Solving equations of signature $(p,p,2)$ with coefficients over number fields

TL;DR

This paper applies the modular method to study solutions of the equation Aa^p+Bb^p=Cc^2 over number fields, providing asymptotic and effective bounds for solutions, especially over quadratic fields.

Contribution

It extends the modular approach to equations of signature (p,p,2) over number fields, establishing asymptotic results and explicit bounds in specific quadratic cases.

Findings

Proves an asymptotic result for solutions over certain number fields.

Provides explicit bounds for solutions over specific quadratic fields.

Verifies an S-unit condition for an infinite family of real quadratic fields.

Abstract

Using the modular method, we study solutions to the Diophantine equation $$Aa^p+Bb^p=Cc^2$$ over number fields. We first prove an asymptotic result for general number fields satisfying an appropriate $S$-unit condition by assuming some standard conjectures in the case of fields that are not totally real. Specifically, we verify that this condition holds for an infinite family of real quadratic fields. Outside the asymptotic setting, we also obtain effective results. In particular, for the equation $$a^p+db^p=c^2$$ over $K= \mathbb{Q}(\sqrt{-d})$ with $d \in \{3, 11, 19, 43, \}$ and $K= \mathbb{Q}(\sqrt d)$ with $d \in \{3, 5, 11, 13, 19, 29\}$, we find explicit bounds (depending on $d$) such that no non-trivial solutions of a certain type exist whenever $p$ exceeds these bounds.