ℚ-Curves, Hecke characters and some Diophantine equations

Pacetti, Ariel; Villagra Torcomian, Lucas

doi:10.1090/mcom/3759

$\mathbb {Q}$-Curves, Hecke characters and some Diophantine equations
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Math. Comp. 91 (2022), 2817-2865 Request permission

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 $Gal_\mathbb {Q}$, therefore (by Serre’s conjectures) corresponds to a newform in $S_2(\Gamma _0(n),\varepsilon )$ for explicit values of $n$ and $\varepsilon$. 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 $p$ of both equations for new values of $d$.

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