On ternary Diophantine equations of signature $ p,p,2 $ over number fields

Temel Bulgular & Veri:We prove that there is a constant $B_K$ depending only on $K$ such that for any prime exponent $p>B_K$ the Fermat type equation $x^p+y^p=z^2$ with $x,y,z\in O_K$ does not have certain type of solutions.

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Sonuç & Katkı:Our main tools in the proof are modularity, level lowering, and image of inertia comparisons.

🔬Tür:Araştırma Makalesi

📍Kapsam:Genişletilmiş Akademik Örneklem

⚙️Değişkenler:Fermat equation · generalized Fermat equation · S-units

Abstract

Let $K$ be a totally real number field with narrow class number one and $O_K$ be its ring of integers. We prove that there is a constant $B_K$ depending only on $K$ such that for any prime exponent $p>B_K$ the Fermat type equation $x^p+y^p=z^2$ with $x,y,z\in O_K$ does not have certain type of solutions. Our main tools in the proof are modularity, level lowering, and image of inertia comparisons.

Anahtar kelimeler

Fermat equation generalized Fermat equation S-units modularity

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Bu Makaleye Atıf Yap

IŞIK, E., KARA, Y., KARAKURT, E. Ö. (2020). On ternary Diophantine equations of signature $ p,p,2 $ over number fields. Turkish Journal of Mathematics, Cilt: 44 - Sayı: 4, 1197–1211. https://ajindex.com/articles/185520

Kaynakça (2)

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