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arXiv · 1108.2526

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

Melanie Matchett Wood
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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 Lalín) 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.

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Melanie Matchett Wood. 2011-08-11. The distribution of the number of points on trigonal curves over $\F_q$. https://arxiv.org/abs/1108.2526

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