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

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Ariel Pacetti and Lucas Villagra Torcomian
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## 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 $d$.## References

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## Additional Information

**Ariel Pacetti**- Affiliation: Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
- MR Author ID: 759256
- Email: apacetti@ua.pt
**Lucas Villagra Torcomian**- Affiliation: FAMAF-CIEM, Universidad Nacional de Córdoba. C.P.: 5000 Córdoba, Argentina
- ORCID: 0000-0003-2525-5694
- Email: lucas.villagra@unc.edu.ar
- Received by editor(s): January 13, 2022
- Received by editor(s) in revised form: April 20, 2022
- Published electronically: August 4, 2022
- Additional Notes: The first author was partially supported by FonCyT BID-PICT 2018-02073 and by the Portuguese Foundation for Science and Technology (FCT) within project UIDB/04106/2020 (CIDMA). The second author was supported by a CONICET grant.
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 2817-2865 - MSC (2020): Primary 11D41, 11F80
- DOI: https://doi.org/10.1090/mcom/3759
- MathSciNet review: 4473105