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We give a short determination of the distribution of the number of-rational points on a random trigonal curve over, in the limit as the genus of the curve goes to infinity. In particular, the expected number of points is, contrasting with recent analogous results for cyclic  p -fold covers ofand plane curves that have an expected number of points of  q +1 (by the work of Kurlberg, Rudnick, Bucur, David, Feigon and Lalín) and curves that are complete intersections that have an expected number of points &lt; q +1 (by work of Bucur and Kedlaya). We also give a conjecture for the expected number of points on a random  n -gonal curve with full  Sn    monodromy based on function field analogs of Bhargava's heuristics for counting number fields.

The Distribution of the Number of Points on Trigonal Curves over q