Galois Groups Via Atkin-Lehner Twists

Using Serre’s proposed complement to Shih’s Theorem, we obtain PSL_2(F_p) as a Galois group over Q for at least 614 new primes p. Under the assumption that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for 3/8 of the primes not covered by previous results; it would also suffice to assume a certain (plausible, and perhaps tractable) conjecture concerning class numbers of quadratic fields. The key issue is to understand rational points on Atkin-Lehner twists of X_0(N). In an appendix, we explore the existence of local points on these curves.