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

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

2011-08-11

arxiv.org/abs/1108.2526v1

Mathematics

Number Theory

Scientific paper

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 $

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