Geodesic
On ternary Diophantine equations of signature $(p,p,\text{3})$ over number fields
Canadian journal of mathematics, Tome 75 (2023) no. 4, pp. 1293-1313

Voir la notice de l'article provenant de la source Cambridge

DOI

In this paper, we prove results about solutions of the Diophantine equation $x^p+y^p=z^3$ over various number fields using the modular method. First, by assuming some standard modularity conjecture, we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Second, we show that there is an explicit bound such that the equation $x^p+y^p=z^3$ does not have a particular type of solution over $K=\mathbb {Q}(\sqrt {-d})$, where $d=1,7,19,43,67$ whenever p is bigger than this bound. During the course of the proof, we prove various results about the irreducibility of Galois representations, image of inertia groups, and Bianchi newforms.
DOI : 10.4153/S0008414X22000311
Mots-clés : Diophantine equations, modular method, Galois representations 
Isik, Erman; Kara, Yasemin; Ozman, Ekin. On ternary Diophantine equations of signature $(p,p,\text{3})$ over number fields. Canadian journal of mathematics, Tome 75 (2023) no. 4, pp. 1293-1313. doi: 10.4153/S0008414X22000311
@article{10_4153_S0008414X22000311,
     author = {Isik, Erman and Kara, Yasemin and Ozman, Ekin},
     title = {On ternary {Diophantine} equations of signature $(p,p,\text{3})$ over number fields},
     journal = {Canadian journal of mathematics},
     pages = {1293--1313},
     year = {2023},
     volume = {75},
     number = {4},
     doi = {10.4153/S0008414X22000311},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X22000311/}
}
TY  - JOUR
AU  - Isik, Erman
AU  - Kara, Yasemin
AU  - Ozman, Ekin
TI  - On ternary Diophantine equations of signature $(p,p,\text{3})$ over number fields
JO  - Canadian journal of mathematics
PY  - 2023
SP  - 1293
EP  - 1313
VL  - 75
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/S0008414X22000311/
DO  - 10.4153/S0008414X22000311
ID  - 10_4153_S0008414X22000311
ER  - 
%0 Journal Article
%A Isik, Erman
%A Kara, Yasemin
%A Ozman, Ekin
%T On ternary Diophantine equations of signature $(p,p,\text{3})$ over number fields
%J Canadian journal of mathematics
%D 2023
%P 1293-1313
%V 75
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/S0008414X22000311/
%R 10.4153/S0008414X22000311
%F 10_4153_S0008414X22000311

Cité par Sources :