of Lie type (including families such as projective special linear groups and other Chevalley and Steinberg groups over finite fields) constitute a large class of groups for which the Inverse Galois Problem is far from completely resolved.

The authors explicitly note that, beyond specific successes, the problem remains unknown for most simple groups of Lie type, underscoring a broad area of ongoing research in inverse Galois theory.