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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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$\mathbb {Q}$-Curves, Hecke characters and some Diophantine equations
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by Ariel Pacetti and Lucas Villagra Torcomian HTML | PDF
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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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