Asymptotic Fermat for signatures (p,p,2)$(p,p,2)$ and (p,p,3)$(p,p,3)$ over totally real fields

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.