The distribution of the number of points on trigonal curves over $\F_q$

TL;DR

This paper determines the distribution of the number of rational points on random trigonal curves over finite fields as their genus grows, revealing a specific expected value and contrasting it with other types of algebraic curves.

Contribution

It provides the first explicit distribution and expected number of points for trigonal curves over finite fields in the large genus limit, extending understanding of rational points on algebraic curves.

Findings

Expected number of points on trigonal curves is $q+2-rac{1}{q^2+q+1}$

Distribution of points differs from cyclic covers and complete intersections

Conjecture for n-gonal curves with full symmetric monodromy

Abstract

We give a short determination of the distribution of the number of $\F_q$-rational points on a random trigonal curve over $\F_q$, in the limit as the genus of the curve goes to infinity. In particular, the expected number of points is $q+2-\frac{1}{q^2+q+1}$, contrasting with recent analogous results for cyclic $p$-fold covers of $\mathbb P^1$ and plane curves which have an expected number of points of $q+1$ (by work of Kurlberg, Rudnick, Bucur, David, Feigon and Lal\'in) and curves which are complete intersections which have an expected number of points $<q+1$ (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 $S_n$ monodromy based on function field analogs of Bhargava's heuristics for counting number fields.