$\mathbb{Q}$-curves, Hecke characters and some Diophantine equations

TL;DR

This paper develops a method using $ ext{Q}$-curves and Hecke characters to analyze and prove the non-existence of solutions to certain Diophantine equations involving fourth and sixth powers for large primes.

Contribution

It constructs explicit Hecke characters to relate Frey curves to modular forms, enabling systematic non-existence proofs for solutions of specific Diophantine equations.

Findings

Proves non-existence of primitive solutions for large primes p

Constructs explicit Hecke characters linked to Frey curves

Provides a systematic approach for similar Diophantine equations

Abstract

In this article we study the equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$ for positive square-free values of $d$. A Frey curve over $\mathbb{Q}(\sqrt{-d})$ is attached to each primitive solution, which happens to be a $\mathbb{Q}$-curve. Our main result is the construction of a Hecke character $\chi$ satisfying that the Frey elliptic curve representation twisted by $\chi$ extends to $\text{Gal}_\mathbb{Q}$, therefore (by Serre's conjectures) corresponds to a newform in $S_2(n,\varepsilon)$ for explicit values of $n$ and $\varepsilon$. Following some well known results and elimination techniques (together with some improvements) it provides a systematic procedure to study solutions of the above equations and allows us to prove non-existence of non-trivial primitive solutions for large values of $p$ of both equations for new values of $d$.