%0 Journal Article
%T Statistics for traces of cyclic trigonal curves over finite fields
%A Alina Bucur
%A Chantal David
%A Brooke Feigon
%A Matilde Lal¨ªn
%J Mathematics
%D 2009
%I arXiv
%R 10.1093/imrn/rnp162
%X 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.
%U http://arxiv.org/abs/0907.5434v2