Local points on twists of $X(p)$ with applications

Abstract

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. We give a complete classification of when $X_E^-(p)(\mathbb Q_\ell)$ is non-empty, for all primes $\ell\neq p$; our result also includes $\ell=p$ in most cases when $E$ is semistable at $p$. We give two different applications. First, we classify CM curves $E/\mathbb Q$ where the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for infinitely many $p$. Assuming the Frey--Mazur conjecture, we prove that for at least $60\%$ of rational elliptic curves $E$, the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for at least $50\%$ of primes $p$. Secondly, we introduce a new technique to the elimination stage of the modular method and apply it to show that $x^3+y^3=5^\alpha z^p$ has no non-trivial primitive solutions for various primes $p$ satisfying $(\alpha/p)=-1$. Moreover, as a by-product of our work, we simplify the assumptions of several local symplectic criteria due to the first author and Alain Kraus.