I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes.
Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly infinite) and $O_S$ the holomorphy ring of $S$, i.e., $O_S:=\bigcap_{P\in S} O_P$ (being $O_P$ the valuation ring of the place $P$).
Give necessary and sufficient conditions for $O_S$ to be a Principal Ideal Domain.
Many thanks in advance!
G.