ℚ-Curves, Hecke characters and some Diophantine equations

Abstract

In this article we study the equations x 4 + d y 2 = z p x^4+dy^2=z^p and x 2 + d y 6 = z p x^2+dy^6=z^p for positive square-free values of d d . A Frey curve over Q ( − d ) \mathbb {Q}(\sqrt {-d}) is attached to each primitive solution, which happens to be a Q \mathbb {Q} -curve. Our main result is the construction of a Hecke character χ \chi satisfying that the Frey elliptic curve representation twisted by χ \chi extends to G a l Q Gal_\mathbb {Q} , therefore (by Serre’s conjectures) corresponds to a newform in S 2 ( Γ 0 ( n ) , ε ) S_2(\Gamma _0(n),\varepsilon ) for explicit values of n n and ε \varepsilon . Following some well known results and elimination techniques (together with some improvements) our result 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 p of both equations for new values of d d .