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

The inverse Galois problem for orthogonal groups