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

Abstract

Let $K$ be a totally real number field and consider a Fermat-type equation $A{a}^p+B{b}^q=C{c}^r$ 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 Işik, 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 $q\ge 5$ 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.

arXiv commentarXiv admin note: text overlap with arXiv:1307.3162 by other authors