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

TL;DR

This paper investigates Fermat-type equations over totally real fields with signatures $(p,p,2)$ and $(p,p,3)$, extending previous results through modularity and level lowering techniques to establish non-existence of solutions under certain conditions.

Contribution

It generalizes recent work by applying modularity and inertia methods to Fermat equations over totally real fields with specific ramification properties.

Findings

No primitive solutions for $a^p+b^p=c^2$ under specified conditions for large primes p.

Extension of non-existence results to broader classes of totally real fields.

Generalization of previous results by Işık, Kara, and Ozman.

Abstract

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