Please use this identifier to cite or link to this item:
http://hdl.handle.net/10773/35070
Title: | ℚ-Curves, Hecke characters and some Diophantine equations |
Author: | Pacetti, Ariel Villagra Torcomian, Lucas |
Keywords: | Q-curves Fermat equations |
Issue Date: | 2022 |
Publisher: | American Mathematical Society |
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 $\Q(\sqrt{-d})$ is attached to each primitive solution, which happens to be a $\Q$-curve. Our main result is the construction of a Hecke character $\chi$ satisfying that the Frey elliptic curve representation twisted by $\chi$ extends to $\Gal_\Q$, 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) it 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 $d$. |
Peer review: | yes |
URI: | http://hdl.handle.net/10773/35070 |
DOI: | 10.1090/mcom/3759 |
ISSN: | 0025-5718 |
Publisher Version: | https://www.ams.org/journals/mcom/2022-91-338/S0025-5718-2022-03759-5/home.html |
Appears in Collections: | CIDMA - Artigos AGG - Artigos |
Files in This Item:
File | Description | Size | Format | |
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QcurvesMC_v1.pdf | 556.1 kB | Adobe PDF |
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