  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.
