Galois groups via Atkin-Lehner twists

Abstract:

Using Serre’s proposed complement to Shih’s Theorem, we obtain $PSL_2(\mathbb {F}_p)$ as a Galois group over $\mathbb {Q}$ for at least $614$ new primes $p$. Assuming that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for $\frac {3}{8}$ of the primes that were 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.