An application of the symplectic argument to some Fermat-type Equations

TL;DR

This paper improves the density results for Fermat-type equations by applying a symplectic criterion to elliptic curves, increasing the known set of exponents for which no non-trivial solutions exist.

Contribution

It introduces a new criterion involving symplectic isomorphisms of elliptic curve torsion modules to enhance density bounds for Fermat-type equations.

Findings

Density of non-solvable exponents increased from 1/4 to 3/8 for the first equation.

Density of non-solvable exponents increased from 1/8 to 3/8 for the second equation.

Method leverages recent advances in elliptic curve symplectic criteria.

Abstract

Let $p$ be a prime number. In the early 2000s, it was proved that the Fermat equations with coefficients \[3x^p + 8y^p + 21z^p =0\quad \text{ and } \quad 3x^p + 4y^p + 5z^p=0 \] do not admit non-trivial solutions for a set of exponents $p$ with Dirichlet density ${1/4}$ and ${1/8}$, respectively. In this note, using a recent criterion to decide if two elliptic curves over $\mathbb{Q}$ with certain types of additive reduction at 2 have symplectically isomorphic $p$-torsion modules, we improve these densities to ${3/8}$.