On the solutions of the generalized Fermat equation over totally real number fields

Let K be a totally real number field, and OK be the ring of integers of K. In this article, we study the asymptotic solutions of the generalized Fermat equation, i.e., Axᵖ+Byᵖ+Czᵖ=0 over K of prime exponent p, where A,B,C ∈ OK \0\ with $ABC$ is even (in the sense that P| ABC, for some prime ideal P of OK with P |2). For certain class of fields K, we prove that the equation Axᵖ+Byᵖ+Czᵖ=0 has no asymptotic solution in K³ (resp., of certain type in K³), under some assumptions on $A,B,C$ (resp., for all A,B,C ∈ OK \0\ with $ABC$ is even). We also present several purely local criteria of K such that Axᵖ+Byᵖ+Czᵖ=0 has no asymptotic solutions in K³.