Q-Curves, Hecke characters and some Diophantine equations

Abstract

In this article we study the equations x4 + dy2 = zp and x2 + dy6 = zp for positive square-free values of d. A Frey curve over Q( √ −d) is attached to each primitive solution, which happens to be a Q-curve. Our main result is the construction of a Hecke character χ satisfying that the Frey elliptic curve representation twisted by χ extends to GalQ, therefore (by Serre’s conjectures) corresponds to a newform in S2(Γ0(n), ε) for explicit values of n and ε. 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 nontrivial primitive solutions for large values of p of both equations for new values of d.