On the Fermat-type equation $x^3+y^3=z^p$

Abstract

We prove that the Fermat-type equation $x^3+y^3=z^p$ has no solutions $(a,b,c)$ satisfying $abc\ne 0$ and gcd$(a,b,c)=1$ when –3 is not a square mod $p$. This improves to approximately 0.844 the Dirichlet density of the set of prime exponents to which the previous equation is known to not have such solutions.

For the proof we develop a criterion of independent interest to decide if two elliptic curves with certain type of potentially good reduction at 2 have symplectically or anti-symplectically isomorphic $p$-torsion modules.