Non-trivial solutions of Aap + Bbp = Cc3 over number fields

KARA Y., Nomden S., Özman E.

Journal of Number Theory, cilt.286, ss.108-130, 2026 (SCI-Expanded, Scopus)

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 286
  • Basım Tarihi: 2026
  • Doi Numarası: 10.1016/j.jnt.2026.02.003
  • Dergi Adı: Journal of Number Theory
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.108-130
  • Anahtar Kelimeler: Diophantine equations, Galois representations, Modular method
  • Boğaziçi Üniversitesi Adresli: Evet

Özet

In this paper, we investigate solutions to the Diophantine equation Aap+Bbp=Cc3 over number fields using the modular method. Assuming certain standard modularity conjectures, we first establish an asymptotic result for general number fields satisfying an appropriate S -unit condition. In particular, we verify that this condition holds for several imaginary quadratic fields. Beyond the asymptotic setting, we also obtain an effective result. Specifically, for the equation ap+dbp=c3 over K=Q(−d) with d∈{7,19,43,67}, we determine an explicit bound (depending on d ) such that no non-trivial solutions of a certain type exist whenever p exceeds this bound.