Revisiting the equation $x^2+y^3=z^p$

Abstract

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parameterizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. The work of Freitas--Naskr\k{e}cki--Stoll uses the modular method to show that all primitive non-trivial solutions of the Fermat-type equation $x^2 + y^3 = z^p$ give rise to rational points on $X_E^-(p)$ with $E \in \{27a1,54a1,96a1,288a1,864a1,864b1,864c1 \}$. Using a criterion classifying the existence of local points due to the first two authors, we show that, for $E$ any of the curves with conductor 864 and certain primes $p \equiv 19 \pmod{24}$, we have $X_E^-(p)(\mathbb Q_\ell) = \emptyset$. Furthermore, for each $E$ in the list and any $p$, we prove that either $X_E^-(p)$ can be discarded using the same criterion, or it cannot be discarded using purely local information.