Asymptotic Fermat for signatures (p, p, 2)$(p, p, 2)$ and (p, p, 3)$(p, p, 3)$ over totally real fields. Issue 4 (13th September 2022)

• Record Type: Journal Article
• Authors: Mocanu, Diana
• Is Part Of: Mathematika. Volume 68:Issue 4(2022)
• Page Start: 1233
• Page End: 1257
• Publication Date: 2022-09-13
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1112/mtk.12162
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract: Let K be a totally real number field and consider a Fermat‐type equation A a p + B b q = C c r $Aa^p+Bb^q=Cc^r$ over K . We call the triple of exponents ( p, q, r ) $(p, q, r)$ the signature of the equation. We prove various results concerning the solutions to the Fermat equation with signature ( p, p, 2 ) $(p, p, 2)$ and ( p, p, 3 ) $(p, p, 3)$ using a method involving modularity, level lowering and image of inertia comparison. These generalize and extend the recent work of Işik, Kara and Özman. For example, consider K a totally real field of degree n with 2 ∤ h K + $2 \nmid h_K^+$ and 2 inert. Moreover, suppose there is a prime q ⩾ 5 $q\geqslant 5$ which totally ramifies in K and satisfies gcd ( n, q − 1 ) = 1 $\gcd (n, q-1)=1$, then we know that the equation a p + b p = c 2 $a^p+b^p=c^2$ has no primitive, non‐trivial solutions ( a, b, c ) ∈ O K 3 $(a, b, c) \in \mathcal {O}_K^3$ with 2 | b $2 | b$ for p sufficiently large.