Function fields of class number 1 (reviewed)

There are exactly 8 isomorphism classes of curves over finite fields with trivial class group. Of these, 7 were identified by Madan-Queen in 1972 [10.4064/aa-20-4-423-432, MR:0306160]; it was claimed by Leitzel-Madan-Queen in 1975 [0.1016/0022-314X(75)90004-9, MR:0369326] that this list is complete. The 8th example was found by Stirpe in 2014 [10.1016/j.jnt.2014.02.016, MR:3227356], and completeness of the resulting list was proved independently by Mercuri-Stirpe [arXiv:1406.5365, 10.1016/j.jnt.2015.02.008, MR:3339577] and Shen-Shi [arXiv:1412.3505, 10.1016/j.jnt.2015.02.005, MR:3339578].