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.