  Let $m$ be a positive integer, $q$ be a prime power, and $\mathrm{PG}(2,q)$
be the projective plane over the finite field $\mathbb F_q$. Finding complete
$m$-arcs in $\mathrm{PG}(2,q)$ of size less than $q$ is a classical problem in
finite geometry. In this paper we give a complete answer to this problem when
$q$ is relatively large compared with $m$, explicitly constructing the smallest
$m$-arcs in the literature so far for any $m\geq 8$. For any fixed $m$, our
arcs $\mathcal A_{q,m}$ satisfy $|\mathcal A_{q,m}|-q\rightarrow -\infty$ as
$q$ grows. To produce such $m$-arcs, we develop a Galois theoretical machinery
that allows the transfer of geometric information of points external to the
arc, to arithmetic one, which in turn allows to prove the $m$-completeness of
the arc.
