On ternary Diophantine equations of signature (p,p,3) over number fields
Canadian Journal of Mathematics, cilt.75, sa.4, ss.1293-1313, 2023 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 75 Sayı: 4
- Basım Tarihi: 2023
- Doi Numarası: 10.4153/s0008414x22000311
- Dergi Adı: Canadian Journal of Mathematics
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, DIALNET
- Sayfa Sayıları: ss.1293-1313
- Anahtar Kelimeler: Diophantine equations, Galois representations, modular method
- Boğaziçi Üniversitesi Adresli: Evet
Özet
In this paper, we prove results about solutions of the Diophantine equation xp+yp=z3 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 xp+yp=z3 does not have a particular type of solution over K=ℚ(√-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.