  Let $K$ be a number field or a function field in one variable over a finite
field, and let $K^{sep}$ be a separable closure of $K$. Let $C/K$ be a smooth,
complete, connected curve. We prove a strong theorem of Fekete-Szego type for
adelic sets $E = \prod_v E_v$ on $C$, showing that under appropriate conditions
there are infinitely many points in $C(K^{sep})$ whose conjugates all belong to
$E_v$ at each place $v$ of $K$. We give several variants of the theorem,
including two for Berkovich curves, and provide examples illustrating the
theorem on the projective line, and on elliptic curves, Fermat curves, and
modular curves.
