Fermat's conjecture over arithmetic function fields

Let $N$ be a positive integer, and let $F_N$ be the Fermat curve of degree $N$ over ${\mathbb Z}$, defined by

$$F_N=\operatorname{Proj}\bigl({\mathbb Z}[X,Y,Z]/(X^N+Y^N-Z^N)\bigr).$$

Let $K$ be an arithmetic function field, meaning a finitely generated field over ${\mathbb Q}$. Say that $F_N$ has Fermat's property over $K$ if

$$\{(x,y)\in K^2\mid x^N+y^N=1\}\subseteq(\{0\}\cup\mu(K))^2,$$

where

$$\mu(K)=\{x\in K\mid \exists\ n\in{\mathbb Z}_{\geqslant1}\text{ such that }x^n=1\}$$

is the group of roots of unity in $K$.

Fermat's conjecture over an arithmetic function field. For every arithmetic function field $K$, there exists a positive integer $N_0$ depending on $K$ such that $F_N$ has Fermat's property over $K$ for all $N\geqslant N_0$.

The corollary established in the paper shows only that the set of positive integers $N$ for which $F_N$ has Fermat's property over $K$ has natural density one. The conjecture asks for the stronger eventual assertion that every sufficiently large degree has the property; it is presented as an open question.