  The purpose of this paper is to exhibit infinite families of conjugate
projective curves in a number field whose complement have the same abelian
fundamental group, but are non-homeomorphic. In particular, for any $d>3$ we
find Zariski tuples parametrized by the $d$-roots of unity up to complex
conjugation. As a consequence, for any divisor $m$ of $d$, $m\neq 1,2,3,4,6$,
we find arithmetic Zariski $\frac{\phi(m)}{2}$-tuples with coefficients in the
corresponding cyclotomic field. These curves have abelian fundamental group and
they are distinguished using a linking invariant.
