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

Paper

Abstract

Let

be an elliptic curve and

a prime. The modular curve

parameterizes elliptic curves with

-torsion modules anti-symplectically isomorphic to

. The work of Freitas--Naskręcki--Stoll uses the modular method to show that all primitive non-trivial solutions of the Fermat-type equation

give rise to rational points on

with

. Using a criterion classifying the existence of local points due to the first two authors, we show that, for

any of the curves with conductor 864 and certain primes

, we have

. Furthermore, for each

in the list and any

, we prove that either

can be discarded using the same criterion, or it cannot be discarded using purely local information.