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

TL;DR

This paper proves the non-existence of primitive solutions to the Fermat-type equation $x^3 + y^3 = z^p$ under specific modular conditions, enhancing the density of primes for which solutions are absent.

Contribution

It introduces a new criterion to determine symplectic or anti-symplectic isomorphisms of elliptic curve $p$-torsion modules, applied to Fermat-type equations.

Findings

No solutions when $-3$ is not a square mod $p$

Density of primes with no solutions increased to approximately 0.844

Develops a criterion for elliptic curve $p$-torsion isomorphisms

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.