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.
Categories: math.NT