The inverse Galois problem for orthogonal groups

Abstract

We prove many 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 Ω 2 n + 1 ( p ) \Omega _{2n+1}(p) and P Ω 4 n + ( p ) \operatorname {P}\!\Omega _{4n}^+(p) both occur as the Galois group of a Galois extension of the rationals for all integers n ≥ 2 n\geq 2 and all primes p ≥ 5 p\geq 5 . 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.