Statistics for traces of cyclic trigonal curves over finite fields
Abstract
We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over a field of q elements as the curve varies in an irreducible component of the moduli space. We show that for q fixed and g increasing, the limiting distribution of the trace of the Frobenius equals the sum of q+1 independent random variables taking the value 0 with probability 2/(q+2) and 1, e^{(2pi i)/3}, e^{(4pi i)/3} each with probability q/(3(q+2)). This extends the work of Kurlberg and Rudnick who considered the same limit for hyperelliptic curves. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to p-fold covers of the projective line.
Keywords
Cite
@article{arxiv.0907.5434,
title = {Statistics for traces of cyclic trigonal curves over finite fields},
author = {Alina Bucur and Chantal David and Brooke Feigon and Matilde Lalín},
journal= {arXiv preprint arXiv:0907.5434},
year = {2010}
}
Comments
30 pages, added statement and sketch of proof in Section 7 for generalization of results to p-fold covers of the projective line, the final version of this article will be published in International Mathematics Research Notices