Let $E$ be a rational elliptic curve and $p$ an odd prime. The modular curve $X_E^{-}(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. In this talk, I present my recent work with Nuno Freitas on a complete classification for when these curves admit points everywhere locally. We will see two applications of this result. Firstly, I will show how to construct counterexamples to Hasse’s principle of the shape $X_E^{-}(p)$ for fixed $E$ and infinitely many primes $p$. Secondly, I will present an application of the modular method together with our results to prove certain generalized Fermat equations have no non-trivial coprime solutions.