The inverse Galois problem for orthogonal groups

TL;DR

This paper advances the inverse Galois problem by demonstrating that many orthogonal groups over finite fields can be realized as Galois groups over the rationals, using elliptic curve twists and monodromy results.

Contribution

It proves new cases of the inverse Galois problem for orthogonal groups, specifically for Omega_{2n+1}(p) and POmega_{4n}^+(p), employing elliptic curve families and monodromy techniques.

Findings

Omega_{2n+1}(p) occurs as Galois group over rationals

POmega_{4n}^+(p) occurs as Galois group over rationals

Utilizes elliptic curve twists and monodromy results

Abstract

We prove many new cases of the Inverse Galois Problem for those simple groups arising from orthogonal groups over finite fields. For example, we show that the finite simple groups Omega_{2n+1}(p) and POmega_{4n}^+(p) both occur as the Galois group of a Galois extension of the rationals for all integers n>1 and all primes p>3. We obtain our representations by studying families of twists of elliptic curves and using some known cases of the Birch and Swinnerton-Dyer conjecture along with a big monodromy result of Hall.