Simple Lie Groups of type An as Galois groups over Q

TL;DR

This paper constructs explicit infinite series of simple Lie groups of type A as Galois groups over Q, expanding the known realizations with large degree fields and novel group types.

Contribution

It provides the first explicit infinite series of such groups realized as Galois groups over Q with large degree fields and distinct from classical projective linear groups.

Findings

Realized PSL(n, q) and PSU(n, q) as Galois groups over Q.

Achieved explicit constructions with arbitrarily large degree fields.

Groups do not coincide with PGL(n, q) or PGU(n, q).

Abstract

In this paper, we utilize our previous results on mod p monodromy of cyclic coverings of the projective line to realize a large series of groups of the form PSL(n, q) and PSU(n, q) as Galois groups over Q. We achieve for the first time a fully explicit infinite series of such groups where simultaneously the field can have arbitrarily large degree over the prime field and the group does not coincide with PGL(n, q) or PGU(n, q), respectively.