Asymptotic Fermat for signature $(4,2,p)$ over number fields

TL;DR

This paper investigates solutions to the equation $x^4 - y^2 = z^p$ over number fields using the modular method, providing asymptotic results under certain conjectures and unconditional results in specific cases.

Contribution

It extends the modular approach to asymptotic solutions of a specific Diophantine equation over number fields, including unconditional results for certain totally real fields.

Findings

Asymptotic solutions under standard conjectures for fields with complex embeddings.

Unconditional results for totally real fields with odd narrow class number.

Effective asymptotic results when elliptic curve modularity is known.

Abstract

Let $K$ be a number field. Using the modular method, we prove asymptotic results on solutions of the Diophantine equation $x^4-y^2=z^p$ over $K$, assuming some deep but standard conjectures of the Langlands programme when $K$ has at least one complex embedding. On the other hand, we give unconditional results in the case of totally real extensions having odd narrow class number and a unique prime above $2$. When modularity of elliptic curves over $K$ is known, for example when $K$ is real quadratic or the $r$-layer of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, effective asymptotic results hold.