The Inverse Galois Problem asks whether every finite group occurs as the Galois group of some extension of the rationals. This problem is still wide open, even in the special case of simple groups. In this talk, we will explain why each of the simple groups PSL_2(F_p) occurs as a Galois group; these are the "simplest" simple groups for which the Inverse Galois Problem was not completely settled. Our Galois extensions will arise by studying the etale cohomology of a well- chosen elliptic surface. An important role will also be played by the L-functions of certain elliptic curves.